Handbook of Optics

HANDBOOK OF OPTICS ABOUT THE EDITORS Editor-in-Chief: Dr. Michael Bass is professor emeritus at CREOL, The College o...

Author: Michael Bass | Casimer DeCusatis | Jay Enoch | Vasudevan Lakshminarayanan | Guifang Li | Carolyn MacDonald | Virendra Mahajan | Eric Van Stryland

446 downloads 3317 Views 11MB Size Report

This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!
Report copyright / DMCA form

DOWNLOAD PDF

HANDBOOK OF OPTICS

ABOUT THE EDITORS

Editor-in-Chief: Dr. Michael Bass is professor emeritus at CREOL, The College of Optics and Photonics, University of Central Florida, Orlando, Florida. Associate Editors: Dr. Casimer M. DeCusatis is a distinguished engineer and technical executive with IBM Corporation. Dr. Jay M. Enoch is dean emeritus and professor at the School of Optometry at the University of California, Berkeley. Dr. Vasudevan Lakshminarayanan is professor of Optometry, Physics, and Electrical Engineering at the University of Waterloo, Ontario, Canada. Dr. Guifang Li is a professor at CREOL, The College of Optics and Photonics, University of Central Florida, Orlando, Florida. Dr. Carolyn MacDonald is a professor at the University at Albany, and director of the Center for X-Ray Optics. Dr. Virendra N. Mahajan is a distinguished scientist at The Aerospace Corporation. Dr. Eric Van Stryland is a professor at CREOL, The College of Optics and Photonics, University of Central Florida, Orlando, Florida.

HANDBOOK OF OPTICS Volume I Geometrical and Physical Optics, Polarized Light, Components and Instruments THIRD EDITION

Sponsored by the OPTICAL SOCIETY OF AMERICA

Michael Bass

Editor-in-Chief

CREOL, The College of Optics and Photonics University of Central Florida Orlando, Florida

Virendra N. Mahajan

Associate Editor

The Aerospace Corporation El Segundo, California

New York Chicago San Francisco Lisbon London Madrid Mexico City Milan New Delhi San Juan Seoul Singapore Sydney Toronto

Copyright © 2010 by The McGraw-Hill Companies, Inc. (with the exception of Chapters 14 and 15, copyright © Russell A. Chipman). All rights reserved. Except as permitted under the United States Copyright Act of 1976, no part of this publication may be reproduced or distributed in any form or by any means, or stored in a database or retrieval system, without the prior written permission of the publisher. ISBN 978-0-07-162925-6, MHID 0-07-162925-4 The material in this eBook also appears in the print version of this title. ISBN: P/N 978-0-07-163598-1 of set 978-0-07-149889-0. MHID: P/N 0-07-163598-X of set 0-07-149889-3. All trademarks are trademarks of their respective owners. Rather than put a trademark symbol after every occurrence of a trademarked name, we use names in an editorial fashion only, and to the benefit of the trademark owner, with no intention of infringement of the trademark. Where such designations appear in this book, they have been printed with initial caps. McGraw-Hill eBooks are available at special quantity discounts to use as premiums and sales promotions, or for use in corporate training programs. To contact a representative please e-mail us at [email protected]. TERMS OF USE This is a copyrighted work and The McGraw-Hill Companies, Inc. (“McGraw-Hill”) and its licensors reserve all rights in and to the work. Use of this work is subject to these terms. Except as permitted under the Copyright Act of 1976 and the right to store and retrieve one copy of the work, you may not decompile, disassemble, reverse engineer, reproduce, modify, create derivative works based upon, transmit, distribute, disseminate, sell, publish or sublicense the work or any part of it without McGraw-Hill’s prior consent. You may use the work for your own noncommercial and personal use; any other use of the work is strictly prohibited. Your right to use the work may be terminated if you fail to comply with these terms. THE WORK IS PROVIDED “AS IS.” McGRAW-HILL AND ITS LICENSORS MAKE NO GUARANTEES OR WARRANTIES AS TO THE ACCURACY, ADEQUACY OR COMPLETENESS OF OR RESULTS TO BE OBTAINED FROM USING THE WORK, INCLUDING ANY INFORMATION THAT CAN BE ACCESSED THROUGH THE WORK VIA HYPERLINK OR OTHERWISE, AND EXPRESSLY DISCLAIM ANY WARRANTY, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO IMPLIED WARRANTIES OF MERCHANTABILITY OR FITNESS FOR A PARTICULAR PURPOSE. McGraw-Hill and its licensors do not warrant or guarantee that the functions contained in the work will meet your requirements or that its operation will be uninterrupted or error free. Neither McGraw-Hill nor its licensors shall be liable to you or anyone else for any inaccuracy, error or omission, regardless of cause, in the work or for any damages resulting therefrom. McGraw-Hill has no responsibility for the content of any information accessed through the work. Under no circumstances shall McGraw-Hill and/or its licensors be liable for any indirect, incidental, special, punitive, consequential or similar damages that result from the use of or inability to use the work, even if any of them has been advised of the possibility of such damages. This limitation of liability shall apply to any claim or cause whatsoever whether such claim or cause arises in contract, tort or otherwise.

Information contained in this work has been obtained by The McGraw-Hill Companies, Inc. (“McGraw-Hill”) from sources believed to be reliable. However, neither McGraw-Hill nor its authors guarantee the accuracy or completeness of any information published herein, and neither McGraw-Hill nor its authors shall be responsible for any errors, omissions, or damages arising out of use of this information. This work is published with the understanding that McGraw-Hill and its authors are supplying information but are not attempting to render engineering or other professional services. If such services are required, the assistance of an appropriate professional should be sought.

COVER ILLUSTRATIONS

Left: Poincaré sphere describing light’s polarization states is shown floating in front of a depolarized field of polarization ellipses, with linearly and circularly polarized fields propagating on its left and right, respectively. See Chaps. 12 and 15. Middle: Triplet lens developed for photographic applications that can zero out the primary aberrations by splitting the positive lens of a doublet into two and placing one on each side of the negative lens. See Chap. 17. Right: Micrographs of different optical storage media showing the straight and narrow tracks with 1.6-μm spacing between adjacent tracks. The recorded information bits appear as short marks along each track. See Chap. 35.

This page intentionally left blank.

CONTENTS

Contributors xvii Brief Contents of All Volumes xix Editors’ Preface xxv Preface to Volume I xxvii Glossary and Fundamental Constants

xxix

Part 1. Geometrical Optics Chapter 1. General Principles of Geometrical Optics 1.3

Douglas S. Goodman 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.10 1.11 1.12 1.13 1.14 1.15 1.16 1.17 1.18 1.19 1.20

Glossary / 1.3 Introduction / 1.7 Fundamentals / 1.8 Characteristic Functions / 1.13 Rays in Heterogeneous Media / 1.18 Conservation of Étendue / 1.22 Skew Invariant / 1.23 Refraction and Reflection at Interfaces between Homogeneous Media / Imaging / 1.26 Description of Systems of Revolution / 1.32 Tracing Rays in Centered Systems of Spherical Surfaces / 1.35 Paraxial Optics of Systems of Revolution / 1.37 Images About Known Rays / 1.43 Gaussian Lens Properties / 1.44 Collineation / 1.56 System Combinations: Gaussian Properties / 1.63 Paraxial Matrix Methods / 1.65 Apertures, Pupils, Stops, Fields, and Related Matters / 1.74 Geometrical Aberrations of Point Images: Description / 1.85 References / 1.92

1.23

Part 2. Physical Optics Chapter 2. Interference John E. Greivenkamp 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10

2.3

Glossary / 2.3 Introduction / 2.3 Waves and Wavefronts / 2.3 Interference / 2.5 Interference by Wavefront Division / 2.14 Interference by Amplitude Division / 2.19 Multiple Beam Interference / 2.28 Coherence and Interference / 2.36 Applications of Interference / 2.42 References / 2.42 vii

viii

CONTENTS

Chapter 3. Diffraction Arvind S. Marathay and John F. McCalmont 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10

Glossary / 3.1 Introduction / 3.1 Light Waves / 3.2 Huygens-Fresnel Construction / 3.4 Cylindrical Wavefront / 3.13 Mathematical Theory of Diffraction / 3.21 Stationary Phase Approximation / 3.29 Vector Diffraction / 3.32 Acknowledgments / 3.38 References / 3.38

Chapter 4. Transfer Function Techniques 4.1 4.2 4.3 4.4 4.5 4.6

Glenn D. Boreman

4.1

Glossary / 4.1 Introduction / 4.1 Definitions / 4.2 MTF Calculations / 4.3 MTF Measurements / 4.6 References / 4.8

Chapter 5. Coherence Theory William H. Carter 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 5.10 5.11

3.1

5.1

Glossary / 5.1 Introduction / 5.1 Some Elementary Classical Concepts / 5.2 Definitions of Coherence Functions / 5.4 Model Sources / 5.9 Propagation / 5.13 Spectrum of Light / 5.19 Polarization Effects / 5.22 Applications / 5.22 References / 5.23 Additional Reading / 5.26

Chapter 6. Coherence Theory: Tools and Applications Gisele Bennett, William T. Rhodes, and J. Christopher James 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9

6.1

Glossary / 6.1 Introduction / 6.2 Key Definitions and Relationships / 6.2 Propagation, Diffraction, and Scattering: Enhanced Backscatter and the Lau Effect Image Formation: Lukosz-Type Super-Resolving System / 6.9 Efficient Sampling of Coherence Functions / 6.10 An Example of When Not to Use Coherence Theory / 6.12 Concluding Remarks / 6.13 References / 6.13

Chapter 7. Scattering by Particles Craig F. Bohren 7.1 7.2 7.3 7.4 7.5 7.6

Glossary / 7.1 Introduction / 7.2 Scattering: An Overview / 7.3 Scattering by Particles: Basic Concepts and Terminology / 7.4 Scattering by an Isotropic, Homogeneous Sphere: The Archetype / Scattering by Regular Particles / 7.14

7.11

/ 6.5

7.1

CONTENTS

7.7 7.8

Computational Methods for Nonspherical Particles References / 7.17

/

ix

7.15

Chapter 8. Surface Scattering Eugene L. Church 8.1

and Peter Z. Takacs 8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.8 8.9 8.10

Glossary of Principal Symbols / 8.1 Introduction / 8.2 Notation / 8.2 The Fresnel-Kirchhoff Approximation / 8.5 The Rayleigh-Rice (RR) or Small-Perturbation Approximation Effects of Finite Illumination Area / 8.12 Surface Statistics / 8.12 Surface Finish Specification / 8.16 Retrospect and Prospect / 8.17 References and Endnotes / 8.18

/

8.9

Chapter 9. Volume Scattering in Random Media 9.1

Aristide Dogariu and Jeremy Ellis 9.1 9.2 9.3 9.4 9.5 9.6

Glossary / 9.1 Introduction / 9.2 General Theory of Scattering Single Scattering / 9.4 Multiple Scattering / 9.8 References / 9.18

/ 9.3

Chapter 10. Optical Spectroscopy and Spectroscopic Lineshapes Brian Henderson 10.1 10.2 10.3 10.4 10.5 10.6 10.7 10.8 10.9 10.10 10.11 10.12

10.1

Glossary / 10.1 Introductory Comments / 10.2 Theoretical Preliminaries / 10.3 Rates of Spectroscopic Transition / 10.4 Lineshapes of Spectral Transitions / 10.6 Spectroscopy of One-Electron Atoms / 10.7 Multielectron Atoms / 10.10 Optical Spectra and the Outer Electronic Structure / 10.12 Spectra of Tri-Positive Rare Earth Ions / 10.16 Vibrational and Rotational Spectra of Molecules / 10.18 Lineshapes in Solid State Spectroscopy / 10.22 References / 10.27

Chapter 11. Analog Optical Signal and Image Processing 11.1

Joseph W. Goodman 11.1 11.2 11.3 11.4 11.5 11.6 11.7 11.8 11.9 11.10 11.11

Glossary / 11.1 Introduction / 11.1 Fundamental Analog Operations / 11.2 Analog Optical Fourier Transforms / 11.3 Spatial Filtering / 11.5 Coherent Optical Processing of Synthetic Aperture Radar Data Coherent Optical Processing of Temporal Signals / 11.8 Optical Processing of Two-Dimensional Images / 11.12 Incoherent Processing of Discrete Signals / 11.17 Concluding Remarks / 11.20 References / 11.20

/

11.6

x

CONTENTS

Part 3. Polarized Light Chapter 12. Polarization Jean M. Bennett 12.1 12.2 12.3 12.4 12.5 12.6 12.7 12.8 12.9

12.3

Glossary / 12.3 Basic Concepts and Conventions / 12.4 Fresnel Equations / 12.6 Basic Relations for Polarizers / 12.14 Polarization by Nonnormal-Incidence Reflection (Pile of Plates) / 12.15 Polarization by Nonnormal-Incidence Transmission (Pile of Plates) / 12.18 Quarter-Wave Plates and Other Phase Retardation Plates / 12.24 Matrix Methods for Computing Polarization / 12.27 References / 12.30

Chapter 13. Polarizers Jean M. Bennett 13.1 13.2 13.3 13.4 13.5 13.6 13.7 13.8 13.9 13.10 13.11 13.12 13.13 13.14

Glossary / 13.1 Prism Polarizers / 13.2 Glan-Type Prisms / 13.8 Nicol-Type Prisms / 13.15 Polarizing Beam-Splitter Prisms / 13.18 Feussner Prisms / 13.22 Noncalcite Polarizing Prisms / 13.23 Dichroic and Diffraction-Type Polarizers / 13.24 Non-Normal-Incidence Reflection and Transmission Polarizers Retardation Plates / 13.43 Variable Retardation Plates and Compensators / 13.53 Half-Shade Devices / 13.56 Miniature Polarization Devices / 13.57 References / 13.58

13.1

/

13.33

Chapter 14. Mueller Matrices Russell A. Chipman 14.1 14.2 14.3 14.4 14.5 14.6 14.7 14.8 14.9 14.10 14.11 14.12 14.13 14.14 14.15 14.16 14.17 14.18 14.19 14.20 14.21 14.22 14.23 14.24 14.25 14.26

Glossary / 14.1 Conventions / 14.3 Objectives / 14.3 Stokes Parameters and Mueller Matrices / 14.4 The Stokes Parameters and the Poincaré Sphere / 14.4 Mueller Matrices / 14.6 Sequences of Polarization Elements / 14.7 Polarization Elements’ Properties in the Mueller Calculus / 14.7 Rotation of an Element About the Optical Axis / 14.8 Nonpolarizing Mueller Matrices / 14.8 Mueller Matrices of Ideal Polarizers / 14.8 Retarder Mueller Matrices / 14.11 Retarder Mueller Matrices Ambiguities and Retarder Space / 14.14 Transmittance and Diattenuation / 14.16 Polarizance / 14.18 Mueller Matrices of Diattenuators / 14.18 Normalizing a Mueller Matrix / 14.19 Coordinate System for the Mueller Matrix / 14.19 Mueller Matrices for Refraction / 14.20 Mueller Matrices for Reflection / 14.21 Conversion between Mueller Matrices and Jones Matrices / 14.22 Nondepolarizing Mueller Matrices and Mueller-Jones Matrices / 14.24 Homogeneous and Inhomogeneous Polarization Elements / 14.25 Mueller Matrices Near the Identity Matrix, Weak Polarization Elements / Matrix Roots of Nondepolarizing Mueller Matrices / 14.27 Depolarization and the Depolarization Index / 14.30

14.1

14.26

CONTENTS

14.27 14.28 14.29 14.30 14.31 14.32 14.33 14.34 14.35

Degree of Polarization Surfaces and Maps / 14.31 The Depolarization Index / 14.32 The Average Degree of Polarization / 14.32 Determining Mueller Matrix Properties / 14.33 Generators for Depolarization / 14.33 Interpretation of Arbitrary Mueller Matrices, the Polar Decomposition of Mueller Matrices Physically Realizable Mueller Matrices / 14.40 Acknowledgments / 14.42 References / 14.43

Chapter 15. Polarimetry Russell A. Chipman 15.1 15.2 15.3 15.4 15.5 15.6 15.7 15.8 15.9 15.10 15.11 15.12 15.13 15.14 15.15 15.16 15.17 15.18 15.19 15.20 15.21 15.22 15.23 15.24 15.25 15.26 15.27 15.28 15.29 15.30 15.31 15.32 15.33 15.34 15.35 15.36 15.37 15.38 15.39 15.40 15.41 15.42 15.43 15.44 15.45 15.46 15.47 15.48

Glossary / 15.1 Objectives / 15.3 Polarimeters / 15.3 Light-Measuring Polarimeters / 15.3 Sample-Measuring Polarimeters / 15.4 Complete and Incomplete Polarimeters / 15.4 Polarization Generators and Analyzers / 15.4 Classes of Polarimeters / 15.5 Time-Sequential Measurements / 15.5 Polarization Modulation / 15.5 Division of Aperture / 15.5 Division of Amplitude / 15.5 Spectropolarimeters / 15.6 Imaging Polarimeters / 15.6 Definitions / 15.6 Stokes Vectors and Mueller Matrices / 15.8 Phenomenological Definition of the Stokes Vector / 15.9 Polarization Properties of Light Beams / 15.9 Mueller Matrices / 15.11 Data Reduction for Light-Measuring Polarimeters / 15.11 Sample-Measuring Polarimeters for Measuring Mueller Matrix Elements / 15.13 Polarimetric Measurement Equation and Polarimetric Data-Reduction Equation / 15.14 Dual Rotating Retarder Polarimeter / 15.16 Incomplete Sample-Measuring Polarimeters / 15.16 Nonideal Polarization Elements / 15.17 Elliptical and Circular Polarizers and Analyzers / 15.17 Common Defects of Polarization Elements / 15.19 Polarization Modulators, Retardance Modulators / 15.20 Rotating Retarders / 15.20 Photo-Elastic Modulators / 15.21 Liquid Crystal Retarders / 15.21 Electro-Optical Modulators / 15.23 Magneto-Optical Modulators / 15.23 Fiber Squeezers / 15.24 Polarimeter Design Metrics / 15.24 Singular Value Decomposition Examples / 15.26 Polarimeter Error Analysis / 15.27 The Mueller Matrix for Polarization Component Characterization / 15.28 Retro-Reflection Testing and Correction for Supplemental Optics / 15.28 Applications of Polarimetry / 15.29 Ellipsometry and Generalized Ellipsometry / 15.30 Liquid Crystal Cell and System Testing / 15.32 Polarization Aberrations / 15.35 Remote Sensing / 15.37 Polarization Light Scattering / 15.38 Ophthalmic Polarimetry / 15.39 Acknowledgments / 15.41 References / 15.41

/

xi

14.39

15.1

xii

CONTENTS

Chapter 16. Ellipsometry Rasheed M. A. Azzam 16.1 16.2 16.3 16.4 16.5 16.6 16.7 16.8 16.9 16.10

16.1

Glossary / 16.1 Introduction / 16.2 Conventions / 16.3 Modeling and Inversion / 16.4 Transmission Ellipsometry / 16.10 Instrumentation / 16.10 Jones-Matrix Generalized Ellipsometry / 16.19 Mueller-Matrix Generalized Ellipsometry / 16.19 Applications / 16.21 References / 16.21

Part 4. Components Chapter 17. Lenses R. Barry Johnson 17.1 17.2 17.3 17.4 17.5 17.6 17.7 17.8 17.9 17.10 17.11 17.12 17.13 17.14 17.15 17.16 17.17 17.18 17.19 17.20 17.21 17.22

Glossary / 17.3 Introduction / 17.4 Basics / 17.5 Stops and Pupils / 17.8 F-Number and Numerical Aperture / 17.9 Magnifier or Eye Loupe / 17.9 Compound Microscopes / 17.10 Field and Relay Lenses / 17.10 Aplanatic Surfaces and Immersion Lenses / 17.10 Single Element Lens / 17.12 Landscape Lenses and the Influence of Stop Position / Two-Lens Systems / 17.20 Achromatic Doublets / 17.22 Triplet Lenses / 17.26 Symmetrical Lenses / 17.26 Double-Gauss Lenses / 17.27 Petzval Lenses / 17.28 Telephoto Lenses / 17.29 Inverted or Reverse Telephoto Lenses / 17.29 Performance of Representative Lenses / 17.29 Rapid Estimation of Lens Performance / 17.36 Bibliography / 17.40

17.3

17.17

Chapter 18. Afocal Systems William B. Wetherell 18.1 18.2 18.3 18.4 18.5 18.6 18.7 18.8

Glossary / 18.1 Introduction / 18.2 Gaussian Analysis of Afocal Lenses / 18.2 Keplerian Afocal Lenses / 18.7 Galilean and Inverse Galilean Afocal Lenses / 18.15 Relay Trains and Periscopes / 18.17 Reflecting and Catadioptric Afocal Lenses / 18.19 References / 18.23

Chapter 19. Nondispersive Prisms William L. Wolfe 19.1 19.2 19.3 19.4 19.5

18.1

Glossary / 19.1 Introduction / 19.1 Inversion, Reversion / 19.2 Deviation, Displacement / 19.2 Summary of Prism Properties / 19.2

19.1

CONTENTS

19.6 19.7

Prism Descriptions / 19.2 References / 19.29

Chapter 20. Dispersive Prisms and Gratings George J. Zissis 20.1 20.2 20.3 20.4 20.5 20.6

xiii

Glossary / 20.1 Introduction / 20.1 Prisms / 20.2 Gratings / 20.3 Prism and Grating Configurations and Instruments References / 20.15

/

20.1

20.4

Chapter 21. Integrated Optics Thomas L. Koch, Frederick J. Leonberger, and Paul G. Suchoski 21.1 21.2 21.3 21.4 21.5 21.6 21.7 21.8

Glossary / 21.1 Introduction / 21.2 Device Physics / 21.3 Integrated Optics Materials and Fabrication Technology Circuit Elements / 21.21 Applications of Integrated Optics / 21.31 Future Trends / 21.39 References / 21.41

21.1

/ 21.13

Chapter 22. Miniature and Micro-Optics Tom D. Milster and Tomasz S. Tkaczyk 22.1 22.2 22.3 22.4 22.5 22.6 22.7 22.8 22.9 22.10 22.11 22.12 22.13

22.1

Glossary / 22.1 Introduction / 22.2 Uses of Micro-Optics / 22.2 Micro-Optics Design Considerations / 22.2 Molded Microlenses / 22.8 Diamond Turning / 22.15 Lithography for Making Refractive Components / 22.18 Monolithic Lenslet Modules / 22.25 Distributed-Index Planar Microlenses / 22.26 Micro-Fresnel Lenses / 22.31 Liquid Lenses / 22.37 Other Technologies / 22.42 References / 22.47

Chapter 23. Binary Optics Michael W. Farn and Wilfrid B. Veldkamp 23.1 23.2 23.3 23.4 23.5 23.6 23.7

Glossary / 23.1 Introduction / 23.2 Design—Geometrical Optics / 23.2 Design—Scalar Diffraction Theory / 23.10 Design—Vector Diffraction Theory / 23.13 Fabrication / 23.14 References / 23.17

Chapter 24. Gradient Index Optics Duncan T. Moore 24.1 24.2 24.3 24.4 24.5

23.1

Glossary / 24.1 Introduction / 24.1 Analytic Solutions / 24.2 Mathematical Representation / Axial Gradient Lenses / 24.3

24.2

24.1

xiv

CONTENTS

24.6 24.7 24.8 24.9 24.10

Radial Gradients / 24.5 Radial Gradients with Curved Surfaces Shallow Radial Gradients / 24.7 Materials / 24.8 References / 24.9

/ 24.7

Part 5. Instruments Chapter 25. Cameras Norman Goldberg 25.1 25.2 25.3 25.4 25.5 25.6 25.7 25.8 25.9 25.10 25.11 25.12 25.13 25.14 25.15 25.16 25.17 25.18 25.19 25.20

Glossary / 25.3 Introduction / 25.3 Background / 25.4 Properties of the Final Image / 25.5 Film Choice / 25.5 Resolving Fine Detail / 25.5 Film Sizes / 25.6 Display / 25.6 Distributing the Image / 25.7 Video Cameras / 25.7 Instant Pictures / 25.8 Critical Features / 25.8 Time Lag / 25.8 Automation / 25.10 Flash / 25.16 Flexibility through Features and Accessories / Advantages of Various Formats / 25.17 Large Format: A Different World / 25.18 Special Cameras / 25.20 Further Reading / 25.26

25.16

Chapter 26. Solid-State Cameras Gerald C. Holst 26.1 26.2 26.3 26.4 26.5 26.6 26.7 26.8 26.9 26.10 26.11 26.12 26.13

25.3

26.1

Glossary / 26.1 Introduction / 26.2 Imaging System Applications / 26.3 Charge-Coupled Device Array Architecture / 26.3 Charge Injection Device / 26.6 Complementary Metal-Oxide Semiconductor / 26.8 Array Performance / 26.9 Camera Performance / 26.12 Modulation Transfer Function / 26.14 Resolution / 26.15 Sampling / 26.16 Storage, Analysis, and Display / 26.19 References / 26.20

Chapter 27. Camera Lenses Ellis Betensky, Melvin H. Kreitzer, and Jacob Moskovich 27.1 27.2 27.3 27.4 27.5 27.6 27.7

Introduction / 27.1 Imposed Design Limitations / 27.1 Modern Lens Types / 27.2 Classification System / 27.17 Lens Performance Data / 27.24 Acknowledgments / 27.25 Further Reading / 27.25

27.1

CONTENTS

xv

Chapter 28. Microscopes Rudolf Oldenbourg 28.1

and Michael Shribak 28.1 28.2 28.3 28.4 28.5 28.6 28.7

Glossary / 28.1 Introduction / 28.1 Optical Arrangements, Lenses, and Resolution Contrast and Imaging Modes / 28.24 Manipulation of Specimen / 28.54 Acknowledgment / 28.55 References / 28.56

/

28.3

Chapter 29. Reflective and Catadioptric Objectives 29.1

Lloyd Jones 29.1 29.2 29.3 29.4 29.5 29.6 29.7

Glossary / 29.1 Introduction / 29.2 Glass Varieties / 29.2 Introduction to Catadioptric and Reflective Objectives / Field-of-View Plots / 29.34 Definitions / 29.36 References / 29.38

29.2

Chapter 30. Scanners Leo Beiser and R. Barry Johnson 30.1 30.2 30.3 30.4 30.5 30.6 30.7 30.8 30.9 30.10

Glossary / 30.1 Introduction / 30.2 Scanned Resolution / 30.6 Scanners for Remote Sensing / 30.14 Scanning for Input/Output Imaging / 30.25 Scanner Devices and Techniques / 30.34 Scan-Error Reduction / 30.48 Agile Beam Steering / 30.51 References / 30.64 Further Reading / 30.68

Chapter 31. Optical Spectrometers Brian Henderson 31.1 31.2 31.3 31.4 31.5 31.6 31.7 31.8 31.9

31.1

Glossary / 31.1 Introduction / 31.2 Optical Absorption Spectrometers / 31.2 Luminescence Spectrometers / 31.5 Photoluminescence Decay Time / 31.12 Polarization Spectrometers / 31.15 High-Resolution Techniques / 31.23 Light Scattering / 31.30 References / 31.31

Chapter 32. Interferometers Parameswaran Hariharan 32.1 32.2 32.3 32.4 32.5 32.6 32.7 32.8 32.9 32.10

30.1

Glossary / 32.1 Introduction / 32.2 Basic Types of Interferometers / 32.2 Three-Beam and Double-Passed Two-Beam Interferometers Fringe-Counting Interferometers / 32.8 Two-Wavelength Interferometry / 32.9 Frequency-Modulation Interferometers / 32.9 Heterodyne Interferometers / 32.10 Phase-Shifting Interferometers / 32.10 Phase-Locked Interferometers / 32.11

/ 32.7

32.1

xvi

CONTENTS

32.11 32.12 32.13 32.14 32.15 32.16 32.17 32.18

Laser-Doppler Interferometers / 32.12 Laser-Feedback Interferometers / 32.13 Fiber Interferometers / 32.14 Interferometeric Wave Meters / 32.16 Second-Harmonic and Phase-Conjugate Interferometers Stellar Interferometers / 32.19 Gravitational-Wave Interferometers / 32.21 References / 32.22

/

32.17

Chapter 33. Holography and Holographic Instruments Lloyd Huff 33.1 33.2 33.3 33.4 33.5 33.6 33.7 33.8 33.9 33.10

Glossary / 33.1 Introduction / 33.2 Background and Basic Principles / 33.2 Holographic Interferometry / 33.4 Holographic Optical Elements / 33.13 Holographic Inspection / 33.16 Holographic Lithography / 33.22 Holographic Memory / 33.24 Conclusion / 33.25 References / 33.25

Chapter 34. Xerographic Systems Howard Stark 34.1 34.2 34.3 34.4 34.5 34.6 34.7 34.8 34.9

33.1

34.1

Introduction and Overview / 34.1 Creation of the Latent Image / 34.1 Development / 34.5 Transfer / 34.10 Fusing / 34.10 Cleaning and Erasing / 34.10 Control Systems / 34.11 Color / 34.11 References / 34.13

Chapter 35. Principles of Optical Disk Data Storage Masud Mansuripur 35.1 35.2 35.3 35.4 35.5 35.6 35.7 35.8 35.9 35.10 35.11

Index

Introduction / 35.1 Preliminaries and Basic Definitions / 35.2 The Optical Path / 35.7 Automatic Focusing / 35.12 Automatic Tracking / 35.14 Thermomagnetic Recording Process / 35.17 Magneto-Optical Readout / 35.21 Materials of Magneto-Optical Recording / 35.25 Concluding Remarks / 35.28 Further Information / 35.30 Bibliography / 35.31

I.1

35.1

CONTRIBUTORS

Rasheed M. A. Azzam Louisiana (CHAP. 16)

Department of Electrical Engineering, University of New Orleans, New Orleans,

Leo Beiser Consultant, Flushing, New York (CHAP. 30) Gisele Bennett Electro-Optical Systems Laboratory and School of Electrical and Computer Engineering, Georgia Tech Research Institute, Georgia Institute of Technology, Atlanta, Georgia (CHAP. 6) Jean M. Bennett Research Department, Michelson Laboratory, Naval Air Warfare Center, China Lake, California (CHAPS. 12, 13) Ellis Betensky Opcon Associates, Inc., West Redding, Connecticut (CHAP. 27) Craig F. Bohren

Pennsylvania State University, University Park, Pennsylvania (CHAP. 7)

Glenn D. Boreman Florida (CHAP. 4)

CREOL, The College of Optics and Photonics, University of Central Florida, Orlando, Naval Research Laboratory, Washington, D.C. (CHAP. 5)

William H. Carter

Russell A. Chipman College of Optical Sciences, University of Arizona, Tucson, Arizona (CHAPS. 14, 15) Instrumentation Division, Brookhaven National Laboratory, Upton, New York (CHAP. 8)

Eugene L. Church Aristide Dogariu (CHAP. 9) Jeremy Ellis (CHAP. 9)

CREOL, The College of Optics and Photonics, University of Central Florida, Orlando, Florida

CREOL, The College of Optics and Photonics, University of Central Florida, Orlando, Florida

Michael W. Farn

MIT/Lincoln Laboratory, Lexington, Massachusetts (CHAP. 23)

Norman Goldberg

Madison, Wisconsin (CHAP. 25)

Douglas S. Goodman Corning Tropel Corporation, Fairport, New York (CHAP. 1) Joseph W. Goodman Department of Electrical Engineering, Stanford University, Stanford, California (CHAP. 11) John E. Greivenkamp

College of Optical Sciences, University of Arizona, Tucson, Arizona (CHAP. 2)

Parameswaran Hariharan

School of Physics, University of Sydney, Sydney, Australia (CHAP. 32)

Brian Henderson Department of Physics and Applied Physics, University of Strathclyde, Glasgow, United Kingdom (CHAPS. 10, 31) Gerald C. Holst

JCD Publishing, Winter Park, Florida (CHAP. 26)

Lloyd Huff Research Institute, University of Dayton, Dayton, Ohio (CHAP. 33) J. Christopher James Electro-Optical Systems Laboratory, Georgia Tech Research Institute, Georgia Institute of Technology, Atlanta, Georgia (CHAP. 6) R. Barry Johnson Consultant, Huntsville, Alabama (CHAPS. 17, 30) Lloyd Jones


Thomas L. Koch Lehigh University, Bethlehem, Pennsylvania (CHAP. 21) Melvin H. Kreitzer

Opcon Associates, Inc., Cincinnati, Ohio (CHAP. 27)

Deceased. xvii

xviii

CONTRIBUTORS

Frederick J. Leonberger

MIT Center for Integrated Photonic Systems, Cambridge, Massachusetts (CHAP. 21)

Masud Mansuripur


Arvind S. Marathay


John F. McCalmont Air Force Research Laboratory, Sensors Directorate, Wright-Patterson AFB, Ohio (CHAP. 3) Tom D. Milster


Duncan T. Moore The Institute of Optics, and Gradient Lens Corporation, Rochester, New York (CHAP. 24) Opcon Associates, Inc., Cincinnati, Ohio (CHAP. 27)

Jacob Moskovich

Rudolf Oldenbourg Marine Biological Laboratory, Woods Hole, Massachusetts, and Physics Department, Brown University, Providence, Rhode Island (CHAP. 28) William T. Rhodes School of Electrical and Computer Engineering, Georgia Institute of Technology, and Department of Electrical Engineering and Imaging Technology Center, Florida Atlantic University, Boca Raton, Florida (CHAP. 6) Michael Shribak Marine Biological Laboratory, Woods Hole, Massachusetts (CHAP. 28) Howard Stark† Xerox Corporation, Corporate Research and Technology, Rochester, New York (CHAP. 34) Paul G. Suchoski Audigence Inc., Melbourne, Florida (CHAP. 21) Peter Z. Takacs

Instrumentation Division, Brookhaven National Laboratory, Upton, New York (CHAP. 8)

Tomasz S. Tkaczyk Wilfrid B. Veldkamp

Department of Bioengineering, Rice University, Houston, Texas (CHAP. 22) MIT/Lincoln Laboratory, Lexington, Massachusetts (CHAP. 23)

William B. Wetherell† Optical Research Associates, Framingham, Massachusetts (CHAP. 18) William L. Wolfe


George J. Zissis

Environmental Research Institute of Michigan, Ann Arbor, Michigan (CHAP. 20)

†

Retired.

BRIEF CONTENTS OF ALL VOLUMES

VOLUME I. GEOMETRICAL AND PHYSICAL OPTICS, POLARIZED LIGHT, COMPONENTS AND INSTRUMENTS PART 1. GEOMETRICAL OPTICS Chapter 1.

General Principles of Geometrical Optics

Douglas S. Goodman

PART 2. PHYSICAL OPTICS Chapter 2. Chapter 3. Chapter 4. Chapter 5. Chapter 6.

Interference John E. Greivenkamp Diffraction Arvind S. Marathay and John F. McCalmont Transfer Function Techniques Glenn D. Boreman Coherence Theory William H. Carter Coherence Theory: Tools and Applications Gisele Bennett, William T. Rhodes, and J. Christopher James Chapter 7. Scattering by Particles Craig F. Bohren Chapter 8. Surface Scattering Eugene L. Church and Peter Z. Takacs Chapter 9. Volume Scattering in Random Media Aristide Dogariu and Jeremy Ellis Chapter 10. Optical Spectroscopy and Spectroscopic Lineshapes Brian Henderson Chapter 11. Analog Optical Signal and Image Processing Joseph W. Goodman PART 3. POLARIZED LIGHT Chapter 12. Chapter 13. Chapter 14. Chapter 15. Chapter 16.

Polarization Jean M. Bennett Polarizers Jean M. Bennett Mueller Matrices Russell A. Chipman Polarimetry Russell A. Chipman Ellipsometry Rasheed M. A. Azzam

PART 4. COMPONENTS Chapter 17. Chapter 18. Chapter 19. Chapter 20. Chapter 21. Chapter 22. Chapter 23. Chapter 24.

Lenses R. Barry Johnson Afocal Systems William B. Wetherell Nondispersive Prisms William L. Wolfe Dispersive Prisms and Gratings George J. Zissis Integrated Optics Thomas L. Koch, Frederick J. Leonberger, and Paul G. Suchoski Miniature and Micro-Optics Tom D. Milster and Tomasz S. Tkaczyk Binary Optics Michael W. Farn and Wilfrid B. Veldkamp Gradient Index Optics Duncan T. Moore

PART 5. INSTRUMENTS Chapter 25. Chapter 26. Chapter 27. Chapter 28.

Cameras Norman Goldberg Solid-State Cameras Gerald C. Holst Camera Lenses Ellis Betensky, Melvin H. Kreitzer, and Jacob Moskovich Microscopes Rudolf Oldenbourg and Michael Shribak xix

xx


Chapter 29. Chapter 30. Chapter 31. Chapter 32. Chapter 33. Chapter 34. Chapter 35.

Reflective and Catadioptric Objectives Lloyd Jones Scanners Leo Beiser and R. Barry Johnson Optical Spectrometers Brian Henderson Interferometers Parameswaran Hariharan Holography and Holographic Instruments Lloyd Huff Xerographic Systems Howard Stark Principles of Optical Disk Data Storage Masud Mansuripur

VOLUME II. DESIGN, FABRICATION, AND TESTING; SOURCES AND DETECTORS; RADIOMETRY AND PHOTOMETRY PART 1. DESIGN Chapter 1. Chapter 2. Chapter 3. Chapter 4. Chapter 5. Chapter 6. Chapter 7. Chapter 8.

Techniques of First-Order Layout Warren J. Smith Aberration Curves in Lens Design Donald C. O’Shea and Michael E. Harrigan Optical Design Software Douglas C. Sinclair Optical Specifications Robert R. Shannon Tolerancing Techniques Robert R. Shannon Mounting Optical Components Paul R. Yoder, Jr. Control of Stray Light Robert P. Breault Thermal Compensation Techniques Philip J. Rogers and Michael Roberts

PART 2. FABRICATION Chapter 9. Optical Fabrication Michael P. Mandina Chapter 10. Fabrication of Optics by Diamond Turning

Richard L. Rhorer and Chris J. Evans

PART 3. TESTING Chapter 11. Chapter 12. Chapter 13. Chapter 14.

Orthonormal Polynomials in Wavefront Analysis Virendra N. Mahajan Optical Metrology Zacarías Malacara and Daniel Malacara-Hernández Optical Testing Daniel Malacara-Hernández Use of Computer-Generated Holograms in Optical Testing Katherine Creath and James C. Wyant

PART 4. SOURCES Chapter 15. Chapter 16. Chapter 17. Chapter 18. Chapter 19. Chapter 20. Chapter 21. Chapter 22. Chapter 23.

Artificial Sources Anthony LaRocca Lasers William T. Silfvast Light-Emitting Diodes Roland H. Haitz, M. George Craford, and Robert H. Weissman High-Brightness Visible LEDs Winston V. Schoenfeld Semiconductor Lasers Pamela L. Derry, Luis Figueroa, and Chi-shain Hong Ultrashort Optical Sources and Applications Jean-Claude Diels and Ladan Arissian Attosecond Optics Zenghu Chang Laser Stabilization John L. Hall, Matthew S. Taubman, and Jun Ye Quantum Theory of the Laser János A. Bergou, Berthold-Georg Englert, Melvin Lax, Marian O. Scully, Herbert Walther, and M. Suhail Zubairy

PART 5. DETECTORS Chapter 24. Chapter 25. Chapter 26. Chapter 27. Chapter 28.

Photodetectors Paul R. Norton Photodetection Abhay M. Joshi and Gregory H. Olsen High-Speed Photodetectors John E. Bowers and Yih G. Wey Signal Detection and Analysis John R. Willison Thermal Detectors William L. Wolfe and Paul W. Kruse

PART 6. IMAGING DETECTORS Chapter 29. Photographic Films Joseph H. Altman Chapter 30. Photographic Materials John D. Baloga


xxi

Chapter 31. Image Tube Intensified Electronic Imaging C. Bruce Johnson and Larry D. Owen Chapter 32. Visible Array Detectors Timothy J. Tredwell Chapter 33. Infrared Detector Arrays Lester J. Kozlowski and Walter F. Kosonocky PART 7. RADIOMETRY AND PHOTOMETRY Chapter 34. Chapter 35. Chapter 36. Chapter 37. Chapter 38. Chapter 39. Chapter 40.

Radiometry and Photometry Edward F. Zalewski Measurement of Transmission, Absorption, Emission, and Reflection James M. Palmer Radiometry and Photometry: Units and Conversions James M. Palmer Radiometry and Photometry for Vision Optics Yoshi Ohno Spectroradiometry Carolyn J. Sher DeCusatis Nonimaging Optics: Concentration and Illumination William Cassarly Lighting and Applications Anurag Gupta and R. John Koshel

VOLUME III. VISION AND VISION OPTICS Chapter 1. Chapter 2. Chapter 3. Chapter 4. Chapter 5. Chapter 6. Chapter 7. Chapter 8. Chapter 9. Chapter 10. Chapter 11. Chapter 12. Chapter 13. Chapter 14. Chapter 15. Chapter 16. Chapter 17. Chapter 18. Chapter 19. Chapter 20. Chapter 21. Chapter 22. Chapter 23. Chapter 24. Chapter 25.

Optics of the Eye Neil Charman Visual Performance Wilson S. Geisler and Martin S. Banks Psychophysical Methods Denis G. Pelli and Bart Farell Visual Acuity and Hyperacuity Gerald Westheimer Optical Generation of the Visual Stimulus Stephen A. Burns and Robert H. Webb The Maxwellian View with an Addendum on Apodization Gerald Westheimer Ocular Radiation Hazards David H. Sliney Biological Waveguides Vasudevan Lakshminarayanan and Jay M. Enoch The Problem of Correction for the Stiles-Crawford Effect of the First Kind in Radiometry and Photometry, a Solution Jay M. Enoch and Vasudevan Lakshminarayanan Colorimetry David H. Brainard and Andrew Stockman Color Vision Mechanisms Andrew Stockman and David H. Brainard Assessment of Refraction and Refractive Errors and Their Influence on Optical Design B. Ralph Chou Binocular Vision Factors That Influence Optical Design Clifton Schor Optics and Vision of the Aging Eye John S. Werner, Brooke E. Schefrin, and Arthur Bradley Adaptive Optics in Retinal Microscopy and Vision Donald T. Miller and Austin Roorda Refractive Surgery, Correction of Vision, PRK, and LASIK L. Diaz-Santana and Harilaos Ginis Three-Dimensional Confocal Microscopy of the Living Human Cornea Barry R. Masters Diagnostic Use of Optical Coherence Tomography in the Eye Johannes F. de Boer Gradient Index Optics in the Eye Barbara K. Pierscionek Optics of Contact Lenses Edward S. Bennett Intraocular Lenses Jim Schwiegerling Displays for Vision Research William Cowan Vision Problems at Computers Jeffrey Anshel and James E. Sheedy Human Vision and Electronic Imaging Bernice E. Rogowitz, Thrasyvoulos N. Pappas, and Jan P. Allebach Visual Factors Associated with Head-Mounted Displays Brian H. Tsou and Martin Shenker

VOLUME IV. OPTICAL PROPERTIES OF MATERIALS, NONLINEAR OPTICS, QUANTUM OPTICS PART 1. PROPERTIES Chapter 1. Chapter 2. Chapter 3. Chapter 4.

Optical Properties of Water Curtis D. Mobley Properties of Crystals and Glasses William J. Tropf, Michael E. Thomas, and Eric W. Rogala Polymeric Optics John D. Lytle Properties of Metals Roger A. Paquin

xxii


Chapter 5. Chapter 6. Chapter 7. Chapter 8. Chapter 9.

Optical Properties of Semiconductors David G. Seiler, Stefan Zollner, Alain C. Diebold, and Paul M. Amirtharaj Characterization and Use of Black Surfaces for Optical Systems Stephen M. Pompea and Robert P. Breault Optical Properties of Films and Coatings Jerzy A. Dobrowolski Fundamental Optical Properties of Solids Alan Miller Photonic Bandgap Materials Pierre R. Villeneuve

PART 2. NONLINEAR OPTICS Chapter 10. Chapter 11. Chapter 12. Chapter 13. Chapter 14. Chapter 15. Chapter 16. Chapter 17. Chapter 18. Chapter 19.

Nonlinear Optics Chung L. Tang Coherent Optical Transients Paul R. Berman and D. G. Steel Photorefractive Materials and Devices Mark Cronin-Golomb and Marvin Klein Optical Limiting David J. Hagan Electromagnetically Induced Transparency Jonathan P. Marangos and Thomas Halfmann Stimulated Raman and Brillouin Scattering John Reintjes and M. Bashkansky Third-Order Optical Nonlinearities Mansoor Sheik-Bahae and Michael P. Hasselbeck Continuous-Wave Optical Parametric Oscillators M. Ebrahim-Zadeh Nonlinear Optical Processes for Ultrashort Pulse Generation Uwe Siegner and Ursula Keller Laser-Induced Damage to Optical Materials Marion J. Soileau

PART 3. QUANTUM AND MOLECULAR OPTICS Chapter 20. Chapter 21. Chapter 22. Chapter 23.

Laser Cooling and Trapping of Atoms Harold J. Metcalf and Peter van der Straten Strong Field Physics Todd Ditmire Slow Light Propagation in Atomic and Photonic Media Jacob B. Khurgin Quantum Entanglement in Optical Interferometry Hwang Lee, Christoph F. Wildfeuer, Sean D. Huver, and Jonathan P. Dowling

VOLUME V. ATMOSPHERIC OPTICS, MODULATORS, FIBER OPTICS, X-RAY AND NEUTRON OPTICS PART 1. MEASUREMENTS Chapter 1. Chapter 2.

Scatterometers John C. Stover Spectroscopic Measurements Brian Henderson

PART 2. ATMOSPHERIC OPTICS Chapter 3. Chapter 4. Chapter 5.

Atmospheric Optics Dennis K. Killinger, James H. Churnside, and Laurence S. Rothman Imaging through Atmospheric Turbulence Virendra N. Mahajan and Guang-ming Dai Adaptive Optics Robert Q. Fugate

PART 3. MODULATORS Chapter 6. Chapter 7. Chapter 8.

Acousto-Optic Devices I-Cheng Chang Electro-Optic Modulators Georgeanne M. Purvinis and Theresa A. Maldonado Liquid Crystals Sebastian Gauza and Shin-Tson Wu

PART 4. FIBER OPTICS Chapter 9. Chapter 10. Chapter 11. Chapter 12. Chapter 13. Chapter 14.

Optical Fiber Communication Technology and System Overview Ira Jacobs Nonlinear Effects in Optical Fibers John A. Buck Photonic Crystal Fibers Philip St. J. Russell and G. J. Pearce Infrared Fibers James A. Harrington Sources, Modulators, and Detectors for Fiber Optic Communication Systems Elsa Garmire Optical Fiber Amplifiers John A. Buck


Chapter 15. Fiber Optic Communication Links (Telecom, Datacom, and Analog) Casimer DeCusatis and Guifang Li Chapter 16. Fiber-Based Couplers Daniel Nolan Chapter 17. Fiber Bragg Gratings Kenneth O. Hill Chapter 18. Micro-Optics-Based Components for Networking Joseph C. Palais Chapter 19. Semiconductor Optical Amplifiers Jay M. Wiesenfeld and Leo H. Spiekman Chapter 20. Optical Time-Division Multiplexed Communication Networks Peter J. Delfyett Chapter 21. WDM Fiber-Optic Communication Networks Alan E. Willner, Changyuan Yu, Zhongqi Pan, and Yong Xie Chapter 22. Solitons in Optical Fiber Communication Systems Pavel V. Mamyshev Chapter 23. Fiber-Optic Communication Standards Casimer DeCusatis Chapter 24. Optical Fiber Sensors Richard O. Claus, Ignacio Matias, and Francisco Arregui Chapter 25. High-Power Fiber Lasers and Amplifiers Timothy S. McComb, Martin C. Richardson, and Michael Bass PART 5. X-RAY AND NEUTRON OPTICS

Subpart 5.1. Introduction and Applications Chapter 26. Chapter 27. Chapter 28. Chapter 29. Chapter 30. Chapter 31. Chapter 32. Chapter 33. Chapter 34. Chapter 35. Chapter 36.

An Introduction to X-Ray and Neutron Optics Carolyn A. MacDonald Coherent X-Ray Optics and Microscopy Qun Shen Requirements for X-Ray diffraction Scott T. Misture Requirements for X-Ray Fluorescence George J. Havrilla Requirements for X-Ray Spectroscopy Dirk Lützenkirchen-Hecht and Ronald Frahm Requirements for Medical Imaging and X-Ray Inspection Douglas Pfeiffer Requirements for Nuclear Medicine Lars R. Furenlid Requirements for X-Ray Astronomy Scott O. Rohrbach Extreme Ultraviolet Lithography Franco Cerrina and Fan Jiang Ray Tracing of X-Ray Optical Systems Franco Cerrina and M. Sanchez del Rio X-Ray Properties of Materials Eric M. Gullikson

Subpart 5.2. Refractive and Interference Optics Chapter 37. Refractive X-Ray Lenses Bruno Lengeler and Christian G. Schroer Chapter 38. Gratings and Monochromators in the VUV and Soft X-Ray Spectral Region Malcolm R. Howells Chapter 39. Crystal Monochromators and Bent Crystals Peter Siddons Chapter 40. Zone Plates Alan Michette Chapter 41. Multilayers Eberhard Spiller Chapter 42. Nanofocusing of Hard X-Rays with Multilayer Laue Lenses Albert T. Macrander, Hanfei Yan, Hyon Chol Kang, Jörg Maser, Chian Liu, Ray Conley, and G. Brian Stephenson Chapter 43. Polarizing Crystal Optics Qun Shen

Subpart 5.3. Reflective Optics Chapter 44. Chapter 45. Chapter 46. Chapter 47. Chapter 48. Chapter 49. Chapter 50. Chapter 51. Chapter 52. Chapter 53.

Reflective Optics James Harvey Aberrations for Grazing Incidence Optics Timo T. Saha X-Ray Mirror Metrology Peter Z. Takacs Astronomical X-Ray Optics Marshall K. Joy and Brian D. Ramsey Multifoil X-Ray Optics Ladislav Pina Pore Optics Marco Beijersbergen Adaptive X-Ray Optics Ali Khounsary The Schwarzschild Objective Franco Cerrina Single Capillaries Donald H. Bilderback and Sterling W. Cornaby Polycapillary X-Ray Optics Carolyn MacDonald and Walter M. Gibson

xxiii

xxiv


Subpart 5.4. X-Ray Sources Chapter 54. Chapter 55. Chapter 56. Chapter 57. Chapter 58. Chapter 59.

X-Ray Tube Sources Susanne M. Lee and Carolyn MacDonald Synchrotron Sources Steven L. Hulbert and Gwyn P. Williams Laser Generated Plasmas Alan Michette Pinch Plasma Sources Victor Kantsyrev X-Ray Lasers Greg Tallents Inverse Compton X-Ray Sources Frank Carroll

Subpart 5.5. X-Ray Detectors Chapter 60. Introduction to X-Ray Detectors Walter M. Gibson and Peter Siddons Chapter 61. Advances in Imaging Detectors Aaron Couture Chapter 62. X-Ray Spectral Detection and Imaging Eric Lifshin

Subpart 5.6. Neutron Optics and Applications Chapter 63. Neutron Optics David Mildner Chapter 64. Grazing-Incidence Neutron Optics

Mikhail Gubarev and Brian Ramsey

EDITORS’ PREFACE The third edition of the Handbook of Optics is designed to pull together the dramatic developments in both the basic and applied aspects of the field while retaining the archival, reference book value of a handbook. This means that it is much more extensive than either the first edition, published in 1978, or the second edition, with Volumes I and II appearing in 1995 and Volumes III and IV in 2001. To cover the greatly expanded field of optics, the Handbook now appears in five volumes. Over 100 authors or author teams have contributed to this work. Volume I is devoted to the fundamentals, components, and instruments that make optics possible. Volume II contains chapters on design, fabrication, testing, sources of light, detection, and a new section devoted to radiometry and photometry. Volume III concerns vision optics only and is printed entirely in color. In Volume IV there are chapters on the optical properties of materials, nonlinear, quantum and molecular optics. Volume V has extensive sections on fiber optics and x ray and neutron optics, along with shorter sections on measurements, modulators, and atmospheric optical properties and turbulence. Several pages of color inserts are provided where appropriate to aid the reader. A purchaser of the print version of any volume of the Handbook will be able to download a digital version containing all of the material in that volume in PDF format to one computer (see download instructions on bound-in card). The combined index for all five volumes can be downloaded from www.HandbookofOpticsOnline.com. It is possible by careful selection of what and how to present that the third edition of the Handbook could serve as a text for a comprehensive course in optics. In addition, students who take such a course would have the Handbook as a career-long reference. Topics were selected by the editors so that the Handbook could be a desktop (bookshelf) general reference for the parts of optics that had matured enough to warrant archival presentation. New chapters were included on topics that had reached this stage since the second edition, and existing chapters from the second edition were updated where necessary to provide this compendium. In selecting subjects to include, we also had to select which subjects to leave out. The criteria we applied were: (1) was it a specific application of optics rather than a core science or technology and (2) was it a subject in which the role of optics was peripheral to the central issue addressed. Thus, such topics as medical optics, laser surgery, and laser materials processing were not included. While applications of optics are mentioned in the chapters there is no space in the Handbook to include separate chapters devoted to all of the myriad uses of optics in today’s world. If we had, the third edition would be much longer than it is and much of it would soon be outdated. We designed the third edition of the Handbook of Optics so that it concentrates on the principles of optics that make applications possible. Authors were asked to try to achieve the dual purpose of preparing a chapter that was a worthwhile reference for someone working in the field and that could be used as a starting point to become acquainted with that aspect of optics. They did that and we thank them for the outstanding results seen throughout the Handbook. We also thank Mr. Taisuke Soda of McGraw-Hill for his help in putting this complex project together and Mr. Alan Tourtlotte and Ms. Susannah Lehman of the Optical Society of America for logistical help that made this effort possible. We dedicate the third edition of the Handbook of Optics to all of the OSA volunteers who, since OSA’s founding in 1916, give their time and energy to promoting the generation, application, archiving, and worldwide dissemination of knowledge in optics and photonics. Michael Bass, Editor-in-Chief Associate Editors: Casimer M. DeCusatis Jay M. Enoch Vasudevan Lakshminarayanan Guifang Li Carolyn MacDonald Virendra N. Mahajan Eric Van Stryland xxv


PREFACE TO VOLUME I

The third edition of the Handbook of Optics has been completely reorganized, expanded, and updated. The four volumes of the second edition grew to five in the current edition. Each volume is divided into parts, where each part, sometimes referred to as a section, consists of several chapters related to a certain topic. Volumes I and II are devoted primarily to the basic concepts of optics and optical phenomena, sometimes called classical optics. Volume I starts with geometrical optics and continues with physical optics. This includes interference, diffraction, coherence theory, and scattering. A new chapter on tools and applications of coherence theory has been added. A several-chapter section follows devoted to issues of polarized light. The chapter on polarimetry has been updated and its content on the Mueller matrices now appears in a separate chapter by that title. Next there are chapters on components such as lenses, afocal systems, nondispersive and dispersive prisms, and special optics that include integrated, miniature and micro-, binary, and gradient index optics. Finally, there are several chapters on instruments. They include cameras and camera lenses, microscopes, reflective and catadioptric objectives, scanners, spectrometers, interferometers, xerographic systems, and optical disc data storage. There are many other chapters in this edition of the Handbook that could have been included in Volumes I and II. However, page limitations prevented that. For example, in Volume V there is a section on Atmospheric Optics. It consists of three chapters, one on transmission through the atmosphere, another on imaging through atmospheric turbulence, and a third on adaptive optics to overcome some of the deleterious effects of turbulence. The chapters are generally aimed at the graduate students, though practicing scientists and engineers will find them equally suitable as references on the topics discussed. Each chapter has sufficient references for additional and/or further study. The whole Handbook has been retyped and the figures redrawn. The reader will find that the figures in the new edition are crisp. Ms. Arushi Chawla and her team from Glyph International have done an outstanding job in accomplishing this monumental task. Many of the authors updated and proofread their chapters. However, some authors have passed away since the second edition and others couldn’t be located. Every effort has been made to ensure that such chapters have been correctly reproduced. Virendra N. Mahajan The Aerospace Corporation Associate Editor

xxvii


GLOSSARY AND FUNDAMENTAL CONSTANTS

Introduction This glossary of the terms used in the Handbook represents to a large extent the language of optics. The symbols are representations of numbers, variables, and concepts. Although the basic list was compiled by the author of this section, all the editors have contributed and agreed to this set of symbols and definitions. Every attempt has been made to use the same symbols for the same concepts throughout the entire Handbook, although there are exceptions. Some symbols seem to be used for many concepts. The symbol C is a prime example, as it is used for absorptivity, absorption coefficient, coefficient of linear thermal expansion, and more. Although we have tried to limit this kind of redundancy, we have also bowed deeply to custom. Units The abbreviations for the most common units are given first. They are consistent with most of the established lists of symbols, such as given by the International Standards Organization ISO1 and the International Union of Pure and Applied Physics, IUPAP.2 Prefixes Similarly, a list of the numerical prefixes1 that are most frequently used is given, along with both the common names (where they exist) and the multiples of ten that they represent. Fundamental Constants The values of the fundamental constants3 are listed following the sections on SI units. Symbols The most commonly used symbols are then given. Most chapters of the Handbook also have a glossary of the terms and symbols specific to them for the convenience of the reader. In the following list, the symbol is given, its meaning is next, and the most customary unit of measure for the quantity is presented in brackets. A bracket with a dash in it indicates that the quantity is unitless. Note that there is a difference between units and dimensions. An angle has units of degrees or radians and a solid angle square degrees or steradians, but both are pure ratios and are dimensionless. The unit symbols as recommended in the SI system are used, but decimal multiples of some of the dimensions are sometimes given. The symbols chosen, with some cited exceptions, are also those of the first two references.

RATIONALE FOR SOME DISPUTED SYMBOLS The choice of symbols is a personal decision, but commonality improves communication. This section explains why the editors have chosen the preferred symbols for the Handbook. We hope that this will encourage more agreement. xxix

xxx


Fundamental Constants It is encouraging that there is almost universal agreement for the symbols for the fundamental constants. We have taken one small exception by adding a subscript B to the k for Boltzmann’s constant.

Mathematics We have chosen i as the imaginary almost arbitrarily. IUPAP lists both i and j, while ISO does not report on these.

Spectral Variables These include expressions for the wavelength N, frequency v, wave number U, W for circular or radian frequency, k for circular or radian wave number and dimensionless frequency x. Although some use f for frequency, it can be easily confused with electronic or spatial frequency. Some use ` for wave number, but, because of typography problems and agreement with ISO and IUPAP, we P have chosen U ; it should not be confused with the Stefan-Boltzmann constant. For spatial frequencies we have chosen Z and J, although fx and fy are sometimes used. ISO and IUPAP do not report on these.

Radiometry Radiometric terms are contentious. The most recent set of recommendations by ISO and IUPAP are L for radiance [Wcm–2sr–1], M for radiant emittance or exitance [Wcm–2], E for irradiance or incidance [Wcm–2], and I for intensity [Wsr–2]. The previous terms, W, H, N, and J, respectively, are still in many texts, notably Smith4 and Lloyd5 but we have used the revised set, although there are still shortcomings. We have tried to deal with the vexatious term intensity by using specific intensity when the units are Wcm–2sr–1, field intensity when they are Wcm–2, and radiometric intensity when they are Wsr–1. There are two sets to terms for these radiometric quantities, which arise in part from the terms for different types of reflection, transmission, absorption, and emission. It has been proposed that the ion ending indicate a process, that the ance ending indicate a value associated with a particular sample, and that the ivity ending indicate a generic value for a “pure” substance. Then one also has reflectance, transmittance, absorptance, and emittance as well as reflectivity, transmissivity, absorptivity, and emissivity. There are now two different uses of the word emissivity. Thus the words exitance, incidence, and sterance were coined to be used in place of emittance, irradiance, and radiance. It is interesting that ISO uses radiance, exitance, and irradiance whereas IUPAP uses radiance excitance [sic], and irradiance. We have chosen to use them both, i.e., emittance, irradiance, and radiance will be followed in square brackets by exitance, incidence, and sterance (or vice versa). Individual authors will use the different endings for transmission, reflection, absorption, and emission as they see fit. We are still troubled by the use of the symbol E for irradiance, as it is so close in meaning to electric field, but we have maintained that accepted use. The spectral concentrations of these quantities, indicated by a wavelength, wave number, or frequency subscript (e.g., LN) represent partial differentiations; a subscript q represents a photon quantity; and a subscript v indicates a quantity normalized to the response of the eye. Thereby, Lv is luminance, Ev illuminance, and Mv and Iv luminous emittance and luminous intensity. The symbols we have chosen are consistent with ISO and IUPAP. The refractive index may be considered a radiometric quantity. It is generally complex and is indicated by ñ = n – ik. The real part is the relative refractive index and k is the extinction coefficient. These are consistent with ISO and IUPAP, but they do not address the complex index or extinction coefficient.


xxxi

Optical Design For the most part ISO and IUPAP do not address the symbols that are important in this area. There were at least 20 different ways to indicate focal ratio; we have chosen FN as symmetrical with NA; we chose f and efl to indicate the effective focal length. Object and image distance, although given many different symbols, were finally called so and si since s is an almost universal symbol for distance. Field angles are S and H; angles that measure the slope of a ray to the optical axis are u; u can also be sin u. Wave aberrations are indicated by Wijk, while third-order ray aberrations are indicated by Ui and more mnemonic symbols. Electromagnetic Fields There is no argument about E and H for the electric and magnetic field strengths, Q for quantity of charge, T for volume charge density, U for surface charge density, etc. There is no guidance from Refs. 1 and 2 on polarization indication. We chose and || rather than p and s, partly because s is sometimes also used to indicate scattered light. There are several sets of symbols used for reflection transmission, and (sometimes) absorption, each with good logic. The versions of these quantities dealing with field amplitudes are usually specified with lower case symbols: r, t, and a. The versions dealing with power are alternately given by the uppercase symbols or the corresponding Greek symbols: R and T versus T and V. We have chosen to use the Greek, mainly because these quantities are also closely associated with Kirchhoff ’s law that is usually stated symbolically as C = G. The law of conservation of energy for light on a surface is also usually written as C T V 1.

Base SI Quantities length time mass electric current temperature amount of substance luminous intensity

m s kg A K mol cd

meter second kilogram ampere kelvin mole candela

J C V F 7 S Wb H Pa T Hz W N rad sr

joule coulomb volt farad ohm siemens weber henry pascal tesla hertz watt newton radian steradian

Derived SI Quantities energy electric charge electric potential electric capacitance electric resistance electric conductance magnetic flux inductance pressure magnetic flux density frequency power force angle angle

xxxii


Prefixes Symbol F P T G M k h da d c m O n p f a

Name exa peta tera giga mega kilo hecto deca deci centi milli micro nano pico femto atto

Common name trillion billion million thousand hundred ten tenth hundredth thousandth millionth billionth trillionth

Exponent of ten 18 15 12 9 6 3 2 1 –1 –2 –3 –6 –9 –12 –15 –18

Constants

c c1 c2 e gn h kB me NA Re Go U Oo OB

speed of light vacuo [299792458 ms–1] first radiation constant = 2Rc2h = 3.7417749 × 10–16 [Wm2] second radiation constant = hc/k = 0.014838769 [mK] elementary charge [1.60217733 × 10–19 C] free fall constant [9.80665 ms–2] Planck’s constant [6.6260755 × 10–34 Ws] Boltzmann constant [1.380658 × 10–23 JK–1] mass of the electron [9.1093897 × 10–31 kg] Avogadro constant [6.0221367 × 1023 mol–1] Rydberg constant [10973731.534 m–1] vacuum permittivity [Oo–1c –2] Stefan-Boltzmann constant [5.67051 × 10–8 Wm–1 K–4] vacuum permeability [4R × 10–7 NA–2] Bohr magneton [9.2740154 × 10–24 JT–1]

B C C c c1 c2 D E e Ev E E Eg f fc fv

magnetic induction [Wbm–2, kgs–1 C–1] capacitance [f, C2 s2 m–2 kg–1] curvature [m–1] speed of light in vacuo [ms–1] first radiation constant [Wm2] second radiation constant [mK] electric displacement [Cm–2] incidance [irradiance] [Wm–2] electronic charge [coulomb] illuminance [lux, lmm–2] electrical field strength [Vm–1] transition energy [J] band-gap energy [eV] focal length [m] Fermi occupation function, conduction band Fermi occupation function, valence band

General


FN g gth H h I I I I i Im() J j J1() k k k L Lv L L L, M, N M M m m MTF N N n ñ NA OPD P Re() R r S s s So Si T t t u V V x, y, z Z

focal ratio (f/number) [—] gain per unit length [m–1] gain threshold per unit length [m1] magnetic field strength [Am–1, Cs–1 m–1] height [m] irradiance (see also E) [Wm–2] radiant intensity [Wsr–1] nuclear spin quantum number [—] current [A]

1 imaginary part of current density [Am–2] total angular momentum [kg m2 s–1] Bessel function of the first kind [—] radian wave number =2R/N [rad cm–1] wave vector [rad cm–1] extinction coefficient [—] sterance [radiance] [Wm–2 sr–1] luminance [cdm–2] inductance [h, m2 kg C2] laser cavity length direction cosines [—] angular magnification [—] radiant exitance [radiant emittance] [Wm–2] linear magnification [—] effective mass [kg] modulation transfer function [—] photon flux [s–1] carrier (number)density [m–3] real part of the relative refractive index [—] complex index of refraction [—] numerical aperture [—] optical path difference [m] macroscopic polarization [C m–2] real part of [—] resistance [7] position vector [m] Seebeck coefficient [VK–1] spin quantum number [—] path length [m] object distance [m] image distance [m] temperature [K, C] time [s] thickness [m] slope of ray with the optical axis [rad] Abbe reciprocal dispersion [—] voltage [V, m2 kgs–2 C–1] rectangular coordinates [m] atomic number [—]

Greek Symbols C C

absorption coefficient [cm−1] (power) absorptance (absorptivity)

xxxiii

xxxiv


G G G G1 G2 V P Y Y N U U T S, H Z, J H H & E 7

diclectric coefficient (constant) [—] emittance (emissivity) [—] eccentricity [—] Re (G) lm (G) (power) transmittance (transmissivity) [—] radiation frequency [Hz] circular frequency = 2RP [rads−1] plasma frequency [H2] wavelength [Mm, nm] wave number = 1/N [cm–1] Stefan Boltzmann constant [Wm−2K−1] reflectance (reflectivity) [—] angular coordinates [rad, °] rectangular spatial frequencies [m−1, r−1] phase [rad, °] lens power [m−2] flux [W] electric susceptibility tensor [—] solid angle [sr]

Other exp (x) loga (x) ln (x) log (x) 3 0 $ Fx dx tx F(x) Fij

responsivity ex log to the base a of x natural log of x standard log of x: log10 (x) summation product finite difference variation in x total differential partial derivative of x Dirac delta function of x Kronecker delta

REFERENCES 1. Anonymous, ISO Standards Handbook 2: Units of Measurement, 2nd ed., International Organization for Standardization, 1982. 2. Anonymous, Symbols, Units and Nomenclature in Physics, Document U.I.P. 20, International Union of Pure and Applied Physics, 1978. 3. E. Cohen and B. Taylor, “The Fundamental Physical Constants,” Physics Today, 9 August 1990. 4. W. J. Smith, Modern Optical Engineering, 2nd ed., McGraw-Hill, 1990. 5. J. M. Lloyd, Thermal Imaging Systems, Plenum Press, 1972. William L. Wolfe College of Optical Sciences University of Arizona Tucson, Arizona

PART

1 GEOMETRICAL OPTICS


1 GENERAL PRINCIPLES OF GEOMETRICAL OPTICS Douglas S. Goodman Corning Tropel Corporation Fairport, New York

1.1

GLOSSARY (NS) italics prime, unprime A A, A` AB a aO , aB , aI B B B, B ` b C C( , , ) c c cs ct D d d d, d ` d, d ` dH dN dF

indicates nonstandard terminology definition or first usage gradient (v /v x, v /v y , v /v z) before and after, object and image space (not derivatives) auxiliary function for ray tracing area, total field areas, object and image points directed distance from A to B unit axis vector, vectors coefficients in characteristic function expansion matrix element for symmetrical systems auxiliary function for ray tracing arbitrary object and image points binormal unit vector of a ray path interspace (between) term in expansion matrix element for conjugacy characteristic function speed of light in vacuum surface vertex curvature, spherical surface curvature sagittal curvature tangential curvature auxiliary distance function for ray tracing distance from origin to mirror nominal focal distance arbitrary point to conjugate object, image points d AO, d ` AÒ ` axial distances, distances along rays hyperfocal distance near focal distance far focal distance 1.3

1.4

GEOMETRICAL OPTICS

dA ds E E0 E, E ` e ex , ey , ez F F, F ` FN FNm F( ) F(x, y, z) f, f ` G g, g ` h, h `

I, I ` I i, i' L L l, l `

M M m mL mC mE mN mP mO mx, my , mz N N, N ` NA, NA` n n NS O, O `

differential area differential geometrical path length image irradiance axial image irradiance entrance and exit pupil locations eccentricity coefficients for collineation matrix element for front side front and rear focal points F-number F-number for magnification m general function general surface function front and rear focal lengths f PF, f ` P `F ` diffraction order focal lengths in tilted planes ray heights at objects and images, field heights hamiltonian incidence angles unit matrix paraxial incidence angles image space term in characteristic function expansion surface x-direction cosine paraxial invariant principal points to object and image axial points l PO, l ` P Ò ` axial distances from vertices of refracting surface l VO, l ` V Ò ` lagrangian for heterogeneous media lambertian emittance surface z-direction cosine transverse magnification longitudinal magnification angular magnification paraxial pupil magnification nodal point magnification n/n` pupil magnification in direction cosines magnification at axial point magnifications in the x, y, and z directions surface z-direction cosine nodal points numerical aperture refractive index normal unit vector of a ray path nonstandard axial object and image points object space term in expansion

GENERAL PRINCIPLES OF GEOMETRICAL OPTICS

P P, P ` P(C, D; x, y) P `(C `, D `; x `, y `) p p p px , py , pz Q (A , B ; x , y) Q `(A `, B `; x `, y `) q qi qi q, q ` q R r r S S(x , y , x `, y `) s s s, s ` T(A , B ; A `, B `) t t, t ` t t U, U ` u, u ` uM uC u1, u2 , u3 , u4 V V(x; x `) V, V ` v WLMN Wx , Wy , Wz W(Z, J; x, y, z) W'(C, D; x `, y `) W(x, y; C `, D `) x (x, y, z) x(U)

power (radiometric) principal points pupil shape functions period of grating ray vector, optical direction cosine p n r (px, py, pz) pupil radius optical direction cosines pupil shape functions relative to principal direction cosines resolution parameter coordinate for Lagrange equations derivative with respect to a parameter auxiliary functions for collineation unit vector along grating lines matrix element for rear side radius of curvature, vertex radius of curvature ray unit direction vector r (C, D, I) surface normal S (L, M, N) point eikonal V (x , y , z 0 ; x `, y `, z 0` ) geometrical length axial length distances associated with sagittal foci skew invariant angle characteristic function thickness, vertex-to-vertex distance distances associated with tangential foci time tangent unit vector of a ray path meridional ray angles relative to axis paraxial ray angles relative to axis paraxial marginal ray angle paraxial chief ray angle homogeneous coordinates for collineation optical path length point characteristic function vertex points speed of light in medium wavefront aberration term wavefront aberration terms for reference shift wavefront aberration function angle-point characteristic function point-angle characteristic function position vector parametric description of ray path

1.5

1.6

GEOMETRICAL OPTICS

S) x( x(S ) y yM yC yP , y P` z z(T) zsphere zconic z, z ` C, D, I C, D, I C `, D `, I ` C0, D0 A 0` , B0` Cmax , Cmin Dmax, Dmin ' Fx, Fy, Fz $C, $D $x, $y, $z G Gx, Gy Z, J S

M M N [

F

T

U

V

V(C `, D `; x `, y `) Y, Y ` dY

derivative with respect to a parameter second derivative with respect to a parameter meridional ray height, paraxial ray height paraxial marginal ray height paraxial chief ray height paraxial ray height at the principal planes axis of revolution surface sag sag of a sphere sag of a conic focal point to object and image distances z FO, z ` F Ò ` ray direction cosines entrance pupil directions exit pupil direction cosines principal direction of entrance pupil principal direction of exit pupil extreme pupil directions extreme pupil directions n ` cos I ` − n cos I reference point shifts angular ray aberrations shifts surface shape parameter transverse ray aberrations pupil coordinates—not specific ray angle to surface normal marginal ray angle plane tilt angle conic parameter curvature of a ray path wavelength azimuth angle field angle power, surface power azimuth radius of curvature of a ray path distance from axis radial pupil coordinate ray path parameter general parameter for a curve reduced axial distances torsion of a ray path pupil transmittance function reduced angle Y nu, Y ` n ù ` differential solid angle


1.2

1.7

INTRODUCTION

The Subject Geometrical optics is both the object of abstract study and a body of knowledge necessary for design and engineering. The subject of geometrical optics is small, since so much can be derived from a single principle, that of Fermat, and large since the consequences are infinite and far from obvious. Geometrical optics is deceptive in that much that seems simple is loaded with content and implications, as might be suggested by the fact that some of the most basic results required the likes of Newton and Gauss to discover them. Most of what appears complicated seems so because of obscuration with mathematical terminology and excessive abstraction. Since it is so old, geometrical optics tends to be taken for granted and treated too casually by those who consider it to be “understood.” One consequence is that what has been long known can be lost if it is not recirculated by successive generations of textbook authors, who are pressed to fit newer material in a fairly constant number of pages.

The Contents The material in this chapter is intended to be that which is most fundamental, most general, and most useful to the greatest number of people. Some of this material is often thought to be more esoteric than practical, but this opinion is less related to its essence than to its typical presentation. There are no applications per se here, but everything is applicable, at least to understanding. An effort has been made to compensate here for what is lacking elsewhere and to correct some common errors. Many basic ideas and useful results have not found their way into textbooks, so are little known. Moreover, some basic principles are rarely stated explicitly. The contents are weighted toward the most common type of optical system, that with rotational symmetry consisting of mirrors and/or lens elements of homogeneous materials. There is a section “Rays in Heterogeneous Media,” an application of which is gradient index optics discussed in Chap. 24. The treatment here is mostly monochromatic. The topics of caustics and anisotropic media are omitted, and there is little specifically about systems that are not figures of revolution. The section on aberrations is short and mostly descriptive, with no discussion of lens design, a vast field concerned with the practice of aberration control. Because of space limitations, there are too few diagrams.

Terminology Because of the complicated history of geometrical optics, its terminology is far from standardized. Geometrical optics developed over centuries in many countries, and much of it has been rediscovered and renamed. Moreover, concepts have come into use without being named, and important terms are often used without formal definitions. This lack of standardization complicates communication between workers at different organizations, each of which tends to develop its own optical dialect. Accordingly, an attempt has been made here to provide precise definitions. Terms are italicized where defined or first used. Some needed nonstandard terms have been introduced, and these are likewise italicized, as well as indicated by “NS” for “nonstandard.”

Notation As with terminology, there is little standardization. And, as usual, the alphabet has too few letters to represent all the needed quantities. The choice here has been to use some of the same symbols more than once, rather than to encumber them with superscripts and subscripts. No symbol is used in a given section with more than one meaning. As a general practice nonprimed and primed quantities are used to indicate before and after, input and output, and object and image space.

1.8

GEOMETRICAL OPTICS

References No effort has been made to provide complete references, either technical or historical. (Such a list would fill the entire chapter.) The references were not chosen for priority, but for elucidation or interest, or because of their own references. Newer papers can be found by computer searches, so the older ones have been emphasized, especially since older work is receding from view beneath the current flood of papers. In geometrical optics, nothing goes out of date, and much of what is included here has been known for a century or so—even if it has been subsequently forgotten.

Communication Because of the confusion in terminology and notation, it is recommended that communication involving geometrical optics be augmented with diagrams, graphs, equations, and numeric results, as appropriate. It also helps to provide diagrams showing both first-order properties of systems, with object and image positions, pupil positions, and principal planes, as well as direction cosine space diagrams, as required, to show angular subtenses of pupils.

1.3

FUNDAMENTALS

What Is a Ray? Geometrical optics, which might better be called ray optics, is concerned with the light ray, an entity that does not exist. It is customary, therefore, to begin discussions of geometrical optics with a theoretical justification for the use of the ray. The real justification is that, like other successful models in physics, rays are indispensable to our thinking, notwithstanding their shortcomings. The ray is a model that works well in some cases and not at all in others, and light is necessarily thought about in terms of rays, scalar waves, electromagnetic waves, and with quantum physics—depending on the class of phenomena under consideration. Rays have been defined with both corpuscular and wave theory. In corpuscular theory, some definitions are (1) the path of a corpuscle and (2) the path of a photon. A difficulty here is that energy densities can become infinite. Other efforts have been made to define rays as quantities related to the wave theory, both scalar and electromagnetic. Some are (1) wavefront normals, (2) the Poynting vector, (3) a discontinuity in the electromagnetic field,1,2 (4) a descriptor of wave behavior in short wavelength or high frequency limit,3 and (5) quantum mechanically.4 One problem with these definitions is that there are many ordinary and simple cases where wavefronts and Poynting vectors become complicated and/or meaningless. For example, in the simple case of two coherent plane waves interfering, there is no well-defined wavefront in the overlap region. In addition, rays defined in what seems to be a reasonble way can have undesirable properties. For example, if rays are defined as normals to wavefronts, then, in the case of gaussian beams, rays bend in a vacuum. An approach that avoids the difficulties of a physical definition is that of treating rays as mathematical entities. From definitions and postulates, a variety of results is found, which may be more or less useful and valid for light. Even with this approach, it is virtually impossible to think “purely geometrically”—unless rays are treated as objects of geometry, rather than optics. In fact, we often switch between ray thinking and wave thinking without noticing it, for instance in considering the dependence of refractive index on wavelength. Moreover, geometrical optics makes use of quantities that must be calculated from other models, for example, the index of refraction. As usual, Rayleigh5 has put it well: “We shall, however, find it advisable not to exclude altogether the conceptions of the wave theory, for on certain most important and practical questions no conclusion can be drawn without the use of facts which are scarcely otherwise interpretable. Indeed it is not to be denied that the too rigid separation of optics into geometrical and physical has done a good deal of harm, much that is essential to a proper comprehension of the subject having fallen between the two schools.”


1.9

The ray is inherently ill-defined, and attempts to refine a definition always break down. A definition that seems better in some ways is worse in others. Each definition provides some insight into the behavior of light, but does not give the full picture. There seems to be a problem associated with the uncertainty principle involved with attempts at definition, since what is really wanted from a ray is a specification of both position and direction, which is impossible by virtue of both classical wave properties and quantum behavior. So the approach taken here is to treat rays without precisely defining them, and there are few reminders hereafter that the predictions of ray optics are imperfect.

Refractive Index For the purposes of this chapter, the optical characteristics of matter are completely specified by its refractive index. The index of refraction of a medium is defined in geometrical optics as n

speed of light in vacuum c speed of light in medium v

(1)

A homogeneous medium is one in which n is the same everywhere. In an inhomogeneous or heterogeneous medium the index varies with position. In an isotropic medium n is the same at each point for light traveling in all directions and with all polarizations, so the index is described by a scalar function of position. Anisotropic media are not treated here. Care must be taken with equations using the symbol n, since it sometimes denotes the ratio of indices, sometimes with the implication that one of the two is unity. In many cases, the difference from unity of the index of air (1.0003) is important. Index varies with wavelength, but this dependence is not made explicit in this chapter, most of which is implicitly limited to monochromatic light. The output of a system in polychromatic light is the sum of outputs at the constituent wavelengths.

Systems Considered The optical systems considered here are those in which spatial variations of surface features or refractive indices are large compared to the wavelength. In such systems ray identity is preserved; there is no “splitting” of one ray into many as occurs at a grating or scattering surface. The term lens is used here to include a variety of systems. Dioptric or refractive systems employ only refraction. Catoptric or reflective systems employ only reflection. Catadioptric systems employ both refraction and reflection. No distinction is made here insofar as refraction and reflection can be treated in a common way. And the term lens may refer here to anything from a single surface to a system of arbitrary complexity.

Summary of the Behavior and Attributes of Rays Rays propagate in straight lines in homogeneous media and have curved paths in heterogeneous media. Rays have positions, directions, and speeds. Between any pair of points on a given ray there is a geometrical path length and an optical path length. At smooth interfaces between media with different indices rays refract and reflect. Ray paths are reversible. Rays carry energy, and power per area is approximated by ray density.

Reversibility Rays are reversible; a path can be taken in either direction, and reflection and refraction angles are the same in either direction. However, it is usually easier to think of light as traveling along rays in a particular direction, and, of course, in cases of real instruments there usually is such a direction. The solutions to some equations may have directional ambiguity.

1.10

GEOMETRICAL OPTICS

Groups of Rays Certain types of groups of rays are of particular importance. Rays that originate at a single point are called a normal congruence or orthotomic system, since as they propagate in isotropic media they are associated with perpendicular wavefronts. Such groups are also of interest in image formation, where their reconvergence to a point is important, as is the path length of the rays to a reference surface used for diffraction calculations. Important in radiometric considerations are groups of rays emanating from regions of a source over a range of angles. The changes of such groups as they propagate are constrained by conservation of brightness. Another group is that of two meridional paraxial rays, related by the two-ray invariant.

Invariance Properties Individual rays and groups of rays may have invariance properties—relationships between the positions, directions, and path lengths—that remain constant as a ray or a group of rays passes through an optical system.6 Some of these properties are completely general, e.g., the conservation of étendue and the perpendicularity of rays to wavefronts in isotropic media. Others arise from symmetries of the system, e.g., the skew invariant for rotationally symmetric systems. Other invariances hold in the paraxial limit. There are also differential invariance properties.7,8 Some ray properties not ordinarily thought of in this way can be thought of as invariances. For example, Snell’s law can be thought of as a refraction invariant n sin I.

Description of Ray Paths A ray path can be described parametrically as a locus of points x(U), where U is any monotonic parameter that labels points along the ray. The description of curved rays is elaborated in the section on heterogeneous media.

Real Rays and Virtual Rays Since rays in homogeneous media are straight, they can be extrapolated infinitely from a given region. The term real refers to the portion of the ray that “really” exists, or the accessible part, and the term virtual refers to the extrapolated, or inaccessible, part.

Direction At each position where the refractive index is continuous a ray has a unique direction. The direction is given by its unit direction vector r, whose cartesian components are direction cosines (C D I), i.e., r (A , B , G ) 2

2

2

(2)

2

where | r | A B G 1 . The three direction cosines are not independent, and one is often taken to depend implicitly on the other two. In this chapter it is usually I, which is

G (A , B ) 1 A 2 B 2

(3)

Another vector with the same direction as r is p n r n(A , B , G ) ( px , p y , pz ) 2

2

(4)

where |p| n . Several names are used for this vector, including the optical direction cosine and the ray vector.


1.11

Geometrical Path Length Geometrical path length is geometrical distance measured along a ray between any two points. The differential unit of length is (5)

ds dx 2 dy 2 dz 2

The path length between points x1 and x2 on a ray described parametrically by x(U), with derivative x (S ) dx(S )/dS is x2 x 2 ds x2 (6) s(x 1 ; x 2 ) ¯ ds ¯ dS ¯ |x (S )|2 dS x1 x1 dS x1 Optical Path Length The optical path length between two points x1 and x2 through which a ray passes is x2

Optical path length V (x 1 ; x 2 ) ¯ n(x ) ds c ¯ x1

ds c ¯ dt v

(7)

The integral is taken along the ray path, which may traverse homogeneous and inhomogeneous media, and include any number of reflections and refractions. Path length can be defined for virtual rays. In some cases, path length should be considered positive definite, but in others it can be either positive or negative, depending on direction.9 If x0, x1, and x2 are three points on the same ray, then V (x 0 ; x 2 ) V (x 0 ; x 1 ) V (x 1 ; x 2 )

(8)

Equivalently, the time required for light to travel between the two points is Time

x 2 ds optical path length V 1 x 2 ¯ n(x ) ds ¯ c c c x1 x1 v

(9)

In homogeneous media, rays are straight lines, and the optical path length is V n ¯ ds (index) r (distance between the points). The optical path length integral has several interpretations, and much of geometrical optics involves the examination of its meanings. (1) With both points fixed, it is simply a scalar, the optical path length from one point to another. (2) With one point fixed, say x0, then treated as a function of x, the surfaces V(x0; x) constant are geometrical wavefronts for light originating at x0. (3) Most generally, as a function of both arguments V(x1; x2) is the point characteristic function, which contains all the information about the rays between the region containing x1 and that containing x2. There may not be a ray between all pairs of points.

Fermat’s Principle According to Fermat’s principle10–15 the optical path between two points through which a ray passes is an extremum. Light passing through these points along any other nearby path would take either more or less time. The principle applies to different neighboring paths. The optical path length of a ray may not be a global extremum. For example, the path lengths of rays through different facets of a Fresnel lens have no particular relationship. Fermat’s principle applies to entire systems, as well as to any portion of a system, for example, to any section of a ray. In a homogeneous medium, the extremum is a straight line or, if there are reflections, a series of straight line segments. The extremum principle can be described mathematically as follows.16 With the end points fixed, if a nonphysical path differs from a physical one by an amount proportional to F, the nonphysical optical path length differs from the actual one by a quantity proportional to F2 or to a higher order. If the order is three or higher, the first point is imaged at the second-to-first order. Roughly speaking,

1.12

GEOMETRICAL OPTICS

the higher the order, the better the image. A point is imaged stigmatically when a continuum of neighboring paths have the same length, so the equality holds to all orders. If they are sufficiently close, but vary slightly, the deviation from equality is a measure of the aberration of the imaging. An extension of Fermat’s principle is given by Hopkins.17 Ray and wave optics are related by the importance of path length in both.18,19 In wave optics, optical path length is proportional to phase change, and the extremum principle is associated with constructive interference. The more alike the path lengths are from an object point to its image, the less the differences in phase of the wave contributions, and the greater the magnitude of the net field. In imaging this connection is manifested in the relationship of the wavefront aberration and the eikonal. Fermat’s principle is a unifying principle of geometrical optics that can be used to derive laws of reflection and refraction, and to find the equations that describe ray paths and geometrical wavefronts in heterogeneous and homogeneous media. It is one of a number of variational principles based historically on the idea that nature is economical, a unifying principle of physics. The idea that the path length is an extremum could be used mathematically without interpreting the refractive index in terms of the speed of light. Geometrical Wavefronts For rays originating at a single point, a geometrical wavefront is a surface that is a locus of constant optical path length from the source. If the source point is located at x0 and light leaves at time t0, then the wavefront at time t is given by V(x 0 ; x ) c(t t 0 )

(10)

The function V(x; x0), as a function of x, satisfies the eikonal equation 2

2

¤vV ³ ¤vV ³ ¤vV ³ n(x )2 ¥ ¦ v x ´µ ¥¦ v y ´µ ¥¦ v z ´µ

2

(11)

2

| $ V(x ; x 0 )|

This equation can also be written in relativistic form, with a four-dimensional gradient as 0 3(v V /v x i ).20 For constant refractive index, the eikonal equation has some simple solutions, one of which is V n[A (x x 0 ) B( y y0 ) G (z z 0 )], corresponding to a parallel bundle of rays with directions (C, D, I). Another is V n[(x x 0 )2 ( y y0 )2 (z z 0 )2 ]1/2 , describing rays traveling radially from a point (x0, y0, z0). In isotropic media, the rays and wavefronts are everywhere perpendicular to each other, a condition referred to as orthotomic. According to the Malus-Dupin principle, if a group of rays emanating front a single point is reflected and/or refracted any number of times, the perpendicularity of rays to wavefronts is maintained. The direction of a ray from x0 at x is that of the gradient of V(x0; x) p nr V or nA

vV vx

nB

vV vy

nG

vV vz

(12)

In a homogeneous medium, all wavefronts can be found from any one wavefront by a construction. Wavefront normals, i.e., rays, are projected from the known wavefront, and loci of points equidistant therefrom are other wavefronts. This gives wavefronts in both directions, that is, both subsequent and previous wavefronts. (A single wavefront contains no directional information.) The construction also gives virtual wavefronts, those which would occur or would have occurred if the medium extended infinitely. This construction is related to that of Huygens for wave optics. At each point on a wavefront there are two principal curvatures, so there are two foci along each ray and two caustic surfaces.8,21


1.13

The geometrical wavefront is analogous to the surface of constant phase in wave optics, and the eikonal equation can be obtained from the wave equation in the limit of small wavelength.3,4 A way in which wave optics differs from ray optics is that the phase fronts can be modified by phase changes that occur on reflection, transmission, or in passing through foci.

Fields of Rays In many cases the optical direction cosine vectors p form a field, where the optical path length is the potential, and the geometrical wavefronts are equipotential surfaces. The potential changes with position according to dV n A dx n B dy n G dz n r d x p d x

(13)

If dx is in the direction of a ray, then dV/dx n, the maximum rate of change. If dx is perpendicular to a ray, then dV/dx 0. The potential difference between any two wavefronts is x2

(14)

V2 V1 ¯ dV x1

where x1 and x2 are any two points on the respective wavefronts, and the integrand is independent of the path. Other relationships for rays originating at a single point are 0 r p r (n r)

and

0= u ¯ p dx

(15)

3

where the integral is about a closed path. These follow since p is a gradient, Eq. (13). In regions where the rays are folded onto themselves by refraction or reflections, p and V are not single-valued, so there is not a field.

1.4

CHARACTERISTIC FUNCTIONS

Introduction Characteristic functions contain all the information about the path lengths between pairs of points, which may either be in a contiguous region or physically separated, e.g., on the two sides of a lens. These functions were first considered by Hamilton,22 so their study is referred to as hamiltonian optics. They were rediscovered in somewhat different form by Bruns23,24 and referred to as eikonals, leading to a confusing set of names for the various functions. The subject is discussed in a number of books.25–36 Four parameters are required to specify a ray. For example, an input ray is defined in the z 0 plane by coordinates (x, y) and direction (C, D). So four functions of four variables specify how an incident ray emerges from a system. In an output plane z ` 0, the ray has coordinates x ` x `(x , y , A , B ), y ` y `(x , y , A , B ), and directions A ` A `(x , y , A , B ), B ` B `(x , y , A , B ). Because of Fermat’s principle, these four functions are not independent, and the geometrical optics properties of a system can be fully characterized by a single function.32 For any given system, there is a variety of characteristic functions related by Legendre transformations, with different combinations of spatial and angular variables.34 The different functions are suited for different types of analysis. Mixed characteristic functions have both spatial and angular arguments. Those functions that are of most general use are discussed next. The others may be useful in special circumstances. If the regions have constant refractive indices, the volumes over which the characteristic functions are defined can be extended virtually from physically accessible to inaccessible regions. From any of its characteristic functions, all the properties of a system involving ray paths can be found, for example, ray positions, directions, and geometrical wavefronts. An important use of

1.14

GEOMETRICAL OPTICS

the characteristic functions is demonstrating general principles and fundamental limitations. Much of this can be done by using the general properties, e.g., symmetry under rotation. (Unfortunately, it is not always known how closely the impossible can be approached.) Point Characteristic Function The point characteristic function is the optical path integral V (x ; x`) V (x , y , z ; x `, y `, z `) taken as a function of both points x and x `. At point x where the index is n,

n A

tV tx

nB

tV ty

nG

tV tz

or

p V

(16)

tV t y`

n` y `

tV tz`

or

p` ` V

(17)

Similarly, at x `, where the index is n`, n` A `

tV tx`

n` B `

It follows from the above equations and Eq. (4) that the point characteristic satisfies two conditions: n2 | V |2

and

n` 2 | ` V |2

(18)

Therefore, the point characteristic is not an arbitrary function of six variables. The total differential of V is dV (x ; x `) p` dx ` p dx

(19)

“This expression can be said to contain all the basic laws of optics”.36 Point Eikonal If reference planes in object and image spaces are fixed, for which we use z0 and z 0` , then the point eikonal is S(x , y ; x `, y `) V (x , y , z 0 ; x `, y `, z 0` ). This is the optical path length between pairs of points on the two planes. The function is not useful if the planes are conjugate, since more than one ray through a pair of points can have the same path length. The function is arbitrary, except for the requirement37 that v 2S v 2S v 2S v 2S

w0 tx tx` t y t y` tx t y` tx` t y

(20)

The partial derivatives of the point eikonal are

n A

tS tx

nB

tS ty

and

n` A `

tS tx`

n` B `

tS t y`

(21)

The relative merits of the point characteristic function and point eikonal have been debated.37–39 Angle Characteristic The angle characteristic function T(A , B ; A `, B `), also called the eikonal, is related to the point characteristic by T (A , B ; A `, B `) V (x , y , z ; x `, y `, z `) n(A x B y G z )

n`(A `x ` B ` y ` G `z `) Here the input plane z and output plane z` are fixed and are implicit parameters of T.

(22)


1.15

FIGURE 1 Geometrical interpretation of the angle characteristic function for constant object and image space indices. There is, in general, a single ray with directions (C, D, I ) in object space and (C`, D`, I`) in image space. Point O is the coordinate origin in object space, and O` is that in image space. From the origins, perpendiculars to the ray are constructed, which intersect the ray at Q and Q`. The angle characteristic function T (C, D C`, D`) is the path length from Q to Q`.

This equation is really shorthand for a Legendre transformation to coordinates px v V /v x , etc. In principle, the expressions of Eq. (16) are used to solve for x and y in terms of C and D, and likewise Eq. (17) gives x ` and y ` in terms of C` and D `, so T (A , B ; A `, B `) V [x (A , B ), y (A , B ), z ; x `(A `, B `), y `(A `, B `), z `] n[A x (A , B ) B y(A , B ) 1 A 2 B 2 z]

(23)

n`[A `x `(A `, B `) B ` y `(A `, B `) 1 A ` 2 B ` 2 z `] The angle characteristic is an arbitrary function of four variables that completely specify the directions of rays in two regions. This function is not useful if parallel incoming rays give rise to parallel outgoing rays, as is the case with afocal systems, since the relationship between incoming and outgoing directions is not unique. The partial derivatives of the angular characteristic function are ¤ A ³ vT n¥ x z´ , vA ¦ G µ ¤ A` ³ vT n` ¥ x ` z `´ , G` µ vA` ¦

¤ B ³ vT n¥ y z´ vB ¦ G µ

(24)

¤ B` ³ vT n` ¥ y ` z `´ G` µ v B` ¦

(25)

These expressions are simplified if the reference planes are taken to be z 0 and z ` 0. The geometrical interpretation of T is that it is the path length between the intersection point of rays with perpendicular planes through the coordinate origins in the two spaces, as shown in Fig. 1 for the case of constant n and n `. If the indices are heterogeneous, the construction applies to the tangents to the rays. Of all the characteristic functions, T is most easily found for single surfaces and most easily concatenated for series of surfaces.

Point-Angle Characteristic The point-angle characteristic function is a mixed function defined by W (x , y , z ; A `, B `) V (x , y , z ; x `, y `, z `) n`(A `x ` B ` y ` G `z `) T (A , B ; A `, B `) n(A x B y G z)

(26)

1.16

GEOMETRICAL OPTICS

As with Eq. (22), this equation is to be understood as shorthand for a Legendre transformation. The partial derivatives with respect to the spatial variables are related by equations like those of Eq. (16), so n2 |W|2, and the derivatives with respect to the angular variables are like those of Eq. (25). This function is useful for examining transverse ray aberrations for a given object point, since v W/v A `, v W/v B ` give the intersection points (x`, y`) in plane z for rays originating at (x, y) in plane z. Angle-Point Characteristic The angle-point characteristic function is W `(A , B ; x `, y `, z `) V (x , y , z ; x `, y `, z `) n(A x B y G z )

(27)

T (A , B ; A `, B `) n`(A `x ` B ` y ` G `z) Again, this is shorthand for the Legendre transformation. This function satisfies relationships like those of Eq. (17) and satisfies n` 2 | ` W ` |2 . Derivatives with respect to spatial variables are like those of Eq. (21). It is useful when input angles are given and output angles are to be found. Expansions About an Arbitrary Ray If two points on a ray that are not conjugate are taken as coordinate origins, and the z axes of the coordinate systems are taken to lie along the rays, then the expansion to second order of the point eikonal about these points is S(x1 , y1 ; x 2 , y 2 ) v a1x12 b1x1 y1 c1 y12 a2 x 22 b2 x 2 y 2 c 2 y 22

(28)

dx1x 2 ey1 y 2 fx1 y 2 g y1x 2 The other characteristic functions have similar expansions. These expansions have three types of terms, those associated with the input space, the output space, and “interspace” terms. From the coefficients, information about imaging along a known ray is obtained. This subject is treated in the references for the section “Images About Known Rays.” Expansions About the Axis For rotationally symmetric systems, the building blocks for an expansion about the axis are Object space term :

x2 y2

Image space term :

x `2 y `2

or

A `2 B `2

Interspace term:

xx ` yy `

or

AA ` BB `

or

Ax` B y`

or

A2 B2

(29) (30) or

xA ` y B ` (31)

(Here x “between.”) The interspace term combines the variables included in and . The general form can be written as a series C( ,

, )

£

aLMN L

M

N

(32)

L , M ,N

To second order, the expansion is C( ,

, ) a0 a100 a010 a001 a200 2 a020 a110

a101 a011

2

a002 2

(33)


1.17

The constant term is the optical path length between coordinate origins in the two spaces. It is often unimportant, but it does matter if two systems are used in parallel, as in an interferometer. The three first-order terms give the paraxial approximation. For imaging systems, the second-order terms are associated with third-order ray aberrations, and so on.30 It is also possible to expand the characteristic functions in terms of three linear combinations of , , and . These combinations can be chosen so that the characteristic function of an aberration-free system depends on only one of the three terms, and the other two describe the aberrations.26,31,40 Paraxial Forms for Rotationally Symmetric Systems These functions contain one each of the object space, image space, and interspace terms, with coefficients aO , aI , and aB. The coefficients of the object and image space terms depend on the input and output plane locations. That of the interspace term depends on the system power. Point eikonal: S(x `, y `; x , y) a aO (x 2 y 2 ) aB (xx ` yy `) aI (x ` 2 y ` 2 )

(34)

T (A `, B `; A , B ) a aO (A 2 B 2 ) aB (AA ` BB `) aI (A ` 2 B ` 2 )

(35)

Angle characteristic:

Point-angle characteristic: W (x , y ; A `, B `) a aO (x 2 y 2 ) aB (xA ` y B `) aI (A ` 2 B ` 2 )

(36)

Angle-point characteristic: W `(A , B , x `, y `) a aO (A 2 B 2 ) aB (A x ` B y `) aI (x ` 2 y ` 2 )

(37)

The corresponding coefficients in these expressions are different from each other. The familiar properties of paraxial and gaussian optics can be found from these functions by taking the appropriate partial derivatives. Some Ideal Characteristic Functions For a system that satisfies certain conditions, the form of a characteristic function can sometimes be found. Thereafter, some of its properties can be determined. Some examples of characteristic functions follow, in each of which expression the function F is arbitrary. For maxwellian perfect imaging (defined below) by a rotationally symmetric system between planes at z 0 and z` 0 related by transverse magnification m, the point characteristic function, defined for z` w 0, is V (x `, y `, z `; x , y) F (x 2 y 2 ) [(x ` mx )2 ( y ` my)2 z ` 2 ]1/2

(38)

Expanding the expression above for small x, x`, y, y` give the paraxial form, Eq. (34). The form of the point-angle characteristic is W (x , y ; A `, B `) F (x 2 y 2 ) m(n` A `x n` B ` y)

(39)

The form of the angle-point characteristic is W `(A , B ; x `, y `) F (x ` 2 y ` 2 )

1 (n A x ` n B y `) m

(40)

The functions F are determined if the imaging is also stigmatic at one additional point, for example, at the center of the pupil.26,34,40,41 The angular characteristic function has the form T (A , B ; A `, B `) F[(n A mn` A `)2 (n B mn` B `)2 ] where F is any function.

(41)

1.18

GEOMETRICAL OPTICS

For a lens of power F that stigmatically images objects at infinity in a plane, and does so in either direction, S(x , y ; x `, y `) F(xx ` yy `)

and

T (A , B ; A `, B `)

nn` (AA ` BB `) F

(42)

Partially differentiating with respect to the appropriate variables shows that for such a system, the heights of point images in the rear focal plane are proportional to the sines of the incident angles, rather than the tangents.

1.5

RAYS IN HETEROGENEOUS MEDIA

Introduction This section provides equations for describing and determining the curved ray paths in a heterogeneous or inhomogeneous medium, one whose refractive index varies with position. It is assumed here that n(x) and the other relevant functions are continuous and have continuous derivatives to whatever order is needed. Various aspects of this subject are discussed in a number of books and papers.42–49 This material is often discussed in the literature on gradient index lenses50–54 and in discussions of microwave lenses.55–58

Differential Geometry of Space Curves A curved ray path is a space curve, which can be described by a standard parametric description, x(S ) [x(S ), y(S ), z(S )], where U is an arbitrary parameter.46,59–62 Different parameters may be used according to the situation. The path length s along the ray is sometimes used, as is the axial position z. Some equations change form according to the parameter, and those involving derivatives are simplest when the parameter is s. Derivatives with respect to the parameter are denoted by dots, so x (S ) dx(S )/dS [x(S ), y(S ), z(S )]. A parameter other than s is a function of s, so d x(S )/ ds (d x / d S )(d S / ds). Associated with space curves are three mutually perpendicular unit vectors, the tangent vector t, the principal normal n, and the binormal b, as well as two scalars, the curvature and the torsion. The direction of a ray is that of its unit tangent vector t

x (S ) x (s) (A , B , G ) | x (S )|

(43)

The tangent vector t is the same as the direction vector r used elsewhere in this chapter. The rate of change of the tangent vector with respect to path length is ¤ dA dB dG ³ K n t(s) x(s) ¥ , , ¦ ds ds ds ´µ

(44)

The normal vector is the unit vector in this direction n

x(s) x(s)| |

(45)

The vectors t and n define the osculating plane. The curvature K | x(s)| is the rate of change of direction of t in the osculating plane. 2

K2

2

| x (S ) r x(S )|2 ¤ dA ³ ¤ dB ³ ¤ dG ³ | x(s)|2 ¥ ´ ¥ ´ ¥ ´ ¦ ds µ ¦ ds µ ¦ ds µ | x (S )|6

2

(46)


1.19

The radius of curvature is T 1/M. Perpendicular to the osculating plane is the unit binormal vector btrn

x (s) r x(s) | x(s)|

(47)

The torsion is the rate of change of the normal to the osculating plane

T b(s)

x(s)] x(s) x(S )] x(S ) [x (s) r d n(s) [x (S ) r 2 ds |x(s)|2 |x(S ) r x(S )|

(48)

The quantity 1/V is the radius of torsion. For a plane curve, V 0 and b is constant. The rates of change of t, n, and b are given by the Frenet equations: t(s) K n,

n (s) K t T b,

b (s) T n

(49)

In some books, 1/M and 1/V are used for what are denoted here by M and V.

Differential Geometry Equations Specific to Rays From the general space curve equations above and the differential equations below specific to rays, the following equations for rays are obtained. Note that n here is the refractive index, unrelated to n. The tangent and normal vectors are related by Eq. (59), which can be written log n K n ( log n t)t

(50)

The osculating plane always contains the vector n. Taking the dot product with n in the above equation gives

K

v log n n log n b (x r log n) vN

(51)

The partial derivative t/tN is in the direction of the principal normal, so rays bend toward regions of higher refractive index. Other relations46 are n Rx (s) r [ log n r x (s)] b Rx (s) r log n

T

and

0 b n

(x (s) r n) n | n r x (s)|2

(52) (53) (54)

Variational Integral Written in terms of parameter U, the optical path length integral, Eq. (7), is ¤ ds ³ V ¯ nds ¯ ¥n ´ d S ¯ d S ¦ dS µ

(55)

The solution for ray paths involves the calculus of variations in a way analogous to that used in classical mechanics, where the time integral of the lagrangian is an extremum.63 If has no explicit dependence on U, the mechanical analogue to the optics case is of no explicit time dependence.

1.20

GEOMETRICAL OPTICS

Differential Equations for Rays General Differential Equations Because the optical path length integral is an extremum, the integrand satisfies the Euler equations.46 For an arbitrary coordinate system, with coordinates q1, q2, q3 and the derivatives with respect to the parameter qi dqi / d S , the differential equations for the path are 0

d v v d ¤ v ds ³ v

n

dS v qi v qi dS ¥¦ v qi dS ´µ v qi

¤ ds ³ ¥¦n dS ´µ ;

i 1, 2, 3

(56)

Cartesian Coordinates with Unspecified Parameter In cartesian coordinates ds /dS (x 2 y 2 z 2 )1/2 , so the x equation is 0

d § nx d ¤ v ds ³ ds v n vn ¶ n

(x 2 y 2 z 2 )1/2 d S ¥¦ v x d S µ´ d S v x d S ¨©(x 2 y 2 z 2 )1/2 ·¸ vx

(57)

Similar equations hold for y and z. Cartesian Coordinates with Parameter Q s With U s, so ds/dU 1, an expression, sometimes called the ray equation, is obtained.28 n

d ¤ dx(s)³ d 2 x(s) dn[x(s)] dx(s) n n ¥ ´ ds ds ¦ ds µ ds ds 2

(58)

the ray equation can also be written Using dn/ds n x,

n n x ( n x )x

or

log n x ( log n x )x

(59)

Only two of the component equations are independent, since | x | 1 . Cartesian Coordinates with Parameter S ¯ ds /n and n2 x 2 y 2 z 2 , gives44

The parameter S ¯ ds /n, for which ds/dU n

d2x ( 12 n2 ) dS 2

(60)

This equation is analogous to Newton’s law of motion for a particle, F m d 2x/dt 2, so the ray paths are like the paths of particles in a field with a potential proportional to n2(x). This analogy describes paths, but not speeds, since light travels slower where n is greater, whereas the particles would have greater speeds.64,65 Euler Equations for Parameter Q z If U z, then ds / d S (x 2 y 2 1)1/2 and (x , y ; x , y ; z ). This gives45,49 0

¶ tn d ¤ v ds ³ ds t n d § nx

(1 x 2 y 2 )1/2 n

tx dz ¥¦ t x dz ´µ dz t x dz ¨©(1 x 2 y 2 )1/2 ·¸

(61)

with a similar equation for y. The equations can also be written (Refs. 51, app. A, and 66) as ¤tn tn ³ nx (1 x 2 y 2 )¥ x ¦ t x t z µ´

¤tn tn ³ ny (1 x 2 y 2 )¥ y ¦ t y t z µ´

This parameter is particularly useful when n is rotationally symmetric about the z axis.

(62)


1.21

Hamilton’s Equations with Cartesian Coordinates for Parameter Q z A set of Hamilton’s equations can also be written in cartesian coordinates using z as the parameter.45,49 The canonical momenta in cartesian coordinates are the optical direction cosines t t px nA p y nB (63) t x t y The hamiltonian is x yp y n2 (x , y , z ) ( px2 p y2 )

(x , y ; px , p y ; z ) xp

(64)

Hamilton’s equations are dx v

, dz t px

dy v

, dz t p y

dpx v

, dz tx

dp y dz

v

ty

(65)

It is not possible to write a set of Hamilton’s equations using an arbitrary parameter and three canonical momenta, since they are not independent.67 Another equation is v d 1 tn tz dz G t z

(66)

Paraxial Form of Hamilton’s Equations for Q z In the paraxial limit, if n0 is the average index, the above set of equations gives49 d 2 x( z ) 1 t n n0 t x dz 2

d 2 y(z ) 1 tn n0 t y dz 2

(67)

Other Forms A variety of additional differential equations can be obtained with various parameters.67 Time cannot be used as a parameter.68 The equations can also be expressed in a variety of coordinate systems.56,58,69–71

Refractive Index Symmetries When the refractive index has symmetry or does not vary with one or more of the spatial variables, the above equations may simplify and take special forms. If, in some coordinate system, n does not vary with a coordinate qi, so vn/vqi 0, and if, in addition, v/vqi(ds/dU) 0, then

t 0 t qi

and

t v ¤ ds ³ n ¥ ´ constant t q t q ¦ dS µ

(68)

There is an associated invariance of the ray path.44,49,56,58 (This is analogous to the case in mechanics where a potential does not vary with some coordinate.) A more esoteric approach to symmetries involves Noether’s theorem.72,73 There are a number of special cases. If the index is rotationally symmetric about the z axis, n n(x 2 y 2 , z ), then v /v F 0, where F is the azimuth angle, and the constant of motion is analogous to that of the z component of angular momentum in mechanics for a potential with rotational symmetry. The constant quantity is the skew invariant, discussed in the section “Skew Invariant.” If the refractive index is a function of radius, n n(r), there are two constants of motion. The ray paths lie in planes through the center (r 0) and have constant angular motion about an axis through the center that is perpendicular to this plane, so x r p is constant. If the plane is in the x-y plane, then n(Cy − Dx) is constant. This is analogous to motion of a particle in a central force field. Two of the best-known examples are the Maxwell fisheye48,74 for which n(r ) s (1 r 2 ) 1 , and the Luneburg lens,45,75 for which n (r ) 2 r 2 for r a 1 and n 1 for r > 1.

1.22

GEOMETRICAL OPTICS

If n does not vary with z, then nI is constant for a ray as a function of z, according to Eq. (66). If the medium is layered, so the index varies in only the z direction, then nC and nD are constant. If S is the angle relative to the z axis, then n(z)sin S(z) is constant, giving Snell’s law as a special case. The homogeneous medium, where vn/vx vn/vy vn/vz 0, is a special case in which there are three constants of motion, nC, nD, and nI, so rays travel in straight lines.

1.6

CONSERVATION OF ÉTENDUE If a bundle of rays intersects a constant z plane in a small region of size dx dy and has a small range of angles dC dD, then as the light propagates through a lossless system, the following quantity remains constant: n2dx dy dA dB n2dAdA dB n2dA cos Q dW dx dy dpx dp y

(69)

Here dA dx dy is the differential area, dY is the solid angle, and S is measured relative to the normal to the plane. The integral of this quantity

¯ n2dxdy dA dB ¯ n2dA dA dB ¯ n2dAcos QdW ¯ dx dy dpx dp y

(70)

is the étendue, and is also conserved. For lambertian radiation of radiance Le, the total power transferred is P ¯ Len2d A d B dx dy. The étendue and related quantities are known by a variety of names,76 including generalized Lagrange invariant, luminosity, light-gathering power, light grasp, throughput, acceptance, optical extent, and area-solid-angle-product. The angle term is not actually a solid angle, but is weighted. It does approach a solid angle in the limit of small extent. In addition, the integrations can be over area, giving n2d A d B ¯ dA, or over angle, giving n2dA ¯ d A d B . A related quantity is the geometrical vector flux,77 with components ( ¯ dp y dpz , ¯ dpx dpz , ¯ dpx dp y ). In some cases these quantities include a brightness factor, and in others they are purely geometrical. The étendue is related to the information capacity of a system.78 As special case, if the initial and final planes are conjugate with transverse magnification m dx`/ dx dy `/dy, then n2dA dB n` 2m 2dA `dB `

(71)

Consequently, the angular extents of the entrance and exit pupil in direction cosine coordinates are related by n2 ¯

entrance pupil

dA dB n ` 2m 2 ¯

exit pupil

dA ` dB `

(72)

See also the discussion of image irradiance in the section on apertures and pupils. This conservation law is general; it does not depend on index homogeneity or on axial symmetry. It can be proven in a variety of ways, one of which is with characteristic functions.79–81 Phase space arguments involving Liouville’s theorem can also be applied.82–85 Another type of proof involves thermodynamics, using conservation of radiance (or brightness) or the principle of detailed balance.86–89 Conversely, the thermodynamic principle can be proven from the geometrical optics one. 90–92 In the paraxial limit for systems of revolution the conservation of étendue between object and image planes is related to the two-ray paraxial invariant, Eq. (152). Some historical aspects are discussed by Rayleigh.93,94


1.7

1.23

SKEW INVARIANT In a rotationally symmetric system, whose indices may be constant or varying, a skew ray is one that does not lie in a plane containing the axis. The skewness of such a ray is

n(A y B x) px y p y x

(73)

As a skew ray propagates through the system, this quantity, known as the skew invariant, does not change.95–104 For a meridional ray, one lying in a plane containing the axis, 0. The skewness can be written in vector form as

a (x r p)

(74)

where a is a unit vector along the axis, x is the position on a ray, and p is the optical cosine vector at that position. This invariance is analogous to the conservation of the axial component of angular momentum in a cylindrical force field, and it can be proven in several ways. One is by performing the rotation operations on C, D, x, and y (as discussed in the section on heterogeneous media). Another is by means of characteristic functions. It can also be demonstrated that is not changed by refraction or reflection by surfaces with radial gradients. The invariance holds also for diffractive optics that are figures of rotation. A special case of the invariant relates the intersection points of a skew ray with a given meridian. If a ray with directions (C, D) in a space of index n intersects the x 0 meridian with height y, then at another intersection with this meridian in a space with index n `, its height y `, and direction cosine C` are related by nA y n ` A ` y `

(75)

The points where rays intersect the same meridian are known as diapoints and the ratio y`/y as the diamagnification.99

1.8

REFRACTION AND REFLECTION AT INTERFACES BETWEEN HOMOGENEOUS MEDIA

Introduction The initial ray direction is specified by the unit vector r (C, D, I). After refraction or reflection the direction is r ` (C`, D`, I`). At the point where the ray intersects the surface, its normal has direction S (L, M, N). The angle of incidence I is the angle between a ray and the surface normal at the intersection point. This angle and the corresponding outgoing angle I` are given by |cos I | |r S| |A L B M G N |

(76)

|cos I `| |r ` S| |A `L B ` M G `N| In addition |sin I | | r r S|

and

|sin I `| | r ` r S|

(77)

The signs of these expressions depend on which way the surface normal vector is directed. The surface normal and the ray direction define the plane of incidence, which is perpendicular to the vector cross product S r r ( MG NB , NA L G , L B MA ). After refraction or reflection, the outgoing ray is in the same plane. This symmetry is related to the fact that optical path length is an extremum.

1.24

GEOMETRICAL OPTICS

The laws of reflection and refraction can be derived from Fermat’s principle, as is done in many books. At a planar interface, the reflection and refraction directions are derived from Maxwell’s equations using the boundary conditions. For scalar waves at a plane interface, the directions are related to the fact that the number of oscillation cycles is the same for incident and outgoing waves. Refraction At an interface between two homogeneous and isotropic media, described by indices n and n`, the incidence angle I and the outgoing angle I` are related by Snell’s law:105 n` sin I ` n sin I

(78)

If sin I` > 1, there is total internal reflection. Another relationship is n` cos I ` n` 2 n2 sin 2 I n` 2 n2 n2 cos 2 I

(79)

Snell’s law can be expressed in a number of ways, one of which is n[r (r S)S] n`[r ` (r ` S)S]

(80)

Taking the cross product of both sides with S gives another form n`(S r r `) n(S r r)

(81)

A quantity that appears frequently in geometrical optics (but which has no common name or symbol) is ' n` cos I ` n cos I

(82)

' (n r n` r `) S n cos I n` 2 n2 sin 2 I n cos I n` 2 n2 n2 cos 2 I

(83)

It can be written in several ways

In terms of ', Snell’s law is n ` r ` nr 'S

(84)

or n` A ` n A L ' ,

n ` B ` n B M' ,

n` G ` n G N '

(85)

The outgoing direction is expressed explicitly as a function of incident direction by n` r ` n r S[n r S n` 2 n2 (n r S)2 ]

(86)

If the surface normal is in the z direction, the equations simplify to n` A ` n A ,

n` B ` n B ,

n` G ` n` 2 n2 n2G 2

(87)

If D 0, this reduces to n`C` nC, the familiar form of Snell’s law, written with direction cosines, with n ` G ` (n ` 2 n2A 2 )1/2 , corresponding to Eq. (79). Another relation from Eq. (85) is n ` A ` nA n ` B ` nB n ` G ` nG (88) ' L M N All of the above expressions can be formally simplified by using p nr and p` n`r`. For a succession of refractions by parallel surfaces, n1 sin I1 n2 sin I 2 n3 sin I 3 !

(89)


1.25

so the angles within any two media are related by their indices alone, regardless of the intervening layers. Refractive indices vary with wavelength, so the angles of refraction do likewise. Reflection The reflection equations can be derived from those for refraction by setting the index of the final medium equal to the negative of that of the incident medium, i.e., n` −n, which gives ' 2n cos I . The angles of incidence and reflection are equal (90)

I` I The incident and reflected ray directions are related by S r r` S r r

(91)

r ` r (2S r)S r (2 cos I )S

(92)

Another expression is

The components are

A ` A 2 L cos I

B ` B 2 M cos I

G ` G 2 N cos I

(93)

This relationship can be written in terms of dyadics106 as r` (I − SS) · r. This is equivalent to the matrix form107–111

2 LN ³ ¤A³ ¤A `³ ¤1 2 L2 2 LM ¥ B `´ ¥ 2 LM 1 2 M 2 2 MN ´ ¥ B´ ´¥ ´ ¥ G `´ ¥ ¦ µ ¦ 2 L N 2 MN 1 2N 2µ ¦ G µ

(94)

Each column of this matrix is a set of direction cosines and is orthogonal to the others, and likewise for the rows. The matrix associated with a sequence of reflections from plane surfaces is calculated by multiplying the matrices for each reflection. Another relationship is

A ` A B ` B G ` G L M N

(95)

If the surface normal is in the z direction, so (L, M, N) (0, 0, 1), then

A` A

B` B

G ` G

(96)

Reflection by a Plane Mirror: Positions of Image Points If light from a point (x, y, z) is reflected by a plane mirror whose distance from the coordinate origin is d, and whose surface normal has direction (L, M, N), the image point coordinates (x`, y`, z`) are given by ¤ x `³ ¤1 2L2 2LM

2 LN ¥ y `´ ¥ 2 LM 1 2 M 2 2 MN ¥ z `´ ¥ 2 ¥ ´ ¥¥ 2 LN 2 MN 1 2 N ¦ 1µ ¦ 0 0 0

2dL ³ ¤ x³ 2dM´ ¥ y´ ´¥ ´ 2dN ´ ¥ z´ 1 ´µ ¦ 1µ

(97)

This transformation involves both rotation and translation, with only the former effect applying if d 0. It is an affine type of collinear transformation, discussed in the section on collineation. The effect of a series of reflections by plane mirrors is found by a product of such matrices. The transformation can also be formulated in terms of quaternions.110,112

1.26

GEOMETRICAL OPTICS

Diffractive Elements The changes in directions produced by gratings or diffractive elements can be handled in an ad hoc geometrical way113,114

L n` rG` r S n r r S G q p

(98)

Here N is the vacuum wavelength, p is the grating period, q is a unit vector tangent to the surface and parallel to the rulings, and G is the diffraction order. Equations (81) and (91) are special cases of this equation for the 0th order.

1.9

IMAGING

Introduction Image formation is the principal use of lenses. Moreover, lenses form images even if this is not their intended purpose. This section provides definitions, and discusses basic concepts and limitations. The purposes of the geometrical analysis of imaging include the following: (1) discovering the nominal relationship between an object and its image, principally the size, shape, and location of the image, which is usually done with paraxial optics; (2) determining the deviations from the nominal image, i.e., the aberrations; (3) estimating image irradiance; (4) understanding fundamental limitations—what is inherently possible and impossible; and (5) supplying information for diffraction calculations, usually the optical path lengths. Images and Types of Images A definition of image (Webster 1934115) is: “The optical counterpart of an object produced by a lens, mirror, or other optical system. It is a geometrical system made up of foci corresponding to the parts of the object.” The point-by-point correspondence is the key, since a given object can have a variety of different images. Image irradiance can be found only approximately from geometrical optics, the degree of accuracy of the predictions varying from case to case. In many instances wave optics is required, and for objects that are not self-luminous, an analysis involving partial coherence is also needed. The term image is used in a variety of ways, so clarification is useful. The light from an object produces a three-dimensional distribution in image space. The aerial image is the distribution on a mathematical surface, often that of best focus, the locus of points of the images of object points. An aerial image is never the final goal; ultimately, the light is to be captured. The receiving surface (NS) is that on which the light falls, the distribution of which there can be called the received image (NS). This distinction is important in considerations of defocus, which is a relationship, not an absolute. The record thereby produced is the recorded image (NS). The recorded image varies with the position of the receiving surface, which is usually intended to correspond with the aerial image surface. In this section, “image” means aerial image, unless otherwise stated. Object Space and Image Space The object is said to exist in object space; the image, in image space. Each space is infinite, with a physically accessible region called real, and an inaccessible region, referred to as virtual. The two spaces may overlap physically, as with reflective systems. Corresponding quantities and locations associated with the object and image spaces are typically denoted by the same symbol, with a prime indicating image space. Positions are specified by a coordinate system (x, y, z) in object space and (x`, y`, z`) in image space. The refractive indices of the object and image spaces are n and n`.


1.27

Image of a Point An object point is thought of as emitting rays in all directions, some of which are captured by the lens, whose internal action converges the rays, more or less, to an image point, the term “point” being used even if the ray convergence is imperfect. Object and image points are said to be conjugate. Since geometrical optics is reversible, if A` is the image of A, then A is the image of A`.

Mapping Object Space to Image Space If every point were imaged stigmatically, then the entire object space would be mapped into the image space according to a transformation x ` x `(x , y , z )

y ` y `(x , y , z )

z ` z `(x , y , z )

(99)

The mapping is reciprocal, so the equations can be inverted. If n and n` are constant, then the mapping is a collinear transformation, discussed below.

Images of Extended Objects An extended object can be thought of as a collection of points, a subset of the entire space, and its stigmatic image is the set of conjugate image points. A surface described by 0 F(x, y, z) has an image surface 0 F `(x `, y `, z `) F[x(x `, y `, z `), y(x `, y `, z `), z(x `, y `, z `)]

(100)

A curve described parametrically by x(S ) [x(S ), y(S ), z(S )] has an image curve x `(S ) {x `[x(S ), y(S ), z(S )], y `[x(S ), y(S ), z(S ) ] , z `[x(S ), y(S ), z(S )]}

(101)

Rotationally Symmetric Lenses Rotationally symmetric lenses have an axis, which is a ray path (unless there is an obstruction). All planes through the axis, the meridians or meridional planes, are planes with respect to which there is bilateral symmetry. An axial object point is conjugate to an axial image point. An axial image point is located with a single ray in addition to the axial one. Off-axis object and image points are in the same meridian, and may be on the same or opposite sides of the axis. The object height is the distance of a point from the axis, and the image height is that for its image. It is possible to have rotational symmetry without bilateral symmetry, as in a system made of crystalline quartz,116 but such systems are not discussed here. For stigmatic imaging by a lens rotationally symmetric about the z axis, Eq. (99) gives x ` x `(x , z )

y ` y `( y , z )

z ` z `(z )

(102)

Planes Perpendicular to the Axis The arrangement most often of interest is that of planar object and receiving surfaces, both perpendicular to the lens axis. When the terms object plane and image plane are used here without further elaboration, this is the meaning. (This arrangement is more common for manufactured systems with flat detectors, than for natural systems, for instance, eyes, with their curved retinas.)

1.28

GEOMETRICAL OPTICS

Magnifications The term magnification is used in a general way to denote the ratio of conjugate object and image dimensions, for example, heights, distances, areas, volumes, and angles. A single number is inadequate when object and image shapes are not geometrically similar. The term magnification implies an increase, but this is not the case in general. Transverse Magnification With object and image planes perpendicular to the axis, the relative scale factor of length is the transverse magnification or lateral magnification, denoted by m, and usually referred to simply as “the magnification.” The transverse magnification is the ratio of image height to object height, m h`/h. It can also be written in differential form, e.g., m dx`/dx or m $x`/$x. The transverse magnification has a sign, and it can have any value from − c to + c. Areas in such planes are scaled by m2. A lens may contain plane mirrors that affect the image parity or it may be accompanied by external plane mirrors that reorient images and change their parity, but these changes are independent of the magnification at which the lens works. Longitudinal Magnification Along the rotational axis, the longitudinal magnification, mL, also called axial magnification, is the ratio of image length to object length in the limit of small lengths, i.e., mL dz`/dz. Visual Magnification With visual instruments, the perceived size of the image depends on its angular subtense. Visual magnification is the ratio of the angular subtense of an image relative to that of the object viewed directly. Other terms are used for this quantity, including “magnification,” “power,” and “magnifying power.” For objects whose positions can be controlled, there is arbitrariness in the subtense without the instrument, which is greatest when the object is located at the near-point of the observer. This distance varies from person to person, but for purposes of standardization the distance is taken to be 250 mm. For instruments that view distant objects there is no arbitrariness of subtense with direct viewing. Ideal Imaging and Disappointments in Imaging Terms such as perfect imaging and ideal imaging are used in various ways. The ideal varies with the circumstances, and there are applications in which imaging is not the purpose, for instance, energy collection and Fourier transformation. The term desired imaging might be more appropriate in cases where that which is desired is fundamentally impossible. Some deviations from what is desired are called aberrations, whether their avoidance is possible or not. Any ideal that can be approximated must agree in its paraxial limit ideal with what a lens actually does in its paraxial limit. Maxwellian Ideal for Single-Plane Imaging The most common meaning of perfect imaging is that elucidated by Maxwell,117 and referred to here as maxwellian ideal or maxwellian perfection. This ideal is fundamentally possible. The three conditions for such imaging of points in a plane perpendicular to the lens axis are: (1) Each point is imaged stigmatically. (2) The images of all the points in a plane lie on a plane that is likewise perpendicular to the axis, so the field is flat, or free from field curvature. (3) The ratio of image heights to object heights is the same for all points in the plane. That is, transverse magnification is constant, or there is no distortion.


1.29

The Volume Imaging Ideal A more demanding ideal is that points everywhere in regions of constant index be imaged stigmatically and that the imaging of every plane be flat and free from distortion. For planes perpendicular to the lens axis, such imaging is described mathematically by the collinear transformation, discussed later. It is inherently impossible for a lens to function in this way, but the mathematical apparatus of collineation is useful in obtaining approximate results. Paraxial, First-Order, and Gaussian Optics The terms “paraxial,” “first order,” and “gaussian” are often used interchangeably, and their consideration is merged with that of collineation. The distinction is often not made, probably because these descriptions agree in result, although differing in approach. One of the few discussions is that of Southall.118 A paraxial analysis has to do with the limiting case in which the distances of rays from the axis approach zero, as do the angles of the rays relative to the axis. The term first order refers to the associated mathematical approximation in which the positions and directions of such rays are computed with terms to the first order only in height and angle. Gaussian refers to certain results of the paraxial optics, where lenses are black boxes whose properties are summarized by the existence and locations of cardinal points. In the limit of small heights and angles, the equations of collineation are identical to those of paraxial optics. Each of these is discussed in greater detail below. Fundamental Limitations There are fundamental geometrical limitations on optical systems, relating to the fact that a given ray passes through many points and a given point lies on many rays. So the images of points on the same line or plane, or on different planes, are not independent. A set of rays intersecting at several points in object space cannot be made to satisfy arbitrary requirements in image space. Such limitations are best studied by the methods of hamiltonian optics. Stigmatic Imaging If all rays from an object point converge precisely to an image point, the imaging of this point is said to be stigmatic. The optical path lengths of all rays between two such points are identical. A stigmatic image point is located by the intersection of any two rays that pass through the points. An absolute instrument is one which images all points stigmatically.119 For such imaging n D x n ` D x `,

n D y n ` D y `,

n D z n` D z `

(103)

where conjugate length elements are Fx and Fx`, Fy and Fy`, Fz and Fz`

Path Lengths and Conjugate Points All the rays from an object point to its stigmatic image point have the same optical path length. For focal lenses, the paths lengths for different pairs of conjugate points in a plane perpendicular to the axis are different, except for points on circles centered on the axis. For afocal lenses path lengths are nominally the same for all points in planes perpendicular to the axis. For afocal lenses with transverse magnification ±n/n`, path lengths can be the same for all points. In general, the path lengths between different points on an object and image surface are equal only if the shape of the image surface is that of a wavefront that has propagated from a wavefront whose shape is that of the object surface.

1.30

GEOMETRICAL OPTICS

FIGURE 2 The cosine condition. A small area in object space about the origin in the x-y plane is imaged to the region around the origin of the x `-y ` plane in image space. A pair of rays from the origin with direction cosines (C1, D1) and (C2, D2) arrive with (C `1, D `1) and (C `2, D `2). The direction cosines and the transverse magnification in the planes are related by Eq. (104).

The Cosine Condition The cosine condition relates object space and image space ray angles, if the imaging is stigmatic over some area.116,120,121 Let the x-y plane lie in the object surface and the x `-y ` plane in the conjugate surface (Fig. 2). Two rays leaving a point in the object region have direction cosines (C1, D1) and (C2, D2), and the rays on the image side have (A1`, B1`) and (A 2` , B2` ). If the imaging is stigmatic, with local transverse magnification m on the surface, then

m

n(A1 A 2 ) n(B1 B2 ) n`(A1` A 2` ) n`(B1` B2` )

(104)

In the limit as C1 l C2 and D1 l D2, the cosine condition gives m

n dA nd B n` d A ` n` d B `

(105)

This condition also holds in the more general case of isoplanatic imaging, where there is aberration that is locally constant across the region in question.122,123

The Abbe Sine Condition The sine condition or Abbe sine condition119,124 is a special case of the cosine condition for object and image planes perpendicular to the axis in regions about the axis. For a plane with transverse magnification m, let S be the angle relative to the axis made by a ray from an axial object point, and S` be that in image space. If the lens is free of coma m

n sin Q nA nB n` sin Q ` n` A ` n` B `

(106)


1.31

for all S and S `. There are signs associated with S and S `, so that m > 0 if they have the same sign, and m < 0 if the signs are opposite. This equation is sometimes written with m replaced by the ratio of paraxial angles. There is sometimes the implication that S and S` refer only to the extreme angles passing through the lens, when in fact the sine condition dictates that the ratio of the sines is the constant for all angles. For an object at infinity, the sine condition is sin Q `

y f`

or

n` B ` y F

(107)

where y is the height of a ray parallel to the axis, F is the power of the lens, and f ` is the rear focal length. These relationships hold to a good approximation in most lenses, since small deviations are associated with large aberrations. A deviation from this relationship is called offense against the sine condition, and is associated with coma.123,125–128 The sine condition does not apply where there are discontinuities in ray behavior, for example, in devices like Fresnel lenses, or to diffraction-based devices like zone plates.

The Herschel Condition The Herschel condition is a relationship that holds if the imaging is stigmatic for nearby points along the axis.119,129,130 The two equivalent relations are m

n sin 12 Q

n` sin

1 2 Q`

and

mL

n sin 2 ( 12 Q ) n(1 G ) n` sin 2 ( 12 Q `) n`(1 G `)

(108)

The Herschel condition is inconsistent with the sine condition unless m o n /n`. So, in general, stigmatic imaging in one plane precludes that in others.

Sine and Herschel Conditions for Afocal Systems For afocal systems the sine condition and Herschel condition are identical. For rays entering parallel to the axis at y and leaving at y `, they are m

y` y

(109)

That is, the ratio of incoming and outgoing heights is independent of the incoming height. (Ref. 128, chap. 3, “The Sine Condition and Herschel’s Condition”).

Stigmatic Imaging Possibilities For object and image spaces with constant refractive indices, stigmatic imaging is only possible for the entire spaces for afocal lenses with identical transverse and longitudinal magnifications m ±n/n ` and |mL| |m|. Such lenses re-create not only the intersection points, but the wavefronts, since the corresponding optical path lengths are the same in both spaces, Eq. (103). For other lenses with constant object and image space indices, the maxwellian ideal can be met for only a single surface. In addition, a single point elsewhere can be imaged stigmatically.127,131 Nonplanar surfaces can be imaged stigmatically, a well-known example being the imaging of spherical surfaces by a spherical refracting surface, for a particular magnification.119 For systems with spherical symmetry, it is possible that two nonplanar surfaces be stigmatically imaged.132 In addition, various systems with heterogeneous indices can image stigmatically over a volume.

1.32

1.10

GEOMETRICAL OPTICS

DESCRIPTION OF SYSTEMS OF REVOLUTION

Introduction This section is concerned with the optical description of lens and mirror systems that are figures of revolution.133–145 From a mechanical viewpoint, optical systems are comprised of lenses and mirrors. From the point of view of the light, the system is regions of media with different indices, separated by interfaces of various shapes. This section is limited to homogeneous isotropic media. It is further restricted to reflecting and refracting surfaces that are nominally smooth, and to surfaces that are figures of revolution arranged so their axes are collinear, so the entire system is a figure of revolution about the lens axis. (The often-used term “optical axis” is also used in crystallography. Moreover, the axis is often mechanical as well as “optical.”) The lens axis is the z axis of an orthogonal coordinate system, with the x-y plane perpendicular. The distance from a point to the axis is R x 2 y 2 . Along the axis, the positive direction is from left to right.

Terminology A meridian or meridional plane contains the axis, all such planes being equivalent. Meridional planes are planes of bilateral symmetry if the indices are homogeneous and isotropic. Some optical systems are comprised of pieces of surfaces of revolution, in which case it is still useful to discuss the behavior about the axis.

Reflection, Unfolded Diagrams Light passes through refractive systems more or less in the same direction relative to the axis. In reflective and catadioptric systems, the light may change directions. (It may not, in the case of grazing incidence optics.) In order to consider all types of systems in a unified way, pictorially and algebraically, reflections can often be “unfolded,” i.e., represented pictorially as transmission, with mirrors replaced by hypothetical lenses with the same properties, Figs. 3 and 18. Some structures must be taken into account several times in unfolding. For example, a hole may block light at one point along a ray and transmit it at another. (In some considerations, unfolding can be misleading—for instance, those involving stray light.)

Description of Surfaces A surface is an interface between media with different refractive indices—either refracting or reflecting. The surface is the optical element, produced by a lens, which is a mechanical element. Surfaces can be described mathematically in many ways. (For example, conics can be described as loci of points with certain relationships.) In optical instruments, the entire surface is rarely used, and the axial region is usually special, so the description usually begins there and works out. The vertex of a figure of revolution intersects with the axis, and is a local extremum. The plane perpendicular to the axis and tangent to the vertex will be referred to as the vertex plane (NS). A surface can be described by its sag, the directed distance z(T) from the vertex plane to the surface, Fig. 4. The vertex is usually taken to have z(0) 0. The vertex curvature or paraxial curvature c and radius of curvature r are given by 1 v 2 z ( R) c r v R2 R 0

(110)


1.33

FIGURE 3 Example of an unfolded diagram. The two-mirror system above has an unfolded representation below. The reflective surfaces are replaced by thin lens equivalents. Their obstructions and the finite openings are accounted for by dummy elements.

For an arbitrary surface, this curvature is identical to that of the sphere which is a best fit on axis. The sign convention for curvature and radius is that c and r are positive if the center of curvature is to the right of the vertex, as in the case shown in Fig. 4. In general, the curvature is mathematically more foolproof than radius, since curvature can be zero, but it is never infinite, whereas radius is never zero, but may be infinite.

FIGURE 4 Description of a surface of revolution. The distance from the axis is T, and the sag z(T) is the distance from the vertex tangent plane to the surface.

1.34

GEOMETRICAL OPTICS

Spherical Surfaces The spherical surface is the most common in optics, since it is most naturally produced. Spherical is the default, and is assumed when no other mention is made. Aspheres are thought of as deviating from spheres, rather than spheres as special cases of more general forms. The equation for a sphere with radius r, curvature c, and a vertex at z 0 is

R 2 (z r )2 r 2

(111)

The sag is given by z(R) r r 2 R 2 r(1 1 c 2 R 2 )

cR 2

(112)

1 1 c 2R2

The Taylor expansion is 5 7 8 7 9 100 z(R) 12 cR 2 18 c 3 R 4 161 c 5 R 6 128 c R 256 c R !

(113)

At the point (x, y, z) on the surface of the sphere, the surface normal has direction cosines ¤ x y z r³ (L , M , N ) ¥ , , (cx , cy , cz 1) ¦ r r r ´µ

(114)

Conics of Rotation The general form of a conic of rotation about the z axis is r cR 2 z(R) (1 1 Ec 2 R 2 ) E 1 1 Ec 2 R 2

(115)

The value of G determines the type of conic, as given in the table below. It is common in optics to use M, the conic parameter or conic constant, related by

K E 1

or

E 1 K

(116)

Another parameter used to describe conics is the eccentricity e, used in the polar coordinate form for conics about their focus: r (Q ) a / (1 e cos Q ) where e2 − M. In the case of paraboloids, the first form of Eq. (115) breaks down. A cone can be approximated by a hyperbola with M − sec2S, where S is the cone half angle. Conic Type and Value of Parameter Parameter Oblate ellipsoid Sphere Prolate ellipsoid Paraboloid Hyperboloid

G

M

e

G>1 G1 0> d 2/N, means that for these distances z the first Fresnel zone overfills the aperture. The entire aperture contributes to the disturbance at any point in the Fraunhofer pattern. In Fresnel diffraction only relatively small portions of the aperture contribute to any one point in the pattern. In this context, the term Fresnel number is frequently used. It is defined in terms of the product of two ratios. The radius r of the aperture to the wavelength N times the radius of the aperture to the distance b measured from the aperture to the plane of observation: Fresnel number x N

r r 1 d2 L b 4b L

(50)

Thus, the Fresnel number can also be expressed as the ratio of the far-field distance, d 2/N, to the distance b from the aperture. With the definition of the Fresnel zones in Sec. 3.4, these ratios indicate that the Fresnel number equals the number of Fresnel zones that may be drawn within the aperture from a point P at a distance b from the aperture. Thus, well within the Fresnel region, b > 1. In the Fraunhofer region b > d 2/N, N < 1 where the first Fresnel zone overfills the aperture as pointed out before. In Figs. 23 and 24 the theoretical plots of Fraunhofer patterns of a rectangular aperture and a circular aperture, respectively, are shown. In the rectangular case the central maximum has equally spaced zeros on either side, while in the circular case the central maximum is surrounded by unequally spaced concentric dark rings. In both cases the central maximum occurs at the geometrical image of the point source that produced parallel light illumination on the aperture.

FIGURE 22 Optical processing arrangement. Collimated or parallel light is incident normally on plane 1. The system images plane 1 onto plane 3 with a Fourier transform in plane 2.

3.26

PHYSICAL OPTICS

y 1.0 0.9 ys 0.8 0.7 y= 2b

0

xs

sin x x

2

0.6 0.5 0.4

2a 0.3 0.2 0.1

0 1

2

2

3

x

FIGURE 23 Rectangular aperture (in coordinates xs and ys) and one section of the diffraction pattern. Normalized irradiance y plotted against a unitless variable x as discussed in the text. (From Born and Wolf.1)

The unitless variable x shown in the plots is defined as follows. (1) In the case of rectangular aperture, x 2Rap/N, where 2a is the width of the aperture in the xs direction. In the other dimension y 2Rbq/N, and 2b is the dimension in the ys direction. As before, p and q are the direction cosines of the vector joining the center of the aperture to the point of observation. (2) In the case of circular aperture, the unitless radial variable x 2Raw/N, where 2a is the diameter of the aperture in the xs , ys plane and w p 2 q 2 . In the far field the size of the diffraction pattern is very large compared to the aperture that produced it. In the focal plane of the lens, z = f and the size of the diffraction pattern is much smaller than the aperture. In both cases the patterns are in a reciprocal width relationship, that is, if the aperture is narrow in the x s direction compared to ys , the pattern is broader in the x direction compared to the y. A converging spherical lens illuminated by a plane wave produces in the focal plane a Fraunhofer diffraction pattern of the amplitude and phase of the aperture of a circular lens. When the lens has negligible phase errors, the diffraction pattern has a bright disk in the center surrounded by concentric dark rings. This is called an Airy disk and it plays an important role in the Rayleigh criterion of resolving power.

Fraunhofer Diffraction Pattern of a Double Slit The diffraction pattern of two slits may be observed by using the optical arrangement of Fig. 25. The center-to-center separation of the two slits is h. The off-axis point P is in the direction S from the axis as shown in the figure. The maxima and minima are determined according to whether the path

DIFFRACTION

3.27

E 1.0 0.9 0.8 0.7 0.6 y= 0.5

2J1 (x) x

2

0.4 0.3 0.2 0.1 0

1

2

3

4

5

6

7

8

9

x

FIGURE 24 A section of the diffraction pattern of a circular aperture. The normalized irradiance E is plotted against a unitless variable x in the plane of observation as discussed in the text. (From Born and Wolf.1)

difference O1H is an even or odd multiple of a half-wave. Let E0 be the irradiance at the center of the single-slit diffraction pattern. The irradiance distribution in the plane of observation is given by 2

¤ sin A ³ E 4 E0 ¥ (cos D )2 ¦ A ´µ

(51)

where F Rh(sin S)/N. The irradiance at the center of the double-slit pattern is 4E0. The second term, (sin C/C)2, describes the diffraction pattern of a single slit of width 2a. Here C 2Ra(sin S)/N.

FIGURE 25 Arrangement to observe the Fraunhofer diffraction of an aperture consisting of two slits. S: point source. P: point of observation. O1, O2: two slit apertures with center-to-center separation h. (From Rossi.8)

3.28

PHYSICAL OPTICS

D(a)

(a)

sin J cos2 d

(b)

sin J

E E0

(c)

sin J –l a

l a

0

FIGURE 26 (a) Plot of a single-slit diffraction pattern D(C); (b) plot of a two-slit interference pattern; and (c) their product E/E0. (From Rossi.8)

The term (cos F )2 is the interference pattern of two slits. These two patterns as well as their product are sketched in Fig. 26. Diffraction Grating In Fig. 27, an arrangement similar to Fig. 25 permits observation of the Fraunhofer diffraction pattern of a grating, of N parallel and equidistant slits. The center-to-center separation between neighboring slits is h. As in the two-slit case, the Fraunhofer pattern consists of the diffraction due to one slit times the interference pattern of N slits. The irradiance distribution in the plane of observation is given by 2

¤ sin A ³ ¤ sin NI ³ E N 2 E0 ¥ ¦ A ´µ ¥¦ N sin I ´µ

2

(52)

DIFFRACTION

3.29

FIGURE 27 Arrangement to observe the Fraunhofer diffraction of an aperture consisting of N slits. S: slit source. P : point of observation. G: grating. (From Rossi.8)

where I Rh(sin S)/N and N 2E0 is proportional to the irradiance at the center of the N-slit pattern. The term (sin C/C)2 is the single-slit pattern as used with Eq. (51). In the case of multiple slits each slit is very narrow; hence, this pattern is very broad, a characteristic of Fraunhofer diffraction. The interference term (sin NI /N sin I )2 shows prominent maxima when both the numerator and denominator are simultaneously zero; this happens when I Rh(sin S)/N mR, where m is an integer. It leads to the grating equation, namely, h sin Q mL

(53)

There are several, (N 1), subsidiary minima in between the principal maxima. This happens when the numerator is zero but the denominator is not, I mR/N. For the case of N 10, these effects are sketched in Fig. 28, which shows the effect of the product of the diffraction and interference terms. In general, as N increases the subsidiary maxima become more nearly negligible, while the principal maxima become narrower, being proportional to (1/N). The location of the principal maxima other than the zeroth order (m 0) are proportional to the wavelength N. The diffraction grating thus forms an important spectroscopic tool. Further discussion of gratings is given by Petit18 and Gaylord and Moharam.19

3.7

STATIONARY PHASE APPROXIMATION The diffracted field in the Rayleigh-Sommerfeld diffraction theory is given by §1 z exp(i kT) ¶ dx dy Y (x , y , z ) ¯¯ Y (x s , y s , 0) ¨ (1 ikT) A 2 P T T 2 ·¸ s s ©

(54)

where Y (x s , y s , 0) is the field in aperture A. The diffracted field can also be represented by

Y (x , y , z ) ¯¯ Y (L /L , M /L , 0) exp §©i 2P (Lx My Nz )/L ¶¸ d(L /L )d( M /L ) A

(55)

The phase term in this integral is

F(L , M ) x 2P (Lx My Nz )/L

2P Lx My [1 (L2 M 2 )] z L

[

]

(56)

3.30

PHYSICAL OPTICS

–l

0 (c)

(a)

sin J

(b)

sin J

l sin J

FIGURE 28 (a) Irradiance plot of a single-slit diffraction pattern; (b) partial plot of an N 10 slit interference pattern; and (c) their product. (From Rossi.8)

DIFFRACTION

3.31

The special values of L and M that make the first derivatives of the phase zero,

tF tF 0 tL tM

(57)

are L0 o

x r

and

M0 o

y r

(58)

where r x 2 y 2 z 2 . The negative sign is omitted for forward propagation, z 0. The phase is approximated by 1 F(L , M ) y F(L0, M 0) [A (L L0 )2 B( M M 0)2 2G (L L0)( M M 0)] 2 where the higher-order terms are neglected and A , B , and G are the second derivatives evaluated at L L0 and M M 0. These constant coefficients are given by

F(L0 , M 0) kr

(59)

t 2F x2 z2 kr 2 tL z2

(60a)

t 2F y2 z2 kr 2 tM z2

(60b)

Ax Bx and

Gx

t 2F xy t 2F kr 2 t L t M t Mt L z

(60c)

The resulting phase function is used in the double integral to obtain the diffracted field, Y (x , y , z ). The reader may also refer to Ref. 1, app. III, Eq. (20). The above procedure yields the stationary phase approximation for the diffracted field given by

Y (x , y , z ) y

¤ x y³ i z exp(ikr ) Y SP ¥ , ´ L r r ¦r rµ

(61)

where § i 2P ¤ xx s y y s ³ ¶ ¤ x y³ Y SP ¥ , ´ x¯¯ Y (x s , y s , 0) exp ¨ ¥ ´µ · dx s dy s A r ¦r rµ ·¸ ¨© L ¦ The diffracted field on a hemisphere is simply the spatial Fourier transform of the field distribution in the aperture as long as the distance r to the observation point satisfies the far-field condition r

N 2a 2 L

(62)

where N z /r cos(Q ) , the third direction cosine, a is the radius of the aperture, and N is the wavelength of the light incident on the aperture. (Q is measured from the z axis, and as Q increases the far-field condition is weakened.) For observation points not satisfying the far-field condition, the higher-order terms of the stationary phase approximation cannot be neglected. Harvey20 and

3.32

PHYSICAL OPTICS

Harvey and Shack21 have shown that these terms can be considered as aberrations of the spatial Fourier transform of the aperture field on the hemisphere of observation. In the stationary phase approximation, there is no restriction on the direction cosines, L, M, and N. Hence the diffracted field amplitude in Eq. (61) is valid over the entire hemisphere.

3.8 VECTOR DIFFRACTION The popularity of the Fresnel-Kirchhoff diffraction formula in the scalar case stems from the fact that it is widely applicable and relatively easy to use. In the study of electromagnetic diffraction,13,22 a similar formula can be obtained [see Ref. 13, Eq. (9.156)] but it has limited applicability because of the boundary conditions that must be satisfied. These conditions are the ones related to perfectly conducting screens. They are not adequately approximated at optical frequencies. The study with finite conductivity makes for complicated mathematical procedures. From the point of view of instrumental optics the applicability of the theory then is severely limited. In the optical literature, periodic structures such as gratings (both shallow and deep compared to the wavelength) have been studied. Boundary conditions are applied to perfectly conducting grating profiles.18,19 The equation of the grating dictating the angular positions of the diffraction orders such as Eq. (53) continues to apply; the amount of power found in the different orders is significantly different in the vector theory compared to the scalar theory. A special case of interest is discussed in detail by Jackson.13 Consider a plane wave incident at an angle C on a thin, perfectly conducting screen with a circular hole of radius a in the x-y plane. The polarization vector (E field) of the incident wave lies in the x-z plane, which is taken to be the plane of incidence. The arrangement is shown in Fig. 29 where k0 stands for the wave vector of the incident wave and k is used for the diffracted field.

z

a

a

q

Bi

y

f

x Ei

FIGURE 29 Coordinate system and aperture geometry for vector diffraction. C : angle of incidence. The E field is in the xz plane. k0: the wave vector of the incident wave. k: the diffracted field. (From Jackson.13)

DIFFRACTION

3.33

FIGURE 30 Fraunhofer diffraction pattern for a circular opening one wavelength in diameter in a thin-plane conducting screen. The angle of incidence is 45n. (a) Powerper-unit solid angle (radiant intensity) in the plane of incidence and (b) perpendicular to it. The solid (dotted) curve gives the vector (scalar) approximation in each case. (From Jackson.13)

The vector and scalar approximations are compared in Fig. 30. The angle of incidence is equal to 45°and the aperture is one wavelength in diameter, ka R. The angular distribution is shown in Fig. 30 for two cases. Figure 30a shows the distribution of the power per unit solid angle in the plane of incidence which contains the E field and Fig. 30b the distribution for the plane perpendicular to it. Both vector and scalar theories contain the Airy-disk-type distribution; the differences show in the angular distribution. For normal incidence C 0 and ka >> 1 the polarization dependence is unimportant and the diffraction is confined to very small angles in the forward direction (Airy-disk-type distribution) as we found before in Fig. 24 under Fraunhofer diffraction.

The Vector Huygens-Fresnel Secondary Source Ideally, the fundamental model of any diffraction theory of light would retain the simplicity of Huygens’ scalar secondary source and wavefront construction theory but also account for its vector nature. It has been shown that an electromagnetic wavefront can be modeled as a set of fictitious oscillating electric and magnetic surface charge and current densities existing at all points on the wavefront. The vector Huygens secondary source is a unit composed of two fictitious coincident dipoles; one electric and the other magnetic their magnitudes and orientation dictated by the wavefront boundary conditions. The fields of the vector Huygens secondary source are composed of the linear, vector superposition of the fields of these electric and magnetic dipoles. The electric dipole’s axis lies in the plane of the page, is oriented in the vertical direction, and is located at the origin. The magnetic dipole’s axis is perpendicular to the plane of the page and is also located at the origin. The vector of (a) the radiated electric field (Fig. 31a) is tangent to the spherical wavefront (represented by the outer circle) and lies in the plane of the page and (b) the radiated magnetic field (Fig. 31b) is tangent to the spherical wavefront (represented by the outer circle) and is perpendicular to the plane of the page. The Poynting vector (Fig. 31c) points radially outward. The magnitude of the irradiance is proportional to the length of the chord from the origin to the irradiance plot along the radius to the point of tangency. The strength of these vectors is proportional to the length of the chord from the origin to the field plot along the radius to the point of tangency.

3.34

PHYSICAL OPTICS

90 1 120

60 0.8

0.6 30

150 0.4

0.2 Etot() Etot(0)

180

0

0

330

210

240

300 270

FIGURE 31a This figure is the mapping of the EM dipole’s normalized electric field strength on the radiated spherical wavefront in the far zone of the EM dipole as a function of the angle between the direction of observation and kˆ EM which lies in the plane of the page and is in the direction of the vector from the origin to 0 degrees.

The plot is rotationally symmetric about kˆ EM which lies in the plane of the page and is in the direction of the vector from the origin to 0 degrees. The diffracted field at any observation point is the summation of the fields radiated from the electriomagnetic dipoles in the aperture visible to the observation point as given by FJ FJ Mo ˆ J J ¹ exp(ikRs) 3 J

i ª §R r H (rs )¶º E(r ) 2 ¯¯¯ «§©Rˆ s r E(rs )¶¸ r Rˆ s d rs ¸ EPW Rs Eo © s 2L ¬ »

(63)

J J J J J J ¹ exp(ikRs ) 3 J E

i ª H (r ) 2 ¯¯¯ «§©Rˆ s r H (rs )¶¸ r Rˆ s o §©Rˆ s r E(rs )¶¸º d rs EPW Rs M 2L o ¬ »

(64)

DIFFRACTION

3.35

90 1 120

60 0.8

0.6 150

30 0.4

0.2 Htot() Htot(0)

180

0

0

210

330

300

240 270

FIGURE 31b This figure is the mapping of the EM dipole’s normalized magnetic field strength on the radiated spherical wavefront in the far zone of the EM dipole as a function of the angle between the direction of observation and kˆ EM which lies in the plane of the page and is in the direction of the vector from the origin to 0 degrees.

FJ J J In these expressions, EJ(r ) and H (r ) are, respectively, the J F diffracted electric and magnetic fields at JF the point of observation Jr . In the integrands, E(rs ) and H (rs ) are the fields of the wavefront incident on the aperture at point rs and are related by J F H (rs )

Eo F J F nˆ(r ) r E(rs ) Mo s

(65)

F F F F Here nˆ(rs ) is the normal to the wavefront. We have defined Rs r rs and use Rs for the magnitude F and Rˆ s for the unit vector in the direction of Rs . The letters EPW under the volume integral stand for “exposed parts of the primary wavefront.” The integration is restricted to the open areas of the aperture.

3.36

PHYSICAL OPTICS

90 1 120

60 0.8

0.6 150

30 0.4

0.2 () (0)

180

0

0

210

330

300

240 270

FIGURE 31c This figure is the mapping of the EM dipole’s normalized irradiance on the radiated spherical wavefront in the far zone of the EM dipole as a function of the angle between the direction of observation and kˆ EM which lies in the plane of the page and is in the direction of the vector from the origin to 0 degrees.

By use of vector identities, the curly bracket in Eq. (63) may be rewritten in the form ª FJ Mo ˆ J J ¹ §R r H (rs )¶º «§©Rˆ s r E(rs )¶¸ r Rˆ s ¸ Eo © s ¬ » F J FJ J J E> (rs ) E(rs )(cosC ) nˆ(rs )(E@ (rs ))

(66)

F J F J In this expression E> (rs ) is the transverse Fcomponent of E(rs ) perpendicular to the direction Rˆ s and J J E@ (rs ) is the longitudinal component of E(rs ) parallel to the direction Rˆ s . The symbol E stands for ˆ ˆ J the angle between the F unit J vectors Rs and n(rs ). In the special case where the angle E is zero, the curly bracket reduces to 2E> (rs ). For further details and additional references, we refer to McCalmont 23 and Marathay and McCalmont.24 Figure 32 is a sequence of irradiance profiles due to the diffraction of light by a narrow slit based on Eqs. (63) and (64). The sequence describes diffraction from deep within the Fresnel region into the Fraunhofer zone.

Normalized irradiance


DIFFRACTION

1.0 0.8 0.6 0.4 0.2 0.0 –20

–10

0 Detector position (a)

10

20

1.0 0.8 0.6 0.4 0.2 0.0 –50

–40

–30

–20

–10 0 10 Detector position (b)

20

30

40

50


1.0 0.8 0.6 0.4 0.2


0.0 –150

–100

–50

0 50 Detector position (c)

100

150

1.0 0.8 0.6 0.4 0.2 0.0 –4000

–2000

0 Detector position

2000

4000

(d)

FIGURE 32 Flux density profiles along the horizontal x axis on an observation screen at the following distances: (a) 5N, (b) 100N, (c) 500N, and (d) 15,000N. These profiles are due to diffraction by a rectangular slit and are based on Eqs. (63) and (64). The slit is of width 20N. The first zero of the Fraunhofer pattern is at 2.81° from the optical axis. The incident field is a plane wave of unit amplitude, at normal incidence on the plane of the aperture and polarized in the vertical y-direction. The position on the x axis is in terms of wavelengths from the origin of the observation plan.

3.37

3.38

PHYSICAL OPTICS

3.9 ACKNOWLEDGMENTS The chapter is dedicated by Arvind Marathay to his wife Sunita and family and by John McCalmont to his wife Ingrid and family.

3.10

REFERENCES Exhaustive and/or the latest listing of references is not the intent but the following source books were used throughout the chapter. 1. M. Born and E. Wolf, Principles of Optics, 6th ed., Pergamon, Oxford, 1980. 2. F. A. Jenkins and H. E. White, Fundamentals of Optics, McGraw-Hill, New York, 1957. 3. R. S. Longhurst, Geometrical and Physical Optics, Wiley, New York, 1967. 4. B. K. Mathur and T. P. Pandya, Geometrical and Physical Optics, Gopal Printing Press, Kanpur, India, 1972. 5. J. B. Marion and M. A. Heald, Classical Electromagnetic Radiation, 2d ed., Academic Press, New York, 1980. 6. A. S. Marathay and J. F. McCalmont, “On the Usual Approximation Used in the Rayleigh-Sommerfeld Diffraction Theory,” J. Opt. Soc. Am. 21(4):510–516, 2004. 7. J. W. Goodman, Introduction to Fourier Optics, McGraw-Hill, New York, 1968. 8. B. Rossi, Optics, Addison-Wesley, Reading, Massachusetts, 1957. 9. C. L. Andrews, Optics of the Electromagnetic Spectrum, Prentice-Hall, Englewood Cliffs, New Jersey, 1960. 10. E. Hecht and A. Zajac, Optics, Addison-Wesley, Reading, Massachusetts, 1976. 11. M. J. Beran and G. B. Parrent Jr., Theory of Partial Coherence, Prentice-Hall, Englewood Cliffs, New Jersey, 1964. 12. A. S. Marathay, Elements of Optical Coherence Theory, Wiley, New York, 1985. 13. J. D. Jackson, Classical Electrodynamics, Wiley, New York, 1962. 14. A. E. Sommerfeld, Electrodynamics, Academic Press, New York, 1964 (Translation of Uber Theoretische Volesung Physik, Vol. 3, Electrodynamik). 15. W. H. Carter, “Wave Theory of a Simple Lens,” Optica Acta, 20(10):805–826, 1973. 16. K. D. Möller, Optics, University Science Books, Mill Valley, California, 1988. 17. H. Kogelnik and T. Li, “Laser Beams and Resonators,” Appl. Opt. 5:1550–1565, 1966. 18. R. Petit, Electromagnetic Theory of Gratings, Springer-Verlag, New York, 1980. 19. T. K. Gaylord and M. G. Moharam, “Analysis and Applications of Optical Diffraction by Gratings,” Proc. IEEE 73:894–937, 1985. 20. J. E. Harvey, “Fourier Treatment of Near-Field Scalar Diffraction Theory,” Am. J. Phys. 47(11), Nov. 1979. 21. J. E. Harvey and R. V. Shack, “Aberrations of Diffracted Wave Fields,” Appl. Opt. 17:3003–3009, 1978. 22. J. A. Stratton, Electromagnetic Theory, McGraw-Hill, New York, 1941. 23. J. F. McCalmont, A Vector Huygens-Fresnel Model of the Diffraction of Electromagnetic Waves, Ph.D. dissertation, University of Arizona, 1999. 24. A. S. Marathay and J. F. McCalmont, “Vector Diffraction Theory for Electromagnetic Waves,” J. Opt. Soc. Am. 18:2585–2593, 2001.

4 TRANSFER FUNCTION TECHNIQUES Glenn D. Boreman CREOL, The College of Optics and Photonics University of Central Florida Orlando, Florida

4.1

GLOSSARY B CTF e (x) FN F(X , H) f(x, y) G(X , H) g(x, y) H(X , H) h(x, y) @(x ) S(X , H) W D (x ) Q (X , H)

4.2

spot full width contrast transfer function (square wave response) edge response focal ratio Fourier transform of f(x, y) object function Fourier transform of g(x, y) image function Fourier transform of h(x, y) impulse response line response power spectrum detector dimension delta function phase transfer function two-dimensional convolution

INTRODUCTION Transfer functions are a powerful tool for analyzing optical and electro-optical systems. The interpretation of objects and images in the frequency domain makes available the whole range of linearsystems analysis techniques. This approach can facilitate insight, particularly in the treatment of complex optical problems. For example, when several optical subsystems are combined, the overall transfer function is the multiplication of the individual transfer functions. The corresponding analysis, without the use of transfer functions, requires convolution of the corresponding impulse responses. 4.1

4.2

PHYSICAL OPTICS

4.3

DEFINITIONS The image quality of an optical or electro-optical system can be characterized by either the system’s impulse response or its Fourier transform, the transfer function. The impulse response h(x, y) is the two-dimensional image formed in response to a delta-function object. Because of the limitations imposed by diffraction and aberrations, the image quality produced depends on the following: the wavelength distribution of the source; the F-number (FN) at which the system operates; the field angle at which the point source is located; and the choice of focus position. A continuous object f(x, y) can be decomposed, using the sifting property of delta functions, into a set of point sources, each with a strength proportional to the brightness of the object at that location. The final image g(x, y) obtained is the superposition of the individually weighted impulse responses. This result is equivalent to the convolution of the object with the impulse response: f (x , y)

h(x , y) g (x , y)

(1)

where the double asterisk denotes a two-dimensional convolution. The validity of Eq. (1) requires shift invariance and linearity. Shift invariance is necessary for the definition of a single impulse response and linearity is necessary for the superposition of impulse responses. These assumptions are often violated in practice, but the convenience of a transferfunction analysis dictates that we preserve this approach if possible. While most optical systems are linear, electro-optical systems that include a receiver (such as photographic film, detector arrays, and xerographic media) are often nonlinear. A different impulse response (and hence transfer function) is obtained for inputs of different strengths. In optical systems with aberrations that depend on field angle, separate impulse responses are defined for different regions of the image plane. Although h(x, y) is a complete specification of image quality (given a set of optical parameters), additional insight is gained by use of the transfer function. A transfer-function analysis considers the imaging of sinusoidal objects, rather than point objects. It is more convenient than an impulseresponse analysis because the combined effect of two or more subsystems can be calculated by a point-by-point multiplication of the transfer functions, rather than by convolving the individual impulse responses. Using the convolution theorem of Fourier transforms, we can rewrite the convolution of Eq. (1) as a multiplication of the corresponding spectra: F (X , H) r H (X , H) G(X , H)

(2)

where the uppercase variables denote the Fourier transforms of the corresponding lowercase variables: F(X , H) is the object spectrum; G(X , H) is the image spectrum; H(X , H) is the spectrum of the impulse response. As a transfer function, H(X , H) multiplies the object spectrum to yield the image spectrum. The variables Z and J are spatial frequencies in the x and y directions. Spatial frequency is the reciprocal of the crest-to-crest distance of a sinusoidal waveform used as a basis function in the Fourier analysis of an object or image. In two dimensions, a sinusoid of arbitrary orientation has a spatial period along both the x and y axes. The reciprocals of these spatial periods are the spatial frequencies Z and J. Typical units of spatial frequency are cycles/millimeter when describing an image, and cycles/milliradian when describing an object at a large distance. For an object located at infinity, these two representations are related through the focal length of the image-forming optical system:

Xangular[cycles/mrad] 0.001 r X [cycles/mm] r f [mm]

(3)

The function H(X , H) in Eq. (2) is usually normalized to have unit value at zero frequency. This yields a transfer function relative to the response at low frequency, and ignores frequency-independent attenuations, such as losses caused by Fresnel reflection or by obscurations. This normalization is appropriate for most optical systems, because1 the transfer function of an incoherent optical system is proportional to the two-dimensional autocorrelation of the exit pupil, which is maximum at zero frequency. For more general imaging systems (for example, the human eye, photographic film, and electronic imaging systems), the transfer function is not necessarily maximum at the origin, and may be more useful in an unnormalized form.

TRANSFER FUNCTION TECHNIQUES

4.3

With the above normalization, H(X , H) is called the optical transfer function (OTF). In general, OTF is a complex function, having both a magnitude and a phase portion: OTF(X , H) H (X , H) | H (X , H)| exp { j Q (X , H)}

(4)

The magnitude of the OTF, | H (X , H)|, is referred to as the modulation transfer function (MTF), while the phase portion of the OTF, Q (X , H), is referred to as the phase transfer function (PTF). MTF is the magnitude response of the imaging system to sinusoids of different spatial frequencies. This response is described in terms of the modulation depth, a measure of visibility or contrast: M

Amax Amin Amax Amin

(5)

where A refers to a value of the waveform (typically W/cm2 vs position) that describes the object or image. These quantities are nonnegative, so the sinusoids always have a dc bias. Modulation depth is thus a number between 0 and 1. The effect of the finite-size impulse response is that the modulation depth in the image is less than that in the object. This attenuation is usually more severe at high frequencies. MTF is the ratio of image modulation to object modulation, as a function of spatial frequency: MTF(X , H)

M image (X , H) M object (X , H)

(6)

PTF describes the relative phases with which the various sinusoidal components recombine in the image. A linear phase such as PTF x 0X corresponds to a shift of the image by an amount x 0 , each frequency component being shifted the amount required to reproduce the original waveform at the displaced location. For impulse responses that are symmetric about the ideal image point, the PTF exhibits phase reversals, with a value of either 0 or R radians as a function of spatial frequency. A general impulse response that is real but not even yields a PTF that is a nonlinear function of frequency, resulting in image degradation. Linearity of PTF is a sensitive test for aberrations (such as coma) which produces asymmetric impulse responses, and is often a design criterion.

4.4

MTF CALCULATIONS OTF can be calculated from wave-optics considerations. For an incoherent optical system, the OTF is proportional to the two-dimensional autocorrelation of the exit pupil. This calculation can account for any phase factors across the pupil, such as those arising from aberrations or defocus. A change of variables is required for the identification of an autocorrelation (a function of position in the pupil) as a transfer function (a function of image-plane spatial frequency). The change of variables is

X

x Ldi

(7)

where x is the autocorrelation shift distance in the pupil, N is the wavelength, and di is the distance from the exit pupil to the image. A system with an exit pupil of full width D has an image-space cutoff frequency consistent with Eq. (7):

Xcutoff

1 (L FN)

(8)

where FN equals (focal length)/D for a system with the object at infinity, and di/D for a system operating at finite conjugates. A diffraction-limited system has a purely real OTF. Diffraction-limited MTFs represent the best performance that a system can achieve, for a given FN and N, and accurately describe systems with

4.4

PHYSICAL OPTICS

Sm

A B

C

D

So

FIGURE 1 (A) Diffraction-limited MTF for system with circular pupil (no obscuration: So/Sm 0). (B) through (D) are diffraction-limited MTF for a system with an annular pupil: (B) So/Sm 0.25; (C) So/Sm 0.5; (D) So/Sm = 0.75. (Adapted from Ref. 4, p. 322.)

negligible aberrations, whose impulse-response size is dominated by diffraction effects. A diffractionlimited system with a square exit pupil of dimensions D r D has a linear MTF along Z or J: ¤ X ³ X MTF ¥ ´ 1 X X ¦ cutoff µ cutoff

(9)

For a system with a circular exit pupil of diameter D, the MTF is circularly symmetric, with Z profile:2 ª ¤ X ³ 2 1 ¤ X ³ X MTF ¥ «cos ¥

¦ Xcutoff ´µ P ¦ Xcutoff ´µ Xc utoff ¬ 0

1/ 2 ¹

§ ¤ X ³ 2¶ ¨1 · ¨ ¥¦ Xcutoff ´µ · © ¸

º »

if X a Xcutoff

(10)

if X Xcutoff

Equation (10) is plotted in Fig. 1, along with MTF curves obtained for annular pupils, which arise in obscured systems such as Cassegrain telescopes. The plots are functions of the obsuration ratio, and the emphasis at high frequencies has been obtained by an overall decrease in flux reaching the image, proportional to the obscured area. If the curves in Fig. 1 were plotted without normalization to 1 at Z 0, they would all be contained under the envelope of the unobscured diffraction-limited curve. A system exhibiting effects of both diffraction and aberrations has an MTF curve bounded by the diffraction-limited MTF curve as the upper envelope. Aberrations broaden the impulse response, resulting in a narrower and lower MTF, with less integrated area. The effect of defocus on the MTF is shown in Fig. 2. The MTF curves resulting from third-order spherical aberration are shown in Fig 3. MTF results for specific cases of other aberrations are contained in Ref. 3. A geometrical-aberration OTF can be calculated from ray-trace data, without regard for diffraction effects. Optical-design computer programs typically yield a diagram of ray-intersection density in the image plane, a geometrical-optics spot diagram. A geometrical-aberration OTF is calculated by Fourier transforming the spot-density distribution. The OTF thus obtained is accurate if the impulse-response size is dominated by aberration effects. A one-dimensional uniform blur spot of full width B has the following OTF in the Z direction: OTF(X)

sin (PX B) PX B

(11)


4.5

A B

D

C E

FIGURE 2 Diffraction MTF for a defocused system: (A) in focus, OPD 0.0; (B) defocus N/2N sin2 u, OPD N/4; (C) defocus N/N sin2 u, OPD N/2; (D) defocus 3N/2N sin2 u, OPD 3N/4; and (E) defocus 2N/N sin2 u, OPD N. (Adapted from Ref. 4, p. 320.)

which has a zero at Z 1/B, and also exhibits the phase reversals mentioned above. When an MTF has been calculated from ray trace data, an approximation to the total system MTF may be made4 by multiplying the diffraction-limited MTF of the proper FN and N with the ray-trace data MTF. This is equivalent to a convolution of the spot profiles from diffraction and geometrical aberrations. In electronic imaging systems, an electronics subsystem performs signal-handling and signalprocessing functions. The performance characterization of electronic networks by transferfunction techniques is well established. The usual independent variable for these time-domain transfer functions is the temporal frequency f (Hz). To interpret the electronics transfer function in the same units as the image-plane spatial frequency (cycles/mm), the temporal frequencies are divided by the scan velocity (mm/s). For a scanning system, this is the velocity of the instantaneous field of view, referred to as image coordinates. For a staring system, an effective scan velocity is the

A

B

D

C

FIGURE 3 Diffraction MTF for system with third-order spherical aberration (image plane midway between marginal and paraxial foci): (A) LAm 0.0, OPD 0; (B) LAm 4N/N sin2 u, OPD N/4; (C) LAm 8N/N sin2 u, OPD N/2; and (D) LAm 16N/N sin2 u, OPD N. (Adapted from Ref. 4, p. 322.)

4.6

PHYSICAL OPTICS

horizontal dimension of the image plane divided by the video line time. With this change of variables from temporal frequencies to spatial frequencies, the electronics can be analyzed as simply an additional subsystem, with its own transfer function that will multiply the transfer functions of the other subsystems. It should be noted that an electronics transfer function is not bounded by a pupil autocorrelation the way an optical transfer function is. Thus, it need not be maximum at the origin, and can amplify certain frequencies and have sharp cutoffs at others. Thus, the usual normalization of MTF may not be appropriate for analysis of the electronics subsystems, or for the entire imaging system including the electronics. An unavoidable impact of the electronics subsystem is the contribution of noise to the image. This limits the amount of electronic amplification that is useful in recovering modulation depth lost in other subsystems. A useful figure of merit, which has been validated to correlate with image visibility,5 is the area between two curves: the MTF and the noise power spectrum. To facilitate comparison on the same graph, the noise power spectrum is expressed in modulation depth units, and is interpreted as a noise-equivalent modulation depth (the modulation needed for unit signal-to-noise ratio) as a function of spatial frequency. The detector photosensitive area has finite size, rather than being a true point. It thus performs some spatial averaging6 on any irradiance distribution that falls on it. Large detectors exhibit more attenuation of high spatial frequencies than do small detectors. For a detector of dimension W in the x direction, the MTF is MTF(X)

sin (PXW ) PXW

(12)

which has a zero at Z 1/W. This MTF component applies to any system with detectors, and will multiply the MTFs of other subsystems. In electronic imaging systems, the image is typically sampled in both directions. The distance between samples will determine the image-plane spatial frequency at which aliasing artifacts will occur. Care must be taken in the calculation of MTF, because different impulse responses are possible depending on the location of the impulse response with respect to the sampling positions. This violates the assumption of shift-invariance needed for a transfer-function analysis.7 One approach for defining a generalized MTF is to average over all possible positions of the impulse response with respect to the sampling lattice [Eq. (4) in Ref. 8]. Research is still underway on the specification of MTF for sampled-image systems.

4.5

MTF MEASUREMENTS In any situation where the measurement of MTF involves the detection of the image-plane flux, one component of the measurement-system MTF is caused by the finite aperture of the detector, which can be accounted for in the calibration of the instrument by dividing out the detector MTF seen in Eq. (12). When OTF is measured with a point-source object, the image formed by the system under test is the impulse response. The two-dimensional impulse response can be Fourier transformed in two dimensions to yield OTF (Z, J). If an illuminated pinhole is used, it should be as small as possible. However, flux-detection considerations dictate a finite size for any source. The object is small enough not to affect the measurement if its angular subtense is much smaller than the angular subtense of the impulse response, when both are viewed from the aperture stop of the system. For sources of larger extent, a Fourier analysis can be made of the object, and an OTF can be calculated using Eq. (2), over the range of spatial frequencies provided by the source. If higher flux levels are needed to maintain signal-to-noise ratio, a line response can be measured. The system under test is presented with an illuminated line source, which acts as a delta function in one direction and a constant in the other: D (x)1( y). The system forms an image, the


4.7

line response @(x), which is a summation of vertically displaced impulse responses. In general @(x ) w h(x , 0)2. The line response only yields information about one profile of OTF (Z, J). The one-dimensional Fourier transform of the line response produces the corresponding profile of the two-dimensional OTF: >{@(x)} OTF(X , 0). To obtain other profiles of the OTF, the line source is reoriented. Line response data are also available from the response of the system to a point source, using a receiver that integrates the impulse response along one direction: a detector that is long in one dimension and is scanned perpendicularly, or a long slit that is scanned in front of a largearea detector. Another measurement of OTF uses the edge response e (x ), which is the response of the system to an illuminated knife edge. Each line in the open part of the aperture produces a displaced line response, so e (x) is a cumulative distribution, related to the line response as follows: d/dx {e (x )} @(x ), which Fourier transforms to the Z profile of the OTF. The derivative operation increases the effect of noise. Any digital filter used for data smoothing has its own impulse response, and hence its own OTF contribution. The edge response can also be measured by using a scanning knife edge in front of a detector in the image plane, with a point-source or a linesource object. An MTF calculated from a measured profile is the product of a diffraction MTF and a geometricalaberration MTF. When combining the separately-measured MTFs of several optical subsystems, care should be taken to ensure that the diffraction MTF (determined by the aperture stop of the combined system) contributes only once to the calculation. The geometrical-aberration MTFs for each subsystem will cascade if each subsystem operates independently on an irradiance basis, with no partial coherence effects.9 The major exception to this condition occurs when two subsystems are designed to correct for each other’s aberrations, and the MTF of the combined system is better than the individual MTFs would indicate. MTF can also be obtained by the system’s response to a sine-wave target, where the image modulation depth is measured as a function of spatial frequency. PTF can also be measured from the position of the waveform maxima as a function of frequency. Sine-wave targets are available as photographic prints or transparencies, which are suitable for testing visible-wavelength systems. Careful control in their manufacture is exercised10 to avoid harmonic distortions, including a limitation to relatively small modulation depths. Sine-wave targets are difficult to fabricate for testing infrared systems, and require the use of half-tone techniques.11 A more convenient target to manufacture is the three- or four-bar target of equal line and space width, with a binary transmission or reflection characteristic. These are widely used for testing both visible-wavelength and infrared systems. The square-wave response is called the contrast transfer function (CTF) and is not equivalent to the sine-wave response for which MTF is defined. CTF is a function of the fundamental spatial frequency X f inverse of the bar-to-bar spacing) and is measured on the peak-to-valley variation of image irradiance. For any particular fundamental frequency, the measured response to bar targets will be higher than that measured for sinewaves of the same frequency, because additional harmonic components contribute to the modulation. For a square-wave pattern of infinite extent, an analytical relationship exists 12 between CTF(X f ) and MTF(X). Each Fourier component of the square wave has a known transfer factor given by MTF(Z), and the modulation depth as a function of Zf of the resultant waveform can be calculated by Eq. (5). This process yields the following series: CTF(X f )

¹ 4ª 1 1 1 1 «MTF(X f ) MTF(3X f ) MTF(5X f ) MT F(7X f ) MTF(9X f ) º (13) P¬ 3 5 7 9 »

CTFs for the practical cases of three- and four-bar targets are slightly higher than the CTF curve for an infinite square wave. Figure 413 compares the MTF for a diffraction-limited circular-aperture system with CTFs obtained for infinite, three- and four-bar targets. Because of the broad spectral features associated with bar patterns of limited extent, a finite level of modulation is present in the image, even when the fundamental frequency of the bar pattern equals the cutoff frequency of the system MTF.14 The inverse process of expressing the MTF in terms of CTFs is more difficult analytically,

4.8

PHYSICAL OPTICS

1.0 CTF for three-bar target CTF for four-bar target CTF for square-wave target MTF for aberration-free system

Modulation depth

0.8

0.6

0.4

0.2

0.0 0.0

0.2

0.4

0.6

0.8

1.0

1.2

Normalized spatial frequency (/cutoff)

FIGURE 4 Comparison of MTF to CTFs obtained with infinite square wave, fourbar, and three-bar targets for a diffraction-limited system with circular pupil.

since square waves are not an orthogonal basis set for the expansion of sinusoids. A term-by-term series subtraction12 yields the following: MTF(X f )

¹ 1 1 1 Pª 1 «CTF(X f ) CTF(3X f ) CTF(5X f ) CT F(7X f ) CTF(11X f ) º (14) 4¬ 3 5 7 11 »

Narrowband electronic filtering can be used to isolate the fundamental spatial-frequency component for systems where the image data are available as a time-domain waveform. These systems do not require the correction of Eq. (14), because the filter converts bar-target data to sinewave data. The MTF can also be measured by the response of the system to a random object. Laser speckle provides a convenient means to generate a random object distribution of known spatial-frequency content. The MTF relates the input and output spatial-frequency power spectra of the irradiance waveforms: Soutput (X , H) | MTF(X , H)|2 r Sinput (X , H)

(15)

This method is useful in the measurement of an average MTF for sampled-image systems,15 since the speckle pattern has a random position with respect to the sampling sites. A number of interferometric methods have been developed for measuring MTF.16 An interferogram of the wavefront exiting the system is reduced to find the phase map. The distribution of amplitude and phase across the exit pupil contains the information necessary for calculation of OTF by pupil autocorrelation.

4.6

REFERENCES 1. 2. 3. 4.

J. W. Goodman, Introduction to Fourier Optics, McGraw-Hill, New York, 1968, pp. 116–120. J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics, Wiley, New York, 1978, pp. 305–307. V. N. Mahajan, Optical Imaging and Aberrations, SPIE Press, Bellingham, WA, 1998, Part II, chap. 2. W. J. Smith, Modern Optical Engineering, McGraw-Hill, New York, 1966, p. 323.


4.9

5. H. L. Snyder, “Image Quality and Observer Performance,” in L. M. Biberman (ed.), Perception of Displayed Information, Plenum, New York, 1973. 6. G. D. Boreman and A. E. Plogstedt, “Spatial Filtering by a Line-Scanned Nonrectangular Detector-Application to SPRITE Readout MTF,” Appl. Opt. 28:1165−1168, 1989. 7. W. Wittenstein, J. C. Fontanella, A. R. Newbery, and J. Baars, “The Definition of the OTF and the Measurement of Aliasing for Sampled-Imaging Systems,” Optica Acta 29(l):41–50, 1982. 8. S. K. Park, R. Schowengerdt, and M. Kaczynski, “Modulation-Transfer-Function Analysis for Sampled Image Systems,” Appl. Opt. 23:2572, 1984. 9. J. B. DeVelis and G. B. Parrent, “Transfer Function for Cascaded Optical Systems,” J. Opt. Soc. Am. 57:1486–1490, 1967. 10. R. L. Lamberts, “The Production and Use of Variable-Transmittance Sinusoidal Test Objects,” Appl. Opt. 2:273–276, 1963. 11. A. Daniels, G. Boreman, A. Ducharme, and E. Sapir, “Random Transparency Targets for Modulation Transfer Function Measurement in the Visible and Infrared,” Opt. Eng. 34:860–868, 1995. 12. J. W. Coltman, “The Specification of Imaging Properties by Response to a Sine Wave Input,” J. Opt. Soc. Am. 44:468, 1954. 13. G. Boreman and S. Yang, “Modulation Transfer Function Measurement Using Three- and Four-Bar Targets,” Appl. Opt. 34:8050–8052, 1995. 14. D. H. Kelly, “Spatial Frequency, Bandwidth, and Resolution,” Appl. Opt. 4:435, 1965. 15. G. D. Boreman, Modulation Transfer Function in Optical and Electro-Optical Systems, SPIE Press, Bellingham, WA, 2001, chap. 4. 16. D. Malacara, Optical Shop Testing, Wiley, New York, 1978, chap. 3.


5 COHERENCE THEORY William H. Carter

Naval Research Laboratory Washington, D.C.

5.1

GLOSSARY I k p t U u WY x '12(V) $@ $V OY H Y Real ( )

5.2

intensity (use irradiance or field intensity) radian wave number unit propagation vector time field amplitude Fourier transform of U cross-spectral density function spatial vector mutual coherence function coherence length coherence time complex degree of spatial coherence phase radian frequency real part of ( )

INTRODUCTION

Classical Coherence Theory All light sources produce fields that vary in time with highly complicated and irregular waveforms. Because of diffraction, these waveforms are greatly modified as the fields propagate. All light detectors measure the intensity time averaged over the waveform. This measurement depends on the integration

Deceased. The author was a visiting scientist at the Johns Hopkins University Applied Physics Laboratory when this chapter was written.

5.1

5.2

PHYSICAL OPTICS

time of the detector and the waveform of the light at the detector. Generally this waveform is not precisely known. Classical coherence theory1−8 is a mathematical model which is very successful in describing the effects of this unknown waveform on the observed measurement of time-averaged intensity. It is based on the electromagnetic wave theory of light as formulated from Maxwell’s equations, and uses statistical techniques to analyze the effects due to fluctuations in the waveform of the field in both time and space. Quantum Coherence Theory Classical coherence theory can deal very well with almost all presently known optical coherence phenomena; however, a few laboratory experiments require a much more complicated mathematical model, quantum coherence theory9−11 to explain them. This theory is not based on classical statistical theory, but is based on quantum electrodynamics.12−14 While the mathematical model underlying classical coherence theory uses simple calculus, quantum coherence theory uses the Hilbert space formulation of abstract linear algebra, which is very awkward to apply to most engineering problems. Fortunately, quantum coherence theory appears to be essential as a model only for certain unusual (even though scientifically very interesting) phenomena such as squeezed light states and photon antibunching. All observed naturally occuring phenomena outside of the laboratory appear to be modeled properly by classical coherence theory or by an approximate semiclassical quantum theory. This chapter will deal only with the simple classical model.

5.3

SOME ELEMENTARY CLASSICAL CONCEPTS

Analytical Signal Representation Solutions of the time-dependent, macroscopic Maxwell’s equations yield six scalar components of the electric and the magnetic fields which are functions of both time and position in space. As in conventional diffraction theory, it is much more convenient to treat monochromatic fields than it is to deal with fields that have complicated time dependencies. Therefore, each of these scalar components is usually represented at some typical point in space (given with respect to some arbitrary origin by the radius vector x (x, y, z) by a superposition of monochromatic real scalar components. Thus the field amplitude for a typical monochromatic component of the field with radial frequency Y is given by U r (x , W ) U 0 (x ) cos [F(x ) W t ]

(1)

where U0(x) is the field magnitude and H(x) is the phase. Trigonometric functions like that in Eq. (1) are awkward to manipulate. This is very well known in electrical circuit theory. Thus, just as in circuit theory, it is conventional to represent this field amplitude by a “phasor” defined by U (x , W ) U 0 (x )e iF ( x )

(2)

The purpose for using this complex field amplitude, just as in circuit theory, is to eliminate the need for trigonometric identities when adding or multiplying field amplitudes. A time-dependent complex analytic signal (viz., Ref. 15, sec. 10.2) is usually defined as the Fourier transform of this phasor, i.e., c

u(x , t ) ¯ U (x , W )e iWt dW 0

(3)

The integration in Eq. (3) is only required from zero to infinity because the phasor is defined with hermitian symmetry about the origin, i.e., U(–x, Y) U (x, Y). Therefore, all of the information is contained within the domain from zero to infinity. To obtain the actual field component from the analytical signal just take the real part of it. The Fourier transform in Eq. (3) is well defined if the analytical signal represents a deterministic field. However, if the light is partially coherent, then the analytic signal is usually taken to be a stationary random process. In this case the Fourier inverse of

COHERENCE THEORY

5.3

Eq. (3) does not exist. It is then possible to understand the spectral decomposition given by Eq. (3) only within the theory of generalized functions (viz., see Refs. 16, the appendix on generalized functions, and 17, pp. 25–30). Scalar Field Amplitude Each monochromatic component of an arbitrary deterministic light field propagating through a homogeneous, isotropic medium can always be represented using an angular spectrum of plane waves for each of the six scalar components of the vector field. The six angular spectra are coupled together by Maxwell’s equations so that only two are independent.18–20 Any two of the six angular spectra can be used to define two scalar fields from which the complete vector field can be determined. A polarized light field can be represented in this way by only one scalar field.20,21 Thus it is often possible to represent one polarized component of a vector electromagnetic field by a single scalar field. It has also been found useful to represent completely unpolarized light by a single scalar field. In more complicated cases, where the polarization properties of the light are important, a vector theory is sometimes needed as discussed later under “Explicit Vector Representations.” Temporal Coherence and Coherence Time Within a short enough period of time, the time dependence of any light field at a point in space can be very closely approximated by a sine wave (Ref. 15, sec. 7.5.8). The length of time for which this is a good approximation is usually called the coherence time $V. The coherence time is simply related to the spectral bandwidth for any light wave by the uncertainty principle, i.e., $T $W q 1

(4)

For a light wave which is also highly directional within some region of space (like a beam) so that it propagates generally in some fixed direction (given by the unit vector p), the field amplitude is given by u(x , t ) f (p x ct )

(5)

Such a traveling wave will be approximately sinusoidal (and hence coherent) over some coherence length $@ in the direction of p where from Eq. (4) we see that $ @ c $ T y c/$ W

(6)

so that the coherence length varies inversely with bandwidth. Spatial Coherence and Coherence Area The time-dependent waveform for any light field is approximately the same at any point within a sufficiently small volume of space called the coherence volume. The projection of this volume onto a surface is termed a coherence area. If we have a field that, within some region, is roughly directional so that its field amplitude is given by Eq. (5), then the coherence length gives the dimension of the coherence volume in the direction of propagation p, and the coherence area gives the dimensions of the coherence volume normal to this direction. Measurements of Coherence Coherence is usually measured by some form of interferometer that takes light from two test points in a light field, x1 and x2, and then allows them to interfere after introducing a time advance V in the light from x1 relative to that from x2. If the field intensity of the interference pattern is measured as a function of V, then in general it has the form (see Ref. 15, sec. 10.3.1) I (T ) I (x 1 ) I (x 2 ) 2 Real ('12 (T ))

(7)

5.4

PHYSICAL OPTICS

where I(x1) is the intensity at the ith test point, and '12(V) is the mutual coherence function which measures the V advanced correlation between the waveforms at the two test points (as subsequently defined under “Mutual Coherence Function”). There are many interferometers which have been developed to measure '12(V) in this way. One of the earliest techniques was developed by Thompson and Wolf.22 They used a diffractometer to measure the coherence over a surface normal to the direction of propagation for a collimated beam from a partially coherent source. More recently, Carter23 used an interferometer made from a grating and a microscope to similarly measure the coherence of a beam transverse to the direction of propagation.

5.4

DEFINITIONS OF COHERENCE FUNCTIONS

Mutual Coherence Function In an older form of coherence theory1 the principal coherence function was the mutual coherence function defined by24 '12 (T ) y

1

T lc2T

T

¯ T u(x 1, t T )u (x 2, t ) dt

(8)

where u(x, t) represents the complex analytic time-dependent signal at some point x and some time t as defined in Eq. (3). This definition was originally motivated by the fact that the intensity, as actually measured, is precisely this time averaged function with x1 x2 and V 0, and that this function is the most readily measured since it appears directly in Eq. (7). Thus it was clearly possible to measure '12(V) over some input plane, propagate it to an output plane,25 and then find the intensity over the output plane from '12(V). It was assumed in this definition in Eq. (8) that u(x, t) is stationary in time so that '12(V) is only a function of V and not of t. In most of the older literature, sharp brackets were usually used to represent this time average rather than an ensemble average (see Ref. 15, sec. 10.3.1). In the early 1960s it was found to be much more convenient to treat u(x, t) as an ergodic, stationary random process so that Eq. (8) could be replaced by '12 (T ) u (x 1, t T )u (x 2, t )®

(9)

where (everywhere in this chapter) the sharp brackets denote an ensemble average. After the change to ensemble averages the cross-spectral density function (to be defined shortly) became the most used correlation function in the coherence literature, because of the simpler and more general rules for its propagation (as discussed later under “Representations” on p. 5.16).

Complex Degree of Coherence To obtain a function that depends only on the coherence properties of a light field it is often useful to normalize the mutual coherence function in the manner of

G 12 (T )

u(x 1, t T )u (x 2, t )® u(x 1, t )u (x 1, t )®u(x 2, t )u (x 2, t )®

(10)

This is called the complex degree of coherence. It is a properly normalized correlation coefficient, so that I11(0) I22(0) 1. This indicates that the field at a point in space must always be perfectly coherent with itself. All other values of I12(V) are generally complex with an amplitude less than one. This indicates that the fields at two different points, or at the same point after a time delay V, are generally less than perfectly coherent with each other. The magnitude of the complete degree of spatial coherence (from zero to one) is a measure of the mutual coherence between the fields at the two test points and after a time delay V.

COHERENCE THEORY

5.5

Cross-Spectral Density Function Just as in classical diffraction theory, it is much easier to propagate monochromatic light than light with a complicated time waveform. Thus the most frequently used coherence function is the crossspectral density function, WY(x1, x2), which is the ensemble-averaged correlation function between a typical monochromatic component of the field at some point x1 with the complex conjugate of the same component of the field at some other point x2. It may be defined by

D (W W `)WW (x 1, x 2 ) U (x 1, W )U (x 2, W `)®

(11)

The amplitude U(x, Y) for a field of arbitrary coherence is taken to be a random variable. Thus U(x, Y) represents an ensemble of all of the possible fields, each of which is represented by a complex phasor amplitude like that defined in Eq. (2). The sharp brackets denote an ensemble average over all of these possible fields weighted by the probability for each of them to occur. The correlation functions defined by Eqs. (11) and (9) are related by the Fourier transform pairs c

'12 (T ) ¯ WW (x 1, x 2 )e iWT dW 0

(12)

and WW (x 1, x 2 )

1 2P

c

¯ c'12(T )e iWT dT

(13)

which is easily shown, formally, by substitution from Eq. (3) into (9) and then using (11). These relations represent a form of the generalized Wiener-Khintchine theorem (see Ref. 26, pp. 107–108). Complex Degree of Spectral Coherence Because the cross-spectral density function contains information about both the intensity (see “Intensity,” which follows shortly) and the coherence of the field, it is useful to define another coherence function which describes the coherence properties only. This is the complex degree of spectral coherence (not to be confused with the complex degree of spatial coherence, which is a totally different function), which is usually defined by27

MW (x 1, x 2 )

WW (x 1, x 2 )

(14)

WW (x 1, x 1 )WW (x 2, x 2 )

It is easy to show that this function is a properly normalized correlation coefficient which is always equal to unity if the field points are brought together, and is always less than or equal to unity as they are separated. If the magnitude of OY(x1, x2) is unity, it indicates that the monochromatic field component with radial frequency Y is perfectly coherent between the two points x1 and x2. If the magnitude of this function is less than unity it indicates less-than-perfect coherence. If the magnitude is zero it indicates complete incoherence between the field amplitudes at the two test points. For most partially coherent fields the cross-spectral density function has significantly large values only for point separations which keep the two field points within the same coherence volume. This function depends only on the positions of the points and the single radial frequency that the field components at the two points share. Field components of different frequency are always uncorrelated (and therefore incoherent), even at the same point. Spectrum and Normalized Spectrum Recently, the changes in the spectrum of light due to propagation have been studied using coherence theory. It is therefore useful to define the spectrum of light as just the monochromatic intensity

5.6

PHYSICAL OPTICS

(which is just the trace of the cross-spectral density function) as a function of omega, and the spectrum of a primary source as a very similar function, i.e., SU (x , W ) U W (x )U W (x )® WU (x , x )

(15)

SQ (x , W ) RW (x ) RW (x )® WQ (x , x )

where the subscript Q indicates that this is a primary source spectrum, and the subscript U indicates that this is a field spectrum. The spectrum for the primary source is a function of the phasor TY(x) which represents the currents and charges in this source as discussed under “Primary Sources” in the next section. It is also useful to normalize these spectra in the manner s A (x , W )

S A (x , W )

(16)

c

¯0 SA (x , W ) dW

where the subscript A can indicate either U or Q, and the normalized spectrum has the property c

¯0 s A (x , W ) dW 1

(17)

so that it is independent of the total intensity.

Angular Correlation Function A new coherence function, introduced for use with the angular spectrum expansion of a monochromatic component of the field,28 is the angular correlation function defined by W (p1 , p2 ) AW (p1 ) AW (p2 )®

(18)

where AY(pi) is the angular spectrum which gives the complex amplitude of the plane wave component of the field which propagates in the direction given by the unit vector pi. This is related to the cross-spectral density function over the z 0 plane by the equation (p1 , p2 )

1 L4

c

c

c

c

¯ c ¯ c ¯ c ¯ cW (W0)(x 1` , x `2 )e ik(p x ` p x ` )d 2 x 1` d 2 x `2 1

1

2

2

(19)

This is the four-dimensional Fourier transform of the cross-spectral density function over the z 0 plane. It represents a correlation function between the complex amplitudes of two plane wave components of the field propagating in the directions given by the unit vectors p1 and p2, respectively. It can be used to calculate the cross-spectral density function (as described later under “Angular Spectrum Representation”) between any pair of points away from the z 0 plane, assuming that the field propagates in a source-free homogeneous medium. In this chapter we will use single-primed vectors, as in Eq. (19), to represent radius vectors from the origin to points within the z 0 plane, i.e., x` (x`, y`, 0), as shown in Fig. 1. All other vectors, such as x or xp, are to be taken as three-dimensional vectors. Generally s and p are three-dimensional unit vectors indicating directions from the origin, a superscript (0) on a function indicates that it is the boundary condition for that function over the z 0 plane, and a superscript (c) on a function indicates that it is the asymptotic value for that function on a sphere of constant radius R from the origin as R l c.

COHERENCE THEORY

5.7

FIGURE 1 Illustrating the coordinate system and notation used for planar sources in the z 0 plane.

Intensity The intensity is usually considered to be the observable quantity in coherence theory. Originally it was defined to be the trace of the mutual coherence function as defined by Eq. (8), i.e., I (x 1 ) '11(0) y

1

T

¯ T u(x 1 , t )u (x 1 , t )dt

T lc2T

(20)

which is always real. Thus it is the time-averaged square magnitude of the analytic signal. This represents the measurement obtained by the electromagnetic power detectors always used to detect light fields. Since the change to ensemble averages in coherence theory, it is almost always assumed that the analytic signal is an ergodic random process so that the intensity can be obtained from the equation I (x 1 ) '11(0) u(x 1 , t )u (x 1 , t )®

(21)

where the sharp brackets indicate an ensemble average. Usually, in most recent coherence-theory papers, the intensity calculated is actually the spectrum, which is equivalent to the intensity of a single monochromatic component of the field which is defined as the trace of the cross-spectral density function as given by I W (x 1 ) WW (x 1 , x 1 ) U (x 1 , W ) U (x 1 , W )®

(22)

Since different monochromatic components of the field are mutually incoherent and cannot interfere, we can always calculate the intensity of the total field as the sum over the intensities of its monochromatic components in the manner c

I (x 1 ) ¯ I W (x 1 ) dW 0

(23)

Since in most papers on coherence theory the subscript omega is usually dropped, the reader should be careful to observe whether or not the intensity calculated is for a monochromatic component of the field only. If it is, the total measurable intensity can be obtained simply by summing over all omega, as indicated in Eq. (23).

Radiant Emittance In classical radiometry the radiant emittance is defined to be the power radiated into the far field by a planar source per unit area. A wave function with some of the properties of radiant emittance has been defined by Marchand and Wolf using coherence theory [see Refs. 29, eq. (32), and 30].

5.8

PHYSICAL OPTICS

However, because of interference effects, the far-field energy cannot be subdivided into components that can be traced back to the area of the source that produced them. The result is that the radiant emittance defined by Marchand and Wolf is not nonnegative definite (as it is in classical radiometry) except for the special case of a completely incoherent source. Since, as discussed in the next section under “Perfectly Incoherent Source,” perfectly incoherent sources exist only as a limiting case, radiant emittance has not been found to be a very useful concept in coherence theory.

Radiant Intensity In classical radiometry the radiant intensity is defined to be the power radiated from a planar source into a unit solid angle with respect to an origin at the center of the surface. This can be interpreted in coherence theory as the field intensity over a portion of a surface in the far field of the source, in some direction given by the unit vector s, which subtends a unit solid angle from the source. Thus the radiant intensity for a monochromatic component of the field can be defined in coherence theory as J W (s) y W W(c)(R s, R s)R 2 Rlc

(24)

To obtain the total radiant intensity we need only sum this function over all omega.

Radiance In classical coherence theory the radiance function is the power radiated from a unit area on a planar source into a unit solid angle with respect to an origin at the center of the source. In the geometrical optics model, from which this concept originally came, it is consistent to talk about particles of light leaving an area at a specified point on a surface to travel in some specified direction. However, in a wave theory, wave position and wave direction are Fourier conjugate variables. We can have a configuration space wave function (position) or a momentum space wave function (direction), but not a wave function that depends independently on both position and direction. Thus the behavior of a wave does not necessarily conform to a model which utilizes a radiance function.31 Most naturally occurring sources are quasi-homogeneous (discussed later). For such sources, a radiance function for a typical monochromatic component of the field can be defined as the Wigner distribution function32–35 of the cross-spectral density function over the z 0 plane, which is given by BW (x ` , s)

cosQ c c (0) W (x ` x ` /2, x ` x ` /2)e iksx ` d 2 x ` L 2 ¯ c ¯ c W

(25)

where S is the angle that the unit vector s makes with the z axis. For quasi-homogeneous fields this can be associated with the energy radiated from some point x ` into the far field in the direction s. Such a definition for radiance also works approximately for some other light fields, but for light which does not come from a quasi-homogeneous source, no such definition is either completely equivalent to a classical radiance function31 or unique as an approximation to it. Much progress has been made toward representing a more general class of fields using a radiance function.36 In general, waves do not have radiance functions.

Higher-Order Coherence Functions In general, the statistical properties of a random variable are uniquely defined by the probability density function which can be expanded into a series which contains correlation functions of all orders. Thus, in general, all orders of correlation functions are necessary to completely define the statistical properties of the field. In classical coherence theory we usually assume that the partially coherent fields arise from many independent sources so that, by the central limit theorem of statistics,

COHERENCE THEORY

5.9

the probability distribution function for the real and imaginary components of the phasor field amplitude are zero-mean gaussian random variables (See Ref. 8, sec. 2.72e). Thus, from the gaussian moment theorem, the field correlation functions of any order can be calculated from the secondorder correlation functions, for example, U (x 1, W ) U (x 1, W ) U (x 2, W ) U (x 2, W )® I W (x 1 ) I W (x 2 ) |WW (x 1, x 2 )|2

(26)

Thus, for gaussian fields, the second-order correlation functions used in coherence theory completely define the statistical properties of the field. Some experiments, such as those involving intensity interferometry, actually measure fourth- or higher-order correlation functions.37

5.5

MODEL SOURCES

Primary Sources In coherence theory it is useful to talk about primary and secondary sources. A primary source distribution is the usual source represented by the actual charge and current distribution which give rise to the field. For propagation from a primary source through a source-free media, the field amplitude is defined by the inhomogeneous Helmholtz equation [see Ref. 38, eq. (6.57)], i.e., ¤ 2 W2³ 2 ´ U W (x ) 4PRW (x ) c µ ¦¥

(27)

where TY(x) represents the charge-current distribution in the usual manner. A solution to this wave equation gives WU (x 1 , x 2 ) ¯

c

c

c

c

c

¯ ¯ ¯ ¯ ¯

c

WQ (x 1``, x ``2 )K (x 1 , x 1``)K (x 2 , x `2`)d 3 x 1`` d 3 x ``2

c c c c c c

(28)

for the cross-spectral density function, where k Y/c 2R/N K (x , x ``)

e ik|x x ``| | x x `` |

(29)

is the free-space propagator for a primary source, WU(x1, x2) is the cross-spectral density function for the fields (with suppressed Y-dependence), as defined by Eq. (11), and WQ (x 1``, x ``2 ) RW (x 1``)RW (x ``2 )®

(30)

is the cross-spectral density function for the source. This three-dimensional primary source can be easily reduced to a two-dimensional source over the z 0 plane by simply defining the charge current distribution to be

RW (x ) RW` (x , y)D (z )

(31)

where F(z) is the Dirac delta function. Secondary Sources Often, however, it is more convenient to consider a field which arises from sources outside of the region in space in which the fields are of interest. Then it is sometime useful to work with boundary conditions for the field amplitude over some surface bounding the region of interest. In many coherence problems these boundary conditions are called a planar secondary source even though they are actually treated as boundary conditions. For example, most conventional diffraction equations

5.10

PHYSICAL OPTICS

assume the boundary condition over the z 0 plane is known and use it to calculate the fields in the z > 0 half-space. Then the field obeys the homogeneous Helmholtz equation [See Ref. 38, eq. (7.8)], i.e., ¤ 2 W2³ ¥¦ c 2 ´µ U W (x ) 0

(32)

which has the solution WW (x 1 , x 2 ) ¯

c

c

c

¯ ¯ ¯

c

W (W0)(x 1` , x `2 )h(x 1 , x 1` )h (x 2 , x `2 )d 2 x 1` d 2 x `2

c c c c

(33)

where WW(0)(x 1` , x `2 ) is the boundary condition for the cross-spectral density function of the fields over the z 0 plane (as shown in Fig. 1), WY(x1, x2) is the cross-spectral density function anywhere in the z > 0 half-space, and h(x , x `)

1 d e ik|x x `| 2P dz | x x ` |

(34)

is the free-space propagator for the field amplitude. This is a common example of a secondary source over the z 0 plane. Perfectly Coherent Source The definition of a perfectly coherent source, within the theory of partial coherence, is somewhat complicated. The light from an ideal monochromatic source is, of course, always perfectly coherent. Such a field produces high-contrast interference patterns when its intensity is detected by any suitable instrument. However, it is possible that light fields exist that are not monochromatic but can also produce similar interference patterns and therefore must be considered coherent. The ability of light fields to produce interference patterns at a detector is measured most directly by the complex degree of coherence I12(V), defined by Eq. (10). If a field has a complex degree of coherence that has unit magnitude for every value of V and for every point pair throughout some domain D, then light from all points in D will combine to produce high-contrast interference fringes.39 Such a field is defined to be perfectly coherent within D. Mandel and Wolf 27 have shown that the mutual coherence function for such a field factors within D in the manner '12 (T ) Y (x 1 )Y (x 2 )e iWT

(35)

We will take Eq. (35) to be the definition of a perfectly coherent field. Coherence is not as easily defined in the space-frequency domain because it depends on the spectrum of the light as well as on the complex degree of spectral coherence. For example, consider a field for which every monochromatic component is characterized by a complex degree of spectral coherence which has unit magnitude between all point pairs within some domain D. Mandel and Wolf 40 have shown that for such a field the cross-spectral density function within D factors is WW (x 1`` x ``2 ) U (x 1``, W )U (x ``2 , W )

(36)

However, even if Eq. (36) holds for this field within D, the field may not be perfectly coherent [as perfect coherence is defined by Eq. (35)] between all points within the domain. In fact it can be completely incoherent between some points within D.41 A secondary source covering the z 0 plane with a cross-spectral density function over that plane which factors as given by Eq. (36) will produce a field in the z > 0 half-space (filled with free space or a homogeneous, isotopic dielectric) which has a cross-spectral density function that factors in the same manner everywhere in the half-space. This can be easily shown by substitution from Eq. (36) into Eq. (28), using Eq. (29). But, even if this is true for every monochromatic component of the field, Eq. (35) may not hold within the half-space for every point pair, so we cannot say that the field is perfectly coherent there. Perfectly coherent light

COHERENCE THEORY

5.11

sources never actually occur, but sometimes the light from a laser can behave approximately in this way over some coherence volume which is usefully large. Radio waves often behave in this manner over very large coherence volumes.

Quasi-Monochromatic Source In many problems it is more useful not to assume that a field is strictly monochromatic but instead to assume that it is only quasi-monochromatic so that the time-dependent field amplitude can be approximated by u(x , t ) u0 (x , t )e iWt

(37)

where u0(x, t) is a random process which varies much more slowly in time than e–iYt. Then the mutual coherence function and the complex degree of spatial coherence can be usefully approximated by (see Ref. 15, sec. 10.4.1) '12 (T ) '12 (0)e iWT

G 12 (T ) G 12 (0)e iWT

(38)

within coherence times much less than the reciprocal bandwidth of the field, i.e., $ T 1/$ W . In the pre-1960 coherence literature, '12(0) was called the mutual intensity. Monochromatic diffraction theory was then used to define the propagation properties of this monochromatic function. It was used instead of the cross-spectral density function to formulate the theory for the propagation of a partially coherent quasi-monochromatic field. While this earlier form of the theory was limited by the quasi-monochromatic approximation and was therefore not appropriate for very wideband light, the newer formulation (in terms of the cross-spectral density function) makes no assumptions about the spectrum of the light and can be applied generally. The quasi-monochromatic approximation is still very useful when dealing with radiation of very high coherence.42

Schell Model Source Other, more general, source models have been developed. The most general of these is the Schell model source (see Refs. 43, sec. 7.5, and 44), for which we only assume that the complex degree of spectral coherence for either a primary or secondary source is stationary in space, so that from Eq. (14) we have WA (x 1, x 2 ) M A (x 1 x 2 ) WA (x 1, x 1 )WA (x 2, x 2 )

(39)

where the subscript A stands for U in the case of a Schell model secondary source and Q in the case of a Schell model primary source. The Schell model does not assume low coherence and, therefore, can be applied to spatially stationary light fields of any state of coherence. The Schell model of the form shown in Eq. (39) has been used to represent both three-dimensional primary sources45,46 and two-dimensional secondary sources.43,47,48

Quasi-Homogeneous Source If the intensity of a Schell model source is essentially constant over any coherence area, then Eq. (39) may be approximated by WA (x 1, x 2 ) M A (x 1 x 2 )I A[(x 1 x 1 )/2]

(40)

5.12

PHYSICAL OPTICS

where the subscript A can be either U, for the case of a quasi-homogeneous secondary source or Q, for the case of a quasi-homogeneous primary source. This equation is very useful in coherence theory because of the important exact mathematical identity.49 c

c

c

c

¯ c ¯ c ¯ c ¯ c M(W0)(x 1` x `2 )I (W0)[(x 1` x `2 )/2]e ik(x ` p x ` p )dx1` dy1` dx 2` dy2` 1

¯

c

¯

c

c c

1

M (W0)(x ` )e ik( x ` p )dx ` dy ` ¯

c

2

¯

2

c

c c

(41)

I (W0)(x ` )e ik( x ` p )dx ` dy `

where x ` (x 1` x `2 )/2

x ` x 1` x `2

p (p1 p2 )/2

p p1 p2

and

which allows the four-dimensional Fourier transforms that occur in propagating the correlation functions for secondary sources [for example, Eqs. (49) or (54)] to be factored into a product of two-dimensional Fourier transforms. An equivalent identity also holds for the six-dimensional Fourier transform of the cross-spectral density function for a primary source, reducing it to the product of two three-dimensional Fourier transforms. This is equally useful in dealing with propagation from primary sources [for example, Eq. (53)]. This model is very good for representing twodimensional secondary sources with sufficiently low coherence that the intensity does not vary over the coherence area on the input plane.49,50 It has also been applied to primary three-dimensional sources.45,46 to primary and secondary two-dimensional sources,51,52 and to three-dimensional scattering potentials.53,54

Perfectly Incoherent Source If the coherence volume of a field becomes much smaller than any other dimensions of interest in the problem, then the field is said to be incoherent. It is believed that no field can be incoherent over dimensions smaller than the order of a light wavelength. An incoherent field can be taken as a special case of a quasi-homogeneous field for which the complex degree of spectral coherence is approximated by a Dirac delta function, i.e., WA (x 1, x 2 ) I (x 1 )D 2 (x 1 x 2 )

(42)

WQ (x 1, x 2 ) I (x 1 )D 3 (x 1 x 2 ) where the two-dimensional Dirac delta function is used for any two-dimensional source and a three-dimensional Dirac delta function is used only for a three-dimensional primary source. Even though this approximation is widely used, it is not a good representation for the thermal sources that predominate in nature. For example, the radiant intensity from a planar, incoherent source is not in agreement with Lambert’s law. For thermal sources the following model is much better.

Thermal (Lambertian) Source For a planar, quasi-homogeneous source to have a radiant intensity in agreement with Lambert’s law it is necessary for the complex degree of spectral coherence to have the form

M A (x 1 x 2 )

sin (k |x 1 x 2|) k |x 1 x 2|

(43)

COHERENCE THEORY

5.13

to which arbitrary spatial frequency components with periods less than a wavelength can be added since they do not affect the far field.55 It can also be shown that, under frequently obtained conditions, blackbody radiation has such a complex degree of spectral coherence.56 It is believed that most naturally occurring light can be modeled as quasi-homogeneous with this correlation function.

5.6

PROPAGATION

Perfectly Coherent Light Perfectly coherent light propagates according to conventional diffraction theory. By substitution from Eq. (36) into Eq. (33) using Eq. (34) we obtain U W (x )

1 2P

c

1

c

¯ c ¯ c U W(0)(x `) 2P

d e ik|x x `| 2 d x` dz ` | x x ` |

(44)

which is just Rayleigh’s diffraction integral of the first kind. Perfectly coherent light does not lose coherence as it propagates through any medium which is time-independent. For perfectly coherent light propagating in time-independent media, coherence theory is not needed.

Hopkin’s Formula In 1951 Hopkins57 published a formula for the complex degree of spatial coherence for the field from a planar, secondary, incoherent, quasi-monochromatic source after propagating through a linear optical system with spread function h(x , x`), i.e.,

G 12 (0)

1

c

c

I (x 1 )I (x 2 )

¯ c ¯ c I (W0)(x 1` )h(x 1 x 1` )h (x 2 x 1` )d 2 x 1`

(45)

where I W(0) (x `) is the intensity over the source plane. This formula can be greatly generalized to give the complex degree of spectral coherence for the field from any planar, quasi-homogeneous, secondary source58 after transmission through this linear optical system, i.e.,

M12 (x 1, x 2 )

1

c

I (x 1 )I (x 2 )

c

¯ c ¯ c I (W0)(x `)h(x 1, x `)h (x 2, x `)d 2 x `

(46)

provided that the spread function h(x , x`) can be assumed to be constant in its x` dependence over any coherence area in the source plane.

van Cittert–Zernike Theorem Hopkins’ formula can be specialized for free-space propagation to calculate the far-field coherence properties of planar, secondary, quasi-homogeneous sources of low coherence. In 1934 van Cittert59 and, later, Zernike60 derived a formula equivalent to

MW (x 1, x 2 )

1 I (x 1 )I (x 2

c

c

e ik[[|x1 x `| |x 2 x `|]

¯ ¯ I (W0)(x `)| x 1 x ` || x 2 x ` | d 2 x ` ) c c

(47)

for the complex degree of spectral coherence between any pair of points in the field radiated from an incoherent planar source, assuming that the points are not within a few wavelengths of the source. We can obtain Eq. (47) by substitution from Eq. (34) into Eq. (46) and then approximating

5.14

PHYSICAL OPTICS

the propagator in a standard manner [see Ref. 61, eq. (7)]. Assume next, that the source area is contained within a circle of radius a about the origin in the source plane as shown in Fig. 4. Then, if the field points are both located on a sphere of radius R, which is outside of the Rayleigh range of the origin (i.e., | x 1 || x 2 | R ka 2), we can apply the Fraunhofer approximation to Eq. (47) to obtain 1

M12 (x 1, x 2 )

c

c

¯ ¯ I (W0)(x `)e ikkx `(x x )/Rd 2 x ` ) c c 1

I (x 1 )I (x 2

(48)

2

This formula is very important in coherence theory. It shows that the complex degree of spectral coherence from any planar, incoherent source with an intensity distribution IW(0) (x `) has the same dependence on (x1 − x2), over a sphere of radius R in the far field, as the diffraction pattern from a closely related perfectly coherent planar source with a real amplitude distribution proportional to IW(0) (x `) (see Ref. 15, sec. 10.4.2a). Equation (48) can also be applied to a planar, quasi-homogeneous source49 that is not necessarily incoherent, as will be shown later under “Reciprocity Theorem.”

Angular Spectrum Representation Much more general equations for propagation of the cross-spectral density function can be obtained. Equation (33) is one such expression. Another can be found if we expand the fields in a source-free region of space into an angular spectrum of plane waves. Then we find that the crossspectral density function over any plane can be calculated from the same function over a parallel plane using a linear systems approach. For an example, consider the two planes illustrated in Fig. 2. We assume that the cross-spectral density function is known over the z 0 plane in the figure, and we wish to calculate this function over the z d plane. To do this we first take the Fourier transform of the cross-spectral density function over the z 0 plane according to Eq. (19) in (p1, p2 )

1 L4

c

c

c

c

¯ c ¯ c ¯ c ¯ c W W(0)(x 1` , x `2 ) e ik(p x ` p x ` )d 2 x 1` d 2 x `2 1

1

2

2

(49)

to obtain the angular correlation function in which all of the phase differences between the plane wave amplitudes are given relative to the point at the origin. Second, we shift the phase reference

FIGURE 2 Illustrating the coordinate system for propagation of the cross-spectral density function using the angular spectrum of plane waves.

COHERENCE THEORY

5.15

from the origin to the point on the z axis in the output plane by multiplying the angular correlation function by a transfer function, i.e., out (p1, p2 ) in (p1, p2 ) exp [ik(m1 m2 )d]

(50)

where d is the distance from the input to the output plane along the z axis (for back propagation d will be negative) and mi is the third component of the unit vector pi (pi, qi, mi), i 1 or 2, which is defined by mi 1 pi2 qi2 i pi2 qi2 1

if pi2 qi2 a 1

(51)

if pi2 qi2 1

and is the cosine of the angle that pi makes with the z axis for real mi. Finally, to obtain the crossspectral density function over the output plane, we simply take the Fourier inverse of out (p1, p2 ), i.e., W W(d )(x 1 , x 2 ) ¯

c

c

c

¯ ¯ ¯

c

c c c c

out (p1, p2 ) e ik(p1 x1 p2 x 2 )d 2 p1 d 2 p2

(52)

where, in this equation only, we use xi to represent a two-dimensional radius vector from the point (0, 0, d) to a field point in the z d plane, as shown in Fig. 2. This propagation procedure is similar to the method usually used to propagate the field amplitude in Fourier optics. In coherence theory, it is the cross-spectral density function for a field of any state of coherence that is propagated between arbitrary parallel planes using the linear systems procedure. The only condition for the validity of this procedure is that the volume between the planes must either be empty space or a uniform dielectric medium. Radiation Field The cross-spectral density function far away from any source, which has finite size, can be calculated using a particularly simple equation. Consider a primary, three-dimensional source located within a sphere of radius a, as shown in Fig. 3. For field points, x1 and x2 which are much farther from

FIGURE 3 Illustrating the coordinate system used to calculate the cross-spectral density function in the far field of a primary, three-dimensional source.

5.16

PHYSICAL OPTICS

the origin than the Rayleigh range (| x i | ka 2 ) , in any direction, a form of the familiar Fraunhofer approximation can be applied to Eq. (28) to obtain [see Ref. 62, eq. (2.5)] W U(c)(x 1, x 2 )

e ik(|x1| |x 2|) c c c c c c W (x ``, x ``) | x 1 || x 2 | ¯ c ¯ c ¯ c ¯ c ¯ c ¯ c Q 1 2 re

ik[x1 x1``/|x 1| x 2 x 2`` /|x 2 |]

3

(53)

3

d x 1`` d x ``2

Thus the cross-spectral density function of the far field is proportional to the six-dimensional Fourier transform of the cross-spectral density function of its sources. A very similar expression can also be found for a two-dimensional, secondary source distribution over the z 0 plane, as illustrated in Fig. 4. If the sources are all restricted to the area within a circle about the origin of radius a, then the cross-spectral density function for all points which are outside of the Rayleigh range (| x i | ka 2 ) from the origin and also in the z > 0 half-space can be found by using a different form of the Fraunhofer approximation in Eq. (33) [see Ref. 62, eq. (3.3)] to get W U(c)(x 1, x 2 )

1 z1 z 2 e ik(|x1| |x 2|) c c c c (0) W (x ` , x ` ) L 2 | x 1 | | x 2 | | x 1 || x 2 | ¯ c ¯ c ¯ c ¯ c U 1 2 re

ik[x 1 x 1` /|x1| x 2 x 2` /|x 2 |]

2

(54)

2

d x 1` d x `2

Because of both their relative simplicity (as Fourier transforms) and great utility, Eqs. (53) and (54) are very important in coherence theory for calculating both the radiant intensity and the far-field coherence properties of the radiation field from any source. Representations Several equations have been described here for propagating the cross-spectral density function. The two most general are Eq. (33), which uses an expansion of the field into spherical Huygens wavelets,

FIGURE 4 Illustrating the coordinate system for calculating the far field cross-spectral density function for a planar, secondary source distribution in the z 0 plane. Single-primed coordinates indicate radius vectors from the origin to points within the z 0 plane.

COHERENCE THEORY

5.17

and Eqs. (49), (50), and (52), which use an expansion of the field into an angular spectrum of plane waves. These two formulations are completely equivalent. Neither uses any approximation not also used by the other method. The choice of which to use for a particular calculation can be made completely on a basis of convenience. The far-field approximations given by Eqs. (53) and (54), for example, can be derived from either representation. There are two bridges between the spherical and plane wave representations, one given by Weyl’s integral, i.e., e ik|x x ``| i c c 1 ik[ p( x x ``)q( y y ``)m|z z ``|] e dp dq | x x `` | L ¯ c ¯ c m

(55)

where m 1 p2 q2

if p 2 q 2 a 1, (56)

i p2 q2 1

if p 2 q 2 1

and the other by a related integral

1 d e ik|x x ``| o 1 c c ik[ p( x x ``)q( y y ``)m|z z ``|] e dp dq 2P dz | x x `` | L 2 ¯ c ¯ c

(57)

which can be easily derived from Eq. (55). In Eq. (57) the ± sign holds according to whether (z z ``) ?0. With these two equations it is possible to transform back and forth between these two representations.

Reciprocity Theorem The radiation pattern and the complex degree of spectral coherence obey a very useful reciprocity theorem for a quasi-homogeneous source. By substituting from Eq. (40) into Eq. (53), and using Eq. (24) and the six-dimensional form of Eq. (41), it has been shown that the radiant intensity in the direction of the unit vector s from any bounded, three-dimensional, quasi-homogeneous primary source distribution is given by [see Ref. 45, eq. (3.11)] J W (s) J 0 ¯

c

c

¯ ¯

c

c c c

MQ (x `` )e ikx `` s d 3 x ``

(58)

where MQ (x ``) is the (spatially stationary) complex degree of spectral coherence for the source distribution as defined in Eq. (40), and J0 ¯

c

c

¯ ¯

c

I (x `` ) d 3 x ``

c c c Q

(59)

where I Q (x ``) is the intensity defined in Eq. (40). Note that the far-field radiation pattern depends, not on the source intensity distribution, but only on the source coherence. We also find from this calculation that the complete degree of spectral coherence between any two points in the far field of this source is given by [see Ref. 45, eq. (3.15)]

M (c)u(R1s1, R2 s 2 )

e ik( R1 R2 ) c c c I (x ``)e ikx `` (s1 s 2 )d 3 x `` J 0 ¯ c ¯ c ¯ c Q

(60)

Note that the coherence of the far field depends, not on the source coherence, but rather on the source intensity distribution, I Q (x ``). Equation (60) is a generalization of the van Cittert–Zernike theorem to three-dimensional, primary quasi-homogeneous sources, which are not necessarily incoherent. Equation (59) is a new theorem, reciprocal to the van Cittert–Zernike theorem, which was discovered by Carter and Wolf.45,49 Equations (58) and (60), taken together, give a reciprocity

5.18

PHYSICAL OPTICS

relation. For quasi-homogeneous sources, far-field coherence is determined by source intensity alone and the far-field intensity pattern is determined by source coherence alone. Therefore, coherence and intensity are reciprocal when going from the source into the far field. Since most sources which occur naturally are believed to be quasi-homogeneous, this is a very useful theorem. This reciprocity theorem has been found to hold much more generally than just for three-dimensional primary sources. For a planar, secondary source, by substitution from Eq. (40) into Eq. (54) and then using Eq. (41), we obtain49 J W (s) J 0` cos 2 Q ¯

c

¯

c

c c

M (U0)(x ` )e ikx ` s d 2 x `

(61)

where Q is the angle that s makes with the z axis, and where c c I (0)(x ` )

c c U

d 2 x `

(62)

1 c c ( 0) I (x ` )e ikx ` (s1 s 2 ) d 2 x ` J 0` ¯ c ¯ c U

(63)

J 0` ¯

¯

and we also obtain

M (Uc) )(R1s1, R2 s 2 )

Very similar reciprocity relations also hold for scattering of initially coherent light from quasihomogeneous scattering potentials,53,54 and for the scattering of laser beams from quasi-homogeneous sea waves.63 Reciprocity relations which apply to fields that are not necessarily quasi-homogeneous have also been obtained.64

Nonradiating Sources One additional important comment must be made about Eq. (58). The integrals appearing in this equation form a three-dimensional Fourier transform of MQ (x ``). However, J W (s), the radiant intensity that this source radiates into the far field at radial frequency Y, is a function of only the direction of s, which is a unit vector with a constant amplitude equal to one. It then follows that only the values of this transform over a spherical surface of unit radius from the origin affect the far field. Therefore, sources which have a complex degree of spectral coherence, MQ (x ``), which do not have spatial frequencies falling on this sphere, do not radiate at frequency W . It appears possible from this fact to have sources in such a state of coherence that they do not radiate at all. Similar comments can also be made in regard to Eq. (53) which apply to all sources, even those that are not quasi-homogeneous. From Eq. (53) it is clear that only sources which have a cross-spectral density function with spatial frequencies (with respect to both x 1`` and x 2`` ) on a unit sphere can radiate. It is believed that this is closely related to the phase-matching condition in nonlinear optics.

Perfectly Incoherent Sources For a completely incoherent primary source the far-field radiant intensity, by substitution from Eq. (42) into Eq. (58), is seen to be the same in all directions, independent of the shape of the source. By substitution from Eq. (40) into Eq. (28), and then using Eqs. (14), (29), and (42), we find that the complex degree of spectral coherence for this incoherent source is given by

MU (x 1, x 2 )

1 I (x 1 )/(x 2 )

c

c

c

¯ c ¯ c ¯ c

I Q (x ``)

e ik[|x1 x ``| |x 2 x ``|] 3 d x `` | x 1 x `` || x 2 x `` |

(64)

Thus it is only the source intensity distribution that affects the field coherence. Comparison of this equation with Eq. (47) shows that this is a generalization of the van Cittert–Zernike theorem to primary

COHERENCE THEORY

5.19

sources. It is clear from this equation that the complex degree of spectral coherence depends only on the shape of the source and not the fact that the source is completely incoherent. For a completely incoherent, planar, secondary source the radiant intensity is given by J W (s) J 0` cos 2Q

(65)

independent of the shape of the illuminated area on the source plane, where S is the angle that s makes with the normal to the source plane. This can be proven by substitution from Eq. (42) into Eq. (61). Note that such sources do not obey Lambert’s law. The far-field coherence, again, depends on the source intensity as given by the van Cittert–Zernike theorem [see Eq. (47)] and not on the source coherence.

Spectrum For a quasi-homogeneous, three-dimensional primary source the spectrum of the radiation S(Uc)(R s, W ) at a point Rs in the direction s (unit vector) and at a distance R from the origin, in the far field of the source, can be found, as a function of the source spectrum S(Q0)(W ), by substitution from Eqs. (24) and (15) into Eq. (58) to get S(Uc)(R s, W )

c 3SQ (W ) 3

W R

2

c

c

c

¯ c ¯ c ¯ c MQ (x `` , W )e ikx ``s d 3(kx `` )

(66)

where we explicitly indicate the dependence of the complex degree of spectral coherence on frequency. Notice that the spectrum of the field is not necessarily the same as the spectrum of the source and, furthermore, that it can vary from point to point in space. The field spectrum depends on the source coherence as well as on the source spectrum. A very similar propagation relation can be found for the far-field spectrum from a planar, secondary source in the z 0 plane. By substitution from Eqs. (24) and (15) into Eq. (61) we get S(Uc)(R s, W )

SU(0)(W ) c c (0) M (x ` , W )e ikx ` s d 2 (kx ` ) R 2 ¯ c ¯ c U

(67)

This shows that the spectrum for the field itself is changed upon propagation from the z 0 plane into the far field and is different in different directions s from the source.65

5.7

SPECTRUM OF LIGHT

Limitations The complex analytic signal for a field that is not perfectly coherent, as defined in Eq. (3), is usually assumed to be a time-stationary random process. Therefore, the integral c

¯ c u(x , t ) e iWt dt

(68)

does not converge, so that the analytic signal does not have a Fourier transform. Therefore, it is only possible to move freely from the space-time domain to the space-frequency domain along the path shown in Fig. 5. Equation (3) does not apply to time-stationary fields within the framework of ordinary function theory.

5.20

PHYSICAL OPTICS

Forbidden

(, )

(, w)

FT

G12 (t)

w(1, 2)

FIGURE 5 Illustrating the transformations which are possible between four functions that are used in coherence theory.

Coherent Mode Representation Coherent mode representation (see Wolf 66,67) has shown that any partially coherent field can be represented as the sum over component fields that are each perfectly self-coherent, but mutually incoherent with each other. Thus the cross-spectral density function for any field can be represented in the form WA (x 1, x 2 ) £ LnF A,n (x 1 )F A ,n (x 2 )

(69)

n

where F A,n (x i ) is a phasor amplitude for its nth coherent component. This representation can be used either with primary (A Q), or secondary (A U) sources. The phasor amplitudes F A,n (x i ) and the complex expansion coefficients Nn in Eq. (69) are eigenfunctions and eigenvalues of WA(x1, x2), as given by the equation c

c

c

¯ c ¯ c ¯ cWA (x 1, x 2 )F A,n(x 2 )d 3x 2 Ln(W )F A,n(x 1)

(70)

Since WA (x 1, x 2 ) is hermitian, the eigenfunctions are complete and orthogonal, i.e.,

£F A,n(x 1)F A ,n(x 2 ) D 3(x1 x 2 )

(71)

n

and c

c

c

¯ c ¯ c ¯ c F A,n(x 1)F A ,m (x 1) d 3x 1 Dnm

(72)

where Fnm is the Kronecker delta function, and the eigenvalues are real and nonnegative definite, i.e., Real {Ln (W )} q 0 Imag {Ln (W )} 0

(73)

This is a very important concept in coherence theory. Wolf has used this representation to show that the frequency decomposition of the field can be defined in a different manner than was done in Eq. (3). A new phasor, UA (x , W ) £ an (W )F A,n (x , W ) n

(74)

COHERENCE THEORY

5.21

where an(Y) are random coefficients such that an (W )am (W )® Ln (W )D nm

(75)

can be introduced to represent the spectral components of the field. It then follows 66 that, if the F A,n (x i ) are eigenfunctions of the cross-spectral density function, WA (x 1, x 2 ) then the cross-spectral density function can be represented as the correlation function between these phasors at the two spatial points x1 and x2, i.e., WA (x 1, x 2 ) U A (x 1, W )U A (x 2, W )®

(76)

in a manner very similar to the representation given in Eq. (11) in respect to the older phasors. Notice, by comparison of Eqs. (11) and (76), that the phasors UA (x i , W ) and UA (x i , W ) are not the same. One may formulate coherence theory either by defining u(x, t) in Fig. 5 and then moving in a counterclockwise direction in this figure to derive the correlation functions using the WienerKhintchene theory or, alternatively, defining UA (x i , W ) and moving in a clockwise direction to define the correlation functions using the coherent mode expansion.

Wolf Shift and Scaling law From Eqs. (66) and (67) it is clear that the spectrum of a radiation field may change as the field propagates. It is not necessarily equal to the source spectrum as it is usually assumed. This brings up an important question. Why do the spectra for light fields appear so constant, experimentally, that a change with propagation was never suspected? Wolf 65 has provided at least part of the answer to this question. He discovered a scaling law that is obeyed by most natural fields and under which the normalized spectrums for most fields do remain invariant as they propagate. We can derive this scaling law by substitution from Eqs. (66) and (67) into Eq. (16). We then find that, if the complex degree of spectral coherence for the source is a function of kx_ only, i.e.,

M A (x , W ) f (kx )

(77)

(so that this function is the same for each frequency component of the field, provided that the spatial separations of the two test points are always scaled by the wavelength), then the normalized spectrum in the far field is given by s(Uc)(R s, W )

SQ (W )/W 3 c

¯0

w f (R s)

(78)

SQ (W )/W 3 dW

if SQ(Y) is the complex degree of spectral coherence for a primary, quasi-homogeneous source [see Additional Reading, Ref. 3, eq. (65)], and s(Uc)(R s, W )

S(U0)(W ) c

S(0)(W ) dW 0 U

¯

s(U0)(W ) w f (R s)

(79)

if s(U0)(W ) is the normalized spectrum for a secondary, quasi-homogeneous source [see Additional Reading, Ref. 3, eq. (51)]. In each case the field spectrum does not change as the field propagates. Since the cross-spectral density function for a thermal source [see Eq. (43)] obeys this scaling law, it is not surprising that these changes in the spectrum of a propagating light field were never discovered experimentally. The fact that the spectrum can change was verified experimentally only after coherence theory pointed out the possibility.68–72

5.22

5.8

PHYSICAL OPTICS

POLARIZATION EFFECTS

Explicit Vector Representations As discussed earlier under “Scalar Field Amplitude,” the scalar amplitude is frequently all that is required to treat the vector electromagnetic field. If polarization effects are important, it might be necessary to use two scalar field amplitudes to represent the two independent polarization components. However, in some complicated problems it is necessary to consider all six components of the vector field explicitly. For such a theory, the correlation functions between vector components of the field become tensors,73,74 which propagate in a manner very similar to the scalar correlation functions.

5.9 APPLICATIONS Speckle If coherent light is scattered from a stationary, rough surface, the phase of the light field is randomized in space. The diffraction patterns observed with such light displays a complicated granular pattern usually called speckle (see Ref. 8, sec. 7.5). Even though the light phase can be treated as a random variable, the light is still perfectly coherent. Coherence theory deals with the effects of time fluctuations, not spatial variations in the field amplitude or phase. Despite this, the same statistical tools used in coherence theory have been usefully applied to studying speckle phenomena.75,76 To treat speckle, the ensemble is usually redefined, not to represent the time fluctuations of the field but, rather, to represent all of the possible speckle patterns that might be observed under the conditions of a particular experiment. An observed speckle pattern is usually due to a single member of this ensemble (unless time fluctuations are also present), whereas the intensity observed in coherence theory is always the result of a weighted average over all of the ensemble. To obtain the intensity distribution over some plane, as defined in coherence theory, it would be necessary to average over all of the possible speckle patterns explicitly. If this is done, for example, by moving the scatterer while the intensity of a diffraction pattern is time-averaged, then time fluctuations are introduced into the field during the measurement; the light becomes partially coherent; and coherence theory can be properly used to model the measured intensity. One must be very careful in applying the coherence theory model to treat speckle phenomena, because coherence theory was not originally formulated to deal with speckle.

Statistical Radiometry Classical radiometry was originally based on a mechanical treatment of light as a flux of particles. It is not totally compatible with wave theories. Coherence theory has been used to incorporate classical radiometry into electromagnetic theory as much as has been found possible. It has been found that the usual definitions for the radiance function and the radiant emittance cause problems when applied to a wave theory. Other radiometric functions, such as the radiant intensity, have clear meaning in a wave theory.

Spectral Representation It was discovered, using coherence theory, that the spectrum of a light field is not the same as that of its source and that it can change as the light field propagates away from its source into the radiation field. Some of this work was discussed earlier under “Perfectly Incoherent Sources” and “Coherent Mode Representation.” This work has been found very useful for explaining troublesome experimental discrepancies in precise spectroradiometry.71

COHERENCE THEORY

5.23

Laser Modes Coherence theory has been usefully applied to describing the coherence properties of laser modes.77 This theory is based on the coherent mode representation discussed under “Spectrum of Light.”

Radio Astronomy Intensity interferometry was used to apply techniques from radio astronomy to observation with light.37 A lively debate ensued as to whether optical interference effects (which indicate partial coherence) could be observed from intensity correlations.78 From relations like Eq. (26) and similar calculations using quantum coherence theory, it quickly became clear that they could.79,80 More recently, coherence theory has been used to model a radio telescope42 and to study how to focus an instrument to observe emitters that are not in the far field of the antenna array. 81 It has been shown82 that a radio telescope and a conventional optical telescope are very similar, within a coherence theory model, even though their operation is completely different. This model makes the similarities between the two types of instruments very clear.

Noncosmological Red Shift Cosmological theories for the structure and origin of the universe make great use of the observed red shift in the spectral lines of the light received from distant radiating objects, such as stars.83 It is usually assumed that the spectrum of the light is constant upon propagation and that the observed red shift is the result of simple Doppler shift due to the motion of the distant objects away from the earth in all directions. If this is true, then clearly the observed universe is expanding and must have begun with some sort of explosion, called “the big bang.” The size of the observable universe is estimated based on the amount of this red shift. A new theory by Wolf 84–87 shows that red shifts can occur naturally without Doppler shifts as the light propagates from the source to an observer if the source is not in thermal equilibrium, i.e., a thermal source as discussed earlier under “Thermal (Lambertian) Source.” The basis of Wolf ’s theory was discussed in this papers.84,85

5.10

REFERENCES 1. L. Mandel and E. Wolf, “Coherence Properties of Optical Fields,” Rev. Mod. Phys. 37, 1965, pp. 231–287. 2. L. Mandel and E. Wolf, Selected Papers on Coherence and Fluctuations of Light, vol. I, Dover, New York, 1970. Reprinted in Milestone Series, SPIE press, Bellingham, Wash., 1990. 3. L. Mandel and E. Wolf, Selected Papers on Coherence and Fluctuations of Light, vol. II, Dover, New York, 1970. Reprinted in Milestone Series, SPIE press, Bellingham, Wash., 1990. 4. M. J. Beran and G. B. Parrent, Theory of Partial Coherence, Prentice-Hall, Engelwood Cliffs, N.J., 1964. 5. B. Crosignani and P. Di Porto, Statistical Properties of Scattered Light, Academic Press, New York, 1975. 6. A. S. Marathay, Elements of Optical Coherence Theory, Wiley, New York, 1982. 7. J. Perina, Coherence of Light, Reidel, Boston, 1985. 8. J. W. Goodman, Statistical Optics, Wiley, New York, 1985. 9. R. J. Glauber, “The Quantum Theory of Optical Coherence,” Phys. Rev. 130, 1963, pp. 2529–2539. 10. R. J. Glauber, “Coherent and Incoherent States of the Radiation Field,” Phys. Rev. 131, 1963, pp. 2766–2788. 11. C. L. Mehta and E. C. G. Sudarshan, “Relation between Quantum and Semiclassical Description of Optical Coherence,” Phys. Rev. 138, B274–B280, 1965. 12. W. Louisel, Radiation and Noise in Quantum Electronics, McGraw-Hill, New York, 1964. 13. W. H. Louisell, Quantum Statistical Properties of Radiation, Wiley, New York, 1973.

5.24

PHYSICAL OPTICS

14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44.

M. Sargent, M. O. Scully, and W. E. Lamb, Laser Physics, Addison-Wesley, Reading, MA, 1974. M. Born and E. Wolf, Principles of Optics, Pergamon Press, New York, 1980. H. M. Nussenvieg, Causality and Dispersion Relations, Academic Press, New York, 1972. A. M. Yaglom, An Introduction to the Theory of Stationary Random Functions, Prentice-Hall, Engelwood Cliffs, N.J., 1962. G. Borgiotti, “Fourier Transforms Method in Aperture Antennas Problems,” Aha Frequenza 32, 1963, pp. 196–205. D. R. Rhodes, “On the Stored Energy of Planar Apertures,” IEEE Trans. AP14, 1966, pp. 676–683. W. H. Carter, “The Electromagnetic Field of a Gaussian Beam with an Elliptical Cross-section,” J. Opt. Soc. Am. 62, 1972, pp. 1195–1201. W. H. Carter, “Electromagnetic Beam Fields,” Optica Acta 21, 1974, pp. 871–892. B. J. Thompson and E. Wolf, “Two-Beam Interference with Partially Coherent Light,” J. Opt. Soc. Am. 47, 1957, pp. 895–902. W. H. Carter, “Measurement of Second-order Coherence in a Light Beam Using a Microscope and a Grating,” Appl. Opt. 16, 1977, pp. 558–563. E. Wolf, “A Macroscopic Theory of Interference and Diffraction of Light from Finite Sources II: Fields with Spatial Range and Arbitrary Width,” Proc. Roy. Soc. A230, 1955, pp. 246–265. G. B. Parrent, “On the Propagation of Mutual Coherence,” J. Opt. Soc. Am. 49, 1959, pp. 787–793. W. B. Davenport and W. L. Root, An Introduction to the Theory of Random Signals and Noise, McGraw-Hill, New York, 1960. L. Mandel and E. Wolf, “Spectral Coherence and the Concept of Cross-spectral Purity,” J. Opt. Soc. Am. 66, 1976, pp. 529–535. E. W. Marchand and E. Wolf, “Angular Correlation and the Far-zone Behavior of Partially Coherent Fields,” J. Opt. Soc. Am. 62, 1972, pp. 379–385. E. W. Marchand and E. Wolf, “Radiometry with Sources of Any State of Coherence,” J. Opt. Soc. Am. 64, 1974, pp. 1219–1226. E. Wolf, “The Radiant Intensity from Planar Sources of any State of Coherence,” J. Opt. Soc. Am. 68, 1978, pp. 1597–1605. A. T. Friberg, “On the Existence of a Radiance Function for Finite Planar Sources of Arbitrary States of Coherence,” J. Opt. Soc. Am. 69, 1979, pp. 192–198. E. Wigner, “Quantum Correction for Thermodynamic Equilibrium” Phys. Rev. 40, 1932, pp. 749–760. K. Imre, E. Ozizmir, M. Rosenbaum, and P. F. Zweifer, “Wigner Method in Quantum Statistical Mechanics,” J. Math. Phys. 8, 1967, pp. 1097–1108. A. Papoulis, “Ambiguity Function in Fourier Optics,” J. Opt. Soc. Am. 64, 1974, pp. 779–788. A. Walther, “Radiometry and Coherence,” J. Opt. Soc. Am. 63, 1973, pp. 1622–1623. J. T. Foley and E. Wolf, “Radiometry as a Short Wavelength Limit of Statistical Wave Theory with Globally Incoherent Sources,” Opt. Commun. 55, 1985, pp. 236–241. R. H. Brown and R. Q. Twiss, “Correlation between Photons in Two Coherent Beams of Light,” Nature 177, 1956, pp. 27–29. J. D. Jackson, Classical Electrodynamics, Wiley, New York, 1962. C. L. Mehta and A. B. Balachandran, “Some Theorems on the Unimodular Complex Degree of Optical Coherence,” J. Math. Phys. 7, 1966, pp. 133–138. L. Mandel and E. Wolf, “Complete Coherence in the Space Frequency Domain,” Opt. Commun. 36, 1981, pp. 247–249. W. H. Carter, “Difference in the Definitions of Coherence in the Space-Time Domain and in the SpaceFrequency Domain,” J. Modern Opt. 39, 1992, pp. 1461–1470. W. H. Carter and L. E. Somers, “Coherence Theory of a Radio Telescope,” IEEE Trans. AP24, 1976, pp. 815–819. A. C. Schell, The Multiple Plate Antenna, doctoral dissertation, M.I.T., 1961. A. C. Schell, “A Technique for the Determination of the Radiation Pattern of a Partially Coherent Aperture,” IEEE Trans. AP-15, 1967, p. 187.

COHERENCE THEORY

5.25

45. W. H. Carter and E. Wolf, “Correlation Theory of Wavefields Generated by Fluctuating Three-dimensional, Primary, Scalar Sources: II. Radiation from Isotropic Model Sources,” Optica Acta 28, 1981, pp. 245–259. 46. W. H. Carter and E. Wolf, “Correlation Theory of Wavefields Generated by Fluctuating Three-dimensional, Primary, Scalar Sources: I. General Theory,” Optica Acta 28, 1981, pp. 227–244. 47. E. Collett and E. Wolf, “Is Complete Coherence Necessary for the Generation of Highly Directional Light Beams?” Opt. Letters 2, 1978, pp. 27–29. 48. W. H. Carter and Bertolotti, “An Analysis of the Far Field Coherence and Radiant Intensity of Light Scattered from Liquid Crystals,” J. Opt. Soc. Am. 68, 1978, pp. 329–333. 49. W. H. Carter and E. Wolf, “Coherence and Radiometry with Quasi-homogeneous Planar Sources,” J. Opt. Soc. Am. 67, 1977, pp. 785–796. 50. W. H. Carter and E. Wolf, “Inverse Problem with Quasi-Homogeneous Sources,” J. Opt. Soc Am. 2A, 1985, pp. 1994–2000. 51. E. Wolf and W. H. Carter, “Coherence and Radiant Intensity in Scalar Wave Fields Generated by Fluctuating Primary Planar Sources,” J. Opt. Soc. Am. 68, 1978, pp. 953–964. 52. W. H. Carter, “Coherence Properties of Some Isotropic Sources,” J. Opt. Soc. Am. Al, 1984, pp. 716–722. 53. W. H. Carter and E. Wolf, “Scattering from Quasi-homogeneous Media,” Opt. Commun. 67, 1988, pp. 85–90. 54. W. H. Carter, “Scattering from Quasi-homogeneous Time Fluctuating Random Media,” Opt. Commun. 77, 1990, pp. 121–125. 55. W. H. Carter and E. Wolf, “Coherence Properties of Lambertian and Non-Lambertian Sources,” J. Opt. Soc. Am. 65, 1975, pp. 1067–1071. 56. J. T. Foley, W. H. Carter, and E. Wolf, “Field Correlations within a Completely Incoherent Primary Spherical Source,” J. Opt. Soc. Am. A3, 1986, pp. 1090–1096. 57. H. H. Hopkins, “The Concept of Partial Coherence in Optics,” Proc. Roy. Soc. A208, 1951, pp. 263–277. 58. W. H. Carter, “Generalization of Hopkins’ Formula to a Class of Sources with Arbitrary Coherence,” J. Opt. Soc. Am. A2, 1985, pp. 164–166. 59. P. H. van Cittert, “Die wahrscheinliche Schwingungsverteilung in einer von einer Lichtquelle direkt oder mittels einer Linse beleuchteten Ebene,” Physica 1, 1934, pp. 201–210. 60. F. Zernike, “The Concept of Degree of Coherence and Its Application to Optical Problems,” Physica 5, 1938, pp. 785–795. 61. W. H. Carter, “Three Different Kinds of Fraunhofer Approximations. I. Propagation of the Field Amplitude,” Radio Science 23, 1988, pp. 1085–1093. 62. W. H. Carter, “Three Different Kinds of Fraunhofer Approximations. II. Propagation of the Cross-spectral Density Function,” J. Mod. Opt. 37, 1990, pp. 109–120. 63. W. H. Carter, “Scattering of a Light Beam from an Air-Sea Interface,” Applied Opt. 32, 3286–3294 (1993). 64. A. T. Friberg and E. Wolf, “Reciprocity Relations with Partially Coherent Sources,” Opt. Acta 36, 1983, pp. 417–435. 65. E. Wolf, “Invariance of the Spectrum of Light on Propagation,” Phys. Rev. Letters 56, 1986, pp. 1370–1372. 66. E. Wolf, “New Theory of Partial Coherence in the Space-frequency Domain. Part I: Spectra and Cross Spectra of Steady-state Sources,” J. Opt. Soc. Am. 72, 1982, pp. 343–351. 67. E. Wolf, “New Theory of Partial Coherence in the Space-frequency Domain. Part II: Steady-state Fields and Higher-order Correlations,” J. Opt. Soc. Am. A3, 1986, pp. 76–85. 68. G. M. Morris and D. Faklis, “Effects of Source Correlation on the Spectrum of Light,” Opt. Commun. 62, 1987, pp. 5–11. 69. M. F. Bocko, D. H. Douglass, and R. S. Knox, “Observation of Frequency Shifts of Spectral Lines Due to Source Correlations,” Phys. Rev. Letters 58, 1987, pp. 2649–2651. 70. F. Gori, G. Guattari, and C. Palma, “Observation of Optical Redshift and Blueshifts Produced by Source Correlations,” Opt. Commun. 67, 1988, pp. 1–4. 71. H. C. Kandpal, J. S. Vashiya, and K. C. Joshi, “Wolf Shift and Its Application in Spectro-radiometry,” Opt. Commun. 73, 1989, pp. 169–172. 72. G. Indebedouw, “Synthesis of Polychromatic Light Sources with Arbitrary Degree of Coherence: From Experiments,” J. Mod. Opt. 36, 1989, pp. 251–259.

5.26

PHYSICAL OPTICS

73. W. H. Carter, “Properties of Electromagnetic Radiation from a Partially Correlated Current Distribution,” J. Opt. Soc. Am. 70, 1980, pp. 1067–1074. 74. W. H. Carter and E. Wolf, “Far-zone Behavior of Electromagnetic Fields Generated by Fluctuating Current Distributions,” Phys. Rev. 36A, 1987, pp. 1258–1269. 75. J. C. Dainty, “The Statistics of Speckle Patterns,” in E. E. Wolf, Progress in Optics, North Holland, Amsterdam, 1976. 76. J. C. Dainty, Laser Speckle and Related Phenomena, Springer Verlag, Berlin, 1975. 77. E. Wolf and G. S. Agarwal, “Coherence Theory of Laser Resonator Modes,” J. Opt. Soc. Am. A1, 1984, pp. 541–546. 78. E. M. Purcell, “The Question of Correlation Between Photons in Coherent Light Rays,” Nature 178, 1956, pp. 1449–1450. 79. R. H. Brown and R. Q. Twiss, “Interferometry of the Intensity Fluctuations in Light: I. Basic Theory: the Correlation between Photons in Coherent Beams of Radiation,” Proc. Roy. Soc. 242, 1957, pp. 300–324. 80. R. H. Brown and R. Q. Twiss, “Interferometry of the Intensity Fluctuations in Light: II. An Experimental Test of the Theory for Partially Coherent Light,” Proc. Roy. Soc. 243, 1957, pp. 291–319. 81. W. H. Carter, “Refocusing a Radio Telescope to Image Sources within the Near Field of the Antenna Array,” NRL Report no. 9141, August 17, 1988. 82. W. H. Carter, “A New Theory for Image Formation with Quasi-homogeneous Sources and Its Application to the Detection and Location of ICBM Launches,” NRL Report no. 9336 (September 17, 1991). 83. J. V. Narliker, “Noncosmological Redshifts,” Space Sci. Revs. 50, 1989, pp. 523–614. 84. E. Wolf, “Non-cosmological Redshifts of Spectral Lines,” Nature 326, 1987, pp. 363–365. 85. E. Wolf, “Red Shifts and Blue Shifts of Spectral Lines Emitted by Two Correlated Sources,” Phys. Rev. Letters 58, 1987, pp. 2646–2648. 86. A. Gamliel and E. Wolf, “Spectral Modulation by Control of Source Correlations,” Opt. Commun. 65, 1988, pp. 91–96. 87. E. Wolf, J. T. Foley, and F. Gori, “Frequency Shifts of Spectral Lines Produced by Scattering from Spatially Random Media,” J. Opt. Soc. Am. A6, 1989, pp. 1142–1149.

5.11 ADDITIONAL READING 1. W. H. Carter, “Difference in the Definition of Coherence in the Space-Time Domain and in the SpaceFrequency Domain,” J. Mod. Opt. 39, 1992, pp. 1461–1470. 2. W. H. Carter, “Coherence Properties of Some Broadband Highly Coherent Light Fields,” J. Opt. Soc. Am. A10, 1993, pp. 1570–1578. 3. W. H. Carter, “A Study of the Propagation of Optical Spectra Using a Wavelength Independent Formulation of Diffraction Theory and Coherence Theory,” J. Mod. Opt. 40, 1993, pp. 2433–2449.

6 COHERENCE THEORY: TOOLS AND APPLICATIONS Gisele Bennett Electro-Optical Systems Laboratory and School of Electrical and Computer Engineering Georgia Tech Research Institute Georgia Institute of Technology Atlanta, Georgia

William T. Rhodes School of Electrical and Computer Engineering Georgia Institute of Technology, and Department of Electrical Engineering and Imaging Technology Center Florida Atlantic University Boca Raton, Florida

J. Christopher James Electro-Optical Systems Laboratory Georgia Tech Research Institute Georgia Institute of Technology Atlanta, Georgia

6.1

GLOSSARY I k u ' J O x

Fourier transform operator Intensity wave number complex field amplitude mutual coherence function mutual intensity function complex degree of coherence spatial vector

6.1

6.2

PHYSICAL OPTICS

6.2

INTRODUCTION The formalisms of coherence theory have been extensively developed over the past century, with roots extending back much further. An understanding of key concepts of the coherence properties of light waves and of their accompanying mathematical models can be exploited in specific applications to extract information about objects (e.g., stellar interferometry, microscopy1,2), to encrypt signals transmitted by optical fiber (coherence modulation), to explain puzzling phenomena observed in nature (sunlight-produced speckle, enhanced backscatter), and to prevent significant errors in the measurement of optical quantities (measurement of the reflectivity of diffuse surfaces), among other examples. There are cases where using a coherence-theory–based analysis provides not only insight into understanding system behavior but simplifies the analysis through reduced computational steps. In Chap. 5, Carter provides a compendium of basic definitions from coherence theory, introduces mathematical models for light sources of fundamental importance, discusses how the coherence properties of optical wave fields evolve under propagation, and shows how the spectral properties of light waves relate to their coherence properties.3 He also notes briefly several areas in which coherence theory has been exploited to advance significantly the understanding of optical phenomena. References 1-8 in Chap. 5 provide a suitable introduction to most aspects of classical (i.e., nonquantum) coherence theory. Since the publication of Vol. I of this Handbook series, Mandel and Wolf have written a comprehensive review of coherence theory and a brief discussion of specific applications.4 Other informative sources include a compilation of early papers on coherence edited by Mandel and Wolf.5 The formal structures of coherence theory can be intimidating to the nonexpert trying to ascertain, for example, the effect of a particular light source on a given optical system. One has only to review the myriad of defined quantities or look at a six- or seven-dimensional integral expression describing an image-forming system to understand why. Nevertheless, the basic concepts of coherence theory, and certain of their key formulas, can be helpful even to the nonexpert when they are applied to the understanding of a variety of optical systems and phenomena. That said, it should be noted that sometimes the application of coherence theory formalisms only serves to obfuscate the operation of a given system. One objective of this chapter is to provide guidance as to when coherence theory can help and when it might serve largely to confuse. We note that coherence theory is not generally needed if the optical wave fields of concern can be modeled as being monochromatic or quasi-monochromatic. The subject of interferometry is easily treated without the inclusion of coherence theory if the source of light is a laser operating in a single mode, and optical coherence tomography is easily understood without appeal to much more than an understanding of temporal coherence theory.6–10 It is with this latter point in mind that we emphasize on analyses involving the spatial coherence properties of the light.

6.3

KEY DEFINITIONS AND RELATIONSHIPS We begin with the presentation of a set of tools that are critical to the analysis of certain applications. Although the cross-spectral density function3,4 is often preferred in the modeling of the spatiotemporal coherence properties of light, in a discussion of the applications of coherence theory where physical understanding is important we find it preferable to work instead with the mutual coherence function, defined by11 '12 (T ) u(x 1 , (t T ))u (x 2 , t )®

(1)

where u(x, t) denotes the complex analytic signal associated with the scalar wave amplitude at position x = (x, y, z AND WHERE THE ANGULAR BRACKETS s DENOTE A SUITABLE TIME AVERAGE

The nature of the time average is discussed later in this chapter. Note that in Chap. 5 Carter uses angular brackets to denote ensemble averages rather than time averages.

COHERENCE THEORY: TOOLS AND APPLICATIONS

6.3

This function has the advantage of relating directly to quantities that can be measured by an easily visualized Young’s two-pinhole interferometer. In theoretical developments u(x, t) is usually treated as an ergodic random process, and the time average is replaced with an ensemble average. However, because such conditions are not satisfied in all applications of interest, we maintain specific reference to time averages. The mutual intensity function, unlike the complex amplitude itself, is a measurable quantity. The temporal frequency bandwidth of optical signals typically far exceeds the bandwidth of any optical detector, thus preventing the direct measurement of u(x, t). As discussed by Greivencamp, a Young’s two-pinhole interferometer can be used, at least conceptually, in the measurement process. 12 In many cases we can assume the light to be quasi-monochromatic.13 The narrowband condition $ L L is satisfied, where $ L denotes the spectral bandwidth of the light and where L is the mean wavelength, and the coherence length of the light,3 $ l L 2 /$ L , is much greater than the maximum path length difference encountered in the passage of light from a source of interest to the measurement plane of concern. Under such conditions11 the mutual coherence function is given by '12 (T ) J12 e iWt

(2)

where Y is the mean angular frequency of the radiation and where J12, the mutual intensity function, is given by J12 '12 (0)

(3)

Note that J12 depends on the spatial coordinates x1 and x2 but not on V. Effect of Transmissive Planar Object In most of the applications considered in this chapter we are concerned with the behavior of J12 in a two-dimensional systems framework. Assume, therefore, that x1 and x2 represent the coordinates of a pair of points in a plane of constant z. The complex amplitude u(x, t) can then be written as u(x, y, t), and J12 can be written in the form J(x1, y1; x2, y2). If the wave field uinc(x, y, t) is incident upon a thin transmissive object of complex transmittance t(x, y), the transmitted wave utrans(x, y, t) has amplitude uinc(x, y, t)t(x, y), and it is easily shown that the corresponding mutual intensity is given by J trans (x1 , y1 ; x 2 , y 2 ) J inc (x1 , y1 ; x 2 , y 2 ) t (x1 , y1 )t (x 2 , y 2 )

(4)

Of particular interest later is the case when the object is a thin spherical lens. The transmittance function of such a lens is given, in the paraxial approximation and ignoring an accompanying pupil function, by § k ¶ t lens (x , y) exp ¨ i (x 2 y 2 )· 2 f © ¸

(5)

where f denotes the focal length of the lens and where k 2R/N. Assuming a lens that is not the limiting aperture in a system and substitution in Eq. (4) yields ª k J trans (x1 , y1 ; x 2 , y 2 ) J inc (x1 , y1 ; x 2 , y 2 ) exp « i §© x12 x 22 y12 y 22 ¬ 2f

¶¸¹º»

(6)

Effect of a General Linear System If the wave amplitude u(x, y, t) is input to a space-invariant two-dimensional linear optical system with spatial impulse response h(x, y), the output wave amplitude has the form

6.4

PHYSICAL OPTICS

uout (x , y ; t ) uin (x , y ; t ) h(x , y) , where denotes a two-dimensional convolution operation. The relationshipfor the corresponding mutual intensity functions is given by

J out (x1 , y1 ; x 2 , y 2 ) J in (x1 , y1 ; x 2 , y 2 ) h(x1 , y1 )

h (x 2 , y 2 )

(7)

The cascade of two-dimensional convolutions in this equation can, if desired, be rewritten in the form of a four-dimensional convolution of J in (x1 , y1 ; x 2 , y 2 ) with a separable four-dimensional kernel: J out (x1 , y1 ; x 2 , y 2 ) J in (x1 , y1 ; x 2 , y 2 ) [h(x1 , y1 )h (x 2 , y 2 )]

(8)

Propagation of the Mutual Intensity It is a straightforward matter to show how the mutual intensity of a wave field propagates.14–16 In most applications, concern is with propagation in the Fresnel (paraxial) regime, in which case h(x, y) in Eq. (7), denoted now by hz(x, y), is given by hz (x , y)

§ ikz(x 2 y 2 ) ¶ exp(ikz ) exp ¨ · 2z iL z © ¸

(9)

where z is the plane-to-plane propagation distance. The factor ( i) 1 exp(ikz ) can often be ignored. If we assume the source distribution to be planar, quasi-monochromatic, and spatially incoherent [the latter condition implying that J12 Go I (x 1 )D (x 1 x 2 ), where I denotes the optical intensity of the wave field and where Io is a constant that depends on characteristics of the source at the scale of a wavelength] with optical intensity Io(x, y), it can be shown by combining Eqs. (8) and (9) that the mutual intensity a distance z away from the source is given by J z (x1 , y1 ; x 2 , y 2 )

§ i 2P ¶ exp( i F ) c c ($ xX $ y H)· dX dH I (X , H)exp ¨ (L z )2 ¯ c ¯ c o © Lz ¸

exp( i F ) ˆ ¤ $ x $ y ³ , I (L z )2 o ¥¦ L z L z ´µ

(10)

where Iô (x , y) denotes the two-dimensional Fourier transform of the source-plane intensity, $x x1 x2 , $y y1 y2, and where

F

P § 2 2 x y x12 y12 ¶¸ Lz © 2 2

(11)

Equation (10) is a compact statement of the van Cittert-Zernike theorem. If the wave field corresponding to Jz in Eq. (10) is incident on a thin spherical lens of focal length f z , then, through Eqs. (10) and (4), the quadratic phase factors of Eq. (11) are removed and the resulting mutual intensity has the simple form (1/L z )2 Iô ($ x /L z , $ y /L z ). This distribution, it is noted, is a function of $x and $y alone, that is, it is a function only of the vector separation between the two sample points x1 and x2. The Issue of Time Averages The nature of the time average implied by the angular brackets in Eq. (1) will depend on the particular situation of concern. If, for example, u(x, t) corresponds to the wave field produced by a white light source, adequate averaging is obtained over a time interval of roughly a picosecond (10 12 s). If, on the other hand, the light source is a stabilized laser with a temporal frequency bandwidth of several tens of kilohertz, the period over which the average is evaluated might be milliseconds in duration. Often the bandwidth of a detector determines the appropriate duration of the averaging process. If we are viewing white-light interference fringes by eye, the integration interval can be anywhere

Although this expression does not appear in the standard textbooks on coherence theory, it is easily derived.


6.5

from a picosecond to perhaps a thirtieth of a second. If the same fringes are being observed with a megahertz-bandwidth measurement system, on the other hand, integration for times greater than a microsecond is inappropriate.

6.4 PROPAGATION, DIFFRACTION, AND SCATTERING: ENHANCED BACKSCATTER AND THE LAU EFFECT Coherence theory can provide a useful tool for describing certain phenomena encountered in connection with wave field propagation, scattering, and diffraction by gratings. Two examples of cases where a coherence theory approach is especially enlightening are the phenomenon of enhanced backscatter (EBS) and the Lau effect, as discussed in the following subsections.

Enhanced Backscatter Enhanced backscatter is observed when a laser beam, after passing through a fine-grained moving diffuser, is reflected back through the same moving diffuser. As illustrated in Fig. 1a, the far field contains, in addition to a broad background irradiance distribution, a tightly focused spot of light, corresponding to the diffraction-limited focusing of the partially recollimated incident beam. This result contrasts with that obtained with two different moving diffusers, as shown in Fig. 1b, in which case there is no focused light spot. Although a ray-optics model can provide some insight

Single moving diffuser

Collimated beam

Observed farfield intensity

Mirror Diffractionlimited

(, 0)

(a)

Moving (different) diffusers

Observed farfield intensity

Collimated light beam

(, 0)

(b) FIGURE 1 Scattering by moving diffusers: (a) enhanced backscatter (partial phase conjugation scattering) produced by double passage of light through the same diffuser and (b) conventional scattering produced by two different diffusers.

6.6

PHYSICAL OPTICS

into this counter-intuitive phenomenon, coherence theory allows us to develop much greater understanding. Consider the case where the incident wave field is quasi-monochromatic and of arbitrary spatial coherence properties. To simplify the math we work with a one-dimensional model. The moving diffuser is represented by a thin dynamic random phase screen with complex amplitude transmittance H(x, t). Transmission of the incident wave u(x, t) through the diffuser results in wave field u(x, t)H(x, t), which propagates a distance z to be scattered a second time by an identical random phase screen. The doubly-scattered wave amplitude u'(x, t) is given by17 c

u `(x , t ) F(x , t )¯ u(X , t )F(X , t )hz (x X)dX

c

(12)

where hz(x) represents the wave propagation kernel of Eq. (9). The mutual intensity of the doublyscattered wave field is calculated as J `(x1 , x 2 ) u `(x1 , t )u ` (x 2 , t )® c c §¨F(x1 , t )¯ u(X , t )F(X , t )hz (x1 X)dX¶· §¨F (x 2 , t )¯ u (H, t )F (H, t )hz (x 2 H)dH¶·

c

c © ¸© ¸

¯

c

¯

c

(13)

u(X , t )u (H, t )®F(x1 , t )F (x 2 , t )F(X , t )F (H, t )®hz (x1 X)hz (x 2 H)dX dH

c c

where in expressing the effect of the time average it has been assumed that u and H vary independently. To simplify this expression we note that u(X , t )u (H, t )® is the mutual intensity J(Z,J) of the input wave field. Additionally, if H x, t) represents a spatially stationary, delta-correlated scatterer with suitably large excursions in phase, the second term in brackets in the integral of Eq. (13) can be modeled by F(x1 , t ) F (x 2 , t ) F(X , t ) F (H, t )® G 2[D (x1 x 2 ) D (X H) D (x1 H) D (X x 2 )]

(14)

where c

G ¯ F(X , t )F (0, t )®dX

c

(15)

Making the appropriate substitutions in Eq. (13) yields J `(x1 , x 2 ) a(x1 )D (x1 x 2 ) b(x1 , x 2 ) J (x1 , x 2 )

(16)

a(x1 ) G 2 I (x1 ) | hz (x1 )|2

(17)

where

b(x1 , x 2 ) G

2

hz (x1 x 2 )hz (x1 x 2 )

(18)

and where the function I(x) in Eq. (17) is the optical intensity of the incident wave. If the onedimensional form of the Fresnel propagation kernel is substituted for hz(x), hz (x )

1 iL z

§ iP ¶ exp ¨ x 2 · ©Lz ¸

(19)

calculation of a(x1) and b(x1, x2) yields c

J `(x1 , x 2 ) D (x1 x 2 )G 2 ¯ I inc (X)dX G 2 J inc (x1 , x 2 )

c

(20)


6.7

The doubly-scattered wave, u'(x, t), can thus be thought of as consisting of two components. One component has the characteristics of a spatially incoherent wave and produces a general background glow in the far field. The second component, on the other hand, effectively replicates coherence properties of the incident wave, having mutual intensity J inc(x1, x2), the complex conjugate of that of the incident wave. A wave field with mutual intensity J inc(x1, x2) behaves like a time-reversed (or “backward-propagating”) version of a wave field with mutual intensity Jinc(x1, x2). Thus, if the incident wave is diverging, the doubly-scattered wave will contain, in addition to an incoherent component, a coherence-replicated component that is converging at the same rate. For the case illustrated in Fig. 1a, since the incident beam is collimated, the coherence-replicated component of the doubly-scattered wave also behaves like a collimated light beam. It should be noted that evaluation of the time average for the term u(X , t )u (H, t )® in Eq. (13) may be satisfactorily complete in a fraction of a microsecond—that is, after an interval large compared to the reciprocal bandwidth of the light incident on the diffuser—whereas calculation of the second brackets term, F(x1, t )F (x 2, t )F(X , t )F (H, t )®, may require milliseconds or even seconds, depending on how rapidly the random phase screen evolves with time. What is essential is that H(x, t) evolves satisfactorily over the duration of the time average. In the terminology of random processes, we require that H(x, t) goes through a sufficient number of realizations as to provide a good statistical average of the bracketed quantity. The enhanced backscatter phenomenon can be exploited, at least in theory, in the imaging of diffuser-obscured objects. Let the mirror in the system of Fig. 1a be replaced by a planar object with amplitude reflectance r(x), assuming that the incident wave is monochromatic and planar. Through an analysis quite similar to that above, one can show that the mutual intensity of the doubly-scattered wave field Jds(x1, x2) again contains two components, a coherent one and a spatially incoherent one. The coherent component, which can be measured by interferometric means, is proportional to the modulus of the Fresnel transform of the object reflectance function18 ¤ x x ³ J dscoh (x1 , x 2 ) sK 2 rz /2 ¥ 1 2 ´ ¦ 2 µ

2

(21)

where rz /2 (x ), defined by c § ik(x H)2 ¶ (22) rz /2 (x ) ¯ r(H) exp ¨ · dH

c z © ¸ is proportional to the wave field that would result from illuminating the object with a normally incident plane wave and propagating the reflected wave a distance z /2. The resulting distribution is the object function blurred by Fresnel diffraction. This Fresnel-blurred image is less distinct than a normal image of the object; however, it is significantly less blurred than would be a conventional image obtained through the diffuser.

Lau Effect In the Lau effect, an incoherently illuminated amplitude grating with grating constant d is followed by a second identical grating at a distance z0.19 If the second grating is followed by a converging spherical lens, fringes appear in the back focal plane of the lens whenever the distance z0 is an integral multiple of d2/2N. The experimental geometry for observing this effect is illustrated in Fig. 2. If white light illumination is used, colored fringes are observed. Although the effect can be explained in terms of geometrical optics and scalar diffraction theory, 20 a much more elegant explanation results from a straightforward application of coherence theory.21–25 The essence of a coherence-based analysis using the mutual intensity function (expressed in one-dimensional coordinates for simplicity) is as follows: The incoherently illuminated

6.8

PHYSICAL OPTICS

Grating 1

Grating 2

Lens

Observation plane

0 FIGURE 2

Geometry for examination of the Lau effect.

grating constitutes a periodic spatially incoherent source with period d with an intensity function described by c ¤ 2x ³ I 0 (x ) rect ¥ ´ £ D (x md) ¦ d µ m c

(23)

From the van Cittert-Zernike theorem [Eq. (10)] we know that the mutual intensity function of the propagated wave field arriving at the second grating can be expressed as J1(x1, x 2 )

c ¹ ¤ 2x ³ exp( i J ) ª «rect ¥ ´ £ D (x md)º 2 d (L z 0 ) ¦ µ m c » ¬

u

$x L z0

(24)

¤ d $x ³ c ¤ mL z 0 ³ ¶ exp( i J ) § d D ¥$x sinc ¥ · £ 2 ¨2 ´ d ´µ ·¸ (L z 0 ) ¨© ¦ 2L z 0 µ m c ¦

where denotes the fourier transform operation. The presence of the Dirac delta functions in the above expression indicates that the propagated wave reaching the second grating is spatially coherent only for transverse separations that are integer multiples of the distance Nz0 /d. If the second grating is treated as a transmissive planar object with a transmission function analytically equivalent to the previously defined intensity function I0(x), Eq. (4) can be used to describe the mutual intensity of the light leaving the grating, obtaining c c ¶ § ¤ 2x 2x ³ J 2 (x1 , x 2 ) ¨rect ¥ 1 , 2 ´ £ £ D (x1 kd , x 2 ld)· d µ k c l c ¦ d ·¸ ¨©

(25)

¤ d $x ³ c ¤ mL z 0 ³ ¶ exp( i J ) § d r s inc ¨ ¥¦ 2L z ´µ £ D ¥¦ $ x d ´µ · (L z 0 )2 ¨© 2 ·¸ 0 m c The first half of the above equation describes a two-dimensionally periodic spatial filter in the coherence domain with a period equal to d in both directions acting upon the mutual intensity of the light incident upon the grating. Maximum coherence (and hence maximum interference fringe-forming capability) is preserved when the period of the one-dimensional Dirac delta function in the second half of the equation (i.e., the nonzero coherence components of the incident light) is equal to the period of the two-dimensional Dirac delta function describing the coherence filter or d Nz0/d. A grating separation of z0 d2/N meets this condition which is a special case of Lau’s more general condition.


6.9

6.5 IMAGE FORMATION: LUKOSZ-TYPE SUPER-RESOLVING SYSTEM It was in the analysis of imaging systems that coherence theory saw its first major application. Frits Zernike, in his classic 1938 paper on “The concept of degree of coherence and its application to optical problems,” showed how the mutual coherence function could be employed in the wave-optics analysis of systems of lenses and apertures in such a way that modification of the system through extension, for example, through the addition of a field lens, did not necessarily require a return to the starting point in the analysis, that is, to the source of illumination.16 In this section we illustrate the utility of Zernike’s approach through application to a particular super-resolving imaging system. In the 1960s Lukosz published a pair of papers on super-resolution imaging systems that continue to interest and intrigue people.26,27 Lukosz-type systems can be used to increase the spacebandwidth product—in contemporary terms, the number of pixels—that can be transmitted from the object plane to the image plane with a system of a given numerical aperture. The intriguing nature and popularity of Lukosz-type imaging systems notwithstanding, they are surprisingly difficult to analyze in detail, and Lukosz’s original papers present challenging reading for one who wants to understand quickly just how to model the systems mathematically. We show in this section how coherence theory tools can be used to good effect in the analysis of a super-resolution imaging system of our invention that is closely related to those described by Lukosz.28 Closely related systems are described in Refs. 29 and 30. The system of interest, illustrated in Fig. 3, consists of a quasi-monochromatic object, a pair of identical time-varying diffusers, a pair of lenses, and a pinhole aperture. The complex wave amplitude following Mask 2 (one-dimensional notation is used for simplicity) for an arbitrary pupil (aperture) function is given by c

(26)

u 3` (x , t ) M(x , t )¯ M( X , t )u inc ( X , t )hsys (x X)dX

c

where hsys(x) is the complex-amplitude point spread function for the imaging system and M(x, t) is the complex amplitude transmittance of the masks. The wave field incident on Mask 1 is related to the object complex amplitude by uinc(x) uobj(x)hd(x), where hd(x) is the Fresnel kernel appropriate for propagation through a distance d. Because of the dynamic (moving) diffusers and the effect of the small aperture, calculation of the corresponding optical intensity tells us virtually nothing about the object distribution. The mutual intensity in the output plane, on the other hand, can tell us a great deal about the object. The mutual intensity of the wave incident on Mask 1 contains information about the mutual intensity of the object distribution through the relationship: J inc (x1, x 2 ) J obj(x1, x 2 ) hd (x1 ) hd (x 2 )

Object

Aperture

Mask 1

inc (x, t) d P1 FIGURE 3

(27)

Mask 2

3 (x, t) f

f

f P2

Geometry for Lukosz-type super-resolution optical system.

f P3

6.10

PHYSICAL OPTICS

Because of the reversibility of the wave propagation phenomenon (manifested by the absence of nulls in the wave propagation transfer function), the mutual intensity of the object may be inferred from the mutual intensity of this incident wave field through J obj(x1, x 2 ) J inc (x1, x 2 ) hd (x1 ) hd (x 2 )

(28)

It is the nature of the system of Fig. 3 that this information can be transmitted to the output plane. As was done in the enhanced backscatter analysis, let the temporal behavior of the incident complex wave amplitude and the masks be statistically independent so that u inc (x , t )M(x , t )® u inc (x , t )®M(x , t )®. In addition, assume that the masks are statistically homogenous random phase screens that satisfy the fourth-moment theorem, as reflected by Eq. (14), so that the time average with respect to the diffusers within the integral simplifies to a product of secondorder moments and the autocorrelation function of the mask is sufficiently narrow that it can be modeled by a delta function within an integral.31 Exploiting these conditions and exploiting the sifting property of the delta function, we can calculate the mutual intensity of the wave immediately following the second mask, that is, in plane 3: ¤x x ³

J 3 (x1 , x 2 ) K 2 J inc (x1 , x 2 ) Pˆ ¥ 1 2 ´ ¦ Lf µ

2

(29) ¤ x X³ ¤ x X³ K G (x1 x 2 )¯ I inc ( X)Pˆ ¥ 1 ´ Pˆ ¥ 2 ´ dX

c ¦ Lf µ ¦ Lf µ c

where I(x) is the autocorrelation of the mask function and where Iinc is the optical intensity of the incident wave field. The first term on the right in this equation is proportional to | Pˆ ()|2, the modulus of the imaging system impulse response (Pˆ denotes the Fourier transform of the pupil function associated with the aperture), times the mutual intensity of the wave field incident on Mask 1. The second term corresponds to a spatially incoherent wave field. Through interferometric measurements, it is possible to infer the mutual intensity of the incident wave field, that is, to determine Jinc and thus Jobj.

6.6

EFFICIENT SAMPLING OF COHERENCE FUNCTIONS Several schemes for optical imaging rely on the measurement or transmission of the spatiotemporal coherence function produced at an aperture by the waves from a distant object.32,33 If the object is planar and spatially incoherent, this coherence function, described by the van Cittert-Zernike theorem,34 is, to within a known quadratic phase factor, a function of the vector separation ($x, $y) between pairs of points in the measurement plane, as illustrated in Eq. (10). As a consequence, the number of measurements required to characterize the function in a Nyquist sense is comparatively small. It is not necessary to sample the optical wave field at all possible pairs of points on a grid but, rather, only at a single pair of points for a given spacing ($x, $y). For nonplanar objects, on the other hand, the coherence function has a more general form, 35 and the number of samples required of the coherence function necessarily increases. In order to keep the measurement time as small as possible, an efficient sampling strategy is desirable. Nonuniform sampling grids have been shown to reduce the total number of samples required to unambiguously characterize the complex amplitude associated with a quasi-monochromatic optical wave field in certain cases.36,37 In the following analysis it is shown that a nonuniform sampling scheme can also be effectively applied to the problem of sampling the mutual intensity produced in an aperture by a quasi-monochromatic, spatially incoherent three-dimensional object distribution. The analysis presented is of brief out of necessity, missing details being presented elsewhere.38


6.11

Ix, z

x

x D

Wx

Wz FIGURE 4 The optical coherence of the wave field generated by the source intensity distribution I(x, z) is sampled in a plane some distance away from the center of the rectangular area bounding I(x, z).

The source object, described by optical intensity I(x, z), is assumed to be contained within a rectangular area of transverse width Wx and longitudinal depth Wz centered about the origin of the coordinate system, as shown in Fig. 4 (only one transverse dimension is considered for simplicity, extension to two being straightforward). If Fresnel-regime conditions are satisfied, the mutual intensity in the measurement plane can be shown to be given by the equation J (x1, x 2 ) ¯

1

c c

¯

c c L (z Z )

¹ ª P §©(x1 X)2 (x 2 X)2 ¶¸ºdX dZ I (X , Z )exp «j L

Z ( z ) » ¬

(30)

This function is conveniently expressed in terms of parameters x (x1 x 2 )/2 and $x (x1 x 2 ), with the result J (x , $ x ) ¯

1

c c

¯

c c L (z Z )

¶ § 2P $ x I (X , Z )exp ¨ j (x X)· dX dZ L Z ( z

) © ¸

(31)

Criteria for both uniform and nonuniform sampling of this function are most easily determined if it is Fourier transformed, once with respect to the sample separation $x and once with respect to sample-pair center coordinate x : c c Jˆ$x (x , N ) ¯ ¯

1

c c L (z Z )

c c Jˆx ($ v , $ x ) ¯ ¯

1

c c L (z Z )

§ (x X) ¶ I (X , Z ) D ¨N dX dZ (z Z ) ·¸ L ©

§ 2P $ xX ¶ § $x ¶ I (X , Z )exp ¨ j · D ¨$ v L(z Z ) · dX dZ © L (z Z ) ¸ © ¸

(32)

(33)

If the finite support of the object in x and z is taken into account, these functions can be shown to have regions of support bounded by lines given by the equations

1

N

1 ¤ Wz ³ ¤ Wx ³ z o ´ ¥x o 2µ ¦ 2 ´µ L ¥¦

$N

$ x ¤ Wz ³ zo ´ 2µ L ¥¦

1

(34)

For distances z satisfying the condition z Wx , the corresponding information bandwidths are well approximated by the expressions BN

Wx Lz

B$ N

Wz $ x Lz 2

(35)

6.12

PHYSICAL OPTICS

A straightforward analysis based on the standard Nyquist criterion shows that uniform sample locations should be located according to the equations (letting D denote the size of the aperture in the measurement plane) xm

mL z 2D

$ xn

nL z D Wx

(36)

whereas a generalized Nyquist analysis, appropriate for nonuniform sampling, yields the results xm,n

mWx z nWz

$ xn

nL z Wx

(37)

where the double subscripts m, n denote the sample order. The number of samples required is found by counting all valid indices (m,n), given the extent D of the measurement area. A comparison of the two sampling regimes shows that nonuniform sampling reduces the number of samples required by a factor [(1 D/Wx )(1 D 2 /L z )] 1 and that the minimum separation between measurement points is increased by a factor (1 D/Wx ). This latter difference is beneficial because a constraint on the measurement apparatus is relaxed. Numerical demonstration based on system specifications given in Ref. 35 yields excellent results.

6.7 AN EXAMPLE OF WHEN NOT TO USE COHERENCE THEORY Coherence theory can provide an extremely valuable and sometimes indispensible tool in the analysis of certain phenomena. On the other hand, it can be applied unnecessarily to certain optical systems where it tends to obfuscate rather than clarify system operation. An example is given by the Koehler-illumination imaging system modeled in Fig. 5. In this system, a uniform, spatially incoherent source is imaged by condenser lens 1 into the pupil plane of the imaging optics, the latter formed by lenses 2 and 3. Beginning with the source distribution, one can calculate, through a succession of operations based on relationships presented in Sec. 6.3, the coherence functions appropriate for the object plane, the pupil plane, and the image plane. The effect on the irradiance distributions in these planes can through this means be investigated. But is this the best way to proceed? In fact, if only the image-plane irradiance is of ultimate interest, it is not, the method being unnecessarily complicated. Frits Zernike made an important but sometimes forgotten observation in his 1938 paper on the concept of the degree of coherence.16 A coherence-function–based analysis of a system consisting of a cascade of subsystems, such as that just considered, can be advantageous if an objective is to calculate the irradiance of the wave field in more than a single plane. Thus, with regard to the system of Fig. 5, if calculations of the object illumination, the pupil-plane irradiance, and the image distribution are all

Is xy

xy

oxy I xy

xs

f

f

FIGURE 5

f

f

f

Koehler-illumination imaging model.

f


6.13

desired, then, assuming a quasi-monochromatic source, an analysis based on the plane-to-plane propagation of the mutual intensity of the light is the proper choice. In each plane of interest, the irradiance can be calculated from the mutual intensity without loss of the ability to propagate that mutual intensity function further in the system. If, on the other hand, the objective is simply the calculation of the imageplane irradiance distribution, given the source, object, and pupil functions, then, so long as the source is spatially incoherent, the precoherence theory approach to analysis is preferable. The image plane irradiance produced by light from a single source point is calculated, with an integration over the source distribution following. In many cases, this form of analysis provides greater insight into the operation of the system under investigation. For example, in the case of Koehler illumination, it shows that the source distribution should be uniform and that its image should overfill the aperture stop (pupil) by an amount proportional to the spatial-frequency bandwidth of the object wave amplitude transmittance function. Such insight is not readily available through a coherence-theory–based analysis.

6.8

CONCLUDING REMARKS Some important points are summarized here: 1. Although it often appears in the analysis of optical systems, the complex amplitude associated with an optical wave field cannot be measured directly because of the very high frequencies of the wave field oscillations. 2. The optical intensity of the wave, which involves a time average, can be measured. 3. The mutual intensity of the wave can also be determined from measurements made using an interferometer. 4. Knowledge of the optical intensity of the wave in a given plane of an optical system does not, in general allow calculation of the mutual intensity and/or optical intensity of the wave in subsequent planes. (An exception is when the wave in that plane is spatially incoherent.) 5. By way of contrast, knowledge of the mutual intensity of the wave in a given plane does allow calculation of the mutual intensity and/or the optical intensity of the wave in subsequent planes. 6. If circular complex gaussian statistics for the wave field can be assumed, third- and higher-order moments of the wave field can be inferred from the second-order statistics—that is, from the mutual intensity—of the wave.11,31 Taken together, these statements imply that the mutual intensity of the quasi-monochromatic wave field in an optical system conveys all of the information that is measurable—and, in a valid sense, meaningful—through the system. Restating point (5) from this perspective, if the mutual intensity of a wave in one plane of an optical system is known, the mutual intensity—and, hence, the information content of the wave—can be determined in subsequent planes of that system.

6.9

REFERENCES 1. H. H. Hopkins, “Applications of Coherence Theory in Microscopy and Interferometry,” J. Opt. Soc. Am. 47 (6), June 1957, pp. 508–526. 2. C. Paul Montgomery, J. M. Lussert, P. Vabre, and D. Benhaddou, “Super-Resolution 3D Optical Imaging of Semiconductors Using Coherence Microscopy,” SPIE Proc. Vol. 2412 Three-Dimensional Microscopy: Image Acquisition and Processing II, T. Wilson and C. J. Cogswell, eds., March 1995, pp. 88–94. 3. M. Bass ed., Handbook of Optics, Vol. 1, McGraw-Hill, Inc., New York, 1995, Chapter 4. 4. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics, Cambridge University Press, Cambridge, 1995. 5. L. Mandel and E. Wolf, eds., “Selected Papers on Coherence and Fluctuations of Light, (1850–1966),” SPIE Press Book, MS19, 1990.

6.14

PHYSICAL OPTICS

6. H. Park, M. Chodorow, and R. Kompfner, “High Resolution Optical Ranging System,” App. Opt. 20, 1981, pp. 2389–2394. 7. R. Youngquist, S. Carr, and D. E. N. Davies, “Optical Coherence-Domain Reflectometry: A New Optical Evaluation Technique,” Opt. Lett. 12, 1987, pp. 158–160. 8. H. H. Gilgen, R. P. Novak, R. P. Salathe, W. Hodel, and P. Beaud, “Submillimeter Optical Reflectometry,” J. of Lightwave Tech. 7, 1989, pp. 1225–1233. 9. B. E. Bouma and G. J. Tearney, Handbook of Optical Coherence Tomography, Informa Health Care, New York, 2001. 10. M. E. Brezinski, Optical Coherence Tomography: Principles and Applications, Academic Press, New York, 2006. 11. J. W. Goodman, Statistical Optics, Wiley, New York, 1985. 12. M. Bass, ed., Handbook of Optics, Vol. 1, McGraw-Hill, Inc., New York, 1995, Chapter 2. 13. A. S. Marathay, Elements of Optical Coherence Theory, Wiley, New York, 1982, pp. 71–73. 14. G. B. Parrent Jr., “On the Propagation of Mutual Coherence,” J. Opt. Soc. Am. A. 49, 1959, pp. 787–793. In: Selected Papers on Coherence and Fluctuations of Light, eds. L. Mandel and E. Wolf, Vol 1. Dover Publications, Inc. New York. 15. L. Mandel and E. Wolf, “Some Properties of Coherent Light,” J. Opt. Soc. Am. A. 51, 1961, pp. 815–819. In: Selected Papers on Coherence and Fluctuations of Light, eds. L. Mandel and E. Wolf, Vol 1. Dover Publications, Inc. New York. 16. F. Zernike, “The Concept of Degree of Coherence and Its Application to Optical Problems,” Physica 5, 1938, pp. 785–795. In: Selected Papers on Coherence and Fluctuations of Light, eds. L. Mandel and E. Wolf, Vol 1. Dover Publications, Inc. New York. 17. W. T. Rhodes and G. Welch, “Determination of a Coherent Wave Field after Double Passage through a Diffuser,” J. Opt. Soc. Am. A. 9(2), Feb 1992, pp. 341–343. 18. G. Welch and W. T. Rhodes, “Imaging Diffuser-Obscured Objects by Making Coherence Measurements on Enhanced Backscatter Radiation,” J. Opt. Soc. Am. A. 9(4), April 1992, pp. 539–542. 19. E. Von Lau, “Beugungserscheinungen an Doppelrastern, ” Ann. Phys. 6, 1948, pp. 417. 20. J. Jahns and A. W. Lohmann, “The Lau Effect (a Diffraction Experiment with Incoherent Illumination)” Opt. Comm. 28(3), 1979, pp. 263–267. 21. F. Gori, “Lau Effect and Coherence Theory,” Opt. Comm. 31, 1979, pp. 4–8. 22. R. Sudol and B. J. Thompson, “An Explanation of the Lau Effect Based on Coherence Theory,” Opt. Comm. 31, 1979, pp. 105–110. 23. R. Sudol and B. J. Thompson, “Lau Effect: Theory and Experiment,” Appl. Opt. 20, 1981, pp. 1107–1116. 24. S. Chitralekha, K. V. Avudainayagam, and S. V. Pappu, “Role of Spatial Coherence on the Rotation Sensitivity of Lau Fringes: An Experimental Study,” Appl. Opt. 28, 1989, pp. 345–349. 25. S. Cartwright, “Incoherent Optical Processing: A Coherence Theory Approach,” Appl. Opt. 23, 1984, pp. 318–323. 26. W. Lukosz, “Optical Systems with Resolving Powers Exceeding the Classical Limit,” J. Opt. Soc. Am. A. 56 (11), November 1966, pp. 1463–1472. 27. W. Lukosz, “Optical Systems with Resolving Powers Exceeding the Classical Limit. II,” J. Opt. Soc. Am. A. 57, July 1967, pp. 932–941. 28. G. Welch, “Applications of Coherence to Enhanced Backscatter and Superresolving Imaging Systems,” PhD dissertation. Georgia Institute of Technology, Atlanta, GA, 1995. 29. A. Cunha and E. N. Leith, “One-Way Phase Conjugation with Partially Coherent Light and Superresolution,” Opt. Let. 13(12), 1988, pp. 1105–1107. 30. P. C. Sun and E. N. Leith, “Superresolution by Spatial-Temporal Encoding Methods,” Appl. Opt. 31(23), 1992, pp. 4857–4862. 31. I. S. Reed, “On a Moment Theorem for Complex Gaussian Processes,” IRE Trans. Inf. Theory IT-8, 1962, pp. 194–195. 32. D. L. Marks, R. A. Stack, and D. J. Brady, “Three-Dimensional Coherence Imaging in the Fresnel Domain,” Appl. Opt. 38(8), 1999, pp.1332–1342. 33. G. Welch and W. T. Rhodes, “Image Reconstruction by Spatio-Temporal Coherence Transfer.” In: Free-Space Laser Communication and Laser Imaging, D. G. Voelz and J. C. Ricklin, eds. (Proc. SPIE, Vol. 4489. 2001).


6.15

34. J. W. Goodman, Statistical Optics, Wiley, New York, 1985, Chapter 5. 35. J. Rosen and A. Yariv, “General Theorem of Spatial Coherence: Application to Three-Dimensional Imaging,” J. Opt. Soc. Am. A. 13(10), 1996, pp. 2091–2095. 36. A. Vanderlugt, “Optimum Sampling of Fresnel Transforms,” Appl. Opt. 29(23), 1990, pp. 3352–3361. 37. F. S. Roux, “Complex-Valued Fresnel-Transform Sampling,” Appl. Opt. 34(17), 1995, pp. 3128–3135. 38. J. C. James, G. Welch, and W. T. Rhodes, “Nonuniform Coherence Sampling for Fields Produced by Incoherent Three-Dimensional Sources,” J. Opt. Soc. Am. A. 20, April 2003, pp. 668–677.


7 SCATTERING BY PARTICLES Craig F. Bohren Pennsylvania State University University Park, Pennsylvania

7.1

GLOSSARY a a n, b n C Dn E ex f G h I I, Q, U, V j k m N n Pn p Q r S W W X

radius scattering coefficients cross section logarithmic derivative, d /dR [lnY n (R)] Electric field strength unit vector in the x direction NU projected particle area thickness irradiance Stokes parameters running index imaginary part of the refractive index, 2R/N relative complex refractive index number running index associated Legendre functions of the first kind phase function, normalized differential scattering cross section efficiencies or efficiency factors distance element of the amplitude-scattering matrix volume power scattering amplitude 7.1

7.2

PHYSICAL OPTICS

x C S N Pn Tn Y, X 7 Y || > Re

size parameter, ka absorption coefficient angle wavelength Pn1 /sin Q dPn1 /dQ Riccati-Bessel functions solid angle radian frequency Parallel perpendicular real part of

Subscripts abs absorbed ext extinction sca scattered

7.2

INTRODUCTION Light scattering by particles plays starring and supporting roles on a variety of stages: astronomy, cell biology, colloid chemistry, combustion engineering, heat transfer, meteorology, paint technology, solid-state physics—the list is almost endless. The best evidence of the catholicity of scattering by particles is the many journals that publish papers about it. Scattering by single particles is the subject of monographs by van de Hulst,1 Deirmendjian,2 Kerker,3 Bayvel and Jones,4 Bohren and Huffman,5 Barber and Hill,6 and of a collection edited by Kerker.7 Two similar collections contain papers on scattering by atmospheric particles8 and by chiral particles9 (ones not superposable on their mirror images); scattering by chiral particles is also treated by Lakhtakia et al.10 Papers on scattering by particles are included in collections edited by Gouesbet and Gréhan11 and by Barber and Chang.12 Within this Handbook scattering by particles is touched upon in Chap. 9, “Volume Scattering in Random Media,” in this volume and Chap. 3, “Atmospheric Optics,” in Vol. V. A grand feast is available for those with the juices to digest it. What follows is a mere snack. A particle is an aggregation of sufficiently many molecules that it can be described adequately in macroscopic terms (i.e., by constitutive parameters such as permittivity and permeability). It is a more or less well-defined entity unlike, say, a density fluctuation in a gas or a liquid. Single molecules are not particles, even though scattering by them is in some ways similar (for a clear but dated discussion of molecular scattering, see Martin13). Scattering by single particles is discussed mostly in the wave language of light, although multiple scattering by incoherent arrays of many particles can be discussed intelligibly in the photon language. The distinction between single and multiple scattering is observed more readily on paper than in laboratories and in nature. Strict single scattering can exist only in a boundless void containing a lone scatterer illuminated by a remote source, although single scattering often is attained to a high degree of approximation. A distinction made less frequently is that between scattering by coherent and incoherent arrays. In treating scattering by coherent arrays, the wave nature of light cannot be ignored: phases must be taken into account. But in treating scattering by incoherent arrays, phases may be ignored. Pure water is a coherent array of water molecules; a cloud is an incoherent array of water droplets. In neither of these arrays is multiple scattering negligible, although the theories used to describe them may not explicitly invoke it.

SCATTERING BY PARTICLES

7.3

The distinction between incoherent and coherent arrays is not absolute. Although a cloud of water droplets is usually considered to be an incoherent array, it is not such an array for scattering in the forward direction. And although most of the light scattered by pure water is accounted for by the laws of specular reflection and refraction, it also scatters light—weakly yet measurably—in directions not accounted for by these laws.13 A single particle is itself a coherent array of many molecules, but can be part of an incoherent array of many particles, scattering collectively in such a way that the phases of the waves scattered by each one individually are washed out. Although this section is devoted to single scattering, it must be kept in mind that multiple scattering is not always negligible and is not just scaled-up single scattering. Multiple scattering gives rise to phenomena inexplicable by single-scattering arguments.14

7.3

SCATTERING: AN OVERVIEW Why is light scattered? No single answer will be satisfactory to everyone, yet because scattering by particles has been amenable to treatment mostly by classical electromagnetic theory, our answer lies within this theory. Although palpable matter may appear to be continuous and is often electrically neutral, it is composed of discrete electric charges. Light is an oscillating electromagnetic field, which can excite the charges in matter to oscillate. Oscillating charges radiate electromagnetic waves, a fundamental property of such charges with its origins in the finite speed of light. These radiated electromagnetic waves are scattered waves, waves excited or driven by a source external to the scatterer: an incident wave from the source excites secondary waves from the scatterer; the superposition of all these waves is what is observed. If the frequency of the secondary waves is (approximately) that of the source, these waves are said to be elastically scattered (the term coherently scattered is also used). Scientific knowledge grows like the accumulation of bric-a-brac in a vast and disorderly closet in a house kept by a sloven. Few are the attempts at ridding the closet of rusty or obsolete gear, at throwing out redundant equipment, at putting things in order. For example, spurious distinctions are still made between reflection, refraction, scattering, interference, and diffraction despite centuries of accumulated knowledge about the nature of light and matter. Countless students have been told that specular reflection is localized at smooth surfaces, and that photons somehow rebound from them. Yet this interpretation is shaky given that even the smoothest surface attainable is, on the scale of a photon, as wrinkled as the back of a cowboy’s neck. Photons conceived of as tiny balls would be scattered in all directions by such a surface, for which it is difficult even to define what is meant by an angle of incidence. Why do we think of reflection occurring at surfaces rather than because of them whereas we usually do not think of scattering by particles in this way? One reason is that we can see the surfaces of mirrors and ponds. Another is the dead hand of traditional approaches to the laws of specular reflection and refraction. The empirical approach arrives at these laws as purely geometrical summaries of what is observed— and a discreet silence is maintained about underlying causes. The second approach is by way of continuum electromagnetic theory: reflected and refracted fields satisfy the Maxwell equations. Perhaps because this approach, which also yields the Fresnel formulas, entails the solution of a boundaryvalue problem, reflected and refracted fields are mistakenly thought to originate from boundaries rather than from all the illuminated matter they enclose. This second approach comes to grips with the nature of light but not of matter, which is treated as continuous. The third approach is to explicitly recognize that reflection and refraction are consequences of scattering by discrete matter. Although this scattering interpretation was developed by Paul Ewald and Carl Wilhelm Oseen early in this century, it has diffused with glacial slowness. According to this interpretation, when the optically smooth interface between optically homogeneous dissimilar media is illuminated, the reflected and refracted waves are superpositions of vast numbers of secondary waves excited by the incident wave.

7.4

PHYSICAL OPTICS

Thus reflected and refracted light is, at heart, an interference pattern of scattered light. Doyle 15 showed that although the Fresnel equations are obtained from macroscopic electromagnetic theory, they can be dissected to reveal their microscopic underpinnings. No optics textbook would be complete without sections on interference and diffraction, a distinction without a difference: there is no diffraction without interference. Moreover, diffraction is encumbered with many meanings. Van de Hulst1 lists several: small deviations from rectilinear propagation; wave motion in the presence of an obstacle; scattering by a flat particle such as a disk; an integral relation for a function satisfying the wave equation. To these may be added scattering near the forward direction and by a periodic array. Van de Hulst stops short of pointing out that a term with so many meanings has no meaning. Even the etymology of diffraction is of little help: it comes from a Latin root meaning to break. There is no fundamental distinction between diffraction and scattering. Born and Wolf 16 refer to scattering by a sphere as diffraction by a sphere. I leave it as a penance for the reader to devise an experiment to determine whether a sphere scatters light or diffracts it. The only meaningful distinction is that between approximate theories. Diffraction theories obtain answers at the expense of obscuring the physics of the interaction of light with matter. For example, an illuminated slit in an opaque screen may be the mathematical source but it is not the physical source of a diffraction pattern. Only the screen can give rise to secondary waves that yield the observed pattern. Yet generations of students have been taught that empty space is the source of the radiation diffracted by a slit. To befuddle them even more, they also have been taught that two slits give an interference pattern whereas one slit gives a diffraction pattern. If we can construct a mathematical theory (diffraction theory) that enables us to avoid having to explicitly consider the nature of matter, all to the good. But this mathematical theory and its quantitative successes should not blind us to the fact that we are pretending. Sometimes this pretense cannot be maintained, and when this happens a finger is mistakenly pointed at “anomalies,” whereas what is truly anomalous is that a theory so devoid of physical content could ever give adequate results. A distinction must be made between a physical process and the superficially different theories used to describe it. There is no fundamental difference between specular reflection and refraction by films, diffraction by slits, and scattering by particles. All are consequences of light interacting with matter. They differ only in their geometries and the approximate theories that are sufficient for their quantitative description. The different terms used to describe them are encrustations deposited during the slow evolution of our understanding of light and matter.

7.4

SCATTERING BY PARTICLES: BASIC CONCEPTS AND TERMINOLOGY A single particle can be considered a collection of tiny dipolar antennas driven to radiate (scatter) by an incident oscillating electric field. Scattering by such a coherent array of antennas depends on its size and shape, the observation angle (scattering angle), the response of the individual antennas (composition), and the polarization state and frequency of the incident wave. Geometry, composition, and the properties of the illumination are the determinants of scattering by particles. Perhaps the only real difference between optics and electrical engineering is that electrical engineers can measure amplitudes and phases of fields whereas the primary observable quantity in optics is the time-averaged Poynting vector (irradiance), an amplitude squared. Several secondary observables are inferred from measurements of this primary observable. Consider, for example, a single particle illuminated by a beam with irradiance Ii . The total power scattered by this particle is Wsca. Within the realm of linear optics, the scattered power is proportional to the incident irradiance. This proportionality can be transformed into an equality by means of a factor Csca: Wsca C sca I i

(1)

For Eq. (1) to be dimensionally homogeneous Csca must have the dimensions of area, hence Csca has acquired the name scattering cross section.


7.5

Particles absorb as well as scatter electromagnetic radiation. The rate of absorption Wabs by an illuminated particle, like scattered power, is proportional to the incident irradiance: Wabs Cabs I i

(2)

where Cabs is the absorption cross section. The sum of these cross sections is the extinction cross section: Cext C sca Cabs

(3)

Implicit in these definitions of cross sections is the assumption that the irradiance of the incident light is constant over lateral dimensions large compared with the size of the illuminated particle. This condition is necessarily satisfied by a plane wave infinite in lateral extent, which, much more often than not, is the source of illumination in light-scattering theories. The extinction cross section can be determined (in principle) by measuring transmission by a slab populated by N identical particles per unit volume. Provided that multiple scattering is negligible, the incident and transmitted irradiances Ii and It are related by I t I i e NCexth

(4)

where h is the thickness of the slab. Only the sum of scattering and absorption can be obtained from transmission measurements. To separate extinction into its components requires additional measurements. Equation (4) requires that all particles in the slab be identical. They are different if they differ in size, shape, composition, or orientation (incident beams are different if they differ in wavelength or polarization state). Equation (4) is generalized to a distribution of particles by replacing NCext with

£ N jCext, j

(5)

j

where j denotes all parameters distinguishing one particle from another. Instead of cross sections, normalized cross sections called efficiencies or efficiency factors, Qsca, Qabs, and Qext, often are presented. The normalizing factor is the particle’s area G projected onto a plane perpendicular to the incident beam. No significance should be attached to efficiency used as shorthand for normalized cross section. The normalization factor is arbitrary. It could just as well be the total area of the particle or, to honor Lord Rayleigh, the area of his thumbnail. Proper efficiencies ought to be less than unity, whereas efficiencies for scattering, absorption, and extinction are not so constrained. Moreover, some particles—soot aggregates, for example—do not have well-defined cross-sectional areas. Such particles have cross sections for scattering and absorption but the corresponding efficiencies are nebulous. If any quantity deserves the designation efficiency it is the cross section per particle volume W. Equation (4) can be rewritten to display this: I t I i e f h(Cext /U )

(6)

where f NW is the total volume of particles per unit slab volume. For a given particle loading, specified by f h (volume of particles per unit slab area), transmission is a minimum when Cext/W is a maximum. Each way of displaying extinction (or scattering) versus particle size or wavelength of the incident beam tells a different story. This is illustrated in Fig. 1, which shows the scattering cross section, scattering efficiency, and scattering cross section per unit volume of a silicate sphere in air illuminated by visible light. These curves were obtained with van de Hulst’s simple anomalous diffraction approximation1 (all that is anomalous about it is that it gives such good results). Each curve yields a different answer to the question, what size particle is most efficient at scattering light? And comparison of Figs. 1c and 2 shows that scattering by a particle and specular reflection are similar.

PHYSICAL OPTICS

100 90

Scattering cross section

80 70 60 50 40 30 20 10 0

0

1

2 Radius (m) (a)

3

4

4

3 Scattering efficiency

7.6

2

1

0

0

1

2 Radius (m) (b)

3

4

FIGURE 1 Scattering of visible light by a silicate sphere calculated using the anomalous diffraction approximation: (a) scattering cross section; (b) scattering efficiency (cross section normalized by projected area); and (c) volumetric scattering cross section (cross section per unit particle volume).


FIGURE 1

(Continued)

FIGURE 2 Reflected power per unit incident irradiance and unit volume of a silicate slab normally illuminated by visible light (reflectance divided by slab thickness).

7.7

7.8

PHYSICAL OPTICS

At sufficiently large distances r from a scatterer of bounded extent, the scattered field Es decreases inversely with distance and is transverse: Es ~

e ik(r z ) XE

ikr

(kr 1)

(7)

where k 2R/N is the wave number of the incident plane harmonic wave E i e x E , E E0 exp(ikz) propagating along the z axis. The vector-scattering amplitude is written as X as a reminder that the incident wave is linearly polarized along the x axis. Here and elsewhere the time-dependent factor exp( iYt) is omitted. The extinction cross section is related in a simple way to the scattering amplitude; Cext

4P Re{(X e x )Q 0 } k2

(8)

This remarkable result, often called the optical theorem, implies that plane-wave extinction depends only on scattering in the forward direction S 0, which seems to contradict the interpretation of extinction as the sum of scattering in all directions and absorption. Yet extinction has two interpretations, the second manifest in the optical theorem: extinction is interference between incident and forward-scattered waves. The scattering cross section is also obtained from the vector-scattering amplitude by an integration over all directions: | X |2 d7 4P k 2

C sca ¯

(9)

At wavelengths far from strong absorption bands, the scattering cross section of a particle small compared with the wavelength satisfies (approximately) C sca s

U2 L4

(ka l 0)

(10)

where a is a characteristic linear dimension of the particle. This result was first obtained by Lord Rayleigh in 1871 by dimensional analysis (his paper is included in Ref. 8). The extinction cross section of a particle large compared with the wavelength approaches the limit Cext l 2G

(ka l c)

(11)

The fact that Cext approaches twice G instead of G is sometimes called the extinction paradox. This alleged paradox arises from the expectation that geometrical optics should become a better approximation as a particle becomes larger. But all particles have edges because of which extinction by them always has a component unaccounted for by geometrical optics. This additional component, however, may not be observed because it is associated with light scattered very near the forward direction and because all detectors have finite acceptance angles. Measured extinction is theoretical extinction reduced by the scattered light collected by the detector. No particle scatters light equally in all directions; isotropic scatterers exist only in the dreams of inept theorists. The angular dependence of scattering can be specified by the differential scattering cross section, written symbolically as dC sca /d7 as a reminder that the total scattering cross section is obtained from it by integrating over all directions: C sca ¯

4P

dC sca d7 d7

(12)

The normalized differential scattering cross section p p

1 dC sca C sca d 7

(13)


7.9

FIGURE 3 Scattering of unpolarized visible light by spheres of radii 0.01, 0.1, and 0.2 Mm calculated according to the Rayleigh-Gans approximation.

is sometimes called the phase function. This coinage of astronomers (after the phases of astronomical bodies) confuses those who are perplexed by phase attached to a quantity from which phase in the usual optical sense is absent. To add to the confusion, the phase function is sometimes normalized to 4R instead of to unity. A salient characteristic of scattering by particles is strong forward-backward asymmetry. Small metallic particles at far infrared wavelengths provide one of the few examples in which backscattering is larger than forward scattering. Except for very small particles, scattering is peaked in the forward direction; the larger the particle, the sharper the peak. Examples are given in Fig. 3, which shows differential scattering cross sections for unpolarized visible light illuminating spheres of various radii. These curves were obtained using the Rayleigh-Gans approximation,1,3,5 valid for particles optically similar to the surrounding medium. Forward scattering is much greater than backscattering even for a sphere as small as 0.2 Mm. A simple explanation of forward-backward asymmetry follows from the model of a scatterer as an array of N antennas. If we ignore mutual excitation (the antennas are excited solely by the external source), the total scattered field is the sum of N fields, the phases of which, in general, are different except in the forward direction. Scattering by noninteracting scatterers in this direction—and only in this direction—is in-phase regardless of their separation and the wavelength of the source. Thus as N increases, the scattered irradiance increases more rapidly in the forward direction than in any other direction. Particles are miniature polarizers and retarders: they scatter differently the orthogonal components into which incident fields can be resolved. Similarly, an optically smooth surface can be both a polarizer and retarder. Just as polarization changes upon reflection are described by decomposing electric fields into components parallel and perpendicular to the plane of incidence, it is convenient to introduce a scattering plane, defined by the directions of the incident and scattered waves, for describing scattering by particles. The incident plane wave is transverse, as is the scattered field at large distances. Thus these fields can be decomposed into two orthogonal components, one parallel, the other perpendicular to the scattering plane. The orthonormal basis vectors are denoted by e|| and e> and form a right-handed

7.10

PHYSICAL OPTICS

FIGURE 4 Vibration ellipses at various scattering angles for light scattered by a sphere much smaller than the wavelength of the incident right-circularly polarized light.

triad with the direction of propagation ep (of either the incident or scattered waves): e>xe|| ep. Incident and scattered fields are specified relative to different basis vectors. With this decomposition the relation between fields can be written1,5 ¤ E||s ³ e ik(r z ) ¤S2 ¥ E ´ ikr ¥¦S ¦ > sµ 4

S3³ ¤ E||i ³ S1´µ ¥¦ E> i´µ

(14)

where i and s denote incident and scattered, respectively. The elements of this amplitude scattering matrix (or Jones matrix) are complex-valued functions of the scattering direction. If a single particle is illuminated by completely polarized light, the scattered light is also completely polarized but possibly differently from the incident light, and differently in different directions. An example is given in Fig. 4, which shows vibration ellipses of light scattered by a small sphere. The polarization state of the scattered light varies from right-circular (the polarization state of the incident light) in the forward direction, to linear (perpendicular to the scattering plane) at 90n, to left-circular in the backward direction. Just as unpolarized light can become partially polarized upon specular reflection, scattering of unpolarized light by particles can yield partially polarized light varying in degree and state of polarization in different directions. Unlike specular reflection, however, an ensemble of particles can transform completely polarized incident light into partially polarized scattered light if all the particles are not identical. Transformations of polarized light upon scattering by particles are described most conveniently by the scattering matrix (or Mueller matrix) relating scattered to incident Stokes parameters:1,5 ¤ Is ³ ¥Q ´ 1 ¥ s´ 2 2 U k r ¥ s´ ¥¦ V ´µ s

¤ S11 S12 ¥S S ¥ 21 222 S ¥ 31 S32 ¥¦S S42 41

S13 S23 S33 S43

S14 ³ ¤ I i ³ S24´ ¥Qi ´ ´¥ ´ S34´ ¥U i´ S44´µ ¥¦ Vi ´µ

(15)

The scattering matrix elements Sij for a single particle are functions of the amplitude-scattering matrix elements. Only seven of these elements are independent, corresponding to the four amplitudes and three phase differences of the Sj . The scattering matrix for an ensemble of particles is the sum of matrices for each of them provided they are separated by sufficiently large random distances. Although all 16 matrix elements for an ensemble can be nonzero and different, symmetry reduces the number of matrix elements. For example, the scattering matrix for a rotationally and mirror symmetric ensemble has the form ¤S11 ¥S ¥ 12 ¥0 ¥¦ 0

S12 S22 0 0

0 0 S33

S34

0³ 0´ ´ S34´ S44´µ

(16)


7.5

7.11

SCATTERING BY AN ISOTROPIC, HOMOGENEOUS SPHERE: THE ARCHETYPE An isotropic, homogeneous sphere is the simplest finite particle, the theory of scattering to which is attached the name of Gustav Mie.17 So firm is this attachment that in defiance of logic and history every particle under the sun has been dubbed a “Mie scatterer,” and Mie scattering has been promoted from a particular theory of limited applicability to the unearned rank of general scattering process. Mie was not the first to solve the problem of scattering by an arbitrary sphere.18 It would be more correct to say that he was the last. He gave his solution in recognizably modern notation and also addressed a real problem: the colors of colloidal gold. For these reasons, his name is attached to the sphere-scattering problem even though he had illustrious predecessors, most notably Lorenz.19 This is an example in which eponymous recognition has gone to the last discoverer rather than to the first. Mie scattering is not a physical process; Mie theory is one theory among many. It isn’t even exact because it is based on continuum electromagnetic theory and on illumination by a plane wave infinite in lateral extent. Scattering by a sphere can be determined using various approximations and methods bearing little resemblance to Mie theory: Fraunhofer theory, geometrical optics, anomalous diffraction, coupled-dipole method, T-matrix method, etc. Thus, is a sphere a Mie scatterer or an anomalous diffraction scatterer or a coupled-dipole scatterer? The possibilities are endless. When a physical process can be described by several different theories, it is inadvisable to attach the name of one of them to it. There is no distinct boundary between so-called Mie and Rayleigh scatterers. Mie theory includes Rayleigh theory, which is a limiting theory strictly applicable only as the size of the particle shrinks to zero. Even for spheres uncritically labeled “Rayleigh spheres,” there are always deviations between the Rayleigh and Mie theories. By hobbling one’s thinking with a supposed sharp boundary between Rayleigh and Mie scattering, one risks throwing some interesting physics out the window. Whether a particle is a Mie or Rayleigh scatterer is not absolute. A particle may be graduated from Rayleigh to Mie status merely by a change of wavelength of the illumination. One often encounters statements about Mie scattering by cylinders, spheroids, and other nonspherical particles. Judged historically, these statements are nonsense: Mie never considered any particles other than homogeneous spheres. Logic would seem to demand that if a particle is a Mie scatterer, then Mie theory can be applied to scattering by it. This fallacious notion has caused and will continue to cause mischief, and is probably the best reason for ceasing to refer to Mie particles or Mie scatterers. Using Mie theory for particles other than spheres is risky, especially for computing scattering toward the backward direction. More often than not, a better term than Mie or Rayleigh scattering is available. If the scatterers are molecules, molecular scattering is better than Rayleigh scattering (itself an imprecise term):20 the former term refers to an agent, the latter to a theory. Mie scatterer is just a needlessly aristocratic name for a humble sphere. Wherever Mie scatterer is replaced with sphere, the result is clearer. If qualifications are needed, one can add small or large compared with the wavelength or comparable to the wavelength. Briefly, the solution to the problem of scattering by an arbitrary homogeneous sphere illuminated by a plane wave can be obtained by expanding the incident, scattered, and internal fields in a series of vector-spherical harmonics. The coefficients of these expansion functions are chosen so that the tangential components of the electric and magnetic fields are continuous across the surface of the sphere. Thus this scattering problem is formally identical to reflection and refraction because of interfaces, although the sphere problem is more complicated because the scattered and internal fields are not plane waves. Observable quantities are expressed in terms of the coefficients an and bn in the expansions of the scattered fields. For example, the cross sections are infinite series: 2P c £(2n 1)Re{an bn} k 2 n1

(17)

2P c £(2n 1)(| an |2 | bn |2 ) k 2 n1

(18)

Cext C sca

7.12

PHYSICAL OPTICS

If the permeability of the sphere and its surroundings are the same, the scattering coefficients can be written an

[Dn (mx)/m n/x]Y n (x) Y n 1(x) [Dn (mx )/m n/x]Xn (x ) Xn 1(x )

(19)

[mDn (mx) n/x]Y n (x ) Y n 1(x ) [mDn (mx) n/x]Xn (x ) Xn 1(x )

(20)

bn

[n and Zn are Riccati-Bessel functions and the logarithmic derivative Dn (R)

d ln Y n (R) dR

(21)

The size parameter x is ka, where a is the radius of the sphere and k is the wavenumber of the incident light in the surrounding medium, and m is the complex refractive index of the sphere relative to that of this (nonabsorbing) medium. Equations (19) and (20) are one of the many ways of writing the scattering coefficients, some of which are more suited to computations than others. During the Great Depression mathematicians were put to work computing tables of trigonometric and other functions. The results of their labors now gather dust in libraries. Today, these tables could be generated more accurately in minutes on a pocket calculator. A similar fate has befallen Mie calculations. Before fast computers were inexpensive, tables of scattering functions for limited ranges of size parameter and refractive index were published. Today, these tables could be generated in minutes on a personal computer. The moral is to give algorithms rather than only tables of results, which are mostly useless except as checks for someone developing and testing algorithms. These days it is not necessary to reinvent the sphere: documented Mie programs are readily available. The first widely circulated program was published as an IBM report by Dave in 1968, although it no longer seems to be available. A Mie program is given in Ref. 5. Reference 6 includes a diskette containing scattering programs for spheres (and other particles). Wiscombe21,22 suggested techniques for increasing the speed of computations, as did Lentz,23 whose method makes use of continued fractions. Wang and van de Hulst24 recently compared various scattering programs. The primary tasks in Mie calculations are computing the functions in Eqs. (19) and (20) and summing series like Eqs. (17) and (18). Bessel functions are computed by recurrence. The logarithmic derivative, the argument of which can be complex, is usually computed by downward recurrence. [n(x) and Zn(x) can be computed by upward recurrence if one does not generate more orders than are needed for convergence, approximately the size parameter x. When a program with no logical errors falls ill, it often can be cured by promoting variables from single to double precision. Cross sections versus radius or wavelength convey physical information; efficiencies versus size parameter convey mathematical information. The size parameter is a variable with less physical content than its components, the whole being less than the sum of its parts. Moreover, cross section versus size parameter (or its inverse) is not equivalent to cross section versus wavelength. Except in the fantasy world of naive modelers, refractive indices vary with wavelength, and the Mie coefficients depend on x and m, wavelength being explicit in the first and implicit in the second. The complex refractive index is written dozens of different ways, one of which is n + ik (despite the risk of confusing the imaginary part with the wavenumber). The quantities n and k are called optical constants. But just as the Lord Privy Seal is neither a lord nor a privy nor a seal, optical constants are neither optical nor constant. Few quantities in optics are more shrouded in myth and misconception than the complex refractive index. The real part for any medium is often defined as the ratio of the velocity of light c in free space to the phase velocity in the medium. This definition, together with notions that nothing can go faster than c, has engendered the widespread misconception that n must be greater than unity. But n can take on any value, even zero. The phase velocity is not the velocity of any palpable object or of any signal, hence is not subject to speed limits enforced by the special relativity police. The least physically relevant property of a refractive index is that it is a ratio of phase velocities. A refractive


7.13

index is a response function (or better, is simply related to response functions such as permittivity and permeability): it is a macroscopic manifestation of the microscopic response of matter to a periodic driving force. When we turn to the imaginary part of the refractive index, we enter a ballroom in which common sense is checked at the door. It has been asserted countless times that an imaginary index of, say, 0.01 corresponds to a weakly absorbing medium (at visible and near-visible wavelengths). Such assertions are best exploded by expressing k in a more physically transparent way. The absorption coefficient C is

A

4P k L

(22)

The inverse of C is the e-folding distance (or skin depth), the distance over which the irradiance of light propagating in an unbounded medium decreases by a factor of e. At visible wavelengths, the e-folding distance corresponding to k 0.01 is about 5 Mm. A thin sliver of such an allegedly weakly absorbing material would be opaque. When can a particle (or any object) be said to be strongly absorbing? A necessary condition is that A d 1, where d is a characteristic linear dimension of the object. But this condition is not sufficient. As k increases, absorption increases—up to a point. As k approaches infinity, the absorption cross section of a particle or the absorptance of a film approaches zero. One of the most vexing problems in scattering calculations is finding optical constants dispersed throughout dozens of journals. Palik25 edited a compilation of optical constants for several solids. The optical constants of liquid water over a broad range were compiled by Hale and Querry;26 Warren27 published a similar compilation for ice. For other materials, you are on your own. Good hunting! For small x and \m\x, the extinction and scattering efficiencies of a sphere are approximately ª m2 1 ¹ Qext 4 x Im « 2 º ¬m 2 » 8 m2 1 Qsca x 4 2 3 m 2

(23)

2

(24)

These equations are the source of a nameless paradox, which is disinterred from time to time, a corpse never allowed eternal peace. If the sphere is nonabsorbing (m real), Eq. (23) yields a vanishing extinction cross section, whereas Eq. (24) yields a nonvanishing scattering cross section. Yet extinction never can be less than scattering. But note that Eq. (23) is only the first term in the expansion of Qext in powers of x. The first nonvanishing term in the expansion of Qsca is of order x4. To be consistent, Qext and Qsca must be expanded to the same order in x. When this is done, the paradox vanishes. The amplitude-scattering matrix elements for a sphere are S1 £

2n 1 (a P b T ) n(n 1) n n n n

(25)

S2 £

2n 1 (a T b P ) n(n 1) n n n n

(26)

n

n

where the angle-dependent functions are

Pn

Pn1 sin Q

Tn

dPn1 dQ

(27)

and Pn1 are the associated Legendre functions of the first kind. The off-diagonal elements of the amplitude-scattering matrix vanish, because of which the scattering matrix is block-diagonal

7.14

PHYSICAL OPTICS

and S12 S21, S43 −S34, S44 S33. Thus, when the incident light is polarized parallel (perpendicular) to the scattering plane, so is the scattered light, a consequence of the sphere’s symmetry.

7.6

SCATTERING BY REGULAR PARTICLES The field scattered by any spherically symmetric particle has the same form as that scattered by a homogeneous, isotropic sphere; only the scattering coefficients are different. One such particle is a uniformly coated sphere. Scattering by a sphere with a single layer was first treated by Aden and Kerker.28 Extending their analysis to multilayered spheres is straightforward.29 New computational problems arise in going from uncoated to coated spheres. The scattering coefficients for both contain spherical Bessel functions, which are bounded only if their arguments are real (no absorption). Thus, for strongly absorbing particles, the arguments of Bessel functions can be so large that their values exceed computational bounds. This does not occur for uncoated spheres because the only quantity in the scattering coefficients with complex argument is the logarithmic derivative, a ratio of Bessel functions computed as an entity instead of by combining numerator and denominator, each of which separately can exceed computational bounds. It is not obvious how to write the scattering coefficients for a coated sphere so that only ratios of possibly large quantities are computed explicitly. For this reason the applicability of the coated-sphere program in Ref. 5 is limited. Toon and Ackerman,30 however, cast the coated-sphere coefficients in such a way that this limitation seems to have been surmounted. Bessel functions of large complex argument are not the only trap for the unwary. A coated sphere is two spheres. The size parameter for the outer sphere determines the number of terms required for convergence of series. If the inner sphere is much smaller than the outer, the various Bessel functions appropriate to the inner sphere are computed for indices much greater than needed. More indices are not always better. Beyond a certain number, round-off error can accumulate to yield terms that should make ever smaller contributions to sums but may not. Scattering by spheres and by infinitely long circular cylinders illuminated normally to their axes are in some ways similar. Spherical Bessel functions in the sphere-scattering coefficients correspond to cylindrical Bessel functions in the cylinder-scattering coefficients. Unlike a sphere, however, an infinitely long cylinder cannot be enclosed in a finite volume. As a consequence, the field scattered by such a cylinder decreases inversely as the square root of distance r instead of inversely as r (for sufficiently large r). Infinite particles may be mathematically tractable but they are physically unrealizable. In particular, cross sections for infinite cylinders are infinite. But cross sections per unit length of infinite cylinders are finite. Such cross sections may be applied to a finite cylindrical particle by multiplying its length by the cross section per unit length of the corresponding infinite particle. If the aspect ratio (length/diameter) of the finite particle is sufficiently large, what are vaguely called “end effects” may be negligible. Because no exact theory for a finite cylinder exists, the aspect ratio at which differences between finite and infinite cylinders become negligible is not known with certainty, although the value 10 is bruited about. Nevertheless, there always will be differences between scattering by finite and infinite particles, which may or may not be of concern depending on the application. A physical difference between scattering by spheres and by cylinders is that cross sections for cylinders depend on the polarization state of the incident plane wave. But normally incident light illuminating an infinite cylinder and polarized perpendicular (parallel) to the plane defined by the incident wave and the cylinder axis excites only scattered light polarized perpendicular (parallel) to the plane defined by the scattered wave and the cylinder axis. Obliquely incident linearly polarized light can, however, excite scattered light having both copolarized and cross-polarized components. Obliquely illuminated uncoated cylinders pose no special computational problems. Coated cylinders, however, pose the same kinds of problems as coated spheres and are even more difficult to solve. Toon and Ackerman’s30 algorithm for coated spheres is based on the fact that spherical Bessel functions can be expressed in a finite number of terms. Because cylindrical Bessel functions cannot


7.15

be so expressed, this algorithm cannot be extended to coated cylinders, for which Bessel functions must be computed separately rather than as ratios and can have values beyond computational bounds. Even if such bounds are not exceeded, problems still can arise. Although Barabás31 discussed in detail scattering by coated cylinders, Salzman and Bohren32 found that his computational scheme is unsuitable when absorption is large. They attempted, with only partial success, to write programs for arbitrary-coated cylinders. Care must be taken in computing Bessel functions. The often-used Miller algorithm can be inadequate for large, complex arguments. The simplest nonspherical, finite particle is the spheroid, prolate, or oblate. Because the scalar wave equation is separable in spheroidal coordinates, scattering by spheroids can be solved in the same way as for spheres and cylinders. The expansion functions are based on spheroidal rather than spherical or cylindrical wave functions. Asano and Yamamoto33 were the first to solve in this way the problem of scattering by an arbitrary spheroid. Although Asano34 subsequently published an extensive set of computations based on this solution, it has not seen widespread use, possibly because of the intractability and exoticness of spheroidal functions. Computational experience with spheroids and even simpler particles such as spheres and cylinders leads to the inescapable conclusion that hidden barriers lie between a mathematical solution to a scattering problem and an algorithm for reliably and quickly extracting numbers from it.

7.7 COMPUTATIONAL METHODS FOR NONSPHERICAL PARTICLES The widespread notion that randomly oriented nonspherical particles are somehow equivalent to spheres is symptomatic of a failure to distinguish between the symmetry of an ensemble and that of its members. Considerable effort has been expended in seeking prescriptions for equivalent spheres. This search resembles that for the Holy Grail—and has been as fruitless. From extensive studies of scattering by nonspherical particles, Mugnai and Wiscombe35 concluded that “after examining hundreds of nonspherical results and observing that they all cluster relatively close together, relatively far from the equivolume spheres (except at forward angles), we have come to regard nonspherical particles as normal, and spheres as the most unrepresentative shape possible—almost a singularity.” This serves as a warning against using Mie theory for particles of all shapes and as a spur to finding methods more faithful to reality. We now turn to some of these methods. Keep in mind that no matter how different they may appear on the surface, they are all linked by the underlying Maxwell equations. The T-matrix method is based on an integral formulation of scattering by an arbitrary particle. It was developed by Waterman, first for a perfect conductor,36 then for a particle with less restricted properties.37 It subsequently was applied to scattering problems under the name extended boundary condition method (EBCM).38 Criticism of the T-matrix method was rebutted by Varadan et al.,39 who cite dozens of papers on this method applied to electromagnetic scattering. Another source of papers and references is the collection edited by Varadan and Varadan.40 Reference 6 is accompanied by a diskette containing T-matrix programs. Linearity of the field equations and boundary conditions implies that the coefficients in the spherical harmonic expansion of the field scattered by any particle are linearly related to those of the incident field. The linear transformation connecting these two sets of coefficients is called the T (for transition) matrix. The T-matrix elements are obtained by numerical integration. Computational difficulties arise for particles with high absorption or large aspect ratios. These limitations of the original T-matrix method have been surmounted somewhat by Iskander et al.,41 whose extension is dubbed the iterative extended boundary condition method. Although the T-matrix method is not restricted to axisymmetric particles, it almost exclusively has been applied to spheroids and particles defined by Chebyshev polynomials.35,42,43 Despite its virtues, the T-matrix method is not readily grasped in one sitting. Another method, variously called the Purcell-Pennypacker,44 coupled-dipole,45 digitized Green’s function46 method and

7.16

PHYSICAL OPTICS

discrete dipole approximation,47 is mathematically much simpler—the most complicated function entering into it is the exponential—and physically transparent. Although originally derived by heuristic arguments, the coupled-dipole method was put on firmer analytical foundations by Lakhtakia.48 In this method, a particle is approximated by a lattice of N dipoles small compared with the wavelength but still large enough to contain many molecules. The dipoles often are, but need not be, identical and isotropic. Each dipole is excited by the incident field and by the fields of all the other dipoles. Thus the field components at each site satisfy a set of 3N linear equations. These components can be calculated by iteration44,49 or by inverting the 3N r 3N coefficient matrix.45 The coefficient matrix for only one particle orientation need be inverted. This inverse matrix then can be used to calculate scattering for other orientations.50 A disadvantage of matrix inversion is that the number of dipoles is limited by computer storage. Arrays of coupled dipoles were considered long before Purcell and Pennypacker entered the scene. More than half a century ago Kirkwood51 treated a dielectric as an array of molecules, the dipole moment of each of which is determined by the external field and by the fields of all the other molecules. What Purcell and Pennypacker did was to apply the coupled-dipole method to absorption and scattering by optically homogeneous particles. They bridged the gap between discrete arrays and continuous media with the Clausius-Mosotti theory. Because this theory, like every effective-medium theory, is not exact, critics of their method have pronounced it guilty by association. But the Clausius-Mosotti theory is merely the effective-medium theory that Purcell and Pennypacker happened to use. Whatever flaws their method may have, one of them is not that it is forever chained to the ghosts of Clausius and Mosotti. Alleged violations of the optical theorem are easily remedied by using the exact expression for the polarizability of a finite sphere,52 which in no way changes the structure of the method. Draine47 applied this method (under the label discrete dipole approximation) to extinction by interstellar grains, obtaining the field components with the conjugate gradient method. An outgrowth of his paper is that by Flatau et al.,53 who considered scattering by rectangular particles. Goedecke and O’Brien46 baptized their version of the digitized Green’s function method and applied it to scattering of microwave radiation by snowflakes.54 Varadan et al.55 applied the method to scattering by particles with anisotropic optical constants. It also has been applied to scattering by helices,56 by a cylinder on a reflecting surface,57 and extended to intrinsically optically active particles.58 Although Yung’s analysis59 of a large (15,600) array of dipoles representing a sphere suggests that there are no intrinsic limitations to the coupled-dipole method, it is plagued with practical limitations, most notably its inability to treat particles (especially compact ones or ones with large complex refractive indices) much larger than the wavelength of the illumination. Chiapetta,60 then Singham and Bohren,61 reformulated the coupled-dipole method, expressing the total field at each dipole as the sum of the incident field and the fields scattered once, twice, and so on by all the other dipoles. Although this formulation is appealing because each term in the scattering-order series has a simple physical interpretation, the series can diverge. The greater the refractive index, the smaller the particle for which the series diverges. For a particle of given volume, fewer terms are needed for convergence the more the particle departs from sphericity. The greater the average separation between dipoles, the weaker the average interaction. Except for improvements and refinements52,62,63 that increase accuracy and speed but do not remove barriers imposed by particle size and composition, the coupled-dipole method has not changed much since it first was used by Purcell and Pennypacker. It is not, of course, limited to optically homogeneous particles. It can be applied readily to aggregates of small particles. Indeed, it is best suited to aggregates with low fractal dimension. Berry and Percival64 considered scattering by fractal aggregates using what they call the mean-field approximation, which is essentially the Rayleigh-Gans approximation, in turn a form of the scattering-order formulation of the coupleddipole method in which the dipoles are excited only by the incident field. The arbitrary border separating electrical engineering from optics is never more obvious than when it comes to methods for computing scattering. The engineers have theirs, the optical scientists have theirs, and rarely do the twain meet. In the hope of promoting smuggling, even illegal immigration, I must at least mention two methods that fall almost exclusively in the domain of electrical engineering: the method of moments and the finite-difference time-domain technique (FDTD).


7.17

Anyone interested in the method of moments must begin with Harrington’s book,65 a focal point from which paths fan out in all directions. As its name implies, the FDTD technique is applied to what electrical engineers call the time domain (as opposed to the frequency domain, which most optical scientists inhabit even though they may not know it) and is explicitly labeled a finite-difference method (all methods for particles other than those of simple shape entail discretization in one form or another). Papers by Yee,66 Holland et al.,67 Mur,68 Luebbers et al.,69 and references cited in them will get you started on the FDTD technique. When pondering the welter of species and subspecies of methods keep in mind that the differences among them and their ranges of validity are probably smaller than their adherents think or are willing to admit. There is no method that will happily compute scattering of arbitrary waves by particles of arbitrary size and composition in a finite amount of time. Moreover, each method, whatever its merits and demerits, often requires a tedious climb up a learning curve.

7.8

REFERENCES 1. H. C. van de Hulst, Light Scattering by Small Particles, Wiley, New York, 1957. Reprinted by Dover, New York, 1981. 2. D. Deirmendjian, Electromagnetic Scattering on Polydispersions, Elsevier, New York, 1969. 3. M. Kerker, The Scattering of Light and Other Electromagnetic Radiation, Academic, New York, 1969. 4. L. P. Bayvel and A. R. Jones, Electromagnetic Scattering and Its Applications, Elsevier, London, 1981. 5. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles, Wiley-Interscience, New York, 1983. 6. P. W. Barber and S. C. Hill, Light Scattering by Particles: Computational Methods, World Scientific, Singapore, 1990. 7. Selected Papers on Light Scattering, SPIE vol. 951, pts. 1 and 2, Milton Kerker (ed.), SPIE Optical Engineering Press, Bellingham, Wash., 1988. 8. Selected Papers on Scattering in the Atmosphere, SPIE vol. MS 7, Craig F. Bohren (ed.), SPIE Optical Engineering Press, Bellingham, Wash., 1989. 9. Selected Papers on Natural Optical Activity, SPIE vol. MS 15, Akhlesh Lakhtakia (ed.), SPIE Optical Engineering Press, Bellingham, Wash., 1990. 10. A. Lakhtakia, V. K. Varadan, and V. V. Varadan, Time-Harmonic Electromagnetic Fields in Chiral Media, Springer-Verlag, Berlin, 1989. 11. Optical Particle Sizing: Theory and Practice, Gérard Gouesbet and Gérard Gréhan (eds.), Plenum, New York, 1988. 12. Optical Effects Associated with Small Particles, P. W. Barber and R. K. Chang (eds.), World Scientific, Singapore, 1988. 13. W. H. Martin, “The Scattering of Light in One-Phase Systems,” in Colloid Chemistry: Theoretical and Applied, vol. I, Jerome Alexander (ed.), Chemical Catalog Co., New York, 1926, pp. 340–352. 14. C. F. Bohren, “Multiple Scattering of Light and Some of Its Observable Consequences,” Am. J. Phys. 55:524 (1987). 15. W. T. Doyle, “Scattering Approach to Fresnel’s Equations and Brewster’s Law,” Am. J. Phys. 53:463 (1985). 16. Max Born and E. Wolf, Principles of Optics, 3d ed., Pergamon, Oxford, 1965. 17. P. Lilienfeld, “Gustav Mie: The Person,” Appl. Opt. 30:4696 (1991). 18. N. A. Logan, “Survey of Some Early Studies of the Scattering of Plane Waves by a Sphere,” Proc. IEEE 53:773 (1965). 19. Helge Kragh, “Ludvig Lorenz and Nineteenth Century Optical Theory: The Work of a Great Danish Scientist,” Appl. Opt. 30:4688 (1991). 20. A. T. Young, “Rayleigh Scattering,” Phys. Today 35:2 (1982). 21. W. J. Wiscombe, “Mie Scattering Calculations: Advances in Techniques and Fast, Vector-Speed Computer Codes,” NCAR/TN-140 + STR, National Center for Atmospheric Research, Boulder, Colo., 1979.

7.18

PHYSICAL OPTICS

22. W. J. Wiscombe, “Improved Mie Scattering Algorithms,” Appl. Opt. 19:1505 (1980). 23. W. J. Lentz, “Generating Bessel Functions in Mie Scattering Calculations Using Continued Fractions,” Appl. Opt. 15:668 (1976). 24. Ru T. Wang and H. C. van de Hulst, “Rainbows: Mie Computations and the Airy Approximation,” Appl. Opt. 30:106 (1991). 25. E. W. Palik (ed), Handbook of Optical Constants of Solids, Academic, New York, 1985. 26. G. M. Hale and M. Querry, “Optical Constants of Water in the 200-nm to 200-Mm Wavelength Region,” Appl. Opt. 12:555 (1973). 27. S. G. Warren, “Optical Constants of Ice from the Ultraviolet to the Microwave,” Appl. Opt. 23:1206 (1984). 28. A. L. Aden and M. Kerker, “Scattering of Electromagnetic Waves from Two Concentric Spheres,” J. Appl. Phys. 22:601 (1951). 29. R. Bhandari, “Scattering Coefficients for a Multilayered Sphere: Analytic Expressions and Algorithms,” Appl. Opt. 24:1960 (1985). 30. O. B. Toon and T. P. Ackerman, “Algorithms for the Calculation of Scattering by Stratified Spheres,” Appl. Opt. 20:3657 (1981). 31. M. Barabás, “Scattering of a Plane Wave by a Radially Stratified Tilted Cylinder,” J. Opt. Soc. Am. A4:2240 (1987). 32. G. C. Salzman and C. F. Bohren, “Scattering by Infinite Coated Cylinders Illuminated at Oblique Incidence,” TCN 90203, U.S. Army Research Office, Research Triangle Park, N.C. 33. S. Asano and G. Yamamoto, “Light Scattering by a Spheroidal Particle,” Appl. Opt. 14:29 (1975). 34. S. Asano, “Light Scattering Properties of Spheroidal Particles,” Appl. Opt. 18:712 (1979). 35. A. Mugnai and W. Wiscombe, “Scattering from Nonspherical Chebyshev Particles, 3. Variability in Angular Scattering Patterns,” Appl. Opt. 28:3061 (1989). 36. P. C. Waterman, “Matrix Formulation of Electromagnetic Scattering,” Proc. IEEE 53:805 (1965). 37. P. C. Waterman, “Symmetry, Unitarity, and Geometry in Electromagnetic Scattering,” Phys. Rev. D3:825 (1971). 38. P. Barber and C. Yeh, “Scattering of Electromagnetic Waves by Arbitrarily Shaped Dielectric Bodies,” Appl. Opt. 14:2864 (1975). 39. V. V. Varadan, A. Lakhtakia, and V. K. Varadan, “Comments on Recent Criticism of the T-Matrix Method,” J. Acoust. Soc. Am. 84:2280 (1988). 40. Acoustic, Electromagnetic and Elastic Wave Scattering—Focus on the T-Matrix Approach, V. K. Varadan and V. V. Varadan (eds.), Pergamon, New York, 1980. 41. M. F. Iskander, A. Lakhtakia, and C. H. Durney, “A New Procedure for Improving the Solution Stability and Extending the Frequency Range of the EBCM,” IEEE Trans. Antennas Propag. AP-31:317 (1983). 42. A. Mugnai and W. Wiscombe, “Scattering from Nonspherical Chebyshev Particles, 1. Cross Sections, SingleScattering Albedo, Asymmetry Factor, and Backscattered Fraction,” Appl. Opt. 25:1235 (1986). 43. W. J. Wiscombe and A. Mugnai, “Scattering from Nonspherical Chebyshev Particles, 2. Means of Angular Scattering Patterns,” Appl. Opt. 27:2405 (1988). 44. E. M. Purcell and C. R. Pennypacker, “Scattering and Absorption of Light by Nonspherical Dielectric Grains,” Astrophys. J. 186:705 (1973). 45. Shermila Brito Singham and G. C. Salzman, “Evaluation of the Scattering Matrix of an Arbitrary Particle Using the Coupled-Dipole Approximation,” J. Chem. Phys. 84:2658 (1986). 46. G. H. Goedecke and S. G. O’Brien, “Scattering by Irregular Inhomogeneous Particles via the Digitized Green’s Function Algorithm,” Appl. Opt. 27:2431 (1988). 47. B. T. Draine, “The Discrete-Dipole Approximation and Its Application to Interstellar Graphite Grains,” Astrophys. J. 333:848 (1988). 48. Akhlesh Lakhtakia, “Macroscopic Theory of the Coupled Dipole Approximation Method,” Opt. Comm. 79:1 (1990). 49. S. D. Druger, M. Kerker, D.-S. Wang, and D. D. Cooke, “Light Scattering by Inhomogeneous Particles,” Appl. Opt. 18:3888 (1979).


7.19

50. M. K. Singham, S. B. Singham, and G. C. Salzman, “The Scattering Matrix for Randomly Oriented Particles,” J. Chem. Phys. 85:3807 (1986). 51. J. G. Kirkwood, “On the Theory of Dielectric Polarization,” J. Chem. Phys. 4:592 (1936). 52. C. E. Dungey and C. F. Bohren, “Light Scattering by Nonspherical Particles: A Refinement to the CoupledDipole Method,” J. Opt. Soc. Am. A8:81 (1991). 53. P. J. Flatau, G. L. Stephens, and B. T. Draine, “Light Scattering by Rectangular Solids in the Discrete-Dipole approximation: A New Algorithm Exploiting the Block-Toeplitz Structure,” J. Opt. Soc. Am. A7:593 (1990). 54. S. G. O’Brien and G. H. Goedecke, “Scattering of Millimeter Waves by Snow Crystals and Equivalent Homogeneous Symmetric Particles,” Appl. Opt. 27:2439 (1988). 55. V. V. Varadan, A. Lakhtakia, and V. K. Varadan, “Scattering by Three-Dimensional Anisotropic Scatterers,” IEEE Trans. Antennas Propag. 37:800 (1989). 56. S. B. Singham, C. W. Patterson, and G. C. Salzman, “Polarizabilities for Light Scattering from Chiral Particles,” J. Chem. Phys. 85:763 (1986). 57. M. A. Taubenblatt, “Light Scattering from Cylindrical Structures on Surfaces,” Opt. Lett. 15:255 (1990). 58. Shermila Brito Singham, “Intrinsic Optical Activity in Light Scattering from an Arbitrary Particle,” Chem. Phys. Lett. 130:139 (1986). 59. Y. L. Yung, “Variational Principle for Scattering of Light by Dielectric Particles,” Appl. Opt. 17:3707 (1978). 60. P. Chiapetta, “Multiple Scattering Approach to Light Scattering by Arbitrarily Shaped Particles,” J. Phys. A 13:2101 (1980). 61. Shermila Brito Singham and C. F. Bohren, “Light Scattering by an Arbitrary Particle: The Scattering Order Formulation of the Coupled-Dipole Method,” J. Opt. Soc. Am. A5:1867 (1988). 62. Shermila Brito Singham and C. F. Bohren, “Hybrid Method in Light Scattering by an Arbitrary Particle,” Appl. Opt. 28:517 (1989). 63. J. J. Goodman, B. T. Draine, and P. J. Flatau, “Application of Fast-Fourier Transform Techniques to the Discrete-Dipole Approximation,” Opt. Lett. 16:1198 (1991). 64. M. V. Berry and I. C. Percival, “Optics of Fractal Clusters Such as Smoke,” Opt. Acta 33:577 (1986). 65. R. F. Harrington, Field Computation by Moment Methods, Robert E. Krieger, Malabar, Fla., 1987. Reprint of 1968 edition. 66. K. S. Yee, “Numerical Solution of Initial Boundary Value Problems Involving Maxwell’s Equations in Isotropic Media,” IEEE Trans. Antennas Propag. AP14:302 (1966). 67. R. Holland, L. Simpson, and K. S. Kunz, “Finite Difference Analysis of EMP Coupling to Lossy Dielectric Structures,” IEEE Trans. Electromag. Compat. EMC22:203 (1980). 68. G. Mur, “Absorbing Boundary Conditions for the Finite Difference Approximation of the Time Domain Electromagnetic Field Equations,” IEEE Trans. Electromag. Compat. EMC23:377 (1981). 69. R. Luebbers, F. P. Hunsberger, K. S. Kunz, R. B. Standler, and M. Schneider, “A Frequency-Dependent Finite-Difference Time-Domain Formulation for Dispersive Materials,” IEEE Trans. Electromag. Compat. EMC32:222 (1990).


8 SURFACE SCATTERING Eugene L. Church and Peter Z. Takacs Instrumentation Division Brookhaven National Laboratory Upton, New York

8.1

GLOSSARY OF PRINCIPAL SYMBOLS Ao A, B, C BRDF C(V) D(V) fxy, fx g, g` Jn Kn Lo Lx, Ly N P RC, rC RCS S S

T Z(x, y) Z(x) 1D, 2D z® C, D T U V

illuminated area model parameters bidirectional reflectance distribution function autocovariance function or ACF structure function spatial frequencies Strehl parameter or Rayleigh index ordinary Bessel function modified Bessel function illuminated length lengths of illuminated area fractal index incident or scattered power Fresnel reflection coefficients radar cross-section two-sided power spectral density (PSD) one-sided profile PSD topothesy topographic surface roughness surface profile dimensionalities ensemble average linear-polarization parameters correlation length root-mean-square roughness lag variable 8.1

8.2

PHYSICAL OPTICS

8.2

INTRODUCTION Imperfect surface finish degrades optical performance. The connection between the two is electromagnetic scattering theory, which is the subject of this chapter. Because of space limitations we limit our attention to the elemental case of simple highly reflective surfaces such as those used in x-ray imaging and other high-performance applications. This chapter is divided into two parts. The first part discusses elementary physical optics and first-order perturbation theory of the scattering of electromagnetic waves by a randomly rough surface, ending with Eq. (35). The second part discusses the interpretation of scattering measurements in terms of surface finish models, and the related question of surface-finish specification. Because many manufactured surfaces show an inverse-power-law behavior, we have concentrated on models that show this behavior. Two types of surfaces are considered throughout—1D or grating-like surfaces, and isotropic 2D surfaces. The field of topographic surface scattering has been highly developed in radio physics, optics, and instrumentation over the past 50 years. There is a staggering volume of literature in each area, expressed in a bewildering variety of notations, which are summarized in Sec. 8.3. Important indepth texts in these fields are the early reviews of Beckmann and Spizzichino1 and the Radar CrossSection Handbook2, the more recent work of Ishimaru3, Fung4, the textbooks of Nieto-Vesperinas5 and Voronovich6, and most recently, the publications of Maradudin et al.7. Scattering measurements are discussed in the works of Stover8 and Germer9, and profile measurements using various optical techniques, including the Long Trace Profiler (LTP) are considered by Takacs et al.10 A number of important subjects have been omitted in the present review. These include instrumentation, statistical estimation, the effects of detrending, figure error, standards, and multilayer surfaces. Discussions of some of these and related subjects can be found in Chap. 11, “Analog Optical and Image Processing,” in this volume and Chap. 4, “Imaging through Atmospheric Turbulence,” Chap. 44, “Reflective Optics,” and Chap. 46, “X-Ray Mirror Metrology,” in Vol. V.

8.3

NOTATION The scattering geometry is sketched in Fig. 1. Note that the angles of incidence, Si and Sf , are always positive, and that the azimuthal angle Lf is understood to be Lf Li with Li 0. Specular reflection occurs at Sf Si and Lf 0, and backscatter at Sf Si and Lf R. The initial and final wavenumbers are ¤ sin (Qi )³ 2P ¥ ki 0 ´ L ¥ ´ ¦ cos (Qi )µ

¤ sin (Q f ) cos (J f )³ 2P ¥ kf sin (Q f ) sin (J f )´ ´ L ¥ ¥¦ cos (Qi ) ´µ

(1)

and the spatial frequency vectors are

f

k f ki 2P

¤ sin (Q ) cos (J ) ¤f ³

sin (Qi )³ f f x 1¥ ´ sin (Q f ) sin (J f ) ¥ f y´ ¥ ´ ¥ ´ L ¥¦ f ´µ ¥¦ cos (Q ) cos (Q ) ´µ f i z

¤f ³ f xy ¥ x ´ ¦ f yµ

(2)

These can be viewed as a generalization of the grating equation for first-order diffraction from a grating with the spatial wavelength d 1/f.

SURFACE SCATTERING

8.3

FIGURE 1 Coordinate system for 1D and 2D scattering. The x-y plane is the surface plane, with air/vacuum above and reflective material below. The planes of incidence and scattering are defined by the z axis and the initial and final directions of propagation.

The Jacobian relating configuration, wavenumber, and frequency space is cos (Q f )dW f

1 dk dk L 2 dfx df y k2 x y

dW f sin (Q f )dQ f dJ f

(3)

The two orthogonal states of linear polarization have a variety of notations in the literature: s (senkrecht) h (horizontal) TE E > (perpendicular) l Dirichlet (soft) boundary condition p((parallel) v (vertical) TM H ||(parallel)

(4)

l Neumann (hard) boundary condition Designation of initial (C) to final (D) states of polarization: ABA AA lB We use the optical or “p, s” notation hereafter. Circular and elliptically polarized results are obtained by taking appropriate linear combinations of these linear forms. Unpolarized results are obtained by summing over the final states of polarization and averaging over the initial states. The polarization vector is a unit vector in the direction of the electric vector. A natural sign convention is sˆ zˆ r kˆ

pˆ kˆ r sˆ

kˆ sˆ r pˆ

(5)

but this is not universal. One can make a distinction between scattering from a surface that is rough in both the x and y directions, and a grating like surface that is rough only along the x axis. The first generates a 2D or bistatic scattering pattern in the upper hemisphere, while the second scatters only into an infinitesimal region about the plane of incidence. Randomly-rough 1D surfaces are of special interest for research purposes since they are easier to fabricate with prescribed statistical properties than are 2D surfaces.11–13 In the following discussions include 1D and 2D results side by side, wherever practical.

8.4

PHYSICAL OPTICS

The distribution of the reflected and scattered power has a wide variety of notations in the optics and radar literature. For a 2D scatterer 1 ¤ dP ³ Pi ¥¦ dW f ´µ

§ E 1 f lim ¨r 2 Ao cos (Qi ) r lc ¨ Ei A lB © ( 2 D)

2¶

· · ¸

( 2 D) ˆ (k f , kˆ i ) f BA

2

Ao cos (Qi )

2 D) ˆ ( 2 D) ˆ (k f , kˆ i ) G BA S B(2AD)(kˆ f , kˆ i ) RCS(BA (k f , kˆ i ) Ao cos (Qi ) 4P Ao cos (Qi ) 4P cos (Qi )

(6)

ˆ ˆ D) ˆ ( 2 D) ˆ DSC(A2l B (k f , k i ) cos (Q f ) BRDFA l B (k f , k i ) cos (Qi )

Ao D{R}(2 D) IAB (Q)(2 D) L2

and for a 1D scatterer 1 ¤ dP ³ Pi ¥¦ dQ f ´µ

§ E 1 f lim ¨r Lo cos (Qi ) r lc ¨ Ei A lB © (1D)

f (1D)(kˆ f , kˆ i ) · 2 BA · Pk Lo cos (Qi ) ¸

2¶

2

(1D) ˆ 1D) ˆ (1D) ˆ S BA (k f , kˆ i ) RCS(BA (k f , kˆ i ) G BA (k f , kˆ i ) Lo cos (Qi ) 2P Lo cos (Qi ) 2P cos (Qi )

(7)

ˆ ˆ D) ˆ (1D) ˆ DSC(A1l B (k f , k i ) cos (Q f ) BRDFA l B (k f , k i ) cos (Qi )

Lo D{R}(1D) IAB (Q)(1D) L

In Eqs. (6) and (7), Pi is the power incident on the surface; that is, Ao cos (SK) |Ei|2/(2 Jo), where Ao is the total illuminated area of the surface for a 2D scatterer, and Ao 1 rLo for a 1D or grating-like scatterer. The f ’s here are the scattering amplitudes for spherical waves in 2D and cylindrical waves in 1D. U is the bistatic cross-section, RCS is the bistatic radar cross-section, I is the bistatic scattering coefficient, DSC is the differential scattering cross-section, and BRDF is the all-important radiometric quantity, the bidirectional reflectance distribution function. This is the quantity we will focus on in the detailed calculations described later in the chapter. The cross-sections U and RCS have the dimensions of area and length or “width.” In practice, these are frequently converted to dimensionless forms by normalizing them to the appropriate power of the radiation wavelength or some characteristic physical dimension of the scattering object. On the other hand, I and BRDF are dimensionless. The former is more appropriate for isolated objects, and the latter for distributed surface roughness. The I ’s used here are the radar crosssection normalized to the illuminated surface area or length, although alternative definitions appear in the literature. In Eq. (7), D{T} and I(Q) are quantities used in the books of Beckmann and Spizzichino1 and Nieto-Vesperinas.5 At this point we do not distinguish deterministic and ensemble-average quantities, or coherent and incoherent scattering. These distinctions are discussed with respect to specific model calculations later in the chapter. These full bistatic-scattering expressions may contain more information than is needed in practice. For example, scatterometry measurements8,9 are usually performed in the plane of incidence (Lf 0 or R), and surface isotropy is checked by rotating the sample to intermediate values of L between measurements. Similarly, the radar literature is frequently concerned only with measurements in the backscatter (retroscatter) direction, (Sf Si, Lf R).

SURFACE SCATTERING

8.5

In principle, relating the BRDF to topographic surface features is a straightforward application of Maxwell’s equations, but approximations are frequently necessary to get practically useful results. The best known of these approximations are the physical-optics (Fresnel-Kirchhoff) and the smallperturbation (Rayleigh-Rice) methods. These are discussed in Secs. 8.4 and 8.5.

8.4 THE FRESNEL-KIRCHHOFF APPROXIMATION Introduction The Fresnel-Kirchhoff or physical-optics approximation is also known as the tangent-plane or Huygens wave approximation. It is valid when the surface roughness has a large radius of curvature, that is, it is locally flat, but places no direct restrictions on the surface height or slope. It is an inherently paraxial approximation, and in its simplest form discussed below [Eq. (8)] omits multiple scattering and shadowing effects. As a result, it does not generally satisfy the conservation of energy. In this approximation the ensemble-average value of the BRDF is in 2D: BRDFA(2lDB) ®FK

1 1 Q FK )(f xy )(2 D) ® L 2 cos (Qi )cos (Q f ) A lB (8a)

( 2 D)

)(f xy )

§ 1 ¶ ® ¯ dT xy exp [i 2P f xy T xy ] exp ¨ (2P f z )2 D(T xy )®·

c 2 © ¸ c

and in 1D: 1 1 Q FK )( f x )(1D) ® L cos (Qi )cos (Q f ) A lB

BRDFA(1lD)B ®FK

(8b) (1D)

)( f x )

§ 1 ¶ ® ¯ dT x exp[i 2P f x T x ] exp ¨ (2P f z )2 D(T x )®·

c 2 © ¸ c

In these expressions the Q carries the information on the polarization of the radiation and the properties of the surface material, and the real quantity carries the information about the statistical properties of the surface roughness. This occurs through the dependence on the ensemble- average surface structure function, D(T xy )® D(| T xy |)® [Z (rxy T xy ) Z (rxy )]2 ®

(9)

where rxy is the position vector in the surface plane. The existence of requires that the surface roughness have statistically stationary first differences. In the limit of a perfectly smooth surface, D(V) vanishes and the *’s become a delta function in the specular direction. The exponential dependence on the roughness structure function in Eq. (8) is a consequence of the usual assumption that the height fluctuations, Z(x, y), have a gaussian bivariate distribution. Nongaussian effects, which become manifest only for rough surfaces, are discussed in the literature.11,13 The polarization-materials factor can be written approximately as 2

QAFKlB AAFKlB RA lB (Q ) AsFK AFK ls pl p

(1 cos (Qi ) cos (Q f )) cos (J f ) sin (Qi ) sin (Q f )

AsFK AFK sin (J f ) pl s lp

cos (Qi ) cos (Q f )

(10)

8.6

PHYSICAL OPTICS

Although these results are correct for a perfect reflector, R 1, the dependence on the surface reflectivity for an imperfect reflector is quite complicated.2,4 Here we follow custom by arbitrarily factoring out a reflection coefficient R in the first line, where the R values are the Fresnel intensity reflectivities, RA lA (Q ) | rA (Q )| 2

RA lB (Q )

1 (r (Q ) rB (Q )) 2 A

2

(11)

and the r’s are the amplitude reflection coefficients rs (Q )

M cos (Q ) ME sin2 (Q )

rp (Q )

M cos (Q ) ME sin2 (Q )

E cos (Q ) ME sin2 (Q ) E cos (Q ) ME sin2 (Q )

(12)

For simplicity in presentation we cavalierly evaluate these reflectivity factors at the local angle of incidence. That is, R(S) R(Sloc), where Sloc \Sf cos(Lf ) Si \/2 in the plane of incidence.2 The material parameters G and O are the electric permeability and magnetic permittivity of the surface relative to vacuum. In the case of a perfect electrical conductor (PEC), Glc and the R’s are unity. In the forward-scattering direction, Lf2 —the ensemble-averages of the 1D and 2D power spectral densities of the surface roughness. These are the Fourier transforms of the corresponding covariance functions: c

S( f x )(1D) ® ¯ dT x exp [i 2P f x T x ] C(T x )®

c

c

(18)

S(f xy )(2 D) ® ¯ dT xy exp [i 2P f xy T xy ] C(T xy )®

c

These are discussed in detail in Sec. 8.7. As a final note, the incoherent terms reduce to somewhat simpler forms when observations are made in the forward direction, Lf2 > 1, the coherent term vanishes for a gaussian bivariate distribution, and we revert to Eq. (8) which then involves the limit of the scattering integral: lim § 1 ¶¹ ª c dT exp [i 2P f xy T xy ] exp ¨ (2P f z )2 D(T xy )®·º D(T xy )® l c «¯ c xy 2 © ¸» ¬

(20)

8.8

PHYSICAL OPTICS

which is determined by the indicial behavior of . In the special case where the structure function is isotropic and quadratic lim 2 D(T xy )® S M T xy2 T lo

(21)

where UM is the dimensionless root mean square (rms) value of the surface gradient of a 2D surface or the rms value of surface slope of a 1D surface. It is easy to see that this quadratic dependence leads to values of that are proportional to N2 in 2D and N1 in 1D, in which case the is independent of the radiation wavelength. In other words, a quadratic structure function leads to geometric-optics results. In fact, the resulting scattering pattern is a mapping of the slope distribution of the surface roughness, including the doubling of the deflection angle on reflection.1,2 The form of the scattering pattern depends on the form of the bivariate distribution involved. For example, a gaussian bivariate distribution leads to a gaussian pattern, while a gamma distribution leads to scattering patterns involving modified Bessel functions.13 If the indicial behavior of is not quadratic, the form of the ’s depend on N and the elegant geometrical-optics limits are not achieved. Mathematically, this occurs because such surfaces are not differentiable at V0 and so have no well-defined “slope.” Fractal Surfaces Introduction

Fractal surfaces have structure functions of the form D(T xy )® fractal T

2

T xy

N

T

(22)

0 N 2

where T is a length parameter called the topothesy.16,17 Physically, T is the separation of surface points whose connecting chord has an rms slope of unity. Because Eq. (22) is proportional to T 2-N where N < 2, a perfectly smooth surface occurs when T l0. On the other hand, because N 2 is excluded, very rough fractal surfaces do not lead to a geometrical-optics result. The number N is the fractal index, which is related to the Hausdorff or Hausdorff-Besicovitch dimension D`,16,17 D ` (4 N )/2 1 D` 2

N 4 2D` (23)

0 N 2

D` can also be expressed in terms of the Hurst dimension or coefficient, but the connection depends on the dimensionality of the problem.18 Fractal Forms of the Scattering Integrals In the case of statistically isotropic surfaces, structure functions of the form of Eq. (22) lead to expressions for the incoherent scattering integrals which may be written as § T c xy )(f xy )(2 D) ® 2P ¯ T xy dT xy J 0 (2P f xyT xy ) S 2 exp ¨ 0 ¨ R © § T c )( f x )(1D) ® ¯ dT x exp [i 2P f x T x ] S 2 exp ¨ x

c ¨ R ©

N¶

· · ¸

(24)

N¶

· · ¸

where

S 1 (dimensionless)

§ ¶ 2 R¨ 2· ( 2 P f T ) z © ¸

1/ N

T

(25)

SURFACE SCATTERING

8.9

are “pseudo” roughness and correlation length parameters. These do not play true physical roles in the case of fractals, but are artificial quantities introduced here to enable us to write Eq. (24) in form of a Fourier transform of a covariance function that will appear later as the power exponential (PEX) model. The integrals in Eq. (24) are symmetric bell-shaped functions of fxy and fx that are flat at low spatial frequencies and fall off with inverse-power-law tails at high spatial frequencies. The 2D expression cannot be expressed in terms of known functions, while the 1D form can be written in terms of centered symmetric Lévy stable distributions of order C, that is, LC(X), where:17 )( f x )(1D) ® 2P S 2 R LN (Y )

Y 2P f x R

(26)

Nolan19 gives the computer program “STABLE” for this in terms of general stable distribution functions fNolan: )( f x )(1D) ® 2P S 2 R f Nolan (x |A , B , G , D , k)

(27)

x | f x |, A N , B 0, G 1 /(2P R), D 0, k 0 or 1 On the other hand, the low- and high-frequency limits of the 1D and 2D forms can be expressed in simple closed form. In the 1D case: )(| f x | 1/2P R)(1D) ® S 2 R 2 '((1 N )/N ) )(| f x | 1/2P R)(1D) ® S 2 R 2 N P

'((1 N )/2) N '((2 N )/2) (2P | f x |R)N 1

(28a)

and in the isotropic 2D case: )(| f xy | 1/2P R)(2 D) ® S 2 R 2 P '((2 N )/N ) )(| f xy | 1/2P R)(2 D) ® S 2 R 2 2 N 1 P

N '((2 N )/2) '((2 N )/2) (2P | f xy | R)N 2

(28b)

It follows from the Fourier transform nature of Eq. (8) that the areas under each of these curves in frequency space is simply U 2. This plus the asymptotic properties of the ’s given above capture most of the physical properties of the of interest. The case N 1, D` 3/2 is called the Brownian fractal, which falls between the Cantor set (D` 0.63093 . . .) and the Sierpinski gasket (D` 1.5850 . . .). The Brownian fractal has the virtue of leading to the simple analytic expressions for the diffraction integrals valid for all spatial frequencies: )( f x )(1D) ® N 1

2S 2 R 1 (2P f x R)2

)( f xy )(2 D) ® N 1

2P S 2 R 2 [1 (2P f xy R)2 ]3/2

(29)

where, for fractal surfaces, U and T are given by Eq. (25).

8.5 THE RAYLEIGH-RICE (RR) OR SMALL-PERTURBATION APPROXIMATION Results The small-perturbation method is an alternative to the Fresnel-Kirchhoff method discussed above. Its first-order form was originally derived by Rice20 using the Rayleigh hypothesis, and hence the name Rayleigh-Rice. Peake2,21 was the first to derive the expression for an arbitrary

8.10

PHYSICAL OPTICS

surface material and the results have been rederived many times in the literature. These RayleighRice results have been extended to higher orders22–24, and in its more general form is called the small-perturbation method. The lowest-order perturbation theory results are2 D) BRDF(f xy )(A2l B

D) BRDF( f x )A(1l B

RR incoherent RR incoherent

16P 2 cos (Qi ) cos (Q f ) QARRlB S(f xy )(2 D) ® L4

(30)

16P 2 3 cos (Qi ) cos (Q ) f QARRlB DA ,B S( f x )(1D) ® L

The RR form of the coherent term is complicated and can be found in the literature.5,25 The Q’s are the material-polarization factors similar to those appearing in the FK calculations. In contrast with the approximation made in the FK case, however, they do not separate into distinct angular and reflectivity factors. In particular,2 QARRlB AARRlB AsRR ls

2

[M 1][B(Qi )B(Q f )cos (J f ) M sin (Qi )sin (Q f )] M 2 (E 1)cos (J f ) [B(Qi ) M cos (Qi )][B(Q f ) M cos (Q f )]

E (M 1)B(Qi ) M (E 1)B(Q f )

AsRR lp

[B(Qi ) M cos (Qi )][B(Q f ) E cos (Q f )]

M(E 1)B(Qi ) E(M 1)B(Q f )

ARR pl s

[B(Qi ) E cos (Qi )][B(Q f ) M cos (Q f )]

ARR pl p

sin (J f )

(31)

sin (J f )

(E 1)[E sin (Qi )sin (Q f ) B(Qi )B(Q f )cos (J f )] E 2 (M 1)cos (J f ) [B(Qi ) E cos (Qi )][B(Q f ) E cos (Q f )]

where B(Q ) E M sin 2 (Q ) and index of refraction E M . The full G–O dependencies are useful for checking the duality of the results. The QRR’s are closely related to the Fresnel reflection coefficients in Eqs. (11) and (12). For nonmagnetic materials, O1 and QsRR Rs (Qi )Rs (Q f ) cos 2 (J f ) ls (32) QARRlB (specular) RA (Qi ) DA ,B On the other hand, for a perfectly reflecting (PEC) surface,

QsRR ls

2

cos (J f )

QsRR lp

§ sin (J f ) ¶ ¨ · ¨© cos (Q f ) ¸·

§ cos (J f ) sin (Qi ) sin (Q f ) ¶ Q RR ¨ · pl p cos (Qi ) cos (Q f ) ·¸ ¨©

2

2

(33) § sin (J f ) ¶ Q pRRls ¨ · ¨© cos (Qi ) ·¸

SURFACE SCATTERING

8.11

In the special case of a perfectly reflecting surface measured in the plane of incidence we get the elegant results: RR

BRDFA(2lDB)

incoherent RR

BRDFA(1lD)B QsRR 1 ls

incoherent

16P 2 cos (Qi ) cos (Q f ) QARRlB S( f x , 0)(2 D) ® L4

16P 2 cos (Qi ) cos (Q f ) QARRlB DA ,B S( f x )(1D) ® L3

QsRR Q RR 0 lp pl s

¤ 1 sin (Qi ) sin (Q f )³ Q RR ´ pl p ¥ ¦ cos (Qi ) cos (Q f ) µ

(34)

2

which may be compared with the FK result in Eq. (19) with R(S) 1. Again, the reader is alerted to the fundamental distinction between <S(fx, 0)(2D)> and <S(fx)(1D)>, appearing here. They may be mathematically related for an isotropically rough surface, but they are never equal. Comparison of RR and FK Results The Rayleigh-Rice results are inherently a smooth-surface approximation for a statistically stationary random surface, so that the proper comparison is with the Fresnel-Kirchhoff results for slightly rough surfaces in Eq. (17) and the RR results in Eq. (30). In the limit of paraxial scattering, Sf ySI and Lf y, the two sets of results become identical and can be written in the common form:

BRDFA(2lDB)

BRDFA(1lD)B

ª 1 A B¹ 2 16P 2 2 l 4 cos (Qi ) RA (Qi ) «¤ sin (J f )³ º S( f x , f y )(2 D) ® incoherent L A B w ¥ ´ cos (Q ) i µ » ¬¦ FK, RR

FK, RR incoherent

l

(35a)

16P 2 cos 2 (Qi ) RA (Qi ) DA ,B S( f x )(1D) ® L3

where RC(Si) is given by Eq. (11). In the case of fractal surfaces the paraxial results are

BRDFA(2lDB)

ª A B¹ 1 2 1 l 2 RA (Qi ) «¤ sin (J f )³ º )( f x , f y )(2 D) ® incoherent L ¥ cos (Q ) ´ A w B i µ » ¬¦ FK, RR

(35b)

1 R (Q ) D )( f x )(1D) ® L A i A ,B where the ’s are given in Eqs. (24) et seq. Later, in “The J-K model” section we discuss the highfrequency forms of the that follow from these results. A nice feature of the paraxial results in Eq. (35) is that they satisfy the conservation of energy. In the case of statistically stationary surfaces this occurs via BRDFA(1lD)B

FK, RR

incoherent

dP

l

L2

¯ dW s dW s R(Qi ) ¯ dfxy BRDF

( 2 D) FK, RR ® total

(1 g ) g 1

L dP (1D) RR (1 g ) g 1 tottal ¯ dQs dQs R(Qi ) ¯ df x BRDF ®FK,

(36)

8.12

PHYSICAL OPTICS

where the terms (1–g) and g come from the coherent and incoherent components. The differences between the FK and RR results that appear at larger deflection angles are attributed to the inherently paraxial approximation of the FK calculations. In particular, the FK results for rougher surfaces do not satisfy the conservation of energy due to their neglect of multiple scattering and shadowing effects.26 On the other hand, the RR or lowest-order perturbation theory results are known to have intrinsic limitations at grazing angles, especially for p-polarized radiation.27 The RR results also may not satisfy the conservation of energy because of the phenomenon of roughness-induced absorption.28

8.6

EFFECTS OF FINITE ILLUMINATION AREA The discussion above has assumed that the illuminated surface area is infinite, which is the meaning of the limits o c in the integrals, , appearing in Eqs. (8) et seq. In general, the effects of the finite rectangular illumination area, Lx r Ly, can be taken into account by convolving the infinite-illumination forms of the with the system-response (SR) or point-spread function 2

Lx L y § sin(P f x Lx ) ¶ § sin(P f y L y ) ¶ ¨ · L 2 ¨© P f x Lx ·¸ ¨© P f y L y ·¸

System response ( f x , f y )

2

(37)

which becomes F(fx)F(fy) in the limit Lx, Ly >> N.29 When the L’s are much larger than the correlation length of the surface roughness, the incoherent scattering pattern is a broad and “smooth” function of spatial frequency and is unaffected by the convolution with the relatively sharp system response function. This argument breaks down when there are sharp features in the . An example of this occurs in the smooth-surface limit of statistically stationary surfaces, which exhibit a delta-function coherent-scattering peak as in Eq. (17). That delta function is then smeared into the expected sinc2 pattern by convolution with the system response. Fractal scattering does not display a separable coherent component but does become increasingly bunched in the specular direction when the topothesy is sufficiently small. In that limit the observed scattering is again affected significantly by convolution with the system response. The non-vanishing width of the system response function plays an important role in the discussion of surface-finish specification in Sec. 8.8.

8.7

SURFACE STATISTICS

Second-Order Statistical Functions The ensemble-average scattered power can be written in terms of three second-order statistical functions: the structure function, , the surface autocovariance function (ACF), , and the power spectral density (PSD) <S(f )>. Equation (18) gives the PSD in terms of the ACF, but it can also be written directly in terms of the surface profile according to S( f x )(1D) ®

lim 1 Lo l c Lo

Lo / 2

¯ L /2 o

2

dx exp[i 2P f x x] Z (x ) (38)

lim 1 S(f xy )(2 D) ® Ao l c Ao

2

¯A drxy exp [i 2P fxy rxy ] Z (rxy ) o

SURFACE SCATTERING

8.13

where c is [L4], while <S(1D)> is [L3]. What is not true is that the 1D form is a simple slice of the 2D form; that is, <S(fx)(1D)> does not equal <S(fx, 0)(2D)>. Instead, the 1D form can be derived from the 2D form by integration, c

S( f x )(1D) ® ¯ df y S( f x , f y )(2 D) ®

c

(41)

but the 2D form cannot be derived from the 1D form without providing further information about the 2D form. This is usually given in terms of its symmetry properties. Incidentally, it follows from Eqs. (35) and (41) that the 1D scattering pattern equals the 2D pattern integrated over an long, narrow slit parallel to the y axis in Fig. 1. If the surface is statistically isotropic, <S(1D)> and <S(2D)> are related by the integral transforms S( f xy )(2 D) ®

S( f x )(1D) ® 2 ¯

1 P

c

¯f

xy

df x d S( f x )(1D) ® f x2 f xy2 df x

c

f xy df xy

fx

f xy2 f x2

(42) S( f xy )(2D) ®

The first is the inverse-Abel or “half derivative” transform and the second is the Abel transform or “half integral” transform.31

8.14

PHYSICAL OPTICS

The transforms in Eq. (42) are useful since they allow profile measurements, which are inherently 1D, to be translated into the 2D spectra which appear in scatterometry and practical applications.32 They also permit the transformation of the high-frequency behavior of the spectra of one dimensionality to be transformed into the high-frequency behavior of the other without knowledge of their low-frequency behavior.32 An important example of this is the case of inverse-power-law high-frequency tails, S( f xy )(2 D) ®

J f xym

S( f x )(1D) ®

'[1/2]'[(m 1)/2] J K '[m/2]

K | f x |m 1

(43)

m2

which is the essence of the so-called J-K model discussed in the “The J-K Model” section.

Finish Models General Remarks The magnitudes of the spatial frequencies |f | appearing in the PSDs discussed above cover the range from 0 to c. Real world measurements, however, include only a lesser range of spatial frequencies, fmin, is C(T )® ABC 2P

2 A 2 C /2 ¤ | T |³ 2P ´ B '[C /2] ¥¦ Bµ

(C 1)/ 2

¤ | T |³ K (C 1)/2 ¥ 2P ´ Bµ ¦

C 0

(45)

where Kn is a modified Bessel function. 33 For this reason the “ABC” model is also called the K-correlation model.30 When C 2, 4, 6, . . . , it reduces to simple algebraic expressions. For C 2, and 4, for example,

C(T )® ABC |C 2 P

§ A | T |¶ exp ¨ 2P · B B¸ ©

(46)

§ P A | T |¶ § | T |¶ C(T )® ABC |C 4 exp ¨ 2P · ¨1 2P · 2 B B¸ © B¸ © where the C 2 form is the well-known two-sided exponential. When C lc the covariance function in Eq. (45) becomes gaussian. The PEX Model Another three-parameter model with an inverse power-law power spectrum is the power exponential model: §T C(T )®PEX S 2exp ¨ ¨© R

N¶

· ·¸

0 N 2

(47)

This model is identical with the ABC model for N 1 and C 2, but otherwise they are different. Some of the mathematical properties of the 1D and 2D spectra of the PEX model have been given earlier in connection with fractal surfaces. In particular, Eqs. (28a) and (28b) give the low- and high-frequency forms, and Eq. (29) gives explicit results for N 1. Note that the ABC model only requires that C > 1 to be statistically stationary. This admits faster high-frequency falloffs than are permitted for the fractal and PEX models, which translate to 1 < C < 3. This means that surfaces with C q3 only fit the ABC model. The J-K Model In the limit of long correlation lengths, the ABC, PEX, and fractal model each reduces to what we call the J-K model, defined in Eq. (43). This can be viewed as a new two-parameter finish model, where the magnitudes of J and m, or equivalently, K and n, are determined directly from experimental data without reference to any underlying model. On the other hand, the values of J and K can be written in terms of the parameters of the ABC, PEX, or fractal models leading to these high-frequency forms. For the fractal surfaces, for example, K

m 2 '[1/2]'[(m 1)/2] 4 m T '[(4 m)/2] 4 P m 1

S( f x )

K | f x |m 1

(48)

where 2 < m < 4. This allows the topothesy T of a fractal scatterer to be determined from its highfrequency tail. The observation of an inverse-power-law spectrum over a limited bandwidth does not guarantee that the surface is fractal. To do that one would have to observe low-frequency behavior conforming to Eqs. (24) and (25) as well, and confirm the unique wavelength and angular dependencies they imply.34 Until then, power-law scattering over a finite bandwidth can at best be called “fractal-like” rather than fractal.

8.16

8.8

PHYSICAL OPTICS

SURFACE FINISH SPECIFICATION

Performance Measures It is desirable to express performance requirements on reflecting surfaces in terms of surface-finish parameters that can be measured by scatterometry or profilometry. One can concoct a wide variety of image-based performance measures.35 For example, earlier we invoked an ambitious measure of image degradation based on the reduction of the on-axis image intensity, rather than the simpler measure based on the integrated image intensity suggested below.36,37 The simplest practical measure appears to place an upper limit on the average total scattering outside the specular core, E FK, RR ®(2 D) E

FK, RR (1D)

®

c 1 L2 c df xy BRDFA(2lDA) ®FK, RR dW f cos (Q f )BRDFA(2lDA) ®FK, RR R (Qi ) ¯outside R(Qi ) ¯outside core core

c 1 L c df x BRDFA(1lDA) ®FK, RR dQ f cos (Q f )BRDFA(1lDA) ®FK, RR R ( R(Qi ) ¯outside Qi ) ¯outside core core

(49)

For small correlation lengths the BRDF is flat at low frequencies, the omitted parts of the integrations in Eq. (49) can be neglected, and so 2

¤ S³ E FK, RR ®(2 D) E FK, RR ®(1D) ¥ 4P cos (Qi ) ´ g Lµ ¦

(50)

where U is the intrinsic rms roughness of the surface given by Eq. (40). In the earlier literature this result is called the total integrated scatter (TIS). This simple and beautiful expression for the image error permeates the wave-optics and microwave literature.15 In the limit of large correlation lengths, which is required to display inverse-power-law behavior, the BRDFs appearing in Eq. (49) diverge at low spatial frequencies. This leads to very different forms for the ’s, which now depend on the bandwidth limits included in the error calculation. In the J-K language of Eq. (43), 2

2

¤ cos (Qi )³ 2P J Lmxy 2 E (R2DL) ® ¥ 4P xy L ´µ m 2 ¦

¤ cos (Qi )³ 2 K Lmx 2 E (R1DL) ® ¥ 4P x L ´µ m 2 ¦

(51)

where Lxy and Lx are the radii of the excluded regions in Eq. (49). Note that for statistically stationary surfaces the surface becomes perfectly smooth when Ul0, while for fractal surfaces this occurs when T l0. Numerical Illustration In an earlier paper we discussed the measurements of a silicon cylinder made with the LTP profiling instrument at Brookhaven National Laboratory.37 These were fitted to the two-sided profile spectrum S( f x )(1D)

3.32 r 10 9 K M m3 m 1 | fx | | f x |1.61

10 5 | f x | 10 1 Mm 1

(52)

If fractal, this corresponds to T 8.36 r10 3 nm. If the surface is isotropic, the corresponding (2D) spectrum is S( f xy )(2 D)

J f xym

1.45 r 10 9 Mm4 f xy2.61

(53)

SURFACE SCATTERING

8.17

These allow us to evaluate the errors associated with the use of this mirror in 1D and 2D geometries. If N10( 4) Mm (0.1 nm), S(R/2) 10 3 (i.e., 1 mrad glancing incidence), and Lx Lo 0.1 m, E (2 D) ® 26.5%

E (1D) ® 19.3%

(54)

The corresponding errors at N0.6328 Mm and at normal incidence are smaller than these by a factor of 0.025. These results indicate that for the parameters used, this mirror would perform marginally well as a glancing-incidence x-ray mirror, but very well as a normal-incidence mirror at visible wavelengths, both in 1D and 2D applications. Statistical Fluctuations The discussion to this point has been concerned with ensemble average quantities . Individual, deterministic measurements of U will fluctuate about this average by an amount measured by the dimensionless quantity

G U2

(U U ®)2 ® U 2 ®

1 U ® 2 U ® 2

(55)

The limit IU l0 indicates a perfect measurement while IU l1 is particularly bad. An important example of this is the inverse-power-law sum that appears in the periodogram and effective mean-square roughness in Eqs. (49) through (51) for the J-K model: c

U (n) £ j 1

1 jn

GU

Z (2n) Z (n)

(56)

where Z(n) is the Riemann zeta function. For the brownian fractal, n 2, for example, IU (2/5) 0.632 . . . . The Tchebycheff inequality then indicates that the sum U may be rather broadly distributed about except for n near unity. IU vanishes when n l1 because the number of degrees of freedom included in the sum becomes infinite, and it approaches unity in the opposite limit of n lc, since the sum then includes only a single term with two degrees of freedom, or a “single speckle.” If the fluctuations in are unacceptably large, the lowest-order statistical description must be replaced by a deterministic version. Steps in this direction have been taken by Mori et al., Mimura et al., and Yamauchi et al.38–40, for very high-performance x-ray mirrors. They have reported very accurate profile measurements of highly polished mirrors which they have correlated with image quality measurements made using the 1-km beam line at SPring-8. Unfortunately, the discussion of this interesting and important work lies outside the scope of the present chapter.

8.9

RETROSPECT AND PROSPECT This chapter outlines methods for understanding the performance of mirror surfaces in terms of statistical models of their surface topography. We illustrate this making use of simple models that exhibit an inverse-power-law fall off at high spatial frequencies. Obvious follow-on steps are to expand the database, to relate residual roughness to finishing methods, to explore different and composite models, and examine measurement errors and instrumental effects in metrology.41–44 It is pointed out that the utility of certain statistical parameters may be limited by broad confidence limits. This suggests that a deterministic approach may be desirable for demanding applications, especially involving one-of-a-kind surfaces. Even so, the statistical approach provides deep insight into the physics involved, offers a pre-screening methodology, and will remain the lingua franca of this wide and diverse field in the future.

8.18

8.10

PHYSICAL OPTICS

REFERENCES AND ENDNOTES 1. P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces, Pergamon Press, New York, NY, 1963. 2. G. T. Ruck, D. E. Barrick, W. D. Stuart, and C. K. Krichbaum, Radar Cross Section Handbook, 2 vols., Pergamon Press, New York, NY, 1970. Especially D. E. Barrick, Chap. 9, “Rough Surfaces.” 3. A. Ishimaru, Electromagnetic Wave Propagation, Radiation and Scattering, Prentice-Hall Inc., Englewood Cliffs, NJ, 1991; A. Ishimaru, Wave Propagation and Scattering in Random Media, 2 vols., Academic Press, New York, NY, 1978. 4. A. K. Fung, Microwave Scattering and Emission Models and their Application, Artech House, Boston, MA, 1994. 5. M. Nieto-Vesperinas, Scattering and Diffraction in Physical Optics, 2d ed., World Scientific, Hackensack, NJ, 2006. 6. A. G. Voronovich, Wave Scattering from Rough Surfaces, 2d ed, Springer Verlag, New York, NY, 1999. See also Chaps. 2 and 4 in Ref. 7 below. 7. A. A. Maradudin, ed., Light Scattering and Nanoscale Surface Roughness, Springer Verlag, New York, NY, 2007. 8. J. C. Stover, Optical Scattering; Measurement and Analysis, 2d ed., SPIE: Optical Engineering Press, Bellingham, WA, 1995. 9. T. A. Germer, “Measuring Interfacial Roughness by Polarized Optical Scattering,” in Ref. 7 above, and references therein. 10. P. Z. Takacs, S. K. Feng, E. L. Church, S. Qian, and W. –M. Liu, “Long Trace Profile Measurements on Cylindrical Aspheres,” Proc. SPIE 966:354–364, 1988; see also P. Z. Takacs, K. Furenlid, R. DeBiasse, and E. L. Church, “Surface Topography Measurements over the 1 Meter to 10 Micrometer Spatial Period Bandwidth,” Proc. SPIE 1164:203–211, 1989. 11. K. A. O’Donnell and E. R. Mendez, “Experimental Study of Scattering from Characterized Random Surfaces,” J. Opt. Soc. Am. A 4:1194–1205, 1987; See also Chap. 9 in Ref. 7 above. 12. E. R. Mendez and D. Macías, “Inverse Problems in Optical Scattering,” Chap. 16 in Ref. 7 above. 13. M. J. Kim, E. R. Mendez, and K. A. O’Donnell, “Scattering from Gamma-Distributed Surfaces,” J. Mod. Opt. 34:1107–1119, 1987. 14. H. Cramér and M. R. Leadbetter, Stationary and Related Stochastic Processes, Dover Pubs. Mineola, NY, 2004. 15. M. Born and E. Wolf, Principles of Optics, 4th ed., Pergamon, New York, NY, 1970. Chap. IX, Sec. 9.1 derives the deterministic “Strehl” criterion for focusing optics. These considerations are very well known to the visible-optics community. X-ray-opticians, on the other hand, often refer to rms slope errors, which is the square root of the bandwidth-limited value of the second moment of the power-spectral density. 16. M. V. Berry, “Difractals,” J. Phys. A: Math. Gen. 12:781–797, 1979. 17. I. Simonsen, D. Vandembroucq, and S. Roux, “Wave Scattering from Self-Affine Surfaces,” Phys. Rev. E 61:5914–5917, 2000; see also ibid “Electromagnetic Wave Scattering from Conducting Self-Affine Surfaces: An Analytic and Numerical Study,” J. Opt. Soc. Am. A 18:1101–1111, 2001. 18. B. B. Mandelbrodt, The Fractal Geometry of Nature, Freeman, New York, NY, 1975. The connection between the Hausdorff and Hurst dimensions is given as D` H` 1 dimensionality. We prefer the descriptor D` because of its direct connection with the geometric scaling properties of the profile. See Ref. 16 for specifics. 19. J. P. Nolan, An Introduction to Stable Distributions, to be published. The program STABLE is available from J. P. Nolan’s Web site: academic2.american.edu/~jpnolan. 20. S. O. Rice, “Reflection of Electromagnetic Waves from Slightly Rough Surfaces,” Commun. Pure Appl. Math. 4:361–378, 1951. 21. W. H. Peake, “Theory of Radar Return from Terrain,” IRE Conv. Rec. 7(1):34–41, 1959. 22. K. A. O’Donnell, “Small-Amplitude Perturbation Theory for One-Dimensionally Rough Surfaces,” Chap. 5 in Ref. 7; see also K. A. O’Donnell, “High Order Perturbation Theory for Light Scattering from a Rough Metal Surface,” J. Opt. Soc Am. A 18:1507–1518, 2001. 23. G. Berginc, “Small-Amplitude Perturbation Theory for Two-Dimensional Surfaces”, Chap. 6 in Ref. 7 above. 24. J. T. Johnson, “Third Order Small Perturbation Method for Scattering from Dielectric Rough Surfaces,” J. Opt. Soc. Am. 16:2720–2736, 1999; see also J. T. Johnson, “Computer Simulations of Rough Surface Scattering,” Chap. 7 in Ref. 7 above.

SURFACE SCATTERING

8.19

25. G. R. Valenzuela, short communication, Trans. IEEE, ca. 1972. Precise title and reference unavailable. 26. L. Tsang, J. A. Kong, and R. T. Shin, Theory of Microwave Remote Sensing, John Wiley and Sons, New York, NY, 1985. 27. D. Barrick and R. Fitzgerald, “The Failure of ‘Classic’ Perturbation Theory at a Rough Neumann Boundary near Grazing,” IEEE Trans. On Antennas and Propagation 48:1452–1460, 2000. 28. E. L. Church and J. C. Stover, “Roughness Effects on Emissivity,” unpublished notes, 1997. 29. The literature frequently uses a gaussian system response function, which leads to algebraic simplifications with no essential loss of generality. 30. E. L. Church and P. Z. Takacs, “The Optimal Estimation of Finish Parameters,” Proc. SPIE.1530:71–85, 1992. 31. A. Erdelyi, Higher Transcendental Functions, The Batemann Manuscript Project. Tables of Integral Transforms, 2 vols., McGraw-Hill, New York, NY, 1953. In particular, q.v. Weyl fractional calculus. 32. E. L. Church, T. A. Leonard, and P. Z. Takacs, “The Prediction of BRDFs from Surface Profile Measurements,” Proc. SPIE.1165:136–150, 1991. 33. I. S. Gradshteyn, I. M. Ryzhik, and A. Jeffrey, eds., Table of Integrals, Series and Products, 5th ed. Academic Press, Inc., Boston, MA, 1980. In particular, entries 6.565.4, 6.576.7, 6.699.12, and 8.432.5. 34. I. A. Popov, L. A. Glushchenko, and J. Uozumi, “The Study of Fractal Structure of Ground Glass Surface by Means of Angle Resolved Scattering of Light,” Optics Comm. 203:191–196, 2002. 35. E. L. Church and P. Z. Takacs, “Measures of Surface Quality,” unpublished notes, 2007. 36. E. L. Church and P. Z. Takacs, “Specification of Surface Figure and Finish in Terms of System Performance,” Appl. Opt. 32:3344–3353, 1993. 37. E. L. Church and P. Z. Takacs, “Specification of Glancing- and Normal-Incidence X-Ray Mirrors,” Opt. Eng. 34:353–359, 1995; erratum, ibid 34:3348, 1995. 38. Y. Mori, K. Yamauchi, K. Yamamura, et al., “Sub-Micron Focusing of Hard X-Ray Beam by Elliptically Figured Mirrors for Scanning X-Ray Microscopy,” Proc. SPIE. 4782:58–64, 2002, and references therein. 39. H. Mimura, K. Yamauchi, K. Yamamura, et al., “Image Quality Improvement in Hard X-Ray Projection Microscope Using Total Reflection Mirror Optics,” J. Synchrotron Rad. 11:343–346, 2004; see also, H. Mimura, H. Yumoto, S. Matsuyama, et al., “Relative Angle Determinable Stitching Interferometry for Hard X-Ray Reflective Optics,” Rev. Sci. Instrum. 76:045102, 2005. 40. K. Yamauchi, K. Yamamura, H. Mimura, et al., “Wave-Optical Evaluation of Interference Fringes and Wavefront Phase in a Hard-X-Ray Beam Totally Reflected by Mirror Optics,” Appl. Opt. 44:6927–6932, 2005. 41. J. M. Bennett, “Characterization of Surface Roughness,” Chap. 1 in Ref. 7 above. 42. E. L. Church, “Fractal Surface Finish,” Appl. Opt. 27:1518–1526, 1988. 43. V. V. Yashchuk, E. L. Church, M. Howells, W. R. McKinney, and P. Z. Takacs, “21st Century Metrology for Synchrotron Radiation Optics Is Understanding How to Specify and Charcterize Optics Completely,” SRI-2006 III Workshop on Optical Metrology, Daegu Korea, 28 May 2006. 44. V. V. Yashchuk, S. C. Irick, A. A. McDowell, W. R. McKinney, and P. Z. Takacs, “Air Convection Noise of Pencil-Beam Interferometer for Long Trace Profiler,” Lawrence Berkeley National Laboratory Notes, 2008; Also, V. V. Yashchuk, “Positioning Errors of Pencil-Beam Interferometers for Long Trace Profilers,” ibid., 2008.


9 VOLUME SCATTERING IN RANDOM MEDIA Aristide Dogariu and Jeremy Ellis CREOL, The College of Optics and Photonics University of Central Florida Orlando, Florida

9.1

GLOSSARY A C c D D E e F f G g I I J k KB KK–M la ls l

N n P q

transversal area correlation function speed of light diffusion constant degree of polarization field amplitude polarization direction scattering form amplitude forward-scattering amplitude pair correlation function asymmetry parameter field intensity Stokes vector diffuse flux wave vector Boltzman constant effective absorption coefficient absorption length scattering length transport mean free path number of particles refractive index scattering phase function scattering wavevector 9.1

9.2

PHYSICAL OPTICS

r, R S S s

9.2

SK– M T t Tab U V v G J S N Oa Os T U Y 7

vector position static structure factor scattering potential specific direction effective scattering coefficient absolute temperature time transmission coefficient diffuse energy density volume volume fraction dielectric constant shear viscosity scattering angle wavelength of light absorption coefficient scattering coefficient number density scattering cross-section frequency solid angle

INTRODUCTION Electromagnetic radiation impinging on matter induces oscillating charges that can be further regarded as secondary sources of radiation. The morphological details and the microscopical structure of the probed medium determine the frequency, intensity, and polarization properties of this re-emitted (scattered) radiation. This constitutes the basis of a long history of applications of light scattering as a characterization tool in biology, colloid chemistry, solid state physics, and so on. A substantial body of applications deals with light scattering by particles. These smaller or larger ensembles of molecules have practical implications in many industries where they are being formed, transformed, or manipulated. Since the early works of Tyndall and Lord Rayleigh, 1 the study of light scattering by molecules and small particles has been consistently in the attention of many investigators. Classical reviews of the field are the books by van de Hulst,2 Kerker, 3 Bayvel and Jones,4 Bohren and Huffman; 5 we also note the recent survey of techniques and theoretical treatments by Jones.6 Why and how the light is scattered by small particles has already been described in Chap. 7, “Scattering by Particles,” by Craig F. Bohren in this volume. Discussions on subjects related to light scattering can also be found in chapters like Chap. 5, “Coherence Theory,” by William H. Carter and Chap. 12, “Polarization,” by Jean M. Bennett. For the topic of this chapter, light scattering by individual particles constitutes the building block of more complicated physical situations. When the three-dimensional extent of the medium that scatters the light is much larger than the typical size of a local inhomogeneity (scattering center), the physical process of wave interaction with matter can be classified as volume scattering. In this regime, the measured radiation originates from many different locations dispersed throughout the volume. Depending upon the structural characteristics of the medium, various scattering centers can act as secondary, independent sources of radiation (incoherent scattering) or they can partially add their contributions in a collective manner (coherent scattering). Another situation of interest happens when, for highly disordered systems, light is scattered successively at many locations throughout the volume (multiple scattering). All these three aspects of volume scattering will be discussed in this chapter.

VOLUME SCATTERING IN RANDOM MEDIA

9.3

In practice, particles very rarely exist singly and, depending on the illuminated volume or the volume seen by the detection system, scattering by a large number of particles needs to be considered. The simplest situation is that of incoherent scattering. When the fields scattered by different centers are completely independent, the measured intensity results from an intensity-based summation of all individual contributions. The ensemble of particles is described by temporal and spatial statistics which does not show up in scattering experiments; one can say that the volume scattering does not resolve the spatial arrangement of scattering centers. When the scattering centers are sufficiently close, the phases of wavelets originating from individual scattering centers are not independent. This is the case of collective or coherent scattering. One faces the problem of expanding the scattering theories to ensembles of particles that can have certain degree of spatial or temporal correlations. A transition sets in from independent to coherent scattering regime. The situation is common for gels or composites which scatter light due to local inhomogeneities of the refractive index with length scales of the order of wavelength and where spatial correlations between the scattering centers are encountered. This is the basis of one of the most successful application of volume scattering: the observation of structural characteristics of inhomogeneous systems. In the case of highly disordered systems, the light propagation can be subject of scattering at many different locations within the probed volume and a multiple-scattering regime sets in. For a long time, the intensity and phase fluctuations determined by multiple light scattering were regarded as optical “noise” that degrades the radiation by altering its coherence, broadening the beam, and decreasing its intensity. Experimentalists were trying to avoid it as much as possible and the development of comprehensive theories was not sufficiently motivated. Over the last two decades, however, remarkable advances in fundamental understanding and experimental methodologies proved that multiple scattering of waves is a source for unexplored physics leading to essentially new applications. New phenomena have been discovered and a series of experimental techniques have been implemented using particular coherent, polarization, temporal, and spectral properties of multiple scattered light. This revival of interest has been stimulated by the use of highly coherent sources in remote sensing and, especially, by considerable advances in solid-state physics. Many features of multiple scattered light are common to other classical waves like sound, heat, or microwaves but several analogies with electron transport phenomena have been at the core of this renewed interest in the propagation of optical waves in random systems.7 There is also another situation, which is often encountered in optics, when waves propagate through media with abrupt changes in their optical properties. Waves passing through inhomogeneous media with defined boundaries usually suffer surface scattering. In principle, scattering at rough surfaces can be considered as a limiting case of wave propagation and it is significant in various practical situations; this topic has been separately discussed in Chap. 8, “Surface Scattering,’’ by Eugene L. Church and Peter Z. Takacs in this volume.

9.3 GENERAL THEORY OF SCATTERING The schematic of a typical scattering experiment is depicted in Fig. 1, where a plane wave E 0 with the wavevector k = W /c is incident on a spatially random medium occupying a finite volume V. Light is scattered by local inhomogeneities of the dielectric constant E (r) and a basic theory of scattering aims at providing the link between the experimentally accessible intensity Is (R) = | E s (R)|2 and the microscopic structure of the random medium. The starting point of the theory is to describe the total electric field E(r) as a summation of the incoming and scattered fields and to consider that it satisfies the equation ( 2 k 2 )E(r) = 4P S(r)E(r), where S(r) represents a generic scattering potential. This equation can be converted into an integral one and, for R sufficiently far from the scattering volume where the associated Green function simplifies, one eventually obtains the general result3,8

E s (R) =

e ikR k 2 R 4P

¯(V )dr{ k s r[k s r (E (r) 1) E(r)]}e

ik r s

(1)

9.4

PHYSICAL OPTICS

FIGURE 1 Schematic representation of the incident k and scattered ks beams in a generic scattering experiment.

This expression represents the scattered field as an outgoing spherical wave that depends on the direction and magnitude of the total field inside the scattering volume V. Approximate solutions can be obtained for the case of weak fluctuations of the dielectric constant. One can expand the field E(r) = E 0 (r) E1(r) E 2 (r) ... in terms of increasing orders of the scattering potential and use successive approximations of E(r) in Eq. (1) to obtain the so-called Born series. In the spirit of a first iteration, one replaces E(r) with E0(r) and obtains the first Born approximation that describes the regime of single scattering. Alternatively, one can write E(r) =exp [9(r)] and develop the series solution for 9(r) in terms of increasing orders of the scattering potential. This is the Rytov’s series of exponential approximations, an alternative to the algebraic series representation of the Born method. The two approaches are almost equivalent, however, preference is sometimes given to Rytov’s method because an exponential representation is believed to be more appropriate to describe waves in line-of-sight propagation problems.9–11 It is worth pointing out here that Eq. (1) can be regarded as an integral equation for the total field and, because the total field is a superposition of the incident field and contributions originating from scattering from all volume V, this generic equation includes all possible multiple scattering effects.8,12

9.4

SINGLE SCATTERING

Incoherent Scattering When a sparse distribution of scattering centers is contained in the volume V, all the scattering centers are practically exposed only to the incident field, E(r) = E 0 (r) = E 0e 0e ikr where E 0 is the field magnitude, e 0 is the polarization direction, and k is the wave vector. A considerable simplification is introduced when the magnitude of the scattered field is much smaller than that of the incident field, such that the total field inside the medium can be everywhere approximated with the incident field. This is the condition of the first Born approximation for tenuous media that practically neglects multiple scattering effects inside the scattering volume V. From Eq. (1), it follows that E s (R) = E 0

e ikR k 2 R 4P

¯(V )dr[e s (E (r) 1) e0]e

ik r s

(2)


9.5

For instance, in the case of a system of N identical particles, Eq. (2) is evaluated to give E s (R) = E 0

e ikR k 2 N ik s rj e R 4P £ j =1

¯(Vj)dr[e s (E (r) 1) e0]e

ik r s

(3)

where Vj is the volume of the jth particle located at rj . In terms of the scattering wavevector q = k s k , the integral in Eq. (3) has the meaning of single-scattering amplitude of the jth particle, F (q) =| f | B(q), and depends on the forward scattering amplitude f and a scattering amplitude normalized such that B(0) =1. As explained in Chap. 7 in this volume, the ratio between the refractive index of the particles np and of the suspending medium ns determine the scattering strength f of an individual scatterer while its shape and size are accounted for in B(q). For a collection of discrete scattering centers the total scattered intensity of Eq. (3) factorizes like:13 2

N ¤ e ikR k 2 ³

i q( r r ) i j I (q) = ¥ Ei NF 2 (q) £ e : NP(q)S(q) ´ ¦ R 4P µ i , j =1

(4)

where we separated the single-scattering form factor P(q) from the interference function or the

i q ( r r ) i j . The structure factor quantifies the phase-dependent static structure factor S(q) = £ iN, j =1e contributions due to different locations of scattering centers. When an ensemble average is taken over the volume V and after separating out the diagonal terms, the static structure factor can be written as13,14 S(q) =1 e

i q ( r r ) i j

® =1 R ¯G(r)e iqr dr

(5)

in terms of the pair-correlation function G(r) (where r is the vectorial distance between two particles), describing the statistical properties of the spatial arrangement of scattering centers. It is through S(q) that the link is made between the statistical mechanics description of the inhomogeneities and the measurable quantities in a scattering experiment. As can be seen from Eq. (5), for the case of particles separated by distances much larger than the wavelength, S(q) becomes unity; the situation corresponds to G(r) x 1, i.e., constant probability to find scattering centers anywhere in the scattering volume. This regime characterized by S(q) =1 is also called the incoherent case of volume scattering where I(q) is a simple, intensity-based summation of individual contributions originating from different scattering centers.

Coherent Scattering For higher volume fractions of particles, the pair-correlation function depends on both the particle size and their concentration. The estimation of the pair-correlation functions—and, therefore, the evaluation of an explicit form for the structure factor—is a subject of highest interest and is usually approached through various approximations.15 Typical structure factors are shown in Fig. 2 for increasing volume fractions of spherical particles with radius r0. These results are based on the Percus-Yevick approximation,16,17 which has the advantage of being available in a closed mathematical form but the trend shown in Fig. 2 is rather general. Note that the strength of the interparticle interactions is practically measured by the magnitude of the first peak in S(q).13,14 In general, wave-scattering experiments can be used to infer the pair-correlation function of the scattering centers in a random medium and, meanwhile, the characteristic of an individual scattering event. In an ideal experiment where one has access to all the values of I(q) and the singlescattering phase function P(q) is known, Eq. (3) can be inverted by standard methods.17,18 Capitalizing on conventional types of single-scattering form factors, various inversion schemes have been implemented to extract the structure factor from the values of angular-resolved intensity.19

PHYSICAL OPTICS

9.6

FIGURE 2 Typical structure factors corresponding to systems of identical spherical particles with increasing volume fractions as indicated. The horizontal line at S(q) 1 corresponds to the independent scattering approximation.

Convergence and stability in the presence of noise are the major requirements for a successful inversion procedure and a subsequent Fourier analysis to provide a description of the pair-correlation function. Of course, for systems of nonspherical particles or when the system exhibits structural anisotropies, a fully vectorial inverse problem needs to be approached. The factorization approximation of Eq. (4) has found numerous applications in opticalscattering experiments for characterization of colloidal, polymeric, and complex micellar systems. Numerous static and dynamic techniques were designed to probe the systems at different length and time scales. For example, reaction kinetics or different phase transitions have been followed on the basis of angular and/or temporal dependence of the scattered intensity.19 Another significant body of applications deals with light-scattering studies of aggregation phenomena.20–22 We should not conclude this section without mentioning here the similarities between volume light scattering and other scattering-based procedures such as x-ray and neutron scattering. Of course, the “scattering potentials’’ and the corresponding length scales are different in these cases but the collective scattering effects can be treated in a similar manner. From a practical viewpoint, however, light scattering has the appealing features of being noninvasive and, most of the time, easier to implement. The average power scattered by a single particle is usually evaluated using the physical concept of total scattering cross-section S = k 4 ¯02kP(q)qdq .2,3,18 For a system with a number density R = N /V of scattering centers, the regime of scattering is characterized by a scattering length ls =1/RS . When the extent over which the wave encounters scattering centers is less than this characteristic scattering length ls, we deal with the classical single-scattering regime. Note that the system of particles can be in the single-scattering regime and exhibit both independent or collective scattering. From the general theory of scattering it follows that the details of the scattering-form factor depend on the size of scattering particle compared to the wavelength. The deviation of the scattering-form factor from an isotropic character is characterized by the asymmetry parameter 2k g = cos (Q )® =1 2q 2 ® /(2k)2, where q 2 ® = k 2 ¯0 p(q)q3dq.2,18 As we have seen in the preceding section, when the particles are closely packed the scattering centers are not independent, i.e., they are sufficiently close that the fields scattered by different centers are partially in phase, collective scattering is considered through the static structure factor S(q). The effect of wave coupling to an individual particle can therefore be isolated from the statistical mechanics description of the particle locations in the volume. In this regime, one can consider the dispersion of correlated particles as a collection of pseudo-scattering centers, equivalent particles, that are characterized by a modified single-scattering form factor P(q)S(q). There is no interaction


9.7

between these fictitious particles and a corresponding single-scattering phase function can also be defined to be: j P (q)

k 2 | f |2 | B(q)|2 S(q)

(6)

2k

| f |2 | B(q)|2 S(q))q dq

¯0

The asymmetry parameter of these pseudo-particles can be written as 2k

g 1

¯0

| f |2 | B(q)|2 S(q)q3dq (7)

2k

2k 2 ¯ | f |2 | B(q)|2 S(q)q dq 0

The equivalent-particle concept is illustrated in Fig. 3, where both the phase function and asymmetry parameters are presented for the specific case of silica particles suspended in water. This simplifying representation of coherent scattering effects is useful to further interpret complex, multiple scattering phenomena. Dynamic Scattering So far, we limited the discussion to the case where the scattering potential varies across the volume V but, at a certain location, it remains constant in time. However, many physical systems are such that the volume distribution of scattering centers fluctuates in time and, therefore, gives rise to temporal fluctuations of the scattered radiation, i.e., to dynamic scattering. A complete analysis of light scattering involves the autocorrelation function of the dielectric constant fluctuations E (r, 0), E(r, t )® that manifests itself in the statistics of the temporal fluctuations of the measured light intensity. Evaluation of such correlation functions requires knowledge of the transport properties of the volume medium and is based on statistical mechanics and many-body theory. In spite of the rather complex phenomenology, different photon correlation techniques have been successfully implemented in studies of reacting systems, molecular dynamics, or atmospheric scintillations. 1 g

P(q)

0.5 ~ g ~ P(q)

1

10

100

0

0

0.2

0.4

(deg)

(a)

(b)

0.6

FIGURE 3 (a) Single scattering form factor P(q) corresponding to silica particles of 0.476 Mm placed in water and illuminated with N0.633 Mm and the form factor P(q) of an equivalent particle corresponding to a collection of such particles at the volume fraction T 0.5. (b) Values of the asymmetry parameter for one silica particle and for a colection of of particles with an increasing volume fraction as indicated.

9.8

PHYSICAL OPTICS

The benchmarks of the dynamic light scattering have been set in the seventies.23,24 An interesting and useful particularity stems from the fact that, based on dynamic scattering, one can actually measure mechanical properties in the scattering volume without knowledge of the refractive index. Random motion of scattering centers induces small Doppler frequency shifts which, in turn, produce an overall broadening of the incident spectrum of light. The detection methods for such spectral changes depend primarily on the time scales of interest and range from high-resolution spectroscopy for very fast phenomena to various mixing or beating techniques for processes slower than about 1 ms. Based on comparing the scattering signal with itself at increasing time intervals, photon correlation spectroscopy (PCS) has emerged as a successful technique for the study of volume distributions of small particles suspended in fluids.25 Practically, one deals with the time-dependent fluctuations of the speckle pattern and the goal is to determine the temporal autocorrelation function E(0)E (t )®, which probes the microscopic dynamics. For instance, in the simple case of a monodisperse and noninteracting system of brownian particles, E(0)E (t )® is a single exponential that depends on the diffusion constant D = kBT /6PHr0. Knowing the absolute temperature T and the shear viscosity J of the solvent, one can infer the particle radius r0. Similar to the case of static scattering discussed previously, refinements can be added to the analysis to account for possible dynamic structuring, polydispersivity as well as asphericity effects.19,23,26

9.5

MULTIPLE SCATTERING When optical waves propagate through media with random distributions of the dielectric constant or when they encounter extended regions containing discrete scatterers or random continuum, one needs to solve a wave equation in the presence of large number of scattering centers; this is a difficult task and, as we will discuss, a series of simplifying approaches have been proposed. A survey of multiple scattering applications is presented by van de Hulst.27

Effective-Medium Representation Some physical insight is given by a simple model, which describes the wave attenuation due to scattering and absorption in terms of an effective dielectric constant. Without explicitly involving multiple scattering, one considers the wave propagation through an homogeneous effective-medium which is defined in terms of averaged quantities. The effective permittivity E eff is calculated by simply considering the medium as a distribution of spheres with permittivity G embedded in a continuum background of permittivity E 0. If a wave with the wavelength much larger than the characteristic length scales (size of inhomogeneity and mean separation distance) propagates through a volume random medium, the attenuation due to scattering can be neglected; therefore, a frequencyindependent dielectric constant will appropriately describe the medium. Based on an induced dipoles model, the Maxwell-Garnett mixing formula relates the effective permittivity to the volume fraction v of spheres E eff = E(1 2va)/(1 va), where a = (E E 0 )/(E 2E 0 ).28 In recent developments, the wave propagation through highly scattering media has been described by including multiple scattering interactions in a mean-field approach. This is simply done by considering that the energy density is uniform when averaged over the correlation length of the microstructure.29 Through this effective medium, a “coherent’’ beam propagates with a propagation constant that includes the attenuation due to both absorption and scattering. In general, the propagation of the coherent beam is characterized by a complex index of refraction associated with the effective medium and a nontrivial dispersion law can be determined by resonant scattering.30 It is worth mentioning that, in the long-wavelength limit, the attenuation due to scattering is negligible and the classical mixture formula for the effective permittivity applies.


9.9

FIGURE 4 The field at r is a summation of the incident filed E0 and contributions Ej from all other scattering centers; in turn, the effective field on each particle consists of an incident contribution E0j and contributions Eis from all other scattering centers.

Analytical Theory of Multiple Scattering A rigorous description of multiple light-scattering phenomena can be made if statistical considerations are introduced for quantities such as variances and correlation functions for E(r) and general wave equations are subsequently produced. The advantage of an analytical theory is that the general formulation does not require a priori assumptions about the strength of individual scattering events nor about the packing fraction of scattering centers. The drawback, however, is that, in order to deal with the complexity of the problem, one needs to use quite involved approximations and, sometimes, rather formal representations. Multiple Scattering Equations As shown in Fig. 4, when the field E0 is incident on a random distribution of N scattering centers located at r1 , r2 , ... , rN throughout the scattering volume V, the total field at one particular location inside V is the sum of the incident wave and the contributions from all the other particles N

E =E 0 £ E j

(8)

j =1

the field scattered from the jth particle depends on the effective field incident on this particle and its characteristics (scattering potential) Sj N ¤ ³ E j = S j ¥E 0j £ E is´ ¦ i =1,i w j µ

(9)

It follows from Eqs. (8) and (9) that the total field can be formally written as N N N ¤ ³ E =E 0 £S j ¥E 0j £ £ Si E is´ µ j =1 ¦ j =1 i =1,i w j

(10)

9.10

PHYSICAL OPTICS

which is a series of contributions from the incident field, single scattering, and increasing orders of multiple scattering.18 In principle, knowing the scattering characteristics Sj of individual centers (this includes strength and location), one can develop the analytical approach by involving chains of successive scattering paths. From the summations in Eq. (10), by neglecting the scattering contributions that contain a scatterer more than once, Twersky has developed an expanded representation of multiple scattering which is practical only for cases of low-order scattering.31 Approximations of Multiple Scattering Rigorous derivation of multiple scattering equations using the Green’s function associated with the multiple-scattering process and a system-transfer-operator approach (the T-matrix formalism) can be found in.32–34 However, in many cases of practical interest, a very large number of scattering centers needs to be involved and it is impossible to obtain accurate descriptions for either the T-matrix or the Green’s function. When large ensembles of scatterers are involved, a statistical description of the multiple scattering equations is appropriate. Probability density functions for finding scattering centers at different locations are determined by radial distribution function in a similar way as discussed in the context of correlated scattering. This expresses both the constraint on the particle locations and the fact that they are impenetrable. By using configuration-averaging procedures, self-consistent integral equations can be obtained for the average and fluctuating parts of the field produced through multiple scattering. When such a procedure is applied to Eq. (10), E ® =E 0 ¯S j E 0j G(rj )drj ¯¯Si S j E 0i G(ri )G(rj )dri drj ...

(11)

a hierarchy of integral equations is generated and successive approximations are obtained by truncating the series at different stages. In fact, this expanded form has a more physically understandable representation in terms of the average field E j ® at the location of a generic particle j: E ® = E 0 ¯S j E j ® G(rj )drj

(12)

Foldy was the first to introduce the concept of configurational averaging and used the joint probability distribution for the existence of a given configuration of scattering centers to average the resulting wave over all possible configurations.35 However, Foldy’s approximation is appropriate for wave propagation in sparse media with a small fractional volume of scatterers. A comprehensive discussion on multiple scattering equations and various approaximations can be found in Tsang’s book34 including the quasi-crystalline approximation36 and coherent potential37 approximations.

Radiative Transfer The drawback of an analytical theory of multiple scattering is that it is too complicated; for systems with volume disorder often encountered in realistic situations, the scattering phenomena depends essentially on the ratio between the characteristic length scales of the system and the radiation wavelength. A statistical description in terms of such characteristic scattering lengths is usually sufficient. In general, the particular location, orientation, and size of a scattering center is irrelevant and the underlying wave character seems to be washed out. Because energy is transported through multiple scattering processes, what matters is only the energy balance. Of course, this approach cannot account for subtle interference and correlation effects but refinements can be developed on the basis of a microscopic interpretation of radiative transfer.38 A comprehensive mathematical description of the nonstationary radiative transport is given by both Chandrasekhar39 and Ishimaru.18 The net effect of monochromatic radiation flow through a medium with a density T of scattering centers is expressed in terms of a modified Stokes vector I(r, s, t ) =[ E1 E1 ® E2 E2 ®, E1 E1 ® E2 E2 ®, Re{ E1 E2 ®}, Im{ E1 E2 ®}]T


9.11

FIGURE 5 Loss-gain balance in a differential volume element; the incident radiance is attenuated by both absorption inside and scattering out of the volume element but it can be increased by ambient scattering and emission within the volume.

This quantity is a vectorial equivalent to the radiance where each element can be defined as the amount of energy in a given state that, at the position r, flows per second and per unit area in the direction s. When radiation propagates over the distance ds, there is a loss of specific intensity due to both scattering and absorption dI = R S sc S abs Ids . In the mean time, there is a gain of specific intensity due to scattering from a generic direction s' into the direction s quantified by the tensoral scattering phase function (Mueller matrix) P(s', s). Also, there could be an increase, G(r, s, t), of specific intensity due to emission within the volume of interest and the net loss-gain balance which is illustrated in Fig. 5, and represents the nonstationary radiative transfer equation:18

§1 t ¶ ¨ c slt t s R(S sc S abs )· I (r, s, t ) = RS sc ¯P(s `, s)(r, s, t )d7 E(r, s, t ) © ¸

(13)

No analytical solution exists for the transfer equation and, in order to solve specific problems, one needs to assume functional forms for both the phase function and the specific intensity. Various methods have been used to approach the transient scalar RTE.40–43 Cheung and Ishimaru44 and Kim et al.45 approached the steady state vector RTE using a Fourier analysis. Vaillon et al.46 used a vector Monte Carlo method to analyze the radiative transfer in a particle-laden semitransparent medium. Jiang et al. presented a model for the atmospheric radiative transfer with polarization for remotesensing applications.47 Ma and Ishimaru used an eigenvalue-eigenfunction technique to solve numerically the vector radiative transfer equation.48 To solve the problem of polarized pulse propagation in random media, Ishimaru et al.49 used the discrete-ordinates method by expanding the Stokes vector in a Fourier series. Successive orders of approximation are obtained by spherical harmonic expansion of the specific intensity; for instance, the so-called P1 approximation is obtained when the diffuse radiance is expressed as a linear combination of an isotropic radiance and a second term modulated by a cosine.18 Diffusion Approximation Perhaps one of the most widely used treatment for multiple light scattering is the diffusion approach. When (i) absorption is small compared to scattering, (ii) scattering is almost isotropic, and (iii) the radiance is not needed close to the source or boundaries

9.12

PHYSICAL OPTICS

then the diffusion theory can be used as an approximation following from the general radiative transfer theory. To get insight into the physical meaning of this approximation it is convenient to define quantities that are directly measurable such as the diffuse energy density (average radiance) U (r, t ) = ¯4P I0 (r, s, t )d7 and the diffuse flux J(r, t ) = ¯4P I(r, s, t )s d7. In the diffusion approximation, the diffuse radiance is approximated by the first two terms of a Taylor’s expansion:8,50 I0(r, s, t ) U (r, t )

3 J(r, t ) s 4P

(14)

and the following differential equation can be written for the average radiance D 2U (r, t ) MaU r, t

t U (r, t ) = S(r, t ) tt

(15)

The isotropic source density is denoted by S(r, t) and D is the diffusion coefficient which is defined in units of length as D=

1 3[Ma Ms (1 g )]

(16)

in terms of the absorption O a and O s scattering coefficients. The diffusion equation is solved subject to boundary conditions and source specifics; most appealing is the fact that analytical solutions can be obtained for reflectance and transmittance calculations. Because the phase function is characterized by a single anisotropy factor, the diffusion approximation provides mathematical convenience. Through renormalization, an asymmetry-corrected scattering cross-section that depends only on the average cosine of scattering angle defines the diffusion coefficient in Eq. (16) and, therefore, an essentially anisotropic propagation problem is mapped into an almost isotropic (diffusive) model. The photon migration approach based on the diffusion approximation has been very successful in describing the interaction between light and complex fluids51 or biological tissues.52,53 It is instructive to note that three length scales characterize the light propagation in this regime: the absorption length la = Ma 1 which is the distance traveled by a photon before it is absorbed, the scattering length ls = Ms 1 which is the average distance between successive scattering events, and the transport mean free path l = ls /(1 g ) that defines the distance traveled before the direction of propagation is randomized. In experiments that are interpreted in the frame of the diffusion approximation, l is the only observable quantity and, therefore, the spatial and temporal resolution are limited by l and l c, respectively. Under appropriate boundary conditions, such as a mixed boundary condition in which the diffuse energy density vanishes linearly on a plane, the steady state diffusion equation can be solved and the photon flux is obtained from Fick’s law.54 Assuming an average energy transport velocity, the path length dependence of the energy flux can be evaulated yielding a path length probability distribution, p(s), which can be regarded as the probability distribution of optical path lengths that correspond to waves that have traveled through the medium along closed loops and have also accumulated a total momentum transfer equal to 4R/N. Low-Order Flux Models for Radiative Transfer In an effort to describe the optical properties of highly scattering materials while reducing the computational difficulties, simplifying flux models have been designed for the radiative energy transport. A volume scattering medium consisting of a collection of scattering centers is described as homogeneous material characterized by effective scattering and absorption properties that are determined by its internal structure. In this approach, the fundamental equation of radiative transfer is based on the balance between the net flux change, the flux input, and flux continuing out in an infinitesimal volume. Assuming two diffusing components, a one-dimensional model based on plane symmetry for unit cross-section has been initially proposed by Schuster.55 One of the most successful extensions of this model is the so-called


9.13

FIGURE 6 The two-flux approximation applied to radiative transfer through a diffusive layer.

Kubelka-Munk theory56 which relates the phenomenological, effective scattering SK M and absorption K K M coefficients to measurable optical properties such as diffuse reflectance or transmittance. The two-flux model, Kubelka-Munk theory, is schematically illustrated in Fig. 6. Diffuse radiation is assumed to be incident on the slab; the diffuse radiant flux in the positive x direction is J while the one returning as a result of scattering is J . The net flux balance at a distance x across an infinitesimal layer of thickness dx is d J = (K K M SK M )J dx sJ dx

(17)

d J = (K K M SK M )J dx sJ dx where the coefficient K K M determines the flux attenuation due to absorption while SK M accounts for the net flux scattered between forward and backward directions. The solution of the simultaneous differential equations of the first order in one dimension is obtained by applying the boundary conditions for the layer shown in Fig. 6. The diffuse reflectance of the substrate at x d can also be accounted for and expressions for the total diffuse reflection and transmission are found for a specific application. In practical applications of the Kubelka-Munk theory, the effective scattering and absorption parameters are inferred by iteration from measurements of diffuse reflectance or transmission. A considerable body of work was dedicated to relate the Kubelka-Munk parameters to microstructure and to incorporate both the single- and multiple-scattering effects. Refinements and higher-order flux models have also been developed. A more accurate model that accounts for the usual condition of collimated incident radiation was elaborated by J. Reichman.57 A four-flux model has been developed that includes certain anisotropy of the scattered radiation.58 A six-flux model was implemented to incorporate the effect of particle shape and interparticles correlation.59 In spite of the fact that it is based on empirical determination of coefficients and that its range of applicability is rather unclear, the simple-to-implement Kubelka-Munk theory makes reasonably good description of experiments and has found applications in areas such as coatings, paper, paints, pigments, medical physics, and atmospheric physics. Specific Effects in Multiple Light Scattering The effects associated with light propagation through multiple scattering media depend on the scale of observation. An interesting analogy exists between optical and electron waves propagation in mesoscopic systems and, based on recent developments in solid state physics, useful insights have

9.14

PHYSICAL OPTICS

been provided for a range of multiple scattering phenomena.7,60 It is clear now that multiple light scattering in volume disorder does not merely scramble a coherent incident wave. In fact, the apparent randomness of the scattered field conceals intriguing and sometimes counter intuitive effects such as the enhancement of the backscattered intensity and a range of correlations and statistics of vector waves. Weak Localization Until recently, the coherent light propagating through random media has been considered to be somehow degraded, losing its coherence properties. However, recent experiments have brought evidence of the enhanced backscattering of light due to interference between the waves taking time-reversed paths,61 an effect which is associated with the more general phenomenon for the weak localization of waves in random media.62,63 First found in solid state physics more than 30 years ago, this phenomenon is applicable to all classical waves. Surveys of the state of the art in the optical counterpart of weak localization can be found in Refs. 64 and 65. When coherent light is scattered by a medium with volume randomness, interference effects between the scattered waves which traveled through the medium along different paths occur. The result is a random pattern of interference called laser speckle. Because the correlations in this granular pattern extend over angular scales of typically 10 3 rad or less, when the individual scatterers are allowed to move over distances of the order of the wavelength or more, the distribution of intensities in the speckle pattern is rapidly averaged out and becomes essentially flat. So far, however, it is well understood and widely recognized that one kind of interference still survives in this average. This is the interference of the waves emerging from the medium in directions close to exact backscattering and which have traveled along the same path but in opposite directions. The pairs of time-reversed light paths have some particularities which can be easily understood in the general context of the waves scattered by random media. In Fig. 7, the main contributions to scattering from a dense distribution of scatterers are presented together with their angular dependence. In the narrow angle of interest, the single scattering I (s) and the ladder term of multiple scattering I (ml) are practically constant. The third, cyclical term I (mc), however, corresponds to the paired (or coherent) scattering channels and, being an interference term, has a definite angular structure.

FIGURE 7 A schematic illustration of the origin of coherent backscattering. The classical contributions to backscattering are I(s) and I(ml); in addition, constructive interference occurs between reversed multiple scattering paths that have the same incident wave vector ki and exit wave vector ki.


9.15

The existence of such a cyclical term (an idea originated from Watson66) is based on the fact that each scattering channel, involving a multitude of scattering centers, has its own coherent channel corresponding to the same sequence of scattering centers but time reversed. In the backward direction, the contribution of I (mc) equals that of I (ml) but its magnitude vanishes quickly away from this direction. The angular profile of I (mc) can be qualitatively described by taking into account the interference between two light paths as shown in Fig. 7. The two time reversed paths have the same initial and final wave vectors, ki and kf , and develop inside the medium through the same scattering centers. Under stationary conditions, the two outgoing waves corresponding to these paths are coherent and can interfere constructively for a special choice of ki and kf . If the positions of the first and the last scatterer in the sequence are ri and rf , respectively, the total phase shift between the two waves is q(ri rf), where q is the momentum transfer ki kf . Close to the backward direction (kf ki), the two waves add coherently and the interference may be described by a weighting factor cos[(k i k f )(ri rf )], which is controlled by the interparticle distance | ri rf |. One can say that the coherence between the time-reversed sequences is lost for angles Q > L /| ri rf | and this actually sets the angular width of the cyclical term I (mc). The detailed angular profile of I (mc) is determined by the probability distribution function for | ri rf | and, based on a diffusive model for light propagation in a semi-infinite medium, an approximate formula was given in Ref. 67:

I (mc ) =[1 e 3.4 ql ]/3.4ql

(18)

It should be noted that the intensities which contribute to classical backscattering, i.e., I(s) and I(ml), correspond to incoherent channels of scattering, they add up on an intensity basis and, upon ensemble average, all angular dependences are washed out as seen in Fig. 7. As can be seen from Eq. (18), the angular shape of the coherent backscattering peak can be used to measure l for a specific multiple scattering medium. Correlations in Speckle Patterns One of the significant discoveries in mesoscopic physics is the phenomenon of conductance fluctuations which arises from correlations between the transmission probabilities for different output and input modes. In multiple light scattering, the same phenomenon shows up in the form of correlations between the intensities of light transmitted in different directions such as the case schematically depicted in Fig. 8. A nice feature of the optical scattering is that in contrast with electronic conductance experiments, one has access to both angular

FIGURE 8 Scattering geometry containing two different wave trajectories.

9.16

PHYSICAL OPTICS

dependence and angular integration of transmittance. A rich assortment of correlation functions of the different transmission quantities can be studied and their magnitudes, decay rates, etc., open up novel possibilities for multiple light scattering-based characterization and tomographic techniques.68,69 These measurements exploit the spatial, spectral, and temporal resolution accessible in optical experiments.70 In a medium with volume disorder, optical waves can be thought to propagate through channels (propagating eigenmodes) defined angularly by one coherence area having the size of a speckle spot. The energy is coupled in and out of the medium through a number of 2RA/N2 such channels where A is the transversal size of the medium. The transmission coefficient Tab of one channel is defined as the ratio between the transmitted intensity in mode b and the incident intensity in mode a at a fixed optical frequency Y. An average transmission through the medium Tab ® can be evaluated by ensemble averaging over different realizations of disorder and, for purely elastic scattering, it can be shown that Tab ® ^ l /L . The motion of scatterers, the frequency shift of the incident wave, and the variation of angle of incidence and/or detection, introduce phase shifts (momentum differences) in different propagating channels. Surprising and novel features are found when one studies carefully how various Tab channels are correlated with one another. A general function Caba `b ` = DTabTa `b ` ® can be designed to evaluate the correlation between changes in the transmission coefficients DTab = Tab Tab ® and, to the lowest order in the disorder parameter 1/kl, it can be shown to be a summation of three different terms.71 The first term C1, short-range correlations, represents the large local intensity fluctuations specific to speckle patterns and exhibits an angular memory effect: when the incident beam is tilted by a small angle, the transmitted speckle pattern will, in average, follow provided that the tilt angle is not too large.72,73 The second term C 2, long-range correlations, arises from paths that cross at a certain scattering site; it is smaller than C 1 by a factor 1/£Tab and decays very slowly with the momentum difference.74 Finally, a uniform positive correlation which is independent of momentum differences is included in C 3, conductance correlations, and it is determined by the less probable case of two crossings in propagation channels. This small correlation term (of the order of 1/£(Tab )2 ) causes just a shift in background, i.e., the spatially averaged intensity in each speckle pattern is always a little darker or brighter than the total intensity averaged over many disorder realizations in the sample. These fluctuations do not decrease when averaging is done over larger and larger spatial regions and they are the optical analogue of the universal conductance fluctuations in electronic systems.71,75 Depolarization A common interpretation of multiple scattering effects assumes depolarization of the incident field. However, when coherent radiation interacts with scattering media, there is always a fixed phase and amplitude relationship between orthogonal electric field components at a given frequency at any point in time and space. Of course, these relationships may vary as a function of time, spatial coordinate, frequency, and material morphology. If, upon ensemble average, there is no correlation between any pair of orthogonal field components, the field is said to be unpolarized. A degree of polarization is defined as D=

I

2 I 22 I32 1

I0

1/2

(19)

where Ii are the ensemble averaged Stokes vector elements. This ensemble average can take the form of spatial, temporal, or frequency average or can also be the result of averaging over different material realizations.76 For a static medium illuminated by polarized light, depolarization occurs when path length differences contributing to a particular point exceed the coherence length of the illuminating light. It is interesting to note that the measured degree of polarization will be a function of the detection geometry, decreasing with increasing the detector’s size and integration time. Based on symmetry considerations, van de Hulst2 finds the Mueller matrix for single scattering on a collection of randomly oriented identical particles each of which has a plane of symmetry to be diagonal with P22 = P33 w P44 . For spheres in exact forward scattering P22 = P44 =1. In multiple scattering


9.17

however, this relation is not true anymore. It is expected that when light of arbitrary incident polarization impinges on an optically thick, multiple scattering medium it emerges diffusely and totally depolarized. When increasing the optical density, the transfer matrix evolves toward that of a total depolarizer which has all elements equal to zero except for P11. This depolarization process will depend on the size parameter of the scattering particles. Owing to a smaller scattering anisotropy for the particles with the size parameter close to 1, the total depolarization stage is reached at higher optical densities than for larger particles. Two different regimes can be identified in terms of optical density d/l : (i) a steep decay for low optical densities, which corresponds to the attenuation of ballistic photons, and (ii) a slower decay for large optical densities, corresponding to the diffusive regime. The effective coefficient of attenuation depends only on the volume fraction of the scattering medium. For an arbitrary input state of polarization, the output state of polarization can be obtained from Iout = PIin . In the case of a diagonal transfer matrix, the renormalized output Stokes vector is T

Iout = §©1 I1P22 I2 P33 I3 P44 ¶¸

(20)

The degree of polarization of the scattered light can be obtained from Eq. (20) using Eq. (19) for any input state. The characteristics of the depolarization will depend both on the input state of polarization and the dominant scattering regime, exhibiting different behavior in the Mie and Rayleigh (ka 4, W is not square and W 1 is not unique; multiple W 1 exist. Sm is overdetermined; there are more equations than unknowns. In the absence of noise, the different W 1 all yield the same Sm. Because noise is always present, the optimum W 1 is desired. The least squares estimate for Sm utilizes a particular matrix inverse, the pseudoinverse WP 1 of W, WP 1 (WT W) 1 WT

(17)

POLARIMETRY

15.13

The optimal estimate of S is (18)

S WP 1 s P (7T s 7) 1 s 7T s0

When W is of rank three or less, the polarimeter is incomplete. The optimal matrix inverse is the pseudoinverse, but only three or fewer properties of the Stokes vector elements are determined; the projection of the Stokes vector onto three or fewer directions is measured. If these directions align with the Stokes basis vectors, then these Stokes vector elements are measured, but in general, linear combinations of elements are measured.

15.21 SAMPLE-MEASURING POLARIMETERS FOR MEASURING MUELLER MATRIX ELEMENTS This section contains a general formulation of Mueller matrix measurement which is an extension of the Stokes vector method of the preceding section. The Mueller matrix is always a function of wavelength, angle of incidence, and location on the sample. These are assumed fixed here for simplicity; this method can be generalized to these more general cases. Figure 2 is a block diagram of a sample-measuring polarimeter. The polarization state generator (PSG) prepares polarization states incident on a sample. The light beam exiting the sample is analyzed by the polarization state analyzer (PSA), and the flux at the detector measured. The objective is to determine several or all of the sample’s Mueller matrix M elements through a sequence q 0, 1 ,..., Q 1 of polarimetric measurements. The polarization generator prepares a set of polarization states with a sequence of Stokes vectors Sq. The Stokes vectors exiting the sample are -s3q. These exiting states are analyzed by the qth polarization state analyzer Aq, yielding the qth measured flux Pq ! Tq s-sSq . Each measured flux is assumed to be a linear function of the sample’s Mueller matrix elements (nonlinear optical interactions such as frequency doubling are not treated by the present formulation). A set of linear equations is developed from the set of polarimetric measurements to solve for the Mueller matrix elements. For example, consider a measurement taken with horizontal linear polarizers as the generator and the analyzer. As Eq. (11) shows, the measured flux only depends on the Mueller matrix elements m0,0, m0,1, m1,0, and m1,1, ¤m0,0 m0 ,1 m0, 2 m0, 3³ ¤1³ ¥m m1,1 m1, 2 m1, 3´ ¥1´ 1 1, 0 ´s P A T s M s S (1, 1, 0, 0)¥ ¥m2, 0 m2,1 m2, 2 m2, 3´ ¥¥0´´ 2 ´ 0 ¥ ¦m3, 0 m3,1 m3, 2 m3, 3µ ¦ µ

m0, 0 m0,1 m1, 0 m1,1 2

(19)

FIGURE 2 A sample-measuring polarimeter consists of a source, polarization state generator (PSG), the sample, a polarization state analyzer (PSA), and the detector. (See also color insert.)

15.14

POLARIZED LIGHT

As another example, the four Mueller matrix elements m0,0, m0,1, m1,0, and m1,1 can be measured using four measurements with ideal horizontal (H) and vertical (V) linear polarizers. Four measurements P0, P1, P2, and P3 are taken with (generator/analyzer) settings of (H/H), (V/H), (H/V), and (V/V), determining the following combinations of Mueller matrix elements, P0 (m0,0 m0,1 m1,0 m1,1 )/ 4

P1 (m0,0 m0,1 m1,0 m1,1 )/4

P2 (m0,0 m0,1 m1,0 m1,1 )/4

P3 (m0,0 m0,1 m1,0 m1,1 )/4

(20)

These four equations are solved for the Mueller matrix elements as ¤m0,0³ ¤ P0 P1 P2 P3³ ¥m ´ ¥ P P P P ´ ¥ 0,1´ ¥ 0 1 2 3´ ¥m1,0´ ¥ P0 P1 P2 P3´ ¥ m ´ ¥¦ P P P P ´µ ¦ 1,1µ 0 1 2 3

(21)

Other Mueller matrix elements are determined using different combinations of generator and analyzer states. The four matrix elements at the corners of a rectangle in the Mueller matrix {m0,0, m0,i, mj,0, mj,i} can be determined from four measurements using a ± i-generator and ± j-analyzer. For example, a pair of right and left circularly polarizing generators and a pair of 45° and 135° oriented analyzers determine elements m0,0, m0,3, m2,0, m2,3. In practice, the data-reduction equations are far more complex than these examples because many more measurements are involved with nonideal polarization elements. The following section contains a systematic method for calculation of data-reduction equations based on calibration data for the generator and analyzer.

15.22 POLARIMETRIC MEASUREMENT EQUATION AND POLARIMETRIC DATA-REDUCTION EQUATION This section develops data-reduction equations to calculate Mueller matrices from arbitrary sequences of measurements. The algorithm uses either ideal or calibrated values for the polarization generator and analyzer vectors. The data-reduction equations are straightforward matrix-vector multiplication on a data vector. This method is an extension of the data-reduction methods presented in “Data Reduction for Light-Measuring Polarimeters”. A Mueller matrix polarimeter takes Q measurements identified by index q 0, 1 , . . . , Q 1. For the qth measurement, the generator produces a beam with Stokes vector Sq and the beam exiting the sample is analyzed by analyzer vector Aq. The measured flux Pq is related to the sample Mueller matrix by

Pq A Tq M3q (aq,0

3

aq,1 aq,2

3

£ £ aq, j m j , k Sq, k j 0 k 0

¤m0,0 m0,1 m0,2 m0,3³ ¤Sq,0³ ¥m m1,1 m1,2 m1,3´ ¥¥ Sq,1´´ ´s aq,3 )s ¥ 1,0 ¥m2,0 m2,1 m2,2 m2,3´ ¥Sq,2´ ´ ¥ ´ ¥m ¦ 3,0 m3,1 m3,2 m3,3µ ¦Sq,3µ

(22)

POLARIMETRY

15.15

16,17 This equation is rewritten First, the Mueller matrix is flattened into F as a vector-vector dot product. a 16 r1 Mueller vector M (m0,0, m0,1, m0,2, m0,3, m1,0,..., m3,3,)T. A 16 r 1 polarimetric measurement vector Wq for the qth measurement is defined as follows:

Wq (w q, 0, 0, w q, 0,1, w q, 0, 2, w q, 0, 3, w q,1,0,.... . . w q,3,3 )T

(23)

(aq, 0 Sq, 0, aq, 0 Sq,1, aq, 0 Sq, 2, aq, 0 Sq, 3, aq,1 Sq, 0,..... , aq, 3 Sq, 3 )T where wq,j,k aq,j sq,k . The qth measured flux is the dot product ¤aq, 0 Sq, 0³ ¤m0, 0³ ¤w q, 0, 0³ ¤m0, 0³ ¥ a S ´ ¥m ´ ¥ w ´ ¥m ´ ¥ q, 0 q,1´ ¥ 0,1´ ¥ q, 0,1´ ¥ 0,1´ ¥aq, 0 Sq, 2´ ¥m0, 2´ ¥w q, 0, 2´ ¥m0, 2´ F ¥ a S ´ ¥m ´ ¥ w ´ ¥m ´ Pq 7q sM ¥ q, 0 q, 3´ s ¥ 0, 3´ ¥ q, 0, 3´ s ¥ 0, 3´ ¥ aq,1 Sq, 0´ ¥m1, 0 ´ ¥ w q,1,0 ´ ¥m1, 0 ´ ´ ¥ ´ ¥ ´ ¥ ¥ ´ ¥ aq,1 Sq,1 ´ ¥ m1,1 ´ ¥ w q,1,1 ´ ¥ m1,1 ´ ¥ " ´ ¥ " ´ ¥ " ´ ¥ " ´ ´ ¥ ´ ¥ ´ ¥ ¥ ´ ¦aq, 3 Sq, 3µ ¦m3, 3µ ¦ w q,3,3µ ¦m3, 3µ

(24)

The full sequence of Q measurements is described by the Q r16 polarimetric measurement matrix W, where the qth row is Wq. The polarimetric measurement equation relates the measurement vector P to the sample Mueller vector as ¤ P0 ³ ¤ w 0, 0,0 F ¥ P ´ ¥ w1, 0,0 P Ws- ¥ 1 ´ ¥ ¥ " ´ ¥ " ¥¦ PQ 1´µ ¥w ¦ Q 1, 0,0

w 0, 0,1 z w 0,3,3 ³ ¤m0,0³ w1, 0,1 z w1,3,3 ´ ¥m0,1´ ´ s¥ ´ " " ´ ¥ " ´ wQ 1, 0,1 z wQ 1,3,3´µ ¥¦m3,3´µ

(25)

If W contains 16 linearly independent rows, all 16 elements of the Mueller matrix can be determined. When Q 16, the matrix inverse is unique and the Mueller matrix elements are determined from the polarimetric data-reduction equation F M W 1 s0 (26) F Often, Q >16,Fand M is overdetermined. The optimal (least-squares) polarimetric data-reduction equation for M uses the pseudoinverse WP 1 of W, F M (WT s7) 1 s7T s0 7P 1 sP

(27)

The advantages of this procedure are as follows: First, this procedure does not assume that the set of states of polarization state generator and analyzer have any particular form. For example, the polarization elements in the generator and analyzer do not need to be rotated in uniform angular increments, but can comprise an arbitrary sequence. Second, the polarization elements are not assumed to be ideal polarization elements or have any particular imperfections. If the polarization generator and analyzer vectors are determined through a calibration procedure, the effects of nonideal polarization elements are corrected in the data reduction. Third, the procedure readily treats overdetermined measurement sequences (more than 16 measurements for the full Mueller matrix), providing a least-squares solution. Finally, a matrix-vector form of data reduction is readily implemented and easily understood.

15.16

POLARIZED LIGHT

Generator

Sample

Analyzer

Source

Detector

Fixed Rotating polarizer retarder S IHg(i)

Rotating Fixed retarder polarizer IHa(i) S

FIGURE 3 The dual rotating retarder polarimeter consists of a source, a fixed linear polarizer, a retarder which rotates in steps, the sample, a second retarder which rotates in steps, a fixed linear polarizer, and the detector. (See also color insert.) (After Ref. 18.)

15.23

DUAL ROTATING RETARDER POLARIMETER The dual rotating retarder Mueller matrix polarimeter shown in Fig. 318 is one of the most common Mueller polarimeters. Light from the source passes first through a fixed linear polarizer, then through a rotating linear retarder, the sample, a rotating linear retarder, and finally through a fixed linear polarizer. In the most common configuration, the polarizers are parallel, and the retarders are rotated in angular increments of five-to-one. This five-to-one ratio encodes all 16 Mueller matrix elements onto the amplitudes and phases of 12 distinct frequencies in the detected signal. This configuration was first described by Azzam16 who provides an explanation of how the ratios one-to-one, two-to-one, threeto-one, and four-to-one all yield incomplete polarimeters. Thus five-to-one is the first integer ratio yielding a complete Mueller matrix polarimeter. The data reduction can be performed using the polarimetric data-reduction matrix method of the preceding section, or alternatively the detected signal can be Fourier analyzed, and the Mueller matrix elements calculated from the Fourier coefficients.19 This polarimeter configuration has several design advantages. Since the polarizers do not move, the polarizer in the generator accepts only one polarization state from the source optics, making the measurement immune to source polarization and polarization aberrations from optics prior to the polarizer. If the polarizer did rotate, and if the beam incident on it were elliptically polarized, a systematic modulation of intensity would be introduced which would require compensation in the data reduction. Similarly, the polarizer in the analyzer does not rotate; only one polarization state is transmitted through the analyzing optics and onto the detector. Any diattenuation in the analyzing optics and any polarization sensitivity in the detector will not affect the measurements. Optimal values for the retardances are near 2R/3 rad (N/3 waveplates).19 If F1 F2 R rad (halfwave linear retarders), only linear states are generated and analyzed, and the last row and column of the sample Mueller matrix are not measured. Hauge and Broch20,21 developed an algorithm to compensate for the linear diattenuation and linear retardance of the retarders. Goldstein and Chipman22 treat five errors, the retardances of the two retarders, and orientation errors of the two retarders and one of the polarizers, in a small angle approximation good for small errors. Chenault, Pezzaniti, and Chipman23 extended this method to larger errors.

15.24 INCOMPLETE SAMPLE-MEASURING POLARIMETERS Incomplete sample-measuring polarimeters do not measure the full Mueller matrix of a sample and thus provide incomplete information regarding the polarization properties of a sample. Often the full Mueller matrix is not needed. For example, many birefringent samples have considerable linear retardance and insignificant amounts of other polarization forms. The magnitude of the retardance

POLARIMETRY

15.17

can be measured, assuming all the other polarization effects are small, using much simpler configurations than a Mueller matrix polarimeter, such as the circular polariscope.5 Similarly, homogeneous and isotropic interfaces, such as dielectrics, metals, and thin films, should only display linear diattenuation and linear retardance aligned with the s-p planes. These interfaces do not need characterization of their circular diattenuation and circular retardance. So most ellipsometers characterize the diattenuation and retardance associated with s and p without providing the full Mueller matrix.1,6,9

15.25

NONIDEAL POLARIZATION ELEMENTS Polarization elements used in polarimetry require a level of characterization beyond what is normally provided by vendors at the time of this writing. For retarders, vendors usually only specify the linear retardance. For polarizers, usually only the two principal transmittances or the extinction ratio is given. For polarization critical applications, this is inadequate. In the following sections, common defects of polarization elements are described. The Mueller calculus is recommended as an appropriate means of describing complex behaviors and shortcomings. For ideal polarization elements, the polarization properties are readily defined. For real polarization elements, the precise description is more complex. Polarization elements such as polarizers, retarders, and depolarizers have three general polarization properties: diattenuation, retardance, and depolarization; a typical element displays some amount of all three. Diattenuation occurs when the intensity transmittance is a function of the incident polarization state.24 The diattenuation D is defined in terms of the maximum Tmax and minimum Tmin intensity transmittances, as D

Tmax Tmin Tmax Tmin

(28)

For an ideal polarizer, D 1. When D 0, all incident polarization states are transmitted with equal attenuation. The quality of a polarizer is often expressed in terms of the related quantity, the extinction ratio E, E

Tmax 1 D Tmin 1 D

(29)

Retardance is the phase difference a device introduces between its eigenpolarizations (eigenstates). For a birefringent retarder with refractive indices n1 and n2 and thickness t, the retardance F expressed in radians is

D

2P | n1 n2 | t L

(30)

Depolarization describes the coupling by a device of incident polarized light into depolarized light in the exiting beam. Depolarization occurs when light transmits through milk or scatters from clouds. Multimode optical fibers generally depolarize the light. Depolarization is intrinsically associated with scattering and a loss of coherence of the polarization state. A small amount of depolarization is associated with the scattered light from all optical components. The depolarization varies as a function of the incident polarization state.25

15.26 ELLIPTICAL AND CIRCULAR POLARIZERS AND ANALYZERS There are few good and convenient circularly or elliptically polarizing mechanisms, whereas linear polarizers are simple, inexpensive, and of high quality. Therefore, most circular and elliptical polarizers incorporate linear polarizers to perform the polarizing, and retarders to convert between

15.18

POLARIZED LIGHT

polarization states. For such compound devices, the distinction between a polarizer and an analyzer is significant. This is illustrated by three examples: (1) a left circular polarizer (and horizontal linear analyzer) constructed from a horizontal linear polarizer LP(0°) followed by a quarter-wave linear retarder with the fast axis oriented at 135°, QWLR(135°) Eq. (31), (2) a left circular analyzer (and horizontal linear polarizer) constructed from a QWLR(45°) followed by a horizontal linear polarizer LP(0°) Eq. (32), and (3) a homogeneous left circular polarizer (and left circular analyzer) constructed from a QWLR(135°), then an LP(0°), followed by a QWLR(45°) Eq. (33). The three Mueller matrix equations and the exiting polarization states for arbitrary incident states are as follows: ¤1 0 ¥0 0 QWLR(135°)s,0(0°)sS ¥ ¥0 0 ¦ 0 1

0 0 1 0

¤1 1 1¥ 0 0 ¥ 2¥ 0 0 ¦ 1 1

0³ ¤1 1´ 1 ¥1 s 0´´ 2 ¥¥0 0µ ¦0 0 0 0 0

1 1 0 0

0 0 0 0

0³ ¤S0³ 0´ ¥ S1´ s¥ ´ 0´´ ¥S2´ 0µ ¥¦S3´µ

(31)

¤ S0 S1 ³ 0³ ¤S0³ 0´ ¥ S1´ 1 ¥ 0 ´ s¥ ´ ¥ ´ 0´´ ¥S2´ 2 ¥ 0 ´ ¥ ´ ¥ 0µ ¦S3µ ¦ S0 S1´µ

0³ ¤ 1 0´ ¥ 0 s 0´´ ¥¥ 0 0µ ¦ 0

0 0 ³ ¤S0³ 0 1´ ¥ S1´ s¥ ´ 1 0 ´´ ¥S2´ 0 0 µ ¥¦S3´µ

¤1 1 ¥1 LP(0°)s QWLR45° s S ¥ 2 ¥0 ¦0

1 1 0 0

0 0 0 0

¤1 1 ¥1 ¥ 2 ¥0 ¦0

0 0 0 0

¤S0 S3³ 0 1³ ¤S0³ 0 1´ ¥ S1´ 1 ¥S0 S3´ s¥ ´ ¥ ´ 0 0 ´´ ¥S2´ 2 ¥ 0 ´ ¥ ´ ¥ 0 0 µ ¦S3µ ¦ 0 ´µ

0 0 0 1

(32)

QWLR(135°)s,0(0°)sQWLR(45°)s3 ¤1 0 ¥0 0 ¥ ¥0 0 ¦0 1 ¤1 1¥ 0 ¥ 2¥ 0 ¦ 1

0 0 1 0 0 0 0 0

0³ ¤1 1´ 1 ¥1 0´´ 2 ¥¥0 0µ ¦0

1 1 0 0

0 0 0 0

0³ ¤1 0´ 1 ¥0 0´´ 2 ¥¥0 0µ ¦0

0 0 0 1

0 0 ³ ¤S0³ 0 1´ ¥ S1´ ¥ ´ 1 0 ´´ ¥S2´ 0 0 µ ¥¦S3´µ

(33)

¤ S0 S3 ³ 0 1³ ¤S0³ 0 0 ´ ¥ S1´ 1 ¥ 0 ´ s¥ ´ ¥ ´ 0 0 ´´ ¥S2´ 2 ¥ 0 ´ ¥¦ S S ´µ 0 1 µ ¥¦S3´µ 0 3

The device in Eq. (31) transmits only left circularly polarized light, because the S0 and S3 have equal magnitude and opposite sign; thus it is a left circular polarizer. However, the transmitted flux (S0 S1)/2 is the flux of horizontal linearly polarized light in the incident beam, making it a horizontal linear analyzer. Similarly, the transmitted flux in Eq. (32), (S0 S3)/2, is the flux of incident left circularly polarized light, making this combination a left circular analyzer. The final polarizer makes the Eq. (32) device a horizontal linear polarizer, although this is not the standard Mueller matrix for horizontal linear polarizers found in tables. Thus an analyzer for a state does not necessarily transmit that state;

POLARIMETRY

15.19

its transmitted flux is proportional to the amount of the analyzed state in the incident beam. The examples in Eqs. (31) and (32) are inhomogeneous polarization elements because the eigenpolarizations are not orthogonal. Equation (31) has left circular and vertically polarized eigenpolarizations. Equation (32) has horizontal and right circularly polarized eigenpolarizations. The characteristics of both devices are different for propagation in opposite directions. The Eq. (33) device is both a left circular polarizer and a left circular analyzer; it has the same characteristics for propagation in opposite directions. The eigenpolarizations are orthogonal, left and right circularly polarized, so this device is a homogeneous left circular polarizer. This is the left circular polarizer Mueller matrix commonly found in tables; however, it is not the most common implementation.

15.27 COMMON DEFECTS OF POLARIZATION ELEMENTS Here we list some common defects found in real polarization elements. 1. Polarizers have nonideal diattenuation since Tmin > 0 and also nonideal transmission since Tmax 0 is the refractive index and k q 0 is the extinction coefficient. The positive directions of p and s before and after reflection form a right-handed coordinate system with the directions of propagation of the incident and reflected waves, Fig. 1. At normal incidence (H0), the p directions in the incident and reflected waves are antiparallel, whereas the s directions are parallel. Some of the consequences of these conventions are as follows: 1. At normal incidence, Rp Rs, T 1, and $ R. 2. At grazing incidence, Rp Rs, T 1, and $ 0. 3. For an abrupt interface between two homogeneous and isotropic semi-infinite media, $ is in the range 0 a $ a R, and 0 a [ a 45°.

16.4

POLARIZED LIGHT

FIGURE 2 Ellipsometric parameters [ and $ of an air/Au interface as functions of the angle of incidence H. The complex refractive index of Au is assumed to be 0.306 – j2.880 at 564-nm wavelength. [, $, and H are in degrees.

As an example, Fig. 2 shows [ and $ vs. H for light reflection at the air/Au interface, assuming N 0.306 j2.880 for Au45 at N 564 nm.

16.4

MODELING AND INVERSION The following simplifying assumptions are usually made or implied in conventional ellipsometry: (1) the incident beam is approximated by a monochromatic plane wave; (2) the ambient or incidence medium is transparent and optically isotropic; (3) the sample surface is a plane boundary; (4) the sample (and ambient) optical properties are uniform laterally but may change in the direction of the normal to the ambient-sample interface; (5) the coherence length of the incident light is much greater than its penetration depth into the sample; and (6) the light-sample interaction is linear (elastic), hence frequency-conserving. Determination of the ratio of complex reflection coefficients is rarely an end in itself. Usually, one is interested in more fundamental information about the sample than is conveyed by T. In particular, ellipsometry is used to characterize the optical and structural properties of the interfacial region. This requires that a stratified-medium model (SMM) for the sample under measurement be postulated that contains the sample physical parameters of interest. For example, for visible light, a polished Si surface in air may be modeled as an optically opaque (semi-infinite) Si substrate which is covered with a SiO2 film, with the Si and SiO2 phases assumed uniform, and the air/SiO2 and SiO2/Si interfaces considered as parallel planes. This is often referred to as the three-phase model. More complexity (and more layers) can be built into this basic SMM to represent such finer details as the interfacial roughness and phase mixing, a damage surface layer on Si caused by polishing, or the possible presence of an outermost contamination film. Effective medium theories46–54 (EMTs) are used to calculate the dielectric functions of mixed phases based on their microstructure and component volume fractions; and the established theory of light reflection by stratified structures 55–60 is employed to calculate the ellipsometric function for an assumed set of model parameters. Finally, values of the model parameters are sought that best match the measured and computed values of T. Extensive data (obtained, e.g., using VASE) is required to determine the parameters of more complicated samples. The latter task, called the

ELLIPSOMETRY

16.5

inverse problem, usually employs linear regression analysis, 61–63 which yields information on parameter correlations and confidence limits. Therefore, the full practice of ellipsometry involves, in general, the execution and integration of three tasks: (1) polarization measurements that yield ratios of complex reflection coefficients, (2) sample modeling and the application of electromagnetic theory to calculate the ellipsometric function, and (3) solving the inverse problem to determine model parameters that best match the experimental and theoretically calculated values of the ellipsometric function. Confidence in the model is established by showing that complete spectra can be described in terms of a few wavelength-independent parameters, or by checking the predictive power of the model in determining the optical properties of the sample under new experimental conditions.27

The Two-Phase Model For a single interface between two homogeneous and isotropic media, 0 and 1, the reflection coefficients are given by the Fresnel formulas1

in which

r01 p (1 S0 0 S1 )/(1 S0 0 S1 )

(9)

r01s (S0 S1 )/(S0 S1 )

(10)

i N i2

i 0, 1

(11)

is the dielectric function (or dielectric constant at a given wavelength) of the ith medium, (12)

Si (i 0 sin 2 F )1/2

and H is the angle of incidence in medium 0 (measured from the interface normal). The ratio of complex reflection coefficients which is measured by ellipsometry is

R [sin F tan F ( sin 2 F)/2 ]/[sin F tan F ( sin 2 F )1/2 ]

(13)

where 1/0. Solving Eq. (13) for gives

1 0 {sin 2 F sin 2 F tan 2 F [(1 R)/(1 R)]2 }

(14)

For light incident from a medium (e.g., vacuum, air, or an inert ambient) of known 0, Eq. (14) determines, concisely and directly, the complex dielectric function 1 of the reflecting second medium in terms of the measured T and the angle of incidence H. This accounts for an important application of ellipsometry as a means of determining the optical properties (or optical constants) of bulk absorbing materials and opaque films. This approach assumes the absence of a transition layer or a surface film at the two-media interface. If such a film exists, ultrathin as it may be, 1 as determined by Eq. (14) is called the pseudo dielectric function and is usually written as 1 ®. Figure 3 shows lines of constant [ and lines of constant $ in the complex plane at F 75°.

The Three-Phase Model This often-used model, Fig. 4, consists of a single layer, medium 1, of parallel-plane boundaries which is surrounded by two similar or dissimilar semi-infinite media 0 and 2. The complex amplitude reflection coefficients are given by the Airy-Drude formula64,65 R (r01N r12N X )/(1 r01N r12N X ) X exp[ j 2P (d /DF )]

N p, s

(15) (16)

16.6

POLARIZED LIGHT

e –30 0

–20

–10

0

10

20

Δ = 165°

tan y = 0.9 –10

30

Δ = 150°

75°

0.8

0.2

90° 0.7

–20

0.3

105° 0.6

e –30

0.4

0.5 120°

–40

–50 135° j = 75° –60 FIGURE 3 Contours of constant tan [ and constant $ in the complex plane of the relative dielectric function of a transparent medium/absorbing medium interface.

rijP is the Fresnel reflection coefficient of the ij interface (ij 01 and 12) for the P polarization, d is the layer thickness, and DF (L /2)(1/S1 )

(17)

where N is the vacuum wavelength of light and S1 is given by Eq. (12). The ellipsometric function of this system is

R ( A BX CX 2 )/(D EX FX 2 ) A r01 p D r011s

B r12 p r01 p r01s r12 s E r12 s r01 p r01s r12 p

(18)

C r12 p r01s r12 s F r12 s r01 p r12 p

(19)

FIGURE 4

Three-phase, ambient-film-substrate system.

ELLIPSOMETRY

4

16.7

IM

3 75

65 2

70

60 55

80

1 85

RE –5

–4

–3

–2

–1

0

1

2

3

4

5

–2

–3

–4

FIGURE 5 Family of constant-angle-of-incidence contours of the ellipsometric function T in the complex plane for light reflection in air by the SiO2/Si film-substrate system at 633-nm wavelength. The contours are for angles of incidence from 30° to 85° in steps of 5°. The arrows indicate the direction of increasing film thickness.66

For a transparent film, and with light incident at an angle H such that 1 0 sin 2 F so that total reflection does not occur at the 01 interface, DF is real, and X, Rp, Rs, and T become periodic functions of the film thickness d with period DF . The locus of X is the unit circle in the complex plane and its multiple images through the conformal mapping of Eq. (18) at different values of H give the constant-angle-of-incidence contours of T. Figure 5 shows a family of such contours66 for light reflection in air by the SiO2–Si system at 633-nm wavelength at angles from 30° to 85° in steps of 5°. Each and every value of T, corresponding to all points in the complex plane, can be realized by selecting the appropriate angle of incidence and the SiO2 film thickness (within a period). If the dielectric functions of the surrounding media are known, the dielectric function 1 and thickness d of the film are obtained readily by solving Eq. (18) for X, X { (B RE ) o [(B RE )2 4(C RF )( A RD)]1/2 }/2(C RF)

(20)

66,67

and requiring that

| X | 1

(21)

Equation (21) is solved for 1 as its only unknown by numerical iteration. Subsequently, d is given by d [ arg( X )/2P ]DF mDF

(22)

where m is an integer. The uncertainty of an integral multiple of the film thickness period is often resolved by performing measurements at more than one wavelength or angle of incidence and requiring that d be independent of N or H.

16.8

POLARIZED LIGHT

FIGURE 6 Light reflection by a multilayer structure.1

When the film is absorbing (semitransparent), or the optical properties of one of the surrounding media are unknown, more general inversion methods68–72 are required which are directed toward minimizing an error function of the form N

f £[(Y im Y ic )2 ($ im $ ic )2 ]

(23)

i 1

where [im, [ic and $im, $ic denote the ith measured and calculated values of the ellipsometric angles, and N is the total number of independent measurements.

Multilayer and Graded-Index Films For an outline of the matrix theory of multilayer systems refer to Chap. 7, “Optical Properties of Films and Coatings,” in Vol. IV. For our purposes, we consider a multilayer structure, Fig. 6, that consists of m plane-parallel layers sandwiched between semi-infinite ambient and substrate media (0 and m 1, respectively). The relationships between the field amplitudes of the incident (i), reflected (r), and transmitted (t) plane waves for the p or s polarizations are determined by the scattering matrix equation73 §Ei ¶ §S11 Ë · ¨S © r ¸ © 21

S12 ¶ §Et ¶ S22 ·¸ ¨© 0 ·¸

(24)

The complex-amplitude reflection and transmission coefficients of the entire structure are given by R Er /Ei S21 /S11

(25)

T Et /Ei 1/S11 The scattering matrix S (Sij ) is obtained as an ordered product of all the interface I and layer L matrices of the stratified structure, S I01L1I12L 2 I( j 1) j L j L mIm(m1)

(26)

ELLIPSOMETRY

16.9

and the numbering starts from layer 1 (in contact with the ambient) to layer m (adjacent to the substrate) as shown in Fig. 6. The interface scattering matrix is of the form §1 Iab (1/t ab ) ¨ ©rab

rab ¶ 1 ·¸

(27)

where rab is the local Fresnel reflection coefficient of the ab[j(j 1)] interface evaluated [using Eqs. (9) and (10) with the appropriate change of subscripts] at an incidence angle in medium a which is related to the external incidence angle H in medium 0 by Snell’s law. The associated interface transmission coefficients for the p and s polarizations are t abp 2(a b )1/2 Sa /(b Sa a Sb )

(28)

t abs 2Sa /(Sa Sb ) where Sj is defined in Eq. (12). The scattering matrix of the jth layer is §Y 1 0 ¶ Lj ¨ j · ¨© 0 Y j ·¸ Y j X 1j /2

(29) (30)

and Xj is given by Eqs. (16) and (17) with the substitution d dj for the thickness, and 1 = j for the dielectric function of the jth layer. Except in Eqs. (28), a polarization subscript P p or s has been dropped for simplicity. In reflection and transmission ellipsometry, the ratios Rr R p / Rs and Rt Tp / Ts are measured. Inversion for the dielectric functions and thicknesses of some or all of the layers requires extensive data, as may be obtained by VASE, and linear regression analysis to minimize the error function of Eq. (23). Light reflection and transmission by a graded-index (GRIN) film is handled using the scattering matrix approach described here by dividing the inhomogeneous layer into an adequately large number of sublayers, each of which is approximately homogeneous. In fact, this is the most general approach for a problem of this kind because analytical closed-form solutions are only possible for a few simple refractive-index profiles.74–76

Dielectric Function of a Mixed Phase For a microscopically inhomogeneous thin film that is a mixture of two materials, as may be produced by coevaporation or cosputtering, or a thin film of one material that may be porous with a significant void fraction (of air), the dielectric function is determined using EMTs.46–54 When the scale of the inhomogeneity is small relative to the wavelength of light, and the domains (or grains) of different dielectrics are of nearly spherical shape, the dielectric function of the mixed phase is given by

h h h va a v b 2 h a 2 h b b 2 h

(31)

where a and b are the dielectric functions of the two component phases a and b with volume fractions Wa and Wb and h is the host dielectric function. Different EMTs assign different values to h. In the Maxwell Garnett EMT, 47,48 one of the phases, say b, is dominant (Wb Wa) and h b. This reduces the second term on the right-hand side of Eq. (31) to zero. In the Bruggeman EMT,49 Wa and Wb are comparable, and h , which reduces the left-hand side of Eq. (31) to zero.

16.10

POLARIZED LIGHT

FIGURE 7 Generic ellipsometer with polarizing optics PO and analyzing optics AO. L and D are the light source and photodetector, respectively.

16.5 TRANSMISSION ELLIPSOMETRY Although ellipsometry is typically carried out on the reflected wave, it is possible to also monitor the state of polarization of the transmitted wave, when such a wave is available for measurement.77–81 For example, by combining reflection and transmission ellipsometry, the thickness and complex dielectric function of an absorbing film between transparent media of the same refractive index (e.g., a solid substrate on one side and an index-matching liquid on the other) can be obtained analytically.79,80 Polarized light transmission by a multilayer was discussed previously under “Multilayer and Graded-Index Films.” Transmission ellipsometry can be carried out at normal incidence on optically anisotropic samples to determine such properties as the natural or induced linear, circular, or elliptical birefringence and dichroism. However, this falls outside the scope of this chapter.

16.6

INSTRUMENTATION Figure 7 is a schematic diagram of a generic ellipsometer. It consists of a source of collimated and monochromatic light L, polarizing optics PO on one side of the sample S, and polarization analyzing optics AO and a (linear) photodetector D on the other side. An apt terminology25 refers to the PO as a polarization state generator (PSG) and the AO plus D as a polarization state detector (PSD). Figure 8 shows the commonly used polarizer-compensator-sample-analyzer (PCSA) ellipsometer arrangement. The PSG consists of a linear polarizer with transmission-axis azimuth P and a linear retarder, or compensator, with fast-axis azimuth C. The PSD consists of a single linear polarizer, that functions as an analyzer, with transmission-axis azimuth A followed by a photodetector D.

FIGURE 8 Polarizer-compensator-sample-analyzer (PCSA) ellipsometer. The azimuth angles P of the polarizer, C of the compensator (or quarter-wave retarder), and A of the analyzer are measured from the plane of incidence, positive in a counterclockwise sense when looking toward the source.1

ELLIPSOMETRY

16.11

i

R

r L

FIGURE 9 Constant C, variable P contours (continuous lines), and constant P C, variable C contours (dashed lines) in the complex plane of polarization for light transmitted by a polarizer-compensator (PC) polarization state generator.1

All azimuths P, C, and A, are measured from the plane of incidence, positive in a counterclockwise sense when looking toward the source. The state of polarization of the light transmitted by the PSG and incident on S is given by

C i [tan C Rc tan(P C )]/[1 Rc tan C tan(P C )]

(32)

where Tc Tcs/Tcf is the ratio of complex amplitude transmittances of the compensator for incident linear polarizations along the slow s and fast f axes. Ideally, the compensator functions as a quarterwave retarder (QWR) and Tc j. In this case, Eq. (32) describes an elliptical polarization state with major-axis azimuth C and ellipticity angle (P C). (The tangent of the ellipticity angle equals the minor-axis-to-major-axis ratio and its sign gives the handedness of the polarization state, positive for right-handed states.) All possible states of total polarization Ei can be generated by controlling P and C. Figure 9 shows a family of constant C, variable P contours (continuous lines) and constant P C, variable C contours (dashed lines) as orthogonal families of circles in the complex plane of polarization. Figure 10 shows the corresponding contours of constant P and variable C. The points R and L on the imaginary axis at (0, 1) and (0, 1) represent the right- and left-handed circular polarization states, respectively. Null Ellipsometry The PCSA ellipsometer of Fig. 8 can be operated in two different modes. In the null mode, the output signal of the photodetector D is reduced to zero (a minimum) by adjusting the azimuth angles P of the polarizer and A of the analyzer with the compensator set at a fixed azimuth C. The choice C ± 45° results in rapid convergence to the null. Two independent nulls are reached for each compensator setting. The two nulls obtained with C 45° are usually referred to as the nulls in zones 2 and 4; those for C 45° define zones 1 and 3. At null, the reflected polarization is linear and is

16.12

POLARIZED LIGHT

FIGURE 10 Constant P, variable C contours in the complex plane of polarization for light transmitted by a polarizer-compensator (PC) polarization state generator.1

crossed with the transmission axis of the analyzer; therefore, the reflected state of polarization is given by

C r cot A

(33)

where A is the analyzer azimuth at null. With the incident and reflected polarizations determined by Eqs. (32) and (33), the ratio of complex reflection coefficients of the sample for the p and s linear polarizations T is obtained by Eq. (2). Whereas a single null is sufficient to determine T in an ideal ellipsometer, results from multiple nulls (in two or four zones) are usually averaged to eliminate the effect of small component imperfections and azimuth-angle errors. Two-zone measurements are also used to determine T of the sample and Tc of the compensator simultaneously.82–84 The effects of component imperfections have been considered extensively.85 The null ellipsometer can be automated by using stepping or servo motors86,87 to rotate the polarizer and analyzer under closed-loop feedback control; the procedure is akin to that of nulling an ac bridge circuit. Alternatively, Faraday cells can be inserted after the polarizer and before the analyzer to produce magneto-optical rotations in lieu of the mechanical rotation of the elements.88–90 This reduces the measurement time of a null ellipsometer from minutes to milliseconds. Large (±90°) Faraday rotations would be required for limitless compensation. Small ac modulation is often added for the precise localization of the null.

Photometric Ellipsometry The polarization state of the reflected light can also be detected photometrically by rotating the analyzer91–95 of the PCSA ellipsometer and performing a Fourier analysis of the output signal I of the linear photodetector D. The detected signal waveform is simply given by I I 0 (1 A cos 2A B sin 2A)

(34)

ELLIPSOMETRY

16.13

and the reflected state of polarization is determined from the normalized Fourier coefficients C and D by

C r [B o (1 A 2 B 2 )1/2 ]/(1 A )

(35)

The sign ambiguity in Eq. (35) indicates that the rotating-analyzer ellipsometer (RAE) cannot determine the handedness of the reflected polarization state. In the RAE, the compensator is not essential and can be removed from the input PO (i.e., the PSA instead of the PCSA optical train is used). Without the compensator, the incident linear polarization is described by

C i tan P

(36)

Again, the ratio of complex reflection coefficients of the sample T is determined by substituting Eqs. (35) and (36) in Eq. (2). The absence of the wavelength-dependent compensator makes the RAE particularly qualified for SE. The dual of the RAE is the rotating-polarizer ellipsometer which is suited for real-time SE using a spectrograph and a photodiode array that are placed after the fixed analyzer.31 A photometric ellipsometer with no moving parts, for fast measurements on the microsecond time scale, employs a photoelastic modulator96–100 (PEM) in place of the compensator of Fig. 8. The PEM functions as an oscillating-phase linear retarder in which the relative phase retardation is modulated sinusoidally at a high frequency (typically 50 to 100 kHz) by establishing an elastic ultrasonic standing wave in a transparent solid. The output signal of the photodetector is represented by an infinite Fourier series with coefficients determined by Bessel functions of the first kind and argument equal to the retardation amplitude. However, only the dc, first, and second harmonics of the modulation frequency are usually detected (using lock-in amplifiers) and provide sufficient information to retrieve the ellipsometric parameters of the sample. Numerous other ellipsometers have been introduced25 that employ more elaborate PSDs. For example Fig. 11 shows a family of rotating-element photopolarimeters25 (REP) that includes the RAE. The column on the right represents the Stokes vector and the fat dots identify the Stokes A

Detector

(a) RA A

A

Detector

A

Detector

(b) RAFA C

(c) RCFA C

A

C

A

Detector

(d) RCA A

Detector

(e) RCAFA FIGURE 11 Family of rotating-element photo-polarimeters (REP) and the Stokes parameters that they can determine.25

16.14

POLARIZED LIGHT

FIGURE 12

Rotating-detector ellipsometer (RODE).102

parameters that are measured. (For a discussion of the Stokes parameters, see Chap. 12 in this volume of the Handbook.) The simplest complete REP, that can determine all four Stokes parameters of light, is the rotating-compensator fixed-analyzer (RCFA) photopolarimeter originally invented to measure skylight polarization.101 The simplest handedness-blind REP for totally polarized light is the rotating-detector ellipsometer102,103 (RODE), Fig. 12, in which the tilted and partially reflective front surface of a solid-state (e.g., Si) detector performs as polarization analyzer.

Ellipsometry Using Four-Detector Photopolarimeters A new class of fast PSDs that measure the general state of partial or total polarization of a quasimonochromatic light beam is based on the use of four photodetectors. Such PSDs employ the division of wavefront, the division of amplitude, or a hybrid of the two, and do not require any moving parts or modulators. Figure 13 shows a division-of-wavefront photopolarimeter (DOWP) 104 for

p

p pp

l

FIGURE 13 Division-of-wavefront photopolarimeter for the simultaneous measurement of all four Stokes parameters of light.104

ELLIPSOMETRY

16.15

FIGURE 14 Division-of-amplitude photopolarimeter (DOAP) for the simultaneous measurement of all four Stokes parameters of light.107

performing ellipsometry with nanosecond laser pulses. The DOWP has been adopted recently in commercial automatic polarimeters for the fiber-optics market.105,106 Figure 14 shows a division-of-amplitude photopolarimeter107,108 (DOAP) with a coated beam splitter BS and two Wollaston prisms WP1 and WP2, and Fig. 15 represents a recent implementation109 of that technique. The multiple-beam-splitting and polarization-altering properties of grating diffraction are also well-suited for the DOAP.110,111 The simplest DOAP consists of a spatial arrangement of four solid-state photodetectors Fig. 16, and no other optical elements. The first three detectors (D0, D1, and D2) are partially specularly reflecting and the fourth (D3) is antirefiection-coated. The incident light beam is steered in such a way that the plane of incidence is rotated between successive oblique-incidence reflections, hence

"$*&"!%&&&&"$

$! &&"$ "$*&"!%&&!$&"$

$!

"##$ "$*$ !#"$* %$ &##$ "&"$ "!&$"$

%#&&! !&" #%"! $! "& %#&&$

'$&$ )( $&$$

'$!& &&"$ $$"$ )&

" #'&$ &" $ FIGURE 15

Recent implementation of DOAP.109

!#$"%%! "'

16.16

POLARIZED LIGHT

FIGURE 16 Four-detector photopolarimeter for the simultaneous measurement of all four Stokes parameters of light.112

the light path is nonplanar. In this four-detector photopolarimeter 112–117 (FDP), and in other DOAPs, the four output signals of the four linear photodetectors define a current vector I [I0 I1 I2 I3]t which is linearly related, I AS

(37)

to the Stokes vector S [S0 S1 S2 S3]t of the incident light, where t indicates the matrix transpose. The 4 × 4 instrument matrix A is determined by calibration115 (using a PSG that consists of a linear polarizer and a quarter-wave retarder). Once A is determined, S is obtained from the output signal vector by S A 1I

(38)

where A 1 is the inverse of A. When the light under measurement is totally polarized (i.e., S02 S12 S22 S32 ), the associated complex polarization number is determined in terms of the Stokes parameters as118

C (S2 jS3 )/(S0 S1 ) (S0 S1 )/(S2 jS3 )

(39)

For further information on polarimetry, see Chap. 15 in this volume.

Ellipsometry Based on Azimuth Measurements Alone Measurements of the azimuths of the elliptic vibrations of the light reflected from an optically isotropic surface, for two known vibration directions of incident linearly polarized light, enable the ellipsometric parameters of the surface to be determined at any angle of incidence. If Si and Sr represent the azimuths of the incident linear and reflected elliptical polarizations, respectively, then119–121 tan 2Qr (2 tanQi tanY cos $)/(tan 2 Y tan 2 Qi )

(40)

A pair of measurements (Si1, Sr1) and (Si2, Sr2) determines [ and $ via Eq. (40). The azimuth of the reflected polarization is measured precisely by an ac-null method using an ac-excited Faraday

ELLIPSOMETRY

16.17

FIGURE 17 Return-path ellipsometer. The dashed lines indicate the configuration for perpendicular-incidence ellipsometry on optically anisotropic samples.126

cell followed by a linear analyzer.119 The analyzer is rotationally adjusted to zero the fundamentalfrequency component of the detected signal; this aligns the analyzer transmission axis with the minor or major axis of the reflected polarization ellipse.

Return-Path Ellipsometry In a return-path ellipsometer (RPE), Fig. 17, an optically isotropic mirror M is placed in, and perpendicular to, the reflected beam. This reverses the direction of the beam, so that it retraces its path toward the source with a second reflection at the test surface S and second passage through the polarizing/analyzing optics P/A. A beam splitter BS sends a sample of the returned beam to the photodetector D. The RPE can be operated in the null or photometric mode. In the simplest RPE,122,123 the P/A optics consists of a single linear polarizer whose azimuth and the angle of incidence are adjusted for a zero detected signal. At null, the angle of incidence is the principal angle, hence $ ±90°, and the polarizer azimuth equals the principal azimuth, so that the incident linearly polarized light is reflected circularly polarized. Null can also be obtained at a general and fixed angle of incidence by adding a compensator to the P/A optics. Adjustment of the polarizer azimuth and the compensator azimuth or retardance produces the null.124,125 In the photometric mode,126 an element of the P/A is modulated periodically and the detected signal is Fourier-analyzed to extract [ and $. RPEs have the following advantages: (1) the same optical elements are used as polarizing and analyzing optics; (2) only one optical port or window is used for light entry into and exit from the chamber in which the sample may be mounted; and (3) the sensitivity to surface changes is increased because of the double reflection at the sample surface.

Perpendicular-Incidence Ellipsometry Normal-incidence reflection from an optically isotropic surface is accompanied by a trivial change of polarization due to the reversal of the direction of propagation of the beam (e.g., right-handed circularly polarized light is reflected as left-handed circularly polarized). Because this change of polarization is not specific to the surface, it cannot be used to determine the properties of the

16.18

POLARIZED LIGHT

FIGURE 18 Normal-incidence rotating-sample ellipsometer (NIRSE).128

reflecting structure. This is why ellipsometry of isotropic surfaces is performed at oblique incidence. However, if the surface is optically anisotropic, perpendicular-incidence ellipsometry (PIE) is possible and offers two significant advantages: (1) simpler single-axis instrumentation of the return-path type with common polarizing/analyzing optics, and (2) simpler inversion for the sample optical properties, because the equations that govern the reflection of light at normal incidence are much simpler than those at oblique incidence.127,128 Like RPE, PIE can be performed using null or photometric techniques.126–132 For example, Fig. 18 shows a simple normal-incidence rotating-sample ellipsometer128 (NIRSE) that is used to measure the ratio of the complex principal reflection coefficients of an optically anisotropic surface S with principal axes x and y. (The incident linear polarizations along these axes are the eigenpolarizations of reflection.) If we define

H (Rxx R yy )/(Rxx R yy )

(41)

H {a2 o j[8a4 (1 a4 ) a22 ]1/2 }/2(1 a4 )

(42)

then

Rxx and Ryy are the complex-amplitude principal reflection coefficients of the surface, and a2 and a4 are the amplitudes of the second and fourth harmonic components of the detected signal normalized with respect to the dc component. From Eq. (41), we obtain

R R yy /Rxx (1 H)/(1 H)

(43)

PIE can be used to determine the optical properties of bare and coated uniaxial and biaxial crystal surfaces.127–130,133

Interferometric Ellipsometry Ellipsometry using interferometer arrangements with polarizing optical elements has been suggested and demonstrated.134–136 Compensators are not required because the relative phase shift is obtained by the unbalance between the two interferometer arms; this offers a distinct advantage for SE. Direct display of the polarization ellipse is possible.134–136

ELLIPSOMETRY

16.7

16.19

JONES-MATRIX GENERALIZED ELLIPSOMETRY For light reflection at an anisotropic surface, the p and s linear polarizations are not, in general, the eigenpolarizations of reflection. Consequently, the reflection of light is no longer described by Eqs. (1). Instead, the Jones (electric) vectors of the reflected and incident waves are related by §Erp ¶ §Rpp ¨ · ¨R ©Ers ¸ ¨© sp

Rps ¶ §Eip ¶ · Rss ·¸ ¨©Eis ·¸

(44)

or, more compactly, E r RE i

(45)

where R is the nondiagonal reflection Jones matrix. The states of polarization of the incident and reflected waves, described by the complex variables Ei and Er of Eqs. (4), are interrelated by the bilinear transformation85,137

C r (Rss C i Rsp )/(Rps C i Rpp )

(46)

In generalized ellipsometry (GE), the incident wave is polarized in at least three different states (Ei1, Ei2, Ei3) and the corresponding states of polarization of the reflected light (Er1, Er2, Er3) are measured. Equation (46) then yields three equations that are solved for the normalized Jones matrix elements, or reflection coefficients ratios,138 Rpp /Rss ( C i 2 C i1H )/( C r1 C r 2 H ) Rps /Rss (H 1)/( C r1 C r 2 H )

(47)

Rsp /Rss ( C i 2 C r1 C i1 C r 2 H )/( C r1 C r 2 H ) H ( C r 3 C r1 )( C i 3 C i 2 )/( C i 3 C i1 )( C r 3 C r 2 ) Therefore, the nondiagonal Jones matrix of any optically anisotropic surface is determined, up to a complex constant multiplier, from the mapping of three incident polarizations into the corresponding three reflected polarizations. A PCSA null ellipsometer can be used. The incident polarization Ei is given by Eq. (32) and the reflected polarization Er is given by Eq. (33). Alternatively, the Stokes parameters of the reflected light can be measured using the RCFA photopolarimeter, the DOAP, or the FDP, and Er is obtained from Eq. (39). More than three measurements can be taken to overdetermine the normalized Jones matrix elements and reduce the effect of component imperfections and measurement errors. GE can be performed based on azimuth measurements alone.139 The main application of GE has been the determination of the optical properties of crystalline materials.138–143

16.8

MUELLER-MATRIX GENERALIZED ELLIPSOMETRY The most general representation of the transformation of the state of polarization of light upon reflection or scattering by an object or sample is described by1 S` MS

(48)

where S and S` are the Stokes vectors of the incident and scattered radiation, respectively, and M is the real 4 × 4 Mueller matrix that succinctly characterizes the linear (or elastic) light-sample interaction.

16.20

POLARIZED LIGHT

y

S

x

y x

C C P

A 5

L

FIGURE 19

D

Dual-rotating-retarder Mueller-matrix photopolarimeter.145

For light reflection at an optically isotropic and specular (smooth) surface, the Mueller matrix assumes the simple form144 §1 a 0 0¶ ä 1 0 0· Mr¨ (49) · ¨0 0 b c · ¨©0 0 c b·¸ In Eq. (49), r is the surface power reflectance for incident unpolarized or circularly polarized light, and a, b, c are determined by the ellipsometric parameters [ and $ as: a cos 2Y

b sin 2Y cos $

and

c sin 2Y sin $

(50)

and satisfy the identity a2 b2 c2 1. In general (i.e., for an optically anisotropic and rough surface), all 16 elements of M are nonzero and independent. Several methods for Mueller matrix measurements have been developed.25,145–149 An efficient scheme145–147 uses the PCSCÀ ellipsometer with symmetrical polarizing (PC) and analyzing (CÀ) optics, Fig. 19. All 16 elements of the Mueller matrix are encoded onto a single periodic detected signal by rotating the quarter-wave retarders (or compensators) C and C ` at angular speeds in the ratio 1:5. The output signal waveform is described by the Fourier series 12

I a0 £(an cos nC bn sin nC )

(51)

n1

where C is the fast-axis azimuth of the slower of the two retarders, measured from the plane of incidence. Table 1 gives the relations between the signal Fourier amplitudes and the elements of the TABLE 1

Relations Between Signal Fourier Amplitudes and Elements of the Scaled Mueller Matrix M^

n

0

an

m11 ` 12 m12 ` 12 m21 ` 14 m22 `

1 0 m14 ` 12 m24 `

bn

2

3

4

5

14 m22 `

14 m43 `

12 m44 `

0

1 m ` 14 m23 ` 2 13

14 m42 `

0

m41 ` 12 m42 `

1 m` 2 12

n

7

an

1 m` 4 43

1 m` 8 22

18 m33 `

1 m` 4 34

1 m ` 14 m22 ` 2 21

14 m34 `

bn

1 m` 4 42

1 m ` 18 m32 ` 8 23

1 m` 4 24

1 m ` 14 m32 ` 2 31

1 m` 4 24

8

6 1 m` 2 44

9

10

11

0 12

1 m` 8 22

18 m33 `

1 m ` 18 m32 ` 8 23

The transmission axes of the polarizer and analyzer are assumed to be set at 0 azimuth, parallel to the scattering plane or the plane of incidence.

ELLIPSOMETRY

16.21

FIGURE 20 Scheme for Mueller-matrix measurement using the fourdetector photopolarimeter.152

Mueller matrix M` which differs from M only by a scale factor. Inasmuch as only the normalized Mueller matrix, with unity first element, is of interest, the unknown scale factor is immaterial. This dual-rotating-retarder Mueller-matrix photopolarimeter has been used to characterize rough surfaces150 and the retinal nerve-fiber layer.151 Another attractive scheme for Mueller-matrix measurement is shown in Fig. 20. The FDP (or equivalently, any other DOAP) is used as the PSD. Fourier analysis of the output current vector of the FDP, I(C), as a function of the fast-axis azimuth C of the QWR of the input PO readily determines the Mueller matrix M, column by column.152,153

16.9 APPLICATIONS The applications of ellipsometry are too numerous to try to cover in this chapter. The reader is referred to the books and review articles listed in the bibliography. Suffice it to mention the general areas of application. These include: (1) measurement of the optical properties of materials in the visible, IR, and near-UV spectral ranges. The materials may be in bulk or thin-film form and may be optically isotropic or anisotropic.3,22,27–31 (2) Thin-film thickness measurements, especially in the semiconductor industry.2,5,24 (3) Controlling the growth of optical multilayer coatings154 and quantum wells.155,156 (4) Characterization of physical and chemical adsorption processes at the vacuum/solid, gas/solid, gas/liquid, liquid/liquid, and liquid/solid interfaces.26,157 (5) Study of the oxidation kinetics of semiconductor and metal surfaces in various gaseous or liquid ambients.158 (6) Electrochemical investigations of the electrode/electrolyte interface.18,19,32 (7) Diffusion and ion implantation in solids.159 (8) Biological and biomedical applications.16,20,151,160

16.10


R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light, North-Holland, Amsterdam, 1987. K. Riedling, Ellipsometry for Industrial Applications, Springer-Verlag, New York, 1988. R. Röseler, Infrared Spectroscopic Ellipsometry, Akademie-Verlag, Berlin, 1990. H. G. Tompkins, and W. A. McGahan, Spectroscopic Ellipsometry and Reflectometry: A User’s Guide, Wiley, New York, 1999. 5. H. G. Tompkins and E. A. Irene (eds.), Handbook of Ellipsometry, William Andrew, Norwich, New York, 2005. 6. R. M. A. Azzam (ed.), Selected Papers on Ellipsometry, vol. MS 27 of the Milestone Series, SPIE, Bellingham, Wash., 1991.

16.22

POLARIZED LIGHT

7. E. Passaglia, R. R. Stromberg, and J. Kruger (eds.), Ellipsometry in the Measurement of Surfaces and Thin Films, NBS Misc. Publ. 256, USGPO, Washington, D.C., 1964. 8. N. M. Bashara, A. B. Buckman, and A. C. Hall (eds.), Recent Developments in Ellipsometry, Surf. Sci. vol. 16, North-Holland, Amsterdam, 1969. 9. N. M. Bashara and R. M. A. Azzam (eds.), Proceedings of the Third International Conference on Ellipsometry, Surf. Sci. vol. 56, North-Holland, Amsterdam, 1976. 10. R. H. Muller, R. M. A. Azzam, and d. E. Aspnes (eds.), Proceedings of the Fourth International Conference on Ellipsometry, Surf. Sci. vol. 96, North-Holland, Amsterdam, 1980. 11. Proceedings of the International Conference on Ellipsometry and Other Optical Methods for Surface and Thin Film Analysis, J. de Physique, vol. 44, Colloq. C10, Les Editions de Physique, Paris, 1984. 12. A. C. Boccara, C. Pickering, and J. Rivory (eds.), Proceedings of the First International Conference on Spectroscopic Ellipsometry, Thin Solid Films, vols. 233 and 234, Elsevier, Amsterdam, 1993. 13. R. W. Collins, D. E. Aspnes, and E. A. Irene (eds.), Proceedings of the 2nd International Conference on Spectroscopic Ellipsometry, Thin Solid Films, vols. 313 and 314, Elsevier, Amsterdam, 1998. 14. M. Fried, K. Hingerl, and J. Humlicek (eds.), Proceedings of the 3rd International Conference on Spectroscopic Ellipsometry, Thin Solid Films, vols. 455 and 456, Elsevier, Amsterdam, 2004. 15. H. Arwin, U. Beck, and M. Schubert (eds.), Proceedings of the 4th International Conference on Spectroscopic Ellipsometry, Wiley-VCH, Weinheim, 2008. 16. G. Poste and C. Moss, “The Study of Surface Reactions in Biological Systems by Ellipsometry,” in S. G. Davison (ed.), Progress in Surface Science, vol. 2, pt. 3, Pergamon, New York, 1972, pp. 139–232. 17. R. H. Muller, “Principles of Ellipsometry,” in R. H. Mueller (ed.), Advances in Electrochemistry and Electrochemical Engineering, vol. 9, Wiley, New York, 1973, pp. 167–226. 18. J. Kruger, “Application of Ellipsometry in Electrochemistry,” in R. H. Muller (ed.), Advances in Electrochemistry and Electrochemical Engineering, vol. 9, Wiley, New York, 1973, pp. 227–280. 19. W.-K. Paik, “Ellipsometric Optics with Special Reference to Electrochemical Systems,” in J. O’M. Bockris (ed.), MTP International Review of Science, Physical Chemistry, series 1, vol. 6, Butterworths, Univ. Park, Baltimore, 1973, pp. 239–285. 20. A. Rothen, “Ellipsometric Studies of Thin Films,” in D. A. Cadenhead, J. F. Danielli, and M. D. Rosenberg (eds.), Progress in Surface and Membrane Science, vol. 8, Academic, New York, 1974, pp. 81–118. 21. R. H. Muller, “Present Status of Automatic Ellipsometers,” Surf. Sci. 56:19–36 (1976). 22. D. E. Aspnes, “Spectroscopic Ellipsometry of Solids,” in B. O. Seraphin (ed.), Optical Properties of Solids: New Developments, North-Holland, Amsterdam, 1976, pp. 799–846. 23. W. E. J. Neal, “Application of Ellipsometry to Surface Films and Film Growth,” Surf. Technol. 6:81–110 (1977). 24. A. V. Rzhanov and K. K. Svitashev, “Ellipsometric Techniques to Study Surfaces and Thin Films,” in L. Marton and C. Marton (eds.), Advances in Electronics and Electron Physics, vol. 49, Academic, New York, 1979, pp. 1–84. 25. P. S. Hauge, “Recent Developments in Instrumentation in Ellipsometry,” Surf. Sci. 96:108–140 (1980). 26. F. H. P. M. Habraken, O. L. J. Gijzeman, and G. A. Bootsma, “Ellipsometry of Clean Surfaces, Submonolayer and Monolayer Films,” Surf. Sci. 96:482–507 (1980). 27. D. E. Aspnes “Characterization of Materials and Interfaces by Visible-Near UV Spectrophotometry and Ellipsometry,” J. Mat. Educ. 7:849–901 (1985). 28. P. J. McMarr, K. Vedam, and J. Narayan, “Spectroscopic Ellipsometry: A New Tool for Nondestructive Depth Profiling and Characterization of Interfaces,” J. Appl. Phys. 59:694–701 (1986). 29. D. E. Aspnes, “Analysis of Semiconductor Materials and Structures by Spectroellipsometry,” SPIE Proc. 946:84–97 (1988). 30. R. Drevillon, “Spectroscopic Ellipsometry of Ultrathin Films: From UV to IR,” Thin Solid Films 163:157– 166 (1988). 31. R. W. Collins and Y.-T. Kim,“Ellipsometry for Thin-Film and Surface Analysis,” Anal. Chem. 62:887A–900A (1990). 32. R. H. Muller, “Ellipsometry as an In Situ Probe for the Study of Electrode Processes,” in R. Varma and J. R. Selman (eds.), Techniques for Characterization of Electrode Processes, Wiley, New York, 1991.

ELLIPSOMETRY

33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50. 51. 52. 53. 54. 55. 56. 57. 58. 59. 60. 61. 62. 63. 64. 65. 66. 67. 68. 69. 70. 71. 72. 73. 74. 75. 76.

16.23

A. Rothen, Rev. Sci. Instrum. 16:26–30 (1945). A. Rothen, in Ref. 7, pp. 7–21. A. C. Hall, Surf. Sci. 16:1–13 (1969). R. M. A. Azzam and N. M. Bashara, Ref. 1, sec. 2.6.1. R. M. A. Azzam and N. M. Bashara, Ref. 1, sec. 1.7. M. M. Ibrahim and N. M. Bashara, J. Opt. Soc. Am. 61:1622–1629 (1971). O. Hunderi, Surface Sci. 61:515–520 (1976). J. Humlíˇcek, J. Opt. Soc. Am. A 2:713–722 (1985). Y. Gaillyová, E. Schmidt, and J. Humlíˇcek, J. Opt. Soc. Am. A 2:723–726 (1985). W. H. Weedon, S. W. McKnight, and A. J. Devaney, J. Opt. Soc. Am. A 8:1881–1891 (1991). J. A. Woollam Co., Lincoln, NE 68508. R. H. Muller, Surf. Sci. 16:14–33 (1969). E. D. Palik, Handbook of Optical Constants of Solids, Academic, New York, 1985, p. 294. L. Lorenz, Ann. Phys. Chem. (Leipzig) 11:70–103 (1880). J. C. Maxwell Garnett, Philos. Trans. R. Soc. London 203:385–420 (1904). J. C. Maxwell Garnett, Philos. Trans. R. Soc. London 205:237–288 (1906). D. A. G. Bruggeman, Ann. Phys. (Leipzig) 24:636–679 (1935). C. G. Granqvist and O. Hunderi, Phys. Rev. B18:1554–1561 (1978). C. Grosse and J.-L. Greffe, J. Chim. Phys. 76:305–327 (1979). D. E. Aspnes, J. B. Theeten and F. Hottier, Phys. Rev. B20:3292–3302 (1979). D. E. Aspnes, Am. J. Phys. 50:704–709 (1982). D. E. Aspnes, Physica 117B/118B:359–361 (1983). F. Abelès Ann. de Physique 5:596–640 (1950). P. Drude, Theory of Optics, Dover, New York, 1959. J. R. Wait, Electromagnetic Waves in Stratified Media, Pergamon, New York, 1962. O. S. Heavens, Optical Properties of Thin Solid Films, Dover, New York, 1965. Z. Knittl, Optics of Thin Films, Wiley, New York, 1976. J. Lekner, Theory of Reflection, Marinus Nijhoff, Dordrecht, 1987. E. S. Keeping, Introduction to Statistical Inference, Van Nostrand, Princeton, 1962, chap. 12. D. W. Marquardt, SIAM J. Appl. Math. 11:431–441 (1963). J. R. Rice, in Numerical Solutions of Nonlinear Problems, Computer Sci. Center, University of Maryland, College Park, 1970. G. B. Airy, Phil. Mag. 2:20 (1833). P. Drude, Annal. Phys. Chem. 36:865–897 (1889). R. M. A. Azzam, A.-R. M. Zaghloul, and N. M. Bashara, J. Opt. Soc. Am. 65:252–260 (1975). A. R. Reinberg, Appl. Opt. 11:1273–1274 (1972). F. L. McCrackin and J. P. Colson, in Ref. 7, pp. 61–82. M. Malin and K. Vedam, Surf. Sci. 56:49–63 (1976). D. I. Bilenko, B. A. Dvorkin, T. Y. Druzhinina, S. N. Krasnobaev, and V. P. Polyanskaya, Opt. Spedrosc. 55:533–536 (1983). G. H. Bu-Abbud, N. M. Bashara and J. A. Woollam, Thin Solid Films 138:27–41 (1986). G. E. Jellison Jr., Appl. Opt. 30:3354–3360 (1991). R. M. A. Azzam and N. M. Bashara, Ref. 1 sec. 4.6, and references cited therein. F. Abelès, Ref. 7, pp. 41–58. J. C. Charmet and P. G. de Gennes, J. Opt. Soc. Am. 73:1777–1784 (1983). J. Lekner, Ref. 60, chap. 2.

16.24

POLARIZED LIGHT

77. 78. 79. 80. 81. 82. 83. 84. 85. 86. 87. 88. 89. 90. 91. 92. 93. 94. 95. 96. 97. 98. 99. 100. 101. 102. 103. 104. 105. 106. 107. 108. 109. 110. 111. 112. 113. 114. 115. 116. 117. 118. 119. 120. 121.

R. M. A. Azzam, M. Elshazly-Zaghloul, and N. M. Bashara, Appl. Opt. 14:1652–1663 (1975). I. Ohlídal and F. Lukeš, Thin Solid Films 85:181–190 (1981). R. M. A. Azzam, J. Opt. Soc. Am. 72:1439–1440 (1982). R. M. A. Azzam, Ref. 11, pp. 67–70. I. Ohlídal and F. Lukeš, Thin Solid Films 115:269–282 (1984). F. L. McCrackin, J. Opt. Soc. Am. 60:57–63 (1970). J. A. Johnson and N. M. Bashara, J. Opt. Soc. Am. 60:221–224 (1970). R. M. A. Azzam and N. M. Bashara, J. Opt. Soc. Am. 62:222–229 (1972). R. M. A. Azzam and N. M. Bashara, Ref. 1, sees. 3.7 and 3.8 and references cited therein. H. Takasaki, Appl. Opt. 5:759–764 (1966). J. L. Ord, Appl. Opt. 6:1673–1679 (1967). A. B. Winterbottom, Ref. 7, pp. 97–112. H. J. Mathiu, D. E. McClure, and R. H. Muller, Rev. Sci. Instrum. 45:798–802 (1974). R. H. Muller and J. C. Farmer, Rev. Sci. Instrum. 55:371–374 (1984). W. Budde, Appl. Opt. 1:201–205 (1962). B. D. Cahan and R. F. Spanier, Surf. Sci. 16:166–176 (1969). R. Greef, Rev. Sci. Instrum. 41:532–538 (1970). P. S. Hauge and F. H. Dill, IBM J. Res. Develop. 17:472–489 (1973). D. E. Aspnes and A. A. Studna, Appl. Opt. 14:220–228 (1975). M. Billardon and J. Badoz, C. R. Acad. Sci. (Paris) 262:1672 (1966). J. C. Kemp, J. Opt. Soc. Am. 59:950–954 (1969). S. N. Jasperson and S. E. Schnatterly, Rev. Sci. Instrum. 40:761–767 (1969). J. Badoz, M. Billardon, J. C. Canit, and M. F. Russel, J. Opt. (Paris) 8:373–384 (1977). V. M. Bermudez and V. H. Ritz, Appl. Opt. 17:542–552 (1978). Z. Sekera, J. Opt. Soc. Am. 47:484–490 (1957). R. M. A. Azzam, Opt. Lett. 10:427–429 (1985). D. C. Nick and R. M. A. Azzam, Rev. Sci. Instrum. 60:3625–3632 (1989). E. Collett, Surf. Sci. 96:156–167 (1980). Lightwave Polarization Analyzer Systems, Hewlett Packard Co., Palo Alto, California 94303. Polarscope, Electro Optic Developments Ltd, Basildon, Essex SS14 3BE, England. R. M. A. Azzam, Opt. Acta 29:685–689 (1982). R. M. A. Azzam, Opt. Acta 32:1407–1412 (1985). S. Krishnan, J. Opt. Soc. Am. A 9:1615–1622 (1992). R. M. A. Azzam, Appl. Opt. 31:3574–3576 (1992). R. M. A. Azzam and K. A. Giardina, J. Opt. Soc. Am. A 10:1190–1196 1993. R. M. A. Azzam, Opt. Lett. 10:309–311 (1985); U.S. Patent 4,681,450, July 21, 1987. R. M. A. Azzam, E. Masetti, I. M. Elminyawi, and F. G. Grosz, Rev. Sci. Instrum. 59:84–88(1988). R. M. A. Azzam, I. M. Elminyawi, and A. M. El-Saba, J. Opt. Soc. Am. A 5:681–689 (1988). R. M. A. Azzam and A. G. Lopez, J. Opt. Soc. Am. A 6:1513–1521 (1989). R. M. A. Azzam, J. Opt. Soc. Am. A 7:87–91 (1990). The Stokesmeter, Gaertner Scientific Co., Chicago, Illinois 60614. P. S. Hauge, R. H. Muller, and C. G. Smith, Surf. Sci. 96:81–107 (1980). J. Monin and G.-A. Boutry, Nouv. Rev. Opt. 4:159–169 (1973). C. Som and C. Chowdhury, J. Opt. Soc. Am. 62:10–15 (1972). S. I. Idnurm, Opt. Spectrosc. 42:210–212 (1977).

ELLIPSOMETRY

122. 123. 124. 125. 126. 127. 128. 129. 130. 131. 132. 133. 134. 135. 136. 137. 138. 139. 140. 141. 142. 143. 144. 145. 146. 147. 148. 149. 150. 151. 152. 153. 154. 155. 156. 157. 158. 159. 160.

16.25

H. M. O’Bryan, J. Opt. Soc. Am. 26:122–127 (1936). M. Yamamoto, Opt. Commun. 10:200–202 (1974). T. Yamaguchi and H. Takahashi, Appl. Opt. 15:677–680 (1976). R. M. A. Azzam, Opt. Acta 24:1039–1049 (1977). R. M. A. Azzam, J. Opt. (Paris) 9:131–134 (1978). R. M. A. Azzam, Opt. Commun. 19:122–124 (1976); Opt. Commun. 20:405–408 (1977) Y. Cui and R. M. A. Azzam, Appl. Opt. 35:2235–2238 (1996). R. H. Young and E. I. P. Walker, Phys. Rev. B 15:631–637 (1977). D. W. Stevens, Surf. Sci. 96:174–201 (1980). R. M. A. Azzam, J. Opt. (Paris) 12:317–321 (1981). R. M. A. Azzam, Opt. Eng. 20:58–61 (1981). R. M. A. Azzam, Appl. Opt. 19:3092–3095 (1980). A. L. Dmitriev, Opt. Spectrosc. 32:96–99 (1972). H. F. Hazebroek and A. A. Holscher, J. Phys. E: Sci. Instrum. 6:822–826 (1973). R. Calvani, R. Caponi, and F. Cisternino, J. Light. Technol. LT4:877–883 (1986). R. M. A. Azzam and N. M. Bashara, J. Opt. Soc. Am. 62:336–340 (1972). R. M. A. Azzam and N. M. Bashara, J. Opt. Soc. Am. 64:128–133 (1974). R. M. A. Azzam, J. Opt. Soc. Am. 68:514–518 (1978). D. J. De Smet, J. Opt. Soc. Am. 64:631–638 (1974); J. Opt. Soc. Am. 65:542–547 (1975). P. S. Hauge, Surf. Sci. 56:148–160 (1976). M. Elshazly-Zaghloul, R. M. A. Azzam, and N. M. Bashara, Surf. Sci. 56:281–292 (1976). M. Schubert, Theory and Applications of Generalized Ellipsometry, in Ref. 5, pp. 637–717. R. M. A. Azzam and N. M. Bashara, Ref. 1, p. 491. R. M. A. Azzam, Opt. Lett. 2:148–150 (1978); U. S. Patent 4, 306, 809, December 22, 1981. P. S. Hauge, J. Opt. Soc. Am. 68:1519–1528 (1978). D. H. Goldstein, Appl. Opt. 31:6676–6683 (1992). A. M. Hunt and D. R. Huffman, Appl. Opt. 17:2700–2710 (1978). R. C. Thompson, J. R. Bottiger, and E. S. Fry, Appl. Opt. 19:1323–1332 (1980). D. A. Ramsey, Thin Film Measurements on Rough Substrates using Mueller-Matrix Ellipsometry, Ph.D. thesis, The University of Michigan, Ann Arbor, 1985. A. W. Dreher, K. Reiter, and R. N. Weinreb, Appl. Opt. 31:3730–3735 (1992). R. M. A. Azzam, Opt. Lett. 11:270–272 (1986). R. M. A. Azzam, K. A. Giardina, and A. G. Lopez, Opt. Eng. 30:1583–1589 (1991). Ph. Houdy, Rev. Phys. Appl. 23:1653–1659 (1988). J. B. Theeten, F. Hottier, and J. Hallais, J. Crystal Growth 46:245–252 (1979). D. E. Aspnes, W. E. Quinn, M. C. Tamargo, M. A. A. Pudensi, S. A. Schwarz, M. J. S. Brasil, R. E. Nahory, and S. Gregory, Appl. Phys. Lett. 60:1244–1247 (1992). R. M. A. Azzam and N. M. Bashara, Ref. 1, sec. 6.3. R. M. A. Azzam and N. M. Bashara, Ref. 1, sec. 6.4. D. E. Aspnes and A. A. Studna, Surf. Sci. 96:294–306 (1980). H. Arwin, Ellipsometry in Life Sciences, in Ref. 5, pp. 799–855.


PART

4 COMPONENTS


17 LENSES R. Barry Johnson Consultant Huntsville, Alabama

17.1

GLOSSARY a, b AChr AST b bf l Co c C1 C2 CC CMAs CMAt Dep do de E ef l ep FN f h Hi K i J1() k

first and second lenses axial chromatic aberration astigmatism factor back focal length scaling factor curvature scaling factor scaling factor conic constant sagittal coma tangential coma diameter of entrance pupil distance from object to loupe distance from loupe to the eye irradiance effective focal length eyepiece F-number focal length height above axis height of ray intercept in image plane shape factor image Bessel function of the first kind 2R/N 17.3

17.4

COMPONENTS

L MP m moc m MTF M n NA o obj P Pi p s SA3 SAC si sot so TPAC t u V W x, y, z D F \ $S S N v H r U V 7

17.2

length magnifying power [cf. linear lateral longitudinal magnification] linear, lateral magnification nodal-point to optical-center magnification linear, longitudinal magnification modulation transfer function factor refractive index numerical aperture object objective partial dispersion principal points sd /fa peak normalized spectral weighting function object to image distance third-order spherical aberration secondary angular spectrum image distance optical tube length object distance transverse primary chromatic aberration thickness slope Abbe number or reciprocal dispersion L-normalized reciprocal object distance 1/soL cartesian coordinates angular blur diameter depth of focus sag angular blur tolerance field of view wavelength spatial frequency lens power radius standard deviation of the irradiance distribution transmission normalized spatial frequency

INTRODUCTION This chapter provides a basic understanding of using lenses for image formation and manipulation. The principles of image formation are reviewed first. The effects of lens shape, index of refraction, magnification, and F-number on the image quality of a singlet lens are discussed in some detail. Achromatic doublets and more complex lens systems are covered next. A representative variety of lenses is analyzed

LENSES

17.5

FIGURE 1

Imaging by a simple lens.

and discussed. Performance that may be expected of each class of lens is presented. The section concludes with several techniques for rapid estimation of the performance of lenses. Refer to Chap. 1 “Geometrical Optics,” in this volume for further discussion of geometrical optics and aberrations.

17.3

BASICS Figure 1 illustrates an image being formed by a simple lens. The object height is ho and the image height is hi with uo and ui being the corresponding slope angles. It follows from the Lagrange invariant that the lateral magnification is defined to be mx

hi ho

(nu)0 (nu)i

(1)

where no and ni are the refractive indices of the medium in which the object and image lie, respectively. By convention, a height is positive if above the optical axis and a ray angle is positive if its slope angle is positive. Distances are positive if the ray propagates left to right. Since the Lagrange invariant is applicable for paraxial rays, the angle nu should be understood to mean n tan u. This interpretation applies to all paraxial computations. For an aplanatic lens, which is free of spherical aberration and linear coma, the magnification can by the optical sine theorem be given by hi ho

mx

no sin uo ni sin ui

(2)

If the object is moved a small distance vso longitudinally, the corresponding displacement of the image vsi can be found by the differential form of the basic imaging equation and leads to an equation analogous to the Lagrange invariant. The longitudinal magnification is then defined as ts mx i t so

(nu 2 )0 (nu 2 )i

(3)

§n ¶ m2 ¨ i · ©no ¸ The following example will illustrate one application of m and m . Consider that a spherical object of radius ro is to be imaged as shown in Fig. 2. The equation of the object is ro2 yo2 z 2, where z is measured along the optical axis and is zero at the object’s center of curvature. Letting the surface sag

17.6

COMPONENTS

FIGURE 2 Imaging of a spherical object by a lens.

as measured from the vertex plane of the object be denoted as \o the equation of the object becomes ro2 (ro Z o )2 yo2 since z ro \o. In the region near the optical axis, Z o2 ro2 , which implies that ro y yo2 /2Z o . The image of the object is expressed in the transverse or lateral direction by yi myo and in the longitudinal or axial direction by Z i mZ o Z om 2 (ni /no ). In a like manner, the image of the spherical object is expressed as ri y (yi)2/2\i . By substitution, the sag of the image is expressed by ri x

no yo2 2niXo

(4)

§n ¶ ro ¨ o · © ni ¸ Hence, in the paraxial region about the optical axis, the radius of the image of a spherical object is independent of the magnification and depends only on the ratio of the refractive indices of the object and image spaces. When an optical system as shown in Fig. 3 images a tilted object, the image will also be tilted. By employing the concept of lateral and longitudinal magnification, it can be easily shown that the

FIGURE 3 Imaging of a tilted object illustrating the Scheimpflug condition.

LENSES

17.7

intersection height of the object plane with the first principal plane P1 of the lens must be the same as the intersection height of the image plane with the second principal plane P2 of the lens. This principle is known as the Scheimpflug condition. The object-image relationship of a lens system is often described with respect to its cardinal points, which are the principal and focal points (conceived by Carl Gauss in 1841) and nodal points (conceived by Johann Listing in 1845). Gauss demonstrated that, so far as paraxial rays are concerned, a lens of any degree of complexity can be replaced by its cardinal points, viz., two principal points and two focal points, where the distances from the principal points to their respective focal points being the focal lengths of the lens. This was the first formal definition of the focal length of a lens system. The properties of the cardinal points and related points are as follows:

s s s

s s

Principal points: the axial intersection points of conjugate planes P1 and P2 where these principal planes are related by unit lateral magnification. A ray incident at height h on P1 will exit P2 at the same height. When ni w no , a ray incident at the first principal point with angle u will exit the second principal point with angle u(ni /no ). Nodal points: conjugate points related by unit angular magnification (m ui /uo). Nodal points are the axial intersection points of conjugate planes N1 and N2 where these nodal planes are related, by application of the Lagrange invariant, lateral magnification (m ni /no ). A ray incident at height h on N1 will exit N2 at h(ni /no ). Focal points: anterior or front (f1) and posterior or rear (f2) focal points are the axial intersections of the respective focal planes, which are not conjugate. Any ray parallel to the optical axis, and entering the lens from the left, will intersect the axis at f2. Any ray parallel to the optical axis, and entering the lens from the right, will intersect the axis at f1. When ni w no, the distance f1 N1 equals the posterior focal length P2 f2 and the distance N2 f2 equals the anterior focal length P1 f1. Antiprincipal points: the axial intersection points of conjugate planes where these antiprincipal planes are related by negative unit lateral magnification. An example is a lens used at m 1 where the object and image planes are located at twice the focal length from the principal points. Antinodal points: conjugate points related by negative unit angular magnification (m ui /uo).

Although imaging can be done solely using ni /no and either the principal point and planes, or the nodal point and planes, it is customary and easier to use a mixed set, that is, principal planes and nodal points. In this manner, the lateral and angular magnifications are both unity. This is particularly useful when performing graphical ray tracing. In addition, the image location and magnification can be determined in the following manner:

s s s s s s

Trace a horizontal ray from the object tip to the first principal plane. Transfer to the second principal plane. Trace the ray through the second focal point. Trace a second ray from the object tip through the first focal point and to the intersection with the first principal plane. Transfer to the second principal plane. Project this ray horizontally until it intersects the first ray.

The intersection is the image height and locates the image plane. Magnification is the ratio of image height to object height. There are several alternative graphical ray-tracing method that can determine image height and location. One alternative is to trace a ray from the object tip to the first nodal point, and then project the ray exiting the second nodal point to intersect with the second ray mentioned above. The focal length of a lens is related to the power of the lens by

F

no ni fo fi

(5)

17.8

COMPONENTS

This relationship is important in such optical systems as underwater cameras, cameras in space, and the like. For example, it is evident that the field of view is decreased for a camera in water. The lens law can be expressed in several forms. If so and si are the distance from the object to the first principal point and the distance from the second principal point to the image, then the relationship between the object and the image is given by

F

ni no si so

(6)

Should the distance be measured with respect to the nodal points, the imaging equation becomes

F

no ni si so

(7)

When the distances are measured from the focal points, the image relationship, known as the Newtonian imaging equation, is given by (8)

f1 f 2 so si

The power of a spherical refracting surface, with curvature c and n being the refractive index following the surface, is given by

F c(n n0 )

(9)

It can be shown that the power of a single thick lens in air is

Fthick F1 F2 F1F2

t n

(10)

where t is the thickness of the lens. The distance from the first principal plane to the first surface is (t/n)H2 f1 and the distance from the second principal point to the rear surface is (t/n)H1 f2. The power of a thin lens (t l 0) in air is given by

Fthin (n 1)(c1 c 2 )

17.4

(11)

STOPS AND PUPILS The aperture stop or stop of a lens is the limiting aperture associated with the lens that determines how large an axial beam may pass through the lens. The stop is also called an iris. The marginal ray is the extreme ray from the axial point of the object through the edge of the stop. The entrance pupil is the image of the stop formed by all lenses preceding it when viewed from object space. The exit pupil is the image of the stop formed by all lenses following it when viewed from image space. These pupils and the stop are all images of one another. The principal ray is defined as the ray emanating from an off-axis object point that passes through the center of the stop. In the absence of pupil aberrations, the principal ray also passes through the center of the entrance and exit pupils. As the obliquity angle of the principal ray increases, the defining apertures of the components comprising the lens may limit the passage of some of the rays in the entering beam thereby causing the stop not to be filled with rays. The failure of an off-axis beam to fill the aperture stop is called vignetting. The ray centered between the upper and lower rays defining the oblique beam is called the chief ray. When the object moves to large off-axis locations, the entrance pupil often has a highly distorted shape, may be tilted, and/or displaced longitudinally and transversely. Due to the vignetting and pupil aberrations, the chief and principal rays may become displaced from one another. In some cases, the principal ray is vignetted.

LENSES

17.9

The field stop is an aperture that limits the passage of principal rays beyond a certain field angle. The image of the field stop when viewed from object space is called the entrance window and is called the exit window when viewed from image space. The field stop effectively controls the field of view of the lens system. Should the field stop be coincident with an image formed within or by the lens system, the entrance and exit windows will be located at the object and/or image(s). A telecentric stop is an aperture located such that the entrance and/or exit pupils are located at infinity. This is accomplished by placing the aperture in the focal plane. Consider a stop placed at the front focal plane of a lens. The stop image or exit pupil is located at infinity and the principal ray exits the lens parallel to the optical axis. This feature is often used in metrology since the measurement error is reduced when compared to conventional lens systems because the centroid of the blur remains at the same height from the optical axis even as the focus is varied.

17.5

F-NUMBER AND NUMERICAL APERTURE The focal ratio or F-number (FN) of a lens is defined as the effective focal length divided by the entrance pupil diameter Dep. When the object is not located at infinity, the effective FN is given by FNeff FNc(1 m)

(12)

where m is the magnification. For example, for a simple positive lens being used at unity magnification (m 1), the FNeff 2FNc. The numerical aperture of a lens is defined as NA ni sin U i

(13)

where ni is the refractive index in which the image lies and Ui is the slope angle of the marginal ray exiting the lens. If the lens is aplanatic, then FNeff

1 2NA

(14)

The T-number of a lens is the effective FN divided by the square root of the transmittance of the lens and is used for radiometric computations; however, the FN should be used when computing depth of focus and depth of field discussed in Sec. 17.21.

17.6

MAGNIFIER OR EYE LOUPE The typical magnifying glass, or loupe, comprises a singlet lens and is used to produce an erect but virtual magnified image of an object. The magnifying power of the loupe is stated to be the ratio of the angular size of the image when viewed through the magnifier to the angular size without the magnifier. By using the thin-lens model of the human eye, the magnifying power (MP) can be shown to be given by MP

25 cm de do Fde do

(15)

where do is the distance from the object to the loupe, de is the separation of the loupe from the eye, and H 1/f is the power of the magnifier. When do is set to the focal length of the lens, the virtual image is placed at infinity and the magnifying power reduces to MP

25 cm f

(16)

17.10

COMPONENTS

Should the virtual image be located at the near viewing distance of the eye (about 25 cm), then MP

25 cm 1 f

(17)

Typically simple magnifiers are difficult to make with magnifying powers greater than about 10r.

17.7

COMPOUND MICROSCOPES For magnifying power greater than that of a simple magnifier, a compound microscope, which comprises an objective lens and an eyepiece, may be used. The objective forms an aerial image of the object at a distance sot from the rear focal point of the objective. The distance sot is called the optical tube length and is typically 160 mm. The objective magnification is MPobj

sot f obj

(18)

The image formed is further magnified by the eyepiece which has a MPep 250 mm/fep. The total magnifying power of the compound microscope is given by MP MPobjMPep

160 250 f obj f ep

(19)

Typically, fep 25 mm, so its MP 10. Should the objective have a focal length of 10 mm, the total magnifying power of the microscope is 16r times 10r, or 160r.

17.8

FIELD AND RELAY LENSES Field lenses are placed at (or near) an image location for the purpose of optically relocating the pupil or to increase the field-of-view of the optical system. For example, a field lens may be used at the image plane of an astronomical telescope such that the field lens images the objective lens onto the eyepiece. In general, the field lens does not contribute to the aberrations of the system except for distortion and field curvature. Since the field lens must be positive, it adds inward curving Petzval. For systems having a small detector requiring an apparent increase in size, the field lens is a possible solution. The detector is located beyond the image plane such that it subtends the same angle as the objective lens when viewed from the image point. The field lens images the objective lens onto the detector. Relay lenses are used to transfer an image from one location to another such as in a submarine periscope or borescope. It is also used as a means to erect an image in many types of telescopes and other such instruments. Often relay lenses are made using two lens groups spaced about a stop, or an image of the system stop, in order to take advantage of the principle of symmetry, thereby minimizing the comatic aberrations and lateral color. The relayed image is frequently magnified.

17.9 APLANATIC SURFACES AND IMMERSION LENSES Abbe called a lens an aplanat that has an equivalent refractive surface which is a portion of a sphere with a radius r centered about the focal point. Such a lens satisfies the Abbe sine condition and implies that the lens is free of spherical and coma near the optical axis. Consequently, the maximum

LENSES

17.11

possible numerical aperture (NA) of an aplanat is unity, or an FN 0.5. In practice, an FN less than 0.6 is difficult to achieve. For an aplanat, FN

1 2 NA

(20)

It can be shown that three cases exist where the spherical aberration is zero for a spherical surface. These are: (1) the trivial case where the object and image are located at the surface, (2) the object and image are located at the center of curvature of the surface, and (3) the object is located at the aplanatic point. The third case is of primary interest. If the refractive index preceding the surface is no and following the surface is ni then the object is located a distance so from the surface as expressed by so

r(no ni ) no

(21)

si

r(no ni ) ni

(22)

and the image is located at

An immersion lens or contact lens can be formed from an aplanatic surface and a plano surface. Figure 4 illustrates a hemispherical magnifier that employs the second aplanatic case. The resultant magnification is ni if in air or ni /no otherwise. A similar magnifier can be constructed by using a hyperhemispherical surface and a plano surface as depicted in Fig. 5. The lateral magnification is ni2. This lens, called an Amici lens, is based upon the third aplanatic case. The image is free of all orders of spherical aberration, third-order coma, and third-order astigmatism. Axial color is also absent from the hemispherical magnifier. These magnifiers are often used as a means to make a detector appear larger and as the first component in microscope objectives.

r n

r/n FIGURE 4 Aplanatic hemispherical magnifier with the object and image located at the center of curvature of the spherical surface. This type of magnifier has a magnification of ni /no which can be used as a contact magnifier or as an immersion lens.

nr FIGURE 5 Aplanatic hyperhemispherical magnifier or Amici lens has the object located at the aplanatic point. The lateral magnification is (ni /n0)2.

17.12

17.10

COMPONENTS

SINGLE ELEMENT LENS It is well known that the spherical aberration of a lens is a function of its shape factor or bending. Although several definitions for the shape factor have been suggested, a useful formulation is

?

c1 c1 c 2

(23)

where c1 and c2 are the curvatures of the lens with the first surface facing the object. By adjusting the lens bending, the spherical aberration can be seen to have a minimum value. The power of a thin lens or the reciprocal of its focal length is given by

F

(n 1)c1 ?

(24)

When the object is located at infinity, the shape factor for minimum spherical aberration can be represented by

?

n(2n 1) 2(n 2)

(25)

The resultant third-order spherical aberration of the marginal ray in angular units is SA 3

n2 (2n 1)? (1 2/n)? 2 16(n 1)2 (FN)3

(26)

n(4n 1) 64(n 2)(n 1)2 (FN)3

(27)

or after some algebraic manipulations, SA 3

where, for a thin lens, the FN is the focal length f divided by the lens diameter, which in this case is the same as entrance pupil diameter Dep. Inspection of this equation illustrates that smaller values of spherical aberration are obtained as the refractive index increases. When the object is located at a finite distance so , the equations for the shape factor and residual spherical aberration are more complex. Recalling that the magnification m is the ratio of the object distance to the image distance and that the object distance is negative if the object lies to the left of the lens, the relationship between the object distance and the magnification is 1 m soF 1 m

(28)

where m is negative if the object distance and the lens power have opposite signs. The term 1/soH represents the reduced or H-normalized reciprocal object distance W, that is, so is measured in units of focal length H 1. The shape factor for minimum spherical aberration is given by

?

n(2n 1) 2(n2 1) ¤ m ³ 2(n 2) n 2 ¥¦ 1 m´µ

(29)

and the resultant third-order spherical aberration of the marginal ray in angular units is SA 3

§ 2 ¤ m ³ 1 n 2 2 ¨n (2n 1)? n ? (3n 1)(n 1)¥ 1 m´ 16(n 1)2 (FN)3 © ¦ µ 2 (3n 2)(n 1)2 ¤ m ³ ¶ 4(n2 1) ¤ m ³ · ?

n n ¦¥ 1 mµ´ ¦¥ 1 mµ´ · ¸

(30)

LENSES

3

17.13

3 =4 =3

2

=2

1

1 = 1.5

0

0

–1

–1

–2 –1.0

Shape factor

Shape factor

2

–2 –0.8

–0.6

–0.4

–0.2

0

Reciprocal object distance FIGURE 6 The shape factor for a single lens is shown for several refractive indexes as a function of reciprocal object distance W where the distance is measured in units of focal length.

where FN is the effective focal length of the lens f divided by its entrance pupil diameter. When the object is located at infinity, the magnification becomes zero and the above two equations reduce to those previously given. Figure 6 illustrates the variation in shape factor as a function of W for refractive indices of 1.5 to 4 for an FN 1. As can be seen from the figure, lenses have a shape factor of 0.5 regardless of the refractive index when the magnification is 1 or W 0.5. For this shape factor, all lenses have biconvex surfaces with equal radii. When the object is at infinity and the refractive index is 4, lenses have a meniscus shape toward the image. For a lens with a refractive index of 1.5, the shape is somewhat biconvex, with the second surface having a radius about 6 times greater than the first surface radius. Since the minimum-spherical lens shape is selected for a specific magnification, the spherical aberration will vary as the object-image conjugates are adjusted. For example, a lens having a refractive index of 1.5 and configured for m 0 exhibits a substantial increase in spherical aberration when the lens is used at a magnification of 1. Figure 7 illustrates the variation in the angular spherical aberration as both a function of refractive index and reciprocal object distance when the lens bending is for minimum spherical aberration with the object located at infinity. As can be observed from Fig. 7, the ratio of the spherical aberration, when m 0.5 and m 0, increases as n increases. Figure 8 shows the variation in angular spherical aberration when the lens bending is for minimum spherical aberration at a magnification of 1. In a like manner, Fig. 9 presents the variation in angular spherical aberration for a convex-plano lens with the plano side facing the image. The figure can also be used when the lens is reversed by simply replacing the object distance with the image distance. Figures 7 to 9 may provide useful guidance in setting up experiments when the three forms of lenses are available. The so-called “off-the-shelf ” lenses that are readily available from a number of vendors often have the convex-plano, equal-radii biconvex, and minimum spherical shapes. Figure 10 shows the relationship between the third-order spherical aberration and coma, and the shape factor for a thin lens with a refractive index of 1.5, stop in contact, and the object at infinity. The coma is near zero at the minimum spherical aberration shape. The shape of the lens as a function of shape factor is shown at the top of the figure.

COMPONENTS

0

0 =4 =3

–0.1

=2 –0.2

–0.2

= 1.5

–0.3

FN = 1

–0.3

–0.4

Lens bending for minimum spherical aberration with object at infinity

–0.4

–0.5 –1.0

Spherical aberration (rad)


–0.1

–0.5 –0.8

–0.6

–0.4

–0.2

0

Reciprocal object distance FIGURE 7 Variation of angular spherical aberration as a function of reciprocal object distance W for various refractive indices when the lens is shaped for minimum spherical aberration with the object at infinity. Spherical aberration for a specific FN is determined by dividing the aberration value shown by (FN)3.

0

0 =4

–0.05

=3 =2

–0.1

–0.1

FN = 1 = 1.5

–0.15

–0.15

–0.2

–0.25 –1.0

Lens bending for minimum spherical aberration for magnification of –1


–0.05 Spherical aberration (rad)

17.14

–0.2

–0.25 –0.8

–0.6

–0.4

–0.2

0

Reciprocal object distance FIGURE 8 Variation of angular spherical aberration as a function of reciprocal object distance W for various refractive indices when the lens is shaped for minimum spherical aberration for a magnification of

1. Spherical aberration for a specific FN is determined by dividing the aberration value shown by (FN)3.

LENSES

0

0 =4 Spherical aberration (rad)

=3 –0.2

–0.2

=2

–0.3

–0.3 FN = 1

= 1.5 –0.4


–0.1

–0.1

–0.4

Convex-plano lens with plano side facing the image

–0.5

–0.5 –1.0

–0.8

–0.6

–0.4

–0.2

0

Reciprocal object distance FIGURE 9 Variation of angular spherical aberration as a function of reciprocal object distance W for various refractive indices when the lens has a convex-plano shape with the plano side facing the object. Spherical aberration for a specific FN is determined by dividing the aberration value shown by (FN)3.

0.01

0.01

0

Aberration (rad)

0

–0.01

–0.01

–0.02

–0.02

–0.03

–0.03

–0.04

–0.04 –2

–1

0 1 Shape factor

2

3

FIGURE 10 Variation of spherical aberration (solid curve) and coma (dashed line) as a function of shape factor for a thin lens with a refractive index of 1.5, stop in contact with the lens, and the object at infinity. The shape of the lens as the shape factor changes is shown at the top of the figure.

17.15

17.16

COMPONENTS

For certain cases, it is desirable to have a single lens with no spherical aberration. A useful form is the plano-convex, with the plano side facing the object, if the convex side is figured as a conic surface with a conic constant of n2. Caution should be exercised when using this lens form at other than infinite object distances; however, imaging at finite conjugates can be accomplished by using two lenses with their plano surfaces facing one another and the magnification being determined by the ratio of the focal lengths. It should be noted that for this lens form, the actual thickness of the lenses is not important and that the inclusion of the conic surface does not alter the focal length. The off-axis performance of a lens shaped for minimum spherical aberration with the object at infinity can be estimated by using the following equations. Assuming that the stop is in contact with the lens, the third-order angular sagittal coma is given by CMA s

Q 16(n 2)(FN)2

(31)

where the field angle S is expressed in radians. The tangential coma is three times the sagittal coma or CMAt 3 · CMAs . The diameter of the angular astigmatic blur formed at best focus is expressed by AST

Q2 FN

(32)

The best focus location lies midway between the sagittal and tangential foci. An estimate of the axial angular chromatic aberration is given by AChr

1 2V(FN)

(33)

where V is the Abbe number of the glass and V (n2 1)/(n3 n1) with n1 n2 n3. If a singlet is made with a conic or fourth-order surface, the spherical aberration is corrected by the aspheric surface, and the bending can be used to remove the coma. With the stop in contact with the lens, the residual astigmatism and chromatic errors remain as expressed by the preceding equations. Figure 11 depicts the shapes of such singlets for refractive indices of 1.5, 2, 3, and 4. Each lens has a unity focal length and an FN of 10. Table 1 presents the prescription of each lens where CC2 is the conic constant of the second surface. The optical center of a thick lens is located where a nodal ray crosses the optical axis of the lens. A nodal ray is aimed at the first nodal point, passes through the lens undeviated (although translated), and appears to emerge from the lens from the second nodal point. It can be shown that the distance from the first surface vertex to the optical center is t /[1 (c1 /c 2 )] where t is the thickness of the lens.

FIGURE 11 Variation of shape of singlets when the spherical aberration is corrected by the conic constant and the coma by the bending .

LENSES

TABLE 1 and Coma

17.17

Prescription of Singlets Corrected for Both Spherical Aberration

Lens

R1

Thickness

R2

Index

a b c d

0.55143 0.74715 0.88729 0.93648

0.025 0.025 0.025 0.025

5.27966 2.90553 1.56487 1.33421

1.5 2.0 3.0 4.0

CC2

673.543 23.2435 0.86904 0.24340

A remarkable property of the optical center is its wavelength independence (n does not appear in the preceding equation). This means that the spatial position of the optical center is fixed, where in contrast, the spatial positions of the six cardinal points are a function of wavelength because of their dependence upon n. The optical center point (plane) is conjugate with the nodal points (planes); however, while the nodal points are related by unit angular magnification, the nodal-point to optical-center magnification (mOC ) is not necessarily unity. In general, mOC is the ratio of the nodal ray slope angles at the first nodal point and the optical center. For a single thick lens, the magnification mOC can be readily shown to be given by mOC (r1 r2 )/[N (r1 r2 ) t (N 1)] All rotationally symmetric lenses have an optical center just as they possess the six cardinal points. Since the optical center is conjugate with N1 and N2, the optical center can justifiably be considered also as a cardinal point. Should the aperture stop be located at the optical center, then the entrance pupil will be located at the first nodal point and the exit pupil will be located at the second nodal point with a unity pupil magnification. This statement is true whether the lens is of symmetrical or unsymmetrical design. When no w ni , the exit pupil magnification will be no /ni rather than unity.

17.11 LANDSCAPE LENSES AND THE INFLUENCE OF STOP POSITION The first lens used for photography was designed in 1812 by the English scientist W. H. Wollaston about a quarter of a century before the invention of photography. He discovered that a meniscus lens with its concave surface toward the object could produce a much flatter image field than the simple biconvex lens commonly used at that time in the camera obscuras. This lens became known as the landscape lens and is illustrated in Fig. 12. Wollaston

FIGURE 12 Landscape lens with the aperture stop located to the left of the lens.

17.18

COMPONENTS

P sp

FIGURE 13 Rays traced at a given obliquity where the intersection of a given ray with the optical axis is P, located a distance sp from the front surface of the lens.

realized that if the stop was placed an appropriate amount in front of the lens and the F-number was made to be modest, the image quality would be improved significantly over the biconvex lens. The rationale for this can be readily seen by considering the influence on the residual aberrations of the lens by movement of the stop. Functionally, the stop allows certain rays in the oblique beam to pass through it while rejecting the rest. By simple inspection, it is clear that the movement of the stop (assuming a constant FN is maintained) will not affect the axial aberrations, while the oblique aberrations will be changed. In order to understand the influence of stop movement on the image quality, a graphical method was devised by R. Kingslake in which he traced a number of rays in the meridional plane at a given obliquity angle as illustrated in Fig. 13. A plot is generated that relates the intercept height of each real ray at the image plane Hi to the distance sp from the intersection of the ray with optical axis P to the front surface of the lens. Each ray can be viewed as the principal ray when the stop is located at the intersection point P. This Hi sp plot provides significant insight into the effect upon image quality incurred by placement of the stop. The shape of the curve provides information about the spherical aberration, coma, tangential field curvature, and distortion. Spherical aberration is indicated by an S-shaped curve, while the curvature at the principal ray point is a gauge of the coma. The coma is zero at inflection points. When the curve is a straight line, both coma and spherical aberration are essentially absent. The slope of the curve at the principal ray point is a measure of the tangential field curvature or the sag of the tangential field, that is, astigmatism. The difference in height of the real and gaussian principal rays in the image plane is distortion. For situations where the curve does not exhibit spherical aberration, it is impossible to correct the coma by shifting the stop. Since a simple meniscus lens has stop position and lens bending as degrees of freedom, only two aberrations can be corrected. Typically, coma and tangential field curvature are chosen to be corrected, while axial aberrations are controlled by adjusting the FN of the lens. The Hi sp plot for the lens shown in Fig. 13 is presented in Fig. 14, where the field angle is 10° and the image height is expressed as a percent of the gaussian image height. The lens has a unity focal length,

LENSES

17.19

FIGURE 14 The image height Hi of each ray traced in Fig. 13 is plotted against the intersection length sp to form the Hi sp plot. Hi is expressed as a percent of the gaussian image height as a direct measure of distortion.

and the lens diameter is 0.275. Table 2 contains the prescription of the lens. Examination of this graph indicates that the best selection for stop location is when the stop is located at sp 0.1505 (left of the lens). For this selection, the coma and tangential astigmatism will be zero since the slope of the curve is zero and an inflection point is located at this stop position. Figure 15 shows the astigmatic field curves which clearly demonstrate the flat tangential image field for all field angles. Other aberrations cannot be controlled and must consequently be tolerated. When this lens is used at F/11, the angular blur diameter is less than 300 Orad. It should be noted that this condition is generally valid for only the evaluated field-angle obliquity and will likely be different at other field angles. Nevertheless, the performance of this lens is often acceptable for many applications. An alternate configuration can be used where the lens is in front of the stop. Such configuration is used to conserve space since the stop would be located between the lens and the image. The optical performance is typically poorer due to greater residual spherical aberration. The principle demonstrated by the Hi sp plot can be applied to lenses of any complexity as a means to locate the proper stop position. It should be noted that movement of the stop will not affect the coma if spherical aberration is absent nor will astigmatism be affected if both spherical aberration and coma have been eliminated.

TABLE 2

Prescription of Landscape Lens Shown in Fig. 13

Surface no.

Radius

Thickness

Index

Comment

1 2 3 4

Infinite

0.45759

0.24887 Infinite

0.15050 0.03419 0.99843

1.0 1.51680 1.0

Stop BK7 Image

17.20

COMPONENTS

FIGURE 15 Astigmatic field curves for the landscape lens having the stop located at the zero slope location on the Hi sp plot in Fig. 14, which is the flat tangential field position. S represents the sagittal astigmatic focus while T indicates the tangential astigmatic focus.

17.12 TWO-LENS SYSTEMS Figure 16 illustrates the general imaging problem where an image is formed of an object by two lenses at a specified magnification and object-to-image distance. Most imaging problems can be solved by using two equivalent lens elements. An equivalent lens can comprise one lens or multiple lenses and may be represented by the principal planes and power of a single thick lens. All distances are measured from the principal points of each equivalent lens element. For simplicity, the lenses shown in Fig. 16 are thin lenses. If the magnification m, object-image distance s, and lens powers Ha and Hb are known, then the equations for s1, s2, and s3 are given by s1

Fb (s s2 ) 1 m mFa Fb

§ 4 §sm(Fa Fb ) (m 1)2 ¶¸ ¶ s · s2 ¨1 o 1 © · 2¨ s 2mFaFb ¸ © s3 s s1 s2 The equation for s2 indicates that zero, one, or two solutions may exist.

(34)

LENSES

17.21

FIGURE 16 lenses.

General imaging problem where the image is formed by two separated

If the magnification and the distances are known, then the lens powers can be determined by

Fa and

s (s1 s2 )(m 1) ms1s2

Fb

s s1(m 1) s2 (s s1 s2 )

(35)

It can be shown that only certain pairs of lens powers can satisfy the magnification and separation requirements. Commonly, only the magnification and object-image distance are specified with the selection of the lens powers and locations to be determined. By utilizing the preceding equations, a plot of regions of all possible lens power pairs can be generated. Such a plot is shown as the shaded region in Fig. 17 where s 1 and m 0.2.

FIGURE 17 Shaded regions indicate all possible power pairs for the two lenses used for imaging. The solution space may be limited by physical considerations such as maximum aperture.

17.22

COMPONENTS

Pa1 Pa2

P1

Focal point

Pb1Pb2

P2

s2

bf l

a

b fab or e f l

First principal plane of system

Second principal plane of system

FIGURE 18 Combination of two thick lenses illustrating the principal points of each lens and the system, the fab or efl, and the bfl. Distances are measured from the principal points with the exception of the bfl.

Examination of this plot can assist in the selection of lenses that may likely produce better performance by, for example, selecting the minimum power lenses. The potential solution space may be limited by placing various physical constraints on the lens system. For example, the allowable lens diameters can dictate the maximum powers that are reasonable. Lines of maximum power can then be plotted to show the solution space. When s1 becomes very large compared to the effective focal length efl of the lens combination, the optical power of the combination of these lenses is expressed by

Fab Fa Fb s2FaFb The effective focal length is

1 Fab

(36)

or f ab

f a fb f a fb s2

(37)

and the back focal length is given by ¤ f s ³ bf l f ab ¥ a 2 ´ ¦ fa µ

(38)

The separation between lenses is expressed by s2 f a fb

f a fb f ab

(39)

Figure 18 illustrates the two-lens configuration when thick lenses are used. The principal points for the lens combination are denoted by P1 and P2, Pa1 and Pa2 for lens a, and Pb1 and Pb2 for lens b. The distance between the principal points of a lens is called a hiatus. With the exception of the back focal length, all distances are measured from the principal points of each lens element or the combined lens system, as shown in the figure. For example, s2 is the distance from Pa2 to Pb1. The bfl is measured from the final surface vertex of the lens system to the focal point.

17.13 ACHROMATIC DOUBLETS The singlet lens suffers from axial chromatic aberration, which is determined by the Abbe number V of the lens material and its FN. A widely used lens form that corrects this aberration is the achromatic doublet as illustrated in Fig. 19. An achromatic lens has equal focal lengths in c and f light.

LENSES

FIGURE 19

17.23

Typical achromatic doublet lens.

This lens comprises two lens elements where one element with a high V-number (crown glass) has the same power sign as the doublet and the other element has a low V-number (flint glass) with opposite power sign. Three basic configurations are used. These are the cemented doublet, broken contact doublet, and the widely airspaced doublet (dialyte). The degrees of freedom are two lens powers, glasses, and shape of each lens. The resultant power of two thin lenses in close proximity, s2 l 0, is Hab Ha Hb and the transverse primary chromatic aberration TPAC is §F F ¶ (40) TPAC yf ab ¨ a b · ©Va Vb ¸ where y is the marginal ray height. Setting TPAC 0 and solving for the powers of the lenses yields

Fa and

Va f ab (Va Vb )

(41)

VbFa Va

(42)

Fb

The bending or shape of a lens is expressed by c c1 c2 and affects the aberrations of the lens. The bending of each lens is related to its power by ca Ha /(na 1) and cb Hb(nb 1). Since the two bendings can be used to correct the third-order spherical and coma, the equations for these aberrations can be combined to form a quadratic equation in terms of the curvature of the first surface c1. Solving for c1 will yield zero, one, or two solutions for the first lens. A linear equation relates c1 to c2 of the second lens. While maintaining the achromatic correction of a doublet, the spherical aberration as a function of its shape (c1) is described by a parabolic curve. Depending upon the choices of glasses, the peak of the curve may be above, below, or at the zero spherical aberration value. When the peak lies in the positive spherical aberration region, two solutions with zero spherical aberration exist in which the solution with the smaller value of c1 is called the left-hand solution (Fraunhofer or Steinheil forms) and the other is called the right-hand solution (Gaussian form). Two additional solutions are possible by reversal of the glasses. These two classes of designs are denoted as crown-in-front and flint-in-front designs. Depending upon the particular design requirements, one should examine all four configurations to select the most appropriate. The spherical aberration curve can be raised or lowered by the selection of the V difference or the n difference. Specifically, the curve will be lowered as the V difference is increased or if the n difference is reduced. As for the thin singlet lens, the coma will be zero for the configuration corresponding to the peak of the spherical aberration curve. Although the primary chromatic aberration may be corrected, a residual chromatic error often remains and is called the secondary spectrum, which is the difference between the ray intercepts in d and c spectral lines. Figure 20a illustrates an F/5 airspaced doublet that exhibits well-corrected spherical light and primary chromatic aberrations and has notable secondary color. The angular secondary spectrum for an achromatic thin-lens doublet is given by SAC

(Pa Pb ) 2(FN)(Va Vb )

(43)

17.24

COMPONENTS

FIGURE 20 An F/5 airspaced doublet using conventional glasses is shown in (a) and exhibits residual secondary chromatic aberration. A similar lens is shown in (b) that uses a new glass to effectively eliminate the secondary color.

where P (nN nc)(nf nc) is the partial dispersion of a lens material. In general, the ratio (Pa

Pb)/(Va Vb) is nearly a constant which means little can be done to correct the SAC. A few glasses exist that allow Pa Pb y 0, but the Va Vb is often small, which results in lens element powers of rather excessive strength in order to achieve achromatism. Figure 20b shows an F/5 airspaced doublet using a relatively new pair of glasses that have a small Pa Pb and a more typical Va V b. Both the primary and secondary chromatic aberration are well corrected. Due to the relatively low refractive index of the crown glass, the higher power of the elements results in spherical aberration through the seventh order. Almost no spherochromatism (variation of spherical aberration with wavelength) is observed. The 80 percent blur diameter is almost the same for both lenses and is 0.007. Table 3 contains the prescriptions for these lenses. TABLE 3 in Fig. 20

Prescriptions for Achromatic Doublets Shown Achromatic Doublet—1

Surface No.

Radius

Thickness

Glass

1 2 3 4

49.331 52.351 43.888

141.706

6.000 4.044 2.000

BK7 517:642 Air SF1 717:295 Air

Achromatic Doublet—2 Surface No.

Radius

Thickness

Glass

1 2 3 4

23.457

24.822

22.516 94.310

6.000 1.059 3.000

FK03 439:950 Air BK7 517:642 Air

LENSES

ya

17.25

yb sd

bf l

b a FIGURE 21

Widely separated achromatic doublet known as the dialyte lens.

When the separation between the lens elements is made a finite value, the resultant lens is known as a dialyte and is illustrated in Fig. 21. As the lenses are separated by a distance sd , the power of the flint or negative lens increases rapidly. The distance sd may be expressed as a fraction of the crown-lens focal length by p sd /fa. Requiring the chromatic aberration to be zero implies that ya2 y2 b 0 f aVa f bVb

(44)

By inspection of the figure and the definition of p, it is evident that yb ya(1 p) from which it follows that fbVb f aVa (1 p)2

(45)

F Fa Fb (1 p)

(46)

The total power of the dialyte is

Solving for the focal lengths of the lenses yields § Vb ¶ f a f ab ¨1 V

p) ·¸ ( a 1 ©

(47)

§ V (1 p) ¶ fb f ab (1 p) ¨1 b Vb ·¸ ©

(48)

The power of both lenses increases as p increases. The typical dialyte lens suffers from residual secondary spectrum; however, it is possible to design an airspaced achromatic doublet with only one glass type that has significantly reduced secondary spectrum. Letting Va Vb results in the former equations becoming fa

pf ab p 1

fb pf ab ( p 1)

sd pf a

bf l f ab ( p 1)

(49)

When fab 0, then p must be greater than unity, which means that the lens is quite long. The focal point lies between the two lenses, which reduces its general usefulness. This type of lens is known as the Schupmann lens, based upon his research in the late 1890s. Several significant telescopes, as well as eyepieces, have employed this configuraton. For fab 0, the lens can be made rather compact and is sometimes used as the rear component of some telephoto lenses.

17.26

COMPONENTS

FIGURE 22

Typical triplet lens.

FIGURE 23 Typical Tessar lens.

17.14 TRIPLET LENSES In 1893, a new type of triplet lens for photographic applications was invented by the English designer H. Dennis Taylor. He realized that the power of two lenses in contact of equal, but opposite, power is zero, as is its Petzval sum. As the lenses are separated, the system power becomes positive since the negative lens contributes less power. The Petzval sum remains zero, since it does not depend upon the marginal ray height. In order to overcome the large aberrations of such a configuration, Taylor split the positive lens into two positive lenses and placed one on each side of the negative lens. A stop is often located between the negative and rear-positive lenses. Figure 22 illustrates a typical triplet lens. The triplet can be used at reasonably large apertures (F/4) and moderately large fields of view (o25°). The triplet has eight degrees of freedom which are the three powers, two airspaces, and three lens bendings. The lens powers and airspaces are used to control the axial and lateral chromatic aberrations, the Petzval sum, the focal length, and the ratio of the airspaces. Spherical aberration, coma, and astigmatism are corrected by the lens bendings. Distortion is usually controlled by the airspace ratio or the choice of glasses. Consequently, the triplet has exactly the number of degrees of freedom to allow correction of the basic aberrations and maintain the focal length. The design of a triplet is somewhat difficult since a change of any surface affects every aberration. The choice of glass is important and impacts the relative aperture, field of view, and overall length. For example, a large $V produces a long system. It should be noted that a triplet corrected for third-order aberrations by using the degrees of freedom almost always leads to a lens with poor performance. A designer normally leaves a certain amount of residual thirdorder aberrations to balance the higher-order terms. The process for thin-lens predesign is beyond the scope of this Handbook; however, it may be found in various references comprising the bibliography. A few years later, Paul Rudolph of Zeiss developed the Tessar, which resembles the triplet, with the rear lens replaced by an achromatic doublet. The Tessar shown in Fig. 23 was an evolution of Rudolph’s anastigmats which were achromatic lenses located about a central stop. The advantage of the achromatic rear component is that it allows reduction of the zonal spherical aberration and the oblique spherical aberration, and reduces the separation of the astigmatic foci at other than the design maximum field angle. Performance of the Tessar is quite good and has generally larger relative apertures at equivalent field angles than the triplet. A variety of lenses were derived from the triplet and the Tessar in which the component lenses were made into doublets or cemented triplets.

17.15

SYMMETRICAL LENSES In the early 1840s, it was recognized that lenses that exhibit symmetry afford various benefits to the lens designer. The first aberration acknowledged to be corrected by the symmetry principle was distortion. It can also be shown that coma and lateral color are necessarily corrected by a symmetrical lens construction. Although the principle of symmetry implies that the lens be

LENSES

17.27

FIGURE 24 The periscopic lens illustrates the earliest form of symmetrical lenses. It is formed by placing two landscape lenses about a central stop. Symmetry removes the aberrations of coma, distortion, and lateral color.

operated at a magnification of 1, the degree to which the aberrations are upset by utilizing the lens at other conjugates is remarkably small. This principle forms the basis of most wide-fieldof-view lenses. One of the earliest symmetrical lenses was the Periscopic (Periskop) lens invented by C. A. Steinheil in 1865. Figure 24 shows an F/11 Periscopic lens constructed from the landscape lens discussed previously. Symmetry corrects for coma and distortion, while the spacing of the lenses and their shapes are selected to produce a flat tangential astigmatic field. Since the stop position for the landscape lens was chosen to yield a flat tangential astigmatic field, essentially no change in the lens separation is necessary even though the Periscopic lens is being used at infinite conjugates. No correction for spherical aberration can be made. When used at other than unit magnification, some optical improvement can be achieved by making the stop slightly asymmetrical and/or having a different shape for the front or rear lens. This lens has continued to find application throughout this century. By 1866, Dallmeyer in England and Steinheil and von Seidel in Germany both invented the Rapid Rectilinear lens that could be used at apertures of up to F/6. The lens has two cemented achromats about a central stop. Use of the doublet allows correction of the axial chromatic and spherical aberrations. Glass selection is of importance in the design. Typically, the $n between the glasses should be large while the $V should be relatively small. The positive lens is located nearest the stop and has the lower refractive index. A notable characteristic of the lens is that the aberrations are reasonably stable over a broad range of object distances. It should be noted that vignetting is often used in these and other lens types to control the higherorder aberrations that are often observed at large field angles. Although a loss in illumination occurs, the gain in resolution is often worthwhile. The airspaced dialyte lens comprises four lenses symmetrically arranged about a central stop. The rear portion of the lens is an achromatic doublet that has five degrees of freedom (an air space, two powers, and two bendings) which may be used to control the focal length, spherical aberration, axial chromatic aberration, astigmatism, and the Petzval sum. With a like pair of lenses mounted in front of the stop, the symmetry corrects the coma, distortion, and lateral color. When used at infinite conjugates, the resultant residuals of the aberrations can be controlled by deviating somewhat from perfect symmetry of the air spaces about the stop. Lenses of this type can provide useful performance with apertures approaching F/4 and fields of view of about o20n or so.

17.16

DOUBLE-GAUSS LENSES In the early 1800s, Gauss described a telescope objective comprising a pair of meniscus lenses with one having positive power and the other negative power. An interesting aspect of his lens is that the spherochromatism is essentially constant. Although this lens found little acceptance, in 1888, Alvan Clark of Massachusetts placed a pair of the Gauss lenses around a central stop to create a high-aperture, widefield-of-view lens. This lens form is known as the Double-Gauss lens and is the basis of almost every

17.28

COMPONENTS

FIGURE 25 Unsymmetrical Double-Gauss or Biotar lens introduced as the Leica Summitar in 1939.

high-aperture lens developed to date. An example of this lens was patented by Richter in 1933 and can cover a field of view of o45n at F/6. In 1896, Paul Rudolph of Zeiss developed the Planar which reduces the often serious oblique spherical aberration and the separation of the astigmatic foci at intermediate field angles. Rudolph placed a buried surface into the thick negative elements to control the chromatic aberration. A buried surface is defined as the interface between two glasses that have the same refractive index nd at the central wavelength, but have significantly different Abbe numbers. Such a surface has no effect upon the monochromatic aberrations or the lens system power, but does allow the inclusion of a wide range of chromatic aberration to compensate for that caused by the rest of the lens. Many Double-Gauss lenses are symmetrical; however, it was discovered that if the lens was made unsymmetrical, then an improvement in performance could be realized. This lens form is often called the Biotar. A large portion of 35-mm camera lenses are based upon this design form or some modification thereof. Figure 25 shows the configuration of the Leica Summitar introduced in 1939. It is the general nature of meniscus lens systems of this type to exhibit little coma, distortion, or lateral color; however, oblique spherical aberration is often observed to increase to significant levels as the field angle increases. Oblique spherical aberration can be recognized in transverse ray plots as the S shape of spherical aberration, but with the S becoming increasingly stronger as the field angle increases. As the aperture is increased beyond about F/8, the outer negative elements must be thickened dramatically and achromatic surfaces must necessarily be included.

17.17

PETZVAL LENSES In 1839, Petzval designed a new type of lens that comprises a front objective with an achromatic, airspaced doublet as the rear elements. The Petzval lens has found great application in projectors and as a portrait lens. Both spherical aberration and coma can be well-corrected, but the lens configuration causes the Petzval sum to be undercorrected, which results in the field of view being limited by the astigmatism. The Petzval-field curves inward and may be corrected by including a field flattener lens in close proximity to the image plane. A typical example of a Petzval lens is shown in Fig. 26.

FIGURE 26

Typical Petzval lens.

LENSES

17.29

17.18 TELEPHOTO LENSES A telephoto lens provides an effective focal length efl that is longer than its overall length sol as measured from the front of the lens to the image plane. The telephoto ratio is defined as sol/efl, thus a lens with a ratio less than one is a telephoto lens. The basic concept of a telephoto lens is illustrated by the dialyte lens configuration in which a negative lens is inserted between the objective lens and the image plane. This concept goes back to Kepler, but Peter Barlow developed the idea in the early 1800s by including a negative achromat in telescopes to increase their magnification. Barlow type lenses are widely used today. As the telephoto ratio is made smaller, the design of the lens becomes more difficult, primarily due to the Petzval sum increasing. When most telephoto lenses are used to view objects that are relatively close, the image quality degrades rapidly due to the typical unsymmetrical lens configuration. Some modern telephoto lenses include one or more elements that move as the lens is focused for the purpose of aberration correction.

17.19

INVERTED OR REVERSE TELEPHOTO LENSES A reverse telephoto lens has a telephoto ratio greater than unity and exhibits a shorter focal length than its overall length, a larger bf l than is provided by normal lenses of the same ef l, lenses with generally large apertures and wide fields of view, and lens elements of physically larger size that allow easier manufacture and handling. The basic configuration has a large negative lens located in front of a positive objective lens. Since the negative lens makes the object appear closer to the objective lens, the resultant image moves beyond the focal point, thereby making the bfl greater than the efl. An extreme form of the reverse telephoto lens is the fish-eye or sky lens. Such lenses have a total field of view of 180° or more. The image formed by these lenses has very large barrel distortion. Recalling that the image height for a distortionless lens on a flat image surface is f tan S, the reverse telephoto lens has mapping relationships such as fS and f sin S. When the barrel distortion is given by f sin S, the illumination across the image will be constant if such effects as vignetting and stop/ pupil distortion are absent. Barrel distortion has the effect of compressing the outer portions of the image toward the central portion, thereby increasing the flux density appropriately. After World War II, the Russian designer M. M. Roosinov patented a double-ended reversetelephoto lens that was nearly symmetrical with large negative lenses surrounding a pair of positive lenses with a central stop. Although the back focal length is quite short, it provides relatively large aperture with a wide field of view and essentially no distortion. Lenses of this type have found significant use in aerial photography and photogrammetry.

17.20

PERFORMANCE OF REPRESENTATIVE LENSES Figures 27 to 38 present the performance of lenses, selected generally from the patent literature, representing a variety of lens types. The measures of performance provided in each figure have been selected for utilization purposes. Diffraction effects have not been included. Each figure is divided into four sections a to d. Section a is a drawing of the lens showing the aperture stop. Section b contains two set of plots. The solid line is for the distortion versus field of view (S) in degrees while the dashed lines show the transmission of the lens versus field of view for three F-numbers. Transmission in this case is one minus the fractional vignetting. No loss for coatings, surface reflection, absorption, and the like is included. The rms diameter of the geometric point source image versus field-of-view for three F-numbers is presented in section c. The spot sizes are in angular units and were calculated for the central wavelength only, that is, monochromatic values. Note that the ordinate is logarithmic. The final section, d, contains angular transverse ray plots in all three colors for both the on-axis and near-extreme field angles with yep being measured in the

Distortion (%)

COMPONENTS

100% t

50

0

0 0

5

10

15

20

25

q (b)

F/11

0.0025 F/22

0.10

–0.0025 q = 25 0.0025 5876 Å 6563 4861

0.01 5

0

10

15

20

25

rad

Rms spot diameter (mrad)

F/11 F/5.6

F/5.6

1.00

F/22

2.5

–2.5

(a)

–0.0025 ep

q (c)

(d)

Distortion (%)

FIGURE 27 Rapid Rectlinear: This lens is an aplanat which is symmetrical with the rear half corrected for spherical aberration and flat tangential field. A compact configuration is realized by having a large amount of coma in each half. Symmetry removes the lens system coma, distortion, and lateral color. This type of lens is one of the most popular camera lenses ever made.

F/22

2.5

F/11

100%

F/5.6

t 50

0 –2.5

0 0

5

(a)

10

15

20

25

q (b) 0.001

1.00 F/5.6

–0.001

F/11

q = 20 0.0005

0.10

F/22

5876 Å 6563 4861

rad


17.30

–0.0005 ep (d)

0.01 0

5

10

15

20

25

q (c) FIGURE 28 Celor : F/5.6 with 50n total field of view. Also known as an airspaced dialyte lens. (After R . Kingslake, Lens Design Fundamentals, Academic Press, New York, 1978, p. 243.)

Distortion (%)

LENSES

F/32

2.5

F/16

100%

F/8

t 50

0

0

–2.5 0

6

12

18

24

30

q (b)

F/8

0.002 F/16 –0.002 F/32

q = 30

0.10

0.0001 5876 Å 6563 4861

rad


(a)

1.00

17.31

–0.0001 ep

(d)

0.01 0

6

12

18

24

30

q (c)

Distortion (%)

FIGURE 29 Symmetrical double anastigmat or gauss homocentric objective: basic form of DoubleGauss lens using a pair of gauss telescope objectives. First patented by Alvan Clark in 1888, USP 399,499. (After R. Kingslake, Lens Design Fundamentals, Academic Press New York, 1978, pp. 224–250.) F/11

2.5

F/5.6

100%

F/2.8

t 50

0

0

–2.5 0

(a)

5

20

25

F/5.6

0.005

F/11 –0.005 q = 20

0.10

0.001 5876 Å 6563 4861

0.01 0

5

10

15

20

25

rad


15 q (b)

F/2.8 1.00

10

–0.001 (d)

q (c) FIGURE 30

Triplet: F/2.8 with 50° total field of view. (Tronnier, USP 3,176,582.)

ep

Distortion (%)

COMPONENTS

F/16

2.5

F/8

100%

F/4

t 50

0

0

–2.5 0

5

10

20

25

F/4

0.001

F/8

–0.001 F/16

0.10

q = 20 0.0005 mr


1.00

15 q (b)

(a)

5876 Å 6563 4861

0.01 0

5

10

15

20

ep (d)

Tessar: F/4 with 50° total field of view. (Tronnier, USP 2,084,714, 1937.)

Distortion (%)

FIGURE 31

–0.0005

25

q (c)

F/8

2.5

100%

F/2 F/4

0

50

–2.5

t

0 0

6

12

18

24

30

q (b)

(a) F/2

0.005

1.00 –0.005

F/4 q = 25 0.10

0.0005

F/8 mr


17.32

5876 Å 6563 4861

–0.0005 ep

0.01 0

6

12

18 q (c)

24

30

(d)

FIGURE 32 Unsymmetrical Double-Gauss: This lens was designed in 1933 for Leitz and was called the Summar. F/2 with 60° total field of view. This lens was replaced by the Leitz Summitar in 1939, due to rapidly degrading off-axis resolution and vignetting. Compare this lens with the lens shown in Fig. 33. (Tronnier, USP 2,673,491.)

Distortion (%)

LENSES

F/5.6

2.5

F/2.8

100%

F/1.4 50

0 –2.5

t

0 0

3

6

(a)

9

12

15

q (b)

F/5.6

1.00

0.005 F/2.8 –0.005 q = 15

0.10

0.001

F/1.4

rad


17.33

5876 Å 6563 4861 0.01 0

3

6

12

9

–0.001 ep (d)

15

q (c)

Distortion (%)

FIGURE 33 Unsymmetrical Double-Gauss: This lens type was designed in 1939 for Leitz and was called the F/2 Summitar. Kodak had a similar lens called the F/1.9 Ektar. A later example of this design form is shown and operates at F/1.4 with 30° total field of view. (Klemp, USP 3,005,379.)

F/8

2.5

100% F/4

F/2

50

0

0

–2.5 0

(a)

3

12

15

F/4 0.005 F/8 –0.005

0.10 = 25

0.001 0.01 0

3

6

9 (c)

12

15

5876 Å 6563 4861

rad


9 (b)

F/2 1.00

6

–0.001 (d)

ep

FIGURE 34 Unsymmetrical Double-Gauss: F/1.75 with 50° total field of view. Similar to the 1949 Leitz F/1.5 Summarit. This lens has a split rear element which produces improved resolution of the field of view and less vignetting than the earlier Summar type lens. (Cook, USP 2,959,102.)

Distortion (%)

COMPONENTS

F/22

2.5

F/11

F/5.6

100%

0

50

–2.5 7

14

21

28

35

(b)

(a) F/5.6

1.00

0.005 F/11

–0.005 = 35

0.10

0.0005

F/22 rad


0 0

5876 Å 6563 4861

–0.0005 ep

0.01 0

7

14

21

28

(d)

35

(c)

Distortion (%)

FIGURE 35 Unsymmetrical Double-Gauss: F/5.6 with 70n field of view. This lens is a variant of the 1933 Zeiss F/6.3 Topogon (USP 2,031,792) and is the Bausch & Lomb Metrogon. The principal difference is the splitting of the front element. (Rayton, USP 2,325,275.) F/11 F/5.6

2.5

100% F/2.8 50

0

0

–2.5

(a)

0

7

14

21

28

35

(b)

F/2.8 F/5.6

1.00

F/11

0.005 –0.005 = 35

0.10

0.005 5876 Å 6563 4861

rad


17.34

–0.005 ep

0.01 0

7

14

21 (c)

28

35

(d)

FIGURE 36 Reverse Telephoto: This lens was developed by Zeiss in 1951 and is known as the Biogon. It operates at F/2.8 with 70n field of view. This lens comprises two reverse-telephoto objectives about a central stop. (Bertele, USP 2,721,499.)

Distortion (%)

LENSES

F/5.6

2.5

100% F/1.4 50

0 –2.5

(a)

0 0

2.4

4.8

7.2

9.6

12

(b)

F/1.4 1.00 F/2.8 F/5.6

0.002 0.10 –0.002 = 12 5876 Å 6563 4861

0.01 0

2.4

4.8

7.2

9.6

12

(c)

0.0005 rad


F/2.8

–0.0005 ep

(d)

Distortion (%)

FIGURE 37 Petzval: Example of Kodak projector lens operating at F/1.4 with 24n total field of view. The front lens group has its power shared between a cemented doublet and a singlet for aberration correction. Note that the aperture stop is located between the front and rear groups rather than the more common location at the front group. Resolution in the region near the optical axis is very good although it falls off roughly exponentially. The limiting aberrations are oblique spherical and cubic coma. (Schade, USP 2,541,484.) F/32

2.5

100%

F/16 0

50

F/8

0

–2.5 0

16

32

48

64

80

(b)

(a)

F/16

1.00

0.01 F/32 –0.01 = 80

0.10 0.001 5876 Å 6563 4861 0.01 0

16

32

48

64

80

mr


F/8

–0.001 ep (d)

(c)

FIGURE 38 Fish-eye: The Hill Sky lens was manufactured by Beck of London in 1924. The lens has moderate resolution and enormous distortion characteristic of this type of lens. (Merte, USP 2,126,126.)

17.35

17.36

COMPONENTS

entrance pupil. The lower right plot shows the axial aberrations while the upper left plot represents the tangential/meridional aberrations and the upper right plot presents the sagittal aberrations. The X included on some of the tangential plots represents the location of the paraxial principal ray which also provides a measure of the distortion. The legend indicating the relationship between line type and wavelength is included. The linear spot size is computed by multiplying the efl by the angular spot size. This value can be compared against the diffraction-limited spot size given by 2.44(N/Dep). If the geometric spot is several times smaller than the diffraction-limited spot, then the lens may be considered to be diffraction-limited for most purposes. If the geometric spot is several times larger, then the lens performance is controlled by the geometric spot size for most applications.

17.21

RAPID ESTIMATION OF LENS PERFORMANCE

Singlet Figure 39 is a nomogram that allows quick estimation of the performance of a single refracting lens, with the stop at the lens, as a function of refractive index N, dispersion V, F-number, and field of view S. Chart A estimates the angular blur diameter D resulting from a singlet with bending for minimum spherical aberration. The angular chromatic blur diameter is given by Chart B. The three rows of FN values below the chart represent the angular blur diameter that contains the indicated 10–2 =5 =3 =2 = 1.5

10–3

F/1.4

F/2

10–4

F/2.8

F/4

10–5

F/5.6

F/8

F/11

Chart A Blur due to spherical aberr. (at “minimum bending”) = 10 = 20 = 40 = 80 = 160 = 320

20 10 5 2.5

100% /1 /2 /4 /8 /16 75–90% /1 /2 /4 /8 /1 /2 /4 40–60% Chart B Blur due to chromatic aberr. /1

/2

/4

/8

/16

/32

/32 /16 /8

/64 /32 /16

/128 /64 /32

/64

Chart C Blur due to astigmatism (stop at lens)

10–2

10–3

10–4

10–5

–angular blur spot (rad) FIGURE 39 Estimation of single lens spot size as a function of refractive index, dispersion, F-number, and field of view. (Smith, Modern Optical Engineering, McGraw-Hill, New York, 1990, p. 458.)

LENSES

17.37

FIGURE 40 Geometric relationships for determining the geometric depth of focus of a lens .

percentage of the total energy. Chart C shows the blur diameter due to astigmatism. Coma for a singlet bent for minimum spherical aberration with the stop at the lens is approximately

Q 16 (N 2) (FN)2

(50)

Depth of Focus The depth of focus of an optical system is expressed as the axial displacement that the image may experience before the resultant image blur becomes excessive. Figure 40 shows the geometric relationship of the angular blur tolerance $S to the depth of focus Fo. If the entrance pupil diameter is Dep and the image distance is si then the depth of focus is

Do

si2 $ Q Dep o si $ Q

(51)

si2 $ Q Dep

(52)

or when F si , the depth of focus becomes

D When si f, then

D f$ Q FN

(53)

The depth of field is distance that the object may be moved without causing excessive image blur with a fixed image location. The distance at which a lens may be focused such that the depth of field extends to infinity is so Dep /$S and is called the hyperfocal distance. If the lens system is diffraction-limited, then the depth of focus according to the Rayleigh criterion is given by L D o (54) 2ni sin 2 ui Diffraction-Limited Lenses It is well known that the shape of the image irradiance of an incoherent, monochromatic point source formed by an aberration-free, circularly-symmetric lens system is described by the Airy pattern § 2 J1(kDepr /2) ¶ E(r ) C0 ¨ · ¨© kDepr ·¸

2

(55)

COMPONENTS

where J1 is the first-order Bessel function of the first kind, Dep is the diameter of the entrance pupil, k is 2R/N, r is the radial distance from the center of the image to the observation point, and C0 is a scaling factor. The angular radius DDL of the first dark ring of the image is 1.22(N/Dep). A common measure for the resolution is Lord Rayleigh’s criterion that asserts that two point sources are just resolvable when the maximum of one Airy pattern coincides with the first dark ring of the second Airy pattern, that is, an angular separation of DDL. Figure 41 presents a nomogram that can be used to make a rapid estimate of the diameter of angular or linear blur for a diffraction-limited system. The modulation transfer function (MTF) at a specific wavelength Nfor a circular entrance pupil can be computed by MTFL 7

2§ arccos 7 7 1 7 2 ¶ ¸ P©

(56)

for 0 a 7 a 1

where 7 is the normalized spatial frequency (v/vco) with the maximum or cut-off frequency vco being be by 1/No FN.

-angular blur spot (rad) 10–1

10–2

54 3

10–3

2

5 4 3

10–4

2

5 4 3

10–5

2

1 21

2 4 8 16 32 10–1

Relative aperture /64 /32 in

/16

10–2

20 10

/8

in

5

/4

2 1 in

B-blur spot diameter (in)

0

10 50

10–2

/2

10–3 Focal length (in)

10–3

/1

Aperture (in) 10–4 Wavelength

17.38

1 2

0.02

2

0.05

0.1

0.2

0.5

1.0

2

5 10

8 32

50 10

–2

54 3

2

10–3

5 4 3

2

10–4

5 4 3

11 2

20

10–4 2 4 8 16 32

2 Wavelength (m) 8

100 32 2 10–5

-angular blur spot (rad) FIGURE 41 Estimation of the spot diameter for a diffraction-limited lens system. The diameter is that of the first dark ring of the Airy disk. (Smith, Modern Optical Engineering, McGraw-Hill, New York, 1990, p. 458.)

LENSES

17.39

FIGURE 42 Variation of $ with NL/NS or b 1 (N) N/N as the (N) 1 as the solid curve and 2 for 2 S dashed curve.

Should the source be polychromatic and the lens system be aberration-free, then the perfectimage irradiance distribution of a point source can be written as 2

§ 2 J1(kDepr /2) ¶ c E(r ) C1 ¯ (L ) ¨ · dL 0 ¨© kDepr ·¸

(57)

where (N) is the peak normalized spectral weighting factor and C1 is a scaling factor. A quick estimation of this ideal irradiance distribution can be made by invoking the central limit theorem to approximate this distribution by a Gaussian function, that is, 2 2 E(r ) y C 2e (r /2S )

(58)

where C2 is a scaling constant and U 2 is the estimated variance of the irradiance distribution. When (N) 1 in the spectral interval NS to NL and zero otherwise with Ns < NL, an estimate of can be written as

S

$L L P Dep

(59)

where $ 1.335 0.625b 0.025b2 0.0465b3 with b (NL/Ns) 1. Should (N) N/NL in the spectral interval NS to NL and zero otherwise, which approximates the behavior of a quantum detector, $ 1. 335 0.65b 0.385b2 0.099b3. The Gaussian estimate residual error is less than a few percent for b 0.5 and remains useful even as b l 0. Figure 42 contains plots of $ for both cases of (N), where the abscissa is NL/NS. A useful estimation of the modulation transfer function for this polychromatic lens system is given by 2

MTF(v ) y e 2(PS v )

(60)

where v is the spatial frequency. This approximation overestimates the MTF somewhat at lower spatial frequencies, while being rather a close fit at medium and higher spatial frequencies. The reason for this is that the central portion of the irradiance distribution is closely matched by the gaussian approximation, while the irradiance estimation beyond several Airy radii begins to degrade, therefore impacting the lower spatial frequencies. Nevertheless, this approximation can provide useful insight into expected performance limits.

17.40

17.22

COMPONENTS

BIBLIOGRAPHY Erfle, H., “Die optische abbildung durch kugelflaechen,” Chap. 3 in S. Czapski und O. Eppenstein, Grundzuege der Theorie der Optischen Instrumente nach Abbe, 3d ed., H. Erfle and H. Boegehold, eds., Leipzig: Barth, 72–134 (1924). Goodman, D. S., “Basic Optical Instruments,” Chap. 4 in Geometrical and Instrumental Optics, Daniel Malacara ed., Methods of Experimental Physics 25, Academic Press, San Diego (1988). Hopkins, R. E., “The Components in the Basic Optical Systems,” Chap. 3 in Geometrical and Instrumental Optics, Daniel Malacara ed., Methods of Experimental Physics 25, Academic Press, San Diego (1988). Hopkins, R. E., “Geometrical Optics,” Chap. 2 in Geometrical and Instrumental Optics, Daniel Malacara ed., Methods of Experimental Physics 25, Academic Press, San Diego (1988). Johnson, R. B., “Basic Lenses,” Chap. 13 in Optical Engineer’s Desk Reference, W. L. Wolfe, ed., SPIE Press, Bellingham (2003). Johnson, R. B. and C. Feng, “A History of IR Lens Designs,” SPIE Critical Reviews CR37:3–18 (1991). Kingslake, R., “Basic Geometrical Optics,” Chap. 6 in Applied Optics and Optical Engineering, 1, Academic Press, New York (1965). Kingslake, R., Lens Design Fundamentals, Academic Press, New York (1978). Kingslake, R., Optical System Design, Academic Press, New York (1983). Kingslake, R., A History of the Photographic Lens, Academic Press, San Diego (1989). Kingslake, R., Optics in Photography, SPIE Press, Bellingham (1992). Kingslake, R., and R. Barry Johnson, Lens Design Fundamentals, 2d ed., Academic Press, New York (2009). Laikin, M., Lens Design, 4th ed., Marcel Dekker, New York (2006). Mahajan, V. N., Aberration Theory Made Simple, SPIE Press, Bellingham (1991). MIL-HDBK-141, Optical Design, Defense Supply Agency, Washington (1962). Mouroulis, R. and J. Macdonald, Geometrical Optics and Optical Design, Oxford University Press, New York (1997). Ray, S. F., Applied Photographic Optics, 2d ed., Focal Press, Oxford (1997). Schroeder, H., “Notiz betreffend die gaussischen hauptpunkte,” Astron. Nachrichten 111:187–188 (1885). Smith, W. J., Practical Optical System Layout, McGraw-Hill, New York (1997). Smith, W. J., Modern Optical Engineering, 3d ed., McGraw-Hill, New York (2000). Smith, W. J., “Camera Lenses,” Chap. 14 in Optical Engineer’s Desk Reference, W. L. Wolfe, ed., SPIE Press, Bellingham (2003). Smith, W. J., Modern Lens Design, 2d ed., McGraw-Hill, New York (2005).

18 AFOCAL SYSTEMS William B. Wetherell

Optical Research Associates Framingham, Massachusetts

18.1

GLOSSARY BFL D ERcp ERk e F, F` FFL h, h` l, l` M m n OR o P, P` R TTL tan C x, y, z $z

back focal length pupil diameter eye relief common pupil position eye relief keplerian exit pupil; eye space focal points front focal length object and image heights object and image distances angular magnification linear, lateral magnification refractive index object relief entrance pupil; object space principal points radius total length slope cartesian coordinates axial separation

Retired.

18.1

18.2

18.2

COMPONENTS

INTRODUCTION If collimated (parallel) light rays from an infinitely distant point source fall incident on the input end of a lens system, rays exiting from the output end will show one of three characteristics: (1) they will converge to a real point focus outside the lens system, (2) they will appear to diverge from a virtual point focus within the lens system, or (3) they will emerge as collimated rays that may differ in some characteristics from the incident collimated rays. In cases 1 and 2, the paraxial imaging properties of the lens system can be modeled accurately by a characteristic focal length and a set of fixed principal surfaces. Such lens systems might be called focusing or focal lenses, but are usually referred to simply as lenses. In case 3, a single finite focal length cannot model the paraxial characteristics of the lens system; in effect, the focal length is infinite, with the output focal point an infinite distance behind the lens, and the associated principal surface an infinite distance in front of the lens. Such lens systems are referred to as afocal, or without focal length. They will be called afocal lenses here, following the common practice of using “lens” to refer to both single element and multielement lens systems. They are the topic of this chapter. The first afocal lens was the galilean telescope (to be described later), a visual telescope made famous by Galileo’s astronomical observations. It is now believed to have been invented by Hans Lipperhey in 1608.1 Afocal lenses are usually thought of in the context of viewing instruments or attachments to change the effective focal length of focusing lenses, whose outputs are always collimated. In fact, afocal lenses can form real images of real objects. A more useful distinction between focusing and afocal lenses concerns which optical parameters are fixed, and which can vary in use. Focusing lenses have a fixed, finite focal length, can produce real images for a wide range of object distances, and have a linear magnification which varies with object distance. Afocal lenses have a fixed magnification which is independent of object distance, and the range of object distances yielding real images is severely restricted. This chapter is divided into six sections, including this introduction. Section 18.3 reviews the gaussian (paraxial) image-forming characteristics of afocal lenses and compares them to the properties of focusing lenses. The importance of the optical invariant in designing afocal lenses is discussed. Section 18.4 reviews the keplerian telescope and its descendants, including both infinite conjugate and finite conjugate variants. Section 18.5 discusses the galilean telescope and its descendants. Thin-lens models are used in the Secs. 18.4 and 18.5 to define imaging characteristics and design principles for afocal lenses. Section 18.6 reviews relay trains and periscopes. The final section reviews reflecting and catadioptric afocal lenses. This chapter is based on an earlier article by Wetherell.2 That article contains derivations of many of the equations appearing here, as well as a more extensive list of patents illustrating different types of afocal lens systems.

18.3

GAUSSIAN ANALYSIS OF AFOCAL LENSES Afocal lenses differ from focusing lenses in ways that are not always obvious. It is useful to review the basic image-forming characteristics of focusing lenses before defining the characteristics unique to afocal lenses.

Focusing Lenses In this chapter, all lens elements are assumed to be immersed in air, so that object space and image space have the same index of refraction. Points in object space and image space are represented by two rectangular coordinate systems (x, y, z) and (x `, y `, z `), with the prime indicating image space. The z- and z` axes form a common line in space, the optical axis of the system. It is assumed, unless noted otherwise, that all lens elements are rotationally symmetric with respect to the optical axis. Under these conditions, the imaging geometry of a focusing lens can be defined in terms of two principal

AFOCAL SYSTEMS

18.3

FIGURE 1

Imaging geometry of focusing lenses.

points P and P`, two focal points F and F`, and a single characteristic focal length f, as shown in Fig. 1. P, P`, F, and F ` all lie on the optical axis. The focal points F and F`, will be the origins for the coordinate systems (x, y, z) and (x `, y`, z `). If the origins are at P and P `, the coordinates will be given as (x, y, s) and (x `, y `, s`), where s z f and s` z ` f. Normal right-hand sign conventions are used for each set of coordinates, and light travels along the z axis from negative z toward positive z`, unless the optical system has internal mirrors. Figure la illustrates the terminology for finite conjugate objects. Object points and image points are assumed to lie in planes normal to the optical axis, for paraxial computations. Object distance is specified by the axial distance to the object surface, z or s, and image distance by z` or s`. The two most commonly used equations relating image distance to object distance are 1 1 1

s` s f

(1)

zz ` f 2

(2)

and

For infinitely distant object points, z` 0 and s ` f, and the corresponding image points will lie in the focal plane at F `. To determine the actual distance from object plane to image plane, it is necessary to know the distance sp between P and P `. The value of sp is a constant specific to each real lens system, and may be either positive [moving object and image further apart than predicted by Eq. (1) or (2)] or negative (moving them closer together).

18.4

COMPONENTS

For rotationally symmetric systems, off-axis object and image coordinates can be expressed by the object height h and image height h`, where h2 x2 y2 and h`2 x`2 y `2. Object height and image height are related by the linear magnification m, where m

h` s ` z ` f h s z f

(3)

Since the product zz` is a constant, Eq. (3) implies that magnification varies with object distance. The principal surfaces of a focusing lens intersect the optical axis at the principal points P and P `. In paraxial analysis, the principal surfaces are planes normal to the optical axis; for real lenses, they may be curved. The principal surfaces are conjugate image surfaces for which m 1.0. This property makes the raytrace construction shown in Fig. la possible, since a ray traveling parallel to the optical axis in either object or image space must intersect the focal point in the conjugate space, and must also intersect both principal surfaces at the same height. In real lenses, the object and image surfaces may be tilted or curved. Planes normal to the optical axis are still used to define object and image positions for off-axis object points, and to compute magnification. For tilted object surfaces, the properties of the principal surfaces can be used to relate object surface and image surface tilt angles, as shown in Fig. 1b. Tilted object and image planes intersect the optical axis and the two principal planes. The tilt angles with respect to the optical axis, u and u`, are defined by meridional rays lying in the two surfaces. The points at which conjugate tilted planes intercept the optical axis are defined by sa and s à, given by Eq. (1). Both object and image planes must intersect their respective principal surfaces at the same height y, where y sa tan u s à tan u `. It follows that tan u ` sa 1 tan u sa` ma

(4)

The geometry of Fig. 1b is known as the Scheimpflug condition, and Eq. (4) is the Scheimpflug rule, relating image to object tilt. The magnification ma applies only to the axial image. The height off axis of an infinitely distant object is defined by the principal ray angle up measured from F or P, as shown in Fig. lc. In this case, the image height is h ` f tan u p

(5)

A focusing lens which obeys Eq. (5) for all values of up within a specified range is said to be distortionfree: if the object is a set of equally spaced parallel lines lying in an object plane perpendicular to the optical axis, it will be imaged as a set of equally spaced parallel lines in an image plane perpendicular to the optical axis, with line spacing proportional to m. Equations (1) through (5) are the basic gaussian imaging equations defining a perfect focusing lens. Equation (2) is sometimes called the newtonian form of Eq. (1), and is the more useful form for application to afocal lens systems.

Afocal Lenses With afocal lenses, somewhat different coordinate system origins and nomenclature are used, as shown in Fig. 2. The object and image space reference points RO and RE are at conjugate image points. Since the earliest and most common use for afocal lenses is as an aid to the eye for viewing distant objects, image space is referred to as eye space. Object position is defined by a right-hand coordinate system (xo , yo , zo ) centered on reference point RO. Image position in eye space is defined by coordinates (xe , ye , ze ) centered on RE. Because afocal lenses are most commonly used for viewing distant objects, their imaging characteristics are usually specified in terms of angular magnification M, entrance pupil diameter Do, and total field of view. Figure 2a models an afocal lens used at infinite conjugates. Object height off axis is

AFOCAL SYSTEMS

18.5

Do De upo

upe RE RO

(a) Infinite conjugate model of afocal lens s

ho1

ho2 RE RO

he1 he2

zo

ze

(b) Finite conjugate model of afocal lens

FIGURE 2

Imaging geometry of focusing lenses.

defined by the principal ray angle upo, and the corresponding image height is defined by upe. Objects viewed through the afocal lens will appear to be magnified by a factor M, where tan upe M tan upo

(6)

If M is negative, as in Fig. 2a, the image will appear to be inverted. [Strictly speaking, since RO and RE are separated by a distance S, the apparent magnification seen by an eye at RE with and without the afocal lens will differ slightly from that indicated by Eq. (6) for nearby objects.]2 The imaging geometry of an afocal lens for finite conjugates is illustrated in Fig. 2b. Since a ray entering the afocal lens parallel to the optical axis will exit the afocal lens parallel to the optical axis, it follows that the linear magnification m relating object height ho and image height he must be invariant with object distance. The linear magnification m is the inverse of the angular magnification M: m

he 1 ho M

(7)

The axial separation $ze of any two images he1 and he2 is related to the separation $zo of the corresponding objects ho1 and ho2 by $ ze m2 $ zo

$ zo M2

(8)

It follows that any convenient pair of conjugate image points can be chosen as reference points RO and RE. Given the location of RO, the reference point separation S, and the magnifications m 1/M, the imaging geometry of a rotationally symmetric distortion-free afocal lens can be given as x e mx o

xo M

ye myo

yo M

ze m2 zo

zo M2

(9)

Equation (9) is a statement that coordinate transformation between object space and eye space is rectilinear for afocal lenses, and is solely dependent on the afocal magnification M and the location

18.6

COMPONENTS

of two conjugate reference points RO and RE. The equations apply (paraxially) to all object and image points independent of their distances from the afocal lens. Any straight line of equally spaced object points will be imaged as a straight line of equally spaced image points, even if the line does not lie in a plane normal to the optical axis. Either RO or RE may be chosen arbitrarily, and need not lie on the axis of symmetry of the lens system, so long as the zo- and ze axes are set parallel to the axis of symmetry. A corollary of invariance in lateral and axial linear magnification is invariance in angular magnification. Equation (6) thus applies to any ray traced through the afocal system, and to tilted object and image surfaces. In the latter context, Eq. (6) can be seen as an extension of Eq. (4) to afocal lenses. The eye space pupil diameter De is of special importance to the design of visual instruments and afocal attachments: De must usually be large enough to fill the pupil of the associated instrument or eye. The object space pupil diameter Do is related to De by De

Do mDo M

(10)

(The more common terminology exit pupil and entrance pupil will be used later in this chapter.)

Subjective Aspects of Afocal Imagery The angular magnification M is usually thought of in terms of Eq. (6), which is often taken to indicate that an afocal lens projects an image which is M-times as large as the object. (See, for example, Fig. 5.88 in Hecht and Zajac.)3 Equation (9) shows that the image height is actually 1/M-times the object height (i.e., smaller than the object when |M| 1). Equation (9) also shows, however, that the image distance is reduced to l/M 2-times the object distance, and it is this combination of linear height reduction and quadratic distance reduction which produces the subjective appearance of magnification. Equation (6) can be derived directly from Eq. (9). tan upe

ye yo /M M tan upo z e z o /M 2

Equation (9) is therefore a more complete model than Eq. (6) for rotationally symmetric, distortionfree afocal lenses. Figure 3 illustrates two subjective effects which arise when viewing objects through afocal lenses. In Fig. 3a for which M 3r, Eq. (9) predicts that image dimensions normal to the optical axis will be reduced by 1/3, while image dimensions along the optical axis will be reduced by 1/9. The image of the cube in Fig. 3a looks three times as tall and wide because it is nine times closer, but it appears compressed by a factor of 3 in the axial direction, making it look like a cardboard cutout. This subjective compression, most apparent when using binoculars, is intrinsic to the principle producing angular magnification, and is independent of pupil spacing in the binoculars. Figure 3a assumes the optical axis is horizontal within the observer’s reference framework. If the axis of the afocal lens is not horizontal, the afocal lens may create the illusion that horizontal surfaces are tilted. Figure 3b represents an M 7r afocal lens whose axis is tilted 10° to a horizontal surface. Equation (6) can be used to show that the image of this surface is tilted approximately 51° to the axis of the afocal lens, creating the illusion that the surface is tilted 41° to the observer’s horizon. This illusion is most noticeable when looking downward at a surface known to be horizontal, such as a body of water, through a pair of binoculars.

AFOCAL SYSTEMS

18.7

!%!"( ×! #

′

′ !")! $#%"(×! #&!#'##$$°$!!")! FIGURE 3

Subjective aspects of afocal imagery.

Afocal Lenses and the Optical Invariant Equations (6) and (7) can be combined to yield he tan upe ho tan upo

(11)

which is a statement of the optical invariant as applied to distortion-free afocal lenses. Neither upo nor upe is typically larger than 35°–40° in distortion-free afocal lenses, although there are examples with distortion where upo l 90°. Given a limit on one angle, Eq. (11) implies a limit on the other angle related to the ratio ho/he Do/De. Put in words, the ratio Do/De cannot be made arbitrarily large without a corresponding reduction in the maximum allowable field of view. All designers of afocal lens systems must take this fundamental principle into consideration.

18.4

KEPLERIAN AFOCAL LENSES A simple afocal lens can be made up of two focusing lenses, an objective and an eyepiece, set up so that the rear focal point of the objective coincides with the front focal point of the eyepiece. There are two general classes of simple afocal lenses, one in which both focusing lenses are positive, and the other in which one of the two is negative. Afocal lenses containing two positive lenses were first described by Johannes Kepler in Dioptrice, in 1611,4 and are called keplerian. Lenses containing a negative eyepiece are called galilean, and will be discussed separately. Generally, afocal lenses contain at least two powered surfaces. The simplest model for an afocal lens consists of two thin lenses.

Thin-Lens Model of a Keplerian Afocal Lens Figure 4 shows a thin-lens model of a keplerian telescope. The focal length of its objective is fo and the focal length of its eyepiece is fe. Its properties can be understood by tracing two rays, ray 1 entering the objective parallel to the optical axis, and ray 2 passing through Fo, the front focal

18.8

COMPONENTS

SF

1 pe

2 RO

po

RE

Eyelens Objective FIGURE 4

Thin-lens model of keplerian afocal lens.

point of the objective. Ray 1 leads directly to the linear magnification m, and ray 2 to the angular magnification M: m

fe f0

M

f o tan upe f e tan upo

(12)

Equation (12) makes the relationship of afocal magnification to the Scheimpflug rule of Eq. (4) more explicit, with focal lengths fo and fe substituting for sa and sa` . The second ray shows that placing the reference point RO at Fo will result in the reference point RE falling on Fe`, the rear focal point of the eyepiece. The reference point separation for RO in this location is SF 2 f e 2 f o 2(1 M ) f e 2(1 m) f o

(13)

Equation (13) can be used as a starting point for calculating any other locations for RO and RE, in combination with Eq. (9). One additional generalization can be drawn from Fig. 4: the ray passing through Fo will emerge from the objective parallel to the optical axis. It will therefore also pass through Fe` even if the spacing between objective and eyepiece is increased to focus on nearby objects. Thus the angular magnification remains invariant, if upo is measured from Fo and upe is measured from Fe`, even when adjusting the eyepiece to focus on nearby objects makes the lens system depart from being strictly afocal. The simple thin-lens model of the keplerian telescope can be extended to systems composed of two real focusing lenses if we know their focal lengths and the location of each lens’ front and rear focal points. Equation (12) can be used to derive M, and SF can be measured. Equation (9) can then be used to compute both finite and infinite conjugate image geometry.

Eye Relief Manipulation The earliest application of keplerian afocal lenses was to obtain magnified views of distant objects. To view distant objects, the eye is placed at RE. An important design consideration in such instruments is to move RE far enough away from the last surface of the eyepiece for comfortable viewing. The distance from the last optical surface to the exit pupil at RE is called the eye relief ER. One way to increase eye relief ER is to move the entrance pupil at RO toward the objective. Most telescopes and binoculars have the system stop at the first surface of the objective, coincident with the entrance pupil, as shown in Fig. 5a.

AFOCAL SYSTEMS

18.9

FIGURE 5

Increasing eye relief ER by moving stop.

In the thin-lens model of Fig. 5a, RO is moved a distance zo fo to place it at the objective. Thus RE must move a distance ze fo/M2 fe /M, keeping in mind that M is negative in this example. Thus for a thin-lens keplerian telescope with its stop at the objective, the eye relief ERk is ER k

( M 1) fe M

(14)

It is possible to increase the eye relief further by placing the stop inside the telescope, moving the location of RO into virtual object space. Figure 5b shows an extreme example of this, where the virtual location of RO has been matched to the real location of RE. For this common-pupil-position case, the eye relief ERcp is ER cp

( M 1) f ( M 1) e

(15)

A price must be paid for locating the stop inside the afocal lens, in that the elements ahead of the stop must be increased in diameter if the same field of view is to be covered without vignetting. The larger the magnitude of M, the smaller the gain in ER yielded by using an internal stop. To increase the eye relief further, it is necessary to make the objective and/or the eyepiece more complex, increasing the distance between Fo and the first surface of the objective, and between the last surface of the eyepiece and Fe`. If this is done, placing RO at the first surface of the objective will further increase ER. Figure 6 shows a thin-lens model of a telephoto focusing lens of focal length ft. For convenience, a zero Petzval sum design is used, for which f1 f and f2 f. Given the telephoto’s focal length ft and the lens separation d, the rest of the parameters shown in Fig. 6 can be defined in terms of the constant C d/ft. The component focal length f, back focal length bf l, and front focal length ff l, are given by f f t C 1/2

bf l f t (1 C 1/2 )

ff l f t (1 C 1/2 )

(16)

and the total physical length ttl and focal point separation sf are given by ttl f t (1 C C 1/2 )

sf f t (2 C )

(17)

18.10

COMPONENTS

FIGURE 6 Zero Petzval sum telephoto lens.

The maximum gain in eye relief will be obtained by using telephoto designs for both objective and eyepiece, with the negative elements of each facing each other. Two cases are of special interest. First, ttl can be minimized by setting C 0.25 for both objective and eyepiece. In this case, the eye relief ERttl is ER ttl 1.5

( M 1) f e 1.5ER k M

(18)

Second, sf can be maximized by setting C 1.0 for both objective and eyepiece. This places the negative element at the focal plane, merging the objective and eyepiece negative elements into a single negative field lens. The eye relief in this case, ERsf , is ER sf 2.0

( M 1) 2.0ER k M

(19)

Placing a field lens at the focus between objective and eyepiece can be problematical, when viewing distant objects, since dust or scratches on the field lens will be visible. If a reticle is required, however, it can be incorporated into the field lens. Equations (14), (18), and (19) show that significant gains in eye relief can be made by power redistribution. In the example of Eq. (18), the gain in ER is accompanied by a reduction in the physical length of the optics, which is frequently beneficial.

Terrestrial Telescopes Keplerian telescopes form an inverted image, which is considered undesirable when viewing earthbound objects. One way to make the image erect, commonly used in binoculars, is to incorporate erecting prisms. A second is to insert a relay stage between objective and eyepiece, as shown in Fig. 7. The added relay is called an image erector, and telescopes of this form are called terrestrial telescopes. (The keplerian telescope is often referred to as an astronomical telescope, to distinguish it from terrestrial telescopes, since astronomers do not usually object to inverted images. Astronomical has become ambiguous in this context, since it now more commonly refers to the very large aperture reflecting objectives found in astronomical observatories. Keplerian is the preferred terminology.) The terrestrial telescope can be thought of as containing an objective, eyepiece, and image erector, or as containing two afocal relay stages. There are many variants of terrestrial telescopes made today, in the form of binoculars, theodolites, range finders, spotting scopes, rifle scopes, and other military optical instrumentation. All are offshoots of the keplerian telescope, containing a positive objective and a positive eyepiece, with intermediate

AFOCAL SYSTEMS

18.11

FIGURE 7 Terrestrial telescope.

relay stages to perform special functions. Patrick5 and Jacobs6 are good starting points for obtaining more information. Field-of-View Limitations in Keplerian and Terrestrial Telescopes The maximum allowable eye space angle upe and magnification M set an upper limit on achievable fields of view, in accordance with Eq. (11). MIL-HDBK-1417 lists one eyepiece design for which the maximum upe 36°. If M 7r, using that eyepiece allows a 5.9° maximum value for upo. It is a common commercial practice to specify the total field of view FOV as the width in feet which subtends an angle 2upo from 1000 yd away, even when the pupil diameter is given in millimeters. FOV is thus given by FOV 6000 tan upo

6000 tan upe M

(20)

For our 7r example, with upe 36°, FOV 620 ft at 1000 yd. For commercial 7 r 50 binoculars (M 7r and Do 50 mm), FOV 376 ft at 1000 yd is more typical. Finite Conjugate Afocal Relays If an object is placed in contact with the front surface of the keplerian telescope of Fig. 5, its image will appear a distance ERk behind the last surface of the eyepiece, in accordance with Eq. (14). There is a corresponding object relief distance ORk M2ERk defining the position of an object that will be imaged at the output surface of the eyepiece, as shown in Fig. 8. ORk and ERk define the portions of object space and eye space within which real images can be formed of real objects with a simple keplerian afocal lens. OR k M ( M 1) f e

(21)

Object relief is enlarged by the power redistribution technique used to extend eye relief. Thus there is a minimum total length design corresponding to Eq. (18), for which the object relief ORttl is OR ttl 1.5 M ( M 1) f e

(22)

and a maximum eye relief design corresponding to Eq. (19), for which ORsf OR sf 2.0 M ( M 1) f e is also maximized.

(23)

18.12

COMPONENTS

FIGURE 8 Finite conjugate keplerian afocal lens showing limits on usable object space and image space.

Figure 9 shows an example of a zero Petzval sum finite conjugate afocal relay designed to maximize OR and ER by placing a negative field lens at the central infinite conjugate image. Placing the stop at the field lens means that the lens is telecentric (principal rays parallel to the optical axis) in both object and eye space. As a result, magnification, principal ray angle of incidence on object and image surface, and cone angle are all invariant over the entire range of OR and ER for which there is no vignetting. Magnification and cone angle invariance means that object and image surfaces can be tilted with respect to the optical axis without introducing keystoning or variation in image irradiance over the field of view. Having the principal rays telecentric means that object and image position can be adjusted for focus without altering magnification. It also means that the lens can be defocused without altering magnification, a property very useful for unsharp masking techniques used in the movie industry. One potential disadvantage of telecentric finite conjugate afocal relays is evident from Fig. 9: to avoid vignetting, the apertures of both objective and eyepiece must be larger than the size of the associated object and image. While it is possible to reduce the diameter of either the objective or the eyepiece by shifting the stop to make the design nontelecentric, the diameter of the other lens group becomes larger. Afocal relays are thus likely to be more expensive to manufacture than focusing lens relays, unless object and image are small. Finite conjugate afocal lenses have been used for alignment telescopes,8 for laser velocimeters,9 and for automatic inspection systems for printed circuit boards.10 In the last case, invariance of magnification, cone angle, and angle of incidence on a tilted object surface make it possible to measure the volume of solder beads automatically with a computerized video system. Finite conjugate afocal lenses are also used as Fourier transform lenses.11 Brief descriptions of these applications are given in Wetherell.2

FIGURE 9 Finite conjugate afocal relay configured to maximize eye relief ER and object relief OR. Stop at common focus collimates principal rays in both object space and eye space.

AFOCAL SYSTEMS

18.13

FIGURE 10

Afocal lens scanner geometry.

Afocal Lenses for Scanners Many optical systems require scanners, and if the apertures of the systems are large enough, it is preferable to place the scanner inside the system. Although scanners have been designed for use in convergent light, they are more commonly placed in collimated light (see Chap. 30, “Scanners,” in this volume, Marshall,12 and Chap. 7 in Lloyd,13 for descriptions of scanning techniques). A large aperture objective can be converted into a high magnification keplerian afocal lens with the aid of a short focal length eyepiece collimator, as shown in Fig. 10, providing a pupil in a collimated beam in which to insert a scanner. For the polygonal scanner shown, given the desired scan angle and telescope aperture diameter, Eq. (11) will define the combination of scanner facet size and number of facets needed to achieve the desired scanning efficiency. Scanning efficiency is the time it takes to complete one scan divided by the time between the start of two sequential scans. It is tied to the ratio of facet length to beam diameter, the amount of vignetting allowed within a scan, the number of facets, and the angle to be scanned. Two limitations need to be kept in mind. First, the optical invariant will place an upper limit on M for the given combination of Do and upo, since there will be a practical upper limit on the achievable value of upe. Second, it may be desirable in some cases for the keplerian afocal relay to have enough barrel distortion so that Eq. (6) becomes upe Mupo

(24)

An afocal lens obeying Eq. (24) will convert a constant rotation rate of the scan mirror into a constant angular scan rate for the external beam. The same property in “f-theta” focusing lenses is used to convert a constant angular velocity scanner rotation rate into a constant linear velocity rate for the recording spot of light. The above discussion applies to scanning with a point detector. When the detector is a linear diode array, or when a rectangular image is being projected onto moving film, the required distortion characteristics for the optical system may be more complex.

Imaging in Binoculars Most commercial binoculars consist of two keplerian afocal lenses with internal prismatic image erectors. Object and image space coordinates for binoculars of this type are shown schematically in Fig. 11. Equation (9) can be applied to Fig. 11 to analyze their imaging properties. In most binoculars, the spacing So between objectives differs from the spacing Se between eyepieces, and So may be either larger or smaller than Se. Each telescope has its own set of reference points, ROL and REL for the left telescope, and ROR and RER for the right. Object space is a single domain with a single origin O. The object point at zo, midway between the objective axes, will be xoL units to the right

18.14

COMPONENTS

FIGURE 11

Imaging geometry of binoculars.

of the left objective axis, and xoR units to the left of the right objective axis. In an ideal binocular system, the images of the object formed by the two telescopes would merge at one point, ze units in front of eye space origin E. This will happen if So MSe , so that xeL xoL/M and xeR xoR/M. In most modern binoculars, however, So MSe , and separate eye space reference points EL and ER will be formed for the left and right eye. As a result, each eye sees its own eye space, and while they overlap, they are not coincident. This property of binoculars can affect stereo acuity2 and eye accommodation for the user. It is normal for the angle at which a person’s left-eye and right-eye lines of sight converge to be linked to the distance at which the eyes focus. (In my case, this linkage was quite strong before I began wearing glasses.) Eyes focused for a distance ze normally would converge with an angle D, as shown in Fig. 11. When So MSe , as is commonly the case, the actual convergence angle D ` is much smaller. A viewer for whom focus distance is strongly linked to convergence angle may find such binoculars uncomfortable to use for extended periods, and may be in need of frequent focus adjustment for different object distances. A related but more critical problem arises if the axes of the left and right telescopes are not accurately parallel to each other. Misalignment of the axes requires the eyes to twist in unaccustomed directions to fuse the two images, and refocusing the eyepiece is seldom able to ease the burden. Jacobs6 is one of the few authors to discuss this problem. Jacobs divides the axes misalignment into three categories: (1) misalignments requiring a divergence D of the eye axes to fuse the images, (2) misalignments requiring a convergence C, and (3) misalignments requiring a vertical displacement V. The tolerance on allowable misalignment in minutes of arc is given by Jacobs as D 7.5/( M 1)

C 22.5/( M 1)

V 8.0/( M 1)

(25)

Note that the tolerance on C, which corresponds to convergence to focus on nearby objects, is substantially larger than the tolerances on D and V.

AFOCAL SYSTEMS

18.15

18.5 GALILEAN AND INVERSE GALILEAN AFOCAL LENSES The combination of a positive objective and a negative eyepiece forms a galilean telescope. If the objective is negative and the eyepiece positive, it is referred to as an inverse galilean telescope. The galilean telescope has the advantage that it forms an erect image. It is the oldest form of visual telescope, but it has been largely replaced by terrestrial telescopes for magnified viewing of distant objects, because of field-of-view limitations. In terms of number of viewing devices manufactured, there are far more inverse galilean than galilean telescopes. Both are used frequently as power-changing attachments to change the effective focal length of focusing lenses.

Thin-Lens Model of a Galilean Afocal Lens Figure 12 shows a thin-lens model of a galilean afocal lens. The properties of galilean lenses can be derived from Eqs. (9), (12), and (13). Given that fe is negative and fo is positive, M is positive, indicating an erect image. If RO is placed at the front focal point of the objective, RE is a virtual pupil buried inside the lens system. In fact, galilean lenses cannot form real images of real objects under any conditions, and at least one pupil will always be virtual.

Field of View in Galilean Telescopes The fact that only one pupil can be real places strong limitations on the use of galilean telescopes as visual instruments when M 1x . Given the relationship $zo M 2 $ze, moving RE far enough outside the negative eyepiece to provide adequate eye relief moves RO far enough into virtual object space to cause drastic vignetting at even small field angles. Placing RE a distance ER behind the negative lens moves RO to the position shown in Fig. 13, SF` units behind RE, where (26)

SF ` ( M 2 1)ER ( M 1)2 f e

In effect, the objective is both field stop and limiting aperture, and vignetting defines the maximum usable field of view. The maximum acceptable object space angle upo is taken to be that for the principal ray which passes just inside Do, the entrance pupil at the objective. If the F-number of the objective is FNob fo /Do, then tan upo

fe 2FNob (MER f e Mf e )

(27)

FIGURE 12

Thin-lens model of galilean afocal lens.

FIGURE 13 Galilean field-of-view limitations.

18.16

COMPONENTS

For convenience, assume ER fe. In this case, Eq. (27) reduces to tan upo

1 2FNob (2 M 1)

(28)

For normal achromatic doublets, FNob q 4.0. For M 3x, in this case, Eq. (28) indicates that upo a 1.43° (FOV a 150 ft at 1000 yd). For M 7x, upo a 0.55° (FOV a 57.7 ft at 1000 yd). The effective field of view can be increased by making the objective faster and more complex, as can be seen in early patents by von Rohr14 and Erfle.15 In current practice, galilean telescopes for direct viewing are seldom made with M larger than 1.5x 3.0x. They are more typically used as power changers in viewing instruments, or to increase the effective focal length of camera lenses.16 Field of View in Inverse Galilean Telescopes For inverse galilean telescopes, where M 1x , adequate eye relief can be achieved without moving RO far inside the first surface of the objective. Inverse galilean telescopes for which upo l 90C are very common in the form of security viewers17 of the sort shown in Fig. 14, which are built into doors in hotel rooms, apartments, and many houses. These may be the most common of all optical systems more complex than eyeglasses. The negative objective lens is designed with enough distortion to allow viewing of all or most of the forward hemisphere, as shown by the principal ray in Fig. 14. Inverse galilean telescopes are often used in camera view finders.18 These present reduced scale images of the scene to be photographed, and often have built in arrangements to project a frame of lines representing the field of view into the image. Inverse galilean power changers are also used to increase the field of view of submarine periscopes and other complex viewing instruments, and to reduce the effective focal length of camera lenses.19 Anamorphic Afocal Attachments Afocal attachments can compress or expand the scale of an image in one axis. Such devices are called anamorphosers, or anamorphic afocal attachments. One class of anamorphoser is the cylindrical galilean telescope, shown schematically in Fig. 15a. Cox20 and Harris21 have patented representative examples. The keplerian form is seldom if ever used, since a cylindrical keplerian telescope would introduce image inversion in one direction. Anamorphic compression can also be obtained using two prisms, as shown in Fig. 15b. The adjustable magnification anamorphoser patented by Luboshez22 is a good example of prismatic anamorphosers. Many anamorphic attachments were developed in the 1950s for the movie industry for use in wide-field cameras and projectors. An extensive list of both types will be found in Wetherell.2

FIGURE 14 Inverse galilean security viewer with hemispheric field of view.

FIGURE 15 Anamorphic afocal attachments.

AFOCAL SYSTEMS

18.17

Equation (9) can be modified to describe anamorphic afocal lenses by specifying separate afocal magnifications Mx and My for the two axes. One important qualification is that separate equations are needed for object and image distances for the x and y planes. In general, anamorphic galilean attachments work best when used for distant objects, where any difference in x-axis and y-axis focus falls within the depth of focus of the associated camera lens. If it is necessary to use a galilean anamorphoser over a wide range of object distances, it may be necessary to add focus adjustment capabilities within the anamorphoser.

18.6

RELAY TRAINS AND PERISCOPES There are many applications where it is necessary to perform remote viewing because the object to be viewed is in an environment hostile to the viewer, or because the object is inaccessible to the viewer without unacceptable damage to its environment. Military applications5 fall into the former category, and medical applications23 fall into the latter. For these applications, instrumentation is needed to collect light from the object, transport the light to a location more favorable for viewing, and dispense the light to the viewing instruments or personnel. Collecting and dispensing optical images is done with focusing lenses, typically. There are three image transportation techniques in common use today: (1) sense the image with a camera and transport the data electronically, (2) transport the light pattern with a coherent fiber optics bundle, and (3) transport the light pattern with a relay lens or train of relay lenses. The first two techniques are outside the scope of this chapter. Relay trains, however, are commonly made up of a series of unit power afocal lenses, and are one of the most important applications of finite conjugate afocal lenses.

Unit Power Afocal Relay Trains Several factors are important in designing relay trains. First, it is desirable to minimize the number of relay stages in the relay train, both to maximize transmittance and to minimize the field curvature caused by the large number of positive lenses. Second, the outside diameter of the relay train is typically restricted (or a single relay lens could be used), so the choice of image and pupil diameter within the relay is important. Third, economic considerations make it desirable to use as many common elements as possible, while minimizing the total number of elements. Fourth, it is desirable to keep internal images well clear of optical surfaces where dust and scratches can obscure portions of the image. Fifth, the number of relay stages must be either odd or even to ensure the desired output image orientation. Figure 16 shows thin-lens models of the two basic afocal lens designs which can be applied to relay train designs. Central to both designs is the use of symmetry fore and aft of the central stop to control coma, distortion, and lateral color, and matching the image diameter Di and stop diameter Ds to

2a

Ds u

1

2b

Dt up

D1

FL (a) Minimum number of lenses FIGURE 16

Basic unit power afocal relay designs.

OB

OB

(b) Minimum tube diameter

FL

18.18

COMPONENTS

maximize the stage length to diameter ratio. In paraxial terms, if Di Ds, then the marginal ray angle u matches the principal ray angle up, in accordance with the optical invariant. If the relay lens is both aplanatic and distortion free, a better model of the optical invariant is Di sin u Ds tan u p

(29)

and either the field of view 2up or the numerical aperture NA n sin u must be adjusted to match pupil and image diameters. For some applications, maximizing the optical invariant which can pass through a given tube diameter Dt in a minimum number of stages is also critical. If maximizing the ratio Di sin u/Dt is not critical, Fig. 16a shows how the number of elements can be minimized by using a keplerian afocal lens with the stop at the common focus, eliminating the need for field lenses between stages. The required tube diameter in this example is at least twice the image diameter. If maximizing Di sin u/Dt is critical, field lenses FL must be added to the objectives OB as shown in Fig. 16b, and the field lenses should be located as close to the image as possible within limits set by obstructions due to dirt and scratches on the field lens surfaces. Symmetry fore and aft of the central stop at 1 is still necessary for aberration balancing. If possible within performance constraints, symmetry of OB and FL with respect to the planes 2a and 2b is economically desirable, making OB and FL identical. For medical instruments, where minimizing tube diameter is critical, variants of the second approach are common. The rod lens design24 developed by H. H. Hopkins25 can be considered an extreme example of either approach, making a single lens so thick that it combines the functions of OB and FL. Figure 17a shows an example from the first of two patents by McKinley.26,27 The central element in each symmetrical cemented triplet is a sphere. Using rod lenses does maximize the optical invariant which can be passed through a given tube diameter, but it does not eliminate field curvature. It also maximizes weight, since the relay train is almost solid glass, so it is most applicable to small medical instruments. If larger diameter relays are permissible, it is possible to correct field curvature in each relay stage, making it possible to increase the number of stages without adding field curvature. Baker28 has patented the lens design shown in Fig. 17b for such an application. In this case, field lens and objective are identical, so that an entire relay train can be built using only three different element forms. Pupil and image diameters are the same, and pupil and image are interchangeable. For purposes of comparison, the two designs shown in Fig. 17 have been scaled to have the same image diameter (2.8 mm) and numerical aperture (0.10), with component focal lengths chosen so that Ds Di. Minimum tube diameter is 4.0 mm for the rod lens and 5.6 mm for the Baker relay. The image radius of curvature is about 20 mm for the rod relay and about 368 mm for the Baker relay (i.e., field curvature is overcorrected). Image quality for the rod relay is 0.011 waves rms on axis and 0.116 waves rms at full field, both values for best focus, referenced to 587-nm wavelength. For the Baker relay, the corresponding values are 0.025 and 0.056 waves rms, respectively. The Baker design used for this comparison was adapted from the cited patent, with modern glasses substituted for types no longer available. No changes were made to the design other than refocusing it and scaling it to match the first-order parameters of the McKinley design. Neither design necessarily represents the best performance which can be obtained from its design type, and both should be evaluated

FIGURE 17

Improved unit power afocal relays.

AFOCAL SYSTEMS

18.19

FIGURE 18

Basic reflecting periscopes.

in the context of a complete system design where, for example, the field curvature of the McKinley design may be compensated for by that of the collecting and dispensing objectives. Comparing the individual relay designs does, however, show the price which must be paid for either maximizing the optical invariant within a given tube diameter or minimizing field curvature. Periscopes Periscopes are relay trains designed to displace the object space reference point RO a substantial distance away from the eye space reference point RE. This allows the observer to look over an intervening obstacle, or to view objects in a dangerous environment while the observer is in a safer environment. The submarine periscope is the archetypical example. Many other examples can be found in the military5 and patent2 literature. The simplest form of periscope is the pair of fold mirrors shown in Fig. 18a, used to allow the viewer to see over nearby obstacles. Figure 18b shows the next higher level of complexity, in the form of a rear-view vehicle periscope patented29 by Rudd.30 This consists of a pair of cylindrical mirrors in a roof arrangement. The cylinders image one axis of object space with the principal purpose of compensating for the image inversion caused by the roof mirror arrangement. This could be considered to be a keplerian anamorphoser, except that it is usually a unit power magnifier, producing no anamorphic compression. Beyond these examples, the complexity of periscopes varies widely. The optics of complex periscopes such as the submarine periscope can be broken down into a series of component relays. The core of a submarine periscope is a pair of fold prisms arranged like the mirrors in Fig. 18a. The upper prism can be rotated to scan in elevation, while the entire periscope is rotated to scan in azimuth, typically. The main optics is composed of keplerian afocal relays of different magnification, designed to transfer an erect image to the observer inside the submarine, usually at unit net magnification. Galilean and inverse galilean power changers can be inserted between the upper prism and main relay optics to change the field of view. Variants of this arrangement will be found in other military periscopes, along with accessories such as reticles or image intensifiers located at internal foci. Optical design procedures follow those for other keplerian afocal lenses.

18.7 REFLECTING AND CATADIOPTRIC AFOCAL LENSES Afocal lenses can be designed with powered mirrors or combinations of mirrors and refractors. Several such designs have been developed in recent years for use in the photolithography of microcircuits. All-reflecting afocal lenses are classified here according to the number of powered mirrors they contain. They will be reviewed in order of increasing complexity, followed by a discussion of catadioptric afocal systems.

18.20

COMPONENTS

FIGURE 19

Reflecting Galilean.

Two-Powered-Mirror Afocal Lenses The simplest reflecting afocal lenses are the variants of the galilean and keplerian telescopes shown in Fig. 19a and 19b. They may also be thought of as afocal cassegrainian and gregorian telescopes. The galilean/cassegrainian version is often called a Mersenne telescope. In fact, both galilean and keplerian versions were proposed by Mersenne in 1636,31 so his name should not be associated solely with the galilean variant. Making both mirrors parabolic corrects all third-order aberrations except field curvature. This property of confocal parabolas has led to their periodic rediscovery,32,33 and to subsequent discussions of their merits and shortcomings.34–36 The problem with both designs, in the forms shown in Fig. 19a and 19b, is that their eyepieces are buried so deeply inside the design that their usable field of view is negligible. The galilean form is used as a laser beam expander,37 where field of view and pupil location is not a factor, and where elimination of internal foci may be vital. Eccentric pupil versions of the keplerian form of confocal parabolas, as shown in Fig. 19c, have proven useful as lens attachments.38 RO, RE, and the internal image are all accessible when RO is set one focal length ahead of the primary, as shown. It is then possible to place a field stop at the image and pupil stops at RO and RE, which very effectively blocks stray light from entering the following optics. Being all-reflecting, confocal parabolas can be used at any wavelength, and such attachments have seen use in infrared designs. Three-Powered-Mirror Afocal Lenses The principle which results in third-order aberration correction for confocal parabolas also applies when one of the parabolas is replaced by a classical cassegrainian telescope (parabolic primary and hyperbolic secondary), as shown in Fig. 20, with two important added advantages. First, with one negative and two positive mirrors, it is possible to reduce the Petzval sum to zero, or to leave a small

FIGURE 20

Three-powered-mirror afocal lenses.

AFOCAL SYSTEMS

18.21

FIGURE 21

Four-powered-mirror afocal lenses.

residual of field curvature to balance higher-order astigmatism. Second, because the cassegrainian is a telephoto lens with a remote front focal point, placing the stop at the cassegrainian primary puts the exit pupil in a more accessible location. This design configuration has been patented by Offner,39 and is more usefully set up as an eccentric pupil design, eliminating the central obstruction and increasing exit pupil accessibility. Four-Powered-Mirror Afocal Lenses The confocal parabola principle can be extrapolated one step further by replacing both parabolas with classical cassegrainian telescopes, as shown in Fig. 21a. Each cassegrainian is corrected for field curvature independently, and the image quality of such confocal cassegrainians can be quite good. The most useful versions are eccentric pupil. Figure 21b shows an example from Wetherell.40 Since both objective and eyepiece are telephoto designs, the separation between entrance pupil RO and exit pupil RE can be quite large. An afocal relay train made up of eccentric pupil confocal cassegrainians will have very long collimated paths. If the vertex curvatures of the primary and secondary mirrors within each cassegrainian are matched, the relay will have zero field curvature, as well. In general, such designs work best at or near unit magnification. Unit Power Finite Conjugate Afocal Lenses The simplest catadioptric afocal lens is the cat’s-eye retroreflector shown in Fig. 22a, made up of a lens with a mirror in its focal plane. Any ray entering the lens will exit parallel to the incoming ray but traveling in the opposite direction. If made with glass of index of refraction n 2.00, a sphere with one hemisphere reflectorized (Fig. 22b) will act as a perfect retroreflector for collimated light entering the transparent hemisphere. Both designs are, in effect, unit power (M 1.00) afocal lenses. Variations on this technique are used for many retroreflective devices. Unit power relays are of much interest in photolithography, particularly for microcircuit manufacturing, which requires very high resolution, low focal ratio unit power lenses. In the Dyson lens,41 shown in Fig. 23a, the powered surfaces of the refractor and the reflector are concentric, with radii R and r given by R n (30) r (n 1)

18.22

COMPONENTS

FIGURE 22

Afocal retroreflector designs.

where n is the index of refraction of the glass. At the center point, spherical aberration and coma are completely corrected. In the nominal design, object and image are on the surface intersecting the center of curvature, displaced laterally to separate object from image sensor (this arrangement is termed eccentric field, and is common to many multimirror lens systems). In practice, performance of the system is limited by off-axis aberrations, and it is desirable to depart from the nominal design to balance aberrations across the field of view.42 The unit power all-reflecting concentric design shown in Fig. 23b is patented43 by Offner.44 It was developed for use in manufacturing microcircuits, and is one of the most successful finite conjugate afocal lens designs in recent years. The spheres are concentric and the plane containing object and image surfaces passes through the common center of curvature. It is an all-reflecting, unit power equivalent of the refracting design shown in Fig. 9. Object and image points are eccentric field, and this is an example of the ring field design concept, where axial symmetry ensures good correction throughout a narrow annular region centered on the optical axis. As with the Dyson lens, having an eccentric field means performance is limited by off-axis aberrations. Correction of the design can be improved at the off-axis point by departing from the ideal design to balance on-axis and off-axis aberrations.45

FIGURE 23 Concentric spheres unit power afocal lenses.

AFOCAL SYSTEMS

18.8

18.23

REFERENCES 1. A. van Helden, “The Invention of the Telescope,” Trans. Am. Philos. Soc. 67, part 4, 1977. 2. W. B. Wetherell, In “Afocal Lenses,” R. R. Shannon and J. C. Wyant (eds.), Applied Optics and Optical Engineering, vol. X, Academic Press, New York, 1987, pp. 109–192. 3. E. Hecht and A. Zajac, Optics, Addison-Wesley, Reading, Mass., 1974, p. 152. 4. H. C. King, The History of the Telescope, Dover, New York, 1979, pp. 44–45. 5. F. B. Patrick, “Military Optical Instruments,” In R. Kingslake (ed.), Applied Optics and Optical Engineering, vol. V, Academic Press, New York, 1969, pp. 183–230. 6. D. H. Jacobs, Fundamentals of Optical Engineering, McGraw-Hill, New York, 1943. 7. Defense Supply Agency, Military Standardization Handbook: Optical Design, MIL-HDBK-141, Defense Supply Agency, Washington, D.C., 1962, section 14, p. 18. 8. A. König, Telescope, U.S. Patent 1,952,795, March 27, 1934. 9. D. B. Rhodes, Scanning Afocal Laser Velocimeter Projection Lens System, U.S. Patent 4,346,990, August 31, 1982. 10. J. C. A. Chastang and R. F. Koerner, Optical System for Oblique Viewing, U.S. Patent 4,428,676, January 31, 1984. 11. A. R. Shulman, Optical Data Processing, Wiley, New York, 1970, p. 325. 12. G. F. Marshall, (ed.), Laser Beam Scanners, Marcel Dekker, Inc., New York, 1970. 13. J. M. Lloyd, Thermal Imaging Systems, Plenum, New York, 1975. 14. M. von Ruhr, Galilean Telescope System, U.S. Patent 962,920, June 28, 1910. 15. H. Erfle, Lens System for Galilean Telescope, U.S. Patent 1,507,111, September 2, 1924. 16. H. Köhler, R. Richter, and H. Kaselitz, Afocal Lens System Attachment for Photographic Objectives, U.S. Patent 2,803,167, August 20, 1957. 17. J. C. J. Blosse, Optical Viewer, U.S. Patent 2,538,077, January 16, 1951. 18. D. L. Wood, View Finder for Cameras, U.S. Patent 2,234,716, March 11, 1941. 19. H. F. Bennett, Lens Attachment, U.S. Patent 2,324,057, July 13, 1943. 20. A. Cox, Anamorphosing Optical System, U.S. Patent 2,720,813, October 18, 1955. 21. T. J. Harris, W. J. Johnson, and I. C. Sandbeck, Wide Angle Lens Attachment, U.S. Patent 2,956,475, October 18, 1960. 22. B. J. Luboshez, Prism Magnification System Having Correction Means for Unilateral Color, U.S. Patent 2,780,141, February 5, 1957. 23. J. H. Hett, “Military Optical Instruments,” In R. Kingslake, (ed.), Applied Optics and Optical Engineering, vol. V, Academic Press, New York, 1969, pp. 251–280. 24. S. J. Dobson and H. H. Hopkins, “A New Rod Lens Relay System Offering Improved Image Quality,” J. Phys. E: Sci. Instrum. 22, 1989, p. 481. 25. H. H. Harris, Optical System Having Cylindrical Rod-like Lenses, U.S. Patent 3,257,902, June 28, 1966. 26. H. R. McKinley, Endoscope Relay Lens, U.S. Patent 5,069,009, October 22, 1991. 27. H. R. McKinley, Endoscope Relay Lens Configuration, U.S. Patent 5,097,359, March 17, 1992. 28. J. G. Baker, Sighting Instrument Optical System, U.S. Patent 2,899,862, August 18, 1959. 29. M. O. Rudd, Two-Mirror System for Periscopic Rearward Viewing, U.S. Patent 4,033,678, July 5, 1977. 30. M. O. Rudd, “Rearview Periscope Comprised of Two Concave Cylindrical Mirrors,” Appl. Opt. 17, 1978, pp. 1687–1696. 31. H. C. King, The History of the Telescope, Dover, New York, 1979, pp. 48–49. 32. S. C. B. Gascoigne, “Recent Advances in Astronomical Optics,” Appl. Opt. 12, 1973, p. 1419. 33. S. Rosin, M. Amon, and P. Laakman, “Afocal Parabolic Reflectors,” Appl. Opt. 13, 1974, p. 741. 34. R. B. Annable, “Still More on Afocal Parabolic Reflectors,” Appl. Opt. 13, 1974, p. 2191. 35. S. Rosin, “Still More on Afocal Parabolic Reflectors,” Appl. Opt. 13, 1974, p. 2192. 36. W. B. Wetherell and M. P. Rimmer, “Confocal Parabolas: Some Comments,” Appl. Opt. 13, 1974, p. 2192.

18.24

COMPONENTS

37. 38. 39. 40. 41. 42. 43. 44. 45.

W. K. Pratt, Laser Communication Systems, Wiley, New York, 1969, p. 53. I. R. Abel, Wide Field Reflective Optical Apparatus, U.S. Patent 3,811,749, May 21, 1974. A. Offner, Catoptric Anastigmatic Afocal Optical System, U.S. Patent 3,674,334, July 4, 1972. W. B. Wetherell, “All-Reflecting Afocal Telescopes,” In “Reflective Optics,” SPIE, vol. 751, 1987, pp. 126–134. J. Dyson, “Unit Magnification Optical System Without Seidel Aberrations,” J. Op. Soc. Am. 49, 1959, pp. 713–716. R. M. H. New, G. Owen, and R. F. W. Pease, “Analytical Optimization of Dyson Optics,” Appl. Opt. 31, 1992, pp. 1444–1449. A. Offner, Unit Power Imaging Cataptric Anastigmat, U.S. Patent 3,748,015, July 24, 1973. A. Offner, “New Concepts in Projection Optics,” Opt. Eng. 14, 1975, p. 130. C. S. Ih and K. Yen, “Imaging and Fourier Transform Properties of an Allspherical Mirror System,” Appl. Opt. 19, 1980, p. 4196.

19

NONDISPERSIVE PRISMS1,2 William L. Wolfe College of Optical Sciences University of Arizona Tucson, Arizona

19.1

GLOSSARY F H Y A, B, C, D, d t x, y C 1, 2

19.2

angular deviation phase radian frequency of rotation prism dimensions time rectangular components angle prism number

INTRODUCTION Prisms of various shapes and sizes are used for folding, inverting, reverting, displacing, and deviating a beam of light, whether it be collimated, converging, or diverging. Prisms, rather than mirrors, are often used for the applications discussed here, since they make use of reflecting coatings at what amounts to an interior surface. The coatings can be protected on their backs by other means, and do not tarnish with age and exposure. Even better, some prisms do not need such coatings if the (internal) angle of incidence exceeds the critical angle. In these applications, chromatism is to be avoided. Thus, the arrangements either make use of perpendicular incidence or compensating angles of incidence. Almost all of these prisms are meant to be used with collimated beams. Most of the operations are somewhat equivalent to the use of plane parallel plates, which displace but do not deviate a 19.1

19.2

COMPONENTS

collimated beam. However, such plates have both chromatism and spherical aberration in a convergent beam. Dispersing prisms are discussed in Chap. 20 by George J. Zissis and polarizing prisms in Chap. 13 by Jean M. Bennett, both in this volume of the Handbook.

19.3

INVERSION, REVERSION A reverted image shifts the image left for right. An inverted image is upside down. A reinverted image or inverted-reverted image does both. The best two references on this subject are the Frankford Arsenal book called Design of Fire Control Optics,1 and Jacobs’ book called Optical Engineering.2 Many of the diagrams shown here have been taken from the former since they provide direct design information as well as descriptions of the performance of the prism.

19.4

DEVIATION, DISPLACEMENT The beam, in addition to being inverted and/or reverted, can also be displaced and/or deviated. Displacement means that the beam has been translated in x or y, but it has not changed the direction in which it was traveling. Deviation indicates that the beam has been caused to change its direction. Deviation is measured in angular measure; displacement in linear measure. If a beam has been deviated, displacement is not important. If a beam has been displaced, it usually has not been deviated. Although the two can occur simultaneously, it seldom happens in optical prisms.

19.5

SUMMARY OF PRISM PROPERTIES Table 1 is a listing of the prisms that are described in this section. The first column provides the name of the prism. The second column indicates whether the image is reverted. The third column indicates whether the image has been inverted. The fourth column indicates how much the image has been displaced as a ratio of the critical prism dimension A. The next column gives the amount of deviation, and the last column provides comments.

19.6

PRISM DESCRIPTIONS Each diagram shows at least one view of the prism and a set of dimensions. The A dimension is a reference dimension. It is always 1.00, and the rest of the dimensions are related to it. The refractive index is almost always taken as 1.5170, a representative value for glass in the visible. Prism dimensions can change somewhat if the refractive index is modestly different from the chosen value. However, if a material like germanium is used, the prism might be drastically different.

NONDISPERSIVE PRISMS

TABLE 1 Prism Right angle Porro A Porro B Abbe A Abbe B Dove, double dove Pechan Amici Schmidt Leman Penta Reversion Wollaston Zeiss Goerz Frank 1 Frank 2 Frank 3 Frank 4 Frank 5 Frank 6 Frank 7 Hastings Rhomboid Risleys Retro D40 D60 D90 D120

19.3

Summary of Prism Properties Reverts

Inverts

Deviates

Comments

Yes No Yes Yes Yes Yes Yes

No Yes Yes Yes Yes Yes No

Displaces —

90

Simplest

1.1A 1.1A 0 0 0

0 0 0 0 0

Binocs Binocs

Yes Yes Yes Yes No Yes No Yes Yes Yes Yes Yes Yes No Yes No Yes No No Yes No No No No

No No Yes Yes No Yes No Yes Yes Yes Yes Yes No Yes Yes No Yes No No Yes No No No No

0 — — 3A — 0 — Design Design — — — — — — — 0 A No No — — — —

0 90 45 0 90 0 90

Parallel light Nonparallel Roof

exactly Nonparallel tracing

115 60 45v 90h 45v 90h 60v 90h 60v 90h 45v 90h 0 0 0–180 180 40–50 50–60 80–100 110–130

Right-Angle Prism Perhaps the simplest of the deviating prisms is the right-angle prism. Light enters one of the two perpendicular faces, as shown in Fig. 1a, reflects off the diagonal face, and emerges at 90° from the other perpendicular face. The beam has been rotated by 90°, and the image has been inverted. If the prism is used in the other orientation, shown in Fig. 1b, then the image is reverted. The internal angle of incidence is 45°, which is sufficient for total internal reflection (as long as the refractive index is greater than 1.42).

Porro Prism A Porro prism, shown in Fig. 2, has a double reflection and may be considered to be two rightangle prisms together. They are often identical. Often, two Porro prisms are used together to invert and revert the image. The incidence angles are the same as with the right-angle prism, so that total internal reflection takes place with refractive indices larger than 1.42. It is a direct-vision prism.

19.4

COMPONENTS

(a)

(b)

A

A

A B

A = 1.00 B = 1.4142A = 1.4142

= 45

d = A = 1.00 d/n = 0.6592 n = 1.5170

FIGURE 1

Right-angle prism.


Δ

R a

B

B

C

D

a

a

L A ° B A Δ Aa C Aa DAa L A a RA FIGURE 2

Porro prism.

a d A a dn n

19.5

19.6

COMPONENTS

Abbe’s version of the Porro prism is shown in Fig. 3. The resultant beam is inverted and reverted and is directed parallel and in the same direction as incident beam.

FIGURE 3

Abbe modification of Porro prisms for binoculars.


19.7

Abbe’s Prisms Two versions of prisms invented by Abbe are shown. They both are direct-vision prisms that revert and invert the image. One version is symmetrical; the other uses three different prism segments. They are shown in Figs. 4 and 5.

L

g

C A

B

a

b

d

A

a A

b

A

b

3A

2A

b

A

A

A

A = 1.00 B = 1.4142A = 1.4142

a = 30 b = 60

a = 0.7071A = 0.7071 b = 0.5774A = 0.5774

C = 1.3094A = 1.3094 L = 3.4644A = 3.4644

g = 90 d = 45

d = 5.1962A = 5.1962 d/n = 3.4253 n = 1.5170

FIGURE 4 Abbe direct-vision prism—A.

19.8

COMPONENTS

2A

L

A

A

B b w A

w a

a

a

b b

b

A

b

A 2A

A

b

A

A A = 1.00 B = 1.1547A = 1.1547 L = 3.4641A = 3.4641

FIGURE 5

a = 135 b = 60 g = 45 w = 30

Abbe direct-vision prism—B.

A a = 0.7071A = 0.7071 b = 0.5773A = 0.5773 d = 5.1962A = 5.1962 d/n = 3.4253 n = 1.5170


19.9

Dove Prism A Dove prism (also known as a Harting-Dove prism) does not deviate or displace an image but it can be used to either invert or revert an image. It must be placed in parallel light. Such a prism is shown in Fig. 6.

A

b a

D E

A

a

a

C B

A = 1.00 B = (A + 2a)

n2

–

sin2a

+ sin a

n2 – sin2a – sin a

= 4.2271 (A + 2a) = 4.6498 C = B – 2a = 4.5498

+1

a

a = 45

a = 0.05

b = 90

d=

n (A + 2a) sin a

FIGURE 6 Harting-Dove prism.

– sin2a – sin a

= 3.3787 (A + 2a) = 3.7165 d/n = 2.4499 n = 1.5170

D = B – 2(A + 2a) = 2.4498 a+A E= = 1.4142 (A + a) = 1.4849 cos a

n2

19.10

COMPONENTS

Double Dove Two Dove prisms are glued together. The length is halved, but the height is doubled. It performs the same functions as a single Dove in almost the same way. It is shown in Fig. 7.

D

D

a a

a a D

D

A/2 A/2

C B

A

a = 45

A = 1.00 B= A 2

n2 – sin2a + sin a n2 – sin2a – sin a

+1

= 2.1136A = 2.1136

d=

nA 2 sin a

= nAC = 1.6893 d/n = 1.1135 n = 1.5170

C = B – A = 1.1136 D=

A = 0.7071A = 0.7071 2 cos a

FIGURE 7

Double Dove prism.

n2 – sin2a – sin a


19.11

Pechan Prism The Pechan prism (shown in Fig. 8) performs the same function as the Dove, but it can do it in converging or diverging beams. The surfaces marked B are silvered and protected. The surfaces bordering the air space are unsilvered.

A

A

b

d B

C

B

a

a b

a

E D

a

A B A C A D A E A FIGURE 8

a b g d

a A d A dn n

Pechan prism.

Amici (Roof) Prism This more complex arrangement of surfaces inverts the image, reverts it, and deviates it 90°. It is shown in Fig. 9. Since this prism makes use of the roof effect, it has the same angles as both the right-angle and Porro prisms, and exhibits total internal reflection for refractive indices larger than 1.42.

19.12

COMPONENTS

a

A

a

A a

A B a A

A

a A

B A

d A dn n

FIGURE 9 Amici (roof) prism.

Schmidt Prism The prism will invert and revert the image, and it will deviate it through 45°. It is shown in Fig. 10. a A/2 A/2

a

B

B A b D

b

C b b

C

g

A = 1.00 B = 1.4142A + 0.5412a = 1.4683 C = 1.0824A = 1.082 D = 1.4142A + 2.3890a = 1.6531

FIGURE 10

Schmidt prism.

a = 45 b = 6730 g = 90

a = 0.10 (chosen at will) b = 1.8478a = 0.1848 c = 0.7071A = 0.7071 d = 3.4142A = 3.4142 d/n = 2.2506 n = 1.5170


19.13

Leman Prism This rather strange looking device, shown in Fig. 11, reverts, inverts, and displaces by 3A, an image.

FIGURE 11

Leman prism.

Penta Prism A penta prism has the remarkable property that it always deviates a beam by exactly 90° in the principal plane. This is akin to the operation of a cube corner. The two reflecting surfaces of the penta prism, shown in Fig. 12, must be reflectorized, as the angles are 22.5° and therefore require a refractive index of 2.62 or greater for total internal reflection. Some penta-prisms are equipped with a roof to revert the image.

19.14

COMPONENTS

C

B A

C

A

A

A B A C A FIGURE 12

a A dn

n

Penta prism.

Reversion Prism This prism operates like an Abbe prism, type A, but does not require parallel light. It is shown in Fig. 13.

Reversion prism.

FIGURE 13


19.15

Wollaston Prism This prism does not invert, revert, or displace. It does deviate a beam by 90°, allowing a tracing to be made. It is shown in Fig. 14.

A

R

R B

A B

P C

A

dR A

B A

dn

CA

n

RA FIGURE 14

Wollaston prism.

19.16

COMPONENTS

Carl Zeiss Prism System This arrangement of three prisms allows the image to be reverted, inverted, and displaced, but not deviated. The amount of displacement is adjustable. The system is shown in Fig. 15.

FIGURE 15

Carl Zeiss prism system.


Goerz Prism System This is an alternate to the Zeiss system. It does the same things. It is shown in Fig. 16.

FIGURE 16

C. P. Goerz prism system.

19.17

19.18

COMPONENTS

Frankford Arsenal 1 This prism, shown in Fig. 17, reverts, inverts, and deviates through 115°.

D

b

A

A A

B

A

C

A B A C A D A FIGURE 17

Frankford Arsenal prism 1.

a

aA bA d

A

dn n


Frankford Arsenal 2 This prism reverts, inverts, and deviates through 60°. It is shown in Fig. 18.

B

C

A

a b

A

A

A A C C

b

A B A CA

FIGURE 18

a


a A b A d

A dn n

19.19

19.20

COMPONENTS

Frankford Arsenal 3 This prism reverts, inverts, and deviates through an angle of 45° upward and 90º horizontally. It is shown in Fig. 19.

F

B E A D

A

G

C A

dn

B A

d A

C A

n

D A E A F A G A FIGURE 19



19.21

Frankford Arsenal 4 This prism reverts the image and deviates it 45° upward and 90° sidewards, like Frankford Arsenal 3. It is shown in Fig. 20.

L

A

B

A

E

A

A

A

C F A

R

D

Silver

A = 1.00 B = 1.4142A = 1.4142 C = 2.4142A = 2.4142 D = 1.0824A = 1.0824 E = 1.7071A = 1.7071 F = 2.4142A = 2.4142 L = 2.7071A = 2.7071 R = A = 1.00 FIGURE 20

A

= 2230 = 45 = 90 = 11230


d = 4.4142A = 4.4142 d/n = 2.9098 n = 1.5170

19.22

COMPONENTS

Frankford Arsenal 5 This prism inverts the image while deviating it 90° sideways and 60° upward. It is shown in Fig. 21.

F A B E A D

A

G A

C

A A

dA

BA

dn

CA

n

D A EA FA G A FIGURE 21



19.23

Frankford Arsenal 6 This prism inverts, reverts, and deviates 90° horizontally and 60° vertically. It is shown in Fig. 22.

G

A F

A

B A A

A D C a

a

E A BA CA

DA E A FA G A H A FIGURE 22


aA d

A

dn n

19.24

COMPONENTS

Frankford Arsenal 7 This prism neither reverts nor inverts, but deviates 90° horizontally and 45° vertically. It is shown in Fig. 23.

L

A B

E

A

A

C

R

A D

A B A C A D A E A L A R A FIGURE 23

d A dn n



19.25

Brashear-Hastings Prism This device, shown in Fig. 24, inverts an image without changing the direction of the beam. Since this is a relatively complicated optical element, it does not see much use.

FIGURE 24

Brashear-Hastings prism.

Rhomboidal Prism A rhomboidal prism, as shown in Fig. 25, displaces the beam without inverting, reverting, deviating, or otherwise changing things. The reflecting analog is a pair of mirrors at 45°.

FIGURE 25

Rhomboidal prism.

Risley Prisms Risley prisms are used in two ways. If they are slightly absorbing, they can be used as variable attenuators by translating one with respect to the other perpendicular to their apexes.3 They can also be rotated to generate a variety of angular deviations.4 A single prism deviates the beam according to its wedge angle and refractive index. If rotated in a circle about an axis perpendicular to its face, it will rotate the beam in a similar circle. A second, identical prism in series with it, as shown in Fig. 26, can double the angle of the beam rotation and generate a circle of twice the radius. If they rotate in opposite directions, one motion is canceled and a line is generated. In fact, all sorts of Lissajous-type figures can be obtained; some are shown in Fig. 27. The equations that govern the patterns are

19.26

COMPONENTS

FIGURE 26

Risley prisms.

FIGURE 27

Risley prism patterns.


19.27

D x D1 cos W1t D 2 cos(W 2t F)

(1)

D y D1 sin W1t D 2 sin(W 2t F)

(2)

where Fx and Fy are the beam deviations, F1 and F2 are the individual prism deviations, Y is the rotation rate, t is time, and H is the phase of the prism position. For relatively monochromatic applications, the prisms can be “fresnelled,” as shown in Fig. 28, and the mirror analogs, shown in Fig. 29, can also be used.

FIGURE 28 Fresnel Risleys.

FIGURE 29

Risely mirrors.

19.28

COMPONENTS

Retroreflectors The familiar reflective cube corner (not corner cube), that sends a ray back in the direction from which it came, has its refractive analog, as shown in Fig. 30. The angles are adjusted so that total internal reflection occurs. The angular acceptance range can be large.

FIGURE 30

Retroreflectors.

General Deviation Prisms Figure 31 shows a 40°-deviation prism. Other angles are obtainable with appropriate changes in the prism manufacture, as shown, for example, in Figs. 32 and 33.

FIGURE 31

40°-deviation prism—D40.


FIGURE 32


FIGURE 33

19.7



F. Arsenal, Design of Fire Control Optics, U.S. Government Printing Office, 1952. D. H. Jacobs, Fundamentals of Optical Engineering, McGraw-Hill, 1943. F. A. Jenkins and H. E. White, Fundamental Optics, 3d ed., McGraw-Hill, 1957. W. L. Wolfe and G. J. Zissis, The Infrared Handbook, U.S. Government Printing Office, 1978.

19.29


20 DISPERSIVE PRISMS AND GRATINGS George J. Zissis Environmental Research Institute of Michigan Ann Arbor, Michigan

20.1

GLOSSARY

20.2

Ap B Dp d E N n p RP r W D I

prism angle prism base angle of minimum deviation grating constant irradiance number of slits refractive index order number resolving power angles prism width angle angle

INTRODUCTION Spectroradiometers (Fig. 1) are radiometers designed specifically to allow determination of the wavelength distribution of radiation. This category of measurement systems usually consists of those in which separation of the radiation into its spectral components, or dispersion, is accomplished by the use of an optical element possessing a known functional dependence on wavelength—specifically, prisms and diffraction gratings. (Interferometers can also provide spectral dispersion as is discussed in Chap. 32, “Interferometers,” by Parameswaran Hariharan.)

20.1

20.2

COMPONENTS

1

1

2 2

Amplifier

Monochromator 1 = collecting optics 1 = entrance slit of monochromator acting also as the field stop 2 = collimating optics = dispersing element (prism or grating) = refocusing optics 2 = exit slit of monochromator = detector element

FIGURE 1

20.3

Display or recorder

Basic spectroradiometer.

PRISMS1,2,3 The wavelength dependence of the index of refraction is used in prism spectrometers. Such an optical element disperses parallel rays or collimated radiation into different angles from the prism according to wavelength. Distortion of the image of the entrance slit is minimized by the use of plane wave illumination. Even with plane wave illumination, the image of the slit is curved because not all of the rays from the entrance slit can traverse the prism in its principal plane. A prism is shown in the position of minimum angular deviation of the incoming rays in Fig. 2. At minimum angular deviation, maximum power can pass through the prism. For a prism adjusted to the position of minimum deviation, r1 r2 Ap /2

(1)

i1 i2 (D p Ap )/2

(2)

and

where

Dp = angle of minimum deviation for the prism Ap = angle of the prism r1 and r2 = internal angles of refraction i1 and i2 = angles of entry and exit

The angle of minimum deviation Dp varies with wavelength. The angular dispersion is defined as dDp/dN, while the linear dispersion is dx /dL F dD p /dL

(3)

where F is the focal length of the camera or imaging lens and x is the distance across the image plane. It can be shown1 that dD p /dL (B /W )(dn/dL )

(4)

DISPERSIVE PRISMS AND GRATINGS

20.3

Top view Ap

i1

Dp i2

r1

r2

Sp Top view B W Slit or line source

Image

Side view

Side view

FIGURE 2 Elementary prism spectrometer schematic. W is the width of the entrance beam; Sp is the length of the prism face; and B is the prism base length.

where

B = base length of the prism W = width of the illumination beam n = index of refraction dx/dN = F(B/W)(dn/dN)

The resolving power RP of an instrument may be defined as the smallest resolvable wavelength difference, according to the Rayleigh criterion, divided by the average wavelength in that spectral region. The limiting resolution is set by diffraction due to the finite beam width, or effective aperture of the prism, which is rectangular. Thus, RPp B(dn/dL )

(5)

If the entire prism face is not illuminated, then only the illuminated base length must be used for B.

20.4

GRATINGS A grating is an n-slit system used in Fraunhofer diffraction with interference arising from division of the incident plane wave front. Thus it is a multiple beam interferometer described by pL d(sin Q sin F )

(6)

20.4

COMPONENTS

where p = order number (= 0, 1, 2,...) of the principal maxima d = the grating constant or spacing (the distance between adjacent slits) H = angle of incidence S = angle of diffraction The most common case is that of normal incidence, that is, F 0, so that pL d sin Q

(7)

and the irradiance distribution is E Eo {sin[(P w sin Q )/L]/[(P w sin Q )/L ]}2

(8)

r {sin[(NP d sin Q )/L]/ sin[(P d sin Q )/L ]}2 where w is the slit width and N is the number of slits or grooves. This equation is often written as E E0[(sin B )/B]2[(sin NG )/ sin G =2

(9)

which can be considered to be E constant r single-slit diffraction function r N-slit interference function

(10)

These considerations are for unblazed gratings. For a diffraction grating, the angular dispersion is given (for constant angle H) by dDg /dL

or

dQ /dL p/(d cos Q )

(11)

The resolving power is given by RPg pN

20.5

(12)

PRISM AND GRATING CONFIGURATIONS AND INSTRUMENTS

Classical There are several basic prism and grating configurations and spectrometer designs which continue to be useful. One of the oldest spectrometer configurations is shown in Fig. 3.1 Reflective interactions and prism combinations are used in Figs. 4, 5, and 6. Dispersion without deviation is realized in Figs. 7 and 8, while half-prisms are used in Fig. 9 in an arrangement which uses smaller prisms but still attains the same beam width. A few classical prism instrumental configurations are shown in Figs. 10, 11, and 12. Multiple-pass prism configurations are illustrated in Figs. 13 and 14.4,5 A well-known example of a single beam double-pass prism infrared spectrometer was the Perkin-Elmer Model 112 instrument shown in Fig. 15. Infrared radiation from a source is focused by mirrors M1 and M2 on the entrance slit S1 of the monochromator. The radiation beam from S1, path 1, is collimated by the off-axis paraboloid M3 and a parallel beam traverses the prism for a first refraction. The beam is reflected by the Littrow mirror M4, through the prism for a second refraction, and focused by the paraboloid, path 2, at the corner mirror M6. The radiation returns along path 3, traverses the prism again, and is returned along path 4 for reflection by mirror M7 to the exit slit S2. By this double dispersion, the radiation is spread out along the plane of S2. The radiation of the frequency interval which passes through S2 is focused by mirrors M8 and


20.5

Illuminated scale 0

5

10

20

P

Collimator lens

Telescope

Slit E FIGURE 3 the telescope.

Bunsen-Kirchhoff spectrometer. An illuminated scale is reflected from the prism face into

M9 on the thermocouple TC. The beam is chopped by CH, near M6, to produce a voltage (at the thermocouple) which is proportional to the radiant power or intensity of the beam. This voltage is amplified and recorded by an electronic potentiometer. Motor-driven rotation of Littrow mirror M4 causes the infrared spectrum to pass across exit slit S2 permitting measurement of the radiant intensity of successive frequencies. Gratings can be used either in transmission or reflection.6 Another interesting variation comes from their use in plane or concave reflection form. The last was treated most completely by Rowland, who achieved a useful combination of focusing and grating action. He showed that the radius of curvature of the grating surface is the diameter of a circle (called the Rowland circle). Any source placed on the

FIGURE 4 Wadsworth constant-deviation, prism-mirror arrangement. The beam enters the prism at minimum deviation and emerges displaced but not deviated from its original direction.

20.6

COMPONENTS

FIGURE 5 Amici prism. The central ray D enters and leaves parallel to the base. The C and F rays are deviated and dispersed. C O A D

E

B

FIGURE 6 Pellin–Broca prism. The prism is equivalent to two 30° prisms, ABC and BED, and one 45° prism, DEC, but is made in one place. The beam shown, entering at minimum deviation, emerges at 90° deviation to its entrance direction. C D

F

FIGURE 7 Zenger prism. The central ray D is undeviated. The C and F rays are deviated and dispersed. C

A

B

A

D

F

FIGURE 8 Wernicke prism. This arrangement is essentially two Zenger prisms, back-to-back.


20.7

FIGURE 9 Young–Thollon half prisms. The passage of a beam at minimum deviation is shown.

circle will be imaged on the circle, with dispersion, if the rulings are made so that d is constant on the secant to the grating-blank (spherical) surface. The astigmatism acts so that a point source on a Rowland circle is imaged as a vertical line perpendicular to the plane of the circle. Rowland invented and constructed the first concave grating mounting, illustrated in Fig. 16.1 If dispersion is sufficiently large, one may find overlapping of the lines from one order with members of the spectra belonging to a neighboring order. Errors and imperfections in the ruling of gratings can produce spurious images which are called “ghosts.” Also, the grooves in a grating can be shaped so as to send more radiation along a preferred direction corresponding to an order other than the zero order. Such gratings are said to be blazed in that order. These issues and many more involved in the production of gratings by ruling engines were thoroughly discussed by Harrison in his 1973 paper “The Diffraction Grating—An Opinionated Appraisal.”7 Six more grating configurations1 which are considered to be “classics” are 1. Paschen-Runge, illustrated in Fig. 17. In this arrangement, one or more fixed slits are placed to give an angle of incidence suitable for the uses of the instrument. The spectra are focused along the Rowland circle P P`, and photographic plates, or other detectors, are placed along a large portion of this circle. 2. Eagle, shown in Fig. 18. This is similar to the Littrow prism spectrograph. The slit and plate holder are mounted close together on one end of a rigid bar with the concave grating mounted on the other end.

Exit slit

Vacuum thermocouple

Entrance slit

Mirror

Littrow mirror Prism

FIGURE 10

Infrared spectrograph of the Littrow-type mount with a rock salt prism.

20.8

COMPONENTS

FIGURE 11 Mirror spectrometer with two choices of the location of the image. Arrangement (b) leads to smaller aberrations than arrangement (a) and is used in the Czerny-Turner mount.

3. Wadsworth, shown in Fig. 19. The Rowland circle is not used in this mounting in which the grating receives parallel light. 4. Ebert-Fastie, shown in Fig. 20. The Ebert-Fastie features a single, spherical, collimating mirror and a grating placed symmetrically between the two slits. The major advantage of the Ebert system is the fact that it is self-correcting for spherical aberration. With the use of curved slits, astigmatism is almost completely overcome.

FIGURE 12 Pfund mirror. The use of a plane mirror to avoid astigmatism in the use of a paraboloidal mirror.


20.9

FIGURE 13

Double-pass monochromator.

FIGURE 14

Source

Perkin-Elmer Model 99 double-pass monochromator.

Window

Window

M2

M1

Sampling area C Window

S2

M9

Window

M8 CH

TC M6

M5 S1

4

1

2

3

M7 Prism

M3

M4

MS

FIGURE 15

MW

Perkin-Elmer Model 112 single-beam double-pass infrared spectrometer.

20.10

COMPONENTS

FIGURE 16 Rowland mounting of the concave grating. The grating plate-holder bar, which slides on the two perpendicular ways, is shown in two positions, GP and G'P'. The slit SI and source S remain fixed.

5. Littrow, shown in Fig. 10. The Littrow system has slits on the same side of the grating to minimize astigmatism. An advantage of the Littrow mount, therefore, is that straight slits can be used. In fact, such slits may be used even for a spherical collimating mirror if the aperture is not too large. Its greatest disadvantage is that it does not correct for spherical aberration—not too serious a defect for long focal-length/small-aperture instruments. If an off-axis parabola is used to collimate the light, aberrations are greatly reduced. 6. Pfund, shown in Figs. 12 and 21. This is an on-axis, Pfund-type grating instrument.5 Incident infrared radiation, focused by a collimating lens on the entrance slit and modulated by a chopper, passes through the central aperture of plane mirror M1. Reflected by the paraboloidal mirror Pl, it emerges as a parallel beam of radiation, which is reflected by mirror M1 to the grating. The grating is accurately located on a turntable, which may be rotated to scan the spectrum. From the grating, the diffracted beam, reflected by mirror M2, is focused by a second paraboloid P2 through the central aperture of mirror M2 to the exit slit. The emerging beam is then focused by the ellipsoidal mirror M3 on the detector. An off-axis, double-pass grating instrument is illustrated in Fig. 22.6 Combinations of prisms and gratings are not uncommon. An illustrative and complex prismgrating, double-monochromator spectrometer designed by Unicam Instruments, Ltd. is shown in Fig. 23.5 The prism monochromator has four interchangeable prisms, and the grating monochromator has two interchangeable gratings. The two monochromators, ganged by cams which are linear in wave number, are driven by a common shaft. The instrument can be used either as a prismgrating double monochromator, or as a prism spectrometer by blanking the grating monochromator. Gratings, prisms, and cams can be automatically interchanged by means of push buttons.


FIGURE 17 Paschen-Runge mounting of the concave grating. Sl is the slit, G is the grating, and S is the light source.

FIGURE 18 Eagle mounting on the concave grating. Sl is the slit, G is the grating, S is the light source, and P is the plate holder.

20.11

20.12

COMPONENTS

FIGURE 19 Wadsworth mounting of the concave grating. Sl is the entrance slit, G is the concave grating, M is the concave mirror, P is the plate holder, and AB is the rail for the plate holder. To minimize aberrations, one must locate the slit close to the grating.

FIGURE 20 Ebert mounting of the plane grating designed by Fastie. Sl is the entrance slit, G is the grating, M is the concave mirror, and P is the photographic plate. The horizontal section is at the top and the vertical section is at the bottom.


20.13

FIGURE 21

On-axis Pfund grating spectrograph.

Magnetically operated slits, programmed by a taped potentiometer, provide a constant energy background. A star-wheel, time-sharing, beam attenuator is used in the double-beam photometer.

Contemporary In recent years there has been more attention paid to total system design and integration for specific purposes and applications, as for analytical atomic and molecular spectroscopy in analytical chemistry. Thus the conventional dispersive elements are often used in the classical configurations with variations. Innovations have come especially in designs tailored for complete computer control; introduction of one- and two-dimensional detector arrays as well as new detector types (especially for signal matching); the use of holographic optical elements either alone or combined with holographic gratings; and special data-processing software packages, displays, and data storage systems. This is the case also for interferometric systems as discussed in Chap. 32, “Interferometers,” by Parameswaran Hariharan.

Grating Entrance slit

Exit slit Detector

FIGURE 22 trograph.

Off-axis, double-pass grating spec-

20.14

COMPONENTS

TABLE 1

Examples of Prism/Grating Spectroradiometers

Manufacturer ARC (Acton Research Corp.), Acton, Mass. ARIES (Acton Research Instrument & Equipment Services Inc.), QEI (Quantum Electronics Instruments Inc.), Concord, Mass. Beckman Instruments Inc., Fullerton, Calif.

CVI Laser Corp., Albuquerque, N. Mex. Cary/Varian Instrument Group, San Fernando, Calif.

CI Systems Ltd., New York City, N.Y. and Israel Infrared Systems, Inc., Orlando, Fla. Instruments SA, Inc., J-Y Optical Systems, Edison, N.J.

LECO Corp., St. Joseph, Mich.

Leeman Labs, Inc., Lowell, Mass.

McPherson, Division of SI Corp., Acton, Mass.

Minirad Systems, Inc., Fairfield, Conn. Optometrics Corp., Ayer, Mass. Optronic Laboratories, Inc., A Subsidiary of Kollmorgen Corp., Orlando, Fla. Oriel Corporation, Stratford, Conn. Perkin-Elmer Corporation, Norwalk, Conn.

Shimadzu Scientific Instruments, Inc., Columbia, Md.

SPEX Industries, Inc., Edison, N.J. Thermo Jarrell Ash Corp., A Subsidiary of Thermo Instrument Systems, Inc., Franklin, Mass.

Comments Czerny-Turner or Rowland systems with triple indexable Vac UV/IR gratings Czerny-Turner variation with double or triple selectable gratings for 165-nm to 40-μm regions DU Series 60 and 70 modular construction, computercontrolled spectrophotometers for analytical applications Digikrom Monochrometers, 1/8-, 1/4-, and 1/2-m Czerny-Turner grating systems, 186 nm–20 μm Cary 1, 3, 4, and 5 spectrophotometers for UV-Vis-IR; double beam, dual chopper/grating Littrow systems; attachments (e.g., reflectance) and applications software CVF spectroradiometers for 0.4 to 20-μm scan CVF spectroradiometer Monochrometers, spectrometers for UV-Vis-IR, holographic gratings in Czerny-Turner or concave aberration-corrected holographic gratings and Rowland mounts; single and double pass; imaging spectrographs ICP (Inductively Coupled Plasma) spectrometer system with Pachen-Runge mount concave grating followed by an echelle and a linear detector array ICP system with a fixed echelle grating followed by a prism with crossed order dispersion and scanned photomultipliers or detector arrays Double/triple monochrometers, spectroradiometers using gratings and/or prisms in Seya-Namioka, Czerny-Turner (C-T), crossed C-T, or Rowland configurations CVF and discrete filters in spectroradiometers for field measurements, 0.2 to 30 μm Monochrometers, prism or grating, Ebert-Fastie systems for UV-Vis-NIR Spectroradiometers, UV-Vis-IR for precision measurements; filter wheels, gratings, and prisms in single/double monochrometer configurations Scanning monochrometers, rotation filter wheels, and detector array instruments Complete sets of UV-Vis-IR spectroscopic systems using gratings and prisms, or FT-IR, with software and hardware for computer control, and accessories for microscopy, reflectance measurement, etc. UV-Vis-NIR spectroscopic systems using holographic gratings in Czerny-Turner mounts in single- and double-beam configurations, computer-controlled, with accessories for analyses UV through IR grating spectrometers, 1/2 and 1/4 m, with CCD or PDA multichannel detectors Monochromators and spectroscopic systems for analyses, UV-Vis-IR with gratings (in 1942 in Wadsworth, then in 1953, Ebert, and now Paschen-Runge and crossed Czerny-Turner mounts); complete systems


20.15

Pfund condenser Golay detector Prism monochromator

Grating monochromator Littrow mirror Magnetic slits

Turntable with prism assembly

Turntable with grating assembly

Glass sector mirrors

Star wheel = 1 Transmittance Reference cell Source

Double beam photometer Sample cell Star wheel = 2 Equalizing

FIGURE 23

Unicam prism-grating double monochromator spectrometer.

Some examples found by a brief look through manufacturers’ literature and journals such as Spectroscopy, Physics Today, Laser Focus, Photonics Spectra, and Lasers & Optronics,8 are presented in Table 1. Most of these systems are designed for analytical spectroscopy with techniques described in many texts such as Robinson’s Atomic Spectroscopy.9

20.6

REFERENCES 1. 2. 3. 4. 5. 6. 7. 8.

9.

R. A. Sawyer, Experimental Spectroscopy, 3d ed. esp. Chapters 4, 6, 7, and 11, Dover Press, New York, 1963. F. A. Jenkins and H. E. White, Fundamentals of Optics, 4th ed., McGraw-Hill, New York, 1976. E. Hecht, Optics, 2d ed., Addison-Wesley, Reading, MA, reprinted April 1988. A. Walsh, “Multiple Monochromators II. Application of a Double Monochromator to Infrared Spectroscopy,” Journal of the Optical Society of America, Optical Society of America, Washington, DC, vol. 42, 1952, p. 95. H. L. Hackforth, Infrared Radiation, McGraw-Hill, New York, 1960, pp. 209, 211, 214. A. H. Nielsen, “Recent Advances in IR Spectroscopy,” Tech. Memo 53-2, Office of Ordnance Research, Durham, NC, December 1953. G. R. Harrison, “The Diffraction Grating—An Opinionated Appraisal,” Applied Optics, vol. 12, no. 9, 1973, p. 2039. Spectroscopy, especially issues of May and June 1990, Aster Publishing Corp., Eugene, OR; Physics Today, Annual Buyers’ Guide, 7 August 1990, American Institute of Physics, 335 East 45th St., New York; Laser Focus World and LFW’s The Buyers’ Guide, 25th ed., 1990, PennWell Publishing Co., Westford, MA; Photonics Spectra and The Photonics Directory, 4 vols., 36th ed., 1990, Laurin Publishing Company Inc., Pittsfield, MA; Lasers & Optronics and L & O’s 1990 Buying Guide, Gorden Publications, Inc., Morris Plains, NJ. J. W. Robinson, Atomic Spectroscopy, Marcel Dekker, Inc., New York, 1990.


21 INTEGRATED OPTICS Thomas L. Koch Lehigh University Bethlehem, Pennsylvania

Frederick J. Leonberger MIT Center for Integrated Photonic Systems Cambridge, Massachusetts

Paul G. Suchoski Audigence Inc. Melbourne, Florida

21.1

GLOSSARY APE CATV CVD CMOS DBR DFB F E FOG Gb/s F H IO IOC Lc LED Mb/s MMIC MZ n OEIC rij PIC RF SOI

annealed proton exchange cable television chemical vapor deposition complementary metal oxide semiconductor distributed Bragg reflector distributed feedback electric field of propagating light fiber optic gyroscope gigabits per second magnetic field of propagating light integrated optics integrated optic circuit coupling length of directional coupler light-emitting diode megabits per second monolithic millimeter-wave integrated circuit Mach-Zehnder index of refraction optoelectronic integrated circuit electro-optic tensor element photonic integrated circuit radio frequency silicon-on-insulator 21.1

21.2

COMPONENTS

tcutoff TE TM VR VLSI WDM D Gm J Scrit , N N PL

21.2

waveguide thickness for cutoff of first odd mode transverse electric mode transverse magnetic mode voltage for R radian phase shift in electro-optic modulator very large scale integration wavelength division multiplexing propagation constant of waveguide mode field amplitude of mode m coupling efficiency between modes critical angle for total internal reflection spatial period of periodic feature along waveguide vacuum wavelength of propagating light photoluminescence peak wavelength of semiconductor

INTRODUCTION The field of integrated optics is concerned with the theory, fabrication, and applications of guided wave optical devices and circuits. These structures guide light along the surface of a wafer typically using dielectric waveguides that confine light to lateral dimensions on the scale of the optical wavelength. Guided wave devices that perform passive operations analogous to classic optics, such as reflecting, beam splitting, attenuating and spectral filtering, can be formed using microelectronicbased fabrication techniques. By fabricating devices in active materials such as ferroelectrics, modulators, and switches based on the classic electro-optic effect can be formed. Compound semiconductors such as GaAs or InP additionally allow for the detection of light, and generation and amplification of light with light-emitting diodes (LEDs), lasers, and optical amplifiers. Extremely compact passive and active optical devices have also recently been demonstrated in silicon, taking explicit advantage of the highly advanced very large scale integration (VLSI) process technology developed for electronics. The monolithic integrate of optically interconnected passive and active devices in a multicomponent circuit is referred to as an integrated optic circuit (IOC) or a photonic integrated circuit (PIC), with the latter term usually applied to active semiconductor-based circuits. In semiconductor materials, purely electronic devices such as transistors can be integrated as well to form what is often referred to as an optoelectronic integrated circuit (OEIC). Progress in the field of integrated optics has been rapid since its inception1 in 1969. Much of this progress is due the availability of increasingly high-quality materials, microelectronic-processing equipment and techniques, and the overall rapid advancement and deployment of fiber optic systems. Interest in integrated optics stems from its numerous advantages over other optical technologies. Integrated optics devices interface efficiently with optical fibers, and can reduce cost in complex circuits by eliminating the need for separate, individual packaging of each circuit element. They also offer smaller size and weight, lower power consumption, improved reliability, and often larger electrical modulation bandwidths compared to their bulk-optic counterparts. The applications for integrated optics are widespread. Generally these applications involve interfacing with single-mode fiber optic systems such as digital and analog communications, but also include RF signal processing using optical techniques, laser beam control, and navigational and biomedical sensors. Integrated optics is viewed in the marketplace as a key enabling technology for high-speed digital optical fiber telecommunications, cable television (CATV) signal distribution, and fiber optic gyroscopes (FOG), and it will certainly continue to have a major impact on progress in broadband information distribution and connectivity. This chapter reviews the integrated optics (IO) field, beginning with a brief review of IO device physics and fabrication techniques. A phenomenological description of IO circuit elements, both passive and active, is given, followed by a discussion of IO applications and system demonstrations.

INTEGRATED OPTICS

21.3

The chapter concludes with a short look at future trends. Due to the brevity of this chapter relative to the work in the field, much of the coverage is necessarily limited. The reader is referred to Refs. 2–17 for more detailed information at a variety of levels.

21.3

DEVICE PHYSICS

Optical Waveguides Central to integrated optics is the concept of guiding light in dielectric waveguide structures with dimensions comparable to the wavelength of the guided light. In this section we present only a brief survey of the relevant physics and analysis techniques used to study their properties. The reader is referred to a number of excellent texts dealing with this topic for a more comprehensive treatment.2–6 A dielectric waveguide confines light to the core of the waveguide by somehow reflecting power back toward the waveguide core that would otherwise diffract or propagate away. While any means of reflection can accomplish this end (e.g., glancing-incidence partial reflections from interfaces between different media can serve as the basis for leaky waveguides), the most common technique employs a 100 percent total internal reflection from the boundary of a high-index core and a lowerindex cladding material. As light propagates down the axis of such a structure, the waveguide cross section can also be viewed as a lens-like phase plate that provides a larger retardation in the core region. Propagation down the guide then resembles a continuous refocusing of light that would otherwise diffract away. The pedagogical structure used to illustrate this phenomenon is the symmetric slab waveguide, composed of three layers of homogeneous dielectrics as shown in Fig. 1. It is well known that propagation in slab structures can be analyzed using either a ray-optics approach, or through the use of interface boundary conditions applied to the simple solutions of Maxwell’s equations in each homogeneous layer of the structure.2–6 In the ray-optics description, the rays represent the phase fronts of two intersecting plane waves propagating in the waveguide core region. Since the steady-state field has a well-defined phase at each point, a finite set of discrete modes arises from the self-consistency condition that, after propagation and two reflections from the core-cladding boundaries, any phase front must rejoin itself with an integral multiple of a 2R phase shift. For a given core thickness, there will be a limited discrete number of propagation angles in the core that satisfy this criterion, with the lower bound on the angle given by the critical angle for total internal reflection, Qcrit sin 1(n0 /n1 ). In general, a thicker and higher-index waveguide core will admit a larger number of confined solutions or bound modes. Figure 1 shows both the fundamental even mode and the first higher order,

FIGURE 1 A symmetric three-layer slab waveguide. The fundamental even and first odd mode are shown.

21.4

COMPONENTS

odd mode. If the dimensions are small enough, only one bound mode for each polarization state will exist and the guide is termed a single-mode waveguide. Care must be exercised to include the angle-dependent phase shift upon total internal reflection, referred to as the Goos-Hanchen shift, that can be viewed as a displaced effective reflection plane.2 The quantity B (2Pn1 /L ) sinQ , referred to as the propagation constant, is the z projection of the wave vector and thus governs the phase evolution of the field along the length of the guide. In addition to the discrete set of bound modes, plane waves can also enter from one side and pass at an angle through such a structure, and form a continuous set of radiation modes. In an asymmetrical structure, some of the radiation modes may be propagating on one side of the guide, but evanescent on the other. From a mathematical point of view, the set of all bound and radiation modes in a nondissipative structure form a complete set for expansion of any electromagnetic field in the structure. Analysis techniques will be discussed in more detail below. The slab waveguide in Fig. 1 employed total internal reflection from an abrupt index discontinuity for confinement. Some fabrication techniques for waveguides, particularly in glasses or electro-optic materials such as LiNbO3, achieve the high-index core by impurity diffusion or implantation, leading to a graded index profile. Here a field solution will usually be required to properly describe the modes and the ray paths become curved, but total internal reflection is still responsible for confinement. Most useful integrated optics devices require waveguide confinement not just in a two-dimensional slab, but in a stripe or channel geometry. While recognizing that the vector nature of the electromagnetic field makes the rigorous analysis of a particular structure quite cumbersome, the reader can appreciate that the same phenomenon of confinement by reflection will be operative in two dimensions as well. Figure 2 shows the cross sections of the most common stripe or channel waveguide types used in integrated optics. Common to the cross section for all these structures is a region on the waveguide axis containing higher index material than the surrounding cladding areas. The diffused waveguide may require a full two-dimensional analysis, but a common technique for the approximate analysis of high-aspect-ratio channel guides such as in Fig. 2a, b, c, and d, is the effective index method.2,3,18 In this technique a slab waveguide analysis is applied sequentially to the two dimensions. First three separate vertical problems are solved to obtain the modal phase index nmod e x B L /2P for a given polarization mode in each lateral region as if it were an infinite air

air

clad

diff

core

sub

(a)

(b)

air

air

core

clad core

sub

sub (c)

FIGURE 2 Various types of channel or stripe waveguides .

(d)

INTEGRATED OPTICS

21.5

FIGURE 3

A general multilayer slab waveguide structure.

slab. These indices are then used as input to a final “effective” slab waveguide problem in the lateral dimension using the opposite polarization boundary conditions. Since the properties of multilayer slab waveguides play an important role in waveguide analysis, a more comprehensive general formulation is outlined below. This task is more tractable using the field solutions of Maxwell’s equations than the ray-optics approach. A general multilayer slab is shown in Fig. 3. Since the wave equation in this case is separable, we need only consider a two-dimensional problem in the y direction perpendicular to the layers, and a propagation direction z. Here we will consider the concept of a mode in such a structure to be quantified in physical terms as a solution to Maxwell’s equations whose sole dependence on the coordinate in the propagation direction z is given by e iB z . This translates to a requirement that the shape of the field distribution in the y direction, perpendicular to layers, remain unchanged with propagation. In this manner we can easily generalize to leaky structures or materials exhibiting loss or gain, where D may be complex to allow for the scaling of the mode amplitude with propagation, but the relative mode profile in the perpendicular y direction still remains constant. These latter solutions are not normalizable or “proper” in the sense of mathematical completeness, but are very useful in understanding propagation behavior in such structures. F F F Since the field in each homogeneous layer m is well known to be e o ikm r , with | km | 2P nm /L for the (generally complex) index of refraction nm, the general solution to the field amplitude in each layer m is

E m ¨©ame iqm y bme iqm y ·¸ e iB z

(1)

where qm x ©2Rnm/N)2 D 2¸. Inspection of the vector Maxwell’s equations reveals that the general vector solution in the multilayer slab can be broken down into the superposition of a TE

21.6

COMPONENTS

(transverse electric) and a TM (transverse magnetic) solution.2,3 The TE (TM) solution is characterized by having only one component of the electric (magnetic) field that points in the x direction, parallel to the layers and perpendicular to the propagation direction z. The mode field amplitude Gm in Eq. (1) refers to the Ex or the Hx field for the TE and TM case, respectively. In a very simple exercise, F for each F of these cases one can successively match boundary conditions for continuous tangential E and H across the interfaces to provide the coefficients am1 and bm1 in each layer m 1 based upon the value of the coefficients in the preceding layer m, §1 ¤ qmG m ³ i(q q ) y e m1 m m ¨ ¥1 q G ´ 2 §am1¶ ¨ ¦ m1 m1 µ ¨b · ¨ ¤ © m1 ¸ ¨ 1 1 qmG m ³ e i(qm1 qm ) ym ¥ ´ ¨© 2 ¦ qm1G m1 µ

1 ¤1 qmG m ³ e i(qm1 qm ) ym ¶· 2 ¥¦ qm1G m1 ´µ · §am ¶ ·¨ · ¤ ³ q G 1 1 m m e i(qm1 qm ) ym · ©bm ¸ ¥ ´ 2 ¦ qm1G m1 µ ·¸

(2)

where ym are the coordinates of the interfaces between layers m and m 1, and G m x 1 for TE modes and G m x nm 2 for TM modes. The wave is assumed evanescently decaying or outward leaking on one initial side of the arbitrary stack of complex-index layers, that is b0 0 on the uppermost layer. When the lowermost “cladding” layer m p is reached, one again demands that only the coefficient bp of the evanescently decaying, or possibly the outward leaking, component be nonzero, which recursively provides the eigenvalue equation ap(D ) 0 for the eigenvalues Dj . Arbitrarily letting a0 1, this can be written explicitly as ¶ §1¶ § m0 a p (B ) [1 0] ¨ Mm (B )· ¨ · 0 ·¸ ©0¸ ¨©m p 1

(3)

where Mm(D ) is the matrix appearing in Eq. (2). This method is essentially the one described for real propagation constants by Kogelnik,2 who further provides important variational and perturbative expressions for determining the changes in propagation constants due to local changes in materials properties as might be used for modulation. In practice Eq. (3) is solved numerically in the form of two equations (for the real and imaginary parts of ap) in two unknowns (the real and imaginary parts of D ). Once the complex solutions D j are obtained using standard root-finding routines, the spatial profiles are easily calculated for each mode j by actually evaluating the coefficients for the solutions using the relations above with a0 1, for example. Application of Eq. (3) to the simple symmetric slab of Fig. 1 with thickness t, and real core index n1 and cladding index n0 can be reduced with some trigonometric half-angle identities to a simple set of equations with intuitive solutions by graphical construction.19 Defining new independent variables r x t /2[(2P n1 /L )2 B 2 ]1/ 2 and s x t /2 [B 2 (2P n0 /L )2 ]1/ 2 , one must simultaneously solve for positive r and s the equation r 2 s 2 (P t /L )2 (n12 n02 )

(4)

and either one of the following equations: s

G 1 ª tan(r ) r « G 0 ¬ cot(r )

(even modes) (odd modes)

(5)

where again G m x 1 for TE modes and G m x nm 2 for TM modes. By plotting the circles described by Eq. (4) and the functions in Eq. (5) in the (r,s) plane, intersections provide the solutions (rj , sj) for mode j, yielding Dj from the definition of either r or s. This construction is shown in Fig. 4 for TE modes, where Eq. (4) has been parametrized with u x (P t /L ) (n12 n02 )1/ 2 . Due to the presence of Im in Eq. (5), the TE and TM modes will have different propagation constants, leading to waveguide birefringence.

INTEGRATED OPTICS

21.7

FIGURE 4 A graphical solution for the symmetric three-layer slab waveguide. For an arbitrary value of the parameter u, solutions are found at the intersections of the circular arcs and the transcendental functions as shown.

It is easy to see from the zero-crossing of Eq. (5) at r R/2 that the cutoff of the first odd mode occurs when the thickness reaches a value t cutoff (L /2)(n12 n02 ) 1/2

(6)

Another important feature of the symmetric slab is the fact that neither the TE nor TM fundamental (even) mode is ever cut off. This is not true for asymmetric guides. More complicated structures are easily handled using Eq. (2), and one can also approximate graded-index profiles using the multi layer slab. However, analytical solutions also exist1 for a number of interesting graded-index profiles, including parabolic, exponential, and “cosh2” index profiles. Once the modes of a waveguide are known, there are many physical phenomena of interest that can be easily calculated. Quite often the propagation constant is required to evaluate the phase evolution of guided-wave components. In other cases the frequency dependence of the propagation constant is required to get the group velocity, v g [t B /t W ] 1, which determines quantities such as the flow of energy or the mode spacing in resonators. In other cases, the designer may need to know the electric field amplitude, or perhaps the fraction of the propagating energy, that lies within a certain layer of the waveguide system. One critical application lies in evaluating how light couples between guided-wave components with different modal structure, or from free space into a guided-wave component. For example, it is easy to show from the mathematical completeness of the waveguide modes2–4 that the energy

21.8

COMPONENTS

efficiency J of the coupling from a field Ginj injected at the input facet of the waveguide into a particular TE mode Gm of a waveguide is given by

H

| ¯ E m ( y)E inj( y)dy |2

(7)

¯| Em( y)|2 dy ¯| Einj( y)|2 dy

Another mathematical formalism, called coupled-mode theory,2,3,9 is one of the most important design tools for the guided-wave device designer and allows for the calculation of the coupling between parallel waveguides as in a directional coupler. It also allows for the evaluation of coupling between distinct modes of a waveguide when a longitudinal perturbation along the propagation direction destroys the exact mode orthogonality. An important example of the latter is when the perturbation is periodic as in a corrugated-waveguide grating where the grating allows for phase matching between modes with different propagation constants. Waveguide gratings are used to form wavelength selective coupling as in Bragg reflectors, distributed feedback lasers, and other grating-coupled devices. The comprehensive treatment of these phenomena is beyond the scope of this chapter, and the reader is referred to the references cited above. In some instances, evaluating the performance of devices where radiation plays a significant role may be tedious using a modal analysis, and numerical techniques such as the beam propagation method (BPM) are used to actually launch waves through the structure to evaluate radiation losses in waveguide bends, branches, or complicated splitters.20 Index of Refraction and Active Index-Changing Mechanisms Index of Refraction Waveguide analysis and design requires precise knowledge of the material index of refraction. One of the most common electro-optic materials is LiNbO3, a uniaxial birefringent crystal whose index can be characterized by providing the wavelength-dependent ordinary and extraordinary indices no and ne. They are given by21 no2, e Ao, e where

Bo, e Do, e L 2

Co , e L 2

(8)

Ao 4.9048 Bo 0.11768 Co 0.027169 Do 0.04750 Ae 4.5820 Be 0.099169 Ce 0.021950 De 0.044432

Glasses are also common substrate materials, but compositions and indices are too varied to list here; indices usually lie in the range of 1.44 to 1.65, and are mildly dispersive. Commonly deposited dielectrics are SiO2 and Si3N4, with indices of ~1.44 and ~2.0 at 1.55 μm. The reader is referred to various tables in the literature for more detail.21 InP and GaAs are by far the most common substrates for IO devices in semiconductors. The usual epitaxial material on GaAs substrates is AlxGa1–x As, which is nearly lattice-matched for all values of x, with GaAs at x 0 providing the narrowest bandgap. For photon energies below the absorption edge, the index of refraction of this material system is given by22 1/2

§ ¤ E 2f E 2 ³ ¶ · nAlGaAs (E , x ) ¨1 G (E 4f E'4 ) 2G (E 2f E'2 )E 2 2G E 4 ln ¥ 2 2´ ¨© ¦ E' E µ ·¸

(9)

INTEGRATED OPTICS

21.9

where E 1.2398/N is the incident photon energy,

G where

Ed 4 E03 (E02 E'2 )

and

(10)

E f (2 E02 E'2 )1/2

E0 (x ) 3.65 0.871x 0.179 x 2 Ed (x ) 36.1 2.45 x E' (x ) 1.424 1.266 x 0.26 x 2

For devices fabricated on InP substrates, common in telecommunications applications for devices in the 1.3-μm and 1.55-μm bands, the most common epitaxial material is a quaternary alloy composition InxGa1 x As y P1 y . In this case the material is only lattice matched for the specific combination y 2.917x, and this lattice-matched alloy can be characterized by its photoluminescence wavelength NPL under low intensity optical excitation. The index of this quaternary allow is given by23 ¤ ³ ´ ¥ A1 A2 ´ ¥ nQ (E , EPL ) ¥ 1 2´ 2 ¤ E ³ ¤ E ³ ´ ¥ 1 ¥ 1 ¥ ¥¦ ¦ EPL E2 µ´ ´µ ¦ EPL E1 µ´

1/2

(11)

where E 1.2398/L and EPL 1.2398/LPL are, respectively, the incident photon energy and photoluminescence peak photon energy for L in micrometer and A1(EPL ), A2 (EPL ), E1 , E2 are fitted parameters given by 2 A1 13.3510 5.4554 EPL 1.2332 EPL

A2 0.7140 0.3606 EPL E1 2.5048 eV

and

E2 0.1638 eV

For application to the binary InP, the value of the photoluminescence peak should be taken as NPL 0.939 μm. Many integrated optics devices rely on active phenomena such as the electro-optic effect to alter the real or imaginary index of refraction. This index change is used to achieve a different device state, such as the tuning of a filter, the switching action of a waveguide switch, or the induced absorption of an electroabsorption modulator. Below, a brief survey is provided of the most commonly exploited index-changing phenomena. Linear Electro-Optic Effect The linear electro-optic or Pockels effect refers to the change in the optical dielectric permittivity experienced in noncentrosymmetric ordered materials that is linear with applied quasi-static electric field. This relation is commonly expressed using the dielectric impermeability (1/n2 )ij x E 0 t Ei /t D j appearing in the index ellipsoid equation for propagation in anisotropic crystals.24 Convention employs symmetry arguments to contract all the (1 / n2 )ij to only six independent values which are then denoted by a single subscript (1/n2)i for i 1,...,6. In the principal axes coordinate system, the impermeability is diagonalized and (1/n2 )i 0 for i 4, 5, 6 in the absence of an applied electric field, with the value of (1/n2)ij providing the inverse square of the index for optical fields polarized along each axis i 1, 2, 3. For an electric field expressed in the

21.10

COMPONENTS

principal axes coordinate system, the changes in the coefficients are then evaluated using the 6 r3 electro-optic tensor r 3 ¤ 1³ $ ¥ 2 ´ £ rij E j ¦ n µ i j 1

i 1,

,6

(12)

With an applied field, the equation for the index ellipsoid in general must be rediagonalized to again yield (1/n2 )i 0 for i 4, 5, 6. This provides a new set of principal axes and the coefficients in the new index ellipsoid equation provide the altered value of the refractive index along each new principal axis. For a particular crystal, symmetry also limits the number of nonzero rij that are possible. In the cubic zinc-blend III-V compounds there are only three equal nonzero components r63 r52 r41 and the analysis is relatively easy. As a specific example, consider a static field E applied along the (001) direction, surface normal to the wafers commonly used for epitaxial growth. The re-diagonalized principal axes in the presence of the field become the (001) direction (z axis), the (011) direction (x axis), and the (011) direction (y axis); the latter two directions are the cleavage planes and are thus common directions for propagation. The respective index values become nx n0 1 n03r41E 2 n y n0 1 n03r41E 2

(13)

nz n0 If we thus consider a slab guide on a (001) substrate and propagation in the (011) direction, the applied field would produce a phase retardation for TE-polarized light of $ F P /L n03r41E L after a propagation length L. With values of r41 ~1.4 r 10 10 cm/V, micron-scale waveguide structures in GaAs or InP lead to retardations in the range of 10°/V·mm. This TE retardation could be used as a phase modulator, or in a Mach-Zehnder (MZ) interferometer to provide intensity modulation. For fields applied in other directions such as the (011), the principal axes are rotated away from the (011) and (011) directions. Propagation along a cleavage direction can then alter the polarization state, a phenomenon that also has device implications as will be discussed in more detail later. In the case of LiNbO3 and LiTaO3, two of the most common integrated optic materials for electro-optic devices, the dielectric tensor is more complex and the materials are also birefringent in the absence of an applied field. There are eight nonzero components to the electro-optic tensor, r22 r12 r61, r51 r42, r13 r23, and r33. For LiNbO3, the largest coefficient is r33 ~ 30.8 r 10 10 cm/V. Both retardation and polarization changes are readily achieved, and the reader is referred to Chap. 7, “Electro-Optic Modulators,” Vol. V of this Handbook or the literature for a comprehensive treatment of the properties of these and other materials.24,25 The electro-optic effect in these materials is associated with field-induced changes in the positions of the constituent atoms in the crystal, and the resulting change in the crystal polarizability. The absorption induced by the conventional electro-optic effect is thus negligible. Below, both free-carrier and field-induced absorption effects that are readily observed in semiconductors will be described. Carrier Effects In semiconductors, other powerful index-changing mechanisms are available related to the interaction of the optical field with the free electrons or holes. The simplest of these is the plasma contribution resulting from the polarizability of the mobile carriers. This mechanism is important in active integrated optical devices in both compound semiconductors and in silicon, where in the latter case it provides one of the only viable technique for modulation. According to the simple Drude model,26 this is given for each carrier species by $n y N e 2 L 2 /(8P 2E 0nc 2m ) in MKS units, where N and m are the carrier concentration and effective mass, e is the electronic charge, and G0 is the free-space permittivity. This can produce index changes approaching $n ~ 0.01 at 1018/cm3 electron-/hole-injected carrier density at 1.5-μm wavelengths. Static index shifts can also be achieved by impurity doping of the semiconductor, and can be exploited for waveguide design. Near the bandgap of the semiconductor, there are additional strong index changes with variations in the

INTEGRATED OPTICS

21.11

–

+

FIGURE 5 Franz-Keldysh effect. Electron can complete transition to the tilted conduction band by tunneling.

carrier density that arise from the associated dramatic changes in the optical loss or gain. Since these correspond to changes in the imaginary index, the Kramers-Kronig relation dictates that changes also occur in the real index. In the spectral region for gain in the semiconductor, these changes are comparable in magnitude and of the same sign as the free-carrier index contribution. These changes are responsible for chirping in the output of modulated semiconductor lasers, and also strongly influence their sensitivity to feedback. They can dramatically impact the static and dynamic performance of semiconductor devices and PICs that employ gain media. In addition to the effects described above arising from changing carrier populations, the electronic transitions that generate the free carriers can be modified by an applied electric field. For optical frequencies close to these transitions, this can give rise both to electroabsorption and to an enhanced electro-optic effect, which shall be termed electrorefraction, due to the Stark effect on the carrier-generating transitions. In bulk material, the induced absorption is termed the Franz-Keldysh effect,27 and can be viewed as a tunneling effect. For an electron in the valence band with insufficient energy to complete a transition to the conduction band, the field can be viewed as adding a potential to the bands that effectively tilts them in space as shown in Fig. 5. If the excited carrier also traversed a short distance down-field from its initial location, it would have sufficient energy to complete the transitions. This distance depends on the tilt, and thus the strength of the applied field. Since carriers can not be precisely localized according to the Heisenberg uncertainty principle, there is a finite amplitude for completing the transition that is an increasing function of electric field. For fields on the order of 105 V/cm, absorption values of ~100 cm 1 can be readily achieved in material that is quite transparent at zero field. According to the Kramers-Kronig relations, in addition to the absorption described above, this will also induce a change in the real index that will be positive below the band edge and can be used for phase modulation. In the case of quantum wells, carrier-induced effects can be enhanced due to the forced carrier proximity arising from confinement in the wells. Excitonic effects, resulting from the coulombic attraction between electrons and holes, produce sharp features in the absorption spectrum near the band gap that can be dramatically altered by an applied electric field. This quantum-confined Stark effect (QCSE) can enhance both electroabsorptive and electrorefractive effects.28,29 This suggests that more compact, lower-drive voltage devices are possible when quantum wells are employed, a fact that has been confirmed experimentally. However, in both the bulk and especially the quantum well case, care must be taken to operate at an optical frequency where strong electroabsorptive or electrorefractive effects are operative but the zero-field background absorption is

21.12

COMPONENTS

not prohibitively high. Another issue that impacts the design of devices based on electroabsorption is the requirement for removal of the photogenerated carriers to prevent screening of the applied field and band-filling which result in saturation. While the vast majority of QCSE devices have been demonstrated in compound semiconductors due to their direct bandgap, recent work has demonstrated that the effect is also operative in quantum wells at the direct bandgap of indirect gap materials such as Ge, providing impetus for further studies of QCSE device applications in the silicon-germanium materials system.30 Thermal Effects In virtually all materials, the optical path length of a given section of waveguide will increase with temperature. This is the combination of both the physical expansion of the material and the change of the index of refraction with temperature. While both are significant, in most integrated-optic materials the latter effect is dominant. In SiO2 on Si, for example, this mechanism provides a useful means of index change, and numbers on the order of $n ~ 10 5/°C are observed. This effect has been used to form a variety of thermo-optic switches and filters, but a significant disadvantage for many applications is the inherent slow speed and high power dissipation. In semiconductors, this index change is more than an order of magnitude larger, and leads to frequency shifts in filter devices and single-longitudinal-mode laser of $f ~10 GHz/°C. Nonlinear Effects Another class of index changes results from the nonlinear optical effects caused by the incident optical fields themselves. This is treated in depth in other portions of this Handbook,31 but two phenomena will be mentioned here. The first is closely related to the electro-optic effect discussed earlier, where now the applied field giving rise to the index change is no longer “quasi-static” but is in fact the optical field itself. The response of the medium is in general not the same at optical frequencies, but the same symmetry arguments and contracted tensor approach is employed. An exemplary phenomenon related to this kind of nonlinearity is second harmonic generation. The polarization resulting from the incident field at Y multiplied by the index oscillating at Y generates second harmonic components at 2Y. This frequency doubling can occur in waveguides, but great care must be taken to achieve phase matching where each locally generated frequencydoubled field propagates in such a way as to add coherently to frequency-doubled fields generated farther along the guide. This requires either that the dispersive properties of the materials and guide geometry allow n(Y) n(2Y), or that the frequency-doubled light radiates away from the guide at an angle to allow phase matching of the z component of the wavevector, or that a periodic domain reversal be introduced into the crystal to allow phase matching. This latter approach, often referred to as quasi-phase matching, has generated considerable interest recently. In this approach, the optic axis of the crystal is periodically reversed with a period equal to N/2 the difference in the index of refraction of the fundamental and second harmonic. To date, the most promising results are in LiTaO3, LiNbO3, and KTP. In LiNbO3, periodic domain reversal has been obtained by the application of 100 μs pulsed electric fields of 24 kV/mm using a 2.8 μm-period segmented electrode that is subsequently removed.32 The domain reversal in LiTaO3 can be obtained on a few-micron scale by proton exchange or electron bombardment. KTP has a higher nonlinear coefficient, but the material is not as well developed. Lower efficiencies have been obtained. This nonlinear mechanism can be used not just for second harmonic generation, but also for sum and difference frequency generation using input signals of different frequencies. A second application of nonlinear optics to integrated structures involves a higher order of nonlinearity referred to four-wave mixing, or in some situations as self-phase modulation. The change in index in these cases arises from the product of two optical fields. If all fields are the same frequency, this is termed degenerate, and if only one field is present, it becomes self-phase modulation with the index change driven by the intensity of the incident wave. This nonlinearity has been of interest in research aimed at creating all-optical logic devices. Here the intensity of either the input signal or a separate gating signal can determine the output port of a Mach-Zehnder or directional coupler switch, for example.33 Recent work has also shown that parametric amplification can be accomplished in guided-wave structures when the waveguide is properly constructed to allow phase matching of the resulting fields,34 suggesting the potential for compact, low-noise amplification in new regions of the spectrum.

INTEGRATED OPTICS

21.13

21.4 INTEGRATED OPTICS MATERIALS AND FABRICATION TECHNOLOGY Ion-Exchanged Glass Waveguides Passive integrated optic devices can be fabricated in certain glass substrates using the ion-exchange technique.35,36 In this fabrication process, a sodium-rich glass substrate is placed in a mixture of molten nitrate salts containing alkali cations, such as Cs, Rb, Li, K, Ag, and Tl. During this process, sodium ions at the surface of the host glass are replaced with the alkali cations, resulting in a local increase in the refractive index. Channel waveguides are realized by selectively masking the glass surface. The index change and the diffusion depth are a function of host material composition, the specific alkali cation being used, the temperature, and the diffusion time. The exchange process can be substantially enhanced by applying an electric field across the wafer while the substrate is immersed in the salt bath . Multimode devices are typically fabricated using thallium ion exchange in borosilicate glasses.37 The high polarizability of the thallium ions results in a large index change (> 0.1) while providing low propagation losses (0.1 dB/cm). However, thallium-sodium ion exchange has two significant drawbacks. Thallium is extremely toxic, and it also has a large ionic radius compared to sodium (1.49 Å compared to 0.95 Å), resulting in low diffusion coefficients and low mobility. It is therefore necessary to process the waveguides at high bath temperatures approaching the glass transition temperature of the host material (500°C) for long diffusion times (10 hours) with high applied electric fields (>100 V/cm) to achieve deep multimode waveguides that efficiently couple to commercially available multimode fiber (50 to 100 Mm core size). Finding suitable masking materials is a challenge. Single-mode devices are typically realized using either Ag Na, K Na, or Cs K exchange.29,30 The first two processes have been extensively studied and are well understood; however, they each appear to have drawbacks. Ag Na exchanged waveguides are somewhat lossy (0.5 dB/cm) due to a tendency for silver reduction in the host material. K Na exchanged waveguides are typically highly stressed and prone to surface scattering that increases the propagation loss. Although not as extensively studied, Cs K exchanged waveguides show great promise. These waveguides are nearly stress free, low loss (< 0.1 dB/cm), reproducible, and can be buried using a two-step exchange process. The two-step process further reduces the propagation loss and results in efficient fiber-waveguide coupling (< 0.1 dB loss per interface).

Thin Film Oxides In recent years there has been substantial interest in IO devices fabricated in thin-film dielectrics on silicon substrates. This is due in part to the excellent surface quality, large-area wafers and mechanical integrity of silicon itself. However, this interest also stems in some cases from the availability of mature silicon-processing technology developed by the electronic integrated circuit industry. IO technology on silicon substrates is usually carried out in SiO2, and there are two generic approaches to the Si/SiO2 fabrication that have proven capable of producing very high performance IO devices. IO devices using both approaches are characterized by waveguides that are extremely low loss and are easily matched in mode characteristics to optical fibers used for transmission, thereby providing very efficient coupling. The first approach borrows more from the technology of optical fiber manufacture than it does from the Si electronics industry.38 Using a technique known as flame hydrolysis (FHD), a “soot” of SiO2 is deposited on a Si wafer to a depth of 50 to 60 μm, followed by a thinner layer of a SiO2/ GeO2 mix to form what will become the high-index waveguide core. This material is consolidated at ~1300°C for several hours down to roughly half its original thickness, and then the waveguide core layer is patterned using reactive ion etching to form square cross-section waveguide cores. Then FHD is again used, followed by more consolidation, to form the upper cladding layers. Typical index differences for the core material are in the range of 0.25 to 0.75 percent, with core dimensions of 6 to 8 μm square.

21.14

COMPONENTS

A measure of the material quality that was available using this approach is given by some of the extraordinary early devices results obtained. IO power splitters were fabricated to sizes of 1 r 128 using seven stages and a total of 127 Y-branch 1 r 2 splitters and a total device length of 5 cm. Total fiber-to-fiber excess loss for this device was 3.2 dB with a standard deviation of 0.63 dB.39 A large variety of devices has been made using this technique. Another technique for Si/SiO2 fabrication employs film-deposition technology borrowed from silicon electronics processing.40 First a base SiO2 layer is deposited using high-pressure steam to a thickness of ~15 μm to prevent leakage to the high-index Si substrate. The waveguide and cladding layers are deposited using low-pressure chemical vapor deposition, either from silane and oxygen, or from tetraethylorthosilane and ammonia. Phosphine is added to increase the index, with guide cores typically containing 6.5 to 8 percent phosphine. The wafer is usually annealed at 1000°C to relieve strain and to densify the films. Waveguide losses below 0.05 dB/cm being reported41 using this technique, and a large variety of devices have been demonstrated using this approach, including splitters, couplers, and WDM devices. One of the interesting features of this approach to fabrication is that it readily lends itself to the inclusion of other thin films common in Si processing. One such film is Si3N4 and this has been used as a high index core for waveguides with much larger core-cladding index step. Such waveguides can generate tightly confined modes that are a much closer match to the modes commonly found in active semiconductor components such as lasers. This feature has been used in a novel mode converter device42 that adiabatically transforms from the smaller mode into the larger, fiber-matched mode commonly employed in Si/SiO2 IOCs. In some instances, slow response active devices have been fabricated in Si/SiO2 technology using thermal effects to achieve local index changes in one arm of a Mach-Zehnder interferometer. This can either be used as a thermo-optic switch or as a tuning element in WDM components. The heating elements in these devices comprises a simple metal-film-resistive heater deposited directly on the upper surface of the wafer. Another characteristic feature of IOCs in Si/SiO2 is a degree of birefringence that results from the compressive stress induced in the film by the Si substrate after cooling down from the hightemperature film deposition or consolidation. Typical amounts of birefringence are nTE nTM 3 r 10 4. This birefringence can cause wavelength shifts with input polarization in WDM components, and techniques to counteract it include stress removal by adding strain-relief grooves in the film, or stress compensation by adding a counteracting stress-inducing film on the surface of the guide, or the explicit introduction of birefringence-compensating waveguide elements. The Si/SiO2 technology has matured to a point where it is commonly used in telecommunications for passive devices such as wavelength demultiplexing circuits that will be discussed later in this chapter, and active thermo-optic device circuits continue to be explored.

Silicon Photonics Fabrication and Materials Recent years have seen a strong increase in research and development of silicon photonics where light is guided in silicon layers that are transparent in the important telecommunications window from 1.2 to 1.6 μm.43–46 The most common approach to waveguide design directly uses silicon-on-insulator (SOI) technology, which provides for a layer of single-crystal silicon with thicknesses in the 0.1- to 5-μm range, separated from the underlying silicon substrate by a buried oxide (BOX) layer of SiO2 with a thickness typically in the 1- to 2-μm range. Typical waveguide structures fabricated in the SOI system are shown in Fig. 6. SOI has become a mainstream VLSI electronics technology offering reduced electrical parasitics for high-performance microprocessor applications, and the most common SOI wafer-fabrication method is the SMARTCUT technique, developed at CEA/LETI in France, and commercialized by Soitec.47 This technique works by implanting hydrogen in a first substrate and growing oxide on a second substrate. After the two substrates are wafer bonded, a rapid annealing step expands the hydrogen, creating an accurate cleave plane and leaving a uniform silicon film on top of the oxide. High-quality SOI wafers are now available to the silicon photonics community in sizes up to 12 in (300 mm), and this materials system immediately brings to bear the unprecedented process

INTEGRATED OPTICS

Large-area rib 3–5 m

Quasi-planar ridge

3–5 m

~1 m

21.15

Strip “Silicon wire”

~200 nm

~400 nm

~200 nm

Si ( = 3.48) SiO2( = 1.44) Si ( = 3.48) FIGURE 6 Typical silicon-on-insulator (SOI) waveguide structures.

precision associated with VLSI complementary metal-oxide-semiconductor (CMOS) technology. This precision is not just in lithography, etching, and feature size, now at the 65-nm node and still shrinking, but also in the remarkable versatility in materials sequencing that allows for exotic dielectrics and multilayer metallizations. One of the distinctive features of the SOI system is the extremely high index contrast available for waveguides, for example, with nSi 3.475 and nSiO2 1.444 at 1.55 μm. As illustrated in Fig. 7, this allows for extremely small waveguides with cross-sectional areas below 0.1 μm2.46 This feature enables extremely compact passive devices, and also enables high-performance active devices by concentrating the mode on the index-changing region. However, the same high index contrast also leads to significant scattering losses from fabrication-induced waveguide sidewall roughness, which becomes increasingly challenging in very small cross-section waveguides. Larger mode devices in SOI have shown losses in the ~0.1 dB/cm range,48 while maximally vertically confined shallow ridge guides have losses of ~0.4 dB/cm,49 and several decibels per centimeter is more typical for most tightly confined “wire” waveguides. The principle active index-changing mechanism used in silicon photonics is the plasma index contribution from free electronics and holes in the silicon. As will be illustrated later in discussions of devices, these free-carrier populations and locations can be controlled in a variety of diode and field-effect structures. The ability to harness the precision of VLSI CMOS technology to predictably

FIGURE 7 Example of losses for high-index-contrast “wire” waveguides: For 6.5-μm radius bends, losses are 0.0043 dB per 180° turn as demonstrated by Vlasov et al., IBM.86

21.16

COMPONENTS

engineer very high confinement waveguides together with precision dynamic control of free carriers in transistor-like device geometries has recently resulted in a number of high-performance modulator designs. While historically silicon has not been viewed as a high-performance active optical material, these and other advances have caused a serious reexamination of the potential of silicon as a powerful medium for PICs.

LiNbO3 and LiTaO3 The majority of the integrated optics R&D from 1975 to 1985 and the majority of the currently commercial integrated optics product offerings utilize LiNbO3 as the substrate material. A number of excellent review papers detailing R&D efforts in LiNbO3 are available.50–54 LiNbO3 is an excellent electro-optic material with high optical transmission in the visible and near infrared, a relatively large refractive index (n 2.15 2.2), and a large electro-optic coefficient (r33 30.8 r 10 10 cm/V). Probably most important, but frequently overlooked, is the widespread availability of high-quality LiNbO3 wafers. Hundreds of tons of LiNbO3 is produced annually for the fabrication of surface acoustic wave (SAW) devices. This large volume has resulted in well-developed crystal growth and wafer-processing techniques. In addition, LiNbO3 wafers are at least an order of magnitude less expensive than they would be if integrated optics was the only user of this material. High-quality 3- and 4-in optical-grade LiNbO3 wafers are now available from multiple vendors. LiNbO3 is a uniaxial crystal which is capable of supporting an extraordinary polarization mode for light polarized along the optic axis (z axis) and an ordinary polarization mode for light polarized in the x-y plane. LiNbO3 is slightly birefringent with ne 2.15 and no 2.20. LiNbO3 devices can be fabricated on x-, y-, and z-cut wafers. Phase modulators, fiber gyro circuits, and Mach-Zehnder interferometers are typically fabricated on x-cut, y-propagating wafers, and operate with the TE (extraordinary) mode. Push-pull devices, such as delta-beta directional coupler switches, are typically fabricated on z-cut, y-propagating wafers and operate with the TM (extraordinary) mode. Both configurations utilize the strong r33 electro-optic coefficient. Devices that require two phasematched modes for operation are typically fabricated on x-cut, z-propagating wafers. The majority of LiNbO3 integrated optic devices demonstrated to date have been fabricated using the titanium in-diffusion process.55 Titanium strips of width 3 to 10 Mm and thickness 500 to 1200 Å are diffused into the LiNbO3 at 950 to 1050nC for diffusion times of 5 to 10 hours.56,57 The titanium diffusion results in a local increase in both the ordinary and extraordinary refractive indices so that both TE and TM modes can be supported for any crystal orientation. Titanium thickness and strip width typically need to be controlled to o 1 percent and o 0.1 Mm, respectively, for reproducible device performance. Due to the high processing temperatures that approach the Curie temperature of LiNbO3, extreme care must be taken to prevent Li2O out-diffusion58,59 and ferroelectric domain inversion, both of which significantly degrade device performance. Photorefractive optical damage60 also needs to be considered when utilizing Ti-diffused devices for optical wavelengths shorter than 1 Mm. Optical damage typically prevents the use of Ti-diffused devices for optical power greater than a few hundred micro Watts at 800-nm wavelength, although the problem can be reduced by utilizing MgO-doped LiNbO3 wafers. Optical damage is typically not a problem at 1300 and 1550 nm for optical powers up to 100 mW. An alternative process for fabricating high-quality waveguides in LiNbO3 is the annealed proton exchange (APE) process.61,62 In the APE process, a masked LiNbO3 wafer is immersed in a proton-rich source (benzoic acid is typically used) at temperatures between 150 and 245nC and times ranging from 10 to 120 minutes. The wafer is then annealed at temperatures between 350 and 400nC for 1 to 5 hours. During the initial acid immersion, lithium ions from the wafer are exchanged with hydrogen ions from the bath in the unmasked region, resulting in a stress-induced waveguide that supports only the extraordinary polarization mode. Proton-exchanged waveguides that are not subjected to further processing are practically useless due to temporal instabilities in the modal propagation constants, high propagation loss, DC drift, and a much-reduced electro-optic coefficient. However, it has been demonstrated62 that proper postannealing results in extremely high quality waveguides that are stable, low-loss, and electro-optically efficient.

INTEGRATED OPTICS

21.17

The APE process has recently become the fabrication process of choice for the majority of applications currently in production. Since the APE waveguides only support the extraordinary polarization mode, they function as high-quality polarizers with polarization extinction in excess of 60 dB.63 As described later in this chapter, high-quality polarizers are essential for reducing the drift in fiber optic gyroscopes and minimizing nonlinear distortion products in analog links. APE waveguides exhibit low propagation losses of 0.15 dB/cm for wavelengths ranging from 800 to 1550 nm. APE LiNbO3 devices exhibit stable performance for optical powers of 10 mW at 800 nm and 200 mW at 1300 and 1550 nm. The APE process can also be used to fabricate devices in LiTaO3 for applications requiring higher optical powers (up to 200 mW) at 800 nm.64 In addition to offering performance advantages, the APE process also appears to be the more manufacturable process. It is relatively easy to scale the APE process so that it can handle 25-wafer lots with no degradation in device uniformity. The fiber pigtailing requirements are also substantially reduced when packaging APE devices since these devices only support a single polarization mode. After the waveguides have been fabricated in the LiNbO3 wafer, electrodes need to be deposited on the surface. One-Mm-thick gold is typically used for lumped-electrode devices while 5-Mm-thick gold is typically used for traveling-wave devices to reduce RF resistive losses. The lift-off process and electron-beam deposition is typically used for lumped-electrode devices while up-plating is typically used for realizing the thicker gold electrodes. Better than 0.5-Mm layer-to-layer registration is required for optimum device performance. As shown in Fig. 10, electrodes on x-cut LiNbO3 are usually placed along side the waveguide so that the horizontal component of the electric field interacts with the propagating TE mode. Electrodes on z-cut LiNbO3 are placed on top of the waveguide so that the vertical component of the electric field interacts with the propagating TM mode. An SiO2 buffer layer (0.1–1 Mm thick) is required between the optical waveguide and the electrode on all z-cut devices to reduce metal-loading loss. A thick (1 Mm) SiO2 buffer layer is also utilized on some x- and z-cut devices to reduce the velocity mismatch between the microwave and optical waves in high-speed traveling-wave modulators. A thin layer of amorphous silicon is also utilized on some z-cut devices to improve device stability over temperature.

III-V Materials and Fabrication Technology In this section we will briefly review some of the epitaxial growth and fabrication techniques that are used to make PICs in III-V materials, with a primary focus on InP-based devices. III-V Epitaxial Crystal Growth The epitaxial growth of III-V optoelectronic materials has evolved during the last several decades from nearly exclusive use of manually controlled liquidphase epitaxial (LPE) growth to a variety of highly versatile computer-automated vapor and beam-growth techniques. These include atmospheric-pressure and low-pressure metal-organic vapor-phase epitaxy (MOVPE), hydride and chloride vapor-phase epitaxy (VPE), molecular beam epitaxy (MBE), chemical beam epitaxy (CBE), and metal-organic molecular beam epitaxy(MOMBE). Detailed descriptions of reactor design and growth chemistry are beyond the scope of this section, and the interested reader is referred to recent texts and conference proceedings for the most current information.65 One of the critical criteria for evaluating crystal growth is the uniformity, both in thickness and in epitaxial composition. Layer thickness changes of several percent can lead to nanometer-scale wavelength changes in grating-based lasers and filter devices. Similarly, compositional changes leading to a 10-nm shift in the photoluminescence peak wavelength of the guide layers, which is not at all uncommon, can also result in nanometer-scale wavelength shifts in distributed feedback (DFB) laser emission wavelengths, in addition to potential undesirable gain-peak mismatches that may result from the LPL shift itself. Proper reactor geometry, sometimes with substrate rotation, have been shown capable of percent-level uniformity both in MOVPE and in the beam techniques. One difficulty associated with latter lies in the ballistic “line-of-sight” growth which prevents regrowth over reentrant mesa geometries or overhanging mask surfaces often encountered in PIC and laser fabrication, while

21.18

COMPONENTS

MOVPE and especially VPE offer outstanding coverage over a wide range of morphologies. Other criteria to be considered are the doping capabilities. The lower growth temperatures associated with MBE, CBE and MOMBE enable very abrupt changes in doping level, and highly concentrated doping sheets that are desirable for high-speed transistors in OEICs, for example. Both the vapor and beam techniques have successfully grown semi-insulating Fe-doped InP, a material that is playing an increasingly pivotal role in photonic devices. The typical PIC processing involves the growth of a base structure that is followed by processing and regrowths. During both the base wafer and regrowths, selective area growth is often employed where a patterned dielectric film is used to prevent crystal growth over protected areas. This film is typically SiO2 or Si3N4 deposited by CVD or plasma-assisted CVD. This technique is readily used with MOVPE, but care must be taken to keep a substantial portion of the field open for growth to avoid the formation of polycrystalline deposits on the dielectric mask. Caution must be exercised during regrowths over mesas or other nonplanar geometries, as well as in the vicinity of masked surfaces. Gross deviations from planarity can occur due to overshoots of crystal growth resulting from crystal-orientation-dependent growth rates on the various exposed surfaces. III-V Etching Technology A fundamental step in III-V PIC processing is mesa etching for definition of the optical waveguides. This is usually accomplished by patterning a stripe etch mask on a base wafer that has a number of epitaxial layers already grown, and removing some of the layers in the exposed regions to leave a mesa comprised of several of the epitaxial layers. The etching process can either be a “wet” chemical etchant, or a “dry” plasma-type etch. Wet etching refers to the use of an acid bath to attack the unprotected regions of a surface. The acids that are commonly used to etch III-V materials66 also tend to significantly undercut a photoresist pattern, and hence photoresist is usually used only in broad-area features or in shallow etches where undercutting is not a concern. For precise geometries such as waveguide stripes, another masking material such as SiO2 or Si3N4 is first deposited and patterned with photoresist and plasma etching, or HF etching (for SiO2). In some instances it is required that the etchants be nonselective, uniformly removing layers regardless of composition. This is usually the case when etching a mesa through a multilayer active region to form a buried heterostructure laser. Br-based etchants, such as bromine in percent-level concentration in methanol, tend to be very good in this regard. This etchant, along with many of the nonselective etchants, will form a reentrant 54.7n (111A) face mesa for stripes along the (011) direction (with a nonundercutting mask) and will form an outward-sloping 54.7n walled mesa for stripes along the (011) direction. Other etchants, with varying degrees of nonselectivity and crystallographic behavior, include mixtures of HBr, CH3COOH, or HCl, CH3COOH, and H2O2.67 In fabricating precise geometries in III-V integrated optic or PIC devices, it is often desirable to remove specific layers while leaving others, or control mesa heights to a very high degree of precision. The combination of material-selective etchants and the inclusion of special etch-stop layers offers a convenient and precise means of achieving this. Hundred-Å-thick layers of InGaAsP can easily halt InP etches even after many microns of etching. Extensive compilations have been made of etches for the InP-based compounds,52 and the most common selective InP etches are HCl-based. Typical mixtures are HCl and H3PO4 in ratios ranging from 3:1 to 1:3, with the lower HCl content leading to less undercutting and slower etch rates. The HCl-based etchants are highly crystallographic in nature,68 and can produce mesas with nearly vertical walls or outward sloping walls, depending on the mesa stripe orientation. A common selective etch for removing InGaAsP or InGaAs while only weakly attacking InP are mixtures of H2SO4, H2O2, and H2O in a ratio of X:1:1 with X typically ranging from 3 to 30. Selectivities in the range of 10:1 and typically much higher are readily achieved. Other selective etchants for InGaAsP are based on HNO3 or mixtures of KOH, K2Fe(CN)6, and H2O. Dry etching techniques, such as reactive ion etching (RIE) or other variants such as chemically assisted reactive ion beam etching (CAIBE), also play a key role in III-V PIC processing. These have often been carried out using Cl2-based mixtures with O2 and Ar,69 while in other cases the reactive chlorine is derived from compounds such as CCl2F2. Excellent results have also been obtained with methane/hydrogen mixtures or ethane/hydrogen.70 In these latter cases Ar is also often used as a

INTEGRATED OPTICS

21.19

sputtering gas to remove interfering redeposited compounds. Reactive ion etching has been used both to form mesa and facet structures as well as in transferring grating patterns into semiconductors through an etch mask. The appeal of reactive ion etching is the lack of mask undercutting that can usually be achieved, allowing very high lateral precision with the promise of reproducible submicron mesa features. In addition, the ability to create vertical wall-etched facets through a variety of different composition epitaxial layers suggests the possibility of integrated resonator or reflecting and coupling structures without the use of gratings. This approach has been used to form corner reflectors,71 square-geometry ring-resonators,72 and a variety of complex waveguide patterns using beam splitters.73 Another recent application has been the use of etched-facet technology to create gratings, not as an interfacial corrugation along the waveguide, but as a grating in the other dimension at the end surface of a waveguide for two-dimensional “free-space” grating spectrometers.74,75 Grating Fabrication Many of the PICs employ corrugated-waveguide grating-based resonators or filters, and the most common technique for fabricating these gratings involves a “holographic” or interferometric exposure using a short wavelength laser source. Here a thin (typically 500 to 1000Å thick) layer of photoresist is spun on a wafer surface and exposed with two collimated, expanded beams from a blue or UV laser at an appropriate angle to form high-contrast fringes at the desired pitch. Since the illuminating wavelength is precisely known, and angles are easily measured in the milliradian range, the typical corrugation in the 2000Å-period range can be fabricated to armstrong-level precision in period. The resist is developed and then functions as an etch mask for the underlying layers. This etching can be either a wet etch (commonly using HBr-based etchants), or a dry reactive ion etch. Commonly used lasers are HeCd at 325 nm or one of the UV lines of an Argon ion laser at 364 nm. Electron-beam lithography has also been successfully applied to the generation of gratings for III-V integrated optic devices. Active-Passive Transitions Compound semiconductors are appealing for PICs in large part due to their ability to emit, amplify, and detect light. However, waveguide elements that perform these functions are not low loss without excitation, and are generally not suitable for providing passive interconnections between circuit elements. One of the most fundamental problems to overcome is the proper engineering and fabrication of the coupling between active waveguides, containing lower bandgap material, and passive waveguides composed of higher bandgap material. Most PICs demonstrated to date have employed some form of butt-coupling, where an active waveguide of one vertical and/or lateral structure mates end-on with a passive waveguide of a different vertical and/or lateral structure. Butt-coupling offers design simplicity, flexibility, and favorable fabrication tolerances. The most straightforward approach for butt-coupling involves the selective removal of the entire active waveguide core stack using selective wet chemical etching, followed by a regrowth of a mated, aligned passive waveguide structure. The principal advantage of such an approach is the independent selection of compositional and dimensional design parameters for the two guides. Another approach to butt-coupling, often called “offset quantum wells” or “active layer removal,” employs a largely continuous passive waveguide structure with a thin active layer (usually of quantum wells) residing on top, which is selectively removed on the portions of the structure which are to be passive. Using material-selective wet chemical etches, the thin MQW stack can be removed with very high reproducibility and precision, and the dimensional control is thus placed in the original computer-automated MOVPE growth of the base wafer. The removal of the thin active layer constitutes only a small perturbation of the continuous guide core constituted by the lower, thicker layer, and efficient coupling can be achieved.76 Two powerful alternatives to the butt-coupling variants discussed above are selective area epitaxy and quantum-well intermixing. These techniques provide for longitudinal modification in the effective bandgap of quantum-well-containing waveguides by altering the dimensions or profile of the quantum well along the length of a waveguide. Selective area epitaxy accomplishes this by the inclusion of growth-inhibiting masks, such as SiO2, laterally adjacent to the waveguide during MOVPE growth. The resulting local increase in vapor-phase reactants, combined with surface diffusion along

21.20

COMPONENTS

the growth-inhibiting mask, leads to an increased growth rate in regions adjacent to the lateral masks. This leads to thicker quantum wells with correspondingly lower bandgaps from quantum confinement, making these regions suitable for gain media or absorption along a waveguide which is transparent in regions with no masks. By controlling the degree of enhancement, regions of transparency, regions suitable for electroabsorption or QCSE, and regions suitable for gain or absorption can all be formed along a guide with remarkably smooth continuity between regions. This technique has been highly successful in PICs comprising an integrated DFB laser and electroabsorption modulator.77 The quantum-well intermixing accomplishes a similar end by implanting regions and diffusing impurities or vacancies into the quantum-well region of a waveguide to intermix the boundaries of the wells and barriers. This intermixing smoothes the rectangular well into a rounded profile and also can change the average composition of the well, leading to a higher bandgap which can be used for longitudinally differentiating gain regions, modulation regions, and transparent regions.78 Although not in as wide commercial use, this method has the advantage that the various regions are large-area materials modifications allowing for standard materials characterization techniques, in contrast to the selective area epitaxy approaches which require difficult characterization of narrow stripe regions. Yet another approach to coupling between two different waveguides employs directional coupling in the vertical plane between epitaxial layers serving as the cores of the two distinct waveguides. This type of vertical coupling can either be accomplished using the principle of intersecting dispersion curves, or through the use of a corrugated-waveguide grating to achieve phase matching. Vertical coupler structures may be useful for wide-tuning applications, since a small change of effective index for one mode can lead to a large change in coupling wavelength.79

Organic Polymers Polymer films are a relatively newer class of materials for integrated optics.80 Polymers offer much versatility, in that molecular engineering permits many different materials to be fabricated; they can be applied by coating techniques to many types of substrates, and their optical and electro-optical properties can be modified in a variety of ways. Applications range from optical interconnects, in which passive guides are used in an optical PC board arrangement, to equivalents of IOCs and OEICs. Polymer devices are also being explored for third-order nonlinear applications. Numerous methods for fabricating polymer waveguide electro-optic devices have been reported. One attractive technique consists of spin-coating a three-layer polymer sandwich over a metal film, often on a semiconductor (Si) substrate. The three polymer layers form a symmetric planar waveguide; the middle layer is electro-optic, due to the presence of a guest molecule that imparts the electro-optic property, or the use of a side-chain polymer. The sample is overcoated with metal and the entire structure is heated near the glass transition temperature and poled at an electric field of typically 150 V/μm. The poling aligns the nonlinear molecules in the middle polymer layer, thereby inducing the Pockels effect and a birefringence. Typical values of index and birefringence are 1.6 and 0.05, respectively. Electro-optic coefficients are in the 16 to 38 pm/V range. Channel waveguides are subsequently defined by a variety of methods. An attractive technique is photobleaching, in which the waveguide region is masked with a metal film and the surrounding area exposed to UV light. This exposure alters the molecules/linking in the active layer, thereby reducing the refractive index and providing lateral confinement. Losses in such guides are typically in the 1dB/cm range. The basic IO modulators have been demonstrated in a variety of polymers. Of particular note are traveling wave modulators with a 3-dB bandwidth of 40 GHz and a low-frequency V pi of 6 V,81 and work illustrating the high-frequency potential of polymers up to modulation frequencies of 110 GHz.82 Relative to LiNbO3, polymer modulators can have higher overlap factors because the lower metal layer provides vertical, well-confined signal fields. However, the relatively low index of polymers and their comparable electro-optic coefficient to LiNbO3 implies a lower electro-optic efficiency. Polymers do provide a better velocity match of optical and electromagnetic velocities which can result in very high frequency performance as described above.

INTEGRATED OPTICS

21.21

For polymers to fulfill their potential, a number of material and packaging issues must be addressed. First, it is highly desirable to develop polymers that overcome the long-term relaxation of the electro-optic effect typical of many of the early reports. Development of polymers with transition temperatures in the 300°C range (so they can withstand the temperatures typical of device processing and packaging) is also highly desirable. Work on polyimide is particularly promising in this area. Recent work has demonstrated encouraging results with devices processed at lower temperatures that exhibit long-term stability at elevated operating temperatures of 85°C.83

21.5

CIRCUIT ELEMENTS

Passive Devices This section provides a phenomenological description of the most common passive and active IO devices. Detailed descriptions of the device theoretical properties as well as typical characteristics can be found in Refs. 2–6. Passive devices form many of the fundamental building blocks for IOCs and PICs, and passive waveguides also comprise the interconnections between devices. Passive devices are defined as those dielectric waveguide structures which involve neither application of electrical signals, generation or detection of light, nor nonlinear optical effects. This section will focus on the most important structures: waveguide bends, polarizers, fiber-to-chip coupling, power splitters, and filters. Waveguide bends, such as those illustrated in Figs. 7, 10, 11, 12, 14, 15, 19, 22, 25, and 26, are needed to laterally offset modulators and other devices, and also to increase device-packing density. Waveguide bends lead to radiation or leakage from the guide, and analytical treatments reveal that the induced losses decrease exponentially with radius of curvature in a manner that is very sensitive to the index difference between the core and cladding of the guide.84,85 The most widely used bend is based on an S-bend geometry described by a raised cosine function.2 This structure minimizes the tendency of light in a dielectric waveguide to “leak” as the guide’s direction is altered by starting with a small bend (large effective bend radius and then increasing the bend rate until the midpoint of the offset, then following the pattern in reverse through the bend completion. Since the index difference between the guide and surrounding dielectric material is often small (10 3 to 10 4) bends must be gradual (effectively a few degrees) to keep losses acceptably (< 0.5 dB) small. In LiNbO3, offsets of 100 Mm require linear distances of typically 3 mm. In semiconductor research device work, designs with high index steps are sometimes used to form small-radius bends and ion beam etching has been utilized to form reflective micromirrors73 at 45n to the guide to create a right-angle bend. While compact, to date these have generally been relatively lossy compared to their bending waveguide counterparts. As noted earlier, silicon photonics carries a very high index contrast of $n ~ 2, which also allows for extremely tight waveguide radii, as illustrated in Fig. 7, without introducing the bending losses that often limit the size of weaker index contrast materials systems.86 This can provide a dramatic increase in the density of possible PICs developed in the SOI system. Polarizers are necessary for polarization-sensitive devices such as many electro-optic modulators and for polarization-sensitive applications such as fiber gyroscopes. Polarizers can be formed on dielectric waveguides that support both TE and TM propagation by forming overlays that selectively couple one polarization out of the guide. For example, a plasmon polarizer formed on LiNbO3 by overcoating the guide with a Si3N4/Au/Ag thin-film sandwich selectively attenuates the TM mode.87 In some materials it is possible to form waveguides that only support one polarization (the other polarization is not guided and radiates into the substrate). By inserting short (millimeter) lengths of such guides in circuits or alternatively forming entire circuits from these polarizing guides, high extinction can be obtained. For example, annealed proton exchange waveguides (APE ) in LiNbO3 exhibit polarization extinction ratios of at least 60 dB.88 Guided wave devices for splitting light beams are essential for most IOCs. Figure 8 illustrates the two common splitters: a directional coupler and a Y junction. The figure illustrates 3-dB coupling (1X2),

21.22

COMPONENTS

FIGURE 8 Passive directional coupler and Y-branch IO splitter devices.

and by cascading such devices and using variations on the basic designs it is possible to fabricate N r N structures. IO splitters of complexity 8 r 8 are commercially available in glass. The operation of the directional coupler is analogous to the microwave coupler and is described by the coupled mode equations. The coupling strength is exponentially dependent of the ratio of the guide spacing and the effective tail length of the guided mode. Thus when guides are far apart (typically greater than 10 Mm in weakly guided glass and ferroelectric devices) as in the left-most portion of the structure in Fig. 8, there is negligible coupling. When the guides are close together (typically a few microns), power will couple to the adjacent guide. The fraction of power coupled is sinusoidally dependent on the ratio of the interaction length to the coupling length Lc. Lc is typically 0.5–10 mm and is defined as the length for full power transfer from an incident guide to a coupled guide. The 3-dB coupling illustrated requires an interaction length of half Lc.2–6 Operation of this device is symmetric; light incident in any one of the four inputs will result in 3-dB splitting of the output light. However, if coherent light is incident on both input guides simultaneously, the relative power out of the two output guides will depend on the phase and power relationship of the incident signals. The Y splitter illustrated in Fig. 8 operates on a modal evolution principle. Light incident on the junction from the left will divide symmetrically so that the fundamental mode of each output branch is excited. Branching circuit design follows closely from the design of waveguide bends. In low index contrast systems, the Y-junction angle is typically a few degrees and the interaction length is a few millimeter. Operation of this device is not symmetric with respect to loss. If coherent light is incident on both guides from the right, the amount of light exiting the single guide will depend on the power and phase relationship of the optical signals as they enter the junction area. If coherent light is only incident in one arm of the junction from the right it will experience a fundamental 3-dB loss in propagation to the left to the single guide. This is due to the asymmetric modal excitation of the junction. (see the next section, “Active Devices”). An extremely important issue in integrated optics is the matching of the waveguide mode to the mode of the fiber coupled to the guide. Significant mode mismatch causes high insertion loss, whereas a properly designed waveguide mode can have coupling loss well under 1 dB. As illustrated in Eq. (7), the proper design requires optimizing the overlap integral of the optical fields in the two media. Often some sort of index matching between the two materials is also employed to minimize additional reflective losses. Figure 9 illustrates the issue with mode profiles in the two transverse directions for a Ti indiffused guide. In general the IO mode is asymmetric relative to the fiber mode. It should be noted that the loss obtained on a pigtailed fiber-chip joint is also highly determined by the precision and stability of the mechanical method of attaching the fiber to the chip. Most techniques use some sort of carrier block for the fiber (e.g., a Si V-groove) and attach the block to the IO chip.

INTEGRATED OPTICS

21.23

er Fib ide

u veg Wa

(a) TM width mode 9-m strip width Fiber 0.5

0

15

10

5

0

5

10

TM width mode 9-m strip width

1.0 Normalized intensity

Normalized intensity

1.0

0.5

0

15

Width (m)

15

10

5

0

5

10

15

Width (m) (b)

FIGURE 9 Mode-matching illustrated by coupling a Ti-indiffused guide to an optical fiber. Mode profiles are shown both in width and depth for the waveguide.

Performance on commercially available devices is typically 1 Gb/s), dispersive pulse distortion can cause severe degradation with directly modulated lasers unless devices which emit only one longitudinal mode are employed. The incorporation of gratings in DFB and DBR lasers has produced sources with exceptional spectral purity and allow multi-gigabit per second transmission over intermediate distances (< 100 km) in conventional fiber. The advent of fiber amplifiers has enabled longer transmission spans without digital regeneration, and here the unavoidable frequency excursions that result from directly modulating even a single-longitudinal-mode laser again lead to pulse spreading and distortion. In these instances, a continuous wave (CW) laser followed by an external modulator is a preferred source. The integrated DFB/electroabsorption modulator, as discussed in Sec. 23.5, provides such a source. These PICs have demonstrated error-free transmission in excess of 500 km in dispersive fiber at 2.5 Gb/s.104 However, even these devices impose a small residual dynamic phase shift on the signal due to electrorefractive effects accompanying the induced absorption in the modulator. This can be especially problematic with typical percent-level antireflection coatings on the output facet, since this will provide phasevarying optical feedback into the laser and further disrupt its frequency stability. The highest performance digital transmission has been achieved using external LiNbO3 MachZehnder modulators to encode a CW semiconductor laser. Modulators similar to that in Fig. 14 have been constructed to allow separate traveling-wave voltages to be applied to each arm of the modulator in a push-pull configuration. This device design can produce a digitally encoded signal with zero residual phase shift or chirp.105 Such a source has only its information-limited bandwidth and generally provides nearly optimized performance in a dispersive environment. The Mach-Zehnder can also be driven to intentionally provide positive or negative chirping to optimize transmission characteristics in dispersive links.106 Semiconductor lasers have also been monolithically integrated with Mach-Zehnder modulators using electrorefraction from the QCSE.107 Recently there has been significant focus on using phase modulation formats such as DPSK, and higher level formats such as differential quadrature phase-shift keyed (DQPSK) modulation where each bit period can contain 2 bits of information. Such formats not only allow higher spectral efficiency, or information per unit bandwidth in the fiber, but also offer advantages in robustness against impairments from fiber nonlinearity which can seriously degrade the performance of longdistance amplified WDM systems. Just as in the case of the simpler integrated laser-modulator, PIC technology provides a powerful vehicle for the precision control of the pathlengths or phase delays required to encode and decode such formats in transmitters and receivers.108,109

Analog Transmission A second application area that is expected to use a large number of IOCs is analog fiber optic links. Analog fiber optic links are currently being used to transmit cable television (CATV) signals at the trunk and supertrunk level. They are also being used for both commercial and military antenna remoting. Analog fiber optic links are being fielded for these applications because of their low distortion, low loss, and low life-cycle cost when compared to more conventional coaxial cable-based transmission systems. An analog fiber optic link using IOCs is typically configured as shown in Fig. 20. A highpower CW solid-state laser, such as a 150-mW diode-pumped YAG laser operating at 1319 nm, is typically used as the source in order to maximize dynamic range, carrier-to-noise ratio, and link gain. An interferometric modulator, such as a Mach-Zehnder interferometer or a Y-fed balanced bridge modulator, is typically used to modulate the light with the applied RF or microwave signal

INTEGRATED OPTICS

Rin reduction

Diode-pumped YAG laser module (includes pump laser and isolator)

RF

Preamplifier

FIGURE 20 optic link.

21.33

Bias control Fiber optic cable

Electro-optic modulator

Linearization network

Standard configuration for externally modulated analog fiber

via the linear electro-optic effect. Current analog links for CATV signal distribution utilize a 1-GHz Y-fed balanced bridge modulator biased at the quadrature point (linear 3-dB point).110,111 A predistortion circuit is required to minimize third-order distortion associated with the interferometric modulator response. The CATV-modulated signal can be transmitted on both output fibers of the device. Analog links for antenna remoting typically fit into one of two categories. Certain applications require relatively narrow passbands in the UHF region while other microwavecarrier applications require very broadband (several GHz) performance. Y-fed balanced bridge modulators biased at the quadrature point are again used to perform the electrical-to-optical conversion, with the electrode structure tailored to the application. In both narrowband and broadband applications, 20- to 30-dB preamplifiers are typically utilized to minimize the noise figure and maximize the RF gain of the link. Two important modulator parameters are the insertion loss and the half-wave drive voltage which both impact the link gain and dynamic range. Fully packaged Y-fed balanced bridge modulators with 2.5 to 4.0 dB insertion loss are now readily achieved in production for both UHF and microwave bandwidths. A trade-off typically needs to be made between half-wave voltage and bandwidth for both lumped-element and traveling-wave electrode structures. Commercially available lumpedelement LiNbO3 interferometric modulators typically have half-wave voltages of approximately 5 V for 600-MHz, 1-dB bandwidths. Commercially available traveling-wave LiNbO3 interferometric modulators typically have half-wave voltages of approximately 8 V for 12-GHz, 3-dB bandwidths. The half-wave voltages of LiNbO3 traveling-wave modulators can be reduced by more than a factor of two using a velocity-matched electrode structure as described in Ref. 93. In order to be used in practical systems, it is critical that the integrated optical modulators have well-behaved, flat frequency responses in the band of interest. Modulators for CATV signal transmission and UHF antenna remoting typically required that the amplitude response and the phase response be flat to o 0.25 dB and o 2°, respectively. The frequency response of an integrated optical modulator is a function of both the device design and packaging parasitics. Care must be exercised in designing modulators since LiNbO3 is both a piezoelectric and an acousto-optic material. Early LiNbO3 modulators typically had 1 to 3 dB of ripple in the passband due to acoustic mode excitation. When packaging lumped-electrode devices, it is also critical to keep terminating resistors close and wire bonds short to minimize stray inductance and capacitance . When packaging travelingwave modulators, it is critical to control the impedance of the launch, the transitions, the device, and the termination. Through proper device design and packaging, it is possible to achieve well-behaved frequency responses in both lumped-electrode and traveling-wave devices as shown in Fig. 21.

21.34

COMPONENTS

FIGURE 21 Frequency response of APE LiNbO3 microwave interferometric modulator for broadband antenna remoting.

An additional issue that impacts IOC modulator design for analog links is harmonic and intermodulation distortion. Most modulators used in analog links are interferometric in nature with a sinusoidal transfer function. By operating the device precisely at the quadrature point, all even harmonics can be suppressed. Second-harmonic distortion less than 75 dBc is easily achieved using an electronic feedback loop for modulator bias. Alternative techniques are being investigated to laser trim one leg of a Mach-Zehnder to bring the modulator to quadrature. Third-order distortion due to the sinusoidal transfer function of the interferometric modulator also poses a problem, but the transfer functions are very well-behaved and predictable, and this distortion can be suppressed to acceptable levels using electronic predistortion or feed-forward techniques. Forty- and eighty-channel CATV transmitters operating at 1300-nm wavelengths with APE LiNbO3 modulators are currently being fielded. Compared to coaxial transmission which requires transmitters every 500 m, the fiber optic systems can transmit over distances up to 50 km without repeaters. Similarly, externally modulated analog fiber optic links are currently being fielded for military and commercial applications. UHF links with 115 dB/Hz2/3 dynamic range, 4-dB noise figure, and unity gain have been demonstrated using commercially available hardware. These systems will maintain this quality of transmission over temperature ranges of 25 to 50oC. Microwave links with 2 to 18 GHz frequency response, 114-dB/Hz2/3 spurious-free dynamic range, and input noise figure of 22 dB can also be achieved using commercially available hardware.

Switching Arrays of IO switches have been proposed and demonstrated for a variety of space switching and time-multiplexed switching (TMS) applications. In space switching, it is generally necessary to formulate the switches in a nonblocking geometry and the reconfiguration time can be relatively slow (seconds or longer). This requirement led to the development of cross-bar switches in which an N r N switch contains N2 IO switches and 2N 1 stages and from 1 to 2N 1 cross points. Typically N 4 in LiNbO3 and in InP. More recently, much attention in IO switch arrays has shifted to the dilated Benes architecture which is only rearrangeably nonblocking but reconfigurable in short (ns) times suitable for TMS, and has the advantage of requiring substantially fewer switches and a constant number 2 log2 N of cross points.

INTEGRATED OPTICS

100

Waveguides

7

15 100

21.35

7

15

8500 FIGURE 22

Architecture of 8 r8 dilated Benes directional coupler switch array.

A schematic of a two-chip 8 r 8 dilated Benes switch making extensive use of crossovers is shown in Fig. 22.112 The performance of switch arrays of the dilated Benes architecture are much more forgiving than crossbar switches to the degradation of individual switches. The device shown contains 48 delta beta switches driven by a single-voltage arrangement. The switching voltage at 1.3 Mm was 9.4 0.2 V, the insertion loss varied from 8 to 11 dB (93 percent of the 256 paths through the switch were within o1 dB), the cross-talk levels in individual switches ranged from 22 to 45 dB, and the reconfiguration time was 2.5 ns. Larger 16 r 16 switches have also been demonstrated.113 An advantage of these types of IO switch arrays is that they are data rate transparent. That is, once the switch is reconfigured, the data stream through the device is simply the passage of light and can easily be multi-gigabit. Crossbar switches are now commercially available and other types of arrays continue to be explored.

Fiber Optic Gyroscopes Another application that may require large quantities of integrated optical circuits is the fiber optic gyroscope (FOG)114–119. A FOG is one form of a Sagnac interferometer, in which a rotation rate results in a phase shift between clockwise- and counterclockwise-propagating optical fields. The most frequently utilized FOG configuration, which was first proposed by workers at Thomson CSF in the mid 1980s,120 is presented in Fig. 23. FOG IOCs are typically fabricated in LiNbO3 using the annealed proton exchange (APE) process66 although titanium-diffused IOCs with surface plasmon polarizers have also been utilized. The IOC performs four primary functions in the fiber gyroscope. First, the Y-junction serves as the loop coupler splitting and recombining the clockwise- and counterclockwise-propagating optical fields. Second, the IOC functions as a high-quality polarizer. Third, a 90° phase dither (at the eigen frequency of the fiber coil) is typically applied to one of the integrated optical phase modulators.

21.36

COMPONENTS

!!% $#! # "$!

$ !

!

#

"$#! !##

FIGURE 23 Standard configuration for fiber optic gyroscope incorporating a three-port integrated optical circuit.

This approach keeps the Sagnac interferometer biased at the 3-dB point where it is linear and most sensitive to rotation. Finally, in a closed-loop FOG configuration, one of the phase modulators functions as a frequency shifter. A serrodyne signal (saw-tooth wave) is applied to the phase modulator to effectively cancel the shift due to the rotation. The output signal from a fiber gyro at rest is the sum of white receiver noise, primarily dependent on the amount of optical power arriving at the detector, and an additional long-term drift of the mean value. The long-term drift in a FOG associated with a residual lack of reciprocity typically limits the sensitivity of the FOG to measure low rotation rates. Another important characteristic of a gyro is the scale factor, which is a measure of the linearity between the actual rotation rate and the gyro response. The critical performance parameters for a FOG IOC are presented in Table 1. The performance of 800- and 1300-nm APE LiNbO3 FOG IOCs that are currently in production is also presented in this table. One application of the FOG is inertial guidance, requiring a FOG with a bias drift < 0.01°/h and a scale factor accuracy < 5 ppm. A 1300-nm LED or an erbium-doped fiber is typically used as the light source. A large coil of polarization-maintaining fiber (typically 1 km of fiber wound in a 15 to 20 cm diameter coil) and precise source spectral stability are required to achieve the desired sensitivity. The fiber is typically wound in a quadrupole configuration to minimize sensitivity to temperature gradients. With recent improvements in optical sources, integrated optics, and fiber coil-winding technology, it is now possible to achieve inertial grade FOG performance over a temperature range of 55 to 95oC.

TABLE 1 Critical Performance Parameters for 800- and 1300-nm APE LiNbO3 FOG IOCs. Listed Values Are Maintained over a Temperature Range of 55 to 95°C and during Vibration up to 15 Grms. Performance Parameter

1300-nm IOCs

800-nm IOCs

Insertion loss (pigtailed) Polarization extinction Y-junction split ratio (pigtailed) Polarization crosstalk at fiber-waveguide interfaces Optical back reflection Half-wave voltage Residual intensity modulation

3 dB 70 dB 48/52 to 52/48 < 30 dB

4 dB 60 dB 45/55 to 55/45 < 25 dB

< 65 dB 4.5 V 0.02%

< 65 dB 3.5 V 0.05%

INTEGRATED OPTICS

21.37

A second tactical-grade FOG design is more typical to aerospace applications, with bias drift and scale factor accuracy requirements ranging from 0.1 to 10°/h and 10 to 1000 ppm, respectively. These systems are typically designed for operation at 810 to 830 nm to make use of low-cost multimode 830-nm AlGaAs laser diodes as used in consumer electronic products. These systems typically utilize 2to 5-cm-diameter fiber coils with 100 to 200 m of either polarization-maintaining or single-mode fiber. A third very low-cost, low-grade FOG design for automotive navigation is also nearing production. The required bias drift is only 1000°/h, and a closed-loop configuration is unneccessary since the scale factor accuracy is only 0.1 percent. Current designs to achieve low cost include low-performance IOCs, laser, and fiber couplers, fully automated FOG assembly and test procedures, and only approximately 50 m of single-mode fiber. More advanced IOCs, including four-port designs that integrate the source/detector coupler into the IOC, are also being considered to reduce component count.

WDM Systems Wavelength division multiplexing (WDM), by encoding parallel data streams at different wavelengths on the same fiber, offers a technique to increase transmission capacity, or increase networking or switching flexibility, without requiring higher speed electronics to process each channel. As noted earlier, IO is an enabling technology in WDM demultiplexer design. However, due to the large number of component interconnections at the transmit and receive end of a WDM link, PIC technology offers great promise in cost reduction for both WDM transmitters and receivers by additionally eliminating separate packages for lasers, amplifiers, detectors, and the like. One key application of PICs for WDM systems is the stable and efficient longitudinal connection of optical elements required to form a tunable laser. The tunable Bragg laser was briefly discussed earlier in the description of the balanced heterodyne receiver PIC. Figure 24 below shows a more complex sampled-grating distributed Bragg reflector PIC. Here the reflectivity of a grating with

EA modulator

Amplifier

Front mirror

Gain

Phase

Rear mirror

Light out AR SG-DBR laser

FIGURE 24 Tunable sampled-grating distributed Bragg reflector (SG-DBR) laser with integrated semiconductor optical amplifier and electroabsorption modulator.

21.38

COMPONENTS

FIGURE 25 Integrated 40-channel optical channel monitor, comprising 9 different AWG demultiplexers and 40 photodetectors. (By III-V Photonics, Inc.)

sections periodically omitted provides a comb of reflection bands, and if the front and back combs have different spacings in frequency, the laser can tune over a wide range with a small index change because the laser will operate only at a frequency where the front and back reflection bands coincide in frequency.121 Tuning is accomplished by changing the index of the front or back section by current injection, for example. Because the InP materials system readily allows detectors as well as high-quality passives, the AWG demultiplexer discussed earlier can be combined with an integrated detector for each wavelength output port to make a single-chip WDM receiver. Figure 25 illustrates a 40-channel WDM receiver PIC, comprising nine different AWG demultiplexers and 40 photodetectors within a 4.6 r 4.8 mm chip.122 This PIC had only 4-dB on-chip loss and provided less than 35-dB cross talk between different wavelength channels. Figure 26 shows the layout of a WDM transmission PIC, which is perhaps the most sophisticated example of commercially deployed active IO demonstrated to date.123 Following the concept initially introduced by Aiki et al.124 in combining multiple DFB lasers, this PIC includes 10 frequency tunable DFB lasers with wavelength locking function, 10 electroabsorption modulators, 10 monitor detectors, 10 adjustable attenuators, an AWG for low-loss, frequency-selective combining into a single waveguide output port. This PIC provides a 100-Gb/s transmission capacity from a single chip, and together with a matching 10-channel WDM receiver PIC has been commercially deployed in telecommunication networks worldwide.123 These PICs have demonstrated remarkable performance, and offer significant improvements in cost, size, power, and reliability compared to discrete component solutions. Silicon Photonics Transceivers As noted in the earlier discussions of IO materials, silicon has gained significant attention in recent years due to its ability to harness highly advanced CMOS fabrication technologies, combined with the very high index contrast available and effective modulation using the plasma index change from free carriers. This modulation is most effectively done at high speed using either a reversebiased P-N junction, or using a structure similar to a field-effect-transistor (FET) where charge can accumulate on a gate oxide within the optical mode. Both of these designs have been demonstrated

INTEGRATED OPTICS

21.39

10 10 Gb/s Electrical input

Optical output 1... 10 10 10 Gb/s

AWG multiplexer

CH1

DC control & bias

ar r ay DF Ba rra OP y M ar r ay

M

Tu n

ab le

EA

VO Aa

r ra y

CH10

FIGURE 26 Chip architecture for commercially implemented PIC with nearly 50 elements, including 10 frequency tunable DFB lasers with wavelength-locking function, 10 electroabsorption modulators, 10 monitor detectors, 10 adjustable attenuators, and an integrated AWG combiner.

by several organizations125,126 and commercialized with fabrication conducted within a CMOS electronics foundry using nearly standard IC fabrication steps. PICs have been demonstrated that include 10 Gb/s transmitters and recievers, and the ability to include CMOS electronics has also allowed for on-chip modulator drivers, preamplifiers, clock and data-recovery circuits, and even pseudo-random bit-stream generators for on-chip testing.46 Commercial silicon photonics transceivers have demonstrated low-power operation, and also 40-Gb/s data rates using integration to achieve parallelism with only one laser source.

21.7

FUTURE TRENDS

Shift from R&D to Manufacturing Integrated optical circuit technology has now advanced from R&D into manufacturing as noted in the examples given. LiNbO3 modulators and integrated DFB/electroabsorption modulators have been critical, high-volume products for more than a decade. There are now several companies producing more complex IO and PIC products in LiNbO3, Si/SiO2, and InP in moderate volumes (several thousand devices each per year) with considerable success in reducing manufacturing costs and achieving high reliability. The majority of the LiNbO3 devices described in this chapter can be fabricated using either existing or slightly modified semiconductor processing equipment. Clean room requirements are not nearly as tight as what is required for VLSI and MMIC wafer fabrication. Production yields of LiNbO3 integrated optical circuits have been obtained well in excess of 95 percent. The majority of the defects were initially mechanical in nature (probes scratching electrodes or fibers damaging polished end faces), but these have been minimized as the processes have become more automated. At this time, it appears that all wafer-processing operations should be easily scalable to wafer batch processing, including the end-face optical polishing. The majority of the cost of an integrated optical circuit is currently associated with fiber attachment, packaging, and final testing. While these operations are

21.40

COMPONENTS

not fundamentally expensive, the limited production volumes do not always justify the capital expenditure required to fully automated processes with robotics and machine vision. The second area that has seen improvement is device reliability. First-generation commercial integrated optic products were plagued by premature failures and poor performance over temperature. The majority of these problems resulted from poor fiber attachment and packaging techniques. These problems have been remedied by carefully selecting compatible material systems and incorporating proven hybrid electronic packaging techniques. For example, commercially available LiNbO3 integrated optical circuits can be thermally cycled hundreds of times from 65 to 125 oC with less than 1 dB variation in insertion loss. The devices can also withstand 30 Grms random vibration testing. Long-term aging effects have been addressed by identifying potential failure mechanisms and then by developing physical models of these failures. LiNbO3 integrated optical circuit reliability has now achieved a level where devices can be certified for operational lifetimes in excess of 25 years for properly designed, fabricated, and assembled devices. Obviously, this level of performance can only be guaranteed by fabricating the devices in well-controlled, welldocumented production environments. In the semiconductor area, reliability certification technology for PICs borrows heavily from the more mature level models and aging methodologies developed for semiconductor lasers. This includes sequences of purges at high currents or voltages combined with high temperatures, together with extended accelerated aging to identify activation energies and aging rates for various degradation mechanisms. Integrated laser modulators have been deployed in high volume with outstanding reliability, also easily in excess of 25 years including wavelength stability for WDM lasers. More complex PICs have also shown extremely high reliability, and as with the LiNbO3 IO devices, any failures that do occur are most often associated with poorly designed packaging of the device rather than the PIC chip itself. This again points to one of the fundamental advantages of PICs, where many individual component packages are eliminated by lithographic connections and only one package is required for the final PIC or IO device.

Advanced Integration and New PIC Applications The most active areas of IO and PIC research today are in telecommunications, datacom, and sensor applications. In InP materials, for example, there has been extensive work on all-optical networking technologies where wavelength-converter PICs have been demonstrated that encode incoming data onto a new wavelength determined by an on-chip tunable laser in conjunction with nonlinear Mach-Zehnder interferometers with SOA gain elements.127,128 The complexity, speed, and wavelength count of high speed WDM transmit and receive PICs continues to grow, with research demonstrations of 1.6 Tb/s from a single chip combining 40 tunable DFB lasers with 40 electroabsorption modulators running at 40 Gb/s each.129 As transmission technology continues to evolve in sophistication and spectral efficiency, work on advanced modulation formats moves forward aggressively. A resurgence of interest in heterodyne-detection technology has shown that many system impairments can be effectively removed in the electronic domain when the information is received linearly in the optical electric field. It is quite likely that information coding in the optical domain will progress to a level that is now commonplace in the world of RF and cell phone technology. Silicon photonics has made remarkable progress in a short period, and whereas true OEIC integration with electronics has only had very limited penetration in III-V materials, the potential for including sophisticated electronics in silicon has provided a strong incentive to explore and commercialize this technology. In datacom, for example, the proximity of driver circuits and modulator devices allows for low power by obviating the usual electrical transmission line and terminating resistor approaches. For sensors, the ability to combine preamplification, analog-to-digital converters, and digital signal processors is extremely compelling, and biochem sensors that can detect frequency shifts in suitably sensitized resonators are now emerging. It is also likely that the mapping of photonic functions into commercial CMOS electronics foundries will become more common, and if suitable design tools become more widely available, the reduced barriers to using silicon photonics solutions may lead to a growth of new market applications.

INTEGRATED OPTICS

21.41

Analogies with massively repetitive logic gates in electronics integration are not particularly good, but there is indeed now a proliferation of applications where stable designs have emerged that require significant numbers of interconnected optical components. In these areas, the improvements in size, cost, power, and reliability are extremely compelling, and today it is safe to say that IO and PIC technologies have earned a secure footing in the world of optics.

REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22.

S. E. Miller, “Integrated Optics: An Introduction,” Bell Labs Tech. J. 48(7):2059–2070 (1969). T. Tamir, ed., Guided-Wave Optoelectronics, 2d ed., New York: Springer-Verlag (1990). R. G. Hunsperger, Integrated Optics: Theory and Technology, Berlin: Springer-Verlag (1985). T. Tamir, ed., Integrated Optics, 2d ed., New York: Springer-Verlag (1979). L. D. Hutchenson, ed., Integrated Optical Circuits and Components, New York: Marcel Dekker (1987). K. Okamoto, Fundamentals of Optical Waveguides, San Diego, Calif.: Academic Press (2000). H. Nishihara, M. Haruna, and T. Suhara, Optical Integrated Circuits, New York: McGraw-Hill (1985). D. Marcuse, Theory of Dielectric Optical Waveguide, 2d ed., San Diego, Calif.: Academic Press Inc., pp. 307–318 (1991). A. Yariv and P. Yeh, “Optical Waves in Crystals,” New York, John Wiley & Sons Inc., pp. 177–201, 425–459 (1984). I. P. Kaminow and S. E. Miller, ed., Optical Fiber Telecommunications II, San Diego, Calif.: Academic Press (1985). I. P. Kaminow and T. L. Koch, ed., Optical Fiber Telecommunications III, San Diego, Calif.: Academic Press (1997). I. P. Kaminow and T. Li, ed., Optical Fiber Telecommunications IV, San Diego, Calif.: Academic Press (2002). Special Issue on Photonic Devices and Circuits, IEEE J. Quantum Electron. QE-27(3) (1991). Special Issue on Integrated Optics, J. Lightwave Technol. 6(6) (1988). Special Issue on Integrated Optics, IEEE J. Quantum Electron. QE-22(6) (1986). Digests of IEEE/OSA Conferences on Integrated Photonics Research, 1990–Present, OSA, Washington, D.C. Digests of IEEE Conference on Integrated and Guided Wave Optics, 1972–1989, OSA, Washington, D.C. G. B. Hocker and W. K. Burns, “Mode Dispersion in Diffused Channel Waveguides by the Effective Index Method,” Appl. Opt. 16:113–118 (1977). J. F. Lotspeich, “Explicit General Eigenvalue Solutions for Dielectric Slab Waveguides,” Appl. Opt. 14:327–335 (1975). M. D. Feit and J. A. Fleck, Jr., “Light Propagation in Graded-Index Optical Fibers,” Appl. Opt. 17:3990–3998 (1978). M. J. Weber, ed., CRC Handbook of Laser Science and Technology, Vol. IV, Optical Materials, Part 2, Boca Raton: CRC Press Inc. (1986). M. A. Afromowitz, “Refractive Index of Ga 1-x Alx As,” Solid State Commun. 15:59–63 (1974).

23. C. H. Henry, L. F. Johnson, R. A. Logan, and D. P. Clarke, “Determination of the Refractive Index of InGaAsP Epitaxial Layers by Mode Line Luminescence Spectroscopy,” IEEE J. Quantum Electron. QE-21:1887–1892 (1985). 24. T. Maldonado, ‘‘Electro-Optics Modulators,’’ in vol. V, chap. 7 of this Handbook. 25. I. P. Kaminow, An Introduction to Electro-Optic Devices, Orlando: Academic Press (1974). 26. T. S. Moss, G. S. Burrell, and B. Ellis, Semiconductor Opto-Electronics, New York: John Wiley & Sons Inc. (1973). 27. K. Tharmalingam, “Optical Absorption in the Presence of a Uniform Field,” Phys. Rev. 130:2204–2206 (1963).

21.42

COMPONENTS

28. D. A. B. Miller, J. S. Weiner, and D. S. Chemla, “Electric-Field Dependence of Linear Optical Properties Quantum Well Structures: Waveguide Electroabsorption and Sum Rules,” IEEE J. Quantum Electron. QE-22:1816–1830 (1986). 29. J. E. Zucker, I. Bar-Joseph, G. Sucha, U. Koren, B. I. Miller, and D. S. Chemla, “Electrorefraction in GaInAs/ InP Multiple Quantum Well Heterostructures,” Electron. Lett. 24:458–460 (1988). 30. Y. -H. Kuo, Y. -K. Lee, Y. Ge, S. Ren, J. E. Roth, T. I. Kamins, D. A. B. Miller, and J. S. Harris, “Strong QuantumConfined Stark Effect in Germanium Quantum-Well Structures on Silicon,” Nature 437:1334–1336 (2005). 31. C. L. Tang, ‘‘Nonlinear Optics,’’ in vol. IV, chap. 10 of this Handbook. 32. M. Yamada, N. Nada, M. Saitoh, and K. Watanabe, “First-Order Quasi-Phase Matched LiNbO3 Waveguide Periodically Poled by Applying an External Field for Efficient Blue Second-Harmonic Generation,” Appl. Phys. Lett. 62:435–436 (1993). 33. H. A. Haus, E. P. Ippen, and F. J. Leonberger, “Nonlinear Optical Waveguide Devices,” in Optical Signal Processing, J. L. Horner (ed.), Orlando, FL: Academic Press, pp. 245–267 (1987). 34. M. A. Foster, A. C. Turner, J. E. Sharping, B. S. Schmidt, M. Lipson, and A. L. Gaeta, “Broad-Band Optical Parametric Gain on a Silicon Photonic Chip,” Nature 441:960–963 (2006). 35. T. K. Findakly, “Glass Waveguides by Ion-Exchange: A Review,” Opt. Eng. 24:244 (1985). 36. R. V. Ramaswamy and R. Srivastava, “Ion-Exchanged Glass Waveguides: A Review,” J. Lightwave Technol. LT-6:984–1002 (1988). 37. T. Izawa and H. Nakogome, “Optical Waveguide formed by Electrically Induced Migration of Ions in Glass Plates,” Appl. Phys. Lett. 21:584 (1972). 38. M. Kawachi, “Silica Waveguides on Silicon and their Applications to Integrated-Optic Components,” Opt. Quant. Electron. 22:391–416 (1990). 39. H. Takahashi, Y. Ohnori, and M. Kawachi, “Design and Fabrication of Silica-Based Integrated-Optic 1 × 128 Power Splitter,” Electron. Lett. 27:2131–2133 (1991). 40. C. H. Henry, G. E. Blonder, and R. F. Kazarinov, “Glass Waveguides on Silicon for Hybrid Optical Packaging,” J. Lightwave Technol. 7:1530–1539 (1989). 41. R. Adar, Y. Shani, C. H. Henry, R. C. Kistler, G. E. Blonder, and N. A. Olsson, “Measurement of Very LowLoss Silica on Silicon Waveguides with a Ring Resonator,” Appl. Phys. Lett. 58(5):444–445 (1991). 42. Y. Shani, C. H. Henry, R. C. Kistler, K. J. Orlowski, and D. A. Ackerman, “Efficient Coupling of a Semiconductor Laser to an Optical Fiber by Means of a Tapered Waveguide on Silicon,” Appl. Phys. Lett. 55:2389–2391 (1989). 43. R. Soref, “The Past, Present, and Future of Silicon Photonics,” IEEE J. Quantum Electron. QE-12:1678–1687 (2006). 44. G. T. Reed and A. P. Knights, Silicon Photonics: An Introduction, West Sussex, England: John Wiley & Sons Ltd. (2004). 45. L. Pavesi and D. L. Lockwood, eds., Silicon Photonics, Berlin Heidelberg: Spinger-Verlag (2004). 46. C. Gunn and T. L. Koch, “Silicon Photonics,” in Optical Fiber Telecommunications V, I. P. Kaminow, T. Li, and A. E. Willner, (eds.), San Diego, Calif.: Academic Press (2008). 47. M. Bruel, “Silicon on Insulator Material Technology,” Electron. Lett. 31:1201–1202 (1995). 48. U. Fischer, T. Zinke, J.-R. Kropp, F. Amdt, and K. Petermann, “0.1 dB/cm Waveguide Losses in Single-Mode SOI Rib Waveguides,” IEEE Phot. Tech. Lett. 8(5):647–648 (1996). 49. M. A. Webster, R. M. Pafchek, G. Sukumaran, and T. L. Koch, “Low-Loss Thin SOI Waveguides and High-Q Ring Resonators,” Paper FTuV3 in Proceedings of the Optical Society of America Annual Meeting, Tucson, (Oct. 18, 2005). 50. R. C. Alferness, “Guided-Wave Devices for Optical Communications,” IEEE J. Quantum Electron. QE-17(6):946–959 (1981). 51. H. F. Taylor, “Applications of Guided-Wave Optics in Signal Processing and Sensing,” Proc. IEEE 75(11):1524–1535 (1987). 52. E. Voges and A. Neyer, “Integrated-Optic Devices on LiNbO3 for Optical Communication,” J. Lightwave Technol. LT-5(9):1229–1238 (1987). 53. L. Thylen, “Integrated Optics in LiNbO3: Recent Developments in Devices for Telecommunications,” J. Lightwave Technol. LT-6(6):847–861 (1988).

INTEGRATED OPTICS

21.43

54. R. C. Alferness, “Waveguide Electrooptic Switch Arrays,” IEEE J. Selected Areas in Communications 6(7):1117–1130 (1988). 55. R. V. Schmidt and I.P. Kaminow, “Metal Diffused Optical Waveguides in LiNbO3,” Appl. Phys. Lett. 15:458–460 (1974). 56. M. Fukuma and J. Noda, “Optical Properties of Ti-Diffused LiNbO3 Strip Waveguides and their Coupling-toa-Fiber Characteristics,” Appl. Opt. 19:591–597 (1980). 57. S. K. Korotky and R. C. Alferness, “Ti:LiNbO3 Integrated Optic Technology,” Integrated Optical Circuits and Components, L. D. Hutcheson, (ed.), New York: Marcel Dekker (1987). 58. R. J. Esdaile, “Closed-Tube Control of Out-Diffusion during Fabrication of Optical Waveguides in LiNbO3,” Appl. Phys. Lett. 33:733–734 (1978). 59. J. L. Jackel, V. Ramaswamy, and S. P. Lyman, “Elimination of Out-Diffused Surface Guiding in TitaniumDiffused LiNbO3,” Appl. Phys. Lett. 38:509–511 (1981). 60. A. M. Glass, “The Photorefractive Effect,” Opt. Eng. 17:470–479 (1978). 61. J. L. Jackel, C. E. Rice, and J. J. Veselka, “Proton Exchange for High-Index Waveguides in LiNbO3,” Appl. Phys. Lett. 47:607–608 (1982). 62. P. G. Suchoski, T. K. Findakly, and F. J. Leonberger, “Stable Low-Loss Proton-Exchanged LiNbO3 Waveguide Devices with No Electrooptic Degradation,” Opt. Lett. 13:1050–1052 (1988). 63. P. G. Suchoski, T. K. Findakly, and F. J. Leonberger, “Low-Loss High Extinction Polarizers Fabricated in LiNbO3 by Proton Exchange,” Opt. Lett. 13:172–174 (1988). 64. T. K. Findakly, P. G. Suchoski, and F. J. Leonberger, “High-Quality LiTaO3 Integrated Optical Waveguides and Devices Fabricated by the Annealed-Proton-Exchange Technique,” Opt. Lett. 13:797–799 (1988). 65. See, for example, Conference Proceedings of 20th International Conference on Indium Phosphide and Related Materials, IEEE, Versailles (2008). 66. R. Clawson, “Reference Guide to Chemical Etching of InGaAsP and In0.53Ga0.47As Semiconductors,” NOSC Tech. Note 1206, San Diego, Calif. (1982). 67. S. Adachi and H. Kawaguchi, “Chemical Etching Characteristics of (001) InP,” J. Electrochem. Soc.: Sol. St. Sci. and Tech. 128:1342–1349 (1981). 68. L. A. Coldren, K. Furuya, B. I. Miller, and J. A. Rentschler, “Etched Mirror and Groove-Coupled GaInAsP/ InP Laser Devices for Integrated Optics,” IEEE J. Quantum Electron. QE-18:1679–1688 (1982). 69. L. A. Coldren and J. A. Rentschler, “Directional Reactive-Ion-Etching of InP with Cl2 Containing Gases,” J. Vac. Sci. Technol.19:225–230 (1981). 70. J. W. McNabb, H. G. Craighead, and H. Temkin, “Anisotropic Reactive Ion Etching of InP in Methane/ Hydrogen Based Plasma,” Paper TuD15 in Tech. Digest of Integrated Photonics Research 1991, Monterey, pp. 26–27. 71. P. Buchmann and H. Kaufmann, “GaAs Single-Mode Rib Waveguides with Reactive Ion-Etched Totally Reflecting Corner Mirrors,” IEEE J. Lightwave Technol. LT-3:785–788 (1985). 72. S. Oku, M. Okayasu, and M. Ikeda, “Low-Threshold CW Operation of Square-Shaped Semiconductor Ringe Lasers (Orbiter Lasers),” IEEE Phot. Tech. Lett. 3:588–590 (1991). 73. W. J. Grande, J. E. Johnson, and C. L. Tang, “AlGaAs Photonic Integrated Circuits Fabricated Using Chemically Assisted Ion Beam Etching,” Paper. OE10.4/ThUU4 in Conf. Digest of IEEE LEOS Annual Meeting, Boston, p. 169 (1990). 74. J. B. D. Soole, A. Scherer, H. P. LeBlanc, N. C. Andreadakis, R. Bhat, and M. A. Koza, “Monolithic InP-Based Grating Spectrometer for Wavelength-Division Multiplexed Systems at 1.5 μm,” Electron. Lett. 27:132–134 (1991). 75. C. Cremer, G. Ebbinghaus, G. Heise, R. Muller-Nawrath, M. Shienle, and L. Stoll, “Grating Spectrograph in InGaAsP/InP for Dense Wavelength Division Multiplexing,” Appl. Phys. Lett. 59:627–629 (1991). 76. T. L. Koch and U. Koren, “Semiconductor Photonic Integrated Circuits,” IEEE J. Quantum Electron. QE-27:641–653 (1991). 77. M. Aoki, M. Suzuki, T. Taniwatari, H. Sano, and T. Kawano, “New Photonic Device Integration by SelectiveArea MOVPE and its Application to Optical Modulator/Laser Integration,” Microwave and Opt. Technol. Lett. 7(3):132–139 (1994). 78. J. H. Marsh, “Quantum Well Intermixing,” Semicon. Sci. Technol. 8:1136–1155 (1993).

21.44

COMPONENTS

79. R. C. Alferness, U. Koren, L. L. Buhl, B. I. Miller, M. G. Young, T. L. Koch, G. Raybon, and C. A. Burrus, “Broadly Tunable InGaAsP/InP Laser Based on a Vertical Coupler Filter with 57-nm Tuning Range,” Appl. Phys. Lett. 60:3209–3211 (1992). 80. E. Van Tomme, P. P. Van Daele, R.G. Baets, and P.E. Lagasse, “Integrated Optic Devies Based on Nonlinear Optical Polymers,” IEEE J. Quantum Electron. QE-27:778 (1991). 81. C. C. Teng, “Traveling-Wave Polymeric Optical Intensity Modulator with More than 40 GHz of 3-dB Electrical Bandwidth,” Appl. Phys. Lett. 60:1538–1540 (1992). 82. D. Chen, H. R. Fetterman, A. Chen, W. H. Steier, L. R. Dalton, W. Wang, and Y. Shi, “Demonstration of 110 GHz Electro-Optic Polymer Modulators,” Appl. Phys. Lett. 70(25):3335–3337 (1997). 83. Y. -H. Kuo, J. Luo, W. H. Steier, and A. K. -Y. Jen, “Enhanced Thermal Stability of Electrooptic Polymer Modulators Using the Diels–Alder Crosslinkable Polymer,” IEEE Phot. Tech. Lett. 18(1):175–177 (2006). 84. D. Marcuse, “Bending Losses of the Symmetric Slab Waveguide,” Bell Syst. Tech. J. 50:2551–2563 (1971). 85. M. Heiblum and J. H. Harris, “Analysis of Curved Waveguides by Conformal Transformation,” IEEE J. Quantum Electron. QE-11:75–83 (1975). 86. Yu. M. Vlasov, IBM Corporation. Private communication. 87. J. N. Polkay and G. L. Mitchell, “Metal-Clad Planar Dielectric Waveguide for Integrated Optics,” J. Opt. Soc. Am. 64:274–279 (1974). 88. P. G. Suchoski, T. K. Findakly, and F. J. Leonberger, “Stable Low-Loss Proton-Exchanged LiNbO3 Waveguide Devices with No Electro-Optic Degradation,” Opt. Lett. 13:1050 (1988). 89. M. K. Smit, “New Focusing and Dispersive Planar Component Based on an Optical Phased Array,” Electron. Lett. 7(24):385–386 (1988). 90. H. Takahashi, S. Suzuki, K. Kato, and I. Nishi, “Arrayed-Waveguide Grating for Wavelength Division Multi/ Demultiplexer with Nanometre Resolution,” Electron. Lett. 26(2):87–88 (1990). 91. C. Dragone, “An N X N Optical Multiplexer Using a Planar Arrangement of Two Star Couplers,” IEEE Photon. Tech. Lett. 3:812–815 (1991). 92. K. Takada, M. Abe, T. Shibata, and K. Okamoto, “5 GHz-Spaced 4200-Channel Two-Stage Tandem Demultiplexer for Ultra-Multi-Wavelength Light Source Using Supercontinuum Generation,” Electron. Lett. 38(12):575–576 (2002). 93. B. E. Little, S. T. Chu, P. P. Absil, J. V. Hryniewicz, F. G. Johnson, F. Seiferth, D. Gill, V. Van, O. King, and M. Trakalo, “Very High-Order Microring Resonator Filters for WDM Applications,” IEEE Phot. Tech. Lett. 16(10):2263–2265 (2004). 94. D.W. Dolfi and T.R. Ranganath, “50 GHz Velocity-Matched Broad Wavelength Ti:LiNbO3 Mach-Zehnder Modulator with Multimode Active Section,” Electron. Lett. 28:1197–1198 (1992). 95. R. A. Becker, C. E. Woodward, F. J. Leonberger, and R. C. Williamson, “Wideband Electrooptic Guided-Wave Analog-to-Digital Converters,” IEEE Proc. 72:802 (1984). 96. R. C. Alferness, “Polarization-Independent Optical Directional Coupler Switch Using Weighted Coupling,” Appl. Phys. Lett. 35:748–750 (1979). 97. F. Heismann, L. L. Buhl, and R. C. Alferness, “Electro-Optically Tunable Narrowband Ti:LiNbO3 Wavelength Filters,” Electron. Lett. 23:572–573 (1987). 98. M. Suzuki, Y. Noda, H. Tanaka, S. Akiba, Y. Kushiro, and H. Isshiki, “Monolithic Integration of InGaAsP/InP Distributed Feedback Laser and Electroabsorption Modulator by Vapor Phase Epitaxy,” IEEE J. Lightwave Technol. LT-5:1277–1285 (1987). 99. H. Soda, M. Furutsu, K. Sato, N. Okazaki, Y. Yamazaki, H. Nishimoto, and H. Ishikawa, “High-Power and High-Speed Semi-Insulating BH Structure Monolithic Electroabsorption Modulator/DFB Laser Light Source,” Electron Lett. 26:9–10 (1990). 100. G. P. Agrawal and N. K. Dutta, Long-Wavelength Semiconductor Lasers, New York: Van Nostrand Reinhold Company Inc. (1986). 101. R. Parke, D. F. Welch, A. Hardy, R. Lang, D. Muhuys, S. O’Brien, K. Dzurko, and D. Scifres, “2.0 W CW, Diffraction-Limited Operation of a Monolithically Integrated Master Oscillator Power Amplfier,” IEEE Phot. Tech. Lett. 5:297–300 (1993). 102. T. L. Koch, F. S. Choa, U. Koren, R. P. Gnall, F. Hernandez-Gil, C. A. Burrus, M. G. Young, M. Oron, and B. I. Miller, “Balanced Operation of an InGaAs/InGaAsP Multiple-Quantum-Well Integrated Heterodyne Receiver,” IEEE Phot. Tech. Lett. 2:577–580 (1990).

INTEGRATED OPTICS

21.45

103. T. L. Koch and U. Koren, “Semiconductor Photonic Integrated Circuits,” IEEE J. Quantum Electron. QE-27:641–653 (1991). 104. P. D. Magill, K. C. Reichman, R. Jopson, R. Derosier, U. Koren, B. I. Miller, M. Young, and B. Tell, “1.3 4BITsKMS 4RANSMISSION 4HROUGH .ON $ISPERSION 3HIFTED &IBER BY $IRECT -ODULATION OF A -ONOLITHIC Modulator/Laser,” Paper PD9 in Technical Digest of OFC ’92, San Jose, CA: OSA, pp. 347–350 (1992). 105. S. K. Korotky, J. J. Vaselka, C. T. Kemmerer, W. J. Minford, D. T. Moser, J. E. Watson, C. A. Mattoe, and P. L. Stoddard, “High-Speed, Low Power Optical Modulator with Adjustable Chirp Parameter,” Paper TuG2 in Tech. Digest of 1991 Integrated Photonics Research Topical Meeting, OSA, p. 53 (1991). 106. H. Gnauck, S. K. Korotky, J. J. Vaselka, J. Nagel, C. T. Kemmerer, W. J. Minford, and D. T. Moser, “Dispersion Penalty Reduction Using an Optical Modulator with Adjustable Chirp,” IEEE Phot. Tech. Lett. 3:916–918 (1991). 107. J. E. Zucker, K. L. Jones, M. A. Newkirk, R. P. Gnall, B. I. Miller, M. G. Young, U. Koren, C. A. Burrus, and B. Tell, “Quantum Well Interfermontric Modulator Monolithically Integrated with 1.55 μm Tunable Distributed Bragg Reflector Laser,” Electron. Lett. 28:1888–1889 (1992). 108. C. R. Doerr, L. Zhang, S. Chandrasekhar, and L. L. Buhl, “Monolithic DQPSK Receiver in InP With Low Polarization Sensitivity,” Phot. Tech. Lett. 19(21):1765–1767 (2007). 109. S. Corzine, P. Evans, M. Kato, G. He, M. Fisher, M. Raburn, A. Dentai, et al., “10-Channel x 40Gb/s per Channel DQPSK Monolithically Integrated InP-Based Transmitter PIC,” Paper PDP18 in the Tech. Digest of the 2008 Optical Fiber Communication Conference (OFC 2008), San Diego, Calif. (2008). 110. G. S. Maurer, G. T. Tremblay, and S. R. McKenzie, “Transmitter Design Alternatives for CATV Distribution in FITL Deployment,” Proc. SPIE OE/Fibers (1992). 111. R. B. Childs and V. A. O’Byrne, “Multichannel AM Video Transmission Using a High Power Nd:YAG Laser and a Linearized External Modulator,” IEEE J. on Selected Areas of Comm. 8:1369 (1990). 112. E. Watson, M. A. Milbrodt, K. Bahadori, M. F. Dautartas, C. T. Kemmerer, D. T. Moser, A. W. Schelling, T. O. Murphy, J. J. Veselka, and D. A. Herr, “A Low-Voltage 8 × 8 TiLiNbO3 Switch with a Dilated-Benes Architecture,” IEEE/OSA J. Lightwave Technol. 8:794–801 (1990). 113. S. S. Bergstein, A. F. Ambrose, B. H. Lee, M. T. Fatehi, E. J. Murphy, T. O. Murphy, G. W. Richards, et al., “A Fully Implemented Strickly Non-Blocking 16 × 16 Photonic Switching System,” Paper PD30 in Tech. Digest of OFC/IOOC ’93, San Jose, Calif, pp. 123–126 (1993). 114. H. Lefevre, The Fiber-Optic Gyroscope, Norwood, MA: Artech House Inc. (1993). 115. R. B. Smith, “Fiber-Optic Gyroscopes 1991: A Bibliography of Published Literature,” SPIE Proc. 1585:464–503 (1991). 116. S. Ezekiel and H.J. Arditty, eds., “Fiber-Optic Rotation Sensors and Related Technologies,” Proc. First International Conference, Springer Series in Optical Sciences 32 (1981). 117. E. Udd, ed., “Fiber Optic Gyros: 10th Anniversary Conference,” Proc. SPIE 719 (1986). 118. S. Ezekiel and E. Udd, eds., “Fiber Optic Gyros: 15th Anniversary Conference,” Proc. SPIE 1585 (1991). 119. R. A. Bergh, H. C. Lefevre, and H. J. Shaw, “An Overview of Fiber-Optic Gyroscopes,” J. Lightwave Technol. LT-2:91–107 (1984). 120. H. C. Lefevre, S. Vatoux, M. Papuchon, and C. Puech, “Integrated Optics: A Practical Solution for the FiberOptic Gyroscope,” Proc. SPIE 719:101–112 (1986). 121. Y. A. Akulova, G. A. Fish, P. -C. Koh, C. L. Schow, P. Kozodoy, A. P. Dahl, S. Nakagawa, et al., This Paper appears in: “Selected Topics in Quantum Electronics,” IEEE J. Sel. Topics In Quant. Electron. JSTQE 8(6):1349–1357 (2002). 122. See, for example, M. K. Smit, “Progress in AWG Design and Technology,” Proc. WPOFC 05 Palermo, Italy, pp. 26–31 (2005). 123. R. Nagarajan, C. Joyner, R. Schneider, Jr. J. Bostak, T. Butrie, A. Dentai, V. Dominic et al., “Large-Scale Photonic Integrated Circuits,” IEEE J. Sel. Topics Quantum Electron. 11:50–65 (2005). 124. Aiki, M. Nakamura, and J. Umeda, “A Frequency-Multiplexing Light Source with Monolithically Integrated Distributed-Feedback Diode Lasers,” IEEE J. Quantum Electron. QE-13:220–223 (1977). 125. A. Huang, C. Gunn, G-L. Li, Y. Liang, S. Mirsaidi, A. Narasimha, and T. Pinguet, “A 10Gb/s Photonic Modulator and WDM MUX/DEMUX Integrated with Electronics in 0.13 μm SOI CMOS,” in Proc. 2006 IEEE International Solid State Circuits Conference, ISSCC 2006, pp. 922–929 (2006). 126. L. Liao, D. Samara-Rubio, M. Morse, A. Liu, D. Hodge, D. Rubin, U. D. Keil, and T. Franck, “High Speed Silicon Mach-Zehnder Modulator,” Opt. Exp. 13:3129–3135 (2005).

21.46

COMPONENTS

127. P. Bernasconi, W. Yang, L. Zhang, N. Sauer, L. Buhl, I. Kang, S. Chandrasekhar, and D. Neilson, “Monolithically Integrated 40Gb/s Wavelength Converter with Multi-Frequency Laser,” Paper PDP16 in Proc. of OFC/NFOEC 2005, OSA (2005). 128. M. L. Masanovic, V. Lal, J. A. Summers, J. S. Barton, E. J. Skogen, L. G. Rau, L. A. Coldren, and D. J. Blumenthal, “Widely Tunable Monolithically Integrated All-Optical Wavelength Converters in InP,” IEEE J. Lightwave Technol. 23(3):1350–1362 (2005). 129. R. Nagarajan, M. Kato, J. Pleumeekers, P. Evans, D. Lambert, A. Chen, V. Dominic, “Single Chip, 40-Channel x 40Gbit/s per Channel, InP Transmitter Photonic Integrated Circuit,” Paper CPDB3 in Proc. of 2006 Conference on Lasers and Electro-Optics/Quantum Electronics and Laser Science Conference and Photonic Applications Systems Technologies, CLEO/QELS 2006, OSA (2006).

22 MINIATURE AND MICRO-OPTICS Tom D. Milster College of Optical Sciences University of Arizona Tucson, Arizona

Tomasz S. Tkaczyk Department of Bioengineering Rice University Houston, Texas

22.1

GLOSSARY A, B, C, D A(r, z) c D d EFL f g h i k k LA l0 M NA n r rmask rm t u Wijk X

constants converging spherical wavefront curvature diffusion constant diffusion depth effective focal length focal length gradient constant radial distance from vertex imaginary conic constants wave number longitudinal aberration paraxial focal length total number of zones numerical aperture refractive index radial distance from optical axis mask radius radius of the mth zone fabrication time slope wavefront function shape factor 22.1

22.2

COMPONENTS

x, y y Z z $ T N S Urms & [ FOV

22.2

Cartesian coordinates height sag optical axis relative refractive difference propagation distance wavelength U rms/2y rms wavefront error phase special function field of view

INTRODUCTION Optical components come in many sizes and shapes. A class of optical components that has become very useful in many applications is called micro-optics. We define micro-optics very broadly as optical components ranging in size from several millimeters to several hundred microns. In many cases, micro-optic components are designed to be manufactured in volume, thereby reducing cost to the customer. The following paragraphs describe micro-optic components that are potentially useful for large-volume applications. The discussion includes several uses of micro-optics, design considerations for micro-optic components, molded glass and plastic lenses, distributed-index planar lenses, micro-Fresnel lenses, laser printing, grayscale lithography, diamond turning and micromilling, and liquid tunable lenses. For further information the reader is directed to Refs. 1–3.

22.3

USES OF MICRO-OPTICS Micro-optics are becoming an important part of many optical systems. This is especially true in systems that demand compact design and form factor. Some optical fiber-based applications include fiber-to-fiber coupling, laser diode-to-fiber connections, LED-to-fiber coupling, and fiber-todetector coupling. Microlens arrays are useful for improving radiometric efficiency in focal-plane arrays, where relatively high numerical aperture (NA) microlenslets focus light onto individual detector elements. Microlens arrays can also be used for wavefront sensors, where relatively low-NA lenslets are required. Each lenslet is designed to sample the input wavefront and provide a deviation on the detector plane that is proportional to the slope of the wavefront over the lenslet area. Microoptics are also used for coupling laser diodes to waveguides and collimating arrays of laser diodes. An example of a large-volume application of micro-optics is data storage, where the objective and collimating lenses are only a few millimeters in diameter.4 Recently micro-optics is also widely used in medical applications like endo-microscopy including confocal, multiphoton microscopy and optical coherence tomography (OCT). It also has a strong position in consumer market in applications like cell phone cameras, Blu-Ray readers, etc.

22.4

MICRO-OPTICS DESIGN CONSIDERATIONS Conventional lenses made with bulk elements can exploit numerous design parameters, such as the number of surfaces, element spacings, and index/dispersion combinations, to achieve performance requirements for NA, operating wavelength, and field of view. However, fabricators of micro-optic

MINIATURE AND MICRO-OPTICS

22.3

lenses seek to explore molded or planar technologies, and thus the design parameters tend to be more constrained. For example, refractive microlenses made by molding, ion exchange, mass transport process resemble single-element optics. Performance of these lenses is optimized by manipulating one or possibly two radii, the thickness, and the index or index distribution. Index choices are limited by the available materials. Distributed-index and graded-index lenses have a limited range of index profiles that can be achieved. Additional performance correction is possible by aspherizing one or both surfaces of the element. This is most efficiently done with the molding process, but molded optics are difficult to produce when the diameter of the lens is less than 1.0 mm. In general, one or two aberrations may be corrected with one or two aspheres, respectively. Due to the single-element nature of microlenses, insight into their performance may be gained by studying the well-known third-order aberrations of a thin lens in various configurations. Lens bending and stop shift are the two parameters used to control aberrations for a lens of a given power and index. Bending refers to distribution of power between the two surfaces, i.e., the shape of the lens, as described in R. Barry Johnson’s Chap. 17, “Lenses.” The shape is described by the shape factor X that is X

C1 C2 C1 C2

(1)

where C1 and C2 are the curvatures of the surfaces. The third-order aberrations as a function of X are shown in Fig. 1. These curves are for a lens with a focal length of 10.0 mm, an entrance pupil diameter of 1.0 mm, field angle u 20° , an optical index of refraction of 1.5, N 0.6328 μm, and the object at infinity. For any given bending of the lens, there is a corresponding stop position that eliminates coma,5 and this is the stop position plotted in the figure. The stop position for which coma is zero is referred to as the natural stop shift, and it also produces the least curved tangential field for the given bending. Because the coma is zero, these configurations of the thin lens necessarily satisfy the Abbe sine condition. When the stop is at the lens (zero stop shift), the optimum shape to eliminate coma is approximately convex-plano (X l) with the convex side toward the object. The optimum shape is a function of the index, and the higher the index, the more the lens must be bent into a meniscus. Spherical aberration is minimized with the stop at the lens, but astigmatism is near its maximum. It is interesting to note that biaspheric objectives for data storage tend toward the convex-plano shape. Astigmatism can be eliminated for two different lens-shape/stop-shift combinations, as shown in Fig. 1. The penalty is an increase in spherical aberration. Note that there is no lens shape for which 3 Aberration (waves) Stop shift (mm)

2 1

Spherical

0 –1 –2 –3

Astigmatism Stop shift

–4 –5 –5 –4 –3 –2 –1 0 1 2 Shape factor, X

3

4

5

FIGURE 1 Third-order aberrations as a function of the shape factor, or bending, of a simple thin lens with focal length 10.0 mm, entrance pupil diameter of 1.0 mm, field angle 20n, n = 1.5, and object at infinity. The stop position shown is the natural stop shift, that is, the position that produces zero coma.

COMPONENTS

5 0.1

4

Field angle (degrees)


5 0.05

3

0.01

0.025

0.002

2 1 0

0

0.1 Numerical aperture (NA) (a)


0.05 0.025

3

0.01

2

0.05 0.1

3

0.2 0.3

0.025

2 0.01

1

0.002 0

0.1 Numerical aperture (NA) (b)

0.2

5

0.1 4

0.4 4

0

0.2

5 Field angle (degrees)

22.4

0.002

1

0.4 0.3

4

0.2

3 0.1 2

0.05 0.01

1 0.002

0

0

0.1 Numerical aperture (NA) (c)

0.2

0

0

0.1 Numerical aperture (NA)

0.2

(d)

FIGURE 2 Contours of normalized rms wavefront deviation, S NUrms/2y, versus field angle and NA, where 2y is the diameter of the stop. The stop is located at the lens. The focus is adjusted to give minimum rms deviation of the wavefront, so effects of Petzval curvature are not included. (a) X 1, n 1.5; (b) X 1, n 1.5; (c) X 1, n 3.0; (d) X 1, n 3.0.

spherical, coma, and astigmatism are simultaneously zero in Fig. 1, that is, there is no aplanatic solution when the object is at infinity. The aplanatic condition for a thin lens is only satisfied at finite conjugates. The plano-convex shape (X –1) that eliminates astigmatism is particularly interesting because the stop location is in front of the lens at the optical center of curvature of the second surface. All chief rays are normally incident at the second surface. Thus, the design is monocentric.6 (obviously, the first surface is not monocentric with respect to the center of the stop, but it has zero power and only contributes distortion.) Two very common configurations of micro-optic lenses are X +1 and X –1 with the stop at the lens. Typically, the object is at infinity. In Fig. 2, we display contours of normalized rms wavefront deviation, S SLrms /2y , versus u and NA, where 2y diameter of the stop. Aberration components in Urms include third-order spherical, astigmatism, and coma. The focus is adjusted to give minimum rms deviation of the wavefront, so effects of Petzval curvature are not included. Tilt is also subtracted. As NA or field angle is increased, rms wavefront aberration increases substantially. The usable field of view of the optical system is commonly defined in terms of Maréchal’s criterion7 as field angles less than those that produce 2 y S /1000L a 0.07 wave. For example, if the optical system operates at 2y 1.0 mm, N 0.6328 Mm, NA 0.1, X +1, n 1.5, and u 2°, the wavefront aberration due to third-order contributions is

S rms

2 yS (1.0 r 10 3 m)(0.015) y 3 0.024 wave 1000L (10 )(0.6328 r 10 6 m/wave )

(2)


22.5

which is acceptable for most situations. Note that the configuration for X –1 yields Urms y 0.079 wave, which is beyond the acceptable limit. When large values of Urms are derived from Fig. 2, care must be taken in interpretation of the result because higher-order aberrations are not included in the calculation. Also, if field curvature is included in the calculation, the usable field of view is significantly reduced. Coma and astigmatism are only significant if the image field contains off-axis locations. In many laser applications, like laser diode collimators, the micro-optic lens is designed to operate on axis with only a very small field of view. In this case, spherical aberration is very significant. A common technique that is used to minimize spherical aberration is to aspherize a surface of the lens. Third-, fifth-, and higher orders of spherical aberration may be corrected by choosing the proper surface shape. In some lens design codes, the shape is specified by Z

ch 2 1 1 (1 k)c 2h 2

Ah 4 Bh6 Ch8 Dh10

(3)

where Z is the sag of the surface, c is the base curvature of the surface, k is the conic constant (k 0 is a sphere, k –1 is a paraboloid, etc.), and h x 2 y 2 is the radial distance from the vertex. The A, B, C, and D coefficients specify the amount of aspheric departure in terms of a polynomial expansion in h. When a plane-parallel plate is inserted in a diverging or converging beam, such as the window glass of a laser diode or an optical disk, spherical aberration is introduced. The amount of aberration depends on the thickness of the plate, the NA of the beam, and to a lesser extent the refractive index of the plate,8 as shown in Fig. 3. The magnitude of all orders of spherical aberration is linearly proportional to the thickness of the plate. The sign is opposite that of the spherical aberration introduced by an X +1 singlet that could be used to focus the beam through the plate. Therefore, the aspheric correction on the singlet compensates for the difference of the spherical aberration of the singlet and the plate. This observation follows the fact that minimum spherical aberration without aspheric correction is achieved with the smallest possible air gap between the lens and the plate. For high-NA singlet objectives, one or two aspheric surfaces are added to correct the residual spherical aberration.

Considerations on High-Performance Miniature Systems

Peak-to-valley (waves)

In recent years, several designs of high-performance miniature optics were implemented.9–13 Since one of the major applications is miniature microscopy, many systems have similar characteristics. The diameter of miniature objectives is in the range of 1 to 8 mm, consists of multiple lenses (upto 10), NA is 0.4 –1.0 and field of view (FOV) 250 to 500 Mm. All these parameters place high requirements on system assembly and fabrication technologies. Examples of manufacturing techniques

5 4

n = 1.5 n = 1.9

3

NA = 0.5

2

NA = 0.4

1 0

NA = 0.3 0

0.2 0.4 0.6 0.8 Window thickness (mm)

NA = 0.2 1

FIGURE 3 Effect of a window on wavefront distortion at N = 830 nm.

22.6

COMPONENTS

A

B

C FIGURE 4 Comparison of designs of miniature NA 1.0, 205 μm FOV microscope objectives. Design A10 is an all glass, spherical lenses system. Design B11 uses a plastic injection molded objective lens made in Zeonex. Design C13 consists of a leading spherical glass lens followed by two aspehrical Zeonex plastic lenses.

include (1) grinding and polishing methods, (2) injection glass and plastic molding, (3) grayscale lithography, and (4) direct diamond turning/micromilling. Grinding and polishing in glass is the most costly technique and requires a large number of components to provide good system correction (lenses are usually spherical). All other mentioned technologies allow easy application of aspherical lenses that significantly reduces the number of parts. On the other hand, these methods often rely on application of photosensitive polymers, solgels, or plastics, which limits selection of materials with different Abbe numbers. This limitation makes correction for chromatic aberrations more difficult. An example of an effective approach is to build a high NA system as a combination of glass spherical lenses and aspherical plastic/polymer lenses. A comparison of three optical designs with single wavelength, NA 1.0, FOV 250 Mm is shown in Fig. 4. Design A10 is an all glass, spherical lenses system. Design B11 uses aspheric plastic injection molded lenses made in Zeonex. Design C13 consists of a leading spherical glass lens followed by two aspherical Zeonex plastic lenses, which can be injection molded or diamond turned. All objectives in Fig. 4 are presented in scale and the clear aperture of the smallest design (C) is 2.75 mm. These designs show progressive decrease of complexity based on material and surface shape choices. A common method used for tolerance analysis of multicomponent systems is Monte Carlo (MC) simulation that statistically adjusts design parameters like radius error, thickness error, surface form error, surface decenter, surface tilt, and refractive index tolerances.11,13 MC can apply a normal distribution of errors and then evaluate the performance of the system based on the rms spot size or wavefront error. Possible compensation parameters can include the object position, image position, and object shape which is for three-dimensional volume imaging in biological applications. The acceptance criteria can be defined based on the percentage of results that have a rms spot size equal to or less than the diffraction limited Airy disk spot size. As mentioned above, one of the most significant issues in building high-performance miniature systems is precision assembly. Several alignment-free approaches were recently proposed to simplify the manufacturing process. They include techniques like: micro-optical table (MOT),14 fabrication of kinematic mechanical features embedded in optical components, 11,15,16 and self centering mounts.13 All these techniques rely on a zero-alignment concept. In practical terms, the


V-shaped channel in mounting slot wall

22.7

(b)

V-shaped channel in mounting slot wall

Silicon spring in mounting slot in deflected position

(c)

(a)

FIGURE 5 (a) Concept of MOT assembly;14 (b) MOT platform made with DXRL process; and (c) NA 0.4 miniature microscope assembled on MOT platform.19

zero-alignment concept translates into assembly errors that are smaller than the tolerances on the performance of the optical system. Very low assembly errors are achieved through positioning features on each optomechanical and optical component. One technology to fabricate mechanical components is deep reactive ion etching (DRIE)17 in silicon. A less expensive and more robust alternative is deep x-ray lithography (DXRL).18 An example of MOT platform fabricated using DXRL technology in nickel-steel alloy is shown in Fig. 5b. The DXRL technology delivers submicron assembly precision and helps with packaging. Spring and grove features in MOT shown in the figure allow inserting optical components fabricated with grayscale lithography on thin glass substrates. Positioning precision is provided by assembly features on both MOT and optics. (Grayscale lithography allows fabrication of lens and mechanical features in one step.) Figure 5 shows (a) the MOT assembly concept, (b) MOT platform made with DXRL process, and (c) assembled NA 0.4 miniature microscope.19 Table 1 provides assembly precision for MOT technology with DRIE and lithographically fabricated opto-mechanics.14 These parameters must be accommodated in optical design. The principle of a kinematic mount fabricated with injection molding or a lithographic process is shown in Fig. 6. Note that V- or U-shaped grooves can be made in the same process as a lens. Precision spheres are then used for stacking consecutive lenses or lens layers, stops, and spacer TABLE 1 Assembly Precision for MOT Technology with DRIE and Lithographically Fabricated Optomechanics14 Position Translation along slot Yaw rotation (left) Yaw rotation (right) Pitch rotation

Measured Position 3 μm o 1 μm

5 arc min o 2 arc min

5 arc min o 2 arc min 0.44n o 0.02n

22.8

COMPONENTS

(a)

(b)

FIGURE 6 (a) The concept of kinematic lens mount and (b) NA 1.0 microscope objective assembled using kinematic mount embedded in plastic injection molded lenses.11

components.11 Figure 6a shows the concept of kinematic mount and Fig. 6b shows NA 1.0 microscope objective assembled using kinematic mounts embedded in plastic injection molded lenses.11 (The design of the system is presented in Fig. 4b.) A kinematic approach was also used to build arrays of complex optical systems. Examples of applications include microchemical chips using microlens arrays16 (used for illumination and detection) and array of high NA microscopes for high throughput digital telepathology.15 Another method to position lens components in the objective is to use a self-centering ring, which engages the lens surface, flexing away as it makes contact, while centering the lens with respect to its optical axis.14 This self-centering ring eliminates any decentration associated with manufacturing error and allows looser lens tolerance. Figure 7 shows a model self-centering lens spring. This approach was used for assembly of NA 1.0 plastic-glass objective13 (see also Fig. 4c).

22.5

MOLDED MICROLENSES Molded micro-optic components have found applications in several commercial products, which include compact disk players, bar-code scanners, and diode-to-fiber couplers. Molded lenses become especially attractive when one is designing an application that requires aspheric surfaces.

FIGURE 7 Model self-centering lens spring for lens alignment.


22.9

Conventional techniques for polishing and grinding lenses tend to be time-expensive and do not yield good piece-to-piece uniformity. Direct molding, on the other hand, eliminates the need for significant grinding or polishing. Another advantage of direct molding is that useful reference surfaces can be designed directly into the mold. The reference surfaces can take the form of flats.20 The reference flats are used to aid in aligning the lens element during assembly into the optical device. Therefore, in volume applications that require aspheric surfaces, molding becomes a cost-effective and practical solution. The molding process utilizes a master mold, which is commonly made by single-point diamond turning and postpolishing to remove tooling marks and thus minimize scatter from the surface. The master can be tested with conventional null techniques, computer-generated null holograms,21 or null Ronchi screens.22 Two types of molding technology are described in the following paragraphs. The first is molded glass technology. The second is molded plastic technology.

Molded Glass One of the reasons glass is specified as the material of choice is thermal stability. Other factors include low birefringence, high transmission over a broad wavelength band, and resistance to harsh environments. Several considerations must be made when molding glass optics. Special attention must be made to the glass-softening point and refractive index.23 The softening point of the glass used in molded optics is lower than that of conventional components. This enables the lenses to be formed at lower temperatures, thereby increasing options for cost-effective tooling and molding. The refractive index of the glass material can influence the design of the surface. For example, a higher refractive index will reduce the surface curvature. Smaller curvatures are generally easier to fabricate and are thus desirable. An illustration is Coming’s glass molding process.23 The molds that are used for aspheric glass surfaces are constructed with a single-point diamond turning machine under strict temperature and humidity control. The finished molds are assembled into a precision-bored alignment sleeve to control centration and tilt of the molds. A ring member forms the outside diameter of the lens, as shown in Fig. 8. The glass material, which is called a preform, is inserted between the molds. Two keys to accurate replication of the aspheric surfaces are forming the material at high glass viscosity and maintaining an isothermal environment. After the mold and preform are heated to the molding temperature, a load is applied to one of the molds to press the preform into shape. After molding, the assembly is cooled to below the glass transformation point before the lens is removed. Optical performance characteristics of the finished lens are determined by the quality of the mold surfaces, the glass material, and the preform volume, which also determines the thickness of the lens when pressed. An alternative process is used at Kodak, Inc., where molded optics are injection molded and mounted into precision lens cells.24 In this process, a tuned production mold can reproduce intricate Sleeve

Molded lens

Molds

Ring

FIGURE 8

Mold for glass optics.

22.10

COMPONENTS

TABLE 2 Preferred and Possible Tolerances for Molded Glass Components24 Preferred

Possible

Center thickness (mm)

10.00 max 0.40 min o0.030 tol

25.00 0.35 o0.015

Diameter (mm)

25.00 max 4.00 min o0.10 tol

50.00 2.00 o0.01

Diameter of lens beyond clear aperture (mm)

2.00

0.50

Surface quality

80 50

40 20

Axis alignment

3 r 10 3 rad

2 r 10 3 rad

Radius (mm)— best fit sphere

5 to c

2 to c

Slope (N/mm)

50 max

100 max

Wavelengths (N) departure from BFS

a250

a500

mounting datum features and extremely well-aligned optics. It can also form a stop, baffle, or a filmplane reference in the system. Table 2 lists preferred and possible tolerances for molded glass components. The Kodak process has been tested with over 50 optical glasses, which include both crowns and flints. This provides a wide index-of-refraction range, 1.51 < n < 1.85, to choose from. Most of the molded glass microlenses manufactured to date have been designed to operate with infrared laser diodes at N 780 to 830 nm. The glass used to make the lenses is transparent over a much broader range, so the operating wavelength is not a significant factor if designing in the visible or near infrared. Figure 9 displays a chart of the external transmission of several optical materials versus wavelength. LaK09 (curve B) is representative of the type of glass used in molded optics. The external transmission from 300 to over 2200 nm is limited primarily by Fresnel losses due to the relatively high index of refraction (n 1.73). The transmission can be improved dramatically with antireflection coatings. Figure 10 displays the on-axis operating characteristics of a Corning 350110 lens, which is used for collimating laser diodes. The rms wavefront variation and effective focal length (EFL) are shown versus wavelength. The highest aberration is observed at shorter wavelengths. As the wavelength increases, the EFL increases, which decreases the NA slightly. Table 3 lists several optical properties of molded optical materials. The trend in molded glass lenses is to make smaller, lighter, and higher NA components.25 Reduction in mass and size allows for shorter access times in optical data storage devices, and higher NA improves storage density in such devices. Molded Plastic Molded plastic lenses are an inexpensive alternative to molded glass. In addition, plastic components are lighter than glass components. However, plastic lenses are more sensitive to temperatures and environmental factors. The most common use of molded plastic lenses is in compact disk (CD) players. Precision plastic microlenses are commonly manufactured with injection molding equipment in high-volume applications. However, the classical injection molding process typically leaves some inhomogeneities in the material due to shear and cooling stresses.26 Improved molding techniques can significantly reduce variations, as can compression molding and casting. The current state of the art in optical molding permits master surfaces to be replicated to an accuracy of roughly one fringe


22.11

100

80

D

C

Transmission (%)

B 60 A A: Polystyrene (1.0 mm) B: Lak09 (10.0 mm) C: Polycarbonate (3.175 mm) D: Fotoform (1.0 mm)

40

20

0.0 200

600

1000 1400 Wavelength (nm)

1800

2200

FIGURE 9 External transmission of several optical materials versus wavelength. (A) Polystyrene 1.0 mm thick, which is used for molded plastic lenses;26 (B) LaK09 10.0 mm thick, which is used for molded glass lenses;105 (C) Polycarbonate 3.175 mm thick, which is used for molded plastic lenses;26 and (D) Fotoform glass 1.0 mm thick, which is used in the production of SMILE lenses.109

0.035

6.40

0.030 6.30 EFL (mm)

RMS

EFL

0.025

6.20 0.020 RMS

0.015 650

875

1100 Wavelength (nm)

1325

6.10 1550

FIGURE 10 On-axis operating characteristics versus wavelength of a Corning 350110 lens, which is a molded glass aspheric used for collimating laser diodes.106

22.12

COMPONENTS

TABLE 3

Properties of Materials Used for Molding Micro-Optics

Property Index (nd) Abbe # (Vd) Density (g/mm3) Max service temp (°C) Thermal expansion coefficient (1E-6 mm/mm °C) Thermal index coefficient (1E-6/°C) Young’s modulus (10E4 kg/cm2) Impact strength Abrasion resistance Cost/lb Birefringence

PMMA (Acrylic)

PMMA (Imide)

SSMA

Polycarbonate

Polystyrene

Zeonex

LaK09

BK7

1.491 57.4 1.19 72 67.9

1.528 48 1.21 142 -

1.564 35 1.09 87 56.0

1.586 30 1.20 121 65.5

1.589 31 1.06 75 50.0

1.525 56.2 1.01 80.0 67

1.734 51.5 4.04 500 5.5

1.517 64.2 2.51 500 7.1

105

-

-

107

-

-

6.5

3

3.02

-

3.30

2.43

3.16

2.2

11.37

83.1

2 4 3 2

-

3 3 2 4

3 3 4 3

4 2 2 5

-

-

1 5 5 1

1 = lowest / 5 = highest.

per 25 mm diameter, or perhaps a bit better.27 Detail as small as 5 nm may be transferred if the material properties and processing are optimum and the shapes are modest. Table 4 lists tolerances of injection-molded lenses.28 The tooling costs associated with molded plastics are typically less than those associated with molded glass because of the lower transition temperature of the plastics. Also, the material cost is lower for polymers than for glass. Consequently, the costs associated with manufacture of molded plastic microlenses are much less than those for molded glass microlenses. The index of refraction for the plastics is less than that for the glass lenses, so the curvature of the surfaces must be greater, and therefore harder to manufacture, for comparable NA. The glass map for molded plastic materials is shown in Fig. 11. The few polymers that have been characterized lie mainly outside the region containing the optical glasses and particularly far from the flint materials.29 Data on index of refraction and Abbe number are particularly difficult to obtain for molded plastic. The material is supplied in pelletized form, so it must first be molded into a form suitable for measurement. The molding process subjects the material to a heating and annealing cycle that potentially affects the optical properties. Typically, the effect of the additional thermal history is to shift the dispersion curve upward or downward, leaving the shape unchanged. A more complete listing of optical plastics and their properties is given in Ref. 26. Additional information

TABLE 4 Injection Molding Tolerances for Plastic Lenses Focal length Radius of curvature Spherical power Surface quality Vertex thickness (in) Diameter (in per in DIA.) Repeatability lens-to-lens

±0.5% ±0.5% 2 to 5 f

60/40 (40/20 possible) ±0.0005 ±0.002–0.0005 0.1–0.3%

Tolerances given in optical fringes abbreviated by “f ”. Vertex-to-edge thickness ratio 4:1 Difficult to mold 3:1 Moderately easy to mold 2:1 Easy to mold


22.13

1.8 1

LaF

SSK

6 8 F

BK BK7

70

K

60

SF

BaSF 10

LaK

KF

1.6 n d

LF

50

40 ud

1.7

Plastic Glass 30

1.5 20

FIGURE 11 Glass map for molded plastic materials, which are shown as triangles in the figure. The few polymers that have been characterized lie mainly outside the region containing the optical glasses and particularly far from the flint materials.27

can be obtained from the Modern Plastics Encyclopedia,30 the Plastics Technology, Manufacturing Handbook and Buyer’s Guide,31 and in John D. Lytle’s Chap. 3, “Polymeric Optics,” in Vol. IV. Changes in dimension or refractive index due to thermal variations occur in both molded glass and molded plastic lenses. However, the effect is more pronounced in polymer optical systems because the thermal coefficients of refractive index and expansion are ten times greater than for optical glasses, as shown in Table 3. When these changes are modeled in a computer, a majority of the optical systems exhibit a simple defocus and a change of effective focal length and corresponding first-order parameters. An experimental study32 was made on an acrylic lens designed for a focal length of 6.171 mm at N 780 nm and 20°C. At 16°C, the focal length changed to 6.133 mm. At 60nC, the focal length changed to 6.221 mm. Thermal gradients, which can introduce complex aberrations, are a more serious problem. Therefore, more care must be exercised in the design of athermalized mounts for polymer optical systems. The transmission of two common optical plastics, polystyrene and polycarbonate, are shown in Fig. 9. The useful transmittance range is from 380 to 1000 nm. The transmission curve is severely degraded above 1000 nm due to C-H vibrational overtone and recombination bands, except for windows around 1300 nm and 1500 nm. Sometimes, a blue dye is added to the resins to make the manufactured part appear “water clear,” instead of slightly yellowish in color. It is recommended that resins be specified with no blue toner for the best and most predictable optical results.26 The shape of the lens element influences how easily it can be manufactured. Reasonable edge thickness is preferred in order to allow easier filling. Weak surfaces are to be avoided because surface-tension forces on weak surfaces will tend to be very indeterminate. Consequently, more strongly curved surfaces tend to have better shape retention due to surface-tension forces. However, strongly curved surfaces are a problem because it is difficult to produce the mold. Avoid clear apertures that are too large of a percentage of the physical surface diameter. Avoid sharp angles on flange surfaces. Use a center/edge thickness ratio less than 3 for positive lenses (or 1/3 for negative lenses). Avoid cemented interfaces. Figure 12 displays a few lens forms. The examples that mold well are C, E, F, and H. Form A should be avoided due to a small edge thickness. Forms A and B should be avoided due to weak rear surfaces. Form D will mold poorly due to bad edge/center thickness ratio. Form G uses a cemented interface, which could develop considerable stress due to the fact that thermal differences may deform the pair, or possibly even destroy the bond. The plastic injection-molding process may cause flow-induced birefringence in molded optics and significantly reduce system performance, as it creates highly localized regions of refractive

22.14

COMPONENTS

A

B

C

D

E

F

G

H

FIGURE 12 Example lens forms for molded plastic lenses. Forms C, E, F, and H mold well. Form A should be avoided due to small edge thickness. Forms A and B should be avoided due to weak rear surfaces. Form D will mold poorly due to bad edge/center ratio. Form G uses a cemented interface, which could develop stress.29

index changes. The effect is more likely for high-power components with small edge thickness. Plane polariscope images of injection-molded lenses are presented in Fig. 13. The very left image is for a lens fabricated in a standard injection-molding process, while the center and right images are for the same lens type but after lowering fill speed and first-stage packing pressure. The center and right image had the same molding parameters, but the position of the optical molding insert pin was corrected. Figure 13 shows how significant birefringence effects may occur in the fabrication process. It also shows that it is possible to reduce birefringence effects through parameter adjustment. These effects should be taken into considerations during lens form design. Note that CA in the picture marks clear aperture of the fabricated lenses.

FIGURE 13 Plane polariscope images of injection-molded lenses. The lens edge thickness is 0.5 mm. The very left lens was fabricated in standard injection-molding process. Center and right lenses were fabricated with lower fill speed and lower first-stage packing pressure. CA is clear aperture of the lens.33


22.15

Since polymers are generally softer than glass, there is concern about damage from ordinary cleaning procedures. Surface treatments, such as diamond films,34 can be applied that greatly reduce the damage susceptibility of polymer optical surfaces. A final consideration is the centration tolerance associated with aspheric surfaces. With spherical optics, the lens manufacturer is usually free to trade off tilt and decentration tolerances. With aspheric surfaces, this tradeoff is no longer possible. The centration tolerance for molded aspherics is determined by the alignment of the mold halves. A common specification is 4 to 6 Mm, although 3 to 4 μm is possible.

22.6

DIAMOND TURNING Ultra precision diamond tool machining is an important technology for fabrication of miniature optical systems. It can be used in two ways: (1) producing molds for glass and plastic injection molding, and (2) for direct part manufacturing. Diamond turning was originally developed for large-scale aspherical optics (reflective and IR optics) for astronomy and military applications.35 Recently, it took an important place in prototyping and fabrication of small-scale systems. Currently diamond turning is capable of fabricating parts with diameters from a fraction of millimeter to over 500 mm. Diamond tool machining is a technology using monocrystal diamond-cutting tools assembled on an ultraprecision numerically controlled lathe. Diamond tools have nanometer precision edges, which allow cutting a variety of materials and produce molds for injection molding (in metals) or components like mirrors (metals) and lenses (infrared materials and plastics). The depth of a single cut can be below 1 μm. Depending on the material, rough passes have cuts of 10 μm (metals) to approximately 100 μm (plastics). The finishing cut is usually in the 1 to 5 μm range. Modern diamond-turning instruments use air bearings and high-pressure oil bearings to eliminate direct contact of moving parts. After application of a counter-balancing mass, it is possible to achieve exceptionally high precision machining. Optical interpolators assembled on current instruments allow for 1-nm position monitoring. Effective positioning precision is in the range 10 to 50 nm. Therefore, the process is capable of producing submicrometer form precision and nanometer range roughness. An important performance parameters is form error which is usually in the range 0.05 to 5 μm, while roughness (Ra) is between 1 and 10 nm. The lowest roughness can be reached for metals, where for high-purity aluminum and nickel-plated aluminum Ra is 1 to 5 nm. Slightly worse Ra of 5 to 10 nm can be obtained for plastics (PMMA. Zeonex), while Ra for infrared crystals is few tens of nanometers. Specifications for diamond-turning machines (for example, UPL 250 from Nanotech) guarantee form error of 0.125 μm over a 75-mm diameter part, and surface roughness of 3 nm (Ra) is achieved for spherical component made with high-purity aluminum. While these parameters are manufacturer specific, it is possible to reach values better than 0.05 μm form error and 1.2 nm roughness (Ra). No polishing is required after diamond cutting, and both form error and roughness depends greatly on the surface size, shape, and material. While diamond-machining technology has many advantages, it also has two major limitations. One drawback is a periodic character of surface roughness of the fabricated part. Improper surface fabrication may result in diffractive effects. Small cutting depth and low feed rate help to mitigate this issue. A second disadvantage is due to material limitations. Diamond turning can machine nonferrous metals, infrared crystals, and selected polymers. It is not possible to machine glass, due to material microcracking. The list of diamond-turnable materials is shown in Table 5. Current diamond-turning machines have up to 5n of freedom to position the part/tool. They are all called (in machining jargon axes) X, Y, Z, C, and B. X, Y, and Z correspond to linear translation of the tool in reference to the part. C axis relates to the angular coordinate of a spindle and in consequence angular position of a machined part. The B axis controls the angular position of the diamond tool. The variety of axes allows producing aspherical and nonrotational arbitrary

22.16

COMPONENTS

TABLE 5 Material Category Metals

Polymers

IR Crystals

Summary of Diamond Turnable Materials1 Material Name

Refractive Index36,37

Abbe Number36,37

Transmission2 (μm)36,37

Aluminum alloys Brass Copper Gold Nickel Silver Acetyl Acrylic (PMMA) Fluoroplastic Polycarbonate Polypropylene Polystyrene Polysulfone Zeonex (polyolefin) Cadmium sulfide

Cadmium telluride Calcium fluoride Cesium iodide Gallium arsenide Germanium Indium antimonite Iridium Lithium niobate

Magnesium Fluoride

Potassium bromide Potassium Phosphate Silicon Sodium chloride Tellurium dioxode

Zinc sellenide Zinc sulfide

Comments

Electroplated

1.491 1.320 1.586

57.4 92 30.1

0.35–2.1 0.30–2.1

Damages tool quickly

1.571 1.634 1.528

41.2 23.5 55.7

0.37–2.1

Possible clouding

0.51–14.8 0.90–30 0.13–12 0.25–62 0.90–17.3 1.80–15 6–25

2.537 (0.023) 2.817 1.562 1.781 4.020 4.052 5.130 2.183 ( 0.076) 1.377 (0.0188) 1.557

3.478 1.531 2.584 (0.288) 2.591 2.536 (0.022)

0.13–7.7 0.20–306

Denotes uniaxial crystal materials Refractive index given for birefringent materials is for ordinary wave. Value in parenthesis is material birefringence $n ne no

1.1–6.5 0.17–18 0.42–4 0.5–20

1

Values in the table are approximate and a general guidance. For IR Crystals transmission is given as a wavelength range of 1-mm-thick sample at 300 K, assuming that the value is larger than 10 percent.36 For polymers transmission is given for 3.2-mm thick sample and at least 10 percent transmission. Note that polymers with listed transmission range have at least 80 percent (3.2-mm sample) transmission in 400 to 1100 nm wavelengths. 2

optical components, like Alvarez plates38 or arrays of lenses. A diagram of a five-axis diamond machine geometry is presented in Fig. 14. Only X and Z axes are required for machining aspherical and axially symmetrical parts. For this two-axis mode, the part is mounted on the vacuum chuck of the rotation spindle, while the diamond tool is mounted on the Z-axis stage. While the spindle rotates with constant speed, X and Z move to provide proper surface height (Z) for the related radius coordinate (X). An example of Zeonex aspherical lenses made for a NA 1.0 miniature microscope cut in this two-axis mode is presented in Fig. 15. The diameter of the lenses is 2.75 mm. Arbitrary (free) form machining can be obtained in two cutting modes: diamond turning with C axis (X, Z, C configuration) and micromilling (X, Y, Z). The B axis is used for changing tool angle and, in effect, allows increasing optical power of the fabricated components. Diamond turning using the C axis is based on moving the diamond-turning tool in the Z direction synchronously with the rotation

22.17

Y axis


Spindle/ C axis

B axis

is

Z ax

X axis

FIGURE 14 A geometry of five-axis diamond-turning machine. (Courtesy of Moore Nanotechnology System, LLC.)

angle of the part controlled with spindle controller (C axis). The C axis is used to change the spindle rate during the cutting process. C-axis machining can be used in slow-slide servo and fast servo modes. The slow-slide servo mode uses the Z stage of the machine to move the diamond tool. Fast-tool servo requires the diamond tool to be mounted on a piezo or voice-coil controller (the controller is placed on the Z stage) and move it with high frequency and small Z amplitude synchronously with angular position of spindle (C axis). The fast-tool servo, depending on the particular design, works at 100 to 700 Hz frequency and within the Z range of approximately 50–70 μm to 500 μm (Precitech, Moore Nanotechnology Systems). Large scanning range of ±3000 μm is possible at low 20-Hz rate (Moore Nanotechnology Systems). The fast-tool servo allows higher fabrication speeds compared with slow-slide servo mode, which in some cases slows the spindle down (during process) below 1 rpm. Depending on the particular machine, spindle-rotation rates can reach upto 10,000 rpm. Common diamond-turning rates are 1000 to 3000 rpm. An example of a lens array fabricated with slow-slide servo

FIGURE 15 Zeonex aspherical lenses made for a NA 1.0 miniature microscope cut in this two-axis mode.

22.18

COMPONENTS

FIGURE 16 An example of a lens array fabricated with slow-slide servo C-axis machining.39

C-axis machining is presented in Fig. 16.39 There is a limitation on the slope angle for diamond turning of array components in C-axis mode. It depends on the primary clearance angle of the diamond tool. For a standard diamond-cutting tool the slope angle limit is aproximately 20n. Some manufactures can supply tools with angles up to 32n, but these tools are very fragile. Note that the slope angle limitation can be mitigated with B axis or in micromilling mode. Micromilling technology requires the diamond tool to be mounted on the spindle and the part now on the Z-axis stage. While the spindle (tool) rotates, it can be moved in X, Y, Z directions in reference to the fabricated part. The cutting time of free-form surfaces with diamond micromilling is significantly longer than for diamond turning in C-axis mode. For that reason, it is more appropriate for small optic applications. On the other hand, micromilling is very effective for fabrication of parts that conform to the shape of the tool. For example, making arrays of spherical lenses can be very fast (1-μm cutting depth/revolution) and deliver high surface quality. Micromilling spindle rates are usually faster than for normal diamond-turning applications and typically are 6000 to 8000 rpm. An example of a lens array cut with micromilling is shown in Fig. 17. Micromilling can produce high-power components, like hemispherical lenses or arrays of such lenses. Research on fabrication of aspherically shaped tools is currently being pursued.40 If successful, it will provide tremendous benefits for micromilling fabrication and will allow fast production of arrays of high-power aspherical lenses.

22.7

LITHOGRAPHY FOR MAKING REFRACTIVE COMPONENTS In recent years lithographic techniques became important for micro-optics fabrication. Since lithography is designed for making small, planar components it is more appropriate for fabrication of miniature and micro components rather than for large optics. While entire micro-optics systems can be made with lithographic processes, fabrication of large-scale devices is limited to phase


22.19

FIGURE 17 A lens array cut with diamond micromilling. (Courtesy of Moore Nanotechnology Systems, LLC.)

correctors/plates. As lithography was derived from electronics, it allows producing components in large quantities (parts can be diced out of the wafer) or arrays of systems. Additional benefits are obtained with use of grayscale lithography, which allows making components of arbitrary shape. In this process, fabrication of aspherical surfaces is as easy as making spherical ones (no rotational symmetry is necessary) and it gives the designer enormous work freedom. One example application of grayscale lithography for making arbitrary surfaces was proposed by J. Rogers in the design of miniature microscope.41 The goal of the design was to remove ghost reflections through lens tilts and to at the same time maintain high performance of NA 0.4 optics. Shapes of lens surfaces made with grayscale lithography corrected for aberrations originating from tilts. Another benefit of using grayscale lithography is making optomechanical (assembly) features in the same process as optics. This combination can significantly improve assembly precision (see also Sec. 22.4). While lithography has many advantages, it also has lens sag limitations (in effect limits lens power) currently between 100 and 150 μm.42 To improve lens parameters research on higher refractive index materials (for example, solgel) is currently being pursued. Grayscale lithography can be divided into two major categories: (1) direct printing and (2) lithography using grayscale masks. Direct printing can be performed using laser, x-ray or e-beam writing in photosensitive material. In this chapter we will concentrate on fabrication using laser direct writing.

Laser Direct Write Fabrication of Micro-optical Elements The maskless optical lithography technique referred to in this section is laser direct write (LDW) technology. Most LDW phototools have been developed by modulating a single laser spot focused by a microscope objective. The exposure media is moved under the single focused spot using a translation stage, as shown in Fig. 18.43 Both rectilinear raster scan and rotational scan techniques have been studied. The rotational scan spins the media on a rotary table, and the objective translates across a radius, as shown in Fig. 19.44 Rectilinear raster scan by means of a translating X-Y stage allows patterning of nonrotational profiles with the advantage of patterning more than one element.43 This method adds a nonrotational pattern structure to the final image.

N= 442 nm HeCd laser

HeNe laser autofocus system

COMPONENTS

CCD camera

22.20

Modulator

Objective Resist-coated substrate Granite x-y stage (air bearings, linear motors) Granite base Active antivibration elements FIGURE 18 Typical laser direct write (LDW) system employing a stable base and x-y scan stage with a modulated laser to expose the photosensitive surface at each scan point.43

The process steps used in a typical application using LDW technology are 1. Calculate the desired two-dimensional surface sag matrix in the lens material, based on the optical wavefront desired in transmission or reflection. In transmission, properties of the optical material used for the micro-optical element must be considered at the design wavelength. Exposure into simple photoresist is often sufficient for many applications, but the pattern may also be milled or etched into an underlying substrate, like fused silica or GaP. 2. If all other process steps are linear, the surface sag matrix is directly proportional to the depth of material removed with a positive-tone photoresist, which is the most common type of photoresist. 3. However, these process steps are rarely linear. Typically, the designer must compensate the surface sag by the nonlinear properties of the photoresist and the etching or milling processes. Nonlinear properties of the photoresist are best calibrated directly on the LDW phototool immediately before exposure. A calibration ramp is shown in Fig. 20, where responses of the exposure system and photoresist to a linear input ramp show a strongly nonlinear behavior. Depending on the milling or etching technique used, the additional correction for differential milling rate (etch rate of the optical material divided by the etch rate of the resist) must be accommodated. In dry-etch system, the differential etch rate is typically 6:1, but may be as high as 100:1 or as low as 1:1.


22.21

Feedback photodetector Microscope

Collimator

Defocus sensor

Argon laser Air-bearing stage

AOM

Objective with actuator He-Ne

Interferometer

Substrate

Linear motor Granite base

AOM driver

Interfacemeter adapter

Rotary encoder

Spindle motor

Radial stage driver

Spindle driver

Air-bearing spindle Autofocus driver

Local bus

Local bus interface

Pixels generator

PC bus FIGURE 19 Polar coordinate writer for fabrication of micro-optical components.44 The substrate rotating on the spindle is exposed with a focused and modulated laser beam along a diameter. A translation stage moves the focused beam along the spindle radius to expose different diameters. Good synchronization between the spindle and laser modulation can produce arbitrary gray-scale profiles.

4. Resist layer is deposited on the substrate, exposed and developed according to the compensated exposure profile. Certain photoresist process steps are required, such as prebake to remove residual solvents and postbake or postdevelop bake, which depend on the chemistry being used. If resist is used as the optical material, there are no more process steps required. 5. If the pattern is transferred into the substrate material, the substrate must be mounted into the milling chamber or etch bath for further processing.

COMPONENTS

Height (Mm)

3.5

0 0%

100% Exposure

FIGURE 20 Response curve of positive-tone photoresist to a linearly increasing exposure signal (0 to 100 percent laser power) that takes into account nonlinearities of the exposure system and the photoresist.

Although exposure in photoresist is the most common method, other materials may be used for direct laser writing. For example, Poleshchuk et al. use a polar coordinate writer to change the absorption of amorphous silicon as a function of position for fabrication of gray-scale masks, as shown in Fig. 21.44 The reduction in optical absorption is caused by an induced crystallization of the amorphous silicon. Other types of LDW phototools have also been proposed. A system using x-y galvanometer mirrors has been constructed to fabricate micro-optical components by photodeposition of amorphous Line spacing (Mm) 1.6 1.0

(a)

0.8 0.6

(b)

Transmission (rel. units)

22.22

1 0.8 (c)

0.6 0.4 200

300

400 Coordinate (Mm)

500

600

FIGURE 21 Directing laser writing of gray-scale patterns in amorphous silicon films using a polar coordinate writer.44


22.23

selenium (a-Se).45 The galvanometer mirrors are used instead of x-y stages to move the focused laser spot across the sample. Yet another system uses digital multi-micromirror device (DMD) array with a 10:1 lithographic reduction camera to expose micro-optical patterns.46 In this system, the DMD is illuminated with a He g-line (436 nm) illumination system, and pixels of the DMD are turned “on” or “off ” to direct light into or away from, respectively, the projection camera. A full image of the DMD array is exposed at one time with this system, thus increasing throughput. Pixels in the “on” state expose the resist. Still another system under development is the high-speed maskless lithography tool (MLT) at the University of Arizona.47 In this system, an 8-bit modulated 370-nm laser beam is reflected from a rotating polygon and focused to a scan line (x dimension) in the photoresist. As the polygon rotates, the photoresist-coated sample is slowly translated with a linear stage perpendicular (y dimension) to the scan line. By the time the scan line is complete, the stage has moved a y distance corresponding to the width of 1 pixel in that dimension and is ready for the next scan line. A twelve-sided polygon rotating at 3000 rpm provides 12,000 pixels across one scan line. Pixel dimensions are 2.1 Mm square, and the spot size is approximately 2.5 Mm. The length of the written pattern in the y direction depends on the range of the stage and the size of the computer buffer memory. One 25 mm r 25 mm substrate can be exposed in approximately 12 seconds.

Lithography with Gray-Scale Masks Lithography with use of gray-scale masks is a three step process: 1. Substrate is coated with photoresist, polymer, or solgel glass. 2. Sample is exposed with UV light through gray-scale mask. 3. Sample is developed to create refractive lens component. For transferring lens shape into glass, an additional step of deep reactive ion etching (DRIE) is required. A conceptual graph presenting the gray-scale lithography process is shown in Fig. 22. The process requires two critical components, which are a gray-scale mask and a photosensitive material with linear relationship between optical density and level of polymerization so the lens height will directly relate to energy delivered to the sample. Note that a nonlinear relation is also possible to use,

UV exposure Grayscale mask Photosensitive material

Darker color coressponds to higher polymerization level

Substrate Development Developed stucture Substrate FIGURE 22 Principle of gray-scale lithography process.

COMPONENTS

TABLE 6 Comparison of Achievable Resolution for Different Type of Gray-Scale Mask3 Minimum Feature Size

Mask Type Photographic masks Halftoning HEBS masks

90 percent, s/T q 0.95, which implies that the falling edge of the zone profile must take no more than 5 percent of the grating period. This is possible with low NA systems, where the zone spacing is large compared to the resolution of the exposure system, but it becomes difficult in high NA systems, where the zone spacing is on the order of several microns. Analysis of the three remaining parameters indicates fairly loose tolerances are acceptable. For J > 98 percent, tolerance on individual parameters are: |d(n 1)/L 1| < 0.25, a/T > 0.50, and $d(n 1)/N < 0.50. Due to the increasing difficulty in fabricating correct zone profiles with decreasing zone width, most MFLs exhibit a variation in diffraction efficiency versus radius. In the center of the zone pattern, where the zone spacing is large, the measured diffraction efficiency can be in excess of 90 percent. At the edge of the zone pattern, where the zone spacing can be on the order of a few wavelengths, the measured diffraction efficiency is much lower. One possible solution to this problem is to use “superzone” construction,74 in which the zone radii are found from a modified form of Eq. (10), that is k( f f 2 rM2 ) 2P Nm

(24)

where N is the superzone number. This results in a maximum thickness of dMAX NN/(nMFL 1). Note that N 1 corresponds to the standard MFL. N 2 implies that zones are spaced at every 4R phase transition boundary instead of at every 2R phase transition boundary. Although this makes the zones wider apart, the surface relief pattern must be twice as thick. Molding provides a potentially valuable process for fabricating large quantities of MFLs economically. MFLs can be produced with conventional injection molding, but due to the large thermal expansion coefficient of polymers, the lenses are sensitive to thermal variations. An alternative MFL molding process is shown in Fig. 38, where a glass substrate is used to avoid large thermal effects.

(a) Master Nt

(b) Electro forming

(c) Stamper

UV-curable resin Substrate UV light (d)

(e) Molded MFL

FIGURE 38 Molding process for a MFL on a glass substrate. First, a master is made by electron-beam lithography, then a stamper is electroformed from the master. MFLs are molded by potting a UV-curable resin between the stamper and the substrate and then exposing through the substrate.53


22.37

Wavefront aberration (N)

0.05

0.04

0.03

0.02

0.01

0

0

20

40

60 80 Temperature (nC)

100

120

FIGURE 39 Wavefront aberration versus substrate temperature for a 0.25 NA molded MFL on a glass substrate designed to operate at N 780 nm.78

First, a master lens is formed with electron-beam or laser writing. A stamper is prepared using conventional nickel electro-forming methods.76 After potting a UV-curable resin between the stamper and the glass substrate, the replica lenses are molded by the photopolymerization (2P) process.77 The wavefront aberration versus temperature for a N 780 nm, NA 0.25, diameter 0.5 mm lens formed with this technique is shown in Fig. 39.78 A variation on this technique is to use the stamper as a substrate in an electron-beam evaporation device.79 Inorganic materials of various refractive indices can be deposited on the stamper, resulting in a thin lens of high refractive index. The high refractive index of a material like ZnS (n 2.35) can be used to lower the maximum thickness requirement of the lens, which makes fabrication of the master with electron-beam writing easier.

22.11

LIQUID LENSES Integrated micro and miniature systems are often limited by fixed geometry, difficult realignment, and tuning capabilities. Therefore, extensive research was performed to develop miniature optical components with adjustable optical power. Even though zoom solutions, like applications of Alvarez-Humphrey plates mounted on microelectro mechanical system (MEMS) actuators, were recently suggested,80 the most successful tunable miniature optic components are based on principles of liquid lenses or liquid crystals. In general, the concept of liquid lenses arises from the fact that it is possible to change the shape of the liquid volume and provide an optical power change. This power change can be done in several ways, including electrowetting,81,82 pressure,83 85 or temperature change.86 Liquid crystal lenses87 are based on changing an electric field to create different crystal orientations and, in consequence, refractive index distribution. One of the most successful approaches for tunable miniature lenses is the concept of an electrowetting lens.81 It was developed primarily for consumer market applications, like cell phone or credit card cameras. The limited amount of space and relatively high cost does not permit using traditional motor-driven systems in these products. Another application of electrowetting lenses is Blu-Ray Disk (BD) recording systems for dual layer disks.

22.38

COMPONENTS

Hydrophobic Incident light Insulating coating fluid Insulator

Electrodes

Glass

(c)

Conducting fluid

(a)

–– + + –– + +S – + – +

+ –– + + –– + + ––

V

(d) (b) FIGURE 40 A-B schematic cross section of electrowetting lens for convex and concave lens, respectively. C-D photographs of electrowetting lenses.81

The principle of an electrowetting lens is presented in Fig. 40. 81 It relies on using two nonmixable liquids, where one is conductive and one nonconductive. An example of conductive liquid is salted water, while a nonconductive example is nonpolar oil. Both liquids must have significantly different refractive index (typically $n 0.15–0.20) and similar density. Refractive index difference is required to provide optical power, while similar density makes the lens insensitive to vibrations and shocks. Liquid is placed in a cylinder where the cylinder wall and bottom are metal coated and act as two electrodes. Note that the cylinder wall is separated from the conductive liquid by an insulating layer. Changing potential between these two electrodes influences the shape of the conductive liquid and creates either a convex or a concave refractive lens. An example of electrowetting camera lens from Philips is presented in Fig. 41. 81 The outer diameter of the lens (Fig. 40) is 4 mm, inner diameter is 3 mm, and height 2.2 mm. Typical switching voltage is in range of 50 to 120 V. Note that a concave lens is a natural state when no voltage is applied (voltage application allows to decrease negative power or obtaining convex lenses). The switching time is about 10 ms. The Philips lens can achieve maximum negative power of 100 dioptres, and positive of 50 dioptres. This corresponds to focus lengths of 10 mm and 20 mm, respectively. Optical power D can be described as D $nC

(25)

where C is lens curvature (reciprocal of lens radius). Electrowetting lenses can provide sufficiently good optical performance. The main imperfections arise from asymmetric cylinder shape and nonuniform coating. Major wave aberrations in the lens are coma and astigmatism. It has been shown that they can be within diffraction limit of 0.07N wavefront error81 within their operating range. Note that work on aspherical, asymmetric electrowetting optical components82 and reconfigurable lens arrays88,89 is currently an active research area. For the purpose of this chapter, however, we limit our discussion of electrowetting to a principle level analysis only.


22.39

FIGURE 41 A prototype of camera module with electrowetting liquid lens.81

Another tunable lens concept is based on using liquids/gels and a pressure change to adjust optical power.83 84 85 In this case diameter of the cylinder containing the gel or liquid is covered with an elastic membrane (most often polydimethylsiloxane [PDMS]). The membrane is pushed out or pulled in by an increase/decrease of liquid pressure, respectively, and can create convex or concave lenses. The principle of the pressure-based liquid lens is presented in Fig. 42. Depending on the liquid and lens diameter, pressure change lenses can obtain approximately 1.0 to 10 mm and 1.0 to 10 mm focal lengths. If the membrane is sufficiently thin, the same equation as used for electrowetted lenses can be used to estimate lens power. The pressure change technique allows easy population of lenses in micro fluidic systems and fabrication of tunable lens arrays.83 84 85 An array of 200-μm (diameter) lenses made on a substrate glass in PDMS is shown in Fig. 43.83 Note that membrane lenses are subject to spherical aberrations, since the membrane shape produces an aspherical form due to lower membrane stiffness in the middle area of a lens that creates a nonuniform curvature. Liquid material used for an array prototype (Fig. 42) was a microscope immersion oil (n 1.51) or UV-curable polymers (Norland 63, n 1.56), but other materials are also possible. While the majority of applications for pressure-based liquid lenses are “lab on a chip” use, they were also recently prototyped for use in optical coherence tomography (OCT). An example of liquid lens and complete head of an OCT probe is shown in Fig. 44.90 Tunable lenses can also implement thermal change of the liquid volume.85 This method is suitable for small millimeter or submillimeter components. It is due to the fact the volume change $V DV$T.

Membrane Membrane

Membrane

Pressure = 0 Liquid

FIGURE 42

Pressure > 0 Liquid

The principle of the pressure-based liquid lens.

Pressure < 0 Liquid

22.40

COMPONENTS

Circular chamber

PDMS membrane

Microlenses

1. The lens array is filled with liquid

2. The array is pressurized to form the lenses

Microfluidic channels 400 Mm

PDMS (a)

GLASS

(b)

(a) Picture of the pressure base microfluidic lens array. (b) Schematic of the microlens array.83

Larger masses require in linear proportion more heat energy. Therefore, this method is more effective for small liquid volumes. Note that D denotes volume expansion coefficient, V and $V material volume and volume change, respectively, and $T is the introduced temperature change. An example of a thermally controlled liquid lens is presented in Fig. 45.86 The liquid chamber is covered with PDMS lens (gives system an initial optical power) made in a two-step replication process. PDMS was applied as a flexible and easily stretchable material. This lens has 1.9-mm clear aperture. The entire package is 8.5 mm r 6.5 mm r 1.5 mm. The lens requires voltage change (for thermal actuation) of 0 to 14 V, which corresponds to temperature change from about 20 to 50°C. In effect, focal length changes from 14.7 to 2.8 mm. Entirely different technology for making tunable lenses is based on the liquid crystal principle87 in which it is possible to create a continuously changing electric field that causes axially symmetric

1.5 mm

m

Fiber optic Pressure

750 Mm

3m

FIGURE 43

LIQUID

Mechanical holder Fluidic channel (a) FIGURE 44

Micro-lens chip (b)

(a) Microfluidic pressure lens and (b) picture of entire OCT head.90


22.41

Filling inlet Lens position Actuator

Thermal sensor

Chamber Conductive pads

Silicon substrate Liquid Glass plate Feedback control Sensing circuit (a) FIGURE 45

(b)

(a) Schematic of an integrated thermal volume change lens. (b) Photo of a complete packaged lens.86

orientation changes of liquid crystals (LC). The crystal orientations results in a distribution of refractive index similar to that obtained in gradient index (GRIN) lenses. LC lenses are being used in machine vision, photonics, and eyeglasses. The LC lens principle can be described following Naumov et al.87 and his design of a liquid crystal lens. The LC is placed between two transparent electrodes deposited on glass substrates. One is the control electrode with distributed resistance much higher than the distributed resistance of the ground electrode. The LC layer placed between these two electrodes acts as a capacitor. In case of application of AC voltage to the single circular contact at the control electrode, the active impedance of the control electrode and the reactive impedance of the capacitor create a distributed voltage divider. This divider results in a distributed AC voltage over the layer of LC relatively close to a parabolic distribution. In consequence the axially symmetric distribution of liquid crystals provides a change of optical power. It is also possible to build cylindrical lenses by using two line contacts at the control electrode. LC lenses are very convenient, due to the fact that they are electrically controlled integrated components. Switching voltage of LC lenses, depending on design, is usually in 0 100 V range. Their major drawback arises from relatively low optical power and polarization effects due to the crystal structure. Liquid crystal lens research is a very broad area. For the purpose of this discussion we concentrate on selected examples. One group of miniature liquid lenses includes relatively large components of 5 to 10 mm in diameter and quite low optical power with focal length between few hundred millimeters and infinity. 87 Multilayer liquid crystal lenses 91 can increase optical power and obtain a range of 93 to 1230 mm. Examples of negative lenses with a glass lens embedded between liquid crystal layers were also demonstrated.92 Most recently, research on aspheric liquid crystal components (like axicons) is also being pursued.93 A second group of liquid crystal lenses are microlenses94 and arrays of microlenses.95,96 The diameter of a single lens is usually few hundred microns (200 to 600 μm) and these lenses can achieve short focal lengths in range ±2 to ±10 mm. To compare various techniques for making tunable miniature or microlenses, a summary of parameters of various lens types are presented in Table 9. Values from literature are rounded to show achievable range rather than detailed numbers. Note that it is possible to find examples of systems going outside values summarized in Table 9. Other tunable liquid lens approaches can be found in literature including adaptive liquid microlenseses activated with different stimuli.97 The optical power of these lenses may potentially be designed in a way so it will adapt to the environment, which means that optical power will change depending on conditions (PH, temperature, etc.). Detailed discussions of these quite new approaches are omitted, as they are in initial research stages.

22.42

COMPONENTS

TABLE 9

Comparison of Parameters of Different Tunable Lenses

Technique

22.12

Common Size of Clear Aperture

Possible Shapes

Focal Range

Range of Tuning Parameter

Switching Time

Electrowetted

0.2–5.0 mm

20 mm–c

10 mm–c

Convex Concave

Voltage 50–120 V

10 ms

Pressure change

0.2–2 mm

1.0–10.0 mm

1.0– 10 mm

Convex Concave

Pressure 0–50 KPa

500 ms– few seconds

Volume change

~2.0 mm, possible Decrease to 300 μm

2.0–15.0 mm

Convex

Temperature 20–50°C

50 seconds

Liquid crystal

5–10 mm 0.2–0.6 mm

100 mm–c 2–10 mm

2– 10 mm

Convex Convex Concave

Voltage 0–100 V

500 ms– few seconds

OTHER TECHNOLOGIES There are several other technologies that are potentially valuable for micro-optic components. Four particularly interesting technologies are melted-resin arrays,98 laser-assisted chemical etching,99 mass transport,100 and drawn preform cylindrical lenses.98 Melted-resin arrays of hemispherical ball lenses are formed with the process shown in Fig. 46. First, an Al film is deposited on a quartz substrate and patterned with holes that serve as aperture Al mask

(a)

Substrate Hardened pedestals

(b)

(c)

Resin cylinders

Thermal cycle

Microlenses (d) FIGURE 46 Process used to form melted-resin microlenses: (a) an AL film is deposited on the substrate and patterned with holes to serve as aperture stops for the array; (b) circular pedestals are formed on top of the aperture holes and hardened; (c) cylinders of resin are developed of the pedestals; and (d) pedestals are melted to form spherical surfaces.98


22.43

UV light Resist cylinders

mask

Resist lenses

resist

FIGURE 47

Principle of fabrication of microlens arrays using photoresist reflow technique.16

stops for the array. Next, circular pedestals are formed on top of the aperture holes. The pedestals are hardened so that they are insoluble and stable for temperatures in excess of 180nC. Cylinders of resin are then developed on top of the pedestals. The device is heated to 140nC to melt the resin. The pedestals serve to confine the melting resin. The lenses form into hemispherical shapes due to surface tension forces. Lens diameters have been demonstrated at 30 μm with good wavefront performance and uniformity.98 Similar to the process for making melted resin arrays is fabrication of reflow lenses.3,16 The graphical representation of reflow process is presented in Fig. 47.16 Figure 48 shows example of fabricated lenslet array.16 Fabrication of reflow lenses consists of two major steps: (1) A layer of photoresist or solgel is exposed through lithographic binary mask and then developed; (2) photoresist cylinders are melted to create surfaces close to spherical shape. The third step of reactive ion etching3,16 or plasma etching101 can be added to transfer lens shapes into a substrate. For the melting step, the sample is heated to the glass temperature of the resist, which is in range of 150 to 200°C. After Sinzinger and Jahns,3 the focal length equation in Eq. (26) of the obtained lens is

f

rc n 1

FIGURE 48 SEM picture of 5-μm diameter microlenses packed hexagonally.16 Lenses were made using reflow process in photoresist.

(26)

22.44

COMPONENTS

where rc is radius of curvature derived from properties of binary cylinders: D2 4 2h

h2 rc

(27)

Note that h is the height of the droplet and corresponds to lens sag and D is diameter of the cylinder. Knowledge of cylinder volume allows also to find a relation between cylinder thickness and the droplet height (sag),3 h¤ h2 ³ t ¥3 4 2 ´ 6¦ D µ

(28)

Reflow lenses require very careful selection of photosensitive material (high viscosity, e.g., Hoechst AZ 4562), cylinder diameter, and its height. If the cylinder structure is too shallow, the melting process may create lenses deviating significantly from a sphere. The ratio between cylinder height and diameter is usually in range of 0.04 to 0.5. To improve quality of fabricated lenses, preshaping of

(a)

(b)

20 Mm

FIGURE 49 SEM photographs showing perspective views of (a) etched multilevel mesa structure and (b) the microlens formed after mass transport.100


22.45

photoresist can be applied. It is also possible to fabricate asymmetric lenses using either preshaping or noncircular masks. The reflow technique can be further extended and combined with a molding process.102,103 After exposure, developing, and melting steps the lens array can be coated with PDMS and cured. In result, one obtains the negative impression of the photoresist array. Due to the fact that PDMS is hydrophobic and elastic such a created mold easily comes of the substrate and can be used in additional steps to create solgel (or other polymer) lens arrays.103 The quality of reflow fabricated lenses is high, resulting in surface roughness in the range of Ra 1 nm and deviation from perfect sphere of N/10 (RMS) can be achieved. By application of aformentioned method103 large sag and NA values can be obtained. Diameters of reflow lenses usually vary between 5 and 2000 μm3,16 and their sag is within 2.5 and 60 to 70 μm. Reported focal lengths were few and 9000 μm3 where longer focal lengths were obtained for larger lens diameters. Laser-assisted chemical etching (LACE) can be used to make arrays of F/0.7–F/10 lenslets with spacings of 50 to 300 μm. Microlenses have been fabricated in glass, silicon, CdTe, and sapphire with 95 percent fill factors and figure quality better than 1/10th wave.99 In the LACE process, a focused laser beam is scanned over a thick layer of photoresist. The irradiance of the laser beam is modulated in order to vary the exposure and thus the thickness of the developed resist. With the proper irradiance mapping, accurate lens profiles can be produced in the developed resist. If lenslet material other than photoresist is required, a pattern can be exposed in the photoresist and transferred into the new material by ion milling. Our discussion of the mass-transport process follows discussion presented in Ref. 100. In the mass-transport process, a multilevel mesa structure is first etched into a semiconductor, as shown in Fig. 49a. The semiconductor must be a binary compound in which the evaporation rate of one element is negligible compared to that of the other. For example, InP has been used successfully. The mesa structure is placed in a furnace at an elevated temperature. Since some surface decomposition occurs with InP, a minimum phosphorus vapor pressure must be maintained in the gas ambient to prevent the sample from being transformed into metallic In. The decomposition produces free In atoms located at the crystal surface, which are in equilibrium with phosphorus in the vapor and InP in the crystal. The concentration of In in the vapor is negligible. The equilibrium concentration of free In atoms increases with increasing positive surface curvature, since the higher surface energy of the high-curvature regions translates into a lower bonding energy in the decomposition process. Consequently, a variation in curvature across the surface will result in diffusion of free In atoms from regions of high positive curvature, where the concentrations are high, to low-curvature regions, where the In-diffused atoms exceed the equilibrium concentration and have to be reincorporated into the crystal by reaction with P to form InP. (The diffusion of P in the vapor phase is presumably much faster than the diffusion of free In atoms on the surface. The latter is therefore assumed to be the rate-limiting process.) The mass transport of InP, resulting from decomposition in high-curvature regions, will continue until the difference in curvature is completely eliminated. After mass transport, a smooth profile is obtained, as shown in Fig. 49. The design of the mesa structure can result in very accurate control of the lens profile, as shown in Fig. 50. Mass-transport lens arrays have been used to collimate arrays of laser diodes, with diffraction-limited performance at NA ~0.5. Very accurate cylindrical lenses can be drawn from performs.104 An SEM photo of an elliptical cylindrical lens is shown in Fig. 51. The first step in the process is to make a preform of a suitable glass material. Since the cross-sectional dimensions of the preform are uniformly reduced (typtically ~50 to 100r) in the fiber drawing process, small manufacturing errors become optically insignificant. Therefore, standard numerically controlled grinding techniques can be utilized to generate a preform of any desired shape. Besides maintaining the preform shape, the drawing process also polishes the fiber. Results are presented98 that demonstrate a 200-μm-wide elliptical cylindrical lens. The SFL6 preform was about 0.75 cm wide. The lens has a nominal focal length of 220 μm at N 800 μm. The lens is diffraction-limited over about a 150-μm clear aperture, or NA ~0.6. The application is to collimate the fast axis of laser diodes.

COMPONENTS

6 GaP lens formation 4

As etched (corrected for broadening)

Stylus surface profiles

After mass transport

2 Height (Mm)

22.46

0 4

2 Ideal lens f = 200 M m

0 –80

–40

0 Position (Mm)

40

80

FIGURE 50 Stylus surface profiles of the multilevel mesa structure and the microlens formed after mass transport (upper half) and the comparison of the measured lens profile with an ideal one (lower half).100

5KV

X183

54.4UM

0009

LLNL

FIGURE 51 Scanning electron microscope photo of an elliptical cylindrical microlens. The lens width is 200 μm.104


22.13

22.47

REFERENCES 1. M. and S. Kufner, Micro-Optics and Lithography, VUB Press, Brussels, 1997. 2. H. N. Herzig, ed., Micro-Optics: Elements, Systems and Applications, Taylor and Francis Ltd., 1997. 3. S. Sinzinger and J. Jahns, Microoptics, Wiley-VCH GmbH & Co. KGaA, Weinheim, Germany, 2003. 4. G. T. Sincerbox, “Miniature Optics for Optical Recording,” Proc. SPIE 935:63–76, 1988. 5. W. J. Smith, Modern Optical Engineering, McGraw-Hill, New York, p. 61, 1966. 6. D. K. Towner, “Scanning Techniques for Optical Data Storage,” Proc. SPIE 695:172–180, 1986. 7. M. Born and E. Wolf, Principles of Optics, Pergamon Press Ltd., Oxford, pp. 468–469, 1980. 8. J. F. Forkner and D. W. Kurtz, “Characteristics of Efficient Laser Diode Collimators,” Proc. SPIE 740:27–35, 1987. 9. A. R. Rouse, A. Kano, J. A. Udovich, S. M. Kroto, and A. F. Gmitro, “Design and Demonstration of a Miniature Catheter for a Confocal Microendoscope,” Appl. Opt. 43(31):5783, 2004. 10. C. Liang, K. B. Sung, R. Richards-Kortum, and M. R. Descour, “Fiber Confocal Reflectance Microscope (FCRM) for In-Vivo Imaging,” Opt. Exp. 9:821–830, 2001. 11. M. D. Chidley, K. Carlson, M. R. Descour, and R. Richards-Kortum, “Design, Assembly, and Optical Bench Testing of a High Numerical Aperture Miniature Injection-Molded Objective for Fiber-Optic Confocal Reflectance Microscopy,” Appl. Opt. 45:2545–2554, 2006. 12. J. D. Rogers, S. Landau, T. Tkaczyk, M. R. Descour, M. S. Rahman, R. Richards-Kortum, A. Kärkkäinen, and T. Christenson, “Performance and Imaging Results for Miniature Integrated Microendoscope,” J Biomed. Opt. 13:(5): 054020-1-6, September/October 2008. 13. R. T. Kester, T. Tkaczyk, M. R. Descour, T. Christenson, and R. Richards-Kortum, “High Numerical Aperture Microendoscope Objective for a Fiber Confocal Reflectance Microscope,” Opt. Exp. 15:2409–2420, 2007. http://www.opticsinfobase.org/abstract.cfm?URIoe-15-5-2409. 14. M. R. Descour, A. H. O. Kärkkäinen, J. D. Rogers, C. Liang, R. S. Weinstein, J. T. Rantala, B. Kilic, et al., “Toward the Development of Miniaturized Imaging Systems for Detection of Pre-Cancer,” IEEE J. Quantum Electron. 38(2), Feb. 2002. 15. A. Olszak and M. Descour, “Microscopy in Multiples,” OE Magazine, May 2005. 16. Ph Nussbaum, R. Völkel, H. P. Herzig, M. Eisner, and S. Haselbeck, “Design, Fabrication and Testing of Microlens Arrays for Sensors and Microsystems,” Pure Appl. Opt. 6:617–636, 1997. 17. M. J. Madou, Fundamentals of Microfabrication: The Science of Miniaturization, 2d ed., CRC, Boca Raton, FL, March 13, 2002. 18. M. Gad-el-Hak, ed., The MEMS Handbook, 2d ed., vol. 2, “MEMS Design and Fabrication,” Chap. 5, “X-Ray Based Fabrication,” CRC Tayor & Francis, Boca Raton, FL, 2006. 19. T. Tkaczyk, J. D. Rogers, M. Rahman, T. Christenson, S. Gaalema, E. Dereniak, R. R. Kortum, and M. R. Descour, “Multi-Modal Miniature Microscope: 4M Device for Bio-Imaging Applications—An Overview of the System,” Proc. SPIE 5959:138–146, Medical Imaging, Warsaw, September 2005. 20. R. O. Maschmeyer, U.S. Patent 4,537,473, 1985. 21. D. Malacara, Optical Shop Testing, John Wiley and Sons, New York, 1978, Chap. 12, pp. 381–397, 1978. 22. G. W. Hopkins and R. N. Shagam, “Null Ronchi Gratings from Spot Diagrams,” Appl. Opt. 16:2602, 1977. 23. M. A. Fitch, “Molded Optics: Mating Precision and Mass Production,” Photonics Spectra, 84–87, October, 1991. 24. T. Aquilina, D. Richards, and H. Pollicove, “Finished Lens Molding Saves Time and Money,” Photonics Spectra, 73–80, September 1986. 25. M. A. Fitch and Y. K. Konishi, “Technical Directions in Molded Optics,” Proc. SPIE 1139:187–190, 1989. 26. D. L. Keyes, “Fundamental Properties of Optical Plastics,” CRC Handbook of Laser Science and Technology, CRC Press Inc., Boca Raton, FL, in press. 27. J. D. Lytle, “Aspheric Surfaces in Polymer Optics,” Proc. SPIE 381:64–68, 1983. 28. D. Keyes, Private Communication. 29. R. M. Altman and J. D. Lytle, “Optical-Design Techniques for Polymer Optics,” Proc. SPIE 237:380–385, 1980.

22.48

COMPONENTS

30. Modern Plastics Encyclopedia, vol 68, McGraw-Hill, New York, 1991/1992. 31. Plastics Technology, Manufacturing Handbook and Buyer’s Guide, vol. 37(8), Bill Communications Inc., New York, 1991/1992. 32. Diverse Optics Inc., Commercial Literature, 1992. 33. M. D. Chidley, T. Tkaczyk, R. Kestera, M. R. Descoura, “Flow-Induced Birefringence: The Hidden PSF Killer in High Performance Injection-Molded Plastic Optics,” Proc. SPIE 6082:60820E, 2006. 34. A. H. Deutchman, R. J. Partyka, and J. C. Lewis, “Dual Ion Beam Deposition of Diamond Films on Optical Elements,” Proc. SPIE 1146:124–134, 1989. 35. C. F. Cheung and W. B. Lee, Surface Generation in Ultra-Precision: Modeling and Practices, Professional Engineering Publishing, UK, 2003. 36. M. J. Weber, Handbook of Optical Materials, CRC Press, Boca Raton, FL, 2003. 37. S. Bäumer, ed., Handbook of Plastic Optics, Wiley-VCH, GmbH & Co. KGaA, Weinheim, Germany 2005. 38. L. W. Alvarez and W. E. Humphrey, “Variable Power Lens and System,” Patent # 3,507,565 United States Patent Office, 1970. 39. A. Y. Yi and L. Li, “Design and Fabrication of a Microlens Array by Use of a Slow Tool Servo,” Opt. Lett. 30(13):1707, 2005. 40. W. C. Sweatt, D. D. Gill, D. P. Adams M. J. Vasile, and A. A. Claudet, “Diamond Milling of Micro-Optics,” Aerospace Conference, IEEE 2006. 41. J. Rogers, T. Tkaczyk, M. Descour, A. Kärkkäinen, and R. Richards-Kortum, “Removal of Ghost Images by Using Tilted Element Optical Systems with Polynomial Surfaces for Aberration Compensation,” Opt. Lett. 31:504–506, 2006. 42. T. Tkaczyk, J. D. Rogers, M. Rahman, T. C. Christenson, S. Gaalema, E. L. Dereniak, R. Richards-Kortum, and M. R. Descour, “Multi-Modal Miniaturized Microscope: Successful Merger of Optical, MEMS, and Electronic Technologies,” Proc. SPIE 6050:605016-1–605016-8, 2005. 43. M. T. Gale, “Fabrication of Continuous-Relief Micro-Optical Elements by Direct Laser Writing in Photoresists,” Opt. Eng. 33(11):3556–3566, 1994. 44. A. G. Poleshchuk, E. G. Churin, V. P. Koronkevich, V. P. Korolkov, A. A. Kharissov, V. V. Cherkashin, V. P. Kiryanov, A. V. Kiryanov, S. A. Kokarev, and A. G. Verhoglyad, “Polar Coordinate Laser Pattern Generator for Fabrication of Diffractive Optical Elements with Arbitrary Structure,” Appl. Opt. 38(8):1295–1301, 1999. 45. I. Baal-Zedaka, S. Hava, N. Mirchin, R. Margolin, M. Zagon, I. Lapsker, J. Azoulay, and A. Peled, “Diffractive Optical Elements Written by Photodeposition,” Appl. Surface Science 208–209(1):226–232, 2003. 46. L. Erdmann, A. Deparnay, F. Wirth and R. Brunner, “MEMS Based Lithography for the Fabrication of Microoptical Components,” in Micromachining Technology for Micrro-Optics and Nano-Optics II, E. G. Johnson and G. P. Nordin eds., Proc. SPIE 5347, SPIE, Bellingham, WA, 2004, pp. 79–84. 47. J. M. Tamkin, B. Bagwell, B. Kimbrough, G. Jabbour, and M. Descour, “High-Speed Gray Sscale Laser Direct Write Technology for Micro-Optic Fabrication,” in Micromachining Technology for Micrro-Optics and Nano-Optics, E. G. Johnson ed., Proc. SPIE 4984, SPIE, Bellingham, WA, 2003, pp. 210–218. 48. J. D. Rogers, A. H. O. Kärkkäinen, T. Tkaczyk, J. T. Rantala, and M. Descour, “Realization of Refractive Microoptics through Grayscale Lithographic Patterning of Photosensitive Hybrid Glass,” Opt. Exp. 12:1294–1303, 2004. 49. L. E. Schmutz, “Hartman Sensing at Adaptive Optics Associates,” Proc. SPIE 779:13–17, 1987. 50. D. D’Amato and R. Centamore, “Two Applications for Microlens Arrays: Detector Fill Factor Improvement and Laser Diode Collimation,” Proc. SPIE 1544:166–177, 1991. 51. Adaptive Optics Associates, Private communication. 52. D. D’Amato, S. Barletta, P. Cone, J. Hizny, R. Martinsen, and L. Schmutz, “Fabricating and Testing of Monolithic Lenslet Module (MLM),” in Optical Fabrication and Testing, pp. 74–77, 1990; Technical Digest Series, vol. 11, Conference ed., June 12–14, 1990, Monterey, California. 53. Adaptive Optics Associates, Commercial Literature. 54. T. D. Milster and J. N. Wong, “Modeling and Measurement of a Micro-Optic Beam Deflector,” to be published in Proc. SPIE 1625. 55. R. K. Luneburg, A Mathematical Theory of Optics, University of California Press, Berkeley, 1964, pp. 164–195.


22.49

56. K. Iga, M. Oikawa, and J. Banno, “Ray Traces in a Distributed-Index Planar Microlens,” Appl. Opt. 21:3451–3455, 1982. 57. M. Oikawa, H. Nemoto, K. Hamanaka, H. Imanishi, and T. Kishimoto, “Light Coupling Characteristics of Planar Microlens,” Proc. SPIE 1544:226–237, 1991. 58. K. Iga and S. Misawa, ‘‘Distributed-Index Planar Microlens and Stacked Planar Optics: A Review of Progress,” Appl. Opt. 25:3388, 1986. 59. K. Iga, Y. Kokobun, and M. Okawa, Fundamentals of Microoptics, Academic Press Inc., Tokyo, 1984. 60. Y. Ohtsuka, “Light-focusing Plastic Rod Prepared from Diallyl Isophthalate-Methyl Methacrylate Copolymerization,” Appl. Phys. Lett. 23:247, 1973. 61. K. Iga and N. Yamanamoto, “Plastic Focusing Fiber for Imaging Applications,” Appl. Opt. 16:1305, 1977. 62. T. Izawa and H. Nakagome, “Optical Waveguide Formed by Electrically Induced Migration of Ions in Glass Plates,” Appl. Phys. Lett. 21:584, 1972. 63. G. D. Khoe, H. G. Kock, J. A. Luijendijk, C. H. J. van den Brekel, and D. Küppers, “Plasma CVD Prepared SiO2/Si3 N4 Graded Index Lenses Integrated in Windows of Laser Diode Packages,” in Technical Digest, Seventh European Conference on Optical Communication, Copenhagen, pp. 7.6-1–7. 6-4, 1981. 64. K. Iga, M. Oikawa, and T. Sanada, Electron. Lett. 17:452, 1981. 65. M. Oikawa, H. Nemoto, K. Hamanaka, and E. Okuda, “High Numerical Aperture Planar Microlenses with Swelled Structure,” Appl. Opt. 29:4077, 1990. 66. O. E. Meyres, “Studies of Transmission Zone Plates,” Am. J. Phys. 19:359, 1951. 67. M. Sussman, “Elementary Diffraction Theory of Zone Plates,” Am. J. Phys. 28:394, 1960. 68. H. Nishihara and T. Suhara, Progress in Optics, North-Holland, New York, vol. 24, pp. 1–37, 1987. 69. M. Young, “Zone Plates and Their Aberrations,” J. Opt. Soc. Am. 62:972–976, 1972. 70. T. D. Milster, R. M. Trusty, M. S. Wang, F. F. Froehlich, and J. K. Erwin, “Micro-Optic Lens for Data Storage,” Proc. SPIE 1499:286–292, 1991. 71. T. Stone and N. George, “Hybrid Diffractive-Refractive Lenses and Achromats,” Appl. Opt. 27:2960–2971, 1988. 72. G. P. Behrmann and J. P. Bowen, “Thermal Effects in Diffractive Lenses,” in Diffractive Optics: Design, Fabrication, and Applications, Technical Digest, Optical Society of America, Washington, D.C., vol. 9, pp. 8–10, 1992. 73. R. Magnusson and T. K. Gaylord, “Diffraction Efficiencies of Thin Phase Gratings with Arbitrary Grating Shape,” J. Opt. Soc. Am. 68:806–809, 1978. 74. J. Futhey and M. Fleming, “Superzone Diffractive Lenses,” in Diffractive Optics: Design, Fabrication, and Applications Technical Digest, Optical Society of America, Washington, D. C., vol. 9, pp. 4–6, 1992. 75. T. Fujita, H. Nishihara, and J. Koyama, “Blazed Gratings and Fresnel Lenses Fabricated by Electron-Beam Lithography,” Opt. Lett. 7:578–580, 1982. 76. R. W. Schneek, “Process Factors for Electroforming Video Disks,” Plating and Surface Finishing, 71(1):38, 1984. 77. K. Goto, K. Mori, G. Hatakoshi, and S. Takahashi, “Spherical Grating Objective Lenses for Optical Disk Pickups,” Proc. International Symposium on Optical Memory, 1987, Jpn. J. Appl. Phys. 26(suppl. 26–24):135, 1987. 78. M. Tanigami, S. Aoyama, T. Yamashita, and K. Imanaka, “Low Wavefront Aberration and High Temperature Stability Molded Micro-Fresnel Lens,” IEEE Photonics Technology Letters 1(11):384–385, 1989. 79. H. Hosokawa and T. Yamashita, “ZnS Micro-Fresnel Lens and Its Uses,” Appl. Opt. 29:5706–5710, 1990. 80. S. Rege, T. Tkaczyk, and M. Descour, “Application of the Alvarez-Humphrey Concept to the Design of a Miniaturized Scanning Microscope,” Opt. Exp. 12:2574–2588, 2004. 81. B. H. W. Hendriks, S. Kuiper, M. A. J. Van Aa, C. A. Renders, and T. W. Tukker, “Electrowetting-Based Variable-Focus Lens for Miniature Systems,” Opt. Rev. 12(3):255–259, 2005. 82. S. Kuiper, B. H. W. Hendriks, R. A. Hayes, B. J. Feenstra, and J. M. E. Baken, “Electrowetting-Based Optics,” Proc. SPIE 5908, SPIE, Bellingham, WA, 2005. 83. N. Chronis, G. L. Liu, K.-H. Jeong, and L. P. Lee, “Tunable Liquid-Filled Microlens Array Integrated with Microfluidic Network,” Opt. Exp. 11(19):2370, 2003. 84. K.-H. Jeong, G. L. Liu, N. Chronis, and L. P. Lee, “Tunable Microdoublet Lens Array,” Opt. Exp. 12(11):2494, 2004.

22.50

COMPONENTS

85. K.-S. Hong, J. Wang, A. Sharonov, D. Chandra, J. Aizenberg, and S. Yang, “Tunable Microfluidic Optical Devices with an Integrated Microlens Array,” J. Micromech. Microeng. 16:1660–1666, 2006. 86. W. Wang and J. Fang, “Design, Fabrication and Testing of a Micromachined Integrated Tunable Microlens,” J. Micromech. Microeng. 16:1221–1226, 2006. 87. A. F. Naumov, M. Yu. Loktev, I. R. Guralnik, and G. V. Vdovin. “Liquid Crystal Adaptive Lenses with Modal Control,” Opt. Lett. 23:992–994, 1998. 88. F. Krogmann, R. Shaik, W. Mönch, and H. Zappe, “Repositionable Liquid Micro-Lenses with Variable Focal Length,” IEEE 707, MEMS 2007, Kobe, Japan, 21–25 January 2007. 89. F. Krogmann, H. Qu, W. Moench, and H. Zappe, “Light-Actuated Push/Pull Manipulation of Liquid Droplets,” IEEE, 2006. 90. K. Aljasem, A. Werber, A. Seifert, and H. Zappe, “Fiber Optic Tunable Probe for Endoscopic Optical Coherence Tomography,” J. Opt. A: Pure Appl. Opt. 10(044012):8, 2008. 91. M. Ye, B. Wang, and S. Sato, “Liquid-Crystal Lens with a Focal Length that is Variable in a Wide Range,” Appl. Opt. 43(35):6407, 2004. 92. B. Wang, M. Ye, and S. Sato, “Lens of Electrically Controllable Focal Length made by a Glass Lens and Liquid-Crystal Layers,” Appl. Opt. 43(17):3420, 2004. 93. A. K. Kirby, P. J. W. Hands, and G. D. Love, “Liquid Crystal Multi-Mode Lenses and Axicons Based on Electronic Phase Shift Control,” Opt. Exp. 15(21):13496, 2007. 94. O. Pishnyak, S. Sato, and O. D. Lavrentovich, “Electrically Tunable Lens Based on a Dual-Frequency Nematic Liquid Crystal,” Appl. Opt. 45(19):4576, 2006. 95. H. Ren, Y.-H. Fan, and S.-T. Wu, “Liquid-Crystal Microlens Arrays Using Patterned Polymer Networks,” Opt. Lett. 29(14), July 15, 2004. 96. C.-C. Cheng, C. A. Chang, C.-H. Liu, and J. A. Yeh, “A Tunable Liquid-Crystal Microlens with Hybrid Alignment,” J. Opt. A: Pure Appl. Opt. 8:S365–S369, 2006. 97. L. Dong, A. K. Agarwal, D. J. Beebe, and H. Jiang, “Adaptive Liquid Microlenses Activated by Stimuliresponsive Hydrogels,” Nature 442, 2006. 98. Z. D. Popovic, R. A. Sprague, and G. A. Neville Connell, “Technique for Monolithic Fabrication of Microlens Arrays,” Appl. Opt. 27:1281–1284, 1988. 99. E. J. Gratrix and C. B. Zarowin, “Fabrication of Microlenses by Laser Assisted Chemical Etching (LACE),” Proc. SPIE 1544:238–243, 1991. 100. V. Diadiuk, Z. L. Liau, and J. N. Walpole, “Fabrication and Characterization of Semiconductor Microlens Arrays,” Proc. SPIE 1354:496–500, 1990. 101. H. W. Choi, E. Gu, C. Liu, C. Griffin, J. M. Girkin, I. M. Watson, and M. D. Dawson, “Fabrication of Natural Diamond Microlenses by Plasma Etching,” J. Vac. Sci. Technol. B 23(1):130–132, Jan/Feb 2005. 102. M. V. Kunnavakkam, F. M. Houlihan, M. Schlax, J. A. Liddle, P. Kolodner, O. Nalamasu, and J. A. Rogers, “Low-Cost, Low-Loss Microlens Arrays Fabricated by Soft-Lithography Replication Process,” Appl. Phys. Lett. 82(8):1152–1154, 2003. 103. X.-C. Yuan, W. X. Yu, M. He, J. Bu, W. C. Cheong, H. B. Niu, and X. Peng, “Soft-Lithography-Enabled Fabrication of Large Numerical Aperture Refractive Microlens Array in Hybrid SiO 2–TiO2 Sol-Gel Glass,” Appl. Phys. Lett. 86:114102, 2005. 104. J. J. Snyder, P. Reichert, and T. M. Baer, “Fast Diffraction-Limited Cylindrical Microlenses,” Appl. Opt. 30:2743–2747, 1991. 105. Ohara Optical Glass Catalog, Ohara Corporation, Somerville, New Jersey, 1990. 106. Corning, Incorporated, Precision Molded Optics Department, Corning, New York, Commercial Literature, 1990. 107. Y. Kokuban and K. Iga, Appl. Opt. 21:1030, 1982. 108. Y. Kokuban, T. Usui, M. Oikawa, and K. Iga, “Wave Aberration Testing System for Microlenses by Shearing Interference Method,” Jap. J. of Appl. Phys. 23(1):101–104, 1984. 109. R. H. Bellman, N. F. Borrelli, L. G. Mann, and J. M. Quintal, “Fabrication and Performance of a One-to-One Erect Imaging Microlens Array for Fax,” Proc. SPIE 1544:209–217, 1991.

23 BINARY OPTICS Michael W. Farn and Wilfrid B. Veldkamp MIT/Lincoln Laboratory Lexington, Massachusetts

23.1

GLOSSARY A C Cm c c(x, y) D f k, l li L, M m P s t Vd x, y, z N J Zi H(x, y)

aspheric describes spherical aberration Fourier coefficients curvature complex transmittance local period focal length running indices paraxial image position direction cosines diffraction order partial dispersion spheric thickness Abbe number Cartesian coordinates wavelength diffraction efficiency paraxial image height phase

23.1

23.2

23.2

COMPONENTS

INTRODUCTION Binary optics is a surface-relief optics technology based on VLSI fabrication techniques (primarily photolithography and etching), with the “binary” in the name referring to the binary coding scheme used in creating the photolithographic masks. The technology allows the creation of new, unconventional optical elements and provides greater design freedom and new materials choices for conventional elements. This capability allows designers to create innovative components that can solve problems in optical sensors, optical communications, and optical processors. The technology has advanced sufficiently to allow the production of diffractive elements, hybrid refractive-diffractive elements, and refractive micro-optics which are satisfactory for use in cameras, military systems, medical applications, and other demanding areas. The boundaries of the binary optics field are not clearly defined, so in this chapter, the concentration will be on the core of the technology: passive optical elements which are fabricated using VLSI technology. As so defined, binary optics technology can be broadly divided into the areas of optical design and VLSI-based fabrication. Optical design can be further categorized according to the optical theory used to model the element: geometrical optics, scalar diffraction theory, or vector diffraction theory; while fabrication is composed of two parts: translation of the optical design into the mask layout and the actual micromachining of the element. The following sections discuss each of these topics in some detail, with emphasis on optical design. For a more general overview, the reader is referred to Refs. 1 for many of the original papers, 2 and 3 for a sampling of applications and research, and 4 to 6 for a lay overview. Directly related areas which are discussed in other chapters but not in this one include microoptics and diffractive optics fabricated by other means (e.g., diamond turning, conventional manufacturing, or optical production), display holography (especially computer-generated holography), mass replication technologies (e.g., embossing, injection molding, or epoxy casting), integrated optics, and other micromachining technologies.

23.3

DESIGN—GEOMETRICAL OPTICS In many applications, binary optics elements are designed by ray tracing and “classical” lens design principles. These designs can be divided into two classes: broadband and monochromatic. In broadband applications, the binary optics structure has little optical power in order to reduce the chromatic aberrations and its primary purpose is aberration correction. The device can be viewed as an aspheric aberration corrector, similar to a Schmidt corrector, when used to correct the monochromatic aberrations and it can be viewed as a material with dispersion an order of magnitude greater than and opposite in sign to conventional materials when used to correct chromatic aberrations. In monochromatic applications, binary optics components can have significant optical power and can be viewed as replacements for refractive optics. In both classes of designs, binary optics typically offers the following key advantages:

s s s s

Reduction in system size, weight, and/or number of elements Elimination of exotic materials Increased design freedom in correcting aberrations, resulting in better system performance Generation of arbitrary lens shapes (including micro-optics) and phase profiles

Analytical Models Representation of a Binary Optics Element As with any diffractive element, a binary optics structure is defined by its phase profile H(x, y) (z is taken as the optical axis), design wavelength N 0, and the surface on which the element lies. For simplicity, this surface is assumed to be planar for the remainder of this chapter, although this is commonly not the case. For example, in many refractive/diffractive systems,

BINARY OPTICS

23.3

the binary optics structure is placed on a refractive lens which may be curved. The phase function is commonly represented by either explicit analytical expression or decomposition into polynomials in x and y (e.g., the HOE option in CODE V). Explicit analytic expressions are used in simple designs, the two most common being lenses and gratings. A lens used to image point (xo, yo, zo) to point (xi, y i, zi) at wavelength N 0 has a phase profile

F( x , y )

2P § z L0 ©¨ o

(x x ) /z ( y y ) /z 1 1

z (x x ) /z ( y y ) /z 1 1 ¶ ¸· 2

o

2

i

i

2 o

2 i

2

o

2

i

2 o

(1)

2 i

where zo and zi are both taken as positive to the right of the lens. The focal length is given by the gaussian lens formula: 1/f 0 1/z i 1/z o

(2)

with the subscript indicating that f 0 is the focal length at N 0. A grating which deflects a normally incident ray of wavelength N 0 to the direction with direction cosines (L, M) is described by

F( x , y )

2P (xL yM ) L0

(3)

Axicons are circular gratings and are described by

F( x , y )

2P ( x 2 y 2 L) L0

(4)

where L now describes the radial deflection. For historical reasons, the polynomial decomposition of the phase profile of the element commonly consists of a spheric term and an aspheric term:

F ( x , y ) FS ( x , y ) F A ( x , y )

(5)

where

F A (x , y)

2P £ akl x k y l L0 £ k l

and the spheric term H S(x, y) takes the form of Eq. (1). Since the phase profiles produced by binary optics technology are not constrained to be spheric, H S(x, y) is often set to zero by using the same object and image locations and the aspheric term alone is used to describe the profile. The binary optics element is then optimized by optimizing the polynomial coefficients akl. If necessary, the aspheric term can be forced to be radially symmetric by constraining the appropriate polynomial coefficients. It is possible to describe the phase profile of a binary optics element in other ways. For example, H(x, y) could be described by Zernike polynomials or could be interpolated from a two-dimensional look-up table.

23.4

COMPONENTS

Ray Tracing by the Grating Equation A binary optics element with phase H(x, y) can be ray traced using the grating equation by modeling the element as a grating, the period of which varies with position. This yields L` L

mL t F 2P t x

M` M

mL t F 2P t y

(6)

(7)

where m is the diffracted order, L, M are the direction cosines of the incident ray, and L`, M` are the direction cosines of the diffracted ray.7 In geometrical designs, the element is usually blazed for the first order (m 1). Note that it is the phase gradient H(x, y) (a vector quantity proportional to the local spatial frequency) and not the phase H(x, y) which appears in the grating equation. The magnitude of the local period is inversely proportional to the local spatial frequency and given by D(x , y) 2P /| F |

(8)

where || denotes the vector magnitude. The minimum local period determines the minimum feature size of the binary optics structure, a concern in device fabrication (see “Fabrication” later in this chapter). Ray Tracing by the Sweatt Model The Sweatt model,8 which is an approximation to the grating equation, is another method for ray tracing. The Sweatt approach models a binary optics element as an equivalent refractive element and is important since it allows results derived for refractive optics to be applied to binary optics. In the Sweatt model, a binary optics element with phase H(x, y) at wavelength N0 is replaced by a refractive equivalent with thickness and refractive index given by t (x , y)

L0 F( x , y ) t0 n0 1 2P

(9)

L (n 1) L0 0

(10)

n(L ) 1

Here, t0 is a constant chosen to make t(x, y) always positive and n0 is the index of the material at wavelength N0. The index n0 is chosen by the designer and as n0 l c , the Sweatt model approaches the grating equation. In practice, a value of n0 10,000 is sufficiently high for accurate results.9 In the special case of a binary optics lens described by Eq. (1), the more accurate Sweatt lens10 can be used. In this case, the element is modeled by two surfaces of curvature co 1 /[(1 n0 )z o ]

(11)

ci 1 /[(1 n0 )z i ]

(12)

and conic constant n02, with the axis of each surface passing through the respective point source. The refractive index is still modeled by Eq. (10).

Aberration Correction Aberrations of a Binary Optics Singlet As a simple example of a monochromatic imaging system, consider a binary optics singlet which is designed to image the point (0, 0, zo) to the point (0, 0, zi)

BINARY OPTICS

23.5

FIGURE 1 Primary aberrations of a binary optics lens.7

at wavelength N0. The phase profile of this lens can be derived from Eq. (1) and the focal length f0 from Eq. (2). Now consider an object point of wavelength N located at (0, Zo, lo). The lens will form an image at (0, Zi, li) (see Fig. 1), with the paraxial image position li and height Zi given by7 1 L 1 li f 0 L0 l0

(13)

Xi /li Xo /lo

(14)

Note that the first equation is just the gaussian lens law but using a wavelength-dependent focal length of f (L ) f 0

L0 L

(15)

The focal length being inversely proportional to the wavelength is a fundamental property of diffractive lenses. In addition, due to the wavelength shift and position change of the object point, the lens will form a wavefront with a primary aberration of 7 1 §¤ 1 1 ³ L ¤ 1 1 ³ ¶ W (x , y) ¨¥ 3 3 ´ ¥ 3 3 ´ ·(x 2 y 2 )2 8 ¨¦ li lo µ L0 ¦ z i z o µ · ¸ ©

1 ¤ 1 1³

X y(x 2 y 2 ) 2li ¥¦ li2 lo2 ´µ i

3 ¤ 1 1³ 2 2 1 ¤ 1 1³ 2 2

X y 2 ¥ ´ Xi x 4li2 ¥¦ li lo ´µ i 4li ¦ li lo µ

(16)

where the ray strikes the lens at (x, y). The first term is spherical aberration, the second is coma, and the last two are tangential and sagittal field curvature. As noted by Welford, all the off-axis aberrations can be eliminated if and only if li lo, a useless configuration. In most systems of interest, the limiting aberration is coma. The performance of the binary optics singlet can be improved by introducing more degrees of freedom: varying the stop position, allowing the binary optics lens to be placed on a curved surface, using additional elements, etc. For a more detailed discussion, see Refs. 1, 7, and 11. Chromatic Aberration Correction Binary optics lenses inherently suffer from large chromatic aberrations, the wavelength-dependent focal length [Eq. (15)] being a prime example. By themselves,

COMPONENTS

they are unsuitable for broadband imaging and it has been shown that an achromatic system consisting only of diffractive lenses cannot produce a real image.12 However, these lenses can be combined successfully with refractive lenses to achieve chromatic correction (for a more detailed discussion than what follows, see Refs. 4, 13, and 14). The chromatic behavior can be understood by using the Sweatt model, which states that a binary optics lens behaves like an ultrahigh index refractive lens with an index which varies linearly with wavelength [let n0 l c in Eq. (10)]. Accordingly, they can be used to correct the primary chromatic aberration of conventional refractive lenses but cannot correct the secondary spectrum. For the design of achromats and apochromats, an effective Abbe number and partial dispersion can also be calculated. For example, using the C, d, and F lines, the Abbe number is defined as Vd [n(Ld ) 1]/[n(L F ) n(LC )]. Substituting Eq. (10) and letting n0 l c yields Vd Ld /(L F LC ) 3.45

(17)

In a similar fashion, the effective partial dispersion using the g and F lines is PgF (L g L F )/(L F LC ) 0.296

(18)

By using these effective values, the conventional procedure for designing achromats and apochromats15 can be extended to designs in which one element is a binary optics lens. Figure 2 plots the partial dispersion PgF versus Abbe number Vd for various glasses. Unlike all other materials, a binary optics lens has a negative Abbe number. Thus, an achromatic doublet can be formed by combining a refractive lens and a binary optics lens, both with positive power. This significantly reduces the lens curvatures required, allowing for larger apertures. In addition, the binary optics lens has a position in Fig. 2 which is not collinear with the other glasses, thus also allowing the design of apochromats with reduced lens curvatures and larger apertures. Monochromatic Aberration Correction For a detailed discussion, the reader is referred to Refs. 1 and 11. As a simple example,4 consider a refractive system which suffers from third-order spherical aberration and has a residual phase given by

Fr ( x , y )

2P C(x 2 y 2 )2 L

(19)

!

23.6

$

#! %

FIGURE 2

"

Partial dispersion vs. Abbe number.14

BINARY OPTICS

23.7

where C describes the spherical aberration. Then, a binary optics corrector with phase

Fb (x , y)

2P C(x 2 y 2 )2 L0

(20)

will completely correct the aberration at wavelength N 0 and will reduce the aberration at other wavelengths to

Fr Fb

2P C(1 L / L0 )(x 2 y 2 )2 L

(21)

The residual aberration is spherochromatism.

Micro-Optics Binary optics technology is especially suited for the fabrication of micro-optics and micro-optics arrays, as shown in Fig. 3. The advantages of binary optics technology include the following:

s s

Uniformity and coherence. If desired, all micro-optics in an array can be made identical to optical tolerances. This results in coherence over the entire array (see Fig. 4). Refractive optics. Binary optics is usually associated with diffractive optics. This is not a fundamental limit but results primarily from fabrication constraints on the maximum achievable depth (typically, 3 μm with ease and up to 20 μm with effort). However, for many micro-optics, this is sufficient to allow the etching of refractive elements. For example, a lens of radius R0

FIGURE 3 96 r 64 array of 51 r 61 μm CdTe microlenses.

23.8

COMPONENTS

FIGURE 4 Micro-optic telescope using (a) coherent arrays and (b) incoherent arrays.

which is corrected for spherical aberration15 and focuses collimated light at a distance z0 (see Fig. 5) has a thickness of t max n § R02 z 02 z 0 ¶ /(n 1) © ¸

(22)

where n is the index of the material.

s s s

Arbitrary phase profiles. Binary optics can produce arbitrary phase profiles in micro-optics just as easily as in macro-optics. Fabricating arrays of anamorphic lenses to correct the astigmatism of semiconductor lasers, for example, is no more difficult than fabricating arrays of conventional spherical lenses. 100 percent fill factor. While many technologies are limited in fill factor (e.g., round lenses on a square grid yield a 79 percent fill factor), binary optics can achieve 100 percent fill factor on any shape grid. Spatial multiplexing. Each micro-optic in an array can be different from its neighbors and the array itself can compose an arbitrary mosaic rather than a regular grid. For example, a binary optics array of individually designed micro-optics can be used to optimally mode-match onedimensional laser arrays to laser cavities or optical fibers.16

Optical Performance Wavefront Quality The wavefront quality of binary optics components is determined by the accuracy with which the lateral features of the element are reproduced. Since the local period (typically several μm) is usually much larger than the resolution with which it can be reproduced (of order 0.1 μm), wavefront quality is excellent. In fact, wavefront errors are typically limited by the optical quality of the substrate rather than the quality of the fabrication.

FIGURE 5

Thickness of a refractive lens.15

BINARY OPTICS

23.9

Diffraction Efficiency The diffraction efficiency of a device is determined by how closely the binary optics stepped-phase profile approximates a true blaze. The theoretical efficiency at wavelength N of an element with I steps designed for use at N0 is4

H(L , I ) sinc (1/I )

sin (IPA ) I sin PA

2

(23)

where sinc (x) sin (R x)/(Rx) and A (L0 /L 1)/I . This result is based on scalar theory, assumes perfect fabrication, and neglects any material dispersion. Figure 6 plots the efficiency J(N, I) for different numbers of steps I; while Table 1 gives the average efficiency over the bandwidth $N for a perfectly blazed element (I l c).4 The efficiency equation is asymmetric in N but symmetric in 1/N. The use of scalar theory in the previous equation assumes that the local period D(x, y) [see Eq. (8)] is large compared to the wavelength. As a rule of thumb, this assumption begins to lose validity when the period dips below 10 wavelengths (e.g., a grating with period less than 10N 0 or a lens faster than F/5) and lower efficiencies can be expected in these cases. For a more detailed discussion, see Ref. 17. The efficiency discussed here is the diffraction efficiency of an element. Light lost in this context is primarily diffracted into other diffraction orders, which can also be traced through a system to determine their effect. As with conventional elements, binary optics elements will also suffer reflection losses which can be minimized in the usual manner.

1.00 0.99

1.0

Continuous 0.95 = 16

0.81

Efficiency (, )

0.8 =8 0.6

=4

0.41 0.4

=2 0.2 0.8

1.0

1.2 /

FIGURE 6

Diffraction efficiency of binary optics.4

1.4

23.10

COMPONENTS

TABLE 1 Average Diffraction Efficiency for Various Bandwidths4

23.4

$N /N 0

H

0.00 0.10 0.20 0.30 0.40 0.50 0.60

1.00 1.00 0.99 0.98 0.96 0.93 0.90

DESIGN—SCALAR DIFFRACTION THEORY Designs via scalar diffraction theory are based on the direct manipulation of the phase of a wavefront. The incident wavefront is generally from a coherent source and the binary optics element manipulates the phase of each point of the wavefront such that the points interfere constructively or destructively, as desired, at points downstream of the element. In this regime, binary optics can perform some unique functions, two major applications being wavefront multiplexing and beam shaping.

Analytical Models In the scalar regime, the binary optics component with phase profile H(x, y) is modeled as a thinphase screen with a complex transmittance of c(x , y) exp[ j F(x , y)]

(24)

The phase screen retards the incident wavefront and propagation of the new wavefront is modeled by the appropriate scalar formulation (e.g., angular spectrum, Fresnel diffraction, Fraunhofer diffraction) for nonperiodic cases, or by Fourier series decomposition for periodic cases. The design of linear gratings is an important problem in the scalar regime since other problems can be solved by analogy. A grating with complex transmittance c(x) and period D can be decomposed into its Fourier coefficients Cm, where Cm

1 D c(x ) exp( j 2P mx /D) dx D ¯0

(25)

c

c( x )

£ Cm exp( j 2Pmx /D)

(26)

m c

The relative intensity or efficiency of the mth diffracted order of the grating is

Hm | Cm |2

(27)

Due to the fabrication process, binary optics gratings are piecewise flat. The grating transmission in this special case can be expressed as c(x) ci for xi < x < xi1, where ci is the complex transmission of step i of I total steps, x0 0, and xI D. The Fourier coefficients then take the form I 1

Cm £ ciD i exp ( j 2P m$ i ) sinc (mD i ) i 0

where D i (x i 1 x i )/D and $ i (x i 1 x i )/(2D).

(28)

BINARY OPTICS

23.11

The sinc term is due to the piecewise flat nature of the grating. If, in addition to the above, the grating transition points are equally spaced, then xi iD/I and Eq. (28) reduces to §1 I 1 ¶ Cm exp( j P m/I ) sinc (m/I ) ¨ £ ci exp( j 2P mi /I )· I © i 0 ¸

(29)

The bracketed term is the FFT of ci, which makes this case attractive for numerical optimizations. If the complex transmittance is also stepped in phase by increments of H0, then ci exp( ji F0 ) and Eq. (29) further reduces to18 Cm exp { j P[(I 1)A m/I ]} sinc (m/I )

sin (IPA ) I sin PA

(30)

where A F0 /(2P ) m/I . This important case occurs whenever a true blaze is approximated by a stepped-phase profile. The efficiency equation [Eq. (23)] is a further specialization of this case.

Wavefront Multiplexers Grating Designs Grating multiplexers (also known as beam-splitter gratings) split one beam into many diffracted beams which may be of equal intensity or weighted in intensity.19 Table 2 shows some common designs. In general, the designs can be divided into two categories: continuous phase and binary. Continuous phase multiplexers generally have better performance, as measured by the total efficiency and intensity uniformity of the diffracted beams, while binary multiplexers are easier to fabricate (with the exception of several naturally occurring continuous phase profiles). Upper bounds for the efficiency of both continuous and binary types are derived in Ref. 20. If the phase is allowed to be continuous or nearly continuous (8 or 16 phase levels), then the grating design problem is analogous to the phase retrieval problem and iterative techniques are commonly used.21 A generic problem is the design of a multiplexer to split one beam into K equal intensity beams. Fanouts up to 1:50 with perfect uniformity and efficiencies of 90–100 percent are typical. The complex transmittance of a binary grating has only two possible values [typically 1 and −1, or exp ( jH0) and exp (−jH0)], with the value changing at the transition points of the grating. By nature, the response of these gratings has the following properties:

s s s

The intensity response is symmetric; that is, Jm J m. The relative intensities of the nonzero orders are determined strictly by the transition points. That is, if the transition points are held constant, then the ratios Jm/Jn for all m, n w 0 will be constant, regardless of the actual complex transmittance values. The complex transmittance values only affect the balance of energy between the zero and nonzero orders.

TABLE 2

Grating Multiplexers of Period D, 0 < x < D

Phase Profile ª0 F( x , y ) « ¬P

x D /2 D /2 x

ª0 x D /2 F( x , y ) « ¬2.01 D /2 x F(x , y) P x /D F(x , y) arctan[2.657 cos (2P x /D)]

J−1

J0

J1

0.41

0

0.41

Binary 1:2 splitter

0.29

0.29

0.29

Binary 1:3 splitter

r 0.31

0.41 0.31

0.41 0.31

Continuous 1:2 splitter Continuous 1:3 splitter

Remarks

23.12

COMPONENTS

Binary gratings are usually designed via the Dammann approach or search methods and tables of binary designs have been compiled.22,23 Efficiencies of 60–90 percent are typical for the 1:K beamsplitter problem. Multifocal Lenses The concepts used to design gratings with multiple orders can be directly extended to lenses and axicons to design elements with multiple focal lengths by taking advantage of the fact that while gratings are periodic in x, paraxial lenses are periodic in (x2 y2), nonparaxial lenses in x 2 y 2 f 02 , and axicons in x 2 y 2 . For gratings, different diffraction orders correspond to plane waves traveling in different directions, but for a lens of focal length f0, the mth diffraction order corresponds to a lens of focal length f0/m. By splitting the light into different diffraction orders, a lens with multiple focal lengths (even of opposite sign if desired) can be designed. As an example, consider the paraxial design of a bifocal lens, as used in intraocular implants. Half the light should see a lens of focal length f0, while the other half should see no lens. This is a lens of focal length f0, but with the light split evenly between the 0 and 1 orders. The phase profile of a single focus lens is given by F(r ) 2P r 2 /(2L0 f 0 ), where r2 x2 y2. This phase, with the 2R ambiguity removed, is plotted in Fig. 7a as a function of r and in Fig. 7b as a function of r2, where the periodicity in r2 is evident. To split the light between the 0 and 1 orders, the blaze of Fig. 7b is replaced by the 1:2 continuous splitter of Table 2, resulting in Fig. 7c. This is the final design and the phase profile is displayed in Fig. 7d as a function of r.

2 Phase (radians)

Phase (radians)

2

0 0.0

0.5

1.0

1.5

0

2.0

0

1

3

4

1.5

2.0

2 Phase (radians)

Phase (radians)

2

0

2 2/2 0 0 (b)

/ 2 0 0 (a)

0

1

2 2

/2 0 0 (c)

3

4

0 0.0

0.5

1.0 / 2 0 0 (d)

FIGURE 7 Designing a bifocal lens: (a) lens with a single focus; (b) same as (a), but showing periodicity in r2; (c) substitution of a beam-splitting design; and (d) same as (c), but as a function of r.

BINARY OPTICS

(a)

(b)

23.13

(c)

FIGURE 8 Reshaping a focused beam: (a) gaussian focus; (b) deterministic beam-shaper; and (c) statistical diffuser.

Beam Shapers and Diffusers In many cases, the reshaping of a laser beam can be achieved by introducing the appropriate phase shifts via a binary optics element and then letting diffraction reshape the beam as it propagates. If the incoming beam is well characterized, then it is possible to deterministically design the binary optics element.24 For example, Fig. 8a shows the focal spot of a gaussian beam without any beamforming optics. In Fig. 8b, a binary optics element flattens and widens the focal spot. In this case, the element could be designed using phase-retrieval techniques, the simplest design being a clear aperture with a R phase shift over a central region. If the beam is not well-behaved, then a statistical design may be more appropriate.25 For example, in Fig. 8c, the aperture is subdivided into randomly phased subapertures. The envelope of the resulting intensity profile is determined by the subaperture but is modulated by the speckle pattern from the random phasing. If there is some randomness in the system (e.g., changing laser wavefront), then the speckle pattern will average out and the result will be a design which reshapes the beam and is robust to variations in beam shape.

Other Devices Other Fourier optics-based applications which benefit from binary optics include the coupling of laser arrays via filtering in the Fourier plane or other means26 the fabrication of phase-only components for optical correlators,27 and the implementation of coordinate transformations.16,28 In all these applications, binary optics is used to directly manipulate the phase of a wavefront.

23.5

DESIGN—VECTOR DIFFRACTION THEORY Binary optics designs based on vector diffraction theory fall into two categories: grating-based designs and artificial index designs. Grating-based designs rely on solving Maxwell’s equations for diffraction by the element. This is practical for periodic structures. Two major methods for this analysis are the expansion in terms of space harmonics (coupled wave theory) and the expansion in terms of modes (modal theory).29 In this category, optical design is difficult since it can be both nonintuitive and computationally intensive. Artificial index designs are based on the following premise. When features on the component are small compared to the wavelength, then the binary optics element will behave as a material of some average index. Two common applications are shown in Fig. 9. In Fig. 9a, the device behaves as an antireflection coating (analogous to anechoic chambers) since, at different depths, the structure has a different average index, continuously increasing from n1 to n2. In Fig. 9b, the regular, subwavelength structure exhibits form birefringence.30 For light polarized with the electric vector perpendicular to the grooves, the effective index is 1 1 1 p 2 (1 p) 2 2 neff n1 n2

(31)

23.14

COMPONENTS

FIGURE 9 Artificial index designs: (a) antireflection layer and (b) form birefringence.

where p is the fraction of total volume filled by material 1. However, for light polarized with the electric vector parallel to the grooves, 2 neff pn12 (1 p)n22

(32)

In both these cases, the period of the structure must be much less than the wavelength in either medium so that only the zero order is propagating.

23.6

FABRICATION

Mask Layout At the end of the optical design stage, the binary optics element is described by a phase profile H(x, y). In the mask layout process, this profile is transformed into a geometrical layout and then converted to a set of data files in a format suitable for electron-beam pattern generation. From these files, a mask maker generates the set of photomasks which are used to fabricate the element. The first step is to convert the phase profile H(x, y) into a thickness profile (see Fig. 10a and b) by the relation t (x , y)

L0 (F mod 2P ) 2P (n0 1)

(33)

where N0 is the design wavelength and n0 is the index of the substrate at N0. The thickness profile is the surface relief required to introduce a phase shift of F(x , y). The thickness varies continuously from 0 to t0, where t 0 L0 /(n0 1)

(34)

is the thickness required to introduce one wave of optical path length difference. To facilitate fabrication, t(x, y) is approximated by a multilevel profile t`(x, y) (Fig. 10c), which normally would require one processing cycle (photolithography plus etching) to produce each thickness level. However, in binary optics, a binary coding scheme is used so that only N processing cycles are required to produce I 2N thickness levels (hence the name binary optics).

(35)

BINARY OPTICS

23.15

(x) 4 2 x

0 (a) t(x)

t0 x

0 (b) t(x) t0

x

0 (c)

FIGURE 10 Translation from H(x, y) to micromachined surface: (a) phase H(x, y); (b) thickness t(x, y); and (c) binary optics profile t`(x, y).

The photomasks and etch depths required for each processing cycle are determined from contours of the thickness t(x, y) or equivalently the phase H (x, y), as shown in Table 3. The contours can be generated in several ways. For simple phase profiles, the contours are determined analytically. Otherwise, the contours are determined either by calculating the thickness at every point on a grid and then interpolating between points31 or by using a numerical contouring method,32 analogous to tracing fringes on an interferogram. To generate the photomasks, the geometrical areas bounded by the contours must be described in a graphics format compatible with the mask vendor (see Fig. 11a and b). Common formats are GDSII and CIF,33 both of which are high-level graphics descriptions which use the multisided polygon (often limited to 200 sides) as the basic building block. Hierarchical constructions (defining structures in terms of previously defined structures) and arraying of structures are also allowed. The photomasks are usually written by electron-beam generators using the MEBES (moving electron beam exposure system) format as input. Most common high-level graphics descriptions can be translated or “fractured” to MEBES with negligible loss in fidelity via existing translation routines. Currently, commercial mask makers can achieve a minimum feature size or “critical dimension” (CD) of 0.8 μm with ease, 0.5 μm with effort, and 0.3 μm in special cases. The CD of a

TABLE 3

Processing Steps for Binary Optics

Layer

Etch Region, Defined by t(x, y)

Etch Region, Defined by F(x , y)

1 2 3 4

0 < t mod (t0) < t0 /2 0 < t mod (t0 /2) < t0 /4 0 < t mod (t0 /4) < t0 /8 0 < t mod (t0 /8) < t0 /16

0 < H mod 2R < R 0 < H mod R < R /2 0 < H mod R /2 < R /4 0 < H mod R /4 < R /8

Etch Depth t0 /2 t0 /4 t0 /8 t0 /16

23.16

COMPONENTS

FIGURE 11 Mask layout descriptions: (a) mathematical description based on thickness contours; (b) high-level graphics description; and (c) MEBES.

binary optics element is determined by the minimum local period [see Eq. (8)] divided by the number of steps, Dmin/I. For lenses, Dmin 2L0 F

(36)

where F is the F-number of the lens; while, for gratings, Dmin is the period of the grating. In MEBES, all geometrical shapes are subdivided into trapezoids whose vertices lie on a fixed rectangular grid determined by the resolution of the electron-beam machine (see Fig. 11c). The resolution (typically 0.05 μm) should not be confused with the CD achievable by the mask maker. In summary, the description of the photomask begins as a mathematical description based on contours of the thickness profile and ends as a set of trapezoids whose vertices fall on a regular grid (see Fig. 11). This series of translations results in the following artifacts. First, curves are approximated by straight lines. The error introduced by this approximation (see Fig. 12) is

D R(1 cos Q /2) R Q 2 /8

(37)

Normally, the maximum allowable error is matched to the electron-beam resolution. Second, all coordinates are digitized to a regular grid. This results in pixelization artifacts (which are usually negligible), analogous to the ziggurat pattern produced on video monitors when plotting gently sloped lines. Finally, the MEBES writing process itself has a preferred direction since it uses electrostatic beam deflection in one direction and mechanical translation in the other. In addition to the digitized thickness profile, photomasks normally include the following features which aid in the fabrication process. Alignment marks34 are used to align successive photomasks, control features such as witness boxes allow the measurement of etch depths and feature sizes without probing the actual device, and labels allow the fabricator to easily determine the mask name, orientation, layer, etc. Micromachining Techniques Binary optics uses the same fabrication technologies as integrated circuit manufacturing.34,35 Specifically, the micromachining of binary optics consists of two steps: replication of the photomasks

FIGURE 12 Quantization angle.

BINARY OPTICS

23.17

pattern into photoresist (photolithography) and the subsequent transfer of the pattern into the substrate material to a precise depth (etching or deposition). The replication of the photomasks onto a photoresist-covered substrate is achieved primarily via contact, proximity, or projection optical lithography. Contact and proximity printing offer lower equipment costs and more flexibility in handling different substrate sizes and substrate materials. In contact printing, the photomask is in direct contact with the photoresist during exposure. Vacuumcontact photolithography, which pulls a vacuum between the mask and photoresist, results in the highest resolution (submicron features) and linewidth fidelity. Proximity printing, which separates the mask and photoresist by 5 to 50 μm, results in lower resolution due to diffraction. Both contact and proximity printing require 1:1 masks. In projection printing, the mask is imaged onto the photoresist with a demagnification from 1r to 20r. Projection printers are suitable for volume manufacturing and can take advantage of magnified masks. However, they also require expensive optics, strict environmental controls, and can only expose limited areas (typically 2 cm r 2 cm). Following exposure, either the exposed photoresist is removed (positive resist) or the unexposed photoresist is removed (negative resist) in a developer solution. The remaining resist serves as a protective mask during the subsequent etching step. The most pertinent etching methods are reactive ion etching (RIE) and ion milling. In RIE, a plasma containing reactive neutral species, ions, and electrons is formed at the substrate surface. Etching of the surface is achieved through both chemical reaction and mechanical bombardment by particles. The resulting etch is primarily in the vertical direction with little lateral etching (an anisotropic etch) and the chemistry makes the etch attack some materials much more vigorously than others (a selective etch). Because of the chemistry, RIE is material-dependent. For example, RIE can be used to smoothly etch quartz and silicon, but RIE of borosilicate glasses results in micropatterned surfaces due to the impurities in the glass. In ion milling, a stream of inert gas ions (usually Ar) is directed at the substrate surface and removes material by physical sputtering. While ion milling is applicable to any material, it is usually slower than RIE. For binary optics designed to be blazed for a single order (i.e., designs based on geometrical optics), the major effect of fabrication errors is to decrease the efficiency of the blaze. There is little or no degradation in the wavefront quality. Fabrication errors can be classified as lithographic errors, which include alignment errors and over/underexposure of photoresist, and etching errors, which include depth errors and nonuniform etching of the substrate. As a rule of thumb, lithographic errors should be held to less than 5 percent of the minimum feature size (105) and sharp cutoff of the excitation wavelengths, coupled with high transmission of the pass band. With transillumination, one can more reliably combine fluorescence with polarization-based microscopy or carry out polarized fluorescence measurements with greater accuracy, since one can avoid the use of dichromatic beam splitters, which tend to be optically anisotropic. Fluorescence microscopy particularly benefits from electronic imaging, especially with the use of low-noise, chilled CCDs as imaging detectors, digital computers to enhance and rapidly process the signal (such as in ratio imaging), and the new fluorescence-conjugated chemical probes that provide incredible sensitivity and selectivity.11,12,132–135

MICROSCOPES

28.49

For imaging specimens that are labeled with more than two or three types of spectrally distinct fluorophores, a technique known as spectral imaging is becoming available. Spectral imaging combines spectroscopy and imaging, measuring the spectral composition of the light recorded at each point of the image. When spectral imaging is applied to fluorescence microscopy, the filter cube is modified as to transmit a broad range of emission wavelengths. A spectrometer placed before the detector samples the emission spectrum at appropriate resolution and intervals (channels) for wavelengths longer than the excitation wavelength. Spectral imaging systems can either be integrated into the microscope (manufacturers include Leica, Nikon, Zeiss) or can be added to an existing stand (manufacturers include Cambridge Research and Instrumentation Inc., Lightform Inc.). Datasets are typically stored as stacks of images, in which each slice corresponds to a wavelength channel. Powerful algorithms reduce an experimental dataset to indicate for each image point the weighted contributions of pure fluorophores whose spectra are stored in a library.136,137

Confocal Microscopy In confocal microscopy, the specimen is scanned point by point either by displacing the specimen (stage scanning) or by scanning a minute illuminating spot (beam scanning), generally in a TV-raster fashion. In either case, the scanning spot is an Airy disk formed by a high-NA objective lens. An exit pinhole is placed conjugate to the spot being scanned so that only the light originating from the scanned spot is transmitted through the exit pinhole. Thus, light originating from other regions of the specimen or optical system is prevented from reaching the photo detector (Fig. 38).138,139 This optical arrangement reduces blurring of the image from out-of-focus light scattering, fluorescence, and the like, and yields exceptionally clear, thin optical sections. The optical sections can then be processed and assembled electronically to yield three-dimensional displays or tilted plane projections. Alternatively, the specimen itself can be scanned through a tilted plane (e.g., by implementing a series of x scans with y, z incremented) to yield a section viewed from any desired orientation, including that normal to the microscope axis.

FIGURE 38 Optical path in simple confocal microscope. The condenser lens C forms an image of the first pinhole A onto a confocal spot D in the specimen S. The objective lens O forms an image of D into the second (exit) pinhole B which is confocal with D and A. Another point, such as E in the specimen, would not be focused at A or B, so that the illumination would be less and, in addition, most of the light g-h scattered from E would not pass the exit pinhole. The light reaching the phototube P from E is thus greatly attenuated compared to that from the confocal point D. In addition, the exit pinhole could be made small enough to exclude the diffraction rings in the image of D, so that the resolving power of the microscope is improved. The phototube provides a signal of the light passing through points D1, D2, D3, etc. (not shown), as the specimen is scanned. D1, D2, D3, etc. can lie in a plane normal to the optical axis of the microscope (as in conventional microscopy), or parallel to it, or at any angle defined by the scanning pattern, so that optical sections can be made at angles tilted from the conventional image plane. Since, in the stage-scanning system, D is a small spot that lies on the axis of the microscope, lenses C and O can be considerably simpler than conventional microscope lenses.138,139

28.50

INSTRUMENTS

The stage-scanning confocal microscope can yield vastly expanded fields of view. Here the image area is not limited by the field of view of the optics but only by the range of movement of the specimen and ability of the photo detector and processor to handle the vast information generated at high speed. Furthermore, the objective lens needs only to be corrected for a narrow field of view on axis.6,138 Laser disk recorders are a form of application that takes advantage of these attributes. The beam-scanning confocal microscope is typically implemented in the reflective or epi-illumination mode. This mode has the advantage that the illuminating beam and the returning light scattered back by the sample pass through the same objective lens and beam-steering devices needed for scanning the sample. The prototype of a modern beam-scanning confocal microscope uses two galvanometric mirrors (one for each dimension of a two-dimensional image) that scan a focused laser beam across a stationary sample field. The backscattered light is collected by the objective and bounces off the same mirrors which “descan” the returning light before it passes through a stationary beamsplitter (to separate the backscattered light from the incoming beam) and a stationary exit pinhole. The exit pinhole is located in a conjugate image plane, while the scanning mirrors are located in positions that are conjugate to the back focal plane of the objective lens. By (indirectly) placing the mirrors into the objective back focal plane, the angular scan of the mirrors is translated into a positional scan of the focused laser beam in the specimen. Beam-scanning microscopes typically require additional relay optics that project the objective back focal plane into the mirror locations. The laser-scanning, epi-illuminating confocal microscope was developed into a practical instrument in the late 1980s and immediately adopted with great enthusiasm for fluorescence imaging in the life sciences. Because laser beams are typically highly collimated, a source or entrance pinhole is commonly omitted in this instrument. The beam splitter combining and separating the illumination and imaging paths is implemented as a dichroic (also called dichromatic) mirror providing high reflectivity at short wavelengths and high transmissivity at longer wavelengths (or vice versa, depending on the particular optical design). A whole industry has evolved around designing and manufacturing dichromatic mirrors that are appropriate for specific fluorescent dyes and combination of dyes. For direct viewing of confocal images in reflective mode a Nipkow disk is used for scanning multiple beams across a stationary sample field. The multiple beams originate in many thousands of pinholes arranged helically on a modified Nipkow disk that is located in the image plane of the objective lens. Thus, a single spinning disk can be made to provide synchronously scanning entrance and exit pinholes.140,141 To overcome the considerable light loss associated with the original designs by Petrán and Kino, Yokogawa Electric Corp. employed a second, coaxially aligned Nipkow disk containing microlenses in its CSU-10 disk confocal scanner (Fig. 39). Each pinhole on the first Nipkow disk has a corresponding microlens on the second Nipkow disk that focuses the laser light into the pinhole. Thus, the light efficiency is increased by a factor equal to the ratio of the microlens to pinhole area. Instead of the 1 percent or so found with conventional Nipkow disk systems, some 40 to 60 percent of the light impinging on the disk containing the microlenses becomes transmitted through the pinholes to illuminate the specimen. Accordingly, the CSU-10 provides a light efficient scan unit that permits direct visual viewing of the confocal image, a great advantage when studying moving objects such as living cells.142,143 In a confocal microscope, the exit pinhole can be made smaller than the diameter of the Airy diffraction image formed by the objective lens so that the Airy disk is trimmed down to regions near its central peak. With this optical arrangement, the unit diffraction pattern that makes up the image turns out to be the square of the Airy pattern given in Eq. (2). Thus, the radius at half maximum of the central bright area (Airy disk) is reduced by a factor of 1.36. (The radial position of the first minimum in both patterns is still equal to rAiry.) The shape of the unit diffraction pattern is thus sharpened so that, compared to nonconfocal imaging, two points which radiate incoherently (as in fluorescence microscopy) can be expected to approach each other by up to a factor of 2 closer to each other before their diffraction patterns encounter the Rayleigh limit. In Fig. 20 the contrast transfer characteristics of a confocal microscope in the coherent imaging mode is compared with the nonconfocal, incoherent imaging mode using the same lenses. Note that the limiting resolution is the same for both imaging modes, while the contrast transfer of the confocal mode increase more steeply for increasing grating periods.

MICROSCOPES

28.51

FIGURE 39 Schematic of the Yokogawa CSU-10 confocal disk scanner. The expanded and collimated laser beam illuminates the upper Nipkow disk containing some 20,000 microlenses. Each microlens focuses the laser beam onto its corresponding pinhole, thus significantly raising the fraction of the illuminating beam that is transmitted by the main Nipkow disk containing the pinhole array. The backscattered or fluorescent light is collected by the objective lens and focused back onto the Nipkow disk containing the array of pinholes, which now act as confocal exit pinholes. A beam splitter located between the first and second Nipkow disk reflects the light toward a camera. A lens projects an image of the pinhole array onto the camera, that acquires a confocal image while the Nipkow disk rotates with high speed. After carefully designing the array pattern and implementing a precise and vibration-free rotation of the Nipkow disks, the confocal disk scanner can produce clean, high-resolution images free of fixed pattern noise. In fluorescence imaging, the camera can be replaced by an ocular for direct viewing of the confocal image. (Schematic provided by Yokogawa Electric Corporation.)

Rather than using confocal optics to eliminate image blurring from out-of-focus planes, one can achieve the same end by computational deconvolution of a stack of serial optical sections obtained by wide field microscopy.49,144,145 While computationally intensive and time consuming, this image restoration method allows one to isolate clean optical sections from a stack of images that can be acquired at higher speed and higher efficiency than with laser-scanning confocal microscopy and in modes of contrast generation typically not accessible to confocal imaging. Alternatively, thin optical sections can be obtained directly with digital enhanced video microscopy using high-NA condenser and objective lenses. Requiring little processing, this approach is especially convenient when many stacks of optical sections have to be acquired at high rates in succession, for example in order to record rapid, three-dimensional changes in microscopic domains over time.

28.52

INSTRUMENTS

Structured Illumination The quest for improved resolution beyond the diffraction limit has led to the development of several methods that modify the illumination pattern in wide-field microscopy. In standard wide field microscopy, the specimen is illuminated using condenser optics that ideally projects a uniform field of light into the specimen. In structured illumination setups, however, a finely patterned illuminating field is projected into the specimen, providing a means for generating optical sections similar to confocal microscopy and for improving resolution. Wilson and colleagues146 first described a simple method of obtaining optical sectioning in a conventional wide-field microscope by projecting a single-spatial-frequency grid pattern onto the object. Images taken at three spatial positions of the grid were processed in real time to produce optically sectioned images that are substantially similar to those obtained with confocal microscopes. The sectioning capability is achieved by superimposing an illumination pattern that is only in focus at a thin section through the specimen, while all other sections of the specimen are illuminated with a more or less blurred version of the pattern. The specimen with the superimposed grid pattern is then imaged with regular wide field optics focused on the grid pattern inside the specimen. Hence, all image features that have the grid pattern imposed on them are located in this specimen section, while image features from other sections of the specimen appear nearly uniformly illuminated. For removing those out-of-focus features and removing the intruding effect of the illumination pattern on the specimen image, three raw sample images are recorded, each with the illumination pattern slightly shifted in position. Subsequently, the raw images are computationally combined to generate an optical section of the sample without the grid pattern noticeable in the image. The company Carl Zeiss has adopted this strategy in its ApoTome slider module for generating optical sections using epi-illumination. Instead with a regular grid pattern, the sample can also be illuminated with a random speckle pattern to provide depth discrimination in thick, fluorescently labeled tissues.147,148 The technique consists of illuminating a sample with a sequence of speckle patterns and displaying the differential intensity variance of the resultant sequence of fluorescence images. The advantage of speckle illumination is that it provides diffraction-limited illumination granularity that is highly contrasted even in scattering media. Structured illumination strategies that go beyond optical sections and provide lateral resolution that exceeds the classical diffraction limit by a factor of 2 or more have been devised by Gustafsson.149 The sample is illuminated with a series of excitation light patterns, which cause normally inaccessible high-resolution information to be encoded into the observed image. The recorded images are linearly processed to extract the new information and produce a reconstruction with twice the normal resolution. Unlike confocal microscopy, the resolution improvement is achieved with no need to discard any of the light arriving from the specimen. In addition to improving the lateral resolution this method can be applied in three dimensions to double the axial as well as the lateral resolution, with true optical sectioning.150 A grating is used to generate three mutually coherent light beams, which interfere in the specimen to form an illumination pattern that varies both laterally and axially. The spatially structured excitation intensity causes normally unreachable high-resolution information to become encoded into the observed images through spatial frequency mixing. This new information is computationally extracted and used to generate a three-dimensional reconstruction with twice as high resolution, in all three dimensions, as is possible in a conventional wide-field microscope. Structured illumination is primarily used in fluorescence microscopy, where in principle it is capable of unlimited resolution. To achieve unlimited resolution, structured illumination has to be combined with a nonlinear dependence of the fluorescence emission rate on the illumination intensity.151,152 As an example of this concept, Gustafsson experimentally demonstrated saturated structured-illumination microscopy, in which the nonlinearity arises from saturation of the excited state. This method can be used in a simple, wide-field (nonscanning) microscope, which uses only a single, inexpensive laser, and requires no unusual photophysical properties of the fluorophore. The practical resolving power is determined by the signal-to-noise ratio, which in turn is limited by photobleaching. Experimental results show that a two-dimensional point resolution of < 50 nm is possible on sufficiently bright and photostable samples.

MICROSCOPES

28.53

Light Field Instead of increasing resolution in a single image plane, it is sometimes desirable to trade lateral resolution for axial resolution in a three-dimensional image stack. To this end, Marc Levoy and colleagues20 have replaced the regular camera on a standard, wide-field microscope by a camera with microlens array, a so-called light field camera or plenoptic camera.153 The array consists of a hundred thousand or more microlenses arranged in a square up to the size of the microscope’s field number. The array is placed in the intermediate image plane of the objective lens. Behind the array the sensor chip is located in the backfocal plane of the microlenses. In other words, the light field camera samples the specimen image on a regular grid at intervals that corresponds to the pitch of the microlens array. At each grid point the camera captures a small subimage of the objective’s back focal plane. Hence, the camera captures a hybrid image of the specimen that is sampled not only in space but also along different directions through the specimen. The raw light field image, when presented to the eye, cannot be directly interpreted since it consists of a multitude of small disk-shaped images (of the objective’s back focal plane) arranged on a regular grid. However, a single light field image is used to reconstruct a multitude of conventional images of a specimen that is viewed along different directions or focused to different object planes.20 These differing views are all based on a single light field image that was captured by a single camera exposure. Hence light field microscopy can be especially useful when imaging three-dimensional structures that change rapidly in time, such as living cells and tissues. Based on a single snapshot one can reconstruct a stack of optical sections that were all recorded at the same point in time, thus avoiding registration problems between sections. However, the versatility of generating different views and optical sections from a single light field image comes at a price. The sacrifice one makes is a reduction in image size. Specifically, if each microlens subimage contains N rN pixels, then the computed images will contain N 2 fewer pixels than if the microlenses were not present. In return, we can compute N2 unique oblique views of the specimen, and we can generate a focal stack containing N slices with nonoverlapping depths of field.20 The recording of light field images is compatible with several contrast modes, including fluorescence and polarized light microscopy. One of the first areas to take advantage of simultaneous optical sections was fluorescence microscopy of functional neuronal tissues and the recording of threedimensional excitation patterns. In polarized light field microscopy, the microlens array generates a hybrid image consisting of an array of small conoscopic images, each sampling a different object area.21 Analysis of the array of conoscopic images reveals the birefringence of each object area as a function of the propagation direction of transmitted light rays. Compared to traditional conoscopy and related methods, the vastly improved throughput and quantitative analysis afforded by the light field LC-PolScope, for example, make it the instrument of choice for measuring three-dimensional birefringence parameters of complex structures. Since light field microscopy was implemented only a few years ago, additional application areas of this new method are likely to emerge in the future.

Aperture Scanning In the aperture-scanning microscope devised by Ellis for phase-contrast microscopy, the tip of a flexible signal optical fiber, illuminated by an Hg arc, makes rapid circular sweeps at the periphery of the condenser aperture.65 This circular, scanning illumination spot replaces the conventional phase annulus in the condenser aperture plane. A quarter-wave plate and absorber, both covering only a small area conjugate to the illuminating spot, spins in synchrony with the fiber at the objective back aperture (or its projected conjugate). Thus, the specimen is illuminated by a narrow, coherent beam that enters the specimen obliquely at high NA, with the azimuth orientation of the beam swinging around and around to generate a full cone of illumination within the integration time of the detector. With this aperture-scanning approach, the specimen is illuminated by a large-NA cone of light which is temporally incoherent, with the phase disk covering only a small fraction of the area normally occupied by the phase ring in conventional phase-contrast systems. The small size of the phase

28.54

INSTRUMENTS

disk, while appropriately reducing the amplitude and introducing the requisite N/4 wave phase retardation to the rays not scattered by the specimen, allows the transmission of a much larger fraction of the scattered rays that carry the high spatial frequency information. The aperture-scanning phase-contrast microscope thus provides a very thin optical section. The image is also virtually free of the phase halo that obscures image detail adjacent to refractile boundaries in conventional phasecontrast microscopy. For polarized light microscopy an aperture scanning scheme was designed and built by Shribak and Oldenbourg using a liquid crystal device in the front focal plane of the condenser lens. 131,154 The liquid crystal device was designed for two functions: (1) to create oblique illumination of the specimen, and (2) to measure the birefringence parameters of microscopic objects for each of four oblique tilt angles of illumination. By measuring the object retardance along each of the four tilted projections, the inclination angle of the optic axis of birefringent objects was revealed, in addition to the orientation or azimuth angle in the plane of focus. The inclination angle of the optic axis is usually not evident from traditional polarized light images (see section on polarized light). Extending this concept, modulation of the transfer functions of the condenser and objective apertures with electro-optical devices should open up intriguing new opportunities. Such modulation eliminates the need for mechanical scanning devices, the spatial distribution of the modulation function can be altered at will, and the amplitude and phase of light passing each point in the aperture can be adjusted rapidly, even coupled dynamically to the image signal through a feedback loop to generate dynamic spatial filters that enhance or select desired features in the image.

28.5

MANIPULATION OF SPECIMEN In addition to viewing microscopic specimens, the light microscope and microscope objectives are also used to project reduced high-intensity images of source patterns into the object plane in order to optically manipulate minute regions of the specimen. Photolithography and laser disk recorders are examples of important industrial applications, which have prompted the design of specially modified objective lenses for such purposes.

Microbeam Irradiation, Ablation Many applications are also found in the biomedical field, initially using UV-transmitting, moderately high NA objectives that are parfocalized for visible light and UV down to approximately 250 nm (Zeiss Ultrafluar, also quartz monochromats from Leitz). In its extreme form, a concentrated image of a circular- or slit-shaped UV or laser source of selected wavelengths is imaged onto a biological specimen to locally ablate a small targeted organelle; for example, a part of a chromosome, the microtubules attached thereto, or tiny segments of cross-striated muscle, are irradiated with the microbeam in order to sever their mechanical connections and, for example, to analyze force transduction mechanisms.155,156 In other cases, oriented chromophores can be selectively altered at the submolecular level, for example, by polarized UV microbeam irradiation. The stacking arrangement of the DNA nucleotide bases (which exhibit a strong UV dichroism, as well as birefringence in visible light) can be selectively altered and disclose the coiling arrangement of DNA molecules within each diffraction-limited spot in the nucleus of living sperm.117 Brief microirradiation of slit- or grid-shaped patterns of UV are used to bleach fluorescent dyes incorporated into membranes of living cells. The time course of recovery of fluorescence into the bleached zone measures the rate of diffusion of the fluorescently tagged molecules in the membrane and reveals unexpected mobility patterns of cell membrane components.157,158 Lasers have become the dominant source for microbeam irradiation experiments in cell and developmental biology and in other application areas. Laser sources can have a wide range of tunable wavelengths (217 to 800 nm), energies, and exposure durations (down to 25 r 10 12).159 They are often used together with sensitizing dyes or fluorescent markers to target specific organelles.160 They can be used in conjunction with versatile beam-shaping optics such as spatial light modulators.161

MICROSCOPES

28.55

Photosensitive and Caged Compounds Selected target molecules within minute regions in living cells can be modified, tagged, or activated by focused beams of light. The target molecules can be naturally photosensitive species such as chlorophyll (which produces oxygen where illuminated with the appropriate visible wavelengths), rhodopsin (which isomerizes and triggers the release of calcium ions and action potentials in retinal cells), or artificially introduced photosensitive reagents such as the drug colchicine (whose antimitotic activity is abolished locally with 366-nm irradiation). Of the photosensitive compounds, the caged compounds have a far-reaching potential. These are compounds that are synthesized so as to “cage” and hide the active chemical group until a photosensitive part of the compound is altered (e.g., by long-wavelength UV irradiation) and unmasks the hidden active group. Thus, by preloading with the appropriate caged compound and irradiating the cell selectively in the region of interest, one can test the role of the uncaged compound. For example, the role of ATP can be tested using caged ATP and ATP analogs; response to subtle increase in calcium ions can be seen using caged calcium or caged calcium chelators.162,163 Likewise, caged fluorescent dyes are irradiated to locally label and follow the transport of subunits within macromolecular filaments in a dividing cell.164 Caged glutamate in brain slices was photolyzed using a holographically generated illumination pattern for simultaneous multispot activation of different dendrites.161 Optical Tweezers Intense laser beams concentrated into a diffraction spot can generate a photon-driven force great enough to capture and suspend small particles whose refractive index differs from its surrounding.165,166 Applied to microscopy, a single swimming bacterium or micrometer-sized organelles in live cells can be trapped and moved about at will at the focus of a near-infrared laser beam focused by an objective lens of high NA. While the energy density concentrated at the laser focus is very high, the temperature of the trapped object remains within a degree or so of its environment; biological targets typically exhibit low absorbance at near-infrared wavelengths and thermal diffusion through water from such minute bodies turns out to be highly effective. Thus, the bacterium continues to multiply while still trapped in the focused spot, and it swims away freely when the laser beam is interrupted. The ability to use “optical tweezers,” not only to capture and move about minute objects but to be able to instantly release the object, provides the microscopist with a unique form of noninvasive, quick-release micromanipulator.167 Optical tweezers are now being used in the investigation of an increasing number of biochemical and biophysical processes, from the basic mechanical properties of biological polymers to the multitude of molecular machines that drive the internal dynamics of the cell. Innovation continues in all areas of instrumentation and technique, with much of this work focusing on the refinement of established methods and on the integration of this tool with other forms of single-molecule manipulation or detection. These developments have important implications for the expanded use of optical tweezers in biochemical research.168

28.6 ACKNOWLEDGMENTS Shinya Inoué was the principal author of the chapter on microscopes in the previous edition of the Handbook of Optics. The authors wish to thank Dr. Inoué for his many suggestions regarding the reorganization of the content, for a new contribution on the single-sideband edge enhancement (SSEE) technique, and for his consent to use much of the previous text and figures for the new edition. In acknowledging contributions to both editions, the authors also wish to thank Gordon W. Ellis of the University of Pennsylvania, late Katsuji Rikukawa, Yoshiyuki Shimizu, and Hiroohi Takenaka from Nikon K. K., Japan, Ernst Keller and Rudi Rottenfusser of Carl Zeiss, Inc., Jan Hinsch

28.56

INSTRUMENTS

of Leica, Inc., Mortimer Abramovitz of Olympus, Inc., and Lee Shuett and Mel Brenner of Nikon, Inc., who all provided helpful information and insights into microscope design. The preparation of this chapter was supported in part by the National Institutes of Health grants R01 EB002583 awarded to R.O. and R01 EB005710 awarded to M.S.

28.7

REFERENCES 1. L. C. Martin, (1966) The Theory of the Microscope, Blackie, London. 2. S. Inoué and K. R. Spring, (1997) Video Microscopy the Fundamentals, Plenum Press, New York. 3. F. Zernike, (1942) “Phase Contrast, a New Method for the Microscopic Observation of Transparent Objects,” Physica 9:686–693. 4. D. Gabor, (1949) “Microscopy by Reconstructed Wavefronts,” Proc. Roy. Soc. London ser A 197:454–487. 5. M. Françon, (1961) Progress in Microscopy, Row, Peterson, Evanston, Ill. 6. T. Wilson and C. Sheppard, (1984) Theory and Practice of Scanning Optical Microscopy, Academic Press, London. 7. M. Pluta, (1988) Advanced Light Microscopy Vol. I: Principles and Basic Properties, Elsevier Science Publishing Co., Amsterdam. 8. M. Pluta, (1989a) Advanced Light Microscopy Vol. II: Specialized Methods, Elsevier Science Publishing Co., Amsterdam. 9. M. Pluta, (1993) Advanced Light Microscopy Vol. III: Measuring Techniques, Elsevier Science Publishing Co., Amsterdam. 10. G. Sluder and D. E. Wolf, (1998) Video Microscopy. In: Methods Cell Biol. (eds. L. Wilson and P. Matsudaira). Academic Press, San Diego. 11. Y.-L. Wang and D. L. Taylor, (1989) Fluorescence Microscopy of Living Cells in Culture. Part A, Academic Press, San Diego. 12. D. L. Taylor and Y.-L. Wang, (1989) Fluorescence Microscopy of Living Cells in Culture. Part B, Academic Press, San Diego. 13. J. B. Pawley, (2006) Handbook of Biological Confocal Microscopy, 3d ed. Springer, New York. 14. B. R. Masters and P. T. C. So, (2008) Handbook of Biomedical Nonlinear Optical Microscopy, Oxford University Press, Oxford. 15. S. Bradbury, P. J. Evennett, H. Haselmann and H. Piller, (1989) Dictionary of Light Microscopy, Oxford University Press, Oxford. 16. H. H. Hopkins and P. M. Barham, (1950) “The Influence of the Condenser on Microscopic Resolution,” Proc. Phys. Soc. London 63 B:737–744. 17. M. Born and E. Wolf, (2002) Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, Cambridge University Press, Cambridge. 18. G. W. Ellis, (1985) “Microscope Illuminator with Fiber Optic Source Integrator,” J. Cell Biol. 101:83a. 19. E. Becker, (1985) Fluorescence Microscopy, Ernst Leitz Wetzlar GmbH, Wetzlar, Germany. 20. M. Levoy, R. Ng, A. Adams, M. Footer and M. Horowitz, (2006) “Light Field Microscopy,” Acm. T. Graphic 25:924–934. 21. R. Oldenbourg, (2008) “Polarized Light Field Microscopy: An Analytical Method Using a Microlens Array to Simultaneously Capture both Conoscopic and Orthoscopic Views of Birefringent Objects,” J. Microsc. 231:419–432. 22. N. H. Hartshorne and A. Stuart, (1964) Practical Optical Crystallography, American Elsevier Publishing Co., Inc., New York, NY. 23. H. E. Keller, (1995) Objective Lenses for Confocal Microscopy. In: Handbook of Biological Confocal Microscopy (ed. J. B. Pawley). 2d ed. Plenum Publ. Corp., New York. 24. Y. Shimizu and H. Takenaka, (1994) “Microscope Objective Design,” Adv. Opt. Electron Microsc. 14:249–334. 25. R. Kingslake, (1978) Lens Design Fundamentals, Academic Press, San Diego.

MICROSCOPES

28.57

26. E. Hecht, (2002) Optics, Pearson/Addison-Wesley, San Francisco, CA. 27. L. W. Smith and H. Osterberg, (1961) “Diffraction Images of Circular Self-Radiant Disks,” J. Opt. Soc. Am. 51:412–414. 28. M. Cagnet, M. Françon and J. C. Thrierr, (1962) Atlas of Optical Phenomena, Springer Verlag, Berlin. 29. E. H. Linfoot and E. Wolf (1956) “Phase Distribution Near Focus in an Aberration-Free Diffraction Image,” Proc. Phys. Soc. B69:823–832. 30. S. Inoué, (1989) Imaging of Unresolved Objects, Superresolution and Precision of Distance Measurement with Video Microscopy. In: Methods Cell Biol. (eds. D. L. Taylor and Y.-L. Wang). Academic Press, New York. 31. R. E. Thompson, D. R. Larson and W. W. Webb, (2002) “Precise Nanometer Localization Analysis for Individual Fluorescent Probes,” Biophys. J. 82:2775–2783. 32. R. J. Ober, S. Ram and E. S. Ward, (2004) “Localization Accuracy in Single-Molecule Microscopy,” Biophys. J. 86:1185–1200. 33. Y. Garini, B. J. Vermolen and I. T. Young, (2005) “From Micro to Nano: Recent Advances in High-Resolution Microscopy,” Curr. Opin. Biotechnol. 16:3–12. 34. S. W. Hell, (2007) “Far-Field Optical Nanoscopy,” Science 316:1153–1158. 35. H. Park, E. Toprak and P. R. Selvin, (2007) “Single-Molecule Fluorescence to Study Molecular Motors,” Q. Rev. Biophys. 40:87–111. 36. R. Heintzmann and G. Ficz, (2007) “Breaking the Resolution Limit in Light Microscopy,” Methods Cell Biol. 81:561–580. 37. A. Yildiz, J. N. Forkey, S. A. McKinney, T. Ha, Y. E. Goldman and P. R. Selvin, (2003) “Myosin V Walks Hand-Over-Hand: Single Fluorophore Imaging with 1.5-nm Localization,” Science 300:2061–2065. 38. E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz and H. F. Hess, (2006) “Imaging Intracellular Fluorescent Proteins at Nanometer Resolution,” Science 313:1642–1645. 39. M. J. Rust, M. Bates and X. Zhuang, (2006) “Sub-Diffraction-Limit Imaging by Stochastic Optical Reconstruction Microscopy (STORM),” Nat. Methods 3:793–795. 40. L. S. Churchman, Z. Okten, R. S. Rock, J. F. Dawson and J. A. Spudich, (2005) “Single Molecule HighResolution Colocalization of Cy3 and Cy5 Attached to Macromolecules Measures Intramolecular Distances through Time,” Proc. Natl. Acad. Sci. U S A, 102:1419–1423. 41. R. Roy, S. Hohng and T. Ha, (2008) “A Practical Guide to Single-Molecule FRET,” Nat. Methods 5:507–516. 42. D. W. Piston and G. J. Kremers, (2007) “Fluorescent Protein FRET: The Good, the Bad and the Ugly,” Trends Biochem. Sci. 32:407–414. 43. S. W. Hell and J. Wichmann, (1994) “Breaking the Diffraction Resolution Limit by Stimulated-Emission— Stimulated-Emission-Depletion Fluorescence Microscopy,” Opt. Lett. 19:780–782. 44. J. L. Harris, (1964) “Diffraction and Resolving Power,” J. Opt. Soc. Am. 54:931–936. 45. E. Betzig, A. Lewis, A. Harootunian, M. Isaacson and E. Kratschmer, (1986) “Near-Field Scanning Optical Microscopy (NSOM); Development and Biophysical Applications,” Biophys. J. 49:269–279. 46. H. K. Wickramasinghe, (1989) “Scanned-Probe Microscopes,” Sci. Am. 261:98–105. 47. I. T. Young, (1989) Image Fidelity: Characterizing the Imaging Transfer Function. In: Methods Cell Biol. (eds. D. L. Taylor and Y.-L. Wang). Academic Press, New York. 48. C. J. Sheppard, (2004) “Defocused Transfer Function for a Partially Coherent Microscope and Application to Phase Retrieval,” J. Opt. Soc. Am. A. Opt. Image Sci. Vis. 21:828–831. 49. D. A. Agard, Y. Hiraoka, P. Shaw and J. W. Sedat, (1989) “Fluorescence Microscopy in Three Dimensions,” Methods Cell Biol. 30:353–377. 50. R. Oldenbourg, S. Inoué, R. Tiberio, A. Stemmer, G. Mei and M. Skvarla, (1996) Standard Test Targets for High Resolution Light Microscopy. In: Nanofabrication and Biosystems: Integrating Material Science, Engineering and Biology (eds. H. C. Hoch, L. W. Jelinsky and H. Craighead). Cambridge University Press, Cambridge, England. 51. E. M. Chamot and C. W. Mason, (1958) Handbook of Chemical Microscopy, John Wiley & Sons, New York. 52. B. Kachar, (1985) “Asymmetric Illumination Contrast: A Method of Image Formation for Video Microscopy,” Science 227:766–768. 53. N. Streibl, (1984) “Phase Imaging by the Transport-Equation of Intensity,” Opt. Commun. 49:6–10.

28.58

INSTRUMENTS

54. C. J. R. Sheppard, (2002) “Three-Dimensional Phase Imaging with the Intensity Transport Equation,” Appl. Opt. 41:5951–5955. 55. E. D. Barone-Nugent, A. Barty and K. A. Nugent, (2002) “Quantitative Phase-Amplitude Microscopy I: Optical Microscopy,” J. Microsc. 206:194–203. 56. D. Paganin, A. Barty, P. J. McMahon and K. A. Nugent, (2004) “Quantitative Phase-Amplitude Microscopy. III. The Effects of Noise,”. J. Microsc. 214:51–61. 57. K. A. Nugent, D. Paganin and A. Barty, (2006) Phase Determination of a Radiation Wave Field. In: U.S. Patent, Number 7,039,553. University of Melbourne, Australia, USA. 58. C. S. Izzard and L. R. Lochner, (1980) “Formation of Cell-to-Substrate Contacts During Fibroblast Motility: An Interference-Reflection Study,” J. Cell Sci. 42:81–116. 59. N. Noda and S. Kamimura, (2008) “A New Microscope Optics for Laser Dark-Field Illumination Applied to High Precision Two Dimensional Measurement of Specimen Displacement,” Rev. Sci. Instrum. 79:023704. 60. F. Zernike, (1958) The Wave Theory of Microscopic Image Formation. In: Concepts of Classical Optics (ed. J. Strong). W.H. Freeman and Co., San Francisco. 61. A. H. Bennett, H. Osterberg, H. Jupnik and O. W. Richards, (1951) Phase Microscopy, Principles and Applications, John Wiley & Sons, Inc., New York. 62. M. Pluta, (1989b) “Simplified Polanret System for Microscopy,” Appl. Opt. 28:1453–1466. 63. R. Hoffman and L. Gross, (1975) “Modulation Contrast Microscopy,” Appl. Opt. 14:1169–1176. 64. G. W. Ellis, (1978) Advances in Visualization of Mitosis In Vivo. In: Cell Reproduction: In Honor of Daniel Mazia, (eds. E. Dirksen, D. Prescott and C. F. Fox). Academic Press, New York, pp. 465–476. 65. G. W. Ellis, (1988) Scanned Aperture Light Microscopy,” Proc. 46th Annual Meeting of the Electron Microscopy Society of America, 48–49. 66. J. Dyson, (1961) “Instrument Classification and Applications,” In: The Encyclopedia of Microscopy (ed. G.L. Clark) Reinhold Pub. Corp., New York, 412–420 67. C. J. Koester, (1961) “Theory and Techniques,” In: The Encyclopedia of Microscopy (ed. G.L. Clark) Reinhold Pub. Corp., New York, 420–434 68. N. H. Hartshorne and A. Stuart, (1960) Crystals and the Polarising Microscope: A Handbook for Chemists and Others, Arnold, London. 69. G. A. Dunn, (1998) “Transmitted-Light Interference Microscopy: A Technique Born before Its Time,” Proc. RMS 33:189–196. 70. W. P. Linnik, (1933) “Apparatus for Interferometric Study of Reflective Specimens with a Microscope (“Microinterferometer”),” C. R. Acad. Sci. USSR 1:18–23 (in Russian and German). 71. A. A. Lebedeff, (1930) “Polarization Interferometer and Its Applications,” Rev. Opt. 9:385. 72. G. V. Rozenberg, (1953) “Interference Microscopy,” Uspekhi Fizicheskikh Nauk 50:271–302 (in Russian). 73. M. Françon, (1964) “Polarization Interference Microscopes,” Appl. Opt. 3:1033–1036. 74. M. J. Jamin, (1868) “Sur un réfracteur differentiel pour la lumière polarisée,” C. R. Acad. Sci. (Paris) 67:814. 75. F. H. Smith, (1952) Interference Microscope. In: U.S. Patent, Number 2,601,175. Francis Hugh Smith, USA. 76. F. H. Smith, (1955) “Microscopic Interferometry,” Research (London), 8:385–395. 77. G. Nomarski, (1960) Interferential Polarizing Device for Study of Phase Objects. In: U.S. Patent, Number 2,924,142. Centre National de la Recherche Scientifique, Paris, France, USA. 78. R. D. Allen, G. B. David and G. Nomarski, (1969) “The Zeiss-Nomarski Differential Interference Equipment for Transmitted Light Microscopy,” Zeitschrift für wissenschaftliche Mikroskopie 69:193–221. 79. M. Shribak and S. Inoue, (2006) “Orientation-Independent Differential Interference Contrast Microscopy,” Appl. Opt. 45:460–469. 80. C. J. Cogswell and C. J. R. Sheppard, (1992) “Confocal Differential Interference Contrast (DIC) Microscopy: Including a Theoretical Analysis of Conventional and Confocal DIC Imaging,” J. Microscopy 165:81–101. 81. C. Preza, D. L. Snyder and J. A. Conchello, (1999) “Theoretical Development and Experimental Evaluation of Imaging Models for Differential-Interference-Contrast Microscopy,” J. Opt. Soc. Am. A. Opt. Image Sci. Vis. 16:185–2199.

MICROSCOPES

28.59

82. R. D. Allen, J. L. Travis, N. S. Allen and H. Yilmaz, (1981) “Video-Enhanced Contrast Polarization (AVEC-POL) Microscopy: A New Method Applied to the Detection of Birefringence in the Motile Reticulopodial Network Allogromia Laticollaris,” Cell Motil. 1:275–289. 83. E. D. Salmon and P. T. Tran, (1998) “High-Resolution Video-Enhanced Differential Interference Contrast (VE-DIC) Light Microscopy,” Methods Cell Biol., 56:153–184. 84. H. Ishiwata, M. Itoh and T. Yatagai, (1996) “Retardation-Modulated Differential Interference Microscope and its Application to 3D Shape Measurement,” Proc. 2873:21–24. 85. G. Holzwarth, S. C. Webb, D. J. Kubinski and N. S. Allen, (1997) “Improving DIC Microscopy with Polarization Modulation,” J. Microsc. 188:249–254. 86. G. M. Holzwarth, D. B. Hill and E. B. McLaughlin, (2000) “Polarization-Modulated Differential-Interference Contrast Microscopy with a Variable Retarder,” Appl. Opt. 39:6288–6294. 87. H. Ishiwata, M. Itoh and T. Yatagai, (2006) “A New Method of Three-Dimensional Measurement by Differential Interference Contrast Microscope,” Opt. Commun. 260:117–126. 88. R. P. Danz, A. Dietrich, C. Soell, Hoyer and M. Wagener, (2006) Arrangement and Method for Polarization-optical Interference Contrast. In: U.S. Patent, Number 7,046,436. Carl Zeiss Jena GmbH (Jena, DE). 89. B. Heise, A. Sonnleitner and E. P. Klement, (2005) “DIC Image Reconstruction on Large Cell Scans,” Microsc. Res. Tech. 66:312–320. 90. C. Preza, (2000) “Rotational-Diversity Phase Estimation from Differential-Interference—Contrast Microscopy Images,” J. Opt. Soc. Am. A-Opt. Image Sci. Vis. 17:415–424. 91. M. R. Arnison, K. G. Larkin, C. J. R. Sheppard, N. I. Smith and C. J. Cogswell, (2004) “Linear Phase Imaging Using Differential Interference Contrast Microscopy,” J. Microsc.-Oxford, 214:7–12. 92. F. Kagalwala and T. Kanade, (2003) “Reconstructing Specimens Using DIC Microscope Images,” IEEE Transactions on Systems Man and Cybernetics Part B-Cybernetics 33:728–737. 93. N. Axelrod, A. Radko, A. Lewis and N. Ben-Yosef, (2004) “Topographic Profiling and Refractive-Index Analysis by Use of Differential Interference Contrast with Bright-Field Intensity and Atomic Force Imaging,” Appl. Opt. 43:2272–2284. 94. M. Shribak, J. LaFountain, D. Biggs and S. Inoue, (2008) “Orientation-Independent Differential Interference Contrast Microscopy and Its Combination with an Orientation-Independent Polarization System,” J Biomed. Opt. 13:014011. 95. J. Dyson, (1950) “An Interferometer Microscope,” Proc. Roy. Soc. London. Series A, Mathematical and Physical Sciences 204:170–187. 96. G. Delaunay, (1953) “Microscope interférentiel A. Mirau pour la mesure du fini des surfaces,” Rev. Optique 32:610–614. 97. G. W. Ellis, (1966) “Holomicrography: Transformation of Image During Reconstruction of a Posteriori,” Science 154:1195–1197. 98. P. Marquet, B. Rappaz, P. J. Magistretti, E. Cuche, Y. Emery, T. Colomb and C. Depeursinge, (2005) “Digital Holographic Microscopy: A Noninvasive Contrast Imaging Technique Allowing Quantitative Visualization of Living Cells with Subwavelength Axial Accuracy,” Opt. Lett. 30:468–470. 99. B. Kemper, D. Carl, J. Schnekenburger, I. Bredebusch, M. Schafer, W. Domschke and G. von Bally, (2006) “Investigation of Living Pancreas Tumor Cells by Digital Holographic Microscopy,” J. Biomed. Opt. 11. 100. A. Stern and B. Lavidi, (2007) “Theoretical Analysis of Three-Dimensional Imaging and Recognition of Micro-Organisms with a Single-Exposure On-line Holographic Microscope,” J. Opt. Soc. Am. A-Opt. Image Sci. Vis. 24:163–168. 101. A. Stadelmaier and J. H. Massig, (2000) “Compensation of Lens Aberrations in Digital Holography,” Opt. Lett. 25:1630–1632. 102. S. S. Kou and C. J. R. Sheppard, (2007) “Imaging in Digital Holographic Microscopy,” Opt. Exp. 15:13640–13648. 103. Y. Kuznetsova, A. Neumann and S. R. J. Brueck, (2007) “Imaging Interferometric Microscopy—Approaching the Linear Systems Limits of Optical Resolution,” Opt. Exp. 15:6651–6663. 104. C. J. R. Sheppard and M. Roy, (2007) Low-Coherence Interference Microscopy. In: Optical Imaging and Microscopy (eds. P. Török and F. J. Kao). 2d ed. Springer-Verlag, Berlin, Heidelberg, New York.

28.60

INSTRUMENTS

105. A. M. Zysk and S. A. Boppart, (2007) Optical Coherence Tomography. In: Optical Imaging and Microscopy (eds. P. Török and F. J. Kao). 2d ed. Springer-Verlag, Berlin, Heidelberg, New York. 106. M. Wojtkowski, R. Leitgeb, A. Kowalczyk, T. Bajraszewski and A. F. Fercher, (2002) “In Vivo Human Retinal Imaging by Fourier Domain Optical Coherence Tomography,” J. Biomed. Opt. 7:457–463. 107. S. R. Chinn, E. A. Swanson and J. G. Fujimoto, (1997) “Optical Coherence Tomography Using a FrequencyTunable Optical Source,” Opt. Lett. 22:340–342. 108. F. Lexer, C. K. Hitzenberger, A. F. Fercher and M. Kulhavy, (1997) “Wavelength-Tuning Interferometry of Intraocular Distances,” Appl. Opt. 36:6548–6553. 109. V. V. Tuchin, L. V. Wang and D. A. Zimnyakov, (2006) Optical Polarization in Biomedical Applications, Springer-Verlag, Berlin, Heidelberg, New York. 110. J. A. Izatt, M. R. Hee, G. M. Owen, E. A. Swanson and J. G. Fujimoto, (1994) “Optical Coherence Microscopy in Scattering Media,” Opt. Lett. 19:590–592. 111. B. M. Hoeling, A. D. Fernandez, R. C. Haskell, E. Huang, W. R. Myers, D. C. Petersen, S. E. Ungersma, R. Y. Wang, M. E. Williams and S. E. Fraser, (2000) “An Optical Coherence Microscope for 3-Dimensional Imaging in Developmental Biology,” Opt. Exp. 6:136–146. 112. S. Inoué, (1952) “Studies on Depolarization of Light at Microscope Lens Surfaces. I. The Origin of Stray Light by Rotation at the Lens Surfaces,” Exp. Cell Res. 3:199–208. 113. S. Inoué and H. Kubota, (1958) “Diffraction Anomaly in Polarizing Microscopes,” Nature 182:1725–1726. 114. H. Kubota and S. Inoué, (1959) “Diffraction Images in the Polarizing Microscope,” J. Opt. Soc. Am. 49:191–198. 115. S. Inoué and W. L. Hyde, (1957) “Studies on Depolarization of Light at Microscope Lens Surfaces II. The Simultaneous Realization of High Resolution and High Sensitivity with the Polarizing Microscope,” J. Biophys. Biochem. Cytol. 3:831–838. 116. M. Shribak, S. Inoué and R. Oldenbourg, (2002) “Polarization Aberrations Caused by Differential Transmission and Phase Shift in High NA Lenses: Theory, Measurement and Rectification,” Opt. Eng. 41:943–954. 117. S. Inoué and H. Sato, (1966) Deoxyribonucleic acid Arrangement in Living Sperm. In: Molecular Architecture in Cell Physiology (eds. T. Hayashi and A. G. Szent-Gyorgyi). Prentice Hall, Englewood Cliffs, NJ. 118. H. G. Jerrard, (1948) “Optical Compensators for Measurement of Elliptical Polarization,” J. Opt. Soc. Am. 38:35–59. 119. R. Oldenbourg, (1999) “Polarized Light Microscopy of Spindles,” Methods Cell Biol. 61:175–208. 120. H. G. Jerrard, (1954) Transmission of Light through Birefringent and Optically Active Media: The Poincaré sphere,” J. Opt. Soc. Am. 44:634–640. 121. A. M. Glazer, J. G. Lewis and W. Kaminsky, (1996) “An Automatic Optical Imaging System for Birefringent Media,” Proc. R. Soc. London A 452:2751–2765. 122. R. D. Allen, J. Brault and R. D. Moore, (1963) “A New Method of Polarization Microscopic Analysis I. Scanning with a Birefringence Detection System,” J. Cell Biol. 18:223–235. 123. J. R. Kuhn, Z. Wu and M. Poenie, (2001) “Modulated Polarization Microscopy: A Promising New Approach to Visualizing Cytoskeletal Dynamics in Living Cells,” Biophys. J. 80:972–985. 124. R. Oldenbourg and G. Mei, (1995) “New Polarized Light Microscope with Precision Universal Compensator,” J. Microsc. 180:140–147. 125. M. Shribak and R. Oldenbourg, (2003a) “Techniques for Fast and Sensitive Measurements of TwoDimensional Birefringence Distributions,” Appl. Opt. 42:3009–3017. 126. R. Oldenbourg, (2005) Polarization Microscopy with the LC-PolScope. In: Live Cell Imaging: A Laboratory Manual (eds. R. D. Goldman and D. L. Spector). Cold Spring Harbor Laboratory Press, Cold Spring Harbor, NY. 127. R. Oldenbourg, (1991) “Analysis of Edge Birefringence,” Biophys. J. 60:629–641. 128. F. D. Bloss, (1981) The Spindle Stage Principles and Practice, Cambridge Universtiy Press, Cambridge. 129. W. Kaminsky and A. M. Glazer, (1996) “Measurement of Optical Rotation in Crystals,” Ferroelectrics 183:133–141. 130. L. A. Pajdzik and A. M. Glazer, (2006) “Three-Dimensional Birefringence Imaging with a Microscope Tilting-Stage. I. Uniaxial Crystals,” J. Appl. Crystallogr. 39:326–337.

MICROSCOPES

28.61

131. M. Shribak and R. Oldenbourg, (2004) “Mapping Polymer Birefringence in Three Dimensions Using a Polarizing Microscope with Oblique Illumination,” SPIE Proc. (Proc. Biophotonics Micro- and NanoImaging) 5462:57–67. 132. D. L. Taylor, M. Nederlof, F. Lanni and A. S. Waggoner, (1992) “The New Vision of Light Microscopy,” Am. Sci. 80:322–335. 133. R. Y. Tsien, (1989) Fluorescent Probes of Cell Signaling. In: Annu. Rev. Neurosci. (eds. W. M. Cowas, E. M. Shooter, C. F. Stevens, and R. F. Thompson). Annual Reviews, Inc., Palo Alto. 134. R. Y. Tsien, (1998) “The Green Fluorescent Protein,” Annu. Rev. Biochem. 67:509–544. 135. J. W. Lichtman and J. A. Conchello, (2005) “Fluorescence Microscopy,” Nat. Methods 2:910–919. 136. R. Neher and E. Neher, (2004) “Optimizing Imaging Parameters for the Separation of Multiple Labels in a Fluorescence Image,” J. Microsc. 213:46–62. 137. Y. Garini, I. T. Young and G. McNamara, (2006) Spectral Imaging: Principles and Applications,” Cytometry Part A 69:735–747. 138. M. Minsky, (1957) Microscopy Apparatus. In: U.S. Patent #3013467 (ed. U. S. P. a. T. Office). USA. 139. S. Inoué, (1990) Foundations of Confocal Scanned Imaging in Light Microscopy. In: Handbook of Biological Confocal Microscopy (ed. J. B. Pawley). Plenum Publ. Corp., New York. 140. G. S. Kino, (1990) Intermediate Optics in Nipkow Disk Microscopes. In: Handbook of Biological Confocal Microscopy (ed. J. B. Pawley). Plenum Press, New York. 141. M. Petrán, M. Hadravsky, D. Egger and R. Galambos, (1968) “Tandem-Scanning Reflected-Light Microscope,” J. Opt. Soc. Am. 58:661–664. 142. S. Inoue and T. Inoue, (2002) “Direct-View High-Speed Confocal Scanner: The CSU-10,” Methods Cell Biol. 70:87–127. 143. E. Wang, C. M. Babbey and K. W. Dunn, (2005) “Performance Comparison between the High-Speed Yokogawa Spinning Disc Confocal System and Single-Point Scanning Confocal Systems,” J. Microsc. 218:148–159. 144. W. A. Carrington, R. M. Lynch, E. D. W. Moore, G. Isenberg, K. E. Fogarty and F. S. Fay, (1995) “Superresolution Three-Dimensional Images of Fluorescence in Cells with Minimal Light Exposure,” Science 268:1483–1487. 145. J. A. Conchello and J. W. Lichtman, (2005) “Optical Sectioning Microscopy,” Nat. Methods 2:920–931. 146. M. A. Neil, R. Juskaitis and T. Wilson, (1997) “Method of Obtaining Optical Sectioning by Using Structured Light in a Conventional Microscope,” Opt. Lett. 22:1905–1907. 147. C. Ventalon and J. Mertz, (2005) “Quasi-Confocal Fluorescence Sectioning with Dynamic Speckle Illumination,” Opt. Lett. 30:3350–3352. 148. C. Ventalon and J. Mertz, (2006) “Dynamic Speckle Illumination Microscopy with Translated versus Randomized Speckle Patterns,” Opt. Exp. 14:7198–7209. 149. M. G. Gustafsson, (2000) “Surpassing the Lateral Resolution Limit by a Factor of Two Using Structured Illumination Microscopy,” J. Microsc. 198:82–87. 150. M. G. Gustafsson, L. Shao, P. M. Carlton, C. J. Wang, I. N. Golubovskaya, W. Z. Cande, D. A. Agard and J. W. Sedat, (2008) “Three-Dimensional Resolution Doubling in Wide-Field Fluorescence Microscopy by Structured Illumination,” Biophys. J. 94:4957–4970. 151. R. Heintzmann, T. M. Jovin and C. Cremer, (2002) “Saturated Patterned Excitation Microscopy—a Concept for Optical Resolution Improvement,” J. Opt. Soc. Am. A-Opt. Image Sci. Vis. 19:1599–1609. 152. M. G. Gustafsson, (2005) “Nonlinear Structured-Illumination Microscopy: Wide-Field Fluorescence Imaging with Theoretically Unlimited Resolution,” Proc. Natl. Acad. Sci. U S A 102:13081–13086. 153. R. Ng, M. Levoy, M. Bredif, G. Duval, M. Horowitz and P. Hanrahan, (2005) Light Field Photography with a Hand-Held Plenoptic Camera. Stanford Tech. Report, CTSR 2005-02. 154. M. Shribak and R. Oldenbourg, (2003b) “Three-Dimensional Birefringence Distribution in Reconstituted Asters of Spisula Oocytes Revealed by Scanned Aperture Polarized Light Microscopy,” Biol. Bull. 205:194–195. 155. R. E. Stephens, (1965) “Analysis of Muscle Contraction by Ultraviolet Microbeam Disruption of Sarcomere Structure,” J. Cell Biol. 25:129–139. 156. M. W. Berns, (1974) Biological Microirradiation, Classical and Laser Sources, Prentice-Hall, Englewood Cliffs.

28.62

INSTRUMENTS

157. D. E. Koppel, D. Axelrod, J. Schlessinger, E. L. Elson and W. W. Webb, (1976) “Dynamics of Fluorescence Marker Concentration as a Probe of Mobility,” Biophys. J. 16:1315–1329. 158. H. G. Kapitza, G. McGregor and K. A. Jacobson, (1985) “Direct Measurement of Lateral Transport in Membranes by Using Time-Resolved Spatial Photometry,” Proc. Natl. Acad. Sci. (USA) 82:4122–4126. 159. M. W. Berns, J. Aist, J. Edwards, K. Strahs, J. Girton, P. McNeill, J. B. Rattner, et al., (1981) “Laser Microsurgery in Cell and Developmental Biology,” Science 213:505–513. 160. V. Magidson, J. Loncarek, P. Hergert, C. L. Rieder and A. Khodjakov, (2007) Laser Microsurgery in the GFP Era: A Cell Biologist’s Perspective,” Methods Cell Biol. 82:239–266. 161. C. Lutz, T. S. Otis, V. Desars, S. Charpak, D. A. Digregorio and V. Emiliani, (2008) “Holographic Photolysis of Caged Neurotransmitters,” Nat. Methods 5:821–827. 162. J. H. Kaplan and G. C. Ellis-Davies, (1988) “Photolabile Chelators for the Rapid Photorelease of Divalent Cations,” Proc. Natl. Acad. Sci. (USA) 85:6571–6575. 163. J. A. Dantzig, M. G. Hibberd, D. R. Trentham and Y. E. Goldman, (1991) Cross-Bridge Kinetics in the Presence of MgADP Investigated by Photolysis of Caged ATP in Rabbit Psoas Muscle Fibres,” J. Physiol. (London) 432:639–680. 164. T. J. Mitchison and E. D. Salmon, (1992) “Poleward Kinetochore Fiber Movement Occurs during Both Metaphase and Anaphase—A in Newt Lung Cell Mitosis,” J. Cell Biol. 119:569–582. 165. A. Ashkin, J. M. Dziedzic, J. E. Bjorkholm and S. Chu, (1986) “Observation of a Single-Beam Gradient Force Optical Trap for Dielectric Particles,” Opt. Lett. 11:288–290. 166. K. Svoboda and S. M. Block, (1994) “Biological Applications of Optical Forces,” Annu. Rev. Biophys. Biomol. Struct. 23:247–285. 167. S. M. Block, (1990) “Optical Tweezers: A New Tool for Biophysics,” In: Noninvasive Techniques in Cell Biology (eds. J. K. Foskett and S. Grinstein). Wiley-Liss, New York. 168. J. R. Moffitt, Y. R. Chemla, S. B. Smith and C. Bustamante, (2008) “Recent Advances in Optical Tweezers,” Annu. Rev. Biochem. 77:205–228.

29 REFLECTIVE AND CATADIOPTRIC OBJECTIVES Lloyd Jones College of Optical Sciences University of Arizona Tucson, Arizona

29.1

GLOSSARY A AN B C c CON D FN GLA h INF k n R RDX RDY STO SUR t THI Z

4th-order aspheric deformation coefficient 4th-order nonsymmetric deformation coefficient 6th-order aspheric deformation coefficient 8th-order aspheric deformation coefficient surface base curvature conic constant 10th-order aspheric deformation coefficient focal ratio glass type radial surface height infinite radius of curvature conic constant index of refraction radius of curvature radius of curvature in the x dimension radius of curvature in the y dimension stop surface surface number element thickness thickness of element or distance to next surface or element surface sag

29.1

29.2

29.2

INSTRUMENTS

INTRODUCTION During the initial stages of an optical design, many optical engineers take advantage of existing configurations that exhibit useful properties. This chapter is a compilation of reflective and catadioptric objective designs that should help inform the reader of available choices and provide reasonable starting solutions. The chapter also includes a cursory introduction to some of the more important topics in system analysis, such as angular and linear blur size, image irradiance, scaling, and stray light control. An extensive list of referenced reading material and brief definitions of terms italicized throughout the text are included.

29.3

GLASS VARIETIES Glasses used in the designs are represented in terms of index of refraction and Abbe number or V number, below. The V number indicates glass dispersion. Most glasses can be obtained from a number of vendors. Glass BK7 F2 F9 FK51 FN11 Germanium LLF1 LAK21 PSK2 Silica Silicon Sapphire SK1 SK2 SK3 SK16 SF5 SF10 UBK7

29.4

Index of Refraction

V Number

1.516 1.620 1.620 1.487 1.621 4.037 1.548 1.640 1.569 1.445 3.434 1.735 1.610 1.607 1.609 1.620 1.673 1.728 1.517

64.2 36.3 38.1 84.5 36.2 117.4 45.8 60.1 63.2 27.7 147.4 15.5 56.5 56.8 58.9 60.3 32.1 28.5 64.3

INTRODUCTION TO CATADIOPTRIC AND REFLECTIVE OBJECTIVES The variety of objectives presented in this chapter is large. Most of the intricate detail relating to each design is therefore presented with the design itself. In the following paragraphs, analysis of the general features of the catadioptric and reflective objectives is undertaken.

Conic Mirrors It is apparent after a brief perusal of the designs that there are many surface types. Among these are the sphere, paraboloid, hyperboloid, prolate ellipsoid, and oblate ellipsoid. The oblate

REFLECTIVE AND CATADIOPTRIC OBJECTIVES

29.3

FIGURE 1 Relative shapes of conic surfaces in two dimensions.

ellipsoid is a prolate ellipsoid turned on its side. The equation of a conic is given by the expression Z

ch 2 1 1 (1 k)c 2h 2

(1)

where Z is the surface sag, k is the conic constant, c is the surface base curvature, and h is the radial height on the surface. The relative shapes of these surfaces are illustrated in Fig. 1. Conic mirrors give perfect geometric imagery when an axial point object is located at one conic focus and the axial point image is located at the other conic focus. Figure 2 illustrates these ray paths. General Aspheres General aspheres are surfaces with fourth- and higher-order surface deformation on top of a flat or curved surface.1 The surface deformation of a rotationally symmetric general asphere is given by the relation Z

ch 2 1 1 (1 k)c 2h 2

Ah 4 Bh6 Ch8 Dh10

(2)

where A, B, C, and D are 4th-, 6th-, 8th-, and 10th-order coefficients that determine the sign and magnitude of the deformation produced by that order. Although general aspheres allow correction of third- and higher-order aberrations and may reduce the number of elements in an optical system, general aspheres are more expensive than spheres or conics. If aspheric deformation is required, conic surfaces should be tried first, especially since a conic offers higher-order correction.2

29.4

INSTRUMENTS

Sphere

Paraboloid

Ellipsoid

Hyperboloid

FIGURE 2 Ray paths for perfect axial imagery.

Obscurations Obscurations that block portions of the entering beam reduce image irradiance and image contrast3,4 in reflective and catadioptric systems. Several methods are used to reduce or eliminate completely the effects of an obscuration (see Fig. 3). Figure 3a illustrates a commonly employed technique for reducing large-mirror obscuration: a small secondary mirror close to the intermediate image moves the larger tertiary mirror out of the beam path. Figure 3b is an illustration of an eccentric pupil system. All elements are symmetric about the same axis and the aperture stop is decentered for a clear light path.

FIGURE 3 Reducing the size of or eliminating an obscuration.


29.5

Figure 3c is an example of an off-axis objective with the field of view biased to direct the center of view away from any intervening elements. All elements and apertures are symmetric about the optical axis. Figure 3d is an illustration of a tilted and decentered-component objective. Each element is rotationally symmetric about its own unique optical axis which may be tilted and/or decentered. The imaging behavior of this system is more complicated to deal with than the imaging behavior of eccentric pupil and off-axis systems. Vector aberration theory5,6 has been developed to properly model the imaging behavior of these systems. Stray Light Suppression Suppression of light diffracted from apertures and obscurations is facilitated with intermediate images and a real and accessible Lyot stop. Figure 4a illustrates a generic refractive configuration with an intermediate image and Lyot stop. Figure 4b illustrates where the diffracted light (shaded region) originates and terminates (at one edge of each aperture, for clarity). A field stop is placed at the focus of the first lens to block diffracted light produced by the front light baffle. To block unwanted objects within the field of view, an occulting disc may be inserted at the focus of the first lens, as is done with a Lyot coronagraph in order to block the sun. By oversizing the field stop slightly, the light diffracted at the field stop falls just outside of the detector area. Following the field stop is a second lens that reimages the intermediate image to the final image and the entrance pupil to the Lyot stop (the shaded region in Fig. 4a illustrates how the entrance pupil is imaged). Undersizing the Lyot stop blocks the light diffracted at the entrance pupil. In this way the Lyot stop becomes the aperture stop of the system. Another application of the Lyot stop in the infrared (assuming the Lyot stop is located exterior to the objective optics) is as a cold stop.7 The cold stop (Fig. 4a) is a baffle that prevents stray infrared light, radiated from the housing, from impinging upon the detector from outside its intended field. Reflective and Catadioptric Objective Designs The objectives to follow are listed according to focal ratio and design type. Objectives have a 20-cm diameter and catadioptric systems are optimized for a wavelength range from 480 to

Baffle

Entrance pupil

Field stop

FIGURE 4

Generic objectives with apertures.

Lyot stop or cold stop

Image

29.6

INSTRUMENTS

680 nm unless otherwise stated. The angles of surface and element tilts are with respect to the horizontal optical axis. Decenters are also with respect to the optical axis. Since many of the designs are aplanatic, anastigmatic, and free of chromatic aberrations, the position of the stop does not affect third-order aberration correction and may be chosen to minimize vignetting, distortion, element size, or stray light contamination. All aberrations mentioned in this section are third-order unless otherwise stated. Definitions of the abbreviated terminology used in the lens data are as follows: SUR Surface number. RDY Surface radius of curvature. A positive value means the center of curvature lies to the right of the surface; negative to the left. THI Thickness of element or distance to next element. The value is positive if the next surface lies to the right of the surface. GLA Glass-type or mirror surface, the latter referred to by the abbreviation REFL. CON Conic constant k. STO Stop surface. INF A surface with an infinite radius of curvature; that is, a flat surface. A, B, C, D The 4th-, 6th-, 8th-, and 10th-order aspheric deformation coefficients in Eq. (2). A potential source of confusion is the terminology used to describe Mangin elements; that is, refractive elements with reflective back surfaces. This is illustrated in design 2 (F/4 Mangin): a ray enters the second refractive surface of the element (surface 2) and travels to the right where it intersects the mirror surface (surface 3). The thickness of surface 2 is therefore positive. The ray is reflected by the mirror surface (surface 3) and travels back through the glass element to surface 2; hence, the notation F9/REFL and the negative surface 3 thickness. Since surface 2 and 4 represent the same surface, the radii are the same. F/4 Paraboloid Objective

Comments A single parabolic mirror objective can be arranged in a variety of forms, the best known being the Newtonian. Here a mirror flat diverts the image away from the light path. A tippedmirror configuration is the Herschelian; a modern version is untipped and employs an eccentric-pupil to give an accessible image. A “backwards” Newtonian, the Pfund has a large flat-mirror primary. The Pfund has a smaller obscuration than the Newtonian and requires no diffraction-inducing support structure for the folding flat. As has been mentioned, the on-axis performance of a paraboloid objective is perfect. Off-axis, coma quickly degrades image quality. For objectives slower than F/11, the easy-to-fabricate spherical mirror gives the same performance as a paraboloid when diffraction is also considered. The paraboloid objective has image quality as good as a Cassegrain (design 3) of equivalent FN and aperture diameter, and is easier to align. The Cassegrain has the advantage of being compact.


29.7

F/4 Mangin SUR 1 2 3 4 5

RDY

THI

GLA

75.15

307.1

123.63

307.1

75.15

1.0 1.4

1.4

1.0

80.48

BK7 F9 F9/REFL BK7

Comments The Mangin8 was invented by a French engineer of the same name to replace difficult-to-fabricate paraboloids in light houses. The objective relies upon the overcorrected spherical aberration produced by the negative first surface to cancel the undercorrected spherical aberration produced by the focusing, reflective surface. The chromatic aberration of the Mangin is reduced by achromatizing with two glasses of different dispersions. Secondary spectrum limits on-axis performance, and coma, one-half that of a spherical mirror, is the primary field-limiting aberration. Kingslake9 takes the reader through the design of a Mangin mirror. Mangin mirrors are susceptible to ghost reflections from the refractive surfaces. Antireflection coatings are usually needed. In some cases the overcorrected chromatic aberration of a Mangin is used to cancel undercorrected chromatic aberration produced by a refractive element. The Schupmann or medial objective10,11 has a positive refractive element with undercorrected chromatic aberration which is annulled by a Mangin element. F/4 Cassegrain

SUR

RDY

THI

GLA

CON

STO 2

45.72

19.2

16 24.035

REFL REFL

1

3.236

Comments The ubiquitous Cassegrain is predominant in situations where a small field of view, high resolution, compact size, long effective focal length, and accessible image are required. The classical Cassegrain is composed of a paraboloid primary and hyperboloid secondary, giving perfect imagery on-axis whenever the primary image coincides with the hyperboloidal focus. Coma and field curvature limit off-axis performance. Many books discuss the first- and third-order properties of Cassegrain objectives. The Rutten,12 Schroeder,13 Korsch,14 and Smith3 texts are among these. F/4 Ritchey-Chretien

SUR

RDY

THI

GLA

CON

STO 2

45.72

19.2

16 24.034

REFL REFL

1.072

3.898

29.8

INSTRUMENTS

Comments The aplanatic Cassegrain or Ritchey-Chretien15 is also corrected for coma, leaving astigmatism and field curvature uncorrected. Both mirrors of the Ritchey-Chretien are hyperboloids. Numerous modern telescope objectives are of Ritchey-Chretien form; among these are the Hubble space telescope and the infrared astronomical satellite IRAS. F/9 Ritchey-Chretien Telescope with Two-Lens Corrector

SUR

RDY

THI

GLA

CON

STO 2 3 4 5 6

2139.7

802.83 67.73 90.39

1925.6 129.1

794.0 853.96 2.54 9.9 1.27 14.39

REFL REFL BK7

1.0778

4.579

BK7

Comments This is a design by Wynne16 for the correction of the Cassegrain focus of a large (350-cm) Ritchey-Chretien. The corrector removes the inherent astigmatism and field curvature of the Ritchey-Chretien. Other Cassegrain focus correctors are discussed by Schulte,17 Rosin,18 and Wilson.19 F/4 Dall-Kirkham

SUR

RDY

STO

45.72

THI

16

GLA

CON

REFL

0.6456

Sixth-order term: 0.593E-10 2

19.2

24.035

REFL

Comments The Dall-Kirkham is another Cassegrain corrected for spherical aberration. The primary is an ellipsoid with sixth-order aspheric deformation and the secondary is spherical. An inverse Dall-Kirkham, or Carlisle, is just the reverse, with a spherical primary. There is zonal spherical aberration without the sixth-order deformation. Five times more coma is produced by the Dall-Kirkham than the classical Cassegrain, seriously limiting the field of view. F/4 Cassegrain with Field Corrector and Spherical Secondary

SUR

RDY

THI

GLA

CON

1 STO 3 4 5 6

94.21

94.29 17.59 8.76

64.15

13.41

27.937 17.72 0.35 0.491 0.6 13.67

REFL REFL Silica

1

Silica


29.9

Comments By adding zero-power refractive correctors, the performance of a reflective objective is substantially enhanced. Zero power is maintained to prevent axial color. Such is the case with this objective similar to one designed by Rosin.20 All third-order aberrations, with the exception of distortion, are corrected. The surfaces, with the exception of the primary, are spherical. One of the most attractive features of this design, in comparison to the Schmidt which will be discussed shortly, is the small size of the refractive elements. Add to this the capability of eliminating any remaining spherical aberration in an assembled objective by adjusting the axial positions of the lenses. Zero Petzval sum and, hence, a flat image (in the absence of astigmatism) is ensured by giving the mirrors the same curvature and the lens elements equal and opposite power. F/15 Spherical-Primary Cassegrain with Reflective Field Corrector

SUR

RDY

THI

GLA

CON

STO 2 3 4

84.03

46.56

17.39

20.87

30.69 36.83

14.77 16.26

REFL REFL REFL REFL

20.97

0.8745

96.62

Comments This well-corrected design from Korsch14 has an easily manufactured spherical primary and is intended for use as a large-aperture telescope objective. Another all-reflective corrector of Gregorian form has been developed for a fast (F/0.6) spherical primary.21 Afocal Cassegrain-Mersenne Telescope

SUR

RDY

THI

GLA

CON

STO 2

100

30

35 40

REFL REFL

1

1

Comments The Mersenne contains two confocal paraboloids working at infinite conjugates. It is aplanatic, anastigmatic, and can be made distortion-free by choosing an appropriate stop location. The utility of confocal mirrors has been emphasized by Baker22 and Brueggeman,23 and is illustrated in the following design. Dual-Magnification Cassegrain SUR

RDY

THI

STO 2 3 4 5 6 7 8 9 10

33.99

10.61 10.486 25.673 48.33 22.68 3.52 4.22 INF INF

11.69 12.76 0.877 0.462 0.798 7.57 1.0 0.377 0.16 0.396

GLA REFL REFL Silicon Germanium Silicon Sapphire

CON

1

1

29.10

INSTRUMENTS

Comments This IR design is related to one introduced by Fjeidsted.24 The system offers two magnifications and fields of view. The high-magnification configuration is with the afocal Mersenne in the optical path. Removing the secondary lets light pass directly to the refractive assembly and a larger field of view is observed. The spectral range is from 3.3 to 4.2 μm. F/3.2 Three-Lens Prime Focus Corrector

SUR

RDY

THI

STO 2 3 4 5 6 7

1494.57

26.98

31.3

53.96

19.0

33.36 236.7

684.08

2.6

22.43

0.586

28.87

2.042

11.65

GLA

CON

REFL UBK7

1

UBK7 UBK7

Comments This is a three-lens corrector for a 250-cm parabolic mirror. The corrector was developed by Wynne16,25 for the region of the spectrum extending from 365 to 1014 nm. It is used to extend the field of a parabolic mirror. Versions for a Ritchey-Chretien primary also exist. The corrector is able to correct spherical aberration, coma, astigmatism, and field curvature while keeping chromatic aberrations under control. The field of view can be extended considerably for smaller apertures. The three-spherical lens corrector is one of the best large-optics prime-focus correctors to come along, both in terms of image quality and ease of fabrication. Other designs have either not performed as well or were heavily dependent on aspheric figuring. This and other prime-focus correctors are surveyed in articles by Gascoigne,26 Ross,27 Meinel,28 Schulte,29 Baker,30 and Wynne.31 F/4 Gregorian

SUR

RDY

THI

GLA

CON

STO 2

24.62 6.4

16 24.1

REFL REFL

1

0.5394

Comments The classical Gregorian is aberration-free on-axis when the paraboloidal mirror image coincides with one of the ellipsoidal-mirror foci; the other focus coincides with the final image. Like the Cassegrain, off-axis image quality is limited by coma and field curvature. The ellipsoidal secondary reimages the entrance pupil to a location between the secondary and final image. Thus, there exists the possibility of unwanted-light suppression at the primary-mirror image and exit pupil. The Gregorian is longer than the Cassegrain and thus more expensive to support and house, but it produces an erect image and the concave secondary is easier to produce. In eccentric-pupil versions it has an accessible prime focus.


29.11

F/4 Aplanatic Gregorian

SUR

RDY

THI

GLA

CON

STO 2

24.62 6.4

16 24.1

REFL REFL

0.989

0.5633

Comments The aplanatic Gregorian is corrected for spherical aberration and coma. Both mirrors are ellipsoids. Astigmatism and field curvature limit off-axis imagery. F/1.25 Flat-Medial-Field Aplanatic Gregorian

SUR

RDY

THI

GLA

CON

STO 2

34.68 6.47

22.806 7.924

REFL REFL

0.767

0.1837

Comments The Gregorian’s field performance is enhanced if image accessibility is sacrificed. This version of the Gregorian14 is aplanatic. A flat medial image is achieved by balancing Petzval curvature with astigmatism, which remains uncorrected. F/1.25 Flat-Medial-Field Aplanatic Gregorian with Spherical Primary

SUR

RDY

THI

GLA

STO 2

42.59 INF

21 46.51

REFL REFL

3

49.84

CON

Tilt: 45°

54.08

REFL

0.078

Comments The field of this objective14 is larger than its cousins, the classical and aplanatic Gregorians, even with the spherical primary. Spherical aberration and coma are corrected, and the medial image is flat. The design has a real intermediate image and exit pupil. The obvious drawback is the size of the secondary in relation to the size of the entrance pupil, which is 15 cm in diameter. Korsch14 analyzes two other designs that are loosely referred to as Gregorians.

29.12

INSTRUMENTS

Afocal Gregorian-Mersenne Telescope

SUR

RDY

THI

GLA

CON

STO 2

50 10

30 40

REFL REFL

1

1

Comments The Gregorian Mersenne, also composed of confocal paraboloids, is aplanatic, anastigmatic, and can be corrected for distortion. The Gregorian-Mersenne has an intermediate image and an accessible exit pupil. F/1.25 Couder

SUR

RDY

THI

GLA

CON

STO 2

142.86 23.08

52.9 7.1142

REFL REFL

6.285

0.707

Comments The Couder,32 composed of two conic mirrors, is corrected for third-order spherical aberration, coma, and astigmatism. Unfortunately, the Couder is long for its focal length and the image is not readily accessible. F/1.25 Aplanatic, Flat-Medial-Image Schwarzschild

SUR

RDY

THI

GLA

CON

STO 2

91.57 23.67

38.17 4.637

REFL REFL

2.156 5.256

Comments The aplanatic, flat-medial-image Schwarzschild33 is similar in appearance to the Couder but the secondary mirror and image locations are different for identical secondary magnifications. F/1.25 Aplanatic, Anastigmatic Schwarzschild

SUR

RDY

THI

GLA

1 2 STO

30.62 80.14 INF

49.44 80.26 24.864

REFL REFL


29.13

Comments The spherical-mirror Schwarzschild33 is aplanatic, anastigmatic, and distortion-free.34 The Schwarzschild relies on the principle of symmetry for its high level of aberration correction and a large field of view. All surfaces have the same center of curvature at the aperture stop. Hence, there are no off-axis aberrations. Spherical aberration is produced but each mirror produces an equal and opposite amount, thus canceling the effect of the aberration. Some higher-order aberrations are also corrected.34 Eccentric portions of this design—above and below the optical axis in the picture—form well-corrected, unobscured designs. Zonal spherical aberration from the mix of third- and higherorder terms limits on- and off-axis performance. An aspheric plate positioned at the center-of-curvature of the mirrors removes this aberration as illustrated in the next design. Wetherell and Rimmer,35 Korsch,14 Schroeder,13 Linfoot,36 and Gascoigne26 offer a general thirdorder analysis of two-mirror systems. The closed-form solutions described provide insight into third-order aberration theory of reflective systems. F/1 Aplanatic, Anastigmatic Schwarzschild with Aspheric Corrector Plate

SUR

RDY

THI

GLA

1 2 STO

24.547 63.92 INF

39.456 64.528

19.098

REFL REFL REFL

A: 0.9998E-7 B: 0.1269E-9

Comments With an aspheric plate at the aperture stop, spherical aberration is eliminated. The only aberrations still remaining are of higher order. To correct these, the mirrors must also be aspherized. Linfoot36 and Abel34 describe this design. F/1.25 Anastigmatic, Flat-Image Schwarzschild

SUR

RDY

THI

GLA

CON

1 STO

69.7 71.35

50.56 61.26

REFL REFL

5.47 0.171

Comments With just two conics, this design type33 achieves aplanatic and anastigmatic performance on a flat image surface. The flat field is attained by making the curvatures of the mirrors equal. Eccentric portions above or below the optical axis form unobscured versions; the design may alternatively be used off-axis. Sasian6,37 and Shafer38 have explored many of this design’s features.

29.14

INSTRUMENTS

F/1.25 Schmidt

SUR

RDY

THI

GLA

STO

1554

1

PSK2

CON

A: 0.2825E-5 B: 0.1716E-8 2 3

INF

52.95

52.33

26.215

REFL

Comments The Schmidt39 also relies on the principle of symmetry; that is, the aperture stop is located at the center of curvature of the spherical mirror and hence the mirror produces no off-axis aberrations. The Schmidt corrector is flat with aspheric deformation figured in to correct the spherical aberration produced by the mirror. It is positioned at the aperture stop because off-axis aberrations are independent of aspheric deformation when an aspheric surface coincides with the stop. Hence the Schmidt plate has no effect on off-axis aberrations, and the benefits of concentricity are preserved. The corrector introduces chromatic variation of spherical aberration (spherochromatism). A small amount of positive power in the corrector introduces undercorrected axial color to reduce the effects of this aberration. Further improvement is obtained by achromatizing the corrector with two glasses of different dispersions. Higher-order aberrations degrade image quality at low focal ratios and large field angles. Kingslake,9 Schroeder,13 Maxwell,40 and Linfoot36 provide additional details of this and other catadioptric objectives. F/1.25 Field-Flattened Schmidt

SUR

RDY

STO

598.7

THI 1.155

GLA PSK2

A: 0.273E-5 B: 0.129E-8 2 3 4 5

INF

52.95

10.35 INF

40.38

24.06

1.49

0.637

REFL PSK2

Comments As is known from third-order aberration theory, a thin element will be nearly aberration-free, except for Petzval curvature, and distortion when it is placed in close proximity to an image. Therefore, by properly choosing the lens power and index to give a Petzval curvature of equal and opposite sign to the Petzval curvature introduced by the other optics, the image is flattened. The image in the Schmidt above has been flattened with the lens near the image plane. The only aberrations introduced by the lens are spherochromatism and lateral color, lateral color being the most noticeable aberration; this can be removed by achromatizing the field-flattening lens. The close proximity of the lens to the image can cause problems with light scattered from areas


29.15

on the lens surfaces contaminated by dirt and dust particles. Clean optics are a must under these circumstances. The field-flattening lens provides two more positive results. First, the lens introduces a small amount of coma which is compensated by moving the Schmidt corrector toward the mirror somewhat, thus reducing the overall length of the objective. Second, oblique spherical aberration, one of the primary field-limiting, higher-order aberrations of the Schmidt, is substantially reduced. Besides its usual function as a telescope or photographic objective, the field-flattened Schmidt has also been used as a spectrograph camera.41 F/1.25 Wright

SUR 1 2

RDY

THI

GLA

INF

699.8

1.0 26.1

BK7 BK7

CON

A: 0.6168E-5 B: 0.5287E-8

53.24

3

26.094

REFL

1.026

Comments The Wright42 is one-half the length of the Schmidt. It also relies on aspheric deformation of the corrector plate for the elimination of spherical aberration. Coma, introduced as the corrector is removed from the center of curvature of the mirror, is cancelled with conic deformation of the mirror; the surface figure is that of an oblate ellipsoid. The remaining astigmatism and Petzval curvature are balanced for a flat medial image. The only on-axis aberration, spherochromatism, is corrected by achromatizing the corrector. F/4 Reflective Schmidt

SUR

RDY

THI

GLA

STO

66752

67.37

REFL

A: 0.5083E-7 2 3

INF

133.97

66.6

66.85

REFL REFL

Comments Another way of defeating chromatic aberration is to eliminate it altogether with a reflective corrector.43 The elements are removed from the light path with a field bias (9°), and hence the objective is off-axis. Spherical aberration, coma, and astigmatism are all corrected. At large field angles, beyond about 12° half-field angle, oblique spherical aberration becomes evident, but otherwise this design provides excellent performance on a curved image over a 24° field of view. In order to avoid severe obstruction of the light path, the full 24° can be taken advantage of only in the plane that extends perpendicular to the picture of the design above. A considerably reduced field of view is allowed in the plane of the picture.

29.16

INSTRUMENTS

F/0.6 Solid Schmidt SUR

RDY

THI

1 2

62.69 103.38

1.69 8.39 A: B: C: D:

3 STO 5 6 7 8

169.36

37.05 INF

45.86

10.295 INF

GLA

CON

BK7

0.1492E-4 0.1988E-7 0.1013E-10 0.35E-12 16.47

16.47

0.3952

0.245

1.136

0.026

BK7 REFL\BK7 BK7 BK7

Comments All monochromatic aberrations, with the exception of distortion, are corrected by the appropriately-named solid Schmidt, a system used mostly as a spectrograph camera.44 All chromatic aberrations, with the exception of lateral color, are corrected. The imaging theory behind the solid Schmidt is expounded by Baker.45 With a refractive index n, the solid Schmidt is n2 times faster than the conventional Schmidt. Focal ratios of F/0.3 have been obtained. Schulte44 offers a design similar to the one given here. F/1.25 Schmidt-Cassegrain SUR

RDY

STO

INF

THI

GLA

0.8

489.574

CON

A: 0.1928E-4 B: 0.298E-7 2 3

2269.1 INF

1.0 16.49

583.303

A: 0.1085E-4 B: 0.2806E-7 4 5 6 7 8 9

55.9 INF 9.1

8.577

8.59

87.44

15.0 10.267 1.2 0.018 0.3 1.317

REFL REFL 489.574

1.077

583.303

Comments The Schmidt-Cassegrain46 represents a successful attempt to resolve the difficulties related to the curved image, considerable length, and awkwardly located image of the Schmidt objective, without destroying the positive attributes of the design. The Schmidt-Cassegrain comes in a wide variety of forms—too many to go into here. Linfoot36 performs an extensive exploration of the design, with one and two aspheric plate correctors. Warmisham47 has gone as far as three. Wayman48 has analyzed a monocentric Schmidt-Cassegrain. In this fast version of the Schmidt-Cassegrain, the corrector is close to the flat secondary. Usually one or both mirrors are aspherics. An achromatized image-flattening lens has been introduced.


29.17

An image-flattening lens is not usually required with a Schmidt-Cassegrain since enough degrees of freedom exist for aberration correction and a flat image. In this case, the secondary mirror is flat and one degree of freedom is lost. Additionally, the primary mirror power introduces a strong Petzval contribution which necessitates a field-flattening lens. The first three digits of the six-digit code in the glass column identify the indices of the materials in the design, in this case plastics. These are 1.489 and 1.583; the Abbe-numbers are given by the last three digits and they are 57.4 and 30.3, respectively. The two plastics have nearly identical thermal coefficients and are very light. Buchroeder49 analyzes designs of this variety with two aspheric correctors. Shafer50 offers a Schmidt-Cassegrain with an aspheric meniscus corrector. Only two elements are used since the secondary mirror surface is on the corrector. Rutten12 has examples of SchmidtCassegrains in a number of configurations. F/3.4 Reflective Schmidt-Cassegrain

SUR

RDY

THI

GLA

STO

INF

92.16

REFL

A: 0.13344E-6 2 3

84 84

CON

AN: 0.1255E-1

26

25.848

REFL REFL

0.3318

Comments The reflective Schmidt-Cassegrain exhibits all the nice properties of the SchmidtCassegrain and, in addition, is achromatic. Schroeder51 points out that, because the corrector is tilted (9° here), adequate aberration correction requires a nonrotationally symmetric corrector plate. The nonaxially symmetric surface deformation in this design is given by Z A[(1 AN )X 2 (1 AN )Y 2 ]2

(3)

where A is the fourth-order symmetric coefficient and AN is the fourth-order nonsymmetric coefficient. The y dimension is in the plane of the picture; the x dimension is perpendicular to it. Since the corrector is tilted by 9°, the reflected rays are deviated by twice that amount. The element spacings (THI) are no longer measured along the horizontal optical axis after reflection off the corrector, but along the line of deviation. The secondary and tertiary are tilted by 18°. F/2 Shafer Relayed Virtual Schmidt SUR

RDY

STO 2 3

320 106.7 INF

THI

GLA

159.82 80.0

68.51

REFL REFL REFL

A: 0.1882E-5 B: 0.1273E-8 C: 0.1757E-12 D: 0.1766E-14 4

63.967

40.774

REFL

Comments Shafer52 has introduced an eccentric-pupil (18-cm stop decenter), virtual Schmidt objective similar to this but with a decentered quaternary mirror. The center of curvature of the spherical primary is imaged by the concave secondary onto the flat Schmidt tertiary mirror.

29.18

INSTRUMENTS

Virtual image

FIGURE 5 Picture of virtual Schmidt with decentered quaternary.

Now the Schmidt plate, which appears to be at the primary center of curvature, is aspherized to produce a well-corrected virtual image, hence the name (see Fig. 5). In this configuration, the Schmidt plate is one-half the size of the primary. The Schmidt plate and the spherical quaternary mirror form a finite conjugate Schmidt system. Thus, the spherical aberration of this mirror is also corrected. Figure 5 shows a pictorial representation of the Shafer design with the last mirror decentered to provide a more accessible image. Since the primary and quaternary mirrors no longer share the same axis of symmetry, a two-axis Schmidt corrector is required to remove the aberrations of both mirrors. The shape of this surface is described by Shafer, for an F/1, unobscured, wide-field design with an intermediate image and Lyot stop. F/2.2 Spherical-Primary Objective that Employs the Schmidt Principle of Correction SUR

RDY

THI

GLA

STO 2

88.07 INF

42.517 2.2

REFL REFL

3 4 5 6 7

4.433

2.527

10.21

2.527

4.433

CON

Tilt: 45° 0.33 9.217

9.217

0.33

0.64

FK51 REFL FK51

0.8631

Comments Baker22 reports on a system where the center of curvature of a large, spherical primary is imaged by a positive lens onto a much smaller mirror where aspheric correction of spherical aberration occurs. A small field offset (0.25°) is required so that the one-to-one relay doesn’t reimage the primary image back onto itself. To avoid overlap, this design is best used over a small or strip field of view. Because of the geometry of the design, coma, astigmatism, image curvature, chromatic aberrations, and distortion are eliminated in addition to the spherical aberration correction from aspheric figuring of the tertiary mirror. Baker22 offers several other interesting designs in his article, including an F/0.8, 10.6-μm, 180° field-of-view Roesch,53 a design that incorporates a Schmidt with a strong negative lens or lens group before the aspheric corrector. The strong divergence produced by this lens reduces the amount of light blocked by the image plane but increases the size of the spherical mirror.


29.19

F/2 Maksutov

SUR

RDY

THI

STO 2 3 4

INF

23.06

25.53

83.9

31.788 3.5 51.09

43.8

GLA

CON

BK7 REFL

Comments The all-spherical Maksutov54 was intended as an inexpensive alternative to the Schmidt at slower speeds. In small sizes it is indeed less expensive. The meniscus corrector is “selfachromatic” when the following relationship is satisfied:

t

n2 (R R ) n 1 2 1

(4)

2

where R1 and R2 are the radii, t is the thickness, and n is the refractive index of the corrector. Bouwers55 also developed a meniscus corrector. All elements of the Bouwers are concentric about the aperture stop. This ensures correction of third-order, off-axis aberrations over a nearly unlimited field of view. In exchange for the wide field, axial color is not well-corrected. F/1.25 Solid Maksutov-Cassegrain

SUR

RDY

THI

STO 2 3 4 5 6 7 8 9 10 11

INF

20.5

25.92 138.58

45.61

51.89 INF 12.026 16.07 INF INF

20.5 0.955 0.0313 15.3

12.973 13.41 0.0475 1.68 0.545 0.394 0.155

GLA

CON

Silica Silica REFL\Silica REFL\Silica Silica Silica

Comments The solid Maksutov-Cassegrain shown here and the solid Schmidt-Cassegrains have been studied extensively by Wynne.56,57 Lateral color is the most consequential aberration left uncorrected.

29.20

INSTRUMENTS

F/1.2 Wide-Field Objective with Maksutov Correction

SUR

RDY

THI

GLA

CON

1 2 3 STO 5

12.467

4.81

3.762 INF 15.98

9.684

1.267

3.679

17.189 10.64

REFL FK51

3.243

REFL

Comments This very wide field imaging system similar to one in Courtes58 is essentially a Maksutov focused on the virtual image of the object produced by the hyperboloidal mirror. Both speed (F/1) and a very wide field of view (80° r 120°) can be achieved with this design form on a flat image but only for small apertures—1.25 cm in this case. Courtes et al.59 describes similar systems with refractive and reflective Schmidt plates instead of a Maksutov corrector. F/1 Gabor

SUR

RDY

THI

GLA

STO 2 3 4 5 6

23.3

25.468

83.33

1.67 9.85

7.71

2 39.5

40

1

0.5

0.942

SKI REFL BK7 SF5

Comments Another meniscus design was invented by Gabor.60 The Gabor is more compact than the Maksutov or Bouwers, and has a smaller focal ratio and field of view. The design shown here began without the field lens. The lens was introduced into the design with the surface closest to the image being concentric about the chief ray and the other surface being aplanatic.61 A surface concentric about the chief ray is free of coma, astigmatism, distortion, and lateral color. The aplanatic surface is free of spherical aberration, coma, and astigmatism with the result that the lens is coma- and astigmatism-free. The spherical aberration produced by the lens is balanced against the spherical aberration produced by the two other elements. The chromatic aberrations were corrected by achromatizing the lens. Shafer62,63 offers interesting suggestions for design with aplanatic and concentric surfaces. Several varieties of field-flattening lens are described. Kingslake9 runs through the design procedure for a Gabor.


29.21

F/4 Schmidt-Meniscus Cassegrain

SUR

RDY

THI

GLA

STO 2 3 4 5 6

787.7 INF

32.69

35.6

81.97

79.5

1.4 32.69 2.62 63.446

21.78 38.65

BK7 BK7 REFL REFL

Comments This system, originally by Bouwers, uses a slightly positive plate to compensate the overcorrected chromatic aberration produced by the meniscus. The Bouwers produces very good quality on a flat image, over a large field of view. Fourth- and sixth-order deformation added to the plate eliminates any residual spherical aberration. Lateral color and oblique spherical aberration affect field performance, although both are small. F/1.2 Baker Super-Schmidt

SUR

RDY

1 2 STO

22.5 19.13

11,783 A: B:

4 5 6 7 8 9 10

135.5 INF

29.31

32.42

55

32.42

29.31

THI

GLA

4.28 19.75 1.4

BK7 F2

0.558E-6 0.244E-8 1.3 23.92 4.28 25.1

25.1

4.28

1.45

SK2 BK7 REFL BK7

Comments The Baker64 super-Schmidt, a design that incorporates both meniscus and Schmidt correction, achieves excellent performance over a wide field of view. The field-limiting aberration of a fast Schmidt, oblique spherical aberration, is controlled by adding a concentric meniscus lens which also introduces overcorrected spherical aberration, thus reducing the amount of overcorrection needed from the Schmidt plate. Since oblique spherical is proportional to the amount of overcorrection in the Schmidt plate, the effect of this aberration is reduced. The most apparent aberration produced by the meniscus is axial color. This is minimized by achromatizing the Schmidt corrector. Spherochromatism is reduced since the magnitudes produced by the Schmidt corrector and meniscus are nearly equal and have opposite signs. Another meniscus element is added to further reduce aberrations.

29.22

INSTRUMENTS

F/1 Baker-Nunn

SUR 1 2

RDY

491.9

115.6

THI 1.06 4.23

GLA LLF1

A: 0.8243E-5 B: 0.1348E-8 STO

125.78

0.64

SK3

A: 0.1158E-4 B: 0.427E-8 C: 0.7304E-11 4

125.78

4.23

A: 0.1158E-4 B: 0.427E-8 C: 0.7304E-11 5

115.6

1.06

LLF1

A: 0.8243E-5 B: 0.1348E-8 6 7

491.87

42.03

36.77

21.961

REFL

Comments The Baker-Nunn 65 was born of work by Houghton 66 during World War II. Houghton wished to find a less expensive alternative to the Schmidt. The result was a zero-power, three-lens combination with easy-to-make spherical surfaces. Spherical aberration and coma can be eliminated for any position of the corrector. The surfaces have equal radii so they can be tested interferometrically against one another using the Newton ring method. Residual spherical aberration that remains after assembly is removed by altering the spacing between the lenses. F/10 Houghton-Cassegrain

SUR

RDY

THI

GLA

STO 2 3 4 5 6

145

172.1

111.9 264.7

129.7

63.94

1.2 0.164 0.639 44.61

43.16 66.84

BK7 BK7 REFL REFL

Comments A two-lens, afocal corrector developed by Houghton and Sonnefeld67 is used here as a corrector for a Cassegrain system. Sigler68 has written on the subject of Cassegrains with


29.23

Houghton, Schmidt, and Maksutov refractive correctors. This Houghton-Cassegrain gives wellcorrected imagery on a curved image surface. An afocal achromatized doublet corrector has also been tried.69 F/3.6 Houghton-Cassegrain

SUR

RDY

THI

GLA

STO 2 3 4 5 6 7 8 9 10

69.64 148.71

61.43

97.53

85.11

97.53 70.44

15.47

15.23

517.29

1.607 3.045 1.607 21.733

21.733 21.733 1.2 0.18 1.3136 11.03

UBK7 LAK21 REFL REFL UBK7 SK16

Comments Another Houghton corrector, with meniscus elements, is utilized in this design by D. Rao.70 The spectral range is 550 to 850 nm. The design is similar to one introduced by Mandler.71 Examples of other Houghton-Cassegrains of this form are studied by Gelles.72 F/1.25 Shenker

SUR

RDY

THI

GLA

STO 2 3 4 5 6 7 8 9 10 11 12

49.42

203.6

34.7

79.25

27

38.87

31.96

38.87 13.73 21.8 7.925 8.56

1.5 5.4 0.863 5.08 0.98 9.32

9.32 8.1 0.39 0.05 0.895 0.856

BK7 BK7 BK7 REFL REFL BK7 BK7

Comments Shenker has studied a large number of variations on the theme of three-element correctors for a Cassegrain. This is related to one of the configurations developed by Shenker.73 Note that the third corrector is also the secondary mirror. Zonal spherical aberration limits performance on-axis. This may be removed by aspherizing one or more surfaces. All elements are of the same glass. Laiken74 has a similar version of this design as well as other catadioptric objectives. Maxwell40 has design examples and catadioptric imaging theory.

29.24

INSTRUMENTS

F/1.25 Mangin-Cassegrain with Correctors

SUR

RDY

THI

STO 2 3 4 5 6 7 8 9 10 11 12 13 14

80.62

102.3

30.43

54.52

30.43

102.3 80.62

526.4 80.62

102.3 11.06

30.43

54.52 52.92

1.64 9.07 2.02

2.02

9.07

1.64

1.01 1.01 1.64 8.32 0.75 2.02 0.5 1.445

GLA BK7 BK7 BK7/REFL BK7 BK7 BK7/REFL BK7 BK7 BK7 SF10

Comments Mangin mirrors are evident in this design by Canzek75 and two elements are used twice. The design has exceptionally good on-axis performance. Lateral color and higher-order aberrations limit the field. F/1.25 Mangin-Cassegrain with Correctors

SUR

RDY

STO 2 3 4 5 6 7 8 9 10 11 12

80.83

325.9

191.4

440.3

31.44

46.13

31.44

440.3 26.97 38.33 8.44 40.87

THI 1.09 8.5 0.728 9.69 1.456

1.456

9.69 10 0.582 0.544 0.728 2.025

GLA FN11 FN11 FN11 FN11/REFL REFL FN11 FN11

Comments Another short and fast catadioptric by Amon76 is shown here. The second corrector is also the secondary mirror.


29.25

F/4 Eisenburg and Pearson Two-Mirror, Three-Reflection Objective

SUR

RDY

THI

GLA

CON

STO 2 3

48.0

14.472

48

17.289 17.289

18.195

REFL REFL REFL

1.05

1.539

1.05

Comments This aplanatic, two-mirror, three-reflection configuration was first introduced by Rumsey.77 The design presented here comes from Eisenburg and Pearson.78 The first and third surface represent the same surface. F/4 Shafer Two-Mirror, Three-Reflection Objective

SUR

RDY

THI

GLA

CON

STO 2 3

106.7 80.01

106.7

80.01 80.01

80.05

REFL REFL REFL

0.4066

5.959

0.4066

Comments Shafer79 has documented numerous versions of multiple-reflection objectives. This is an aplanatic, anastigmatic design with field curvature. For optimum aberration correction, the primary is at the center of curvature of the secondary mirror. Shafer80 suggests a ring field for a flat, accessible image on an annular surface, and a Lyot stop. A simple ring field design is depicted in Fig. 6. Only one field angle S is required, easing the difficulties associated with off-axis aberration correction. The single viewing direction is rotated about the optical axis, forming, in this case, a ring image. In reality, less than half the ring image is used to avoid interference of the image with the entering beam.

FIGURE 6

The ring field system.

29.26

INSTRUMENTS

F/15 Two-Mirror, Three-Reflection Objective

SUR

RDY

THI

GLA

CON

STO 2 3

116.33

22.6

116.33

46.53 46.53

67.05

REFL REFL REFL

1.024

1.0037

1.024

Comments This is another aplanatic, anastigmatic, eccentric-pupil design which gives wellcorrected imagery on a curved image. It has a 30-cm stop decenter. F/15 Yolo SUR

RDY

THI

GLA

CON

STO

1015

160.36

REFL

4.278

Tilt: 3.5° 2

1045.72

208.19

REFL

RDX: 1035.0 Tilt:

9.82° Image tilt: 11.7°

Comments Leonard81,82 invented the Yolo (named after a scenic county in California) so that he could have an achromatic system without obscurations. The result is a tilted and decentered component objective that gives the high contrast of an unobscured refractive objective without the chromatic effects. Spherical aberration is corrected by the hyperboloidal figuring of the first surface. The anamorphism introduced into the secondary (by a warping harness) corrects astigmatism; RDX is the surface radius of curvature perpendicular to the picture. Coma is eliminated by adjusting the curvatures and tilting the secondary. Relatives of the two-mirror Yolo are the Solano, an in-line three-mirror Yolo, or the threedimensional, three-mirror Yolo.83 As in design 28, thickness (THI) is measured along the deviated ray paths. With the angle of reflection known, element decenter may be easily determined. F/15 Schiefspiegler SUR

RDY

THI

GLA

CON

STO

397.2

101.4

REFL

0.607

2

552.5

3

3411

4

INF

Tilt angle: 4.5° 35.84

REFL


BK7

Tilt angle: 50° 111.11

Tilt angle: 50.0529° Image tilt angle: 22.80°


29.27

Comments The Schiefspiegler (“oblique reflector” in German) was introduced about a century ago, and at the time was called a brachyt (or bent). The motivation behind the Schiefspiegler’s design is essentially the same as the Yolo’s. Like the Yolo, elements are tilted and decentered. Coma and astigmatism are corrected by tilting the secondary and corrector lens. The lens is thin and slightly wedged to minimize chromatic effects. Spherical aberration is corrected with the aspheric deformation of the primary. A three-mirror Schiefspiegler, or Trischiefspiegler, has been developed by Kutter.84 This design is all-reflective and completely achromatic. Like the Schiefspiegler, aspheric deformation of the primary corrects spherical aberration; coma and astigmatism are corrected with element tilts. A four-mirror Schiefspiegler was recently introduced by Brunn.85 For more on unusual telescope objectives, see Manly.86 F/8 Catadioptric Herschelian Objective SUR

RDY

STO 2

269.61 INF

THI 1.487 2.147

GLA BK7

Element tilt: 0.35° 3 4

269.61 INF

1.321 151.97

BK7

Element tilt: 5.38° 5

317.26

158.69

REFL


Comments Several centuries ago, Herschel tilted his large parabolic mirror to give him access to the image. A spherical mirror in this design by D. Shafer87 has been tilted for the same reason. Element tilts in the Houghton corrector control the astigmatism introduced by tilting the mirror. The Houghton corrector also eliminates the spherical aberration of the mirror with lens bending. Note the smaller focal ratio of this design compared to either the Yolo or the Schiefspiegler. Other catadioptric Herschelians, as well as Schiefspieglers and Yolos, have been studied by Buchroeder88 and Leonard.82 Tilted, decentered, and unobscured Cassegrains are discussed by Gelles.89 F/4 SEAL

SUR

RDY

THI

GLA

1 2 STO 4

181.2 350.9 INF 350.9

147.8 147.8

147.8 119

REFL REFL REFL REFL

CON

0.404

0.404

Comments For an all-reflective objective, this flat-image design provides an exceptionally wide, unobscured field of view—greater than 90° with a ring field. Referred to as the SEAL,90 it is

29.28

INSTRUMENTS

derived from its cousin the WALRUS;91 a related design has been described by Shafer.92 The SEAL is another Mersenne-Schmidt hybrid: primary and secondary form an inverse-Mersenne; tertiary and quaternary (also the secondary) form a reflective Schmidt. Residual spherical aberration limits the performance, but by aspherizing the flat, this residual aberration is corrected as well. Clearing all obscurations requires at least a 22° field offset. The SEAL shown here is optimized for a 20° strip field although a square, rectangular, annular, or almost any shape field is possible. F/4 Paul Three-Mirror Objective

SUR

RDY

THI

GLA

CON

STO 2 3

117.1

31.38

42.87

42.87 42.87

21.42

REFL REFL REFL

1

.6076

Comments This design is based on work by Paul93 and later by Baker,94 who was looking for an achromatic field corrector for a parabolic primary. Their efforts culminated in a design similar to this one, which combines the properties of an afocal Cassegrain-Mersenne in the first two elements with the properties of an all-reflective Schmidt in the secondary and tertiary elements. Since both modules are corrected for spherical aberration, coma, and astigmatism to third order, the complete system is aplanatic and anastigmatic. Petzval curvatures are equal and opposite so a flat image is achieved. The conic deformation of the secondary is modified to give it an ellipsoidal shape. This gives the required Schmidt aspherization needed to correct the spherical aberration of the tertiary mirror. Other all-reflective designs have been proposed by Meinel21,95 and Baker.22 The Meinel-Shack objective96 exhibits similar performance and offers a more accessible image. F/4 Alternative Paul Three-Mirror Objective

SUR

RDY

THI

GLA

CON

STO 2 3

142.4

39.51

54.69

49.28 49.28

30.7

REFL REFL REFL

1 0.101

Comments This Paul objective has an aspheric tertiary mirror, instead of an aspheric secondary. F/4 Off-Axis, Eccentric-Pupil Paul-Gregorian

SUR

RDY

THI

STO 2 3

INF

158.4 79.2

79.2

118.8 79.2

GLA

CON

REFL REFL

1

1

A: 0.2707E-6 B: 0.117E-9 4

77.53

38.768

REFL


29.29

Comments This eccentric-pupil (22-cm), off-axis (1°) design utilizes a Gregorian-Mersenne module in the primary and secondary mirrors. Spherical aberration produced by the spherical tertiary mirror is corrected by superimposing aspheric deformation on the paraboloid secondary, located at the tertiary mirror center of curvature. With all concave surfaces, field curvature is uncorrected. A real and accessible exit pupil and intermediate image offer possibilities for excellent straylight suppression. As is the case with the virtual Schmidt system, the tertiary mirror may be decentered to provide a more convenient image location. This requires two-axis aspheric deformation of the secondary mirror.97 F/4 Three-Mirror Cassegrain

SUR

RDY

THI

GLA

CON

STO 2 3

39.67

10.66 INF

15.814 21

9.05

REFL REFL REFL

0.9315

2.04

REFL

0.4479

Tilt: 45° 4

13.66

13.651

Comments A design similar to the aplanatic, anastigmatic, flat-image design shown here was conceived by Korsch98 and is described by Williams,99 Korsch,100 and Abel.43 The exit pupil is accessible and an intermediate image exists. A 1° field offset is needed to displace the image from the folding flat. Residual coma limits field performance. Small element tilts and decenters will improve the performance of this design. Three-Mirror Afocal Telescope

SUR

RDY

THI

GLA

CON

STO 2 3

100.725

46.109

74.819

33.514 100

55.56

REFL REFL REFL

1

3.016

1

Comments This 5r afocal design from Smith2 is an eccentric-pupil Cassegrain and a parabolic tertiary combined. The design is aplanatic and anastigmatic. The entrance pupil is decentered by 32 cm. Three-Mirror Afocal Telescope

SUR

RDY

THI

GLA

CON

STO 2 3

240

160

480

200 200

250

REFL REFL REFL

1

9

1

29.30

INSTRUMENTS

Comments A similar design by Korsch14 is also aplanatic and anastigmatic. The entrance pupil is decentered by 20 cm. Other afocal designs are described by Gelles101 and King.102 F/4 Three-Mirror Cassegrain

SUR

RDY

THI

GLA

CON

STO 2 3

59.64

28.63

55.05

18.29 33.74

13.244

REFL REFL REFL

1.134

2.841

5.938

Comments Robb103 has introduced another aplanatic, anastigmatic, flat-image, three-mirror Cassegrain without an intermediate image. F/6.7 Spherical Primary Three-Mirror Objective SUR

RDY

THI

GLA

CON

STO 2 3

429.67

104.16

126.49

149.87 211.14

73.0

REFL REFL REFL

3.617

0.179

Comments Making the largest element in an objective a spherical mirror reduces cost and may enhance performance. This aplanatic, anastigmatic, flat-image, eccentric-pupil design ( 35 cm stop decenter) with an unobscured light path is similar to one described by Korsch14 and another developed for use as an astrometric camera by Richardson and Morbey.104 F/4 Spherical Primary Three-Mirror Objective SUR

RDY

THI

GLA

CON

STO 2 3

194.58

64.42

38.47

79.13 113.68

26.24

REFL REFL REFL

12.952

0.4826

Comments Here is another aplanatic, anastigmatic, flat-field, eccentric-pupil design with a 17-cm stop decenter and large spherical primary. There is an intermediate image and an accessible exit pupil. F/4 Three-Mirror Korsch Objective

SUR

RDY

THI

GLA

CON

1 STO 3

201.67

96.5

172.54

133.36 131.8

200.83

REFL REFL REFL

0.689

1.729


29.31

Comments This off-axis (5°) design by Korsch105 is aplanatic, anastigmatic, and has a flat image. The same configuration has been employed by Pollock106 as a collimator. Characteristics include a large field of view, low pupil magnification, accessible pupils, and an intermediate image. The tertiary in this design is spherical. With reoptimization, the secondary may also be spherical. F/4 Three-Mirror Cook Objective

SUR

RDY

THI

GLA

CON

1 2 3 STO

123.2

37.46

51.89 INF

57.38 57.45

35.87 −15.92

REFL REFL REFL

0.7114

3.824

0.1185

Comments This objective was introduced by Cook.107–109 The aplanatic, anastigmatic, flat-image design shown here has a larger pupil magnification and a smaller field than the previous design. The eccentric-pupil, off-axis design has a −3.2-cm stop decenter and a 5° field bias. A space-based surveillance objective in this configuration has been developed and built by Wang et al.110 F/4 Three-Mirror Wetherell and Womble Objective

SUR

RDY

THI

GLA

CON

1 STO 3

166.19

55.19

82.46

38.78 38.78

65.24

REFL REFL REFL

2.542

0.428 0.133

Comments Another aplanatic, anastigmatic, flat-image, off-axis (9°) design has been introduced by Wetherell and Womble.111 Figosky112 describes a variant of this form to be sent into orbit. The aperture stop is located at the secondary mirror; hence, this mirror is symmetric with respect to the optical axis. F/10 Korsch Three-Mirror, Four-Reflection Objective

SUR

RDY

THI

GLA

CON

STO 2 3 4

66.44

22.15

66.44

44.29

22.15 22.15

22.15 21.96

REFL REFL REFL REFL

1.092

1.295

1.092 0.8684

29.32

INSTRUMENTS

Comments The three-mirror, four-reflection design shown here from Korsch113 is extremely compact for its 200-cm focal length, and the image is accessible. F/1.25 McCarthy

SUR

RDY

THI

GLA

CON

STO 2

81.57 INF

40.21 25.09

REFL REFL

1

REFL REFL REFL

1

Tilt: 45° 3 4 5

48.68

19.15

49.85

29 30.64

65.483

Comments McCarthy114 intended this design, which combines a Cassegrain-Mersenne primary and tertiary mirror with a quaternary and quintenary Schwarzschild arrangement, as a wide stripfield imager. Both the Mersenne and Schwarzschild groups are separately corrected for spherical aberration, coma, and astigmatism. The Petzval curvature of the Mersenne is equal and opposite in sign to the Petzval curvature of the Schwarzschild and hence there is no net Petzval curvature. The quaternary mirror may be moved out from the entering beam with only a slight reduction in performance. F/2.2 Cassegrain Objective with Schwarzschild Relay SUR

RDY

THI

1 2 3 4 STO

51.49

37.37 38.18 29.94 INF

19.01 34.19

10.493 11.27 40.484

GLA REFL REFL REFL REFL

CON

1.048

20.35

1.358

Comments Williams115 describes a technique for optimizing a high-resolution system similar to this one while maintaining proper clearances, obscuration sizes, and packaging requirements. An all-reflective zoom system of the above configuration, developed by Johnson et al.,116 gives a 4r zoom range and a field-of-view range of 1.5 to 6.0°. The Schwarzschild module and image position change with zoom position, while the front Cassegrain module remains fixed. F/4 Altenhof Objective

SUR

RDY

STO

INF

THI

80

GLA

CON

REFL

Tilt: 25° 2 3 4 5

155.64

77.26

34.146

80.65

165.53

38.57 40.367

82.539

REFL REFL REFL REFL

0.0897


29.33

Comments This objective is similar to one designed by Altenhof.117 Intended as a ring field system, the flat primary couples the light incident from a large azimuthal angle (60°) into the system where the spherical primary mirror focuses the light to a poorly corrected image. The three-mirror Offner relay,9,118 a unit magnification relay which is corrected for spherical aberration, coma, and astigmatism, improves the degraded image in the process of reimaging it to a flat focal surface in the form of an annulus. A two-dimensional scene is imaged by rotating the flat mirror about an axis perpendicular to the picture so as to scan the other dimension. A two-dimensional mosaic image can also be produced by building up a series of one-dimensional annular strip images as the imaging system is moved along its trajectory. F/4.5 Shafer Four-Mirror, Unobscured Objective

SUR

RDY

THI

GLA

1

158.1

71.21

REFL

Tilt angle: 16.44° STO

186.8

74.25

REFL

Tilt angle: 20.88° 3

337.4

111.4

REFL

Tilt angle: 24.82° 4

239.1

121.4

REFL


Comments This is a tilted and decentered-component infrared imaging system by David Shafer. Mirror tilts provide an unobscured path and help correct the aberrations. Thickness is measured along the deviated ray paths. With the reflection angle known, element decenter may be easily determined. F/4.5 Shafer Five-mirror, Unobscured Objective

SUR

RDY

THI

GLA

1

239.5

160.2

REFL

2

228.9


REFL

Tilt angle: 9.2° 3

75.94

STO

39.81

5

78.72

37.24

REFL

Tilt angle: 19.01° 39.24

REFL

Tilt angle: 28.82°

74.5

Tilt angle: −40.55° Image tilt angle: −11.28°

REFL

29.34

INSTRUMENTS

Comments Another all-spherical, tilted, and decentered-component infrared imager by Shafer is presented here. The entrance pupil is accessible and there is an intermediate image. A number of variations on this arrangement are described by Shafer.80 Korsch Two- and Three-Mirror Objectives

Comments A new class of eccentric-pupil objectives has been introduced by Korsch.14 Unlike most systems, which are conceived using third-order aberration theory, these systems are based upon the fulfillment of axial stigmatism, the Abbe sine condition, and the Herschel condition; meeting these three conditions guarantees a perfect axial point image, axially perpendicular image area, and axial line element, respectively. Design examples are not given for two reasons. First, rays strike some mirror surfaces at angles greater than 90°, which can cause ray-trace errors. Second, some of the surface shapes are particularly complex and must be entered in design software as a user-defined surface. Design (c) gives perfect imagery on-axis and less than one milliradian resolution at all other points over a 6° field of view, for an aperture diameter equal to F/6.0 (F is focal length).

29.5

FIELD-OF-VIEW PLOTS The plots that follow give rms spot size and angular resolution as a function of half-field of view. The curves have been generated by calculating the resolution for a number of field angles and connecting them with smooth curves. The dashed horizontal line is the Airy disc diameter for 0.55-μm radiation. The numbers in the plots correspond to the designs presented in the previous section. The aperture of each design is 20 cm and the spectral range 480 to 680 nm, unless stated otherwise in the previous section.


32

F/1.25 10

14

20

26

34

64

100

21

15 41

80 27

40 23

15

60 33

24

42

10

40

30 18

63

5

20

RMS Angular diameter (r)

RMS Spot diameter (m)

25

29.35

Blur diameter 0.1

1.0 Half field in degrees

10

FIGURE 7 Field-of-view plots: F/1.25 on a flat image.

It should be kept in mind that these are representative designs: they have usually been optimized for a specific purpose; they are meant to be a starting place for the design of a new system that may have entirely different requirements. Flat-field designs show consistent performance out to the field angle for which the objective is optimized. Beyond this point, the graph leaps upward. Reoptimization is needed if the field of view is to be extended further; a considerable increase in the average rms spot size may occur if this is attempted. The curved-image designs show a quadratic dependence with field angle. Off-axis and eccentric-pupil designs have rectangular fields with most of the field of view in one dimension only. Data plotted for these designs are representative of the larger field. In Figs. 8 and 10, plots for the curved image designs are provided. The curvature of the image is adjusted to give optimum performance. Figures 7 and 9 are for the flat image designs.

F/1.25

25

100 22

37

20

80 31

24 60

15 17

19

20 40

10 36

20

5 Blur diameter 0.1

FIGURE 8

10 Half field in degrees Field-of-view plots: F/1.25 on a curved image.

100



29

INSTRUMENTS


25

F/4 1

6

67

58 57

66

20 2

11

31.25

50 51

56

3

25.00

53

4

60 18.75

15 12

7

35

49 10 13

12.5

39

59 61

5

Blur diameter 6.25

28


29.36

65 0.1


10

FIGURE 9 Field-of-view plots: F/4 on a flat image.

F/4

25

31.25

52

20

25.00 2

4 18.75

15 62 10

44

12.5

43

Blur diameter 6.25

5



13

45 25 0.1 FIGURE 10

29.6


10

Field-of-view plots: F/4 on a curved image.

DEFINITIONS Abbe number: A number that indicates the dispersion of a glass. Low dispersion glasses have a high Abbe number. Abbe sine condition: A condition for zero coma, based on the requirement of equal marginal and paraxial magnifications. See Welford,61 Kingslake,9 or Korsch.14 anastigmatic: A surface or system free of astigmatism. Also stigmatic. aperture stop: The aperture that limits the size of the axial beam passing through the system; the chief ray always passes through its center. aplanatic: A surface or system that is corrected for spherical aberration and coma.


29.37

astigmatism: An aberration that generates two different focal positions for rays in two perpendicular planes centered on the optical axis. These are called the sagittal and tangential planes. axial color: The variation in focal position with wavelength. axial stigmatism: A characteristic of a surface which is able to produce a perfect image of a single point object on-axis. catadioptric: An optical system composed of refractive and reflective elements: catoptric, reflective and dioptric, refractive. chief ray: A ray that passes through the center of the aperture stop and the edge of the image or object. coma: An aberration resulting from the change in magnification with ray height at the aperture, so that rays near the edge of the aperture are focused further from rays near the axis, for the same field point. conic constant: A constant defined by k 2 where is the eccentricity of the conic. distortion: The variation in magnification with field angle. entrance pupil: The image of the aperture stop in object space. The chief ray passes or appears to pass through the center of the entrance pupil. exit pupil: The image of the aperture stop in image space. The chief ray passes or appears to pass through the center of the exit pupil. focal ratio: The effective focal length of an objective divided by its entrance-pupil diameter. Focal ratio is also referred to as the FN, F-number, and speed. field curvature: Image curvature produced by the combined effects of astigmatism and Petzval curvature. When astigmatism is absent, the image surface coincides with the Petzval surface. field stop: An aperture that limits the size of an intermediate or final image. Herschel condition: A condition for invariance of aberrations with change in axial conjugates. See Welford61 and Korsch.14 higher-order aberrations: Aberrations defined by the higher-order terms in the aberration power series expansion. See Welford61 and Schulz.1 lateral color: An aberration that produces a dependence of image size on wavelength; also called chromatic difference of magnification. Lyot stop: A real and accessible image of the aperture stop; used to block stray light. marginal ray: A ray that passes through the center of the object or image and past the edge of the aperture stop. medial image: The image halfway between the sagittal and tangential images. See Welford.61 monocentric system: An optical system in which all surfaces are concentric about the chief ray. oblique spherical aberration: A higher-order aberration that is the variation of spherical aberration with field angle. optical axis: The axis about which all optical elements are symmetric. In tilted and decentered systems, each element has a unique optical axis. Petzval sum: The sum defined by F p£ n where H is element power and n is the index of refraction. The reciprocal of the Petzval sum is the image radius of curvature.

29.38

INSTRUMENTS

secondary magnification: System focal length divided by primary-mirror focal length. secondary spectrum: The difference in focal position between two wavelengths corrected for axial color and one other wavelength which is not. For example, the blue and red focus coincide and the yellow focus is axially displaced. spherical aberration: The only on-axis monochromatic aberration, spherical aberration results from rays at different heights coming to focus at different points along the optical axis. Smith,3 Rutten,12 Kingslake,9 Mackintosh,83 and Welford61 discuss aberrations. Welford specifically addresses aberrations. third-order aberrations: Any of the Seidel aberrations: spherical aberration, coma, astigmatism, Petzval curvature, and distortion. See Welford.61 vignetting: The off-axis clipping of light by apertures in an optical system. virtual image: A real image is visible when a screen is placed at its location. The image is visible because rays from the object converge at the image. A virtual image is not visible when a screen is placed at its location since real rays do not converge. zonal spherical aberration: The incomplete correction of spherical aberration at radial zones in the aperture. For example, spherical aberration could be corrected for rays close to the center and edge of the aperture, but not corrected at other ray heights in the aperture.

29.7

REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23.

G. Schulz, “Aspheric Surfaces,” Progress in Optics XXV:351 (1988). W. Smith, Modern Lens Design, McGraw-Hill, Inc., New York, 1992. W. Smith, Modern Optical Engineering, McGraw-Hill, Inc., New York, 1990. E. Everhart, and J. Kantorski, “Diffraction Patterns Produced by Obstructions in Reflective Telescopes of Modest Size,” Astronomical Journal 64:455 (1959). K. Thompson, “Aberration Fields in Tilted and Decentered Optical Systems,” Ph.D. dissertation, Optical Sciences Center, University of Arizona, 1980. J. Sasian, “Review of Methods for the Design of Unsymmetrical Optical Systems,” SPIE 1396:453 (1990). R. Fischer, “What’s So Different About IR Lens Design?” SPIE Proceedings CR41:117 (1992). A. Mangin, “Memorial de L’officier du Genie (Paris),” 25(2):10, 211 (1876). R. Kingslake, Lens Design Fundamentals, Academic Press, Inc., New York, 1978. L. Schupmann, “Die Medial-Fernrehre,” Leipzig (1899). E. Olsen, “A Schupmann for Amateurs,” in Advanced Telescope Making Techniques, Willmann-Bell, Inc., Richmond, Va., 1986, p. 223. H. Rutten, and M. Venrooij, Telescope Optics, Willmann-Bell, Inc., Richmond, Va., 1988. D. Schroeder, Astronomical Optics, Academic Press, Inc., San Diego, Calif., 1987. D. Korsch, Reflective Optics, Academic Press, San Diego, Calif. 1991. M. Chretien, “Le Telescope de Newton et le Telescope Aplanatique,” Rev. d’Optique (2):49 (1922). C. Wynne, “Field Correctors for Large Telescopes,” Applied Optics 4(9):1185 (1965). D. Schulte, “Anastigmatic Cassegrain Type Telescope,” Applied Optics 5(2):309 (1966a). S. Rosin, “Ritchey-Chretien Corrector System,” Applied Optics 5(4):675 (1966). R. Wilson, “Corrector Systems for Cassegrain Telescopes,” Applied Optics 7(2):253 (1968). S. Rosin, “Corrected Cassegrain System,” Applied Optics 3(1):151 (1964). A. Meinel, M. Meinel, D. Su, and Ya-Nan Wang, “Four-Mirror Spherical Primary Submillimeter Telescope Design,” Applied Optics 23(17):3020 (1984). J. Baker, “Explorations in Design,” SPIE 147:102 (1978). H. Brueggeman, Conic Mirrors, Focal Press, London, 1968.


29.39

24. T. Fjeidsted, “Selectable Field-of-View Infrared Lens,” U.S. Patent 4,453,800, 1984. 25. C. Wynne, “A New Wide-Field Triple Lens Paraboloid Field Corrector,” Monthly Notices Royal Astronomical Society 167:189 (1974). 26. S. Gascoigne, “Recent Advances in Astronomical Optics,” Applied Optics 12:1419 (1973). 27. F. Ross, “Lens Systems for Correcting Coma or Mirrors,” Astrophysics Journal, 81:156 (1935). 28. A. Meinel, “Aspheric Field Correctors for Large Telescopes,” Astrophysical Journal 118:335 (1953). 29. D. Schulte, “Prime Focus Correctors Involving Aspherics,” Applied Optics 5(2):313 (1966). 30. J. Baker, Amateur Telescope Making, Book Three, Scientific American, Inc., 1953, p. 1. 31. C. Wynne, “Field Correctors for Astronomical Telescopes,” Progress in Optics, X:139 (1972). 32. A. Couder, Compt. Rend. Acad. Sci. Paris, 183, II, 1276, p. 45 (1926). 33. K. Schwarzschild, “Untersuchungen zur Geometrischen Optik, II; Theorie der Spiegelteleskope,” Abh. der Konigl. Ges. der Wiss. zu Gottingen, Math.-phys. Klasse, 9, Folge, Bd. IV, No. 2 (1905). 34. I. Abel, and M. Hatch, “The Pursuit of Symmetry in Wide-Angle Reflective Optical Designs,” SPIE 237:271 (1980). 35. W. Wetherell, and P. Rimmer, “General Analysis of Aplanatic Cassegrain, Gregorian and Schwarzschild Telescopes,” Applied Optics 11(12):2817 (1972). 36. E. Linfoot, Recent Advances in Optics, Oxford University Press, London, 1955. 37. J. Sasian, “Design of a Schwarzschild Flat-Field, Anastigmatic, Unobstructed, Wide-Field Telescope,” Optical Engineering 29(1) January:1 (1990). 38. D. Shafer, “An All-Reflective Infrared Target Simulator Design,” October OSA Workshop on Optical Fabrication and Testing, 1988. 39. B. Schmidt, Mitt. Hamburg. Sternwart 7(36) (1932). 40. J. Maxwell, Catadioptric Imaging Systems, Elsevier, New York, 1972. 41. C. Wynne, “Shorter than a Schmidt,” Monthly Notices Royal Astronomical Society 180:485 (1977). 42. F. Wright, “An Aplanatic Reflector with a Flat-Field Related to the Schmidt Telescope,” Astronomical Society of the Pacific 47:300 (1935). 43. I. Abel, “Mirror Systems: Engineering Features, Benefits, Limitations and Applications,” SPIE 531:121 (1985). 44. D. Schulte, “Auxiliary Optical Systems for the Kitt Peak Telescopes,” Applied Optics 2(2):141 (1963). 45. J. Baker, “The Solid Glass Schmidt Camera and a New Type of Nebular Spectrograph,” Proceeding American Philosophical Society 82:323 (1940). 46. J. Baker, “A Family of Flat-Field Cameras, Equivalent in Performance to the Schmidt Camera,” Proceeding American Philosophical Society 82(3):339 (1940). 47. A. Warmisham, British Patent 551,112, 1941. 48. R. Wayman, “The Monocentric Schmidt-Cassegrain Cameras,” Proceeding Physics Society (London), 63:553 (1944). 49. R. Buchroeder, “Catadioptric Designs,” OSC, University of Arizona, Technical Report 68, May, 1971. 50. D. Shafer, “Well-Corrected Two-Element Telescope with a Flat Image,” Optica Acta 28(11):1477 (1981). 51. D. Schroeder, “All-Reflecting Baker-Schmidt Flat-Field Telescopes,” Applied Optics 17(1):141 (1978). 52. D. Shafer, “Design with Two-Axis Aspheric Surfaces,” SPIE 147:171 (1978). 53. M. Roesch, Transactions of the International Astronomical Union VII:103 (1950). 54. D. Maksutov, Journal of the Optical Society of America, 34:270 (1944). 55. A. Bouwers, Achievements in Optics, Elsevier, Amsterdam, 1950. 56. C. Wynne, “Maksutov Spectrograph Cameras,” Monthly Notices Royal Astronomical Society 153:261 (1971). 57. C. Wynne, “Five Spectrograph Camera Designs,” Monthly Notices Royal Astronomical Society 157:403 (1972). 58. G. Courtes, “Optical Systems for UV Space Researches,” From New Techniques in Space Astronomy, D. Reidel Publishing Company, Dordrecht, Holland, 1971. 59. G. Courtes, P. Cruvellier, M. Detaille, and M. Saisse, Progress in Optics vol. XX, 1983, p. 3.

29.40

INSTRUMENTS

60. D. Gabor, British Patent 544,694, 1941. 61. W. Welford, Aberrations of Optical Systems, Adam Hilger Ltd., Bristol, England, 1986. 62. D. Shafer, “Simple Method for Designing Lenses,” SPIE 237:234 (1980). 63. D. Shafer, “Optical Design with Only Two Surfaces,” SPIE 237:256 (1980). 64. J. Baker, “Schmidt Image Former with Spherical Abberation Corrector,” U.S. Patent 2,458,132, 1945. 65. J. Baker, “Correcting Optical System,” U.S. Patent 3,022,708, 1962. 66. J. Houghton, “Lens System,” U.S. Patent 2,350,112 (1944). 67. A. Sonnefeld, “Photographic Objective,” U.S. Patent 2,141,884, 1938. 68. R. Sigler, “Compound Catadioptric Telescopes with All-Spherical Surfaces,” Applied Optics 17(10):1519 (1978). 69. D. Hawkins, and E. Linfoot, “An Improved Type of Schmidt Camera,” Monthly Notices Royal Astronomical Society 105:334 (1945). 70. D. Rao, “Design of a Near Diffraction-Limited Catadioptric Lens,” SPIE 766:220 (1987). 71. W. Mandler, “Reflecting Mirror and Lens System of the Cassegrain Type,” U.S. Patent 2,726,574, 1951. 72. R. Gelles, “A New Family of Flat-Field Cameras,” Applied Optics 2(10):1081 (1963). 73. M. Shenker, “High Speed Catadioptric Objective in Which Three Corrector Elements Define Two Power Balanced Air Lenses,” U.S. Patent 3,252,373, 1966. 74. M. Laiken, Lens Design, Marcel Dekker, Inc., New York, 1991. 75. L. Canzek, “High Speed Catadioptric Objective Lens System,” U.S. Patent 4,547,045, 1985. 76. M. Amon, “Large Catadioptric Objective,” U.S. Patent 3,711,184, 1973. 77. N. Rumsey, “Telescopic System Utilizing Three Axially Aligned Substantially Hyperbolic Mirrors,” U.S. Patent 3,460,886, 1969. 78. S. Eisenburg, and E. Pearson, “Two-Mirror Three-Surface Telescope,” SPIE 751:24 (1987). 79. D. Shafer, “Well Baffled Two-Mirror Corrector for a Parabola,” Applied Optics 16(5):1175 (1977). 80. D. Shafer, “Four-Mirror Unobscured Anastigmatic Telescopes with All-Spherical Surfaces,” Applied Optics 17(7):1072 (1978). 81. A. Leonard, “New Horizons for Tilted-Component Telescopes,” in Advanced Telescope Making Techniques, Willmann-Bell, Inc., Richmond, Va., 1986, p. 110. 82. A. Leonard, “T.C.T.’s (Tilted Component Telescopes),” in Advanced Telescope Making Techniques, WillmannBell, Inc., Richmond, Va., 1986, p. 131. 83. A. Mackintosh, Advanced Telescope Making Techniques, Willmann-Bell, Inc., Richmond, Va., 1986. 84. A. Kutter, “A New Three-Mirror Unobstructed Reflector,” Sky and Telescope, Jan. 1975, p. 46; Feb. 1975, p. 115. 85. M. Brunn, “Unobstructed All-reflecting Telescopes of the Schiefspiegler Type,” U.S. Patent 5,142,417, 1992. 86. P. Manly, Unusual Telescopes, Cambridge University Press, Cambridge, England, 1991. 87. D. Shafer, “A Simple Unobstructed Telescope with High Performance,” Telescope Making 41:4 (TM 41). 88. R. Buchroeder, “Fundamentals of the TST,” OSC, University of Arizona, Technical Report 68, May, 1971. 89. R. Gelles, “Unobscured-Aperture Two-Mirror Systems,” Journal of the Optical Society of America 65(10):1141 (1975). 90. R. Owen, “Easily Fabricated Wide-Angle Telescope,” SPIE 1354:430 (1990). 91. K. Hallam, B. Howell, and M. Wilson, “Wide-Angle Flat Field Telescope,” U.S. Patent 4,598,981, 1986. 92. D. Shafer, “30 Degree F/1.0 Flat Field Three Mirror Telescope,” presented Oct. 31 in San Francisco at Optical Society of America Annual Meeting, 1978. 93. M. Paul, Rev. Opt. 14:169 (1935). 94. J. Baker, “On Improving the Effectiveness of Large Telescopes,” IEEE Transactions AES-5, (2):261 (1969). 95. A. Meinel, and M. Meinel, SPIE 332:178 (1982). 96. A. Meinel, and R. Shack, “A Wide-Angle All-Mirror UV Camera,” Optical Sciences Center Tech. Rep. No. 6, 1966.


29.41

97. D. Shafer, “20 Degree Field-of-View Strip Field Unobscured Telescope with Stray Light Rejection,” presented Oct. 31 in San Francisco at Optical Society of America Annual Meeting, 1978. 98. D. Korsch, “Anastigmatic Three-Mirror Telescope,” U.S. Patent 4,101,195 (1978). 99. S. Williams, “On-Axis Three-Mirror Anastigmat with an Offset Field of View,” SPIE 183:212 (1979). 100. D. Korsch, “Anastigmatic Three-Mirror Telescope,” Applied Optics 16(8):2074 (1977). 101. R. Gelles, “Unobscured Aperture Stigmatic Telescopes,” Optical Engineering 13(6):534 (1974). 102. W. King, “Unobscured Laser-Beam-Expander Pointing System with Tilted Spherical Mirrors,” Applied Optics 13(1):21 (1974). 103. P. Robb, “Three-mirror Telescopes: Design and Optimization,” Applied Optics 17(17):2677 (1978). 104. E. Richardson, and C. Morbey, “Optical Design of an Astrometric Space Telescope,” SPIE 628:197 (1986). 105. D. Korsch, “Wide-Field Three-Mirror Collimator,” U.S. Patent 4,737,021, 1988. 106. D. Pollock, “An All-Reflective, Wide Field of View Collimator,” SPIE 751:135 (1987). 107. L. Cook, “Three-Mirror Anastigmatic Optical Systems,” U.S. Patent 4,265,510, 1981. 108. L. Cook, “Three-Mirror Anastigmat Used Off-Axis in Aperture and Field,” SPIE 183:207 (1979). 109. L. Cook, “The Last Three-Mirror Anastigmat (TMA)?” SPIE Proceedings vol. CR41:310 (1992). 110. D. Wang, L. Gardner, W. Wong, and P. Hadfield, “Space Based Visible All-Reflective Stray Light Telescope,” SPIE 1479:57 (1991). 111. W. Wetherell, and D. Womble, “All-Reflective Three Element Objective,” U.S. Patent 4,240,707, 1980. 112. J. Figosky, “Design and Tolerance Specification of a Wide-Field, Three-Mirror, Unobscured, HighResolution Sensor,” SPIE 1049:157 (1989). 113. D. Korsch, “Two Well-Corrected Four-Mirror Telescopes,” Applied Optics 13(8):1767 (1974). 114. E. McCarthy, “Anastigmatic Catoptric Systems,” U.S. Patent 3,062,101, 1959. 115. S. Williams, “Design Techniques for WFOV High-Resolution Reflecting Optics,” SPIE 147:94 (1978). 116. R. Johnson, J. Hadaway, and T. Burleson, “All-Reflective Four-Element Zoom Telescope: Design and Analysis,” SPIE 1354:669 (1990). 117. R. Altenhof, “The Design of a Large Aperture Infrared Optical System,” SPIE 62:129 (1975). 118. A. Offner, “New Concepts in Projection Mask Aligners,” Optical Engineering 14(2):130 (1975).


30 SCANNERS Leo Beiser Consultant Flushing, New York

R. Barry Johnson Consultant Huntsville, Alabama

30.1

GLOSSARY a A C d D fe f F FL FOV FWHM H I k m m` M n N % P PSF q Q

aperture shape factor; calibrates diffraction angle as function of aperture intensity distribution area capacitance, electrical grating spacing, array element spacing useful beam width in scan direction (see W ); Dm enlarged beam width due to a data bandwidth focal length F-number (f/D) field lens field of view full width (of F ) measured at half maximum intensity vehicle height resolution invariant; adaptation of Lagrange invariant, I 1D 1`D`, normalized intensity scanning constant (see m) scan magnification (d1/d&) (for m constant, m k 1/&) composite magnification optical magnification (image/object) number of facets, refractive index, diffractive order number of resolution elements subtended by 1 or S total number of cells in a phased array radiant power (watts) point spread function (intensity distribution of focused spot) number of cells in array period q-factor (electromechanical) 30.1

30.2

INSTRUMENTS

r R s S t T V W w x y C I F $ G J 1, S & N , T U V

30.2

radius reciprocity failure factor (q1), scanner rpm sensitivity (recording medium) format width, no. of scans per revolution of scanner time time period, optical transfer factor (a1) vehicle velocity, electrical voltage full aperture width, of which D is utilized (see T) width at 1/e2 intensity along-scan direction cross-scan direction angular departure from normal beam landing, spectral power transfer factor, radiation intensity parameter phosphor utilization factor (a1) spot size; if Gaussian, measured across 1/e2 intensity width or at FWHM shift distance chromatic error, ellipticity duty cycle, conversion efficiency (a1), efficiency optical scan angle, angular beamwidth mechanical scan angle, along-track optical scan angle, optical field angle wavelength wavelength (acoustic grating), array period truncation ratio (W/D) Gaussian standard deviation transit time (acoustic), retrace time, dwell time

INTRODUCTION This chapter provides an overview of optical scanning techniques in context with their operational requirements. System objectives determine the characteristics of the scanner which, in turn, influence adjacent system elements. For example, the desired resolution, format, and data rate determine the scanner aperture size, scan angle, and speed, which then influence the associated optics. The purpose of this chapter is to review the diverse options for optical scanning and to provide insight to associated topics, such as scanned resolution and the reduction of spatial errors. This broad perspective is, however, limited to those factors which bear directly on the scanner. Referencing is provided for related system relationships, such as image processing and data display. Topics are introduced with brief expressions of the fundamentals. And, where appropriate, historical and technical origins are referenced. The subject of scanning is often viewed quite differently in two communities. One is classified as remote sensing and the other, input/output scanning. Associated component nomenclature and jargon are, in many cases, different. While their characteristics are expanded in subsequent sections, it is useful to introduce some of their distinctions here. Remote sensing detects objects from a distance, as by a space-borne observation platform. An example is infrared imaging of terrain. Sensing is usually passive and the radiation incoherent and often multispectral. Input/output scanning, on the other hand, is local. A familiar example is document reading (input) or writing (output). Intensive use of the laser makes the scanning active and the radiation coherent. The scanned point is focused via finite-conjugate optics from a local fixed source. While the scanning components may appear interchangeable, special characteristics and operational modes often preclude this option. This is most apparent for diffractive devices such as

SCANNERS

30.3

acousto-optic and holographic deflectors. It is not so apparent regarding the differently filled scanning apertures, imparting important distinctions in resolution and duty cycle. The unification of some of the historically separated parameters and nomenclature is considered an opportunity for this writing. A recent concentration of R&D in the field of agile beam steering is presented in Sec. 30.8. Intensive work has yielded encouragement in the long quest for achieving, for example, the performance of articulated large mirrors while avoiding some of the systemic burdens of size, weight, and inertia. Recently, remarkable work has been done using microelectromechanical systems (MEMS) to make scanners on a small scale. Two particularly interesting devices are presented and are the digital micromirror device (DMD) incorporated into digital light processing (DLP) projectors and gimbal-less two-axis scanning-micromirror devices (GSMD). The DMD is essentially a two-dimensional array binary-state scanner or light switches, while the GSMD is a fully analog dual-axis scanner.

System Classifications The following sections introduce the two principal disciplines of optical scanning, remote sensing, and input/output scanning, in preparation for discussion of their characteristics and techniques. Remote Sensing The applications for passive (noninvasive) remote sensing scanners are varied and cover many important aspects of our lives. A signature representative of the target is obtained to form a signal for subsequent recording or display. This process is operationally distinct from active scanning, as expressed further in this chapter. Table 1 lists typical applications of these techniques. Clearly, remote scanning sensors can be hand held to satellite-borne. A variety of scanning methods has been developed to accomplish the formation of image (or imagelike) data for remote sensing. These methods may be roughly divided into framing, pushbroom, and mechanical. Generally stated, frame scanning requires no physical scan motion and implies that the sensor has a two-dimensional array of detectors which are read out by use of electronic means (e.g., CCD), electron beam, or light beam. Such an array requires an optical system that has two-dimensional wide-angle capability. Pushbroom methods typically employ some external means to move the image of a linear array of detectors along the area to be imaged. Mechanical methods generally include one- and two-dimensional scanning techniques incorporating as few as one detector to multiple- detector arrays. As is the case for pushbroom methods, image formation by one-dimensional mechanical scanning requires that the platform containing the sensor (or in

TABLE 1 Representative Applications of Passive Scanning Sensors Medical Cancer Arthritis Whiplash Industrial Energy management Thermal fault detection Electronic circuit detection Nondestructive testing Scientific Earth resources Weather Astronomy

Government Forest fires Police Smuggling Search and rescue Military Gun sights Night vision Tactical Navigation Missiles Strategic Aircraft ICBM Surveillance

30.4

INSTRUMENTS

some cases the object) be moved to create the second dimension of the image. The latter two methods are discussed further in later sections of this chapter. Mechanical scanners can be configured to perform either one- or two-dimensional scan patterns. In the case of a one-dimensional scanner, the second dimension needed to form an image is most often generated by movement of the sensor platform. A number of optical scanning systems have been invented to satisfy the wide variety of applications. In the early years of passive scanning systems, the entire optical system was often moved in order to produce either a one- or two-dimensional scan pattern. The advent of airborne mapping or reconnaissance electro-optical systems began during the 1950s. Typically, the scanner performed a one-dimensional scan in object-space, as defined under the section “Object-Space and Image-Space Scanners,” orthogonal to the flight direction of the aircraft, while the motion of the aircraft generated the second dimension of the image. The resultant video information is stored on a recording medium such as photographic film, digital tape, and the like. The design and resultant performance of a scanning system are governed by a variety of parameters that are related by trade-off equations and considerations. The selection of the scanner type typically has a strong influence upon the ultimate system performance. In subsequent discussion, the more important parameters related to the scanner selection will be covered. The complexities of the total system design and optimization are not within the scope of this chapter. Input/Output Scanning In contrast to remote sensing, which captures passive radiation, active input/output scanning illuminates an object or medium with a “flying spot,” derived typically from a laser source. Some examples appear in Table 2, divided into two principal functions: input (detecting radiation scattered from the scanning spot) and output (recording or display). Input is modulated by the target to form a signal; output is modulated by a signal. Some merit clarification. Under input is laser radar—a special case of active remote sensing, using the same coherent and flying-spot scanning disciplines as the balance of those exemplified. Earth resources imaging is the recording of remotely sensed image signals. Finally, data/image display denotes the general presentation of information, which could include “hard copy” and/or actively projected and displayed images. Active scanning is synonymous with flying-spot scanning, the discipline most identified with the ubiquitous cathode-ray tube (CRT). While the utilized devices and their performance differ significantly, the distinctions between CRT and laser radiation are primarily their degrees of monochromaticity and coherence, as addressed later in this chapter. Thus, most high-resolution and high-speed flying-spot scanning are now conducted using the laser as a light source. This work in input/output scanning concentrates on the control of laser radiation and the unique challenges encountered in deflecting photons, devoid as they are of the electric and magnetic fields accompanying the electron beam. Reference is provided1–3 for pursuit of the CRT scanning discipline.

TABLE 2

Examples of Input/Output Scanning

Input Image scanning/digitizing Bar-code reading Optical inspection Optical character recognition Optical data readout Graphic arts camera Scanning confocal microscopy Color separation Robot vision Laser radar Mensuration

Output Image recording/printing Color image reproduction Medical image outputs Data marking and engraving Microimage recording Reconnaissance recording Optical data storage Phototypesetting Graphic arts platemaking Earth resources imaging Data/Image display

SCANNERS

30.5

FIGURE 1 Conjugate imaging system, showing scan regions, as determined by position of scanning member relative to the objective lens. Translational (objective) scan and angular (pre/postobjective) scan can occur individually or simultaneously.4

Scanner classification Following the nomenclature introduced in the early 1970s,4,5 laser scanners are designated as preobjective, objective, and postobjective. Figure 1 indicates the scan regions within a general conjugate optical transfer of a fixed reference (object) point Po to a moving focal (image) point Pi. The component which provides principal focusing of the wavefront identifies the objective lens. The scanner can perform two functions (see section “Objective, Preobjective, and Postobjective Scanning” later in this chapter): one is translation of the aperture with respect to the information medium. This includes translation of the lens element(s) or translation of the object, or both, and is identified as an objective scan. The other is angular change of the optical beam with respect to the information medium. Angular scanners are exemplified by plane mirrors on rotating substrates. Although lenses can be added to an angular scanner, it is seldom so configured. The scanner is either preobjective or postobjective. In holographic scanning, however, the hologram can serve as an objective lens and scanner simultaneously. Radial symmetry and scan magnification A basic characteristic of some angular scanners is identified as radial symmetry. When an illuminating beam converges to or diverges from the nodal or rotating axis of an angular scanner, it is said to exhibit radial symmetry.6 The collimated beam which is parallel to the rotating axis is a special case of radial symmetry, in which the illuminating beam propagates to or from a very distant point on the axis. Scanners exhibiting radial symmetry provide unity angular optical change for unity mechanical change. That is, m d1 /d& 1, where 1 is the optical scan angle and is the mechanical change. The parameter m is called the scan magnification, discussed later under “Augmented Resolution, the Displaced Deflector” for Eq. (19). It ranges typically between 1 and approximately 2, depending on the scanner-illumination configuration, per Table 3. In remote sensing, m 1/& k. (See “Compound Mirror Optics Configurations.”) The prismatic polygon (see “Monogon and Polygon Scanners”) exhibits a variable m, depending on the degree of collimation or focusing of the output beam. When collimated, m 2. When focusing, the value of m shifts from 2 according to the composite magnification m ` 2 r /f

(1)

where f and r are according to Fig. 4 and Eq. (19). This is similar to the ratio of angular velocities of the scanned focal point along the arc of a limaçon,5 /& 2¤1 cos & ³ 1 ¥¦ 1 f /r ´µ

(2)

30.6

INSTRUMENTS

TABLE 3 Item

Typical Features of Pyramidal and Prismatic Polygon Scanners Description

1

Input beam directiona

2

Output beam directiona

3

Scan magnificationb (scanning constant) Along-scan error magnification Max. scan angle, 1max (n no. of facets) Output beam rotation about its axisd Aperture shapee (overilluminated) Enlargement of alongscan beam width Error due to axial polygon shiftg Error due to radial polygon shiftg Fabrication cost

3a 3b 4 5 6 7 8 9

Pyramidal

Prismatic

Radially symmetricb (typically parallel to axis) Arbitrary angle to axis (typically perpendicular) 1

Perpendicular to axisc Perpendicular to axisc

1

2b

2R/n

4R/nb

Yes

No

Triangular/keystone

Rectangular

No Yes

Yes f Dm D/cos C No

Yes

Yes

Greater

Lower

2b

a

With respect to rotating axis. b See sections “Radial Symmetry and Scan Magnification” and “Augmenting and Scan Magnification.” Output beam assumed collimated. c All beams typically in same plane perpendicular to axis. See Figs. 26 and 28. d Observable when beam is nonisotropic; e.g., elliptic, polarized, etc. Rotation of isotropic beam normally not perceived. See “Image Rotation and Derotation.” e See Table 4. f C angular departure from normal landing. See “Scanner-Lens Relationships.” g Shift of image focal point in noncollimated light. No error in collimated light.

/& l 2. In holographic scanners which are not radially symNote that when r l 0 or when f l c, 1 metric, m depends on the angles of incidence and diffraction of the input and first-order output beam, m sinQi sinQo L /d

(3)

where Si and So are the input and diffracted angles (with respect to the grating normal) and d is the grating spacing. For example, when Si So 30°, m 1; when Si So 45°, m 2.

30.3

SCANNED RESOLUTION

Remote Sensing Resolution and Data Rates Figure 2 illustrates the scanning geometry for an airborne line-scanning system where the aircraft is flying along a “track” at a height H and velocity V. The total scanned field of view is Smax and the cross-track and along-track instantaneous fields of view are $S and $H, respectively. The direction directly below the aircraft and normal to the scanned surface is called the nadir. The instantaneous field

Cross-track and along-track in remote sensing correspond to along-scan and cross-scan, respectively, in input/output scanning.

SCANNERS

30.7

FIGURE 2 Scanning geometry for an airborne linescanning system with a total scanned FOV of Smax. The aircraft is flying at height H and velocity V. The cross-track and across-track instantaneous FOVs are $S and $L, respectively. (After Wolfe, Proc. IRE, 1958.)

of view is defined as the geometrical projection of the detector with spatial dimensions dct and dat by the optics having a focal length of F. Therefore, $S dct/F and $S dat/F. Figure 2 shows the “bow-tie” distortion of the scanning geometry which will be discussed further under “Image Consequences.” The basic equation relating the aircraft velocity to the angular velocity of the scanning system to produce contiguous scan lines at the nadir is V /H s$ F , where s is the scanning system’s scan rate in scans per second. For a system with n detector elements aligned in the flight direction, V /H ns$ F . The number of resolution elements or pixels in a single scan line is N

Q max $Q

2PQ max 360°$ Q

(4a)

(4b)

where $S is in radians, Smax is the total field of view measured in radians in Eq. (4a) and in degrees in Eq. (4b), for the scanning means employed, taking due regard for the duty cycle given

30.8

INSTRUMENTS

in Eq. (23). The scan rate, in scans per second, may be expressed as a function of the scan mirror speed by s;

RS 60

(5)

where R is the scan mirror rpm and S is the number of scans produced by the scanning mechanism per revolution of the optics. It follows that the number of resolution elements or pixels per second per scan line is ;

N

2PQ max RS 60.360.$ Q

(6)

The angle Smax (in degrees) is determined by the configuration of the scan mirror and is Smax 360 · k/S, where k is the scanning constant or scan magnification and can have values ranging from 1 to 2. The specific value is dependent upon the optical arrangement of the scanner as exemplified in Table 3. The pixel rate may now be written as ;

N

2P kR 60 $ Q

(7)

The information retrieval rate of a system is often expressed in terms of the dwell time V or the data bandwidth fe as ;

fe

1 N 2T 2

(8)

By combining the preceding equations, the data bandwidth for a multiple-detector system can be expressed as fe

P k(V /H ) nS $ Q $ F

(9)

which illustrates clearly the relationship between important system parameters such as fe being inversely proportional to instantaneous field-of-view solid angle ($S$H).

Input/Output Scanning Resolution Criteria, Aperture Shape Factor The resolution of an optical scanner is expressed5,7 by the number N of spots or elements that can be conveyed along a contiguous spatial path. The path is usually (nearly) linear and traversed with uniform velocity. Although the elements F are analogous to the familiar descriptors pixels or pels (picture elements), such identification is avoided, for pixels often denote spatially digitized scan, where each pixel is uniform in intensity and/or color. Active optical scan, on the other hand, is typically contiguous, except as it may be subjected to modulation. Normally, therefore, the scanned spots align and convolve to form a continuous spatial function that can be divided into elements by modulation of their intensity. To avoid perturbation of the elemental point spread function (PSF) by the modulating (or sampling) process, we assume that the scan function is modulated in intensity with a series of (Dirac) pulses of infinitesimal width, separated by a time t such that the spatial separation between spot centers is w vt, where v is the velocity of the

See “Radial Symmetry and Scan Magnification” regarding scan magnification m which represents a more general form of the scanning constant k.

SCANNERS

30.9

scanned beam. It is often assumed that the size F of the thus-established elemental spot corresponds to w; that is, the width of the imaged spot is equal to the spacing from its neighbor. To quantify the number N of such spots, those which exhibit a Gaussian intensity distribution are usually considered overlapping at one of two widths; at their 1/e2 intensity points, or at their 50 percent intensity points [the latter denoted as full width at half maximum (FWHM)]. Their relationship is

D FWHM 0.589D1/e 2

(10)

The resolution N is identified with its measurement criterion, for the same system will convey different apparent N, per Eq. (10). That is, it will count approximately 1. 7 times as many spots at FWHM than at 1/e2 intensity. These distinctions are accommodated8 by their aperture shape factors a. For example, the above Gaussian aperture distribution is represented by shape factors a1/e 2

4 1.27 P

aFWHM 0.589a1/e 2 0.75

(11a) (11b)

When adapted to the applicable equation for spot size

D aF L

(12)

where F f/D is the F-number of the cone converging over the distance f from beam width D, and N is the radiation wavelength, the resulting Gaussian spot size becomes

D1/e 2

4 f L 1.27 F L PD

(13a)

when measured across the 1/e2 intensity points, and

D FWHM 0.75 F L

(13b)

when measured across FWHM. The factor a further accommodates the resolution changes due to changes in aperture shape, as for apodized and truncated Gaussians. Ultimate truncation is that manifest when the illuminating spatial distribution is much larger than the limiting aperture (overillumination or overfilling), forming the uniformly illuminated aperture. Familiar analytic examples are the rectangular and round (or elliptic) apertures, which generate (for the variable x) the normalized intensity distributions [sin(x)/x]2 and [2J1(x)/x]2, respectively, where J1(x) is the first-order Bessel function of the first kind. Figure 3 illustrates† the MTFs9 of several uniformly illuminated apertures. Their intersections with the 0.5 MTF value identifies the spatial frequency at which their modulation is 50 percent. With the rectangular aperture as a reference (its straight line intersects 0.5 MTF at 50 percent of the limit frequency, forming a 1), the intersections of the others with MTF 0.5 yield corresponding spatial frequencies and relative a-values. Since the spatial frequency bandpass is proportional to D/f 1/F, the apertures of the others must be widened by their a-values (effectively lowering their F-numbers) to render equivalent response midrange.

Although the illumination and resulting PSFs are of a coherent wave, scanning forms a sequence of incoherently related intensity measurements of the space-shifting function, yielding an incoherent MTF. † See Figs. 23 and 24 and related discussion for reduced power throughput due to approaching uniform illumination within the aperture.

INSTRUMENTS

$##!"!$#

30.10

a D a D

a D a D

R

D

f

D

f D R f D R f R

#%"#! $&

D f

FIGURE 3 Modulation transfer function versus relative spatial frequencies for uniformly illuminated rectangular, round, keystone, and triangular apertures. Spatial frequency at 50 percent modulation (relative to that of rectangular aperture) determines the a value.5

Table 4 summarizes the aperture shape factors (a) for several useful distributions. 8 Truncated, when applied, is two dimensional. Noteworthy characteristics are 1. Scanning is in the direction of the width D or W. 2. The a-value of 1.25 for the uniformly illuminated round/elliptic aperture corresponds closely to the Rayleigh radius value of 1.22. 3. The Gaussian-illuminated data requires that the width D, measured at the 1/e2 intensity points be centered within the available aperture W. Two conditions are tabulated: untruncated10 (W q 1.7D) and truncation at W D. 4. The Gaussian-illuminated data also provides the a-values for 50 percent MTF, allowing direct comparison with performance of the uniformly illuminated apertures. This data relates to apertures which, if apodized, are truncated two dimensionally. However, onedimensional truncation of a Gaussian beam by parallel boundaries is not uncommon, typical of that for acousto-optic scanning. There, the limiting aperture width W is constant, as determined by the device, while the Gaussian width D is variable.4,5,11,12 Table 5 tabulates the shape factor a for such conditions. TABLE 4

Aperture Shape Factor a

Uniformly Illuminated Shape Rectangular Round/elliptic Keystone Triangular Width D for 50% MTF

Gaussian Illuminated a 1.0 1.25 1.5 1.7 1.7

F (Spot Overlap) @ 1/e2 Intensity @ 12 –Intensity for 50% MTF

a(Untruncated) W > 1.7 D 1.27 0.75 0.85 Beam width D @ 1/e 2, intensity centered within aperture width W

a (Truncated) WD 1.83 1.13 1.38

SCANNERS

30.11

TABLE 5 Aperture Shape Factor a for OneDimensional Truncation of a Gaussian Intensity Distribution Truncation Ratio T W/D

Shape Factor a for 50% MTF

0 0.5 1.0 1.5 2.0

1.0 1.05 1.15 1.35 1.75

W width of aperture. D width of Gaussian beam at 1/e2 intensity points. W and D measured in scan direction.

To relate to data in Table 4, the case of T W/D 0 represents illumination through a narrow slit. This corresponds to the uniformly illuminated rectangular aperture, whence a 1. When T 1, then W D and the parallel barriers truncate the Gaussian beam at its 1/e2 intensity points. Compared to symmetric truncation, this allows more of the Gaussian skirts to contribute to the aperture width, providing a 1.15 versus 1.38. When T 2, the Gaussian beam of half the width of the boundaries is effectively untruncated, halving the resolution (a 1.75 vs. 0.85), but maximizing radiometric throughput. (See Fig. 24, observing nomenclature, where D 2wx and W 2ro.) Fundamental Scanned Resolution The section on “Input/Output Scanning” introduced the two forms of optical scan: translation and angular deflection. Beam translation is conducted by objective scan, while angular deflection is either preobjective or postobjective. Examples of each are provided later in this chapter. The resolution Ns of translational scan, by a beam focused to spot size F executing a scanned path distance S, is simply, Ns

S D

(14)

Extremely high resolutions are practical, but are often limited to moderate speeds and bandwidths. Common implementations provide Ns 3000 to 100,000.

The resolution N of angular scan,† represented schematically in Fig. 4, capable of much higher speeds, is given by 4,5,7 NQ

1 Do aL

(15)

where 1 is the useful deflected optical angle and Do is the effective aperture width at its nodal center, discussed in the next section. Common implementations provide NS 2000 to 30,000. Equation (15) is independent of spot size F and dependent only on the aperture characteristics of Do and a, and the wavelength N. The beam could be converging, collimated, or diverging. When collimated, Do D, the actual width of the illuminated portion of the aperture. When converging or diverging, resolution augmentation occurs (see next section). The numerator of Eq. (9) is a form of the Lagrange invariant,13 expressed in this nomenclature as n 1 D n '1 ' D '

(16)

A high-resolution laser printer provides N = 3000 to 10,000, and a high-resolution graphic arts imager N = 10,000 to 100,000. Derived from Eq. (4a) with sin $1$1 = aN/Do.

†

30.12

INSTRUMENTS

Do

D

p

o

r

f

FIGURE 4 Deflecting element of width D (e.g., mirror of polygon) displaced by radius r from axis o, propagating a converging beam to p over focal distance f. Effective larger aperture Do appears at rotating axis.4

where the primed terms are the refractive index, (small) angular deviation, and aperture width, respectively, in the final image space. For the common condition of n n` in air, the 1D product and resolution N are conserved, invariant with centered optics following the deflector. Augmented Resolution, the Displaced Deflector In general, a scanning system can accumulate resolution N by adding the two processes described previously, augmentation of angular scan with linear translation, forming N NQ N s

(17)

Augmentation occurs, for example, with conventional multielement scanners (such as polygons) having deflecting elements (facets) which are displaced from the rotating axis by a distance r, and whose output beam is noncollimated. One active element (of width D) and its focused output beam is illustrated in Fig. 4. For convenient analysis,6 the deflecting element appears as overilluminated with an incident beam. The resulting resolution equations and focal spot positions are independent of over- or underillumination (see “Duty Cycle”). Augmentation for increased resolution is apparent in Fig. 4, in which the output beam is derived effectively from a larger aperture Do which is located at o. By similar triangles, Do D (1 r/f ), which yields from Eq. (15), N

1D ¤ r ³ 1 aL ¥¦ f ´µ

(18)

This corresponds to Eq. (17), for in the Ns term the aperture D executes a displacement component S r 1, which, with Eq. (12) forms Eq. (14). Following are some noteworthy observations regarding the parenthetic augmentation term: 1. Augmentation goes to zero when r 0 (deflector on nodal axis) or when f c (output beam collimated). 2. Augmentation adds when output beam is convergent ( f positive) and subtracts when output beam is divergent ( f negative). 3. Augmentation adds when r is positive and subtracts when r is negative (output derived from opposite side of axis o). The fundamental or nonaugmented portion of Eq. (18), N 1D/aN, has been transformed to a nomograph, Fig. 5, in which the angle 1 is represented directly in degrees. D/aN is plotted as a radius, particularly useful when aN 1 μm, whereupon D/aN becomes the aperture size D, directly in mm. The set of almost straight bold lines is the resolution N. Multiples of either scale yield corresponding multiples of resolution. Augmenting and scan magnification Equation (18) develops from Fig. 4, assuming that the optimal scan angle 1 is equal to the mechanical angle 1. This occurs only when the scanner exhibits

SCANNERS

90n

80n

12

70n

17, 16,0 000 00 15,0 00 14,0 00 13,0 00 12,0 00 11,0 00 10,000

60n l deg ica , pt le 1 O g an an

n

50

sc

n

40

11 10 9 8

9,000 8,000

7

Normalized aperture

6

D , mm aN Mm

30n

7,000 6,000

20n

5,000

5

4,000

4

3,000 10n

30.13

3

2,000

2

N = 1,000

1

0n

N=0 0 12

11

FIGURE 5

10

9

8

7

6 D/aN

5

Nomograph of resolution equation n

4

3

2

1

0

Q D4 . aL

radial symmetry (see “Radial Symmetry and Scan Magnification”). When, however, m d1/dH w 1, as for configurations represented in the section “Objective, Preobjective, and Postobjective Scanning,” account must be taken of scan magnification m. Thus, the more complete resolution equation is represented by6,14 N where

1D ¤ r ³ 1 aL ¥¦ mf ´µ

per Fig. 4, 1 optical scan angle (active) D scan aperture width N wavelength (same units as D) a aperture shape factor m scan magnification ( d1/d&) & mechanical scan angle about o r distance from o to D f distance from D to p

[see variations for r and f under Eq. (18)].

(19)

30.14

INSTRUMENTS

Considering m 1/& as a constant, another useful form is N

&D ¤ r³ m ´ aL ¥¦ fµ

(20)

whose augmenting term shows a composite magnification m ` m r /f

(21a)

which, for the typical prismatic polygon becomes m ` 2 r /f

(21b)

Duty Cycle The foregoing resolution equations refer to the active portion of a scan cycle. The full scan period almost always includes a blanking or retrace interval. The ratio of the active portion to the full scan period is termed the duty cycle J. The blanking interval can include short overscan portions (straddling the active format), which are used typically for radiometric and timing calibration. The duty cycle is then expressed as

H 1 T /T

(22)

where V is the blanking interval time and T is the full scan period. A reduced duty cycle increases instantaneous bandwidth for a given average data rate. In terms of the scan angle of polygons, for example, it limits the useful component to

Q HQ max

(23)

where Smax is the full available scan angle (see Table 3). Over- and Underillumination (Over- and Underfilling) In overillumination, the light flux encompasses the entire useful aperture. This is usually implemented by illuminating at least two adjacent apertures (e.g., polygon facets) such that the active one is always filled with light flux. This not only allows unity duty cycle, but provides for resolution to be maximized for two reasons: (1) blanking or retrace may be reduced to zero; and (2) the full available aperture width is operative as D throughout scan. The trade-off is the loss of illuminating flux beyond the aperture edges (truncation) and attendant reduction in optical power throughput (see “Coherent Source” under Sec. 30.5, p. 30.26). An alternative is to prescan15 the light flux synchronously with the path of the scanning aperture such that it is filled with illumination during its entire transit. In underillumination, the light flux is incident on a portion of the available aperture, such that this subtense delimits the useful portion D. A finite and often substantive blanking or retrace interval results, thereby depleting the duty cycle, but maximizing the transfer of incident flux to the scanned output.

30.4

SCANNERS FOR REMOTE SENSING

Early Single-Mirror Scanners Early scanning systems comprised an object-space mirror followed by focusing optics and a detector element (or array). The first scanners were simple rotating mirrors oriented typically at 45° to the axis as illustrated in Fig. 6. The rotational axis of the scan mirror lies parallel to the flight direction. In Fig. 6a, the scan efficiency and duty cycle of the oblique or single ax-blade scanner (see monogon under “Monogon and Polygon Scanners”) is quite low since only one scan per revolution (S 1) is generated. The scan efficiency of the wedge or double ax-blade scanner shown in Fig. 6b is twice as great (S 2), although the effective optical aperture is less than half that of the oblique scanner for the same mirror diameter. The scanning constant is k 1 for both types (see “Remote Sensing Resolution and Data Rates” in Sec. 30.3).

30.15

Ax is

Ax is

SCANNERS

(a)

(b)

FIGURE 6 Early forms of scanners for remote sensing. The oblique or single ax-blade scanner is shown in (a) and the wedge or double ax-blade is shown in (b).

Compound Mirror Optics Configurations The aforementioned scanners suffered from a varying optical aperture as a function of view angle. To overcome this difficulty that causes severe variation in the video resolution during a scan line, several new line scanner configurations were developed. Most notable among these was the rotating prism scanner invented by Howard Kennedy16 in the early 1960s and which forms the basis for most of the produced wide-field-of-view line scanners. Figures 7 and 8 illustrate two

FIGURE 7 Basic split-aperture scanner with a threesided scan mirror developed in the early 1960s. This scanner features wide FOV, constant optical aperture versus scan angle, and compact mechanical configuration.

FIGURE 8 Basic split-aperture scanner with a foursided scan mirror developed in the early 1960s. This scanner features wide FOV, constant optical aperture versus scan angle, and compact mechanical configuration.

30.16

INSTRUMENTS

configurations of this scanner. The three-sided scan mirror (SM) shown in Fig. 7 rotates about its longitudinal axis at a high rate and the concomitant folding mirrors (FMs) are arranged such that the scanned flux is directed onto suitable focusing optics (FO) which focuses the flux at the detector D. As may be seen in the drawing of a four-sided scanner shown in Fig. 8, the effective optical aperture is split into two portions such that their sum is a constant value as a function of view angle. The width of each portion varies as the view angle is changed, with the portions being of equal value at the nadir position. The isometric view in Fig. 8 shows a portion of the scanner comprising the scan mirror, one folding mirror, and the focusing mirror. For this design, the number of scans per rotation of the scan mirror is equal to the number of faces on the scan mirror, and the scanning constant is k 2, which is also known as optical doubling (see item 3 of the prismatic polygon in Table 3). Also, two faces of the scan mirror are always used to form the total optical aperture. Another advantage of this scanner configuration is that it produces a compact design for the total scanner system, a major reason for its popularity for over a quarter of a century.

Image Consequences In airborne sensing, it is reasonable to assume that the earth is flat beneath the aircraft. When viewing along the nadir, the detector spatial footprint on the ground is H $S and H $H in the across- and along-track directions, respectively. As the view angle (S) moves away from the nadir, the geometric resolution on the ground changes as illustrated in Fig. 2, which creates the bow-tie pattern. In the cross-track direction, it is easily shown that the footprint dimension is H $S · sec2S, while in the along-track direction, the footprint dimension is H $H · secS. The change in footprint as a function of view angle can be significant. For example, if Smax 120°, then the footprint area at the extremes of the scan line is about eight times greater than at the nadir.

Image Relation and Overlap When a linear array of n detectors is used, it is easily seen that the image of the detector array rotates by exactly the same amount as the view angle if the scanner is pyramidal as shown in Fig. 6. No such rotation occurs for the prismatic polygon, as in the Kennedy scanner, for which each scan comprises n adjacent detector footprints on the ground that form a segmented bow tie. The next scan footprint has significant overlap with the preceding scan(s) for S w 0. A means to compensate for the radiometric difficulties caused by the overlap of scans has been developed.17 In a single detector system, this artifact is easily compensated by electronic means.

Rotating Wedge Scanner Figure 9 shows a simple rotating wedge scanner that allows the generation of a wide variety of scan patterns, including a line scan. By controlling the rotational rates and phasing of the wedges, such patterns as included in Fig. 10 can be realized.18

Circular Scan In some cases, a circular scan pattern has found utility. Typically, the entire optical system is rotated about the nadir with the optical axis inclined at an angle [ to the nadir. Figure 11 depicts an objectplane scanner showing how the aircraft or satellite motion creates contiguous scans. Although the duty cycle is limited, an advantage of such a scanner is that, at a given altitude, the footprint has the same spatial size over the scanned arc.

SCANNERS

30.17

FIGURE 9 Basic geometry of a simple rotating wedge scanner. A wide variety of scan patterns can be produced by controlling the rotational rates and phasing of the wedges. In principal, the detector can view any point within the circular scanned FOV of the scanner. Figure 10 presents typical scan patterns for constant rotational rates. Two-dimensional raster scans can be generated if general positional control of each prism is allowed.

FIGURE 10 Typical scan patterns for a rotating wedge scanner. The ratio of the rotational frequencies of the two prisms is m, the ratio of the prism angles is k, and the phase relation at time zero is L (L = 0 implies the prism apexes are oriented in the same direction). A negative value of m indicates that the prisms are counter-rotating. (After Ref. 18, Fig. 12, p. 12.)

30.18

INSTRUMENTS

Vehicle motion Detector system Scanning mirror

Focusing optics

Detector ground patch CBA

CB A

Portion of scanned ground path utilized

FIGURE 11 Basic configuration of a circular or conical scanner. The normal of the scanning mirror makes an angle [ with the nadir. The scan pattern of the ground forms a series of arcs illustrated by scans A, B, and C. (After Ref. 19, Fig. 8-3, p. 340.)

Pushbroom Scan A pushbroom scanner comprises typically an optical system that images onto the ground a linear array of detectors aligned in the cross-track direction or orthogonal to the flight direction. The entire array of detectors is read out every along-track dwell time which is Vat $H/(V/H). Often, when a serial read-out array is employed, the array is rotated slightly such that the read-out time delay between detectors creates an image that is properly aligned to the direction of motion. Some state-of-the-art arrays can transfer the image data in parallel to a storage register for further processing. The principal advantage of the pushbroom scanner is that no moving parts are required other than the moving platform upon which it is located.

Two-Dimensional Scanners Two-dimensional scanners have become the workhorses of the infrared community during the past two decades even though line scanners still find many applications, particularly in the area of earth resources. Scanners of this category can be classified into three basic groups, namely, object-space scanner, convergent-beam or image-space scanner, and parallel-beam or intermediate space scanner. Figure 12 depicts the generic form of each group.

Object-Space and Image-Space Scanners The earliest two-dimensional scanners utilized an object-space scan mechanism. The simplest optical configuration is a single flat-mirror (see Fig. 12a) that is articulated in such a manner as to form a raster scan. The difficulty with this scan mechanism is that movement of a large mirror with the necessary accuracy is challenging. The size of the mirror aperture when in object space must be greater than that of the focusing optics. By using two mirrors rotating about orthogonal axes, the scan can be generated by using smaller mirrors, although

SCANNERS

30.19

FIGURE 12 The three basic forms of two-dimensional scanners are shown in (a), (b), and (c). The object-space scanner in (a) comprises a scan mirror located in object space where the mirror may be moved in orthogonal angular directions. Image-space or convergent-beam scanner in (b) forms a spherical image surface due to motion of the scan mirror (unless special compensation motion of the scan mirror is provided). The parallel-beam or intermediatespace scanner is illustrated in (c). It is similar to the scanner in (a) except that the scan mirror is preceded by an afocal telescope. By proper selection of the afocal ratio and FOV, the scan mirror and focusing lens become of reasonable size. The scan mirror forms the effective exit pupil. (After Ref. 20, Figs. 7.1 and 7.10.)

the objective optics must have the capability to cover the entire field of view rather than the FOV of the detector. Figure 13 illustrates such a scanner 21 where mirror SM1 moves the beam in the vertical direction at a slow rate while mirror SM2 generates the high-speed horizontal scan. Although the focusing optics is shown preceding the image-space scan mirrors, the optics could be placed following the mirrors which would then be in object space. Although the highF-number or low-numerical-aperture focusing lens before the mirrors must accommodate the FOV, it allows the use of smaller mirror facets. The left-hand side of Fig. 13 shows an integral recording mechanism that is automatically synchronized to the infrared receptor side. This feature is one of the more notable aspects of the configuration and sets the stage for other scanner designs incorporating the integrated scene and display scanner. A disadvantage of this scanner is the large size and weight of the vertical scan mirror, in part, to accommodate both scene and display scan. A variation of the two-mirror object-space scanner is known as the discoid scanner, which produces a raster scan at TV rates.22 Figure 14 depicts the scanner configuration which uses a highspeed, multiple-facet scan mirror SM1 to generate the horizontal scan and a small, oscillating flat mirror SM2 to produce the vertical scan. An advantage of this scanner is that only a single detector is needed to cover the FOV, although a linear array oriented in the scan direction is sometimes used, with time-delay integration, to improve sensitivity. A feature of the “paddle” mirror scanner is the maintenance of a relatively stable aperture on the second deflector without the use of relay optics (see Figs. 21 and 32 and the section on the “Parallel-Beam Scanner”).

30.20

INSTRUMENTS

FIGURE 13 Early slow-scan-rate, image-space scanner where the flux from the scanned scene is imaged by the focusing objective lens FO onto the detector. The scene is scanned by mirrors SM1 (vertical) and SM2 (horizontal). A raster image of the scene is formed by imaging the light source using lenses L1 and L2. The display image and the scanned scene are synchronized by using the same pair of mirrors. The light source is modulated by the output of the detector.

Figure 15 depicts a reflective polygon scanner that generates the high-speed horizontal scan (per facet) by rotation of mirror SM about its rotational axis and the vertical movement of the scan pattern by tilting the spinning mirror about pivots P1 and P2 using cam C and its follower CF.23 The path of the flux from the object reflects from the active facet A of the scan mirror to the folding mirror FM to the focusing mirror FO back through a hole in mirror FM to the detector located in dewar D. Almost

FIGURE 14 Real-time, object-space scanner that has a compact geometry. The exit pupil EP is located on the active facet of the horizontal scan mirror SM1.

SCANNERS

30.21

FIGURE 15 Object-space scanner that generates a raster scan using a single mirror SM which is driven by motor M1. The vertical motion of the mirror is accomplished by tilting the housing containing the scan mirror SM about pivots P1 and P2 using the drive mechanism comprising motor M2, cam C, and CF. The FOV of scanners of this type can exceed 30°.

all scanners of this type exhibit scanned-field distortion; that is the mapping of object to image space is nonrectilinear (e.g., see the target distortion discussion in the section “Image Consequences” on p. 30.16). In general, convergent-beam, image-space scanners suffer from severe defocus over the scanned field of view due to the field curvature produced by the rotation of the scan mirror in a convergent beam. The use of this type scanner is therefore rather limited unless some form of focus correction or curved detector array is employed. A clever invention by Lindberg24,25 uses a high-speed refractive prism and a low-speed refractive prism to create the scanned frame. Figure 16 shows the basic

FIGURE 16 Basic configuration of a refractive prism scanner. The scan is generated by rotation of the prism. As shown, four scans are produced per complete rotation of the prism. By proper selection of the refractive index of the prism, reasonably wide FOV can be realized. Although only one prism is shown, a second prism can be included to produce the orthogonal scan direction.

30.22

INSTRUMENTS

FIGURE 17 Rotating reflective or “soupbowl” scanner. (After Ref. 20, Fig. 7.21, p. 309.)

configuration for a one-dimensional scanner where the cube P is rotated about the axis orthogonal to the page. By proper selection of the refractive index and the geometry of the prism, focus is maintained over a significant and useful field of view. As can be seen from the figure, flux from the object at a given view angle is focused by lens L onto surface I which is then directed to the detector D by the refraction caused by the rotated prism. Numerous commercial and military thermographic systems have utilized this principle for passive scanning. Since the field of view, maximum numerical aperture, optical throughput, and scan and frame rates are tightly coupled together, such scanners have a reasonably constrained design region. Other image-space scanners used in a convergent beam are the “soupbowl” and carousel scanners.20 The soupbowl scanner shown in Fig. 17 uses a rotating array of mirrors to produce a circularly segmented raster scan. The mirror facets may be at the same angle to generate more frames per rotation, given a detector array that has adequate extent to cover the field of view. The facets could also be tilted appropriately with respect to one another to produce contiguous segments of the field of view if a small detector array is employed. Figure 18 illustrates the configuration of the carousel scanner which uses an array of mirrors arranged such that they create essentially a rectangular scan of the field of view. Another scanning means that has been used for certain forward-looking infrared systems (FLIRs) was to mechanically rotate a detector array of angular extent & about the

FIGURE 18 p. 309.)

Rotating reflective carousel scanner. (After Ref. 20, Fig. 7.22,

SCANNERS

30.23

Scanned scene

(a)

Scanned scene

(b)

FIGURE 19 “Windshield wiper” scanner. The circular scan is generated by rotating the anamorphic optics about its optical axis. The detector array is located radially and offset from the optical axis. Two scans are produced for each complete rotation of the optics. The lens is shown in (a) at rotation angle S and in (b) at rotation angle S + 90°. Although the lens shown is focal, the lens could be afocal and followed by a focusing lens.

optical axis of the focusing optics such that one end of the array was located an angular distance &os from the optical axis. The rotating action generated a circular arc scan pattern similar to that of a windshield wiper. The inner radius of the scan pattern is &os and the outer radius is &os &. Clearly, the scan efficiency is rather poor and the necessity to use slip rings or the equivalent to maintain electrical connections to the detector array complicated acceptance of this scanner. The windshield wiper scan can also be generated by rotating only the optics if the optics incorporates anamorphic elements. A pair of cylindrical lenses placed in an afocal arrangement, as illustrated in Fig. 19 at rotational angles S and S 90°, will rotate the beam passing through it at twice the rotational rate of the optics.26 See “Image Rotation in Derotation” in Sec. 30.6. Multiplexed Image Scanning With the advent of detector arrays comprising significant numbers of elements, the use of a single scan mirror became attractive. Figure 20 presents the basic parallel-beam scanner configuration used for the common module FLIR and thermal night sights. The flat scan mirror SM is oscillated with either a sawtooth or a triangular waveform such that the detector array D (comprising 60, 120, or 180 elements) is scanned over the FOV in the azimuthal direction while the extent of the detector covers the elevation FOV. Since the detectors are spaced on centers two detector widths apart, the scan mirror is tilted slightly in elevation every other scan to produce a 2:1 interlaced scan of the FOV. As shown in Fig. 20, the back side of the scan mirror is used to produce a display of the scanned scene by coupling the outputs of the detectors to a corresponding array of LEDs which are projected to the user’s eye by lenses L1, L2, L3, and L4. Parallel-Beam Scanner A more complex two-dimensional, parallel-beam scanner configuration of the type shown in Fig. 12c has been developed by Barr & Stroud and is illustrated in Fig. 21, which incorporates an oscillating mirror SM1, a high-speed polygon mirror SM2 driven by motor M2, and relay optics L1. (See discussion

30.24

INSTRUMENTS

"

!

! !

$

"

!

#!"

! $ FIGURE 20 Basic configuration of the common module scanner. The front side of the flat scan mirror SM is used to direct the flux from the scanned scene to the detector D through the focusing lens. The outputs from the infrared detector array are used to modulate a corresponding LED array. The LED array is collimated by L1 and scanned over image space by the back side of SM. Lenses L2–L4 are used to project the image of the LED array to the observer’s eye.

FIGURE 21 Compact real-time scanner. The horizontal scan mirror SM2 is shown in two positions to illustrate how the field of view is related to location on mirror L1. Mirror L1 serves as a relay mirror of the pupil on mirror SM1 to SM2.

SCANNERS

30.25

FIGURE 22 Extremely compact, real-time scanner. Diamond-turned optics are used throughout this configuration. (After Ref. 28, Fig. 1.)

at end of Sec. 30.5.)27 An afocal telescope is located before the scanner to change the FOV, as is typical of parallel-beam scanners. Another innovative and compact two-dimensional scanner design by Kollmorgen is depicted in Fig. 22 and features diamond-turned fabrication technology for ease of manufacture and alignment of the mirrors and mounts.28 Another parallel-beam scanner that uses a simple scan mirror has been developed.29 The scan mirror is multifaceted with each facet tilted at an angle that positions the detector array in a contiguous manner in elevation. By having the nominal tilt angle of the facets be 45° to the rotation axis, minimal scanned-field distortion is realized.

30.5

SCANNING FOR INPUT/OUTPUT IMAGING

Power Density and Power Transfer Incoherent Source This topic merits introduction as the predecessor to laser scanning—cathoderay tube (CRT), flying-spot scanning and recording.1,2,3 Adaptation to other forms of incoherent sources, such as light-emitting diodes (LEDs) will be apparent. Similarities and contrasts with the handling of coherent sources are expressed. In a CRT, the electron beam power P (accelerating voltage beam current) excites a phosphor of conversion efficiency J and utilization factor I. The resulting radiant power is transferred through an imaging system of optical transmission efficiency T and spectral power transfer C to a photosensitive medium of area a during a time t. The resulting actinic energy density is given by3 E [1 J (joule) 1 W-s (watt-sec) 107 ergs].

HAT G Pt J/cm 2 A

(24)

30.26

INSTRUMENTS

The first four terms are transfer factors (a1) relating to the CRT, but adaptable to other radiant sources. They are determined3 for a principal group of CRT recording phosphors having varying processes of deposition and aluminizing, and for two typical (silver halide) photosensitive spectral responses: noncolor sensitized and orthochromatic. The spectral transfer term C is determined from the relatively broad and nonanalytic spectral characteristics of the CRT phosphors and the photosensors, n

Pi

3

A

Si

i 1 P Smax max m P j

3

j 1 P max

$ Li ( j q i)

(25)

$L j

where the Ps and the Ss are the radiant power and medium sensitivity, respectively, taken at significant equal wavelength increments $N. The optical transfer term T is composed of three principal factors, T TrTfTv , where Tr is the fixed transmission which survives losses due to, for example, reflection and scatter, Tf is the fixed geometric transfer, and Tv is the spectrally variable transmission of, for example, different glass types. The fixed geometric transfer is given by3 Tf

cos 4 & V& 1 4 F 2 ( M 1)2

(26)

The numerator (a1) is a transfer factor due to field angle & and vignetting30 losses, F is the lens F-number, and M is the magnification, image/object. The variable component Tv requires evaluation in a manner similar to that conducted for the C. The resulting available energy density E is determined from Eq. (24) and compared to that required for satisfactory exposure of the selected storage material. Coherent Source Determination of power transfer is much simplified by utilization of a monochromatic (single-line laser) source. Even if it radiates several useful lines (as a multispectral source), power transfer is established with a finite number of relatively simple determinations. Laser lines are sufficiently narrow, compared to the spectral characteristics of most transmission and detection media, so that single point evaluations at the wavelengths of interest are usually adequate. The complexity due to spectral and spatial distributions per Eqs. (25) and (26) are effectively eliminated. In contrast to the incoherent imaging system described above, which suffers a significant geometric power loss represented by Tf of Eq. (26), essentially all the radiant power from the laser (under controlled conditions discussed subsequently) can be transferred to the focal spot. Further, in contrast to the typical increase in radiating spot size with increased electron beam power of a CRT, the radiating source size of the laser remains essentially constant with power variation. The focused spot size is determined (per the section “Resolution Criteria, Aperture Shape Factor”) by the converging beam angle or corresponding numerical aperture or F-number, allowing for extremely high power densities. Thus, a more useful form of Eq. (24) expresses directly the laser power required to irradiate a photosensitive material as P where

sR ¤ A³ watts T ¥¦ t ´µ

(27)

s material sensitivity, J/cm2 R reciprocity failure factor, q1 T optical throughput efficiency, a1 A exposed area, cm2 t time over area, A

The reciprocity failure factor R appears in Eq. (27), since the exposure interval t (by laser) can be sufficiently short to elicit a loss in sensitivity of the photosensitive medium (usually registered by silver halide media). If the A/t value is taken as, for example, an entire frame of assumed uniform

SCANNERS

30.27

y

Wy x O ro

Wx FIGURE 23 Irradiance of a single-mode laser beam, generalized to elliptical, centered within a circular aperture of radius ro.31 Glossary as published; D (as used here) = 2wx and w (as used here) = 2ro.

exposure interval (including blanking), then the two-dimensional values of J must appear in the denominator, for they could represent a significant combined loss of exposure time. The optical throughput efficiency T is a result of loss factors including those due to absorption, reflection, scatter, diffraction, polarization, diffraction inefficiency in acousto-optic and holographic elements, and beam truncation or vignetting. Each requires disciplined attention. While the radiation from (fundamental mode) laser sources is essentially conserved in traversing sufficiently large apertures, practical implementation can be burdensome in scanners. To evaluate the aperture size consistent with throughput power transfer, Figs. 23 and 24 are useful. The data is generalized to 100 L (%)

(95.4)

100 (86.6) 80 (68.3) 60 40

(38.3)

4.0

3.0

(86.5)

6.0 5.0

2.0 (67.5)

1.5

ro = 1.0 Wx

(39.3)

0.75 0.5

20

(11.6) =

0.25 0

1.0

2.0

3.0

4.0

5.0

Wy Wx

6.0

Circular Gaussian beam FIGURE 24 Variations of the encircled energy 100 × L(%) versus the ellipticity G and the ratio ro/wx as a parameter.31 Glossary as published; D (as used here) = 2wo and w (as used here) = 2ro.

30.28

INSTRUMENTS

FIGURE 25 Drum configuration executing two forms of objective scan: (a) lens and its focal point translate with respect to storage medium and (b) storage medium translates with respect to lens during drum rotation.4

elliptic, accommodating the irradiance of typical laser diodes.31,32 Figure 23 shows an irradiance distribution having ellipticity G wx/wy (w @ 1/e2 intensity) apertured by a circle of radius ro. Figure 24 plots the encircled power (percent) versus the ellipticity, with the ratio ro/wx as a parameter. When G 1, it represents the circular Gaussian beam. Another parameter closely related to this efficiency is the aperture shape factor (discussed previously) affecting scanned resolution. (Note: D 2wx and w 2ro.) Objective, Preobjective, and Postobjective Scanning Classification Characteristics The scanner classifications designated as preobjective, objective, and postobjective were introduced previously and represented in Fig. 1 as a general conjugate optical transfer. This section expresses their characteristics. Objective scan (transverse translational) Translation of an objective lens transverse to its axis translates the imaged focal point on the information surface. (Axial lens translation which optimizes focus is not normally considered scanning.) Translation of the information medium (or object) with respect to the objective lens forms the same effect, both termed objective scan. The two forms of objective scan appear in Fig. 25, the configuration of a drum scanner. Preobjective scan (angular) Preobjective scan can provide a flat image field.† This is exemplified by angularly scanning a laser beam into a flat-field or f-S lens,33 as illustrated in Fig. 26, an important technique discussed further under “Pyramidal and Prismatic Facets” and “Flat-Field Objective Optics.”

Objective scan is limited in speed because the translating lens elements or storage medium must execute the desired final scan velocity, limited by the articulation of relatively massive components. † Applies beyond the small scan angle S arctan S, which may be considered linear. Also, no separate dynamic focus to aid forming a flat field.

SCANNERS

30.29

FIGURE 26 Polygon preobjective scan. The rotating polygon reflects and scans the input beam through the angle 1. The flat-field lens transforms this 1 change to (nominally) linear x displacement along a straight scan line. The input beam and the scanned beam reside nominally in the same plane.4

Postobjective scan (angular) Postobjective scan which is radially symmetric per Fig. 27 generates a perfectly circular scan locus. Departure from radial symmetry (e.g., focal point not on the axis of Fig. 27) generates noncircular (e.g., limacçon5) scan, except for when a postobjective mirror with its surface on its axis generates a perfectly circular scan locus, illustrated in Fig. 28. The input beam is focused beyond the axis at point o. Scan magnification m 2.

FIGURE 27 Monogon postobjective scan. Generates a curved image. When input beam is focused on the axis (at point o), then system becomes radially symmetric, and image locus forms section of a perfect circle.4

Placing the objective lens in the output beam and coupling it rigidly to the scanner (e.g., of Fig. 27) maintains the same characteristic. This is identified as objective scan (angular). The scanner and lens may be combined, as in a hologram.

30.30

INSTRUMENTS

FIGURE 28 Postobjective scan with mirror surface on rotating axis. Input beam is focused by objective lens to point o, intercepted by mirror and reflected to scanning circular locus. Scan magnification m d 1/d H 24.

Objective Optics The objective lens converges a scanned laser beam to a moving focal point. The deflector can appear before, at, or after the lens, forming preobjective, objective, and postobjective scanning, respectively (see previous discussion). On-Axis Objective Optics The simplest objective lens arrangement is one which appears before the deflector, as in Fig. 27, where it is required only to focus a monochromatic beam on-axis. The (postobjective) deflector intercepts the converging beam to scan its focal point. Ideally, the process is conducted almost aberrationlessly, approaching diffraction-limited performance. Since the lens operates on-axis only (accommodates no field angle), if the F-number of the converging cone is sufficiently high (see “Resolution Criteria, Aperture Shape Factor”), it can be composed of a single lens element. This simple arrangement can scan a perfectly circular arc [see “Postobjective Scan (Angular)]”, the basis for the elegance of the internal drum scanner and the requirement for adapting the information medium to a curved surface. Flat-Field Objective Optics Almost all other lens configurations are required to form a flat field by transforming the angular scan to a straight line.33 The deflector appears before the lens— preobjective. The most common configuration is similar to that of Fig. 26, as detailed further in “Design Considerations” under “Monogon and Polygon Scanners,” in which the scanned beam is shown collimated. Application is not limited to polygon scanners. Similar lenses adapt to a variety of techniques, including galvanometer, acousto-optic, electro-optic, and holographic scanners. The lens must accept the scanned angle S from the aperture D and converge the beam throughout the scanned field to a best-focus along a straight-line locus. Depending on the magnitudes of S and D, the F-number of the converging cone and the desired perfection of straight-line focus and linearity, the lens assembly can be composed of from 2 to 7 (or more) elements, with an equal number of choices of index of refraction, 4 to 14 (or more) surfaces, and 3 to 8 (or more) lens spacings, all representing the degrees of freedom for the lens designer to accommodate performance. A typical arrangement of three elements is illustrated in Fig. 34.

SCANNERS

30.31

FIGURE 29 High-performance telecentric lens at output of pyramidal polygon scanner, see Fig. 34. (Lens elements shown cut for illustration only.)4

Telecentricity A more demanding arrangement is illustrated in Fig. 29, showing six elements forming a high-performance scan lens in a telecentric configuration.30 Telecentricity is represented schematically in Fig. 30, in which an ideal thin-lens element depicts the actual arrangement of Fig. 29. Interposed one focal length f between the scanning aperture D (entrance pupil) and the flat image surface, the ideal lens transforms the angular change at the input to a translation of the output cone. The chief ray of the ideal output beam lands normal to the image surface. The degree of telecentricity is expressed by the angular departure from normal landing. Telecentricity is applied typically to restrict the spread of the landing beam and/or to retroreflect the probing beam efficiently

Image plane

Entrance aperture D

f

Ideal lens

Translating output cone

Scanned focal point Axis

f

FIGURE 30 Telecentric optical system. Schematic illustration of ideal lens transforming angular scan 1 from aperture D to translational scan landing normal to the image plane.4

30.32

INSTRUMENTS

FIGURE 31 Prismatic polygon in double-pass configuration. Narrow input beam is focused by positive lens, expanded, and picked-off by folding mirror to be launched through flat-field lens (in reverse direction). Beam emerges collimated, illuminates facets and is reflected and scanned by rotating polygon to be reconverged to focus on scan line. Input and output beams are slightly skewed above and below lens axis to allow clear separation by folding mirror. (Flat-field lens elements shown cut for illustration.)4

for internal system calibration. This facility comes dearly, however, for the final lens elements must be at least as wide as the desired scan format. A further requirement is the need to correct the nonlinearity of the simple system of Fig. 30, in which the spot displacement is proportional to the tangent of the scan angle, rather than to the angle directly. As in all scan lenses, compensation to make displacement proportional to scan angle is termed the f-S correction. Double-pass and beam expansion Another variation of the objective lens is its adaptation to double-pass,4,5,33 as depicted in Fig. 31. The lens assembly serves two functions: first, as the collimating portion of a lenticular beam expander 33 and second, upon reflection by the scanner, as a conventional flat-field lens. This not only provides compaction, but since the illuminating beam is normal to the undeflected facet, the beam and facet undergo minimum enlargement, conserving the size of the deflector. A slight skew of the input and output planes, per Fig. 31, avoids obstruction of the input and scanned beams at the folding mirror. An alternate input method is to make the lens array wide enough to include injection of the input beam (via a small mirror) from the side; at an angle sufficiently off-axis to avoid obstruction of the reflected scanned beam.33 This method imposes an off-axis angle and consequential facet enlargement and beam aberration, but allows all beams to remain in the same plane normal to the axis, avoiding the (typically) minor

Beam expansion/compression can also be achieved nonlenticularly with prisms.34 Introduction in 1964 of the phrase “beam expander” by Leo Beiser, and its dissemination to generic form, is summarized in App.1 of Ref. 6.

SCANNERS

30.33

scanned bow which develops in the aforementioned center-skewed method. Other factors relating to increased surface scatter and reflection need be considered. The requirement for beam expansion noted here is fundamental to the formation of the aperture width D which provides a desired scanned resolution. Since most gas lasers radiate a collimated beam which is narrower than that required, the beam is broadened by propagating it through an inverted telescope beam expander, that is, an afocal lens group having the shorter focal length followed by the longer focal length. Operation may be reversed, forming beam compression, as required. In the previously described double-pass system (Fig. 31), the objective lens provides the collimating portion (long-focal-length group) of a beam expander. Conservation of Resolution A most significant role of objective optics following the scanner is its determination of the integrity of scanned format, not of scanned resolution, as discussed under “Input/Output Scanning”. Denoting N as the total number of scanned elements of resolution to be conveyed over a full format width, in first analysis, N is invariant with intervening ideal optics. In reasonably stigmatic systems, the lens determines the size of the spots, not their total number. The number of spots is determined at the deflector, whether it be galvanometer, acousto-optic, electro-optic, polygonal, holographic, phased array, or any other angular scanner. This invariance is expressed as I Q D Q `D `

(28)

an adaptation of the Lagrange invariant [see “Fundamental Scanned Resolution,” Eq.(16)], which is illustrated effectively with telescopic operation. If the scanned beam is directed through a telescope (beam compression), as in Fig. 32, the demagnification of f2/f1 reduces D to D`, but also expands S to S ` by the same paraxial factor, sustaining resolution invariance. If D were the deflecting aperture width and L1 were its objective lens (telecentric in this case), then the image along surface S would exhibit the same number of N spots as would appear if the output beam from D` were focused by another objective lens to another image plane. This schematic represents, effectively, a pupil-transferring optical relay.4,5 (See Chap. 17, “Lenses” and Chap. 18, “Afocal Systems” in this volume.)

L1 L2

S D

F

D`

W/2 C L 1`/2 1/2

f1

FIGURE 32

f1

f2

f2

Illustration of invariance I = 1D = 1`D` with telescopic transfer of scanned angle 1 from aperture D.4

30.34

INSTRUMENTS

Retrace interval = transit time across illuminating beam

Optical aperature

Active scan angle

D

q = 2f 4p h = n

D Input beam

f

q

Reflected scanned beam

W Mirror size (n facets)

FIGURE 33 Prismatic polygon (underilluminated). Input beam perpendicular to axis. Its width D illuminates a portion of facet width W. Rotation through angle & yields scanned angle 1 = 2&, till facet corner encounters beam. Scan remains inactive for fraction D/W, yielding duty cycle J = 1 Dm/W.4

30.6

SCANNER DEVICES AND TECHNIQUES Many of the techniques addressed here for input/output imaging apply equally to remote sensing. Their reciprocal characteristic can be applied effectively by reversing the positions (and ray directions) of the light source(s) and detector(s). Preobjective and postobjective scanning have their counterparts in object-space and image-space scanning. A notable distinction, however, is in the option of underillumination or overillumination of the deflecting aperture in input/output imaging, while the aperture is most often fully subtended in collecting flux from a remote source. This leads to the required attention to aperture shape factor in input/output imaging, which is less of an issue in remote sensing. Another is the need to accommodate a relatively broad spectral range in remote sensing, while input/output operation can be monochromatic, or at least polychromatic. The frequent use of reflective optics in both disciplines tends to normalize this distinction.

Monogon and Polygon Scanners The rotating mirrored polygon is noted for its capacity to render high data rate at high resolution. It is characterized by a multiplicity of facets which are usually plane and disposed in a regular array on a shaft which is rotatable about an axis. When the number of facets reduces to one, it is identified as a monogon scanner. Pyramidal and Prismatic Facets Principal arrangements of facets are termed prismatic (Fig. 33) or pyramidal (see Fig. 34 and “Scanner-Lens Relationship”). Figure 27 is a single-facet pyramidal equivalent, while Fig. 28 is a single-facet prismatic equivalent (common galvanometer mount). The prismatic polygon of Fig. 33 is oriented typically with respect to its objective optics in a manner shown in Fig. 26, while Fig. 34 shows the relationship of the pyramidal polygon to its flatfield lens. The pyramidal arrangement allows the lens to be oriented close to the polygon, while, as in Fig. 26, the prismatic configuration requires space for clear passage of the input beam. Design consideration for this most popular arrangement is provided later in this chapter.

In remote sensing, applying reciprocity, this is the detected beam.

SCANNERS

30.35

FIGURE 34 Pyramidal polygon (overilluminated). Input beam, parallel to axis, is scanned (by 45° pyramidal angle) in plane perpendicular to axis. When input beam illuminates two facets (as shown), one facet is active at all times, yielding (up to) 100 percent duty cycle. Facet width is full optical aperture D, minimizing polygon size for a given resolution, but wasting illumination around unused facet regions. Can operate underilluminated (per Fig. 33) to conserve throughput efficiency, but requires increased facet width (and polygon size) to attain high-duty cycle. (Flat-field lens elements shown cut for illustration.4)

Table 3 lists significant features and distinctions of typical polygon scanners. Consequences of item 3, for example, are that the scan angle of the prismatic polygon is twice that of the pyramidal one for a given rotation. To obtain equal scan angles S of equal beam width D (equal resolutions N) and to provide equal duty cycle (see “Augmenting and Scan Magnification”) at equal scan rates, the prismatic polygon requires twice the number of facets, is almost twice the diameter, and rotates at half the speed of the pyramidal polygon. The actual diameter is determined with regard for the aperture shape factor (previously discussed) and the propagation of the beam cross section (pupil) across the facet during its rotation (see “Design Considerations”). Image Rotation and Derotation When a beam having a round cross section is focused to an isotropic point spread function (psf), the rotation of this distribution about its axis is typically undetectable. If, however, the psf is nonisotropic (asymmetric or polarized), or if an array of 2 or more points is scanned to provide beam multiplexing,6 certain scanning techniques can cause an undesired rotation of the point and the array of points on the image surface. Consider a monogon scanner, per Fig. 27. As shown, the input beam overilluminates the rotating mirror. Thus, the mirror delimits the beam, establishing a rectangular cross section which maintains its relative orientation with respect to the image surface. Thus, if uniformly illuminated, the focal point on the image surface (having in this case a sinc2x · sinc2 y psf, x along-scan and y crossscan) maintains the same orientation along its scanned line. If, however, the input beam is polarized, the axis of polarization of the imaged point will rotate directly with mirror rotation within the rectangular psf. Similarly will be rotation for any radial asymmetry (e.g., intensity or ellipticity) within the aperture, resulting in rotation of the psf. Consider, therefore, the same scanner underilluminated with, for example, an elliptical Gaussian beam (with major axis horizontal). The axis of the imaged elliptic spot (intended major axis vertical)

30.36

INSTRUMENTS

will rotate directly with the mirror. Similarly, if the scanner is illuminated with multiple beams displaced slightly angularly in a plane (to generate an in-line array of spots), the axis of the imaged array will rotate directly with mirror rotation. This effect is transferrable directly to the pyramidal polygon which is illuminated per Fig. 34. It may be considered as an array of mirrors, each exhibiting the same rotation characteristics as the monogon of Fig. 27. The mirrors of Fig. 34 are also overilluminated, maintaining a stationary geometric psf during scan (if uniformly illuminated), but subject to rotation of, for example, polarization within the psf. Similarly, it is subject to rotation of an elliptical beam within the aperture, or of a multiple-beam array. Not so, however, for the mirror mounted per Fig. 28 (galvanometer mount), or for the prismatic polygon of Figs. 26 and 33, which may be considered a multifacet extension of Fig. 28. When the illuminating beam and the scanned beam form a plane which is normal to the axis of mirror rotation, execution of scan does not alter the characteristics of the psf, except for the possible vignetting of the optical aperture and possible alteration of reflection characteristics (e.g., polarization) with variation in incident angle. It is noteworthy that in the prior examples, the angles of incidence remained constant, while the image is subject to rotation; and here, the angles of incidence change, while the image develops no rotation. The distinction is in the symmetry of the scanning system with respect to the illumination. The prior examples (maintaining constant incidence while exhibiting image rotation) are radially symmetric. The latter examples (which vary incidence but execute no image rotation) represent the limit of radial asymmetry. While mirrored optical scanners seldom operate in regions between these two extremes, holographic scanners can, creating possible complications with, for example, polarization rotation. This is manifest in the variation in diffraction efficiency of gratings for variation in p and s polarizations during rotation. (See “Operation in the Bragg Regime.”) Image Derotation Image derotation can be implemented by interposing another image-rotating component in the optical path to cancel that caused by the scanner. The characteristic of an image rotator is that it inverts an image.13 Thus, with continuous rotation, it rotates the image, developing two complete rotations per rotation of the component. It must, therefore, be rotated at half the angular velocity of the scanner. While the Dove prism13 is one of the most familiar components used for image rotation, other coaxial image inverters include13

s s s

Three-mirror arrangement, which simulates the optical path of the Dove prism Cylindrical/spherical lens optical relay Pechan prism, which allows operation in converging or diverging beams

Design Considerations A commonly encountered scanner configuration is the prismatic polygon feeding a flat-field lens in preobjective scan, illustrated schematically in Fig. 26. The size and cost of the flat-field lens (given resolution and accuracy constraints) is determined primarily by its proximity to the scanner and the demand on its field angle. A larger distance from the scanner (pupil relief distance) imposes a wider acceptance aperture for a given scan angle, and a wider scan angle imposes more complex correction for off-axis aberration and field flattening. The pupil relief distance is determined primarily by the need for the input beam (Fig. 26) to clear the edge of the flatfield lens. Also, a wider scan angle reduces the accuracy requirement for pixel placement. Since the scan angle S subtends the desired number N of resolution elements, a wider angle provides a larger angular subtense per element and correspondingly larger allowed error in angle $S for a desired elemental placement accuracy $N. This applies in both along-scan and cross-scan directions, $Sx and $Sy, respectively (see Sec. 30.7). Subsequent consideration of the scanner-lens relationships requires a preliminary estimate of the polygon facet count, in light of its diameter and speed. Its speed is determined by the desired data rates and entails considerations which transcend the optogeometric ones developed here. Diffraction-limited relationships are used throughout, requiring adjustment for anticipated aberration in real systems. The wavelength N is a convenient parameter for buffering the design to accommodate

SCANNERS

30.37

aberration. For example, an anticipated fractional spot growth of 15 percent due to systematic aberration is accommodated by using N 1.15N. Performance characteristics which are usually predisposed are the resolution N (elements per scan), the optical scan angle S, and the duty cycle J. Their interrelationships are presented under “Input/Output Scanning,” notably by Eqs. (15) and (23). The values of N and S for a desired image format width must be deemed practical for the flat-field lens. Following these preliminary judgments, the collimated input beam width D is determined from [see “Fundamental Scanned Resolution” Eq. (15)] D NaL /Q

(29)

where a is the aperture shape factor and N is the wavelength. For noncollimated beams, see “Augmented Resolution, the Displaced Deflector,” notably Eq. (19). The number of facets is determined from Table 3 and Eq. (23), n 4PH /Q

(30)

whereupon it is adjusted to an integer. Scanner-lens relationships The polygon size and related scan geometry into the flat-field lens may now be determined.35 Figure 35 illustrates a typical prismatic polygon and its input and output beams, all in the same plane. One of n facets of width W is shown in three positions: undeflected and in its limit-rotated positions. The optical beams are shown in corresponding undeflected and limit positions, deflected by o S/2. A lens housing edge denotes the input surface of a flat-field lens. Angle I provides clear separation between the input beam and the down-deflected beam or lens housing. The pupil relief distance P (distance ac) and its slant distance Pe (distance bc) are system parameters which establish angle C such that cos C P/Pe. Angle C represents the off-axis illumination on the

P Lens housing

R

o

Dp

Up-deflected facet

d ecte defl Up-

h

Undeflected

Axis Undeflected facet Down-deflected facet

W

Dow

c

In

pu

tb

ea

n-d

eflec

a

ted

a

Θ/2 b

m γ/2

Pe

Θ/2

D

Ds

bg

b

D

Input beam

FIGURE 35 Polygon, beam, and lens relationships. Showing undeflected and limit facet and beam positions.4

30.38

INSTRUMENTS

polygon which broadens the input beam on the facet. The beam width Dm on the facet is widened due to C and due to an additional safety factor t (1 a t a 1.4) which limits one-sided truncation of the beam by the edge of the facet at the end of scan. Applying these factors, the beam width becomes Dm

Dt cosA

(31)

Following Eq. (15), the duty cycle is represented by J 1 − Dm/W, yielding the facet width W Dm /(1 H)

(32)

from which the outer (circumscribed) polygon diameter is developed;35 expressed by Dp

Dt (1 H)sinP /ncos A

(33)

Solution of Eq. (33) or expressions of similar form36 entails determination of C, the angle of offaxis illumination on the facet. This usually requires a detailed layout, similar to that of Fig. 35. Series approximation of cos C allows transformation of Eq. (33) to replace C with more direct dependence on the important lens parameter P (pupil relief distance), yielding, Dp

Dt 1 Q Ds /2 P (1 H)sinP /n 1 Q 2 /8

(34)

where, per Fig. 35, s y 2 is a safety multiplier on D for secure input/output beam separation and clearance. Orientation of the scanner and lens also requires the height h, the normal distance from the lens axis to the polygon center. This is developed35 as h Rc sin(G /2 Q /4)

(35)

where Rc is the radial distance oc, slightly shorter than the outer radius R, approximated to be § 1 ¶ Rc R ¨1 (P /n)2 · © 4 ¸

(36)

Holographic Scanners General Characteristics Almost all holographic scanners comprise a substrate which is rotated about an axis, and utilize many of the concepts representative of polygons. An array of holographic elements disposed about the substrate serves as facets, to transfer a fixed incident beam to one which scans. As with polygons, the number of facets is determined by the optical scan angle and duty cycle (see “Duty Cycle”), and the elemental resolution is determined by the incident beam width and the scan angle [see Eq. (15)]. In radially symmetric systems, scan functions can be identical to those of the pyramidal polygon. While there are many similarities to polygons, there are significant advantages and limitations.6 The most attractive features of holographic scanners are 1. Reduced aerodynamic loading and windage with elimination of radial discontinuities of the substrate 2. Reduced inertial deformation with elimination of radial variations 3. Reduced optical-beam wobble when operated near the Bragg transmission angle Additional favorable factors are 1. Operation in transmission, allowing efficient beam transfer and lens-size accommodation 2. Provision for disk-scanner configuration, with facets disposed on the periphery of a flat surface, designable for replication

SCANNERS

30.39

3. No physical contact during exposure; precision shaft indexing between exposures allows for high accuracy in facet orientation 4. Filtering in retrocollection, allowing spatial and spectral selection by rediffraction 5. Adjustability of focus, size, and orientation of individual facets Some limiting factors are 1. Need for stringent design and fabrication procedures, with special expertise and facilities in diffractive optics, instrumentation, metrology, and processing chemistry. 2. Accommodation of wavelength shift: exposure at one wavelength (of high photosensitivity) and reconstruction at another (for system operation). Per the grating equation6 for first-order diffraction, sinQi sinQo L /d

(37)

where Si and So are the input and diffracted output angles with respect to the grating normal and d is the grating spacing, a plane linear grating reconstructs a collimated beam of a shifted wavelength at a shifted angle. Since wavefront purity is maintained, it is commonly employed,6 although it requires separate focusing optics (as does a polygon). When optical power is added to the hologram (to provide self-focusing), its wavelength shift requires compensation for aberration.6 Further complications arise when intended for multicolor operation, even if plane linear gratings. Further, even small wavelength shifts, as from laser diodes, can cause unacceptable beam misplacements, requiring corrective action.6,37 3. Departure from radial symmetry develops complex interactions which require critical balancing to achieve good scan linearity, scan-angle range, wobble correction, radiometric uniformity, and insensitivity to input beam polarization.6,36 This is especially demanding in systems having optical power in the holograms. 4. Systems which retain radial symmetry to maintain scan uniformity may be limited in Bragg angle wobble reduction, and can require auxiliary compensation, such as anamorphic error correction. Holographic Scanner Configurations A scanner which embodies some of the characteristics expressed above is represented in Fig. 36.38 A cylindrical glass substrate supports an array of equally spaced hololenses which image the input beam incident at o to the output point at P. Since point o intersects the axis, the scanner is radially symmetric, whereupon P executes a circular (arced) scan concentric with the axis, maintaining magnification m S/& 1. A portion of the radiation incident on the image surface is backscattered and intercepted by the hololens, and reflected to a detector which is located at the mirror image o` of point o. The resolution of this configuration is shown to be analogous to that of the pyramidal polygon.6 An even closer analogy is provided by an earlier reflective form illustrated in Fig. 37, emulating the pyramidal polygon, Fig. 34. It scans a collimated beam which is transformed by a conventional flat-field lens to a scanned focused line. This is one of a family of holofacet scanners, 6 the most prominent of which tested to the highest performance yet achieved in combined resolution and speed—20,000 elements per scan at 200 Mpixels/s. This apparatus is now in the permanent collection of the Smithsonian Institution. Operation in the Bragg Regime The aforementioned systems are radially symmetric and utilize substrates which allow derivation of the output beam normal to the rotating axis. While operation with radial asymmetry was anticipated in 1967,39 it was not until operation in the Bragg regime was introduced6,40 that major progress developed in disk configurations. Referring to Fig. 38, the input and output beams I and O appear as principal rays directed to and diffracted from the holographic sector HS, forming angles Si and So with respect to the grating surface normal. For the tilt-error reduction in the vicinity of Bragg operation, the differential in output angle dSo for a differential in hologram tilt angle dC during tilt error $C is given by § cos(Qi $ A ) ¶ dQo ¨1 · dA © cos(Qo $ A ) ¸

(38)

30.40

INSTRUMENTS

P

P

O n

D

O

O

r

FIGURE 36 Transmissive cylindrical holographic scanner. Input beam underilluminates hololens which focuses diffracted beam to image surface. Dashed lines designate optional collection of backscattered radiation for document scanning.6

Hence, when Si So, a small $C is effectively nulled. While the Si and So depart from perfect Bragg symmetry during hologram rotation and scan, the reduction in error remains significant. An analogy of this important property is developed for the tilting of a refractive wedge operating at minimum deviation.6 When Si So 45°, another property develops in the unbowing of the output

Axis

1.50 sphere rad

Input lens

Flat-field lens

FIGURE 37 Reflective holofacet scanner, underilluminated. Flat-field microimage scanner (100 1p/mm over 11-mm format).

SCANNERS

30.41

FIGURE 38 Holographic scanner in Bragg regime; 1i = 1o, in which output angle So is stabilized against tilt error $C of holographic segment HS.

scanned beam: the locus of the output beam resides (almost) in a plane normal to that of the paper over a limited but useful range.6,37 Further, the incremental angular scan for incremental disk rotation becomes almost uniform: their ratio m at small scan angles is shown to be equal to the ratio N/d of the grating equation [see the section on “Radial Symmetry and Scan Magnification” and Eq. (3)].6 At Si So 45°, m N/d 2. This results in the output-scan angle to be 2 larger (in its plane) than the disk-rotation angle. While such operation provides the above attributes, it imposes two practical restrictions. 1. For high diffraction efficiency from relief gratings (e.g., photoresist), the depth-to-spacing ratio of the gratings must be extremely high, while the spacing d N/ 2 must be extremely narrow. This is difficult to achieve and maintain, and difficult to replicate gratings which provide efficient diffraction. 2. Such gratings exhibit a high polarization selectivity, imposing a significant variation in diffraction efficiency with grating rotation (see “Image Rotation and Derotation”). Accommodation of these limitations is provided by reducing the Bragg angle and introducing a bow correction element to straighten the scan line. This is represented in Fig. 39; a high-performance scanner intended for application to the graphic arts. The Bragg angle is reduced to 30°. This reduces the magnification to m 1 N/d (as in radially symmetric systems), increases d to equal N for more realizable deep-groove gratings, and reduces significantly the angular polarization sensitivity of the grating. The elegance of the 45° Bragg configuration has been adapted6 to achieve self-focusing in less demanding tasks (e.g., laser printing). This is exemplified in Fig. 40, which includes a holographic lens to balance the wavelength shift of the laser diode,41,42,43 to shape the laser output for proper illumination of the scanner and to accommodate wavelength shift reconstruction. However, such multifunction systems are compounded by more critical centration requirements6 and balancing of characteristics for achievement of a discrete set of objectives. Galvanometer and Resonant Scanners To avoid the scan nonuniformities which can arise from facet variations (see Sec. 30.7) of polygons or holographic deflectors, one might avoid multifacets. Reducing the number to one, the polygon becomes a monogon. This adapts well to the internal drum scanner (Fig. 27), which achieves a high

30.42

INSTRUMENTS

FIGURE 39 Plane linear grating (hologon) holographic disk scanner. Bragg angle of 30° provides fabricatable and polarization-insensitive grating structure, but requires bow compensation prism. Useful scan beam is in and out of plane of paper. (After Holotek Ltd, Rochester, NY product data.)

duty cycle, executing a very large angular scan within a cylindrical image surface. Flat-field scanning, however, as projected through a flat-field lens, allows limited optical scan angle, resulting in a limited duty cycle from a rotating monogon. If the mirror is vibrated rather than rotated completely, the wasted scan interval may be reduced. Such components must, however, satisfy system speed, resolution, and linearity. Vibrational scanners include the familiar galvanometer and resonant devices4,5,44,45 and the less commonly encountered piezoelectrically driven mirror transducer.5,45

FIGURE 40 Holographic disk scanner with corrective holographic lens, both operating in approximately Bragg regime, providing complex error balancing.42

SCANNERS

30.43

FIGURE 41 Examples of galvanometer and resonant scanner transducers. Fixed field of permanent magnet(s) is augmented by variable field from current through stator coils. (a) Galvanometer: torque rotates iron or magnetic core. Mirror surface (not shown) on extended shaft axis. (b) Resonant scanner: torque from field induced into single-turn armature coil (in plane perpendicular to paper) rotates mirror suspended between torsion bars. One stator coil may be nondriven and used for velocity pick-off.

The Galvanometer Referring to Fig. 41a, a typical galvanometer driver is similar to a torque motor. Permanent magnets provide a fixed field which is augmented (o) by the variable field developed from an adjustable current through the stator coils. Seeking a new balanced field, the rotor executes a limited angular excursion ( o&/2). With the mirror and principal ray per Fig. 28, the reflected beam scans through oS/2, twice that of the rotor. The galvanometer is a broadband device, damped sufficiently to scan within a wide range of frequencies, from zero to an upper value close to its mechanical resonance. Thus, it can provide the sawtooth waveform with a longer active linearized portion and shorter retrace time V. This is represented in Fig. 42 (solid lines) showing rotation angle & versus time. As a broadband device, it can also serve for random access, positioning to an arbitrary location within its access-time limitations. For this feature of waveform shaping, the galvanometer was categorized as a low inertia scanner.5 The Resonant Scanner When damping is removed almost completely, large vibrations can be sustained only very near the resonant frequency of the oscillating system. The resonant scanner is thus characterized by larger angular excursions at a fixed and usually higher frequency, executing near-perfect sinusoidal oscillations. A typical driver configuration is illustrated in Fig. 41b. Figure 42 (dashed lines) shows a sinusoid with the same zero-crossings as those of the sawtooth waveform. Contrary to its popular designation as “low-inertia,” the resonant scanner provides rigid time increments, as though it exhibits a high inertia. While the rotary inertia of the suspension system is low to allow high repetition rates, it permits no random access and no scan waveform shaping, as do the galvanometer, acousto-optic, electro-optic, and other wideband scanners designated as low-inertia devices.5 Suspension Systems In the vibrational scanners, the bearings and suspension systems are the principal determinants of scan uniformity. The galvanometer shaft must be sufficiently stiff and long to inhibit cross-scan wobble. However, to maximize the oscillating frequency, the armature is restricted in size and mass. Fortunately, its reciprocating motion tends to retrace its path (and its perturbations) faithfully over many cycles, making adjacent scans more uniform than if the same shaft rotated completely within the same bearings, as in a motor.

Rotor types include moving iron, moving magnet, or moving coil. Figure 41a illustrates the first two and Fig. 41b exemplifies the moving coil type. Moving magnet types (having NdFeB high-energy magnetic material) can exhibit some advantage in lower inertia and higher torque.

30.44

INSTRUMENTS

0.4

+

Time

–

0.7

0.33

FIGURE 42 Waveforms (& vs. time) of vibrational scanners having same period and zero crossings. Solid line: galvanometer with linearized scan, providing 70 percent duty cycle. Dashed line: Resonant scanner providing 33.3 percent duty cycle (unidirectional) with 2:1 slope change, or 40 percent duty cycle with 3.24:1 slope change. Ratio of maximum slopes: resonant/galvanometer 2.6/1.4

Some bearings are flexure, torsion, or taut-band devices which insert almost no along-scan perturbations.44 Because of their low damping, these suspensions are most often applied to the resonant scanner. When damped, they can serve for the galvanometer, suffering a small sacrifice in bandwidth and maximum excursion, but gaining more uniform scan with very low noise and almost unlimited life. Some considerations are their low radial stiffness and possible coupling perturbation from torsion, shift of the axis of rotation with scan angle, and possible appearance of spurious modes when lightly damped. Most of these factors can be well-controlled in commercial instrument designs. Adaptations and Comparisons Because the resonant scanner oscillates sinusoidally, and we seek typically a linearized scan, some significant adaptations are often required. As illustrated in Fig. 42 (dashed lines), we must select from the sine function a central portion which is sufficiently linear to be linearized further by timing the pixels or extracting them out of memory at a corresponding rate.46 To limit the variation in pixel rate to 2:1 (i.e., velocity at zero crossover will be twice that at the same limit), then the useful excursion must be restricted to 60°/90° or 66.7 percent of its peak angle. When scanning with only one slope of the sinusoid (as for generation of a uniformly spaced raster), this represents a duty cycle of only 33.3 percent. To raise the duty cycle, one must accommodate a greater variation in data rate. If, for example, the useful scan is 80 percent of its full excursion (40 percent when using one slope), then the velocity variation rises to 3.24r. That is, the data rate or bandwidth at crossover is 3.24 times that at the scan limit. Also, its bandwidth at crossover is approximately 2 1/2 times that of the galvanometer, as represented by their relative slopes in Fig. 42 There is a corresponding variation in the dwell time of the pixels, resulting in predictable but significant variation in image exposure or detectivity: 2:1 for 33.3 percent duty cycle and 3 1/4: 1 for 40 percent duty cycle. This may require compensation over the full scan interval, using position sensing and control.45–48 In contrast, the broadband galvanometer with feedback can provide a highly linearized scan44 at a duty cycle of approximately 70 percent.5 Acousto-Optic Scanners Acousto-optic diffraction serves effectively for high-speed low-inertia optical deflection. It can provide random beam positioning within extremely short access times, or generate repetitive linear scans at very high rates, or divide a single beam into multiple beams for multiplexing applications. The tradeoff is, however, relatively low resolution, seldom providing more than N 1000 elements per scan.

SCANNERS

30.45

The principles of acousto-optics were formulated in 193249 and its attributes were applied only 5 years later to the Scophony TV projection system.50 Its potential for laser scanning was explored in the mid-1960s.51,52 While various acousto-optic interactions exist, laser scanning is dominated by operation in the Bragg regime.5,53 Fundamental Characteristics Diffraction from a structure having a periodic spacing , is expressed as sin Si sin So nN/,, where Si and So are the input and output beam angles respectively, n is the diffractive order, and N is the wavelength. Bragg operation requires that Si So SB. In a “thick” diffractor, length L q .2/N, wherein all the orders are transferred efficiently to the first, and the Bragg angle reduces to

QB

1 L 2,

(39)

Per Fig. 43, the grating spacing is synthesized by the wavefront spacing formed by an acoustic wave traveling through an elastic medium. An acoustic transducer at one end converts an electrical drive signal to a corresponding pressure wave which traverses the medium at the velocity vs, whereupon it is absorbed at the far end to suppress standing waves. The varying pressure wave in the medium forms a corresponding variation in its refractive index. An incident light beam of width D is introduced at the Bragg angle (angle shown exaggerated). An electrical drive signal at the center frequency fo develops a variable index grating of spacing , which diffracts the output beam at SB into position b. The drive signal magnitude is adjusted to maximize beam intensity at position b, minimizing intensity of the zero-order beam at position a. When fo is increased to fs fo $f, the grating spacing is decreased, diffracting the output beam through a larger angle, to position c. The small scan angle S is effectively proportional to the change in frequency $f. The scan angle is S N/$, (N/vs)$f. The beam width, traversed by the acoustic wave over the transit time V is D vsV. Substituting into Eq. (15) and accounting for duty cycle per Eq. (22), the resolution of the acousto-optic scanner (total N elements for total $f) is N

T$f (1 T /T ) a

(40)

The V $f component represents the familiar time-bandwidth product, a measure of informationhandling capacity. Deflection Techniques Because the clear aperture width W of the device is fixed, anamorphic optics is often used to illuminate W with an adjusted beam width D—encountering selective truncation by the parallel boundaries of W. The beam height (in quadrature to D) can be arbitrarily narrow to avoid apodization by the aperture. This one-dimensional truncation of the Gaussian beam requires assignment of an appropriate aperture shape factor a, summarized in Table 5. Additional topics in acousto-optic deflection are cylindrical lensing due to linearly swept fs,53 correction for decollimation in random access operation,5 Scophony operation,54 traveling lens or chirp operation,55 correction for color dispersion,56 polarization effects,57 and multibeam operation.58

Electro-Optic (Gradient) Scanners The gradient deflector is a generalized form of beam scanner4,5,59 in which the propagating wavefronts undergo increasing retardation transverse to the beam, thereby changing the wavefront spacing (wavelength) transverse to the beam. To maintain wavefront continuity, the rays (orthogonal trajectories of the wavefronts) bend in the direction of the shorter wavelength. Referring to Fig. 44a, this bend angle S through such a deflection cell may be expressed as

Q ko (dn/dy)l

(41)

30.46

INSTRUMENTS

2

L>Λ

Acoustic absorber c

Incident light

= ( v ) Δf s

D b B = 1 2 Λ

B

Λ a

Acoustic transducer

Output beam position

Electrical drive signal frequency = fs

Traveling acoustic wave velocity = vs wavelength = Λ = vs/fs

Beam condition

Acoustic frequency

a

Undiffracted zero order

fs = zero (or amplitude = 0)

b

Nominal Bragg angle ΘB

fs = fo (center frequency)

c

Added diffraction, scanned through angle Θ

fs = fo + Δf

FIGURE 43 Bragg acousto-optic deflector (angles exaggerated for illustration). Electrical drive signal generates traveling acoustic wave in medium which simulates a thick optical grating. Relationship between the output beam position and the electrical drive frequency is tabulated.4

where n is taken as the number of wavelengths per unit axial length l, y is the transverse distance, and ko is a cell system constant. For the refractive material form in which the wavefront traverses a change $n in index of refraction and the light rays traverse the change in index over the full beam aperture D in a cell of length L, then the relatively small deflection angle becomes4,5

Q ($n/n f )L /D

(42)

where nf is the refractive index of the final medium (applying Snell’s law and assuming sin S S). Following Eq. (15), the corresponding resolution in elements per scan angle is expressed as N ($ n/n f )L /aL

(43)

(for class I materials) $n no3rij E z

(44a)

(for class II materials) $n ne3rij E z

(44b)

The $n is given by

SCANNERS

30.47

1

1

Grad n (y) y

D 2

2

x (a)

1

1 D

na

nb

2

2

(b) FIGURE 44 Equivalent gradient deflectors: (a) basic deflector cell composed of material having grad n(y). Ray 1, propagating through a higher refractive index, is retarded more than Ray 2, tipping the wavefront through angle 1 (including boundary effect). (b) Analogous prismatic cell, in which na > nb, such that Ray 1 is retarded more than Ray 2, tipping the wavefront through angle 1.4

where no,e is the (ordinary, extraordinary) index of refraction, rij is the electro-optic coefficient, and E z V/Z is the electric field in the z direction (see Fig. 45). Methods of Implementation An electroacoustic method of developing a time-dependent index gradient was proposed in 196360 utilizing the (harmonic) pressure variations in an acoustically driven cell (of a transparent elastic material). Although this appears similar to acousto-optic deflection (see “Acousto-Optic Scanners”), it differs fundamentally in that the cell is terminated reflectively rather than absorptively (to support a standing wave). Also, the acoustic wavelength is much longer than the beam width, rather than much shorter for Bragg operation. A method of approaching a linearly varying index gradient utilizes a quadrupolar array of electrodes bounding an electrooptic material;61,62 and is available commercially.63 A continuous index gradient can be simulated by the use of alternating electro-optic prisms.59,64 A single stage biprism is illustrated in Fig. 44b and an iterated array for practical implementation appears in Fig. 45. Each interface imparts a cumulative differential in retardation across the beam. The direction and speed of retardation is controlled by the index changes in the electro-optic material. While resolution is limited primarily by available materials, significant experiment and test is reported for this form of deflector.5 Drive Power Considerations given by5

The electrical power dissipated within the electro-optic material is 1 P PV 2Cf /Q 4

(45)

30.48

INSTRUMENTS

FIGURE 45 Iterated electro-optic prism deflector. Indicating alternating crystallographic (z) axes. Input beam polarization for class 1 electro-optic coefficient (r63) materials.4

where V is the applied (p-p sinusoidal) voltage in volts, C is the deflector capacitance in farads, f is the drive frequency in hertz and Q is the material Q factor [Q 1/loss tangent (tan F) 1/power factor, (Q > 5)]. The capacitance C for transverse electroded deflectors is approximately that for a parallel-plate capacitor of (rectangular) length L, width Y, and dielectric thickness Z (per Fig. 45) C 0.09K LY /Z picofarads

(46)

where k is the dielectric constant of the material (L, Y, Z in centimeters). The loss characteristics of materials which determine their operating Q are often a strong function of frequency beyond 105 Hz. The dissipation characteristics of some electro-optic materials are provided,5,65,66 and a resolution-speed-power figure of merit has been proposed.67 Unique Characteristics Most electro-optic coefficients are extremely low, requiring high drive voltages to achieve even moderate resolutions (to N 100). However, these devices can scan to very high speeds (to 105/s) and suffer effectively no time delay (as do acousto-optic devices), allowing use of broadband feedback for position control.

30.7

SCAN-ERROR REDUCTION High-resolution scanners often depend on precise rotation of a shaft about its axis, said shaft supporting a multiplicity of deflecting elements (facets, mirrors, holograms). The control of angular uniformity of these multielements with respect to the axis, and of the axis with respect to its frame, can create an imposing demand on fabrication procedures and consequential cost. Since uniformity of beam position in the cross-scan direction may not be approached by phasing and timing of the data (as can be the along-scan errors), several noteworthy techniques have been developed to alleviate this burden.

SCANNERS

TABLE 6

30.49

Techniques for Cross-Scan Error Reduction

Available Methods The general field of cross-scan error reduction is represented in Table 6. Fabrication accuracy may be selected as the only discipline, or it may be augmented by any of the auxiliary methods. The active ones utilize high-speed low-inertia (A-O or E-O) or piezoelectric deflectors5,68 or lower-speed (galvanometer) deflectors which are programmed to rectify the beam-position errors. While open-loop programming is straightforward (while accounting for angular magnification/demagnification as a function of the accessed beam size), elegant closed-loop methods may be required to rectify pseudorandom perturbations. This must, however, be cost-effective when compared to the alternatives of increased fabrication accuracy and of the passive techniques. Passive Methods Passive techniques require no programming. They incorporate optical principles in novel configurations to reduce beam misplacement due to angular error in reflection or diffraction. Bragg-angle error reduction of tilted holographic deflectors is discussed in the section, “Operation in the Bragg Regime.” Anamorphic Error Control Anamorphic control, the most prominent treatment, may be applied to any deflector. The basics and operational characteristics6 are summarized here. Separating the nonaugmented portion of the resolution equation [Eq. (19)] into quadrature components and denoting the cross-scan direction as y, then the error, expressed in the number of resolvable elements, is Ny

Q y Dy aL

(47)

where aN is assumed constant, S y is the angular error of the output beam direction, and D y is the height of the beam illuminating the deflector. The objective is to make Ny l 0. Mechanical accuracies determine Sy, while anamorphics are introduced to reduce Dy; usually accomplished with a cylindrical lens focusing the illuminating beam in the y direction upon the deflector. [The quadrature (along-scan) resolution is retained by the unmodified Dx and scan angle Sx.] As Dy is reduced, the y displacement error is reduced. Following deflection, the y-direction scanned spot distribution is restored by additional anamorphics—restoring the nominal converging beam angle (via Fy, the F-number forming the scanning spot in the y direction). The error reduction ratio is R D y` /D y where D y` is the compressed beam height and Dy is the original beam height on the deflector.

(48)

30.50

INSTRUMENTS

Output beam

Output beam

Output beam 135

Mirrored

Input beam

Input beam

Axis

Axis

Input beam

Axis

Mirrored 45 (b)

(a) FIGURE 46

(c)

Monogon scanners employing double-reflection: (a) pentaprism; (b) pentamirror; and (c) open mirror.4

A variety of anamorphic configurations has been instituted, with principal variations in the output region, in consort with the objective lens, to reestablish the nominal Fy while maintaining focused spot quality and uniformity. Double-Reflection Error Control In double-reflection (Table 6), the deflector which creates a crossscan error is reilluminated by the scanned beam in such phase as to tend to null the error. This can be conducted in two forms: internal and external. An internal double-reflection scanner is exemplified by the pentaprism monogon69 in Fig. 46a; a (glass) substrate having two of its five surfaces mirrored. This is an optically stabilized alternate to the 45° monogon of Fig. 27, operating preobjective in collimated light. Tipping the pentaprism cross-scan (in the plane of the paper) leaves the 90° output beam unaffected. A minor translation of the beam is nulled when focused by the objective lens. The pentamirror69 per Fig. 46b, requires, however, significant balancing and support of the mirrors, since any shift in the nominal 45° included angle causes twice the error in the output beam. A stable double-reflector is the open mirror monogon70 of Fig. 46c. Its nominal 135° angle serves identically to maintain the output beam angle at 90° from the axis, independently of cross-scan wobble. With a rigid included angle and simple balancing, it can provide high-speed operation. Two variations which double the duty cycle, as would a two-faceted pyramidal polygon or ax-blade scanner (see “Early Single-Mirror Scanners”) appear in Fig. 47. Figure 47a is effectively two pentamirrors

FIGURE 47 Paired scanners employing double-reflection: (a) paired pertamirror “butterfly” scanner and (b) paired open mirror scanner. 4

SCANNERS

30.51

FIGURE 48 Method of external double-reflection—shown in undeflected position. Components and distances not to scale. Only principal rays shown.4

forming a butterfly scanner71 and Fig. 47b is effectively a pair of open mirrors.72 The absolute angles of each half-section must maintain equality to within half of the allowed error in the output beam. Also, the center section of Fig. 47a must be angularly stable to within one-quarter of the allowed error, because an increased included angle on one side forms a corresponding decrease on the other. Other dynamic considerations involve inertial deformation, and the beam displacements and mirror widths (not shown) to accommodate the distance of the input beam from the axis during rotation. The need for near-perfect symmetry of the multiple double-reflectors can be avoided by transferring the accuracy requirement to an external element that redirects recurrent beam scans. One such form73 is illustrated in Fig. 48. A prismatic polygon illuminated with a collimated beam of required width (only principal rays shown) deflects the beam first to a roof mirror, which returns the beam to the same facet for a second deflection toward the flat-field lens. The roof mirror phases the returned beam such as to null the cross-scan error upon the second reflection. Several characteristics are noteworthy: 1. The along-scan angle is doubled. That is, scan magnification m 4 rather than 2. 2. This normally requires increasing the number of facets to provide the same angle with the same duty cycle. 3. However, during polygon rotation, the point of second reflection shifts significantly along the facet and sacrifices duty cycle. 4. The pupil distance from the flat-field lens is effectively extended by the extra reflections, requiring a larger lens to avoid vignetting. 5. The roof mirror and flat-field lens must be sized and positioned to minimize obstruction of the input and scanned beams. Allow for finite beam widths (see “Scanner-Lens Relationships”).

30.8 AGILE BEAM STEERING A class of low-inertia scanning, called agile beam steering74–77 was developed initially for such challenging tasks as laser radar (LIDAR) and forward-looking infrared (FLIR) systems. Further advancement may allow its extension to more general application. The motivation for this

30.52

INSTRUMENTS

work is to achieve the performance of the scanned mirror while avoiding some of the concomitant burdens of size, weight, and inertia. This has been a long-envisioned goal of many earlier researchers in work having important similarities 5 to the more current activity in agile beam steering. Recent research has harvested new resources such as liquid crystal E-O phase shifters micromachined devices, and microlens techniques assembled in novel configurations. Two principal approaches have dominated investigation and development, viz., the phased array and the decentered microlens array; along with some of their principal variations. Although the basic operation of these two array types differ, they both develop the same form of the steered output wavefronts

Phased Array Beam Steering The directing of radiation in the radio and microwave regions by driving antenna arrays with controlled relative phase was especially familiar to the early radar specialist.78,79 Its adaptation to the optical spectrum, notably as radiated by lasers, was investigated80 in 1964 following the invention of the laser and continued81–83 through the early 1970s. The prospect of altering the direction of a laser beam with small adjustments on a group of radiators appeared very attractive. With the introduction of electrostatically actuated membrane mirror arrays82 in 1971, and the programming of electro-optic crystal arrays83 in 1972, operational utility was affirmed. Beam steering with arrays of mirrors was investigated81 in 1967. Further work was conducted in mirror array beam steering in the infrared region, where mirror reflectance exceeds the transmittance of even the exotic infrared materials, and where the longer wavelength imposes lower requirements on mirror flatness. With the current use of faster acting electro-optic materials and novel design variations, substantive advances have been achieved. The steering of an optical wavefront by phase variation is introduced with the effect of refractive prisms.84,87 Figure 49a illustrates a plane wavefront in air incident parallel to the plane surface of one dielectric wedge. Within the material of refractive index >1, the wavelength is compressed proportionately, while the fronts remain parallel to the incident wavefront. A linearly increasing local phase delay results from the progressive retardation of the wavefronts across the enlarging wedge thickness. Traversing the tilted boundary, the wavelength in air is reexpanded and its angle of propagation is refracted as illustrated. This is exemplified in the iterated prismatic deflector of Fig. 45 given the dynamics of the electro-optic material. To provide a wide aperture, the single prism of Fig. 49a can exhibit substantive bulk. To relieve this, the long wedge profile is divided into an array of smaller wedge increments where each causes a linear phase delay of from 0 to 2R. As illustrated in Fig. 49b, when implemented as described below, it provides the same deflection as the continuous single wedge. Along with the need to accommodate both the slope and refractive index of the material, one must dimension the periods of the wedges such that they form 2R phase differentials (or multiples thereof, i.e., modulo 2R) at the operating wavelength to assemble continuous nonstaircased wavefronts in the near field. This is functionally analogous to the reflective blazed grating, in which high efficiency is achieved when the angle of specular reflection (from the sawtooth slopes of the grating surfaces) coincides with the angle of diffraction at the selected wavelength. Thus, an array formed of such 2R phase differentials exhibits dispersion which limits efficient performance to narrow spectral-band (near-monochromatic) operation. Work conducted to alleviate this limitation is discussed in if “Development of Phased Arrays” subsection. The above examples provide continuous phase retardation by virtue of their linear surface slopes; incremental or continuous. Incremental phase retardation can be controlled in transmission by an array of small electro-optic cells, and in reflection by precise actuation of individual mirrored pistons. An array of refractive phase retarders and the assembly of the radiated wavelets into contiguous wavefronts is represented in Fig. 49c. Operation is similar with pistons, except that the reflective piston requires displacement of only half of the 2R phase-retardation distance. A thin electro-optic retarder having a reflective ground plane requires not only half thickness, but attains a

SCANNERS

Wav e cons front of tant phas e Wavefront propagation direction Dielectric wedge

Wavefront of constant phase (a)

Wav con efront stan o t ph f ase Array period

Blazed grating of dielectric wedges Wavefront of constant phase (b) 1st order wavefront direction of propagation (h0.81-%)

0th order wavefront direction of propagation

Phase delay = 3p/2

Phase delay = p

Array of phase modulators

Phase delay = p/2

Phase delay = 0

Phase delay = 3p/2

Phase delay = p

Phase delay = p/2

Phase delay = 0

Phase delay = 3p/2

Phase delay = p

Phase delay = p/2

Phase delay = 0

Array period

Wavefront of constant phase (c) FIGURE 49 Illustrative steps (a, b, and c) toward optical-phased array beam steering. (Wavefront enters parallel to each bottom surface.) (a) Single dielectric wedge, illustrating familiar refraction of plane wavefront. (b) Synthesis of (a) with array of wedges. Each wedge imparts a 2R phase delay over each array period. (c) Synthesis of (b) with multiple delay elements (4 per 2R period). Output wavelets superpose into wavefronts having idealized efficiency of 81 percent. Greater multiplicity provides higher efficiency. (After Refs. 84 and 87.)

30.53

30.54

INSTRUMENTS

fourfold increase in switching speed which can be useful for a nominally slow liquid crystal retarder. An alternative option, shown in this figure, shows division of the full 2R phase change into four substeps per full cycle, which diffracts 81 percent of the energy into the first order. If more steps are available, then the diffraction efficiency is greater. For example, eight-step cycle attains an otherwise lossless efficiency of 95 percent. Figure 49 is a combination of three separated illustrations 84,87 to unify their progression.74,85,86 The heuristic observations rendered above are affirmed by the considerations which follow.

The Analytic Base The angular beam relationships of the phased array are expressed by the diffraction grating equation,87 sin(Qi ) sin(Qo ) n L /,

(49a)

where Si and So are the input and output beam angles with respect to the grating normal (boresight), n is the diffraction order, L is the free space wavelength, and , is the grating (array) period, per Fig. 49b and c. As in Fig. 49c showing four delay elements per array period, for q delay elements, each separated by a fixed distance d, , qd. Since the number of elements in each period is q 2R/H, where H is the phase shift between elements, then , (2P /F )d is the distance required to assemble a one-wave phase difference. When Si 0, the angle of first-order wave propagation (Fig. 49c) is given by sinQo L /,

(49b)

L /qd LF /2P d

(49c)

The normalized intensity I of the radiation pattern follows the analogy of the one-dimensional microwave phased array79 expressed as

with

I (sin NA /N sinA )2

(50)

A P d/L(sinQ sinQo )

(51)

where S is the angle with respect to the grating normal at which the field in free space is measured, and N is the number of phase shifters in the array. The elemental spacing d provides uniform phase difference H between elements. The efficiency Jq of a linear array having the nominal (blazed) 2R phase resets, illustrated in Fig. 49b and c is expressed by 2

¤ P³ ¥ sin q ´ P Hq ¥ ´ sinc 2 / P q q ´ ¥ ´µ ¥¦

(52)

where q is number of elements per 2R array period. This may be recognized as similar to the Fourier transform13 of a uniformly illuminated linear aperture. 87 Inserting values of q, i.e., 4 and 8, Eq. (52) yields J4 0.81 and J8 0.95, respectively, as indicated earlier. With a reduced q, lower efficiency results from depletion of the main lobe to the sidelobes due to wavefront staircasing.

SCANNERS

30.55

Typical liquid crystal phase retarder elements exhibit a unique loss factor established by the minimum space required to relax its orientation from a 2R phase shift to zero. This “flyback” transition is analogous to the flyback time V of many conventional scanners as expressed in Eq. (22) as J 1 V/T, where T is the full scan period. This represents a time loss burdening high-speed operation. The duty cycle for liquid crystal elements is given by

H, (1 , /,)2

(53)

where the time terms are replaced with , representing the flyback width and , the full 2R width. The expression is squared to denote the radiated intensity rather than the time in Eq. (22), as illustrated by the solid line in Fig. 42. In addition, fill factor accounts for a cell having its operating portion occupy less than its full allotted area while the vignetting factor accounts for the loss of input illumination beyond the boundary of the array. The far-field angular beamwidth SB is expressed as a minor variation to the familiar diffraction relation

Q B aL /D

(54a)

where a is the aperture shape factor87 modifying the beamwidth, discussed in Sec. 30.3. With the full aperture width D Nd,

Q B aL /Nd

(54b)

The Resolution of Phased Arrays Equation (54) denotes the output beamwidth, that is the breadth of the principal lobe of radiation. For scanned resolution (Sec. 30.3), the number N of these adjacent lobes which fill the field-of-view along a linear track represents the number of phase-shifting calls in a linear phased array. Analogous to Eq. (4a), % S/SB

(55)

Half of the full deflected field angle S is represented by the (positive) first diffracted order of grating [i.e., n 1 in Eq. (49)]. When the array is addressed in complementary phase sequence, the same deflection magnitude results in the opposite (n 1) direction. Thus, for typically small So in Eq. (49) and with D %d, the numerator for Eq. (55) becomes

Q

2L 2L % qd qD

(56)

With Q B aL /D providing the denominator of Eq. (55), and accounting for the central boresight position, the steered resolution reduces to % 1

2 % aq

(57a)

which is independent of wavelength. Although Eq. (57) appears to differ from the fundamental Eq. (15) for scanned resolution, substituting Eq. (56) into the fundamental equation and adding one for the boresight beam yields Eq. (57). With a as a relative constant, the ratio of the two variables % and q dominates the total number of elements divided by the number of elements per phase reset. Thus, the number of phase resets is the principal variable which determines the steered resolution and %/q D/,, where , is the array period. Since q is a parameter of the system, its adjustment also affects88 the closeness of the adjacent steering states. When a is unity, it denotes uniform illumination upon a rectangular aperture. This yields a far-field intensity distribution89 of the sinc2(x) function having a main lobe within equispaced null

30.56

INSTRUMENTS

intervals. Rayleigh resolution requires this uniform illumination upon a rectangular aperture, and that the adjacent spots in the far field overlap such that the maximum of each main lobe coincides with the first null of each adjacent one. Further delimiting Eq. (57a) is that it is impractical to form a modulo 2R array in which q is less than three cells in view of the resulting disruption of the ramp wavefronts and the loss in efficiency. Letting a 1 and setting qmin 3, the steered resolution is often expressed as an assumed Rayleigh resolution yielding %max 1 2 % 3

(57b)

A common illumination (laser) is the Gaussian function, with adjustment of the aperture overfill (and/or with complementary Gaussian filtering) to control the intensity distributed across the full aperture width W. The degree of overfill is, however, moderated by the reduction in light throughput due to aperture vignetting. As W is illuminated more uniformly, it may be limited by the appearance of fine structure beyond the main lobe when approaching the appearance of the sinc2 function. To quantify this value of a, different conditions can be considered87 and tabulated, summarizing its value for the Gaussian beam of width D, either falling substantially within the aperture W (untruncated beam), or when the 1/e2 intensity of the input beam occurs at the aperture boundary (truncated beam). These are two typical illumination conditions of most conventional deflectors. For the rectangular aperture of width W > height, illumination with a Gaussian beam primarily in the W or scan direction is further evaluated and tabulated (Table 5) providing data of current interest. A variable beam width D (at 1/e2 intensity) illuminates the full width W of a linear array. Assigning a parameter T W/D, when T 1, the 1/e2 beamwidth matches the full aperture width W. At T 1.5, the array is 1.5 times wider than that of the 1/e2 beamwidth. This terminates the Gaussian function at o3U, where its intensity tapers to a small fraction of its maximum value, representing a practical limit on the narrowness of the input beam. At the other extreme, when R l 0, the input beamwidth D W , extracting near-uniform illumination from the center of the beam, and imposing extreme light loss beyond the aperture. This is the case of a l1. The aperture shape factors for the other two cases are determined; at T 1, a 1.15 and at T 1.5, a 1.35. Related to the topic of resolution is the finesse which is the smallest addressable increment of beam position. Consideration of this factor87 involves (a potentially nonuniform) adjustment of the values of q (number of delay elements per array period). Development of Phased Arrays Work using nematic-phase liquid crystal electro-optic retarders is detailed comprehensively in a 1993 Air Force document.86 The materials are known as types E7 and PTTP-33 liquid crystals, having birefringence $ (ne no ) y 0.2 in the infrared, requiring a cell be only 5 optical waves thick for a full-wave phase shift in transmission and only 2.5 waves thick in reflection. The thinner the cell, the shorter is its reorientation time. Switching speeds in the millisecond range with high-efficiency, diffraction-limited steering have been demonstrated at 10.6 pm with CO2 lasers, and at 1.06 μm and 0.53 μm with Nd:YAG lasers. The cascading of tandem scanners by optical relaying5 is a means for adding the contributions of two or more deflectors with each operating optimally. One-dimensional arrays may be compounded having one for azimuth and one for elevation. Also, individual deflectors requiring excessive spatial separation may be cascaded using relay optics to avoid walk-off of the beam from the second aperture by the action of the first deflector.89 Another approach to tandem arrays,90 named discrete/offset cascade, reduces potential “noise” (beam artifacts) n in the instances of large quantization mismatches when cascading phase-delayed groups. Experiments have demonstrated improved overall diffraction efficiency, along with requiring a reduced number of control lines. (A similar approach was demonstrated with microlens arrays.91) Also significant is the use of an electro-optic phase retarder other than liquid crystal. The material selected was PLZT (lead lanthanum zirconate titanate) which exhibits a large electro-optic coefficient, broadband optical transmission, very fast switching, and good thermal stability,92 and is a well-documented ceramic material, familiar in electro-optic modulation and deflection. Mirrored

SCANNERS

30.57

piston-like phase adjustment is also reviewed,76 and later work93 describes both continuous phase change and binary phase shift. Problems in broadband operation of phased arrays are reviewed74 and early work was directed toward their solution.94,95 A wavelength-independent phase shift is achieved by polarization modulation of chiral smectic liquid crystals (CSLC), providing action similar to the mechanical rotation of a waveplate. However, grating dispersion remains due to wavelength deviation from nominal 2R phase resets, rendering a variation in efficiency Jd similar to Eq. (52)

Hd sinc 2E

(58)

where G is the chromatic error due to mismatch of the nominal 2R phase reset. Not only is energy lost, but side-lobe amplitudes increase and nondiffracted components result in image blurring and interference from sources outside the desired acceptance angle. This dispersion is reduced with the application of achromatic Fourier transform optics,96 as investigated in more recent comprehensive work,97 yielding precautions regarding the ability to reduce dispersion completely and the difficulty in implementation of the technology. More conventional achromatic optics has been applied98 to the decentered lens; the second of the two major techniques for agile beam steering which is discussed next.

The Decentered Lens and Microlens Arrays The decentering of one lens with respect to a matching afocal lens is an alternative to the phased array described above. Although its basic action differs from the phased systems, when smaller lenses are formed into a mating periodic array, the assembly can exhibit some of the characteristics of phased arrays, including functioning as blazed gratings.75 However, single lens-group operation can avoid some of the image faults of array steering as later discussed. Consider Fig. 50a illustrating a pair of afocal lenses (denoted 1 and 2) oriented originally on a common axis, now with lens 2 shifted “downward” through a distance $ (dotted axes). Beyond the focus of lens (left lens), the diverging beam continues into lens 2 shifted angularly off its axis, resulting in deflecting the recollimated output beam through an angle So. Thus, a transverse shift of one lens with respect to the other affects beam steering. The vignetting of the output beam and the related diversion of its residual output flux outside the lens is discussed subsequently. Constraining this simple two-lens technique is its limitation on the width of the lens aperture, consistent with the energy required for rapid $ shift within a reasonable burden on acceleration of its more massive components. Consider combining many lens pairs like lenses 1 and 2 (maintaining the F) to form an array of microlenses, as illustrated in Fig. 50b, and illuminating the group from the left by a single broad beam. The steered waves sum into the total field in a manner similar to those of the prior phased arrays and similar to Fig. 49b. This results in a significant decrease in mass for a given full aperture size and a decrease in shift distance $ for the same steered angle. The effect is a reduction in $ and in the acceleration/deceleration forces required for rapid beam steering. Although the composite wavefront is discontinuous, the segments are tipped at the same slopes such that the output exhibits the characteristics of a blazed grating. When, at the operating wavelength their junctions exhibit 2R phase differentials, a sawtooth pattern is formed typified by a high-diffraction efficiency blazed grating. This technique is satisfactory for small steered angles, where high fill factors remain at the second lens array, and the spurious components are a small residue. At wider steering angles, however, when the vignetting and the disruptive effects of the spurious components become significant, remedies are required. A classic method for the control of such vignetting is to include a field lens99 into the microlens array.75 Fig. 50c illustrates this as a variation of Fig. 50a with a field lens (FL) inserted at the common focal plane of the original two lenses. The bar over the output pair of lenses represents physical connection for simultaneous shift. With equal focal lengths for all lenses, the expanding light cone completely fills the lens pair during $ shift. This technique for realizing output efficiency and spectral quality is directly transferrable to the microlens array of Fig. 50b with an added plane of field

30.58

INSTRUMENTS

Δ

Θ

Δ

Δ

Δ

FIGURE 50 Beam steering with decentered lens of afocal pair shifted through distance $. (a) Single lens pair, showing $-shift deflecting major portion of output beam through angle 1o while upper portion of the beam by-passes lens 2. (b) Microlens arrays operating as in (a), but lighter and with smaller $-shift. The desired outputs accumulate, while the by-pass portions become spurious. (c) Field lens (FL) added to (a) provides constant filling of lens 2. When added to (b), the FLs maintain the wavefront synthesis of a blazed grating. (After Refs. 75 and 87.)

lenses affixed to the output array. The inertia can be accommodated by the force of piezoelectric or electrodynamic drive transducers. Alternatively, the single-element group may be shifted instead. A microlens-field lens design was fabricated and tested100 over a o1.6° field. Larger angles (o17°) have been demonstrated,101 but with loss of beam quality. Further consideration for reduction of the spurious beams during $shift is represented in Fig. 51. The method of Fig. 51a provides75 some tolerance for beam displacement on lens 2 by changing the ratio of focal lengths. The initial condition of f1 f2 is adjusted to f1 /f 2 1. This forms a beam compressor (see Fig. 32) with a compression ratio y 2 : 1 . A similar approach is investigated102 using

SCANNERS

30.59

FIGURE 51 Elimination of spurious radiation over a range of operation with increased focal length ratio f1/f2. (a) Similar to Fig. 50 (b) with f1/f2 y 2. (b) Analogous galilean form with f1/f2 y 2.5 and the beam energy is conserved. However, the spaced output beams represent diffraction from a discontinuous blazed grating. (After Refs. 75, 102 and 87).

a positive-negative lens combination. Method in Fig. 51a employs the equivalent of a Keplerian telescope and method in Fig. 51b that of a Galilean telescope. While the spurious components of Fig. 50b may be abated over its initial range of operation, the fill factor at the second array is reduced. Although the energy is conserved in this reduced light cone, the ideal sawtooth pattern of the blazed grating is disrupted by the truncated sawtooth function. This, in turn, causes its own spurious noise75,102 which limits operation to a small range of $shift. It is propsed102 that the second array be maximally filled ideally by reducing the lens separation in Fig. 51b toward zero. Practically, this is approached with the development of thin binary optics microlens arrays. Binary forms of Fresnel zone patterns are fabricated utilizing high-resolution etching and transfer techniques formed on substrate materials. Hundred percent fill factors of lenslet arrays are attainable, approximating a continuous phase profile in a stepwise manner, to allow achievement of high diffraction efficiency. As presented earlier for a phased array composed of q elements per 2R phase reset [Eq. (52)], the efficiency Jb of a multilevel binary optic of m levels within one width of a Fresnel feature is given by103 Hb sinc 2 (1/m). An experimental system102 utilized such arrays of F/5 microlenses; each having a 0.2 mm diameter. The second lenslet array was spaced from the first by 10 μm, allowing relative translation. This system steered a 6-mm HeNe beam over an 11.5° field using o0.1-mm travel at a 35-Hz sweep rate. Practical mask alignment, etch errors, and transfer errors during fabrication reduced the 95 percent theoretical efficiency to 84 percent and 72 percent for the positive and negative lens arrays, respectively. Overall efficiency of the unsteered beam measured approximately 50 percent. The F/5 system exhibited low efficiency when steered. This is expected to improve with more precise fabrication and operation at lower F-numbers. A variation to the above work was conducted using “phased-arraylike” binary optics.77 A continuous quadratic phase function was sampled at equal intervals of $x, forming a stepwise matching of the continuous phase profile. The $ shifts are conducted in integral increments of $x, where the formerly disruptive region is shown analytically to render a continuous linear phase profile across the full aperture. Experimental binary micro-optics were designed to compare phased-arraylike and microlens arrays, fabricated simultaneously and adjacent on the same quartz substrate. Tests confirmed that the phased-arraylike structure provided approximately. 50 percent increased intensity in the steered mode, and less than 1 percent leakage into the immediate (local) sidelobes. While stronger distant sidelobes developed, they were well separated from the steered mode. As mentioned previously, achromatic optics applied to a single decentered lens system98 reduces dispersion and allows operation in the 2 to 5 μm IR band. The prototype design of Fig. 50c was altered to avoid the high power intensity focus and its potential damage within the field lens FL.

30.60

INSTRUMENTS

1/2

1/2 (a)

(b) FIGURE 52 Adaptation for high power and achromatic operation. (a) Intense power at focus within the (positive) field lens FL of Fig. 50c is avoided by replacing it with a negative field lens (having 1/2 the focal length of the positive lenses). (b) System (to scale) achromatized and optimized for beam deflection of 22.5°. The field lens is shifted downward through distance $m, while the output lens is shifted upward through the greater distance $k. (After Ref. 98.)

Thus, per Fig. 52a, the positive field lens of Fig. 50c was replaced with a negative one, requiring a focal length one-fourth that of the positive lens. The system was achromatized and optimized for o22.5° maximum deflection, as represented in Fig. 52b. The negative field lens and the output lens now require unequal and opposite shift directions to implement scan. The 60-mm diameter output lens requires 30 mm of $ shift. This work demonstrated that for some systems, a single group of three cemented achromatic doublets enables eliminating several problems associated with microlens arrays, such as spurious diffraction, multiple beam orders, blind spots, and large dispersion while accommodating relatively wide scan angles.

Digital Micromirror Device In 1987, the first digital micromirror device (DMD) was created at Texas Instruments.104,105 Larry Hornbeck was granted the first patent for the DMD design in 1991.106 This initial design was the basis for current digital micromirror device chips which were incorporated into digital light processing (DLP) projectors for both visible and infrared applications.107–109,110 Although the DMD is an all-digital spatial light modulator (SLM) rather than the more traditional optical scanners of this

SCANNERS

30.61

chapter, it is briefly discussed as it can be considered a binary-state scanner. Its impact on the display industry has been most significant. A DMD is an array of “light switches” having a MEMS structure that is fabricated by using micromachining CMOS processes over a CMOS static random access memory (SRAM) integrated circuit.110,111 Each light switch has an aluminum mirror that can reflect light in one of two directions depending on the state of the underlying memory cell. With the memory cell in the On state, the mirror rotates to 10°. With the memory cell in the Off state, the mirror rotates to 10°. By combining the DMD with a suitable light source and projection optics, the mirror reflects incident light either into or out of the pupil of the projection lens. Thus, the On state of the mirror appears bright and the Off state of the mirror appears dark. Gray scale is achieved by binary pulsewidth modulation of the incident light while color is achieved by using color filters, either stationary or rotating, in combination with one, two, or three DMD chips. The DMD light switch is a MEMS structure consisting of a mirror that is rigidly connected to an underlying yoke. The yoke in turn is connected by two thin, mechanically compliant torsion hinges to support posts that are attached to the underlying substrate. Electrostatic fields developed between the underlying memory cell and the yoke and mirror cause rotation in the positive or negative rotation direction. The rotation is limited by mechanical stops to typically o10°. The use of semiconductor and MEMS processing technologies allow production of very large arrays of these individually controllable micromirrors having minimal defects and high reliability.112–114

Gimbal-Less Two-Axis Scanning Micromirrors Gimbal-less two-axis scanning-micromirror devices (GSMD) have been recently developed by Mirrorcle Technologies, Inc. and are based on multilevel beam silicon-on-insulator micro-electromechanical (SOI-MEMS) fabrication technology.115 Due to their small scale and electrostatic actuation, these devices require ultralow power and can provide fast optical beam scanning in two axes when compared to the large-scale galvanometer-based optical scanners.116 Laser beams can be deflected to optical scanning angles of up to 32° at very high speeds in both axes. Continuous full-speed operation of the electrostatic actuators that drive the GSMD dissipates less than 1 mW of power. These devices are made entirely of monolithic single-crystal silicon, resulting in excellent repeatability and reliability. The flat, smooth mirror surfaces can be coated with a thin film of metal with desired reflectivity. Larger mirrors can be bonded onto actuators for custom aperture sizes. In contrast to the two-state mirror movement of the DMD, the GSMD mirror movement is fully analog and can maintain a selected tilt angle or move dynamically upon command. At this time, huge arrays of micromirrors comprise typical DMD arrays, while GSMD arrays are presently limited to a few micromirrors; however, GSMD mirrors can be physically much larger than the DMD mirrors. The GSMD are designed and optimized for point-to-point optical beam scanning mode of operation. A steady-state analog actuation voltage results in a steady-stage analog angle of rotation of the micromirror. Specifically, there is a one-to-one correspondence of actuation voltages and resulting angles that is highly repeatable. Positional precision in open-loop driving of the micromirrors is at least 14 bits, that is within 10 μrad. For a particular high-speed 3D tracking and position-measurement application, 16-bit precision has been demonstrated.117 Devices can be operated over a very wide bandwidth from DC (they maintain position at constant voltage) to several kilohertz. Such fast and broadband capability allows nearly arbitrary waveforms such as vector graphics, constant velocity scanning, point-to-point step scanning, and the like.118 The major advantage of the gimbal-less design is the capability to scan optical beams at equally high speeds in both axes. A typical GSMD with a 0.8-mm diameter micromirror achieves angular beam scanning of up to 600 rad/s and has first resonant frequency in both axes above 5 kHz. Devices with larger-diameter micromirrors are correspondingly slower due to the increased inertia. The gimbal-less design combines one-axis electrostatic combdrive-based rotators,115 and allows their operation to be nearly independent of the other axis’ operation without the added inertia of a gimbal frame. A schematic diagram of the conceptual operation of the gimbal-less two-dimensional

30.62

INSTRUMENTS

Mic rom pla irror te

Stage

at

or

B

Rotation actuator A

Rotator A

at io n

Outside linkage

Rotation Inside transformer linkage

Ro t

Ro t

at

x axis 2DOF linkage

Rotator A

ac tu at or B

io

n

ac

tu

y axis

Rotation actuator A Suspended structures

Anchored structures

FIGURE 53 Schematic diagram of a gimbal-less two-axis scanning actuator based on four high aspect ratio rotators connected to the central pedestal by two degrees-of-freedom (2 DOF) linkages. Cross-sectional depiction of device operation.116 (Diagram courtesy of Mirrorcle Technologies, Inc.).

designs is shown in Fig. 53. Two one-axis rotators are utilized for each axis of the overall twodimensional scanner. For the x axis, actuators A and A` are utilized, and for the y axis, actuators B and B`. The inside linkages are designed such that they allow torsion on the axis, specifically during the operation of the orthogonal axis. In other words, each linkage that connects a rotator to the central micromirror is actually designed to be a two degree-of-freedom component. GSMD can also operate in the dynamic, resonant mode. When operated near the resonant frequency, devices give significantly more angle at lower operating voltages and sinusoidal motion. Resonant frequencies are in the range of several kilohertz, although in some cases one of the axes is made exceptionally stiff to achieve 16 kHz and faster actuation for video projection applications.119 The gimbal-less design lends itself inherently to a modular design approach, hence, several types of dual-axis actuator designs are available.120 Each actuator can utilize rotators of arbitrary length, arbitrarily stiff linkages, and arbitrarily positioned mechanical rotation transformers. In addition, the GSMD can have an arbitrarily large mirror diameter. Because of modularity, these devices can be customized for the requirements of a particular application. Silicon mirrors of up to 1.2 mm diameter can be fabricated as an integral (monolithic) part of some GSMDs. Due to the limitations of the fabrication steps of the actuator, the standard mirrors are relatively thick (24 μm.) The inherent properties of the single crystal silicon substrate yield a polished surface with nearly perfect flatness. Larger and customizable mirror sizes and shapes can be utilized by fabricating those separately and assembling them on top of gimbal-less actuators. A SEM image of a 2-mm diameter bonded mirror on an actuator is shown in Fig. 54. Sets of electrostatic actuators optimized for speed, angle, area footprint, or resonant driving are designed and realized in a self-aligned deep reactive ion etching (DRIE) fabrication process.121 Metalized, ultralow inertia, single crystal mirrors stiffened by a backbone of thicker silicon beams (see Fig. 53) are created in a separate fabrication process. The diameter, as well as geometry, of the mirror is selected by customers, in order to optimize the trade-offs between speed, beam size, and scan angle for each individual application. The mirrors are subsequently bonded to the actuators. The modular approach allows either the absolute optimization of a device prior to fabrication, or the ability to economically adapt a small set of fabricated devices for a wide range of applications. Larger sizes up to 3.6 mm are regularly assembled in experiments and applications. Summary of Agile Beam Steering Two basic methods for low-inertia laser beam steering are presented: the phased array and the decentered lens. Although their operating principles differ, as arrays, they both form diffraction

SCANNERS

30.63

FIGURE 54 SEM image of a 2.0-mm diameter micromirror bonded to dual-axis actuators. (Photograph courtesy of Mirrorcle Technologies, Inc.)

gratings yielding output wavefronts having the properties of a blazed grating. While this results in high diffraction-efficiency at a selected wavelength, typical grating dispersion limits broadband operation. Complicating corrective measures have been applied to both array systems to approach broadband operation. The single decentered lens group avoids array dispersion. And, if its increased size and inertia can be tolerated, lenticular dispersion remains controllable with familiar achromatizing techniques. The dominant phased arrays are completely electro-optic, while the microlens array requires very small translations of reasonably low inertia assemblies. An alternative phased array utilizes individual micromirrors requiring minute (N/4) axial displacements. However, difficulty may be encountered in fabrication of high optical-integrity and in providing the high optical-fill-factor of the electro-optic types. Another alternative microlens array is formed of Fresnel-lens-type binary optics. For low F-number lenticules, whose theoretical steering efficiency is high, their minimum feature sizes become miniscule and presently are difficult to fabricate. Auxiliary control facilities can impose burdens of mass, volume, and cost. Phased arrays utilize complex multielement electrical programming, while the lens arrays require small but very precise positioning of their assemblies. Though such additional requirements are generally not detailed in the literature, a comparative analysis121 provided some related observations. The authors preferred a microlens array over the liquid crystal phased array, thereby avoiding the “heavy burden” on electronic control of the many phase-delay elements. An x-y microlens array system was designed and built for test, and was compared to a two-galvanometer x-y system assembled of commercial components. Evaluations confirmed that the microlens system steered faster, consumed lower power, and packaged smaller and lighter. However, no comment appeared on design to minimize mirror inertia and to reduce the bulk of the components and their assembly. Nor was x-y relay optics5 considered to allow minimum-sized mirrors serve to reduce inertia. Also, meriting evaluation is the single mirror suspended and actuated in x-y for precise two-dimensional scan.122,123 Such diverse considerations are invaluable for rating design alternatives for their relative compliance to system requirements.

30.64

INSTRUMENTS

30.9

REFERENCES 1. S. Sherr, Fundamentals of Display System Design, John Wiley & Sons, New York, 1970. 2. S. Sherr, Electronic Displays, John Wiley & Sons, New York, 1979. 3. L. Beiser, “Unified Approach to Photographic Recording from the Cathode-Ray Tube,” Photo. Sci. Engr. 7(3):196–204 (1963). 4. L. Beiser, Laser Scanning Notebook, SPIE Press, Washington, vol. PM13, 1992. 5. L. Beiser, “Laser Scanning Systems,” in Laser Applications, vol. 2. Academic Press, New York, 1974, pp. 53–159. 6. L. Beiser, Holographic Scanning, John Wiley & Sons, New York, 1988. 7. J. D. Zook, “Light Beam Deflector Performance: A Comparative Analysis,” Appl. Opt. 13(4):875–887 (1974). 8. L. Beiser (ed.), “A Guide and Nomograph for Laser Scanned Resolution,” Proc. SPIE 1079:2–5 (1989). 9. G. Boreman, “Modulation Transfer Techniques,” Chap. 32, Handbook of Optics, vol. II, McGraw Hill, New York, 1995. 10. L. R. Dickson, “Characteristics of Propagating Gaussian Beams,” Appl. Opt. 9(8):1854–1861 (1970). 11. J. Randolph and J. Morrison, “Rayleigh-Equivalent Resolution of Acoustooptic Deflection Cells,” Appl. Opt. 10(6):1383–1385 (1971). 12. L. R. Dickson, “Optical Considerations for Acoustooptic Deflectors,” Appl. Opt. 11(10):2196–2202 (1972). 13. L. Levy, Applied Optics, vol. 1, John Wiley & Sons, New York, 1968. 14. L. Beiser, “Generalized Equations for the Resolution of Laser Scanners,” Appl. Opt. 22(20):3149–3151 (1983). 15. J. C. Urbach, T. S. Fisli, and G. K. Starkweather, “Laser Scanning for Electronic Printing,” Proc. IEEE 70(6):597–618 (1982). 16. H. V. Kennedy, Split Image, “High Scanning Rate Optical System with Constant Aperture,” U.S. Patent 3,211,046, 1965. 17. H. B. Henderson, “Scanner System,” U.S. Patent 3,632,870, 1972. 18. W. L. Wolfe, “Optical-Mechancial Scanning Techniques and Devices,” Chap. 10, in W. L. Wolfe and G. Zissis (eds.), The Infrared Handbook, ERIM, Ann Arbor, Mich., 1989; U.S. Office of Naval Research, Washington, Fig. 10-12, p. 10–12. 19. Manual of Remote Sensing, 2nd ed., Amer. Soc. of Photogrammetry, Falls Church, 1983, Figs. 3–8, p. 340. 20. J. M. Lloyd, Thermal Imaging Systems, Plenum Press, New York, 1975, pp. 308–309. 21. R. B. Barnes, “Infrared Thermogram Camera and Scanning Means Theorem,” U.S. Patent 3,287,559, 1966. 22. B. A. Wheeler, “Optical Raster Scan Generator,” U.S. Patent 3,764,192, 1973. 23. R. B. Johnson, “Target-Scanning Camera Comprising a Constant Temperature Source for Providing a Calibrated Signal,” U.S. Patent 3,631,248, 1971. 24. P. J. Lindberg, “A Prism Line-Scanner for High Speed Thermography,” Optica Acta 15:305–316 (1966). 25. P. J. Lindberg, “Scanning Mechanism for Electromagnetic Radiation,” U.S. Patent 3,253,498, 1966. 26. R. B. Johnson, “Image Rotation Device for an Infrared Scanning System or the Like,” U.S. Patent 3,813,552, 1974. 27. P. J. Berry and H. M. Runciman, “Radiation Scanning System,” U.S. Patent 4,210,810, 1980. 28. T. H. Jamieson, “Optical Design of Compact Thermal Scanner,” Proc. SPIE 518:15–21 (1984). 29. J. E. Modisette and R. B. Johnson, “High Speed Infrared Imaging System,” U.S. Patent 4,419,692, 1983. 30. D. C. O’Shea, Elements of Modern Optical Design, John Wiley & Sons, New York, 1985. 31. K. S. Petche, Handbook of Optics, OSA, W. G. Driscoll and W. Vaughn (eds.), McGraw Hill, New York, 1978, pp. 5–10. 32. Y. Li and J. Katz, “Encircled Energy of Laser Diode Beams,” Appl. Opt. 30(30):4283 (1991). 33. R. E. Hopkins and D. Stephenson, “Optical Systems for Laser Scanners,” in G. F. Marshall (ed.), Optical Scanning, Marcel Dekker, New York, 1991. 34. A. B. Marchant, U.S. Patent 4,759,616, 1988; and A. W. Lohman and W. Stork, “Modified Brewster Telescope,” Appl. Opt. 28(7):1318–1319 (1989).

SCANNERS

30.65

35. L. Beiser, “Design Equations for a Polygon Laser Scanner,” in G. F. Marshall and L. Beiser (eds.), Proc. SPIE 1454:60–66 (1991). 36. D. Kessler, “High Resolution Laser Writer,” in L. Beiser (ed.), Proc. SPIE 1079:27–35 (1989). 37. C. J. Kramer, “Holographic Deflectors for Graphic Arts Systems,” in G. F. Marshall (ed.), Optical Scanning, Marcel Dekker, New York, 1991. 38. R. V. Pole and H. P. Wollenmann, “Holographic Laser Beam Deflector,” Appl. Opt. 14(4):976–980 (1975). 39. I. Cindrich, “Image Scanning by Rotation of a Hologram,” Appl. Opt. 6(9):1531–1534 (1967). 40. C. J. Kramer, “Holo-Scanner for Reconstructing a Scanning Light Spot Insensitive to Mechanical Wobble,” U.S. Patent 4,239,326, 1980. 41. D. B. Kay, “Optical Scanning System with Wavelength Shift Correction,” U.S. Patent 4,428,643, 1984. 42. H. Ikeda, et al., “Hologram Scanner,” Fujitsu Sci. Tech J. 23:3 (1987). 43. F. Yamagishi, S. Hasegawa, H. Ikeda, and T. Inagaki, “Lensless Holographic Line Scanner,” Proc. SPIE 615:128–132 (1986). 44. J. Montagu, “Galvanometric and Resonant Low Inertia Scanners,” in G. F. Marshall (ed.), Optical Scanning, Marcel Dekker, New York, 1991. 45. S. Reich, “Use of Electro-Mechanical Mirror Scanning Devices,” in L. Beiser (ed.), SPIE Milestone Series 378:229–238 (1985). 46. D. G. Tweed, “Resonant Scanner Linearization Techniques,” Opt. Eng. 24(6):1018–1022 (1985). 47. F. Blais, “Control of Galvanometers for High Precision Laser Scanning Systems,” Opt. Eng. 27(2):104–110 (1988). 48. G. F. Marshall and J. S. Gadhok, “Resonant and Galvanometer Scanners: Integral Position Sensing,” Photonics Spectra, 155–160 (1991). 49. P. Debye and F. W. Sears, “On the Scattering of Light by Supersonic Waves,” Proc. Natl. Acad. Sci. (USA) 18:409 (1932). 50. F. Okolicsanyi, “The Waveslot, an Optical TV System,” Wireless Eng. 14:536–572 (1937). 51. R. Adler, “Interaction between Light and Sound,” IEEE Spectrum 4(5):42–54 (1967). 52. E. I. Gordon, “Review of Acoustooptical Deflection and Modulation Devices,” Proc. IEEE 54(10):1391–1401 (1966). 53. A. Korpel, “Acousto-Optics,” in R. Kingslake and B. J. Thompson (eds.), Appl. Opt. and Opt. Eng. VI, Academic Press, New York, 1980, pp. 89–141. 54. R. V. Johnson, “Scophony Light Valve,” Appl. Opt. 18(23):4030–4038 (1979). 55. L. Bedamian, “Acousto-Optic Laser Recording,” Opt. Eng. 20(1):143–149 (1981). 56. W. H. Watson and A. Korpel, “Equalization of Acoustooptic Deflection Cells in a Laser Color TV System,” Appl. Opt. 7(5):1176–1179 (1970). 57. M. Gottlieb, “Acousto-Optical Scanners and Modulators,” in G. Marshall (ed.), Optical Scanning, Marcel Dekker, New York, 1991. 58. D. L. Hecht, “Multibeam Acoustooptic and Electrooptic Modulators,” in L. Beiser (ed.), Proc. SPIE 396:2–9 (1983). 59. L. Beiser, “Generalized Gradient Deflector and Consequences of Scan of Convergent Light,” J. Opt. Soc. Am. 57:923–931 (1967). 60. A. J. Giarola and T. R. Billeter, “Electroacoustic Deflection of a Coherent Light Beam,” Proc. IEEE 51:1150 (1963). 61. V. J. Fowler, et al., “Electro-Optical Light Beam Deflection,” Proc. IEEE 52:193 (1964). 62. J. F. Lospeich, “Electro-Optic Light Beam Deflections,” IEEE Spectrum 5(2):45–52 (1968). 63. R. J. Pizzo and R. J. Kocka, Conoptics Inc., Danbury, CT 06810. Available at http://www.conoptics.com/. 64. T. C. Lee and J. D. Zook, “Light Beam Deflection with Electrooptic Prisms,” IEEE J. Quant. Electron. QE-4(7):442–454 (1968). 65. C. S. Tsai and J. M. Kraushaar, “Electro-Optical Multiplexers and Demultiplexers for Time-Multiplexed PCM Laser Communication Systems,” Proc. Electro-Opt. Syst. Des. Conf. 176–182 (1972). 66. A. S. Vasilevskaya, Soviet Phys. Crystallogr. 12(2):308 (1967).

30.66

INSTRUMENTS

67. J. D. Zook and T. C. Lee, “Design of Analog Electrooptic Deflectors,” Proc. SPIE (of 1969) 281 (1970). 68. J. P. Donohue, “Laser Pattern Generator for Printed Circuit Board Artwork Generation,” Proc. SPIE 200:179–186 (1979); also in L. Beiser (ed.), SPIE Milestone Series 378:421–428 (1985). 69. G. K. Starkweather, “Single Facet Wobble Free Scanner,” U.S. Patent 4,475,787, 1984. 70. L. Beiser, “Light Scanner,” U.S. Patent 4,936,643, 1990. 71. G. F. Marshall, T. Vettese, and J. H. Carosella, “Butterfly Line Scanner,” G. F. Marshall and L. Beiser (eds.), Proc. SPIE 1454 (1991). 72. L. Beiser, “Double Reflection Light Scanner,” U.S. Patent 5,114,217, 1992. 73. D. F. Hanson and R. J. Sherman, “Method and Apparatus for Generating Optical Scans,” U.S. Patent 4,433,894, Feb. 28, 1984; and R. J. Garwin, “Optical Alignment Compensation,” U.S. Patent 4,429,948, Feb. 7, 1984. 74. P. F. McManamon, T. A. Dorschner, D. L. Corkum, L. J. Friedman, D. S. Hobbs, M. Holz, S. Liberman, et al., “Optical Phased Array Technology,” Proc. IEEE 84(2):268–298 (Feb. 1996). 75. E. A. Watson, “Analysis of Beam Steering with Decentered Microlens Arrays, Opt. Eng. 32(11):2665–2670 (Nov. 1993). 76. E. A. Watson and A. R. Miller, “Analysis of Optical Beam Steering Using Phased Micromirror Arrays,” SPIE 2687:60–67 (1966). 77. M. W. Farn, “Agile Beam Steering Using Phased Array-Like Binary Optics,” Appl. Opt. 33(22):5151–5158 (Aug. 1994). 78. T. C. Cheston and J. Frank, Radar Handbook, M. I. Skolnick (ed.) McGraw Hill, New York, 1970. 79. M. I. Skolnick, Introduction to Radar Systems, McGraw Hill, New York, 1962. 80. L. W. Procopio, F. A. Jessen, and L. J. Brown., “Laser Phased Arrays,” Proc. 8th Int. Conf. Mil. Electron. Group p. 67; RADC contract AF30(602)-2901 (1964). 81. L. J. Alpet, et al., “Laser Array Techniques,” Quart. Rep. 3A, Air Force contract AF33(615)3918, AD No. 821235 (1967). 82. P. G. Grant, R. A. Meyer, and D. N. Qualkinbush, “An Optical Phased Array Beam Steering Technique,” Proc. Electrc-Opt. Syst. Des. Conf. 259–264 (1971). 83. R. A. Meyer, “Optical Beam Steering Using a Multichannel Lithium Tantalate Crystal,” Appl. Opt. 11(3):613 (Mar. 1972). 84. A. S. Keys, R. L. Fork, T. R. Nelson, and J. P Loehr, “Resonant Tansmissive Modulator Construction for use in Beam Steering Arrays”, in L. Beiser, S. Sagan and G. F. Marshall (eds.), Optical Scanning: Design and Applications, SPIE 3787:115–125 (Jul. 1999). 85. P. F. McManamon and E. A. Watson, “Optical Beam Steering Using Phased Array Technology,” in L. Beiser and S. Sagan (eds.), SPIE 3131:90–98 (1992). 86. T. A. Dorschner, R. C. Sharp, D. P. Resler, L. J. Friedman, and D. C. Hobbs, “Basic Laser Beam Agility Techniques,” Wright Patterson Air Force Base, WL-TR-93-1020 (AD-B-175-883) (1993). 87. L. Beiser, Unified Optical Scanning Technology, John Wiley & Sons, New York, 2003. 88. E. A. Watson, D. T. Miller, and P. F. McManamon, “Applications and Requirements for Nonmechanical Beam Steering in Active Electro-Optic Modulators,” in I. Cindrich, S. H. Lee, and R. L. Sutherland (eds.), SPIE 3633:216–225 (1999). 89. L. Beiser, “Laser Scanning Systems,” Laser Applications, vol. 2, M. Ross (ed.), Academic Press, New York, pp. 55–159, 1974. 90. J. A. Thomas, M. E. Lasher, Y. Fainman, and P. Soltan, “A PLZT-Based Dynamic Diffractive Optical Element for High Speed Random Access Beam Steering,” in L. Beiser and S. Sagan (eds.), SPIE 3131:124–132 (1997). 91. K. M. Flood, W. J. Cassarly, C. Sigg, and M. J. Finlan, “Continuous Wide Angle Beam Steering Using Translation of Binary Microlens Arrays and a Liquid Crystal Phased Array,” SPIE 1211:296–304 (1990). 92. G. H. Haertling, “PLZT Electrooptic Materials and Applications—A Review,” Ferroelectrics 75:25–55 (1987). 93. D. M. Burns, V. M. Bright. S. C. Gustafson and E. A. Watson, “Optical Beam Steering Using Surface Machined Gratings and Optical Phased Arrays,” in L. Beiser and S. Sagan (eds.), SPIE 3131:99–110 (1997). 94. E. A. Watson, P. R. McManamon, L. J. Barnes, and A. J. Carney, “Applications of Dynamic Gratings to Broad Spectral Band Beam Steering,” SPIE 2120 (1994).

SCANNERS

30.67

95. J. E. Stockley, S. A. Serati, G. D. Sharp, P. Wang, K. F. Walsh, and K. M. Johnson, “Broad-Band Beam Steering,” in Optical Scanning Systems: Design and Applications, L. Beiser and S. Sagan (eds.), Proc. SPIE 3131:111–123 (1997). 96. G. M. Morris and D. A. Zweig, “White Light Fourier Transforms,” in J. J. Hovner (ed.), Optical Signal Processing, Chap. 1.2, Academic Press, New York, 1987. 97. P. F. McManamon, E. A. Watson, S. Jianru, and P. J. Bos, “Nonmechanical Steering of the Field of View of Broad Spectral Band Optical Systems,” in Optical Scanning 2005, S. F. Sagan and G. F. Marshall (eds.), SPIE 5873:26–37 (2005). 98. J. L. Gibson, B. D. Duncan, E. A. Watson, and J. S. Loomis, “Wide Angle Decentered Lens Beam Steering for Infrared Countermeasures Applications,” Opt. Eng. 43(10):2312–2321 (Oct. 2004). 99. L. Levi, Applied Optics, vol. 1, John Wiley & Sons, New York 1968. 100. T. D. Milster and J. N. Wang, “Modeling and Measurement of a Micro-Optic Beam Deflector,” in Y. Kohanzadeh, G. W. Lawrence, J. G. McCoy, and H, Weichel (eds.), SPIE 1625:78–83 (1992). 101. E. A. Watson, W. E. Whitaker, C. D. Brewer, and S. R. Harris, “Implementing Optical Phased Array Beam Steering with Cascaded Microlens Arrays,” IEEE Aerospace Conf. Paper No. 5.036, Big Sky, MT (March 2002). 102. W. Goltos and M. Holz, “Agile Beam Steering Using Binary Optics Microlens Arrays,” Opt. Eng. 29(11):1392–1397 (Nov. 1990). 103. G. J. Swanson, “Binary Optics Technology: Theory and Design of Multi-Level Diffractive Optics Elements, Lincoln Labs Tech. Rep. 854 (Aug. 14, 1989). 104. L. J. Hornbeck, “Deformable-Mirror Spatial Light Modulators,” Spatial Light Modulators and Applications I, SPIE Critical Reviews 1150:86 (Aug. 1989). 105. J. Younse, “Mirrors on a Chip,” IEEE Spectrum 30(11):27 (Nov. 1993). 106. L. J. Hornbeck, “Spatial Light Modulator and Method,” U.S. Patent 5,061,049 (1991). 107. P. F. Van Kessel, L. J. Hornbeck, R. E. Meier, and M. R. Douglass, “A MEMS-Based Projection Display,” Proc. IEEE, 86(8):1687–1704 (Aug. 1998). 108. J. M. Younse, “Projection Display Systems Based on the Digital Micromirror Device (DMD),” SPIE Conference on Microelectronic Structures and Microelectromechanical Devices for Optical Processing and Multimedia Applications, Austin, Texas, SPIE Proc. 2641:64–75 (Oct. 1995). 109. J. B. Sampsell, “An Overview of the Digital Micromirror Device (DMD) and Its Application to Projection Displays,” 1993 SID International Symposium Digest of Technical Papers, 24:1012 (1993). 110. L. J. Hornbeck, “Digital Light Processing™ and MEMS: Timely Convergence for a Bright Future,” Plenary Session, SPIE Micromachining and Microfabrication ’95, Austin, TX, October 24, 1995, p. 2. 111. P. F. Van Kessel, L. J. Hornbeck, R. E. Meier, M. R. Douglass, “A MEMS-Based Projection Display,” Proc. IEEE, 86(8):1687–1704, (Aug. 1998). 112. L. J. Hornbeck, “From Cathode Rays to Digital Micromirrors: A History of Electronic Projection Display Technology,” Texas Instruments White Paper, http://www.dlp.com/tech/research.aspx (Sept. 1998). 113. L. J. Hornbeck, “Digital Light Processing™: A New MEMS-Based Display Technology,” Texas Instruments White Paper, http://www.dlp.com/tech/research.aspx (2008). 114. M. R. Douglass, “Development of the Digital Micromirror Device™ (DMD™) Microsystem,” Texas Instruments White Paper, http://www.dlp.com/tech/research.aspx, Accessed on Sept. 2005. 115. V. Milanovic´, “Multilevel-Beam SOI-MEMS Fabrication and Applications,” IEEE/ASME J. Microelectromechanical Systems, 13(1):19–30, (Feb. 2004). 116. V. Milanovic´, D. T. McCormick, and G. Matus, “Gimbal-Less Monolithic Silicon Actuators For Tip-Tilt-Piston Micromirror Applications,” IEEE J. Select Topics in Quantum Electron. 10(3):462–471, (May-Jun. 2004). 117. V. Milanovic´, and W. K. Lo, “Fast and High-Precision 3D Tracking and Position Measurement with MEMS Micromirrors,” 2008 IEEE/LEOS Optical MEMS and Their Applications Conf., Freiburg, Germany, Aug. 12, 2008. 118. V. Milanovic´, K. Castelino, and D. McCormick, “Highly Adaptable MEMS-based Display with Wide Projection Angle,” 2007 IEEE Int. Conf. on Microelectromechanical Systems (MEMS’07), Kobe, Japan, Jan. 25, 2007. 119. V. Milanovic´, “Improved Control of the Vertical Axis Scan for MEMS Projection Displays,” 2007 IEEE/LEOS Optical MEMS and Their Applications Conf., Hualien, Taiwan, Aug. 12, 2007.

30.68

INSTRUMENTS

120. D. T. McCormick, V. Milanovic´, and K. Castelino, “A Modular Custom Aperture Technology for Optimization of MEMS Scanners,” 2005 IEEE/LEOS Optical MEMS and Their Applications Conf., Oulu, Finland, Aug. 2005, pp. 29–30. 121. Devoe, Donald Lad, and Tsai, Lung-Wen, “Process for Fabrication of 3-Dimensional Micromechanisms,” U.S. Patent 6,664,126 (2003). 122. G. F. McDearmon, K. M. Flood, and J. M. Finlan, “Comparison of Conventional and Microlens-Array Agile Beam Steerers,” in L. Beiser and M. Modamedi (eds.), Proc. SPIE 2383:167–178 (Feb. 1995). 123. A. Berta, “Development of High Performance Fast Steering Mirrors,” in Optical Scanning: Design and Applications, L. Beiser, S. Sagan, and G. F. Marshall (eds.), Proc. SPIE 3787:115–125 (Jul. 1999).

30.10

FURTHER READING The following listing augments the text and its references with a select group of publications, many arcane and of historic value, representing substantive work in the field. Barnes, C. W., “Direct Optical Data Readout and Transmission at High Data Rates,” AFAL-TR-65- 45, Wright Pat. A. F. B., AD-460486, 1965. Becker, C. H., “Coherent Light Recording Techniques,” RADC-TR-65-130, AFSC, Griffiss AFB, New York, 1965. Beiser, L., “Laser Beam Scanning for High Density Data Extraction and Storage,” Proc. SPSE Symp. on Photography in Information Storage and Retrieval, Washington, D.C., 1965. Beiser, L., “Laser Beam and Electron Beam Extremely Wideband Information Storage and Retrieval,” Photo. Sci. Eng. 10(4):222 (1966). Beiser, L., “Laser Scanning Systems,” in M. Ross (ed.), Laser Applications, vol. 2, Academic Press, New York, 1974. Beiser, L. (ed.), “Selected Papers on Laser Scanning and Recording,” SPIE Milestone Series, vol. 378, 1985. Beiser, L., Holographic Scanning, John Wiley & Sons, New York, 1988. Bousky, S., “Scanning Techniques for Light Modulation Recording,” Wright Pat. AFB, Ohio, Contr. No. AF33(615)–2632, 1965. Cindrich, I., et al., “Wide Swath Wide Bandwidth Recording,” AFAL-TR-11-361, Wright Pat. AFB, Ohio, DDC No. 13217, AD 513171, 1971. Dubovic, A. S., Bases of Mirror Scanning Theory (trans from Russian), AD-703569, Wright Pat. AFB, F. T. D. HT-23-654-69, 1969. Dubovic, A. S., The Photographic Recording of High Speed Processes, John Wiley & Sons, New York, 1981. Fowler, V. J., “Survey of Laser Beam Deflection Techniques,” Appl. Opt. 5(10):1675 (1966). Johnson, R. B. and W. L. Wolfe (eds), “Selected Papers on Infrared Design,” SPIE Milestone Series, vol. 513, 1985. Katys, G. P., Automatic Scanning (trans. from Russian, 1969), National Tech Info. Service, JPRS 51512, 1970. Korpel, A., et al., “A TV Display Having Acoustic Deflection and Modulation of Coherent Light,” Appl. Opt. 5(10):667 (1966). Klein, D. A., “Laser Imaging Techniques,” Naval Air Syst. Comm. TR-1615 (AIR-52022), Naval Avionics Facility, Indiana 46218, 1970. Laemmel, A. E., “The Scanning Process in Picture Transmission,” Polytech. Inst. of B’klyn, RADC-TN-57-215, DDC AD-131112, 1957. Marshall, G. F. (ed.), Laser Beam Scanning, Marcel Dekker, New York, 1985. Marshall, G. F. (ed.), Optical Scanning, Marcel Dekker, New York, 1991. Newgard, P. M. and Brain, E. E., “Optical Transmission at High Data Rates,” AFAL-TR-67-217, Wright. Pat. AFB, Ohio, 1967. Townsend, C. V. R., “Study of Advanced Scanning Methods,” ASD-TDR-63-595, DDC 412794, Wright Pat. AFB, Ohio, 1963. Wolfe, W. L. and G. J. Zissis (eds.), The Infrared Handbook, ERIM, Ann Arbor, Mich., 1989. Zook, J. D., “Light Beam Deflector Performance,” Appl. Opt. 13:875 (1974).

31 OPTICAL SPECTROMETERS Brian Henderson Department of Physics and Applied Physics University of Strathclyde Glasgow, United Kingdom

31.1

GLOSSARY Aba ao Bif e ED EDC Ehf En EQ E(t) E(Y) ga gb gN h Hso I I(t) j li m MD nY(T)

Einstein coefficient for spontaneous emission Bohr radius Einstein coefficient for transition between initial state |i® and final state | f ® electron charge electric dipole term Dirac Coulomb term hyperfine energy eigenvalues of quantum state n electric quadrupole term electric field at time t electric field at frequency Y degeneracy of ground level degeneracy of excited level gyromagnetic ratio of nucleus Planck’s constant spin-orbit interaction Hamiltonian nuclear spin emission intensity at time t total angular momentum vector given by j l o 1/2 orbital state electron mass magnetic dipole term equilibrium number of photons in a blackbody cavity radiator at angular frequency Y and temperature T 31.1

31.2

INSTRUMENTS

QED Rnl(r ) Re s si T Wab Wba Z C e 2 /4PE 0c &Y &YD Go \(r) OB T(Y) VR Y Yk f |V ` | i ®

31.2

quantum electrodynamics radial wave function Rydberg constant for an infinitely heavy nucleus spin quantum number with the value 1/2 electronic spin absolute temperature transition rate in absorption transition between states a and b transition rate in emission transition from state b to state a charge on the nucleus fine structure constant natural linewidth of transition Doppler width of transition permittivity of free space spin-orbit parameter Bohr magneton energy-density at frequency Y radiative lifetime angular frequency mode k with angular frequency Y matrix element of perturbation V

INTRODUCTION This chapter outlines the physical basis of optical measurements in the wavelength/frequency and time domains. From the multiplicity of different apparatus, only simple examples are given of spectrometers designed for optical absorption, photoluminescence, and radiative decay measurements. Rather more detailed expositions are given of modern developments in laser spectrometers especially where high resolution is possible in both frequency and time domains. Included are specific developments for linewidth measurements down to tens of kilohertz using saturated absorption techniques as well as temporal decay characteristics in the sub-picosecond domain. A description is also given of a multiple resonance spectrometer including optically detected electron spin resonance and optically detected electron nuclear double resonance.

31.3

OPTICAL ABSORPTION SPECTROMETERS

General Principles In optical absorption spectroscopy, electromagnetic radiation in the near-ultraviolet, visible, or near-infrared regions is used to excite transitions between the electronic states. Whereas atoms in low-pressure gas discharges exhibit very sharp lines, electronic centers in molecules and condensed matter display a variety of different bandshapes. In consequence, the absorbed intensity is a function of the photon wavelength (or energy). The most desirable experimental format plots the absorption coefficient C as a function of the radiation frequency v, because v is directly proportional to the energy separation between the states involved in the transition. Nevertheless, optical

OPTICAL SPECTROMETERS

31.3

spectroscopists quote peak positions and linewidths in energy units E (eV, meV), in wave numbers v (in cm 1), in frequency units (v or Y), or in wavelength N [nanometers (nm) or micrometers (Mm)]. The following approximate relationships exist: 1 cm 1 1.24 r 10 4 eV; 1 eV 8066 cm 1; and E(eV) 1.24/N (Mm). Very often the spectrometer output is given in terms of the specimen transmission, T I (v )/I o (v ) expressed as a percentage, or the optical density (or absorbance), OD log10 (1/T ), which are related to the absorption coefficient C by OD log10 (1/T ) A (v )l /2.303

(1)

where l is the thickness of the sample. Typically one measures the absorption coefficient C over the wavelength range 185 to 3000 nm. Since C may be a function of both frequency v and polarization Eˆ , we may use the designation A (v , Eˆ). For a center containing N noninteracting absorbing centers per unit volume, each absorbing radiation at a frequency v and polarization Eˆ , the attenuation of a beam of intensity I o (v , Eˆ) by a solid of thickness l is given by I (v , Eˆ) I o (v , Eˆ)exp[ A (v , Eˆ)l]

(2)

Experimentally I o (v , Eˆ) represents the transmission of the system in the absence of an absorbing specimen. In practice I o (v , Eˆ) and I (v , Eˆ) are measured and the value of the absorption coefficient A (v , Eˆ) at a particular frequency is obtained using the formula

A (v , Eˆ)

I (v , Eˆ) 1 ln o l I (v , Eˆ)

(3)

A (v , Eˆ) has units of cm 1 or m 1. The variation of the absorption coefficient with frequency is difficult to predict. In general, the absorption transition has a finite width, and the absorption strength, ¯ A (v , Eˆ)dv, is related to the density of absorbing centers and to the transition probability. The value of the absorption coefficient in an isotropic material is related to the Einstein A coefficient for spontaneous emission by1 ¤ g ³ c2 1 A (v ) ¥ N a b N b´ Aba G(v ) 8P v 2 n2 ¦ ga µ

(4)

where ga and gb are the statistical weights of the states, G(v) is the lineshape function [defined such that ¯ G(v )dv 1], c/n is the velocity of light in the medium, and n is the refractive index. In Eq. (4), the population densities in the ground and excited states, Na and Nb, respectively, have been assumed to be invariant with time and unaffected by the absorption process. Under conditions of weak excitation we can ignore the small value of Nb, and replace Na by N so that

A (v ) NAba

c 2 1 gb G(v ) NS G(v ) 8P v 2 n2 g a

(5)

where U is the absorption cross section per center. The absorption strength, i.e., the area under the absorption band, is related to U by

¯ A (v)dv NS

(6)

31.4

INSTRUMENTS

If we ignore the refractive index and local field correction factors and assume a gaussian-shaped absorption band, then Nf ab 0.87 r 1017 A (v o )$ v

(7)

where C(vo) is measured in cm 1, $v, the full-width at half maximum absorption, is measured in eV, and N is the number of centers cm 3. Equation (7) is often referred to as Smakula’s formula. To obtain the oscillator strength from the area under the absorption band, one needs an independent determination of the density of centers. For impurity ions in solids, N may be determined by chemical assay or by electron spin resonance.

The Double-Beam Spectrophotometer The first essential of an absorption spectrophotometer is a broadband source: deuterium, hydrogen, xenon, and tungsten lamps are commonly used. Their outputs cover different wavelength ranges: a hydrogen lamp is suitable for wavelengths in the range 150 to 350 nm whereas highpressure xenon lamps have usable light outputs in the range 270 to 1100 nm. For Xe arc lamps the output is relatively continuous in the wavelength range 270 to 800 nm apart from some sharp lines near 450 nm. In the infrared region 800 to 1100 nm, much of the most intense part of the output is in the form of sharp lines. In the arc lamp, radiation is due to the collision of Xe atoms with electrons which flow across the arc. Complete separation of the excited electrons from the atoms leads to ionization and the continuum output. The formation of Xe atoms in excited states leads to the sharp lines in the output from Xe arc lamps. Tungsten filament lamps may also be used in absorption spectrophotometers. The spectral output from such a heated filament is approximately that of a blackbody radiator at a temperature of 2000 K. In consequence, the emission intensity is a smooth function of wavelength with peak at 1500 nm, with the detailed curve following Planck’s thermal radiancy law. Accordingly, the peak in the distribution of light varies with filament temperature (and therefore current through the filament), being determined by L maxT 2.898 r 10 3 mK . This relationship expresses Wien’s wavelength displacement law. Although containing all wavelengths from the ultraviolet into the infrared region, the total output is fairly modest compared with a high-pressure mercury lamp. Accurate measurements of the absorption coefficient at different wavelengths are best made using a double-beam spectrophotometer: a schematic is shown in Fig. 1. The exciting beam from the broadband source passes through a grating monochromator: the resulting narrow band radiation is divided by a beam-splitting chopper into separate monochromatic beams which traverse the sample and a reference channel. Thus the light incident on the sample and that which passes through the reference channel have the same wavelengths and is square-wave modulated (on/ off) at some frequency in the range 1 to 5 kHz. The sample and reference beams are recombined at the phototube, and the magnitude and phase of sample and reference signals are amplified and compared by the lock-in detector. Chopping at a preselected frequency permits narrowband amplification of the detected signal. Thus any noise components in the signal are limited to a narrowband centered at the chopping frequency. The dc output from the lock-in detector is plotted as a function of wavelength using a pen recorder. Alternatively, the signal may be processed using a microcomputer, so that the absorbed intensity may be signaled as the transmission, the optical density [Eq. (1)], or the absorption coefficient [Eq. (3)] of the sample as a function of wavelength N, wave number v , or photon energy (E hv). Ensuring a high light throughput in both sample and reference channels usually limits the resolution of the monochromator used in the spectrophotometer (Fig. 1). In consequence, very narrow absorption lines, &N < 0.1 nm, are normally broadened instrumentally. Note that because in an absorption spectrophotometer one measures the light transmitted by the sample relative to that transmitted by the reference chamber [Eqs. (2) and (3)], the absorption coefficient is independent of the spectral dependencies of the lamp, the monochromator, and the detection system.


31.5

Lamp

Monochromator

Beam-splitting chopper

Wavelength drive

Signal Sample Reference

Reference signal PT Power supply

Chart recorder

Current amplifier Lock-in detector

FIGURE 1 Block diagram of a dual-beam absorption spectrometer.

The measurement is also independent of the polarization properties of the monochromator system. By taking ratios, many nonideal behaviors of the components cancel.

31.4

LUMINESCENCE SPECTROMETERS

General Principles To study luminescence it is necessary to optically pump into the absorption spectrum using highintensity sources. Typical sources used in luminescence spectroscopy, which have broadbands in near ultraviolet and blue regions, include hydrogen and xenon arc lamps. The xenon arc lamp is particularly useful for exciting luminescence in the yellow-red region of the spectrum since xenon does not show interfering sharp line emission in this region. In general, high-pressure mercury (Hg) arc lamps have higher intensities than Xe arc lamps. However, the intensity is concentrated in sharp lines. Consequently, such lamps are utilized mainly with broadband absorbers or in situations that permit the individual lines to suit the absorption lines of the particular sample. In addition, a variety of lasers may be used, including Ar, Kr, He-Ne, and He-Cd lasers which have emissions at fixed wavelengths. Tunable dye lasers can be selected to closely match the absorption bands of particular

INSTRUMENTS

materials. Because of their low intensity, tungsten filament lamps are not normally used in luminescence spectrometers. The light emitted is resolved into its component lines/bands using a monochromator. For medium resolution a 1-m Czerny-Turner monochromator will give a spectral resolution of about 0.02 nm. An order of magnitude lower resolution can be achieved using a grating spectrometer with focal length 0.25 m. The light emerging from the monochromator is detected using an electron multiplier phototube with associated high-voltage power supplies. Gallium arsenide phototubes operate with good quantum efficiency in the range 280 to 860 nm. For measurements in the near-infrared, a lead sulphide cell, cooled germanium photodetector, or special III-V compound photodiode may be used. Under steady-state optical pumping, a steady-state luminescence output is obtained and detected as a photocurrent which is amplified and converted to a voltage signal to be displayed on a pen recorder. Luminescence detection is inherently more sensitive than absorption measurements and sensitivities of 1011 centers cm 3 are routine. Ideally, the excitation source should yield a constant light output at all wavelengths, the monochromator must pass all wavelengths with equal efficiency, and be independent of polarization. In addition, the detector should detect all wavelengths with equal efficiency. Unfortunately, such ideal light sources, monochromators, and phototubes are not available and it is necessary to compromise on the selection of components and to correct for the nonideal response of the luminescence spectrometer. Generally, luminescence spectra are recorded by selecting the excitation wavelength which results in the most intense emission and then scanning the wavelength of the emission monochromator. In consequence, techniques must be developed to allow for the wavelength-dependent efficiency of the emission monochromator and photomultiplier tube. This is not required in absorption spectrophotometers where the ratio of I (v , Eˆ)| I o (v , Eˆ) are used to compute the values of A (v , Eˆ) from Eq. (3). Modern spectrometers use diffraction gratings in monochromators rather than prisms. This results in less interference from stray light and in greater dispersion. Stray light may also be reduced using narrow entrance and exit slits as well as double monochromators (i.e., monochromators incorporating two gratings). Nevertheless, the transmission efficiency of the grating monochromator is a strong function of wavelength, which can be maximized at any given wavelength by choice of the blaze angle: the efficiency is less at other wavelengths as Fig. 2 shows. The stray light levels are to some extent controlled by exit and entrance slits. Smaller slit widths also yield higher resolution as do gratings with greater numbers of grooves per unit area. The efficiency of a grating monochromator also depends upon the polarization of the light. For this reason, the observed fluorescence intensities can be dependent upon the polarization of the emitted radiation. A typical plot of the

31.6

FIGURE 2 Showing how the grating efficiency varies with wavelength for gratings blazed at 300, 500, and 1000 nm.


Holographic grating 1800 g/mm

Efficiency (%)

100

E

80

31.7

Ruled grating, 1200 g/mm, 500 nm blaze

E

60 40 20

E E 300 400 500 600 700 800 900 1000 Wavelength, nm

300 400 500 600 700 800 900 1000 Wavelength, nm

FIGURE 3 Showing the effect of polarization on the efficiency of a ruled grating with 1200 grooves/mm and blazed at 500 nm.

wavelength dependence of the efficiency of a ruled grating as a function of polarization is shown in Fig. 3. As a consequence, the emission spectrum of a sample can be shifted in wavelength and altered in shape by the polarization properties of the monochromator. In modern spectrometers the monochromators can be calibrated using standard lamps and polarizers, the information stored in the memory of the control computer, and the detected intensities corrected at the data processing stage of the experiment. Most manufacturers also provide data sheets describing monochromator performance, and use can be made of such data for approximate corrections to the measured spectra. Care must be taken with polarization anisotropy measurements. Thin-film polarizers have absorption properties which are strongly wavelength-dependent. Precise corrections can be made using computer-controlled facilities with provision for computerized data processing. However, it is preferable to use a Glan-Thompson prism made from quartz or calcite which has good transparency from the ultraviolet into the infrared. Furthermore, the polarization properties are not wavelength-dependent. In general terms, the light signal is detected using a photomultiplier tube in which the photon flux produces an electrical current that is proportional to the light intensity. The basis of the device is the photoelectric effect. Incident photons cause photoelectrons to be emitted from a photocathode with an efficiency dependent upon the incident wavelength. The photocathode is held at a high negative potential of 1000 to 2000 V. The photoelectrons are incident upon a series of dynodes which are also held at negative potentials in order to accelerate electrons toward the next dynode. Each photoelectron arriving at the first dynode chain causes the ejection of a further 10 to 20 electrons, depending on the voltage difference between photocathode and first dynode. This process of electron multiplication and consequent current amplification continues down the dynode chain until a current pulse arrives at the anode. Although the photomultiplier tube responds to individual photons, the individual current pulses are generally detected as an average signal. The anode current must be directly proportional to the light intensity. However, at wavelengths longer than the work function of the photocathode, the photomultiplier tube is no longer sensitive to the incident photons. Thus, different photocathodes are used in different wavelength ranges. For phototubes used in the ultraviolet region, quartz windows are used. For the ultraviolet-visible region (200 to 550 nm) a K-Cs bialkali photocathode may be used; such devices have high quantum efficiency, up to 25 percent between 350 to 500 nm, high gain, and low dark current. Typically, the operating anode currents are of the order of a few microamps, whereas the dark current is in the nanoamp range. A somewhat more useful device, in that the quantum efficiency is almost constant from 300 to 860 nm, uses a GaAs photocathode. For longer wavelength operation, 800 to 1250 nm, a germanium photodiode may be used. In other words, spectroscopic studies over a wide wavelength range may require several different photodetectors to be used. Techniques for correcting for the nonideal wavelength-dependent properties of the monochromator, polarizers, and photomultiplier tubes have been described at length by Lackowicz.2

31.8

INSTRUMENTS

Luminescence Spectrometers Using Phase-Sensitive Detection Where phase-sensitive detection techniques are used, the excitation intensity is switched on and off at a certain reference frequency so that the luminescence intensity is modulated at this same frequency. The detection system is then set to record signals at the reference frequency only. This effectively eliminates all noise signals except those closely centered on the modulation frequency. A typical luminescence spectrometer is shown in Fig. 4. The pumping light is modulated by a mechanical light chopper operating at frequencies up to 5 kHz. A reference signal is taken from the chopper to one channel of a lock-in detector. The magnitude and phase of the luminescence signal is then compared with the reference signal. Because of the finite radiative lifetime of the emission and phase changes within the electronics, the luminescence signal is not in phase with the reference signal. Hence, to maximize the output from the lock-in detector, the phase control of the reference signal is adjusted until input (luminescence) and reference signals to the lock-in detector are in phase. Of course, the phase of the reference signal may also be adjusted so that reference and luminescence signals are in quadrature giving zero output from the lock-in. This method of phase adjusting may enable one to separate the overlapping luminescence bands from different centers. In such experiments, the chopping frequency is adjusted so that there is an appreciable reduction in the luminescence intensity during the “off ” half-cycle. This effectively puts an upper limit on the rate at which the lock-in system can operate.

FIGURE 4 Schematic of a spectrometer for measuring luminescence spectra by phase-sensitive detection techniques.


31.9

The use of a mechanical chopper restricts the maximum modulation frequency to 5 kHz. Essentially, the mechanical chopper consists of a rotating blade of metal into which slots are cut at regular angular intervals. When the excitation beam is incident on the metal section, the excitation intensity at the sample is zero, and when on the slot, the sample receives the full intensity in the excitation beam. If the blade is rotated at a frequency of 1 Hz and there are n slots cut in the blade, then the excitation beam is essentially switched on/off at a rate of n Hz. Obviously, the modulation rate can be increased either by increasing the number of slots cut in the blade and/or by increasing the revolution rate. If the excitation is well-focused onto the chopper, then the modulation is in the form of a square wave with maximum modulation index M 1. Other modulators impose sinusoidal variations in the excitation intensity, at frequencies up to 50 MHz.2 There are various means for providing sinusoidal intensity variations, including Kerr cells and Pockels cells. Both types require high electric fields to obtain the desired modulation and such high driver voltages may interfere with the detection of weak signals. Kerr cells do not transmit ultraviolet light and so may only be used in the visible/near-infrared region. The Pockels cells may be used in the ultraviolet region as well as at visible and infrared wavelengths. They may also be operated at variable frequencies. However, since they require highly collimated light sources for efficient operation, they require a laser for excitation. The ultrasonic Debye-Sears modulator overcomes the experimental difficulties associated with both Pockel cells and Kerr cells. A vibrating quartz crystal is used to set up standing waves in a tank containing an ethanolwater mixture. (The crystal restricts the device to operate at the fundamental and one or two harmonic frequencies only.) The standing waves act as a closely spaced refractive index diffraction grating normal to the incident exciting radiation. A slit permits only the undiffracted light to pass to the sample. The result is a sinusoidally varying light intensity with about 50 percent modulation index. The emission signal is forced to respond to the modulated excitation at the same circular frequency Y as the excitation. However, the detected emission signal is delayed in phase by an angle H relative to the excitation, and with reduced modulation depth. The radiative lifetime may be calculated from the measured phase angle H and demodulation factor m. For a single exponential decay the appropriate relations are3 (8)

tanF WT R and m §©1 W 2T R2 ¶¸

1/ 2

(9)

Even with more complex processes, where several decaying species are present, phase angles and demodulation factors can be measured and used to calculate actual lifetimes.3,4 Phase-sensitive detection techniques may also be used to “time-resolve” overlapping absorption/ luminescence spectra with different decay characteristics. The phase-sensitive detector (PSD) yields a direct-current signal proportional to the modulated amplitude and to the cosine of the phase difference between the detector phase HD and the signal phase H, i.e., I (L , FD ) ms I o (L )cos(FD F )

(10)

where N is the wavelength, Io(N) is the steady-state excitation intensity, and ms is the source modulation index. Now suppose that there are two components A and B with lifetimes T A T B . The modulated emission measured with the PSD results in an unmodulated signal given by I (L , FD ) msA I oA (L )cos(FD F A ) msB I oB (L )cos(FD FB )

(11)

INSTRUMENTS

If the phase-control of the PSD is adjusted so that HD HB + 90°, then the second term in Eq. (11) is zero, and the output intensity is given by (12)

I (L , FD ) msA I oA (L ) sin (FB F A )

In other words, the emission output from species B has been suppressed. Species A can be suppressed at the detector phase angle HD HA + 90°. If we now scan the wavelength, then the consequence of Eq. (12) is that the steady-state spectrum of species A is recorded, i.e., I oA (L ), and conversely for species B. In the example given in Fig. 5a, the steady-state fluorescence of a mixture of indole and dimethylindole dissolved in dodecane is shown.5 With the detector phase angle set to 90° 9.7 and using a modulation frequency of 10 MHz in Fig. 5b, we resolve the indole emission with wavelength maximum at 306 nm. The phase angle of 9.7° corresponds to a radiative lifetime close to the isolated

1.0

Fluorescence intensity

Ethyl/Me2 indole mixture

0.5

0 (a) 1.0 Phase-sensitive fluorescence

31.10

9.7 + 90

28.6 – 90

Detector phase (deg)

Me2 indole 0.5

0 280

Indole 320 360 Wavelength/nm

480

(b) FIGURE 5 (a) Steady-state emission spectra of a mixture of indole and dimethylindole in dodecane and (b) shows the phaseresolved spectra of the indole and dimethylindole.5


!

31.11

!

#

"

FIGURE 6

Schematic representation of a luminescence excitation spectrometer.

methylindole molecules in dodecane (VR 5 ns). The suppression of the indole signal gives the dimethylindole spectrum with peak at 323 nm at a phase angle of 28.6 – 90°, giving the VR value of indole as 9.0 ns.

Luminescence Excitation Spectrometers Some inorganic solids have strong overlapping absorption bands due to nonluminescent centers, which completely overwhelm the absorption spectrum related to a particular luminescence center. These difficulties are overcome by excitation spectroscopy, Fig. 6, in which the intensity of the luminescence output is recorded as a function of the wavelength of the excitation beam. Strong emission at a particular excitation wavelength signals that the emitting center absorbs strongly at that wavelength. In this way it is possible to determine the shape and position of the

31.12

INSTRUMENTS

!

!

FIGURE 7 Luminescence and excitation luminescence spectrum of F2 centers in MgO.

absorption bands which excite the emission process. A low-resolution scanning monochromator is placed immediately after the chopper, and light from its exit slit is then focused onto the sample. This monochromator may be of focal length only 250 mm and have a grating of area 5 cm r 5 cm, ruled with only 600 lines per mm. Alternatively, it may be a double monochromator chosen to reduce stray light levels. In either case, an optical band pass filter may be used in front of the monochromator. Generally, the grating blaze is chosen to give high efficiency in the ultraviolet/ blue/green regions for the excitation monochromator (e.g., gratings blazed at 300 nm or 500 nm), whereas the emission monochromator is chosen to give high efficiency at visible and near-infrared wavelength (i.e., gratings blazed at 500 nm or 750 nm). With such an apparatus, it is possible to distinguish absorption transitions from several centers whose absorption bands partially or completely overlap. The example given in Fig. 7 shows the luminescence pattern emitted by F2 centers in magnesium oxide and the excitation spectrum associated with this emission. Other strong absorption bands due to Fe3 ions and F centers which overlap the F2-absorption bands are strongly discriminated against by selective detection of the F2-center luminescence.

31.5

PHOTOLUMINESCENCE DECAY TIME

Radiative Lifetime In order to measure the radiative lifetime of a transition it is necessary to use a sharp intense pulse of excitation in the absorption band together with some means of recording the temporal evolution of the luminescence signal. Suitable excitation sources include pulsed lasers, flash lamps, or stroboscopes. Laser systems may produce pulses of duration 0.1 to 100 ps; flash lamps and stroboscopes will produce pulses of order 10 8 s and 10 5 s, respectively. A possible spectrometer system is shown in Fig. 8. Usually the luminescence yield following a single excitation pulse is too small for good signal-to-noise throughout the decay period. In consequence, repetitive pulsing techniques are used together with signal averaging to obtain good decay statistics. The pulse reproducibility of the stroboscope is advantageous in the signal averaging process in which the output from the detector is sampled at equally spaced time intervals after each excitation pulse. If the pulse is repeated N times


31.13

!!"!"#

!

!

!

FIGURE 8 Spectrometer for measuring luminescence decay times.

$ !

FIGURE 9 Decay in the intensity of Cr3+ luminescence in MgO. Detection of the broadband luminescence at 790 nm shows two components, one fast (VR = 35 Ms) and one slow (VR = 11.4 ms). Detection of the R-line at 698.1 nm shows a single component with VR = 11.4 ms.

then there is an N1/2 improvement in the signal-to-noise ratio. If a multichannel analyzer is used, the excitation pulse is used to trigger the analyzer, and hence the time between pulses need not be constant. Of course the phase sensitive detection spectrometer may also be used to measure lifetimes, but only down to about 100 Ms. An illustration of the data obtainable using the stroboscope technique is shown in Fig. 9. The luminescence signal detected is the broadband emission with peak at 780 nm from Cr3 ions in orthorhombic symmetry sites in magnesium oxide measured at 77 K. At low Cr3 ion concentration, the radiative lifetime of this luminescence center is 35 Ms. These data show that the evolution of the intensity during the pulse-decay cycle is not necessarily in the form of a single exponential decay. On sampling the emission at times long relative to VR there is a component with characteristic decay time of 11.4 ms, which is the lifetime of Cr3 ions occupying octahedral symmetry sites in magnesium oxide and which emit a characteristic R-line emission at 698.1 nm. This result implies that excitation is being transferred from excited Cr3 ions in octahedral sites to Cr3 in orthorhombic sites. For rather faster decay processes (10 10 – 10 8 s), fast flashlamps are used to excite the luminescence. The gated flashlamps have extremely reproducible pulses, down to 0.8-ns width with repetition rates of up to 50 kHz.2 The usual gases for such lamps are hydrogen, deuterium, and air. Hydrogen has several advantages, not the least being the continuum output in the ultraviolet and visible ranges, with pulse profiles which are independent of wavelength. The combination of pulsesampling techniques and computer deconvolution of the decaying luminescence enables decay times to be measured down to 20 ps. However, judicious choice of photomultiplier tube and careful design of the photomultiplier dynode chain is necessary to eliminate signal noise. It is usual to use coincidence single-photon counting techniques to obtain good decay data.2

31.14

INSTRUMENTS

Picosecond and Sub-Picosecond Relaxation During the past two decades there have been quite remarkable developments in techniques for generating and measuring ultrashort pulses into the femtosecond domain. In semiconductors, a very wide range of ultrafast phenomena are being studied—electronhole plasma formation, exciton and biexciton formation dynamics, hot electron effects, phase-conjugate self-defocusing, and degenerate four-wave mixing. However, one very general optical phenomenon that may be addressed using ultrashort pulses involves nonradiative decay times in nonresonant fluorescence spectra. Such processes include ionic relaxations around a center in decaying from an excited state, sometimes including reorientations of anisotropic centers. Many picosecond phenomena, especially nonradiative decay processes, are studied by excite-and-probe techniques in which light pulses at wavelength N 1 are used to excite a phenomenon of interest, and then a delayed optical pulse at wavelength N 2 interrogates a change of some optical property of this phenomenon. Ideally, two sources of picosecond pulses at different, independently tunable wavelengths are required, which must be synchronized on the picosecond timescale. A convenient experimental system for studying vibrational relaxation at color centers and transition metal ions in ionic crystals is shown in Fig. 10.6 A mode-locked dye laser producing subpicosecond pulses at wavelength N1 is used both to pump in the absorption band and to provide the timing beam. Such pumping leads to optical gain in the luminescence band and prepares the centers in their relaxed state. The CW probe beam, collinear with the pump beam, operates at a longer wavelength, N 2. The probe beam and gated pulses from the pump laser are mixed in a nonlinear optical crystal and a filter allows only the sum frequency of the pump and probe beams, which is detected by a phototube. The photomultiplier tube actually measures the rise in intensity of the probe beam which signals the appearance of gain when the FA(Li)-centers have reached the relaxed excited state. The pump beam is chopped at low frequency to permit phase-sensitive detection. The temporal evolution gain signal is measured by varying the time delay between pump and gating pulses. Although specifically used by Mollenauer et al.6 to probe the relaxation dynamics of color centers, the spectrometer system shown in Fig. 10 is readily adapted to other four-level systems, including transition metal ions.

&( ' " ",

%"%! ,"(' #

)&& $ #%)%' ")'

%&&'

',%())

$)'"(' #

*") $$" $",-'

+ $ ',()"

%! $ #&" '

FIGURE 10 A sub-picosecond pump and probe spectrometer for measuring vibrational relaxation times in excited defects and transition metal ions.


31.6

31.15

POLARIZATION SPECTROMETERS

General Principles The absorbed intensity is sometimes dependent on the optical anisotropy of the sample. Whether or not a transition is allowed in a particular polarization is determined by examining the value of the square of the matrix element b \ K C j \ a ® , see “Optical Spectroscopy and Spectroscopic Lineshapes” (Chap. 10 in this volume), Eqs. (12) and (13), where a and b are the states involved in the transition, K·Cj is the appropriate component of the dipole operator summed over all j electrons involved in the transition. Optical transitions may be linearly or circularly polarized. For an electronic dipole transition, the dipole operator is Ke · CE where K e 3 je rj is summed over the j electrons and CE is the unit electric polarization vector parallel to the E-field of the radiation. The matrix element is evaluated using group theory, which shows how the symmetry properties of the states affect the transition rate.1 From this matrix element the selection rules of the transition are determined. The polarization of the radiation is defined in Fig. 11 by reference to the zˆ direction of the system, which itself is assumed to be parallel to an external perturbation (static electric or magnetic fields) or to unique symmetry direction in a crystal. For the R-and U-senses of linear polarization, the radiation travels in a direction perpendicular to zˆ with its electric field Cˆ E either parallel to zˆ (R-polarization) or perpendicular to zˆ (U-polarization). The electric dipole operators are then given by 3 jerj zˆ 3 je z j for R-polarization and 3j exj or 3j eyj for U-polarization. The xˆ and yˆ directions have been assumed equivalent. In C-polarization the radiation propagates along the unique symmetry axis, zˆ , with Cˆ E anywhere in the x-y plane: in this case the electric dipole operator is also 3j exj or 3j eyj. We define right circularly polarized (RCP) radiation as having electric (and magnetic) polarization vectors which rotate clockwise when viewed from behind the direction of propagation. For electric dipole absorption transitions, the electric dipole operator for RCP light propagating in the z direction is 3 je(x jy)j / 2. Accordingly, in the case of LCP light, where the sense of rotation is anticlockwise, the electric dipole operator is 3 je(x iy)j / 2. Polarized Absorption Although the selection rules of dipole transitions provide for polarized spectra, the optical spectra of atoms in the absence of external fields and of electronic centers with octahedral symmetry in solids are isotropic. Since the unit polarization vector, Cˆ E, has direction cosines cos C, cos D, and

FIGURE 11 Definitions of the senses of C-, R- and U-polarized light beams relative to a local symmetry axis.1

31.16

INSTRUMENTS

m

g a b

(a) e f Luminescence beam e

q

Luminescence beam

q

Incident beam

(b)

FIGURE 12 Showing (a) the orientation of the dipole moment μ relative to the x, y, and z axes and (b) two different geometrical arrangements used for polarized excitation luminescence spectroscopy.

cos I, where the angles C, D, and I are defined in Fig. 12a, the square of the electric dipole matrix element is | b | M x cos A M y cos B M z cos G | a ® |2

(13)

When the center has octahedral symmetry the cross terms in Eq. (13) are zero so that the squared matrix element becomes M x ® 2 cos 2 A M y ® 2 cos 2 B M z ® 2 cos 2 G

(14)

using M x ® b | M x | a ® with similar expressions for M y ® and M z ® . Since in octahedral symmetry Mx ®2 M y ®2 Mz ®2

(15)

| K Cˆ ® |2 becomes M x ® 2 and the strength of the transition is independent of the direction of the polarization of the incident radiation and the direction of propagation. In octahedral solids, the local symmetry of an electronic center may be reduced by the application of an external perturbation or internally through the presence of a nearby defect. In tetragonal symmetry with the z axis parallel to the symmetry axis, the transition probability is again given by


31.17

Eq. (14) but with M x ® 2 M y ® 2 w M z ® 2. Since the transition probability for radiation at polarization Cˆ E is then proportional to M x ® 2 (cos 2 A cos 2 B ) M z ® 2 cos 2 G A B cos 2G

(16)

where A and B are constants, the spectroscopic properties of the center are anisotropic. In terms of the experimental situation referred to in Fig. 12b, in C-polarization the angle I is always R/4 radians, and the intensity is proportional to A. For R-polarization, the angles are C R/4, D R/4 C, and I 0, and the intensity is proportional to B. Similarly, for U-polarization, the intensity is proportional to A. This shows that in tetragonal symmetry a rotation of the polarizer from the I 0 to I R/4 in, for example, the y-z plane determines the magnitudes of A and B. The linear dichroism D is then given by D (B A)/(A B). To illustrate these ideas, Fig. 13 shows the polarization of the 2 S1/2 l 2 P1/2 , 2 P3/2 lines of atomic sodium, i.e., the D1 and D2 absorption lines, in the presence of an applied magnetic field. The Zeeman splittings of energy levels are much smaller than the spin-orbit splitting between the 2P1/2 and 2P3/2 levels. The wave functions are labeled in Fig. 11 by the MJ-values: the relevant ClebschGordan coefficients and theoretical intensities of the transitions for linear and circular polarizations are shown in Fig. 13, as are the theoretical intensities of the 2 S1 2 l 2 P1/2 , 2 P3/2 right circularly polarized (RCP) and left circularly polarized (LCP) absorption transitions. The experimental pattern of lines for R- and U-polarizations are in excellent agreement with the predicted Zeeman pattern. An analysis of the polarization properties of the sample absorption requires that a polarizer be inserted in the light path immediately prior to the sample chamber. For accurate measurements of the absorption anisotropy, the polarizers must be accurately positioned relative to the beam and rotatable. The angle of rotation about the beam must be accurately indexed so that the orientationdependence of the anisotropy may be determined. The polarizer should also be removable since it is unnecessary for measurements with optically isotropic solids. A sample which has different absorption coefficients in different crystallographic directions is said to be dichroic. The dichroism is defined as D

A (P ) A (S ) 1 ¤ I (P ) I (S )³ A (P ) A (S ) l ¥¦ I (P ) I (S )´µ

(17a)

in the limit of small absorption coefficients. Although discussion has focused on radiative absorption transitions via electric dipole transitions, a similar analysis can be made for magnetic dipole transitions. In this case, the phase relationships between the magnetic fields Bx and By are exactly the same as those between Ex and Ey, and the magnetic dipole operator is K m C B ; where K m 3 j (e /2m) r (l 2s) j and CB is the unit vector along the direction of the magnetic field of the radiation. If the absorption transitions used to excite the luminescence are unpolarized, so too will be the resulting luminescence spectrum. However, as discussed above, the absorption spectrum of an atomic system may be made anisotropic by the application of an external field or by using polarized exciting radiation. The resulting emission spectrum will be correspondingly polarized. Absorption and luminescence spectra from optically isotropic solids can also be made anisotropic using similar techniques.

Polarized Absorption/Luminescence Just as the absorption spectra of free atoms and isotropic solids are unpolarized, so too are the luminescence spectra, at least when exciting with unpolarized radiation. This is shown by simple extensions to the arguments leading to Eq. (15) in which the electric dipole operators for luminescence are the complex conjugates of those for the appropriate absorption transitions. In practice, both absorption and emission properties are anisotropic. Although the host crystal may possess a cubic unit cell in which the electronic centers are anisotropic, a regular distribution of equivalent sites will

31.18

INSTRUMENTS

FIGURE 13 Zeeman splittings of 2S1/2 l 2P1/2, 2P3/2 levels of sodium. The electric dipole matrix elements and relative intensities of linearly and circularly polarized absorption transitions are compared with some experimental spectra.1


31.19

still result in isotropic spectra. The use of polarized absorption/luminescence techniques can reveal details of the local site anisotropy. Such methods have been discussed by Henderson and Imbusch1 and in more detail by Feofilev.7 To measure the effects of polarization on the absorption coefficient [i.e., A (v , Eˆ)] it is necessary to place a polarizer immediately before the beam splitter in the double-beam spectrophotometer, Fig. 1. In polarized luminescence measurements, linear polarizers are placed immediately before the sample in the absorption channel and just after the sample in the emission channel of a luminescence excitation spectrometer such as that shown in Fig. 6. The spectrometer may then operate in the “straight through” configuration or the emitted light may be collected in a direction at 90° to the direction of the excitation light, as illustrated in Fig. 13. Note that provision is made for rotatable polarizers in both excitation (S) and detection channels (H), and the measured emission signal will be a function of both S and H. The circular dichroism may be defined in an analogous manner to the linear dichroism, i.e., Eq. (17a). Since circular dichroism has a specific relevance to the Zeeman effect, we use Fig. 14a and consider circularly polarized absorption transitions which are excited between two Kramers doublets. With light propagating along the direction of the magnetic field, the selection rule is that 1> 2 – 1> 2

6

+

–

1> 2 – 1> 2

6 (a)

–

+

Wavelength, (b)

+

– (c) FIGURE 14 Circularly polarized dipole transitions excited between Kramers states.

31.20

INSTRUMENTS

U-polarized light induces $Ms 1 absorption transitions and U-polarized light induces absorption transitions in which $Ms 1. As a result of the Zeeman effect, the absorption peak splits into two overlapping bands (Fig. 14b) centered at different wavelengths with absorption coefficients C± in U- and U -polarizations that are different at particular frequencies. The peaks in the two oppositely polarized bands are separated in energy by (ge gg)OBB, where the g-values refer to the excited e and ground g states. The energy difference corresponds to a frequency splitting $ v ( g e g g )eB 4P m , which for ge gg 2.0 and B 1 T gives a separation between band peaks of 0.04 nm for a band centered at 500 nm. In a magnetic field, the difference [$C(v)] in the absorption coefficients for U and U circularly polarized light is referred to as magnetic circular dichroism (MCD). In the limit of small absorption coefficient the circular dichroism is

$ A (v )

2(I (v ) I (v )) l(I (v ) I (v ))

(17b)

where l is the sample thickness and I(v), I (v) refer to the transmitted intensities of the U and U circularly polarized light at frequency v. In most cases, a splitting of only 0.04 nm would be hard to resolve directly by Zeeman effect measurements on a broadband. However, this Zeeman structure may be resolved by measuring $C(v) as a function of magnetic field, as can be seen from a simple estimate. We approximate the MCD signal, $C(v) for a sample of thickness l, as the product of the magnetic splitting $v with the rate of change of the absorption coefficient with frequency which is given by dA (v )/dv y A (v o )/', for a symmetrical, structureless band. Hence $v

' $ A (v ) r l A (v o )

(18)

In a typical experiment, $C(v)l 105 and C(vo)l ~ 1, hence $v 105'. For a typical broadband, ' 0.25 eV 2000 cm 1 and Eq. (18) yields $v 0.02 cm 1 (i.e., $N ~ 0.05 nm) which is of the same order of magnitude as the Zeeman splitting calculated above. Although the intensity changes, which determine the magnitude of $C(v), may be quite small, they may be assumed very precisely using lock-in techniques. This is done very efficiently by replacing the circular polarizer in the excitation system by a stress-modulated quarter-wave plate, a device which transmits circularly polarized light, the polarization of which is switched between U and U at the vibration frequency of the plate, usually 50 kHz. Using this piezo-optic modulation, MCD signals as low as 10 6 can be measured.8 The MCD signal is strongly dependent on both frequency and temperature. Since at low temperatures the populations N± of the Ms = o1/2 levels of the spin 1/2 ground state are different for a system in thermal equilibrium, the MCD signal [Eq. (17b)] is given by § g M B¶ $ A (v )l A o (v )G($ v ) tan h ¨ B · © 2kT ¸

(19)

In this expression Co(v) and the sample thickness, l, are experimental constants and, in consequence, the MCD signal only varies through the Brillouin function for the s 1/2 ground state [i.e., tan h( g M B B /2k T )]. This MCD signal is paramagnetic, being strongest at high field and low temperature, and measurement of its magnitude probes the ground-state magnetization. In order to test Eq. (19) experimentally, it is best to work at either the positive or negative peak in Fig. 14 and so maximize the MCD signal. Having thus obtained a suitable MCD signal, its variation with temperature and magnetic field can then be measured. Excitation of the Kramers’ system in Fig. 14 with circularly polarized radiation of appropriate frequency results in the circularly polarized emission. The electric dipole operators for RCP and LCP emission are the complex conjugates of those


31.21

for absorption. The consequent change in emission intensity, Eq. (17), is referred to as the magnetic circular polarization (MCP).

Optically Detected Magnetic Resonance In optical absorption spectroscopy, electronic transitions (usually) out of the ground state may result in one of a rich tapestry of possible bandshapes, depending upon the strength of the electronphonon coupling. Photoluminescence measurements involve transitions which originate on an excited electronic state and frequently the terminal state is the electronic ground state. Overlapping absorption and luminescence bands can cause difficulty in assigning individual optical absorption and luminescence bands to particular optical centers. Since the lifetimes of excited states are in the range 10 3 to 10 8 s, it is no trivial matter to measure excited-state electron spin resonance using the microwave detection techniques pioneered in ground-state studies. Geschwind et al.9 developed techniques in which the excited-state ESR was detected optically. In favorable cases this method enables one to correlate in a single experiment ESR spectra in the ground state and in the excited state with particular optical absorption and luminescence spectra. The technique of measuring the effect of resonant microwave absorption on the MCD and/or MCP signal may be termed optically detected magnetic resonance (ODMR). In ODMR measurements involving the MCD signal, microwave-induced transitions between the Zeeman levels of the ground state are detected by a change in intensity of the absorption (i.e., MCD) spectrum. Electron spin resonance transitions in the excited state are signaled by microwave-induced changes in the MCP signal. Figure 15 is a schematic drawing of an ODMR spectrometer. There are three necessary channels: a microwave channel and channels for optical excitation and detection. The microwave system is relatively simple, comprising a klystron or Gunn diode operating at some frequency in the range 8.5 to 50 GHz, followed by an isolator to protect the microwave source from unwanted reflected signals in the waveguide path. The microwave power is then square-wave modulated at frequencies up to 10 kHz, using a PIN diode. A variable attenuator determines the power incident upon the resonant cavity, although for high-power operation a traveling-wave amplifier might be added to the waveguide system. The sample is contained in the microwave cavity, which is designed to allow optical access of the sample by linearly or circularly polarized light traveling either parallel or perpendicular to the magnetic field direction. The cavity is submerged in liquid helium to achieve as large a population difference as possible between the Zeeman levels. The magnetic field is provided either by an electromagnet (B0 2.0 T) or a superconducting solenoid (B0 6.5 T). Radiation from the sample is focused onto the detection system, which in its simplest form consists of suitable filters, a polarizer, and photomultiplier tube. A high-resolution monochromator may be used instead of the filters to resolve sharp features in the optical spectrum. The signal from the phototube is processed using a phase-sensitive detector, or alternatively using computer data collection with a multichannel analyzer or transient recorder. The recorded spectrum is plotted out using a pen recorder as a function of either magnetic field or photon energy (or wavelength). With such an experimental arrangement one may examine the spectral dependence of the ODMR signal on the wavelength of the optical excitation or on the wavelength of the detected luminescence by use of one of the two scanning monochromators. In order to carry out ODMR, microwave radiation of fixed frequency v is introduced while the optical wavelength is kept at the positive or negative peak in Fig. 14c. The magnetic field is then adjusted until the ESR condition, hv gOBB, is satisfied. Since ESR transitions tend to equalize the populations N and N , resonance is observed as a decrease in $C(v), and as the microwave power is increased, the MCD gradually tends to zero. In certain circumstances the ground-state spin polarization may be used to monitor excited-state ESR transitions because of the selectivity of the transitions induced by circularly polarized radiation. This measurement technique is an example of trigger detection where one microwave photon in absorption triggers the detection of one optical photon emitted. The resulting enhancement in sensitivity relative to the normal ESR technique is approximately in the ratio of optical to microwave frequency (i.e., 1015/1010 105). At x-band (y10 GHz), the ESR sensitivity is about 1010 spins per gauss linewidth so that ODMR sensitivity is of order 105

31.22

INSTRUMENTS

Attenuator

Klystron or Gunn diode Power supply PIN Modulation

Power supply

Reference signal

Recorder

Variable attenuator Lock-in detector

Cryostat and superconducting magnet

Preamplifier

Signal Lens Sample in microwave cavity

PMT Polarizer analyzer

Monochromator (2)

Wavelength scan drive

Power supply

Polarizer Lens

Monochromator (1)

Lens

Filter Light source

FIGURE 15 A schematic representation of a spectrometer for measuring optically detected magnetic resonance spectra via circularly polarized absorption or emission transitions.

atoms in the excited state. With the ODMR technique, one may gather information on a wide range of important solid-state processes including spin-lattice and cross relaxation, spin memory, energy transfer, electron-hole recombination, phonon bottlenecks, and spin coherence effects. A major attribute of the ODMR technique is illustrated in Fig. 16, showing the optical characteristics of the ODMR spectrum of F centers in calcium oxide.10 These spectra were measured at 18.7 GHz and 1.6 K with the magnetic field along a crystal 100® direction. A high-pressure xenon discharge lamp and monochromator (M1 in Fig. 15) set at 400 nm was used to excite the fluorescence, which was detected through monochromator M2. The spectrum consists of four equally spaced lines due to an S 1 state of a center with tetragonal symmetry. Then with the magnetic field set at the strongest ODMR line, the excitation wavelength is scanned using monochromator M1 (Fig. 15) over the visible and near-ultraviolet region. A single broad structureless excitation peak is observed


31.23

"

#

!

FIGURE 16 The ODMR spectrum of triplet state of F-centers in CaO.

at 400 nm corresponding to the 1 A1g l1T1u absorption band of the F center (Fig. 16). Subsequently, the excitation monochromator is set at the peak of this excitation band and the same magnetic field while the detecting monochromator (M2 in Fig. 15) is scanned over the fluorescence spectrum. This spectral dependence (Fig. 16) shows a sharp zero-phonon line at a wavelength of 574 nm with an accompanying broad vibronic sideband with peak at 602 nm. In a single experiment, a unique and unambiguous relationship is established between the ESR spectrum, absorption, and fluorescence bands of an intrinsic lattice defect.

31.7

HIGH-RESOLUTION TECHNIQUES Inhomogeneous broadening arises when individual atoms are distinguished by the frequency at which they absorb light. The absorption profile is then the sum of separate absorption lines. In atomic spectroscopy, the major source of the spectral linewidth is Doppler broadening; the frequency shift is ($v/v) ±(vz /c) due to an atom moving with velocity component vz towards () or away from ( ) the observer. At thermal equilibrium, a gaussian lineshape is observed because of the

31.24

INSTRUMENTS

Maxwell-Boltzmann velocity distribution. In solids, the distribution of internal strains is a source of inhomogeneous broadening. Because crystals contain imperfections, electronic centers experience crystal fields which vary slightly from site to site in the crystal; in consequence, zero-phonon lines may have linewidths of order 0.1 to 50 cm 1. The use of narrow-band laser excitation makes it possible to eliminate inhomogeneous broadening and to realize a resolution limited only by the homogeneous width of the transition, which in crystals can vary from kilohertz to gigahertz. This factor of 103 to 104 improvement in resolution enables the spectroscopist to carry out high-resolution studies of the physical properties and electronic structures of centers and of the mechanisms responsible for homogeneous broadening. Contributions to homogeneous width come from population dynamics and random modulation of the optical frequency by phonons and nuclear spins.

Saturated Absorption and Optical Holeburning The experimental basis of recovering the homogeneous width of an inhomogeneously broadened optical spectrum, so-called saturated absorption or optical holeburning (OHB) spectroscopy, is illustrated in Fig. 17. An inhomogeneously broadened line of width 'inh is produced by many narrow components of homogeneous width 'hom 'inh. Each component is centered at a different

FIGURE 17 Optical holeburning (OHB) and fluorescence line narrowing (FLN) of an inhomogeneously broadened spectroscopic line.1


31.25

Chopper Tunable laser

Saturating beam Probe beam

Sample

Lock-in amplifier

FIGURE 18 A spectrometer system for Doppler-free saturation spectroscopy.12

frequency within the inhomogeneous line profile. If a narrow laser line of frequency vL and bandwidth 'L < 'hom is incident upon an atomic assembly having an inhomogeneously broadened linewidth 'inh, the resulting absorption of laser radiation depletes only that subassembly of excited centers whose energies are within 'hom of the laser line frequency vL. Consequently, a “hole” is burned in the lineshape in the neighborhood of vL. Resolution of the homogeneous width requires that 'L 'hom 'inh. In holeburning spectroscopy, the narrow laser linewidth and high power make it possible to maintain a significant fraction of those atoms with transition frequency vL in the excited state, where they no longer contribute to the absorption at this frequency. To observe holeburning experimentally requires that ^5 percent of those centers within the pump laser bandwidth be transferred to the excited state. The first measurements of optical holeburning or saturated absorption spectroscopy in atoms were made by the Stanford group on the HC-line (n 2 l n 3) in hydrogen using a pulsed dye laser.11 A schematic diagram of an appropriate absorption spectrometer is shown in Fig. 18. A strong laser beam interacts with those atoms that are moving with the right velocity to Doppler and shift them into resonance. If the laser beam is intense enough, it tends to equalize the population in the two levels, thereby reducing the intensity. The hole burned in the absorption profile, which extends over the natural width of the transition, is probed by a second beam at the same frequency but lower intensity and traveling in the opposite direction. This beam interacts with atoms having the same velocity but in the opposite direction to the saturating beam. When the laser is tuned to line center, both pump and probe beams interact with atoms moving with zero longitudinal velocity. The probe beam then measures the reduced absorption caused by the saturating beam. In experiments using pulsed lasers, very high intensity is required to achieve saturation and hence there must be very tight focusing and overlap of pump and probe beam. In consequence, CW lasers are preferred in both gas-phase and solid-state spectroscopy. Saturated absorption measurements on atomic hydrogen have been reviewed by Ferguson and Tolchard12 and OHB in solids by Selzer13 and by Yen.14 To burn a hole in a narrow absorption line in crystals requires that the laser be focused onto the sample for periods of order 102 to 103 s, depending upon the specific system. When the laser excitation is switched off, the holes recover on some timescale characteristic of the physical process responsible for holeburning. For short-lived holes the exciting beam is divided using a beam splitter into pump and probe beams. The weaker probe beam passes through an optoacoustic modulator which scans it backward and forward over the hole.13 To observe long-lived holes, the sample is irradiated for a short time in the zero-phonon line with a few hundred milliwatts of single-mode dye laser light with a width of a few megahertz. The shape of the hole is then displayed by reducing the laser intensity to a few milliwatts and scanning the laser over the inhomogeneous line profile. Figure 19 shows an example of holeburning in the 601.28-nm line of Pr3 : LaCl3. The homogeneous width measured in this holeburning experiment is 'hom 10 MHz, which corresponds to a lifetime of 10 ns. There have been many reports of holeburning spectroscopy on transition metal ions, rare-earth ions, and color centers in inorganic materials. For rare-earth ions, holeburning with lifetimes determined by the nuclear spin relaxation processes have been reported to vary from 10 to 103 s. Many measurements are aimed at the mechanisms leading to the homogeneous width of optical transitions.

31.26

INSTRUMENTS

= 601.8 nm LaCl3 : 0.5% Pr3+

Absorbance

5 GHz

1 GHz

Photon energy FIGURE 19 Optical holeburning in the 601.28-nm line of Pr3+ ions in LaCl3. The hole was burned using some 200 W cm−2 of single-frequency laser light with a bandwidth of 2 MHz. The zero-phonon line has an inhomogeneous width of 7.5 GHz. (After Harley and Macfarlane, 1986, unpublished.)

In these cases, techniques have been developed for the detection of coherent transients (e.g., photon echo or free induction decay) because the measurements are made on the timescale of the dephasing and are not affected by spectral diffusion and other such processes.

Polarized Absorption Spectrometers Polarized absorption spectroscopy is a technique related to sub-Doppler absorption spectroscopy. However, in this case use is made of the circularly polarized absorption properties of atomic transitions. In the Wieman-Hansch experiment,11 the probe beam passes through crossed polarizers immediately before and after the sample. If the pump beam is unpolarized, the sample is optically isotropic and no light falls on the detector. However, if the s l p transitions are excited using RCP light, the pump beam induces optical anisotropy in the medium with which it interacts. In consequence, as pump and probe beams are tuned to line center so that both interact with the same class of atoms, the weak probe beam becomes slightly elliptically polarized and light is transmitted through the crossed polarizers. The advantage of the method is a factor of about 103 enhancement in sensitivity relative to saturation absorption spectroscopy. SubDoppler two-photon absorption spectroscopy is also much used in atomic physics.15 The selection rule for two-photon absorption is that $l 0 or 2. In consequence, for l 1 electron atoms S lS and S lD transitions are allowed.


31.27

Laser Stark Spectroscopy of Molecules Sub-Doppler resolution enhancement is also used in studying the heterogeneously broadened rotational/vibrational spectra of molecules. Such spectra are generally observed in the mid-IR region and are studied using a wide variety of gas lasers (e.g., N2O, CO, and CO2). Such laser lines are not usually in exact resonance with the particular molecular transition: laser and molecular transition are brought into register using a variable electric field to tune the molecular system into resonance. In general, parallel-plate Stark cells are used in which free-space propagation of the short-wavelength infrared radiation occurs. This makes it easy to use both perpendicular and parallel polarization configurations in the electric resonance experiments so that both $MJ 0 and $MJ ±1 transitions are observed. The subject of laser Stark spectroscopy has been discussed at length by Duxbury.16 A schematic intracavity laser Stark spectrometer is shown in Fig. 20; the same basic principles are obtained as with optical holeburning spectroscopy. The effects of the saturating laser field are confined to a narrow frequency region centered on the velocity component of those molecules whose absorption is Doppler-shifted into resonance. In a standing wave field, two holes are burned, one on either side of the line center, corresponding to molecules moving toward or away from the detector. The applied electric field is used to tune the two holes to line center where they coalesce to give a sharp dip in the absorption coefficient at line center. Since the resonance method relies on the use of an electric field for tuning, it is necessary both to generate high uniform fields and to study molecules with appreciable Stark tuning coefficients. In order to generate high electric fields, which may approach 90 kV cm 1, narrow electrode spacings from 1 to 4 mm are commonly used. With such narrow gaps, the plates must be flat to one or two fringes of visible light, and must be held accurately parallel. The gas pressure used must also be restricted to the low-pressure region below 100 mtorr. A potential difference of roughly 3000 V may be sustained without electrical breakdown across any gas at a pressure of 100 mtorr and below. The electric field is then modulated at some convenient frequency to permit the use of phasesensitive detection techniques. In order to get above the principal noise region of the electric discharge lasers used in the 5- and 10-μm regions and as pumps for the FIR lasers, it is necessary to use electric field modulation frequencies in the range from 5 to 100 kH. The amplitude of the electric field modulation used to detect the signals is usually small compared to the equivalent electric field linewidth of the transitions. The most common modulation waveform is sinusoidal.

!

!

!!

# $

"!

FIGURE 20 Schematic diagram of an intracavity laser Stark spectrometer. PSD stands for phase sensitive detector, DVM for digital voltmeter, HV for high voltage, and MOD for modulation source.16

Intensity

INSTRUMENTS

10

10

9

9

8

8

7

7

6

6

5

5

4

4

3

3

2

2

1

1

0

0 Frequency/MHz

Frequency/MHz

(a)

Intensity

31.28

(b)

10

10

9

9

8

8

7

7

6

6

5

5

4

4

3

3

2

2

1

1

0

0 Frequency/MHz

Frequency/MHz

(c)

(d)

FIGURE 21 Lineshapes which occur when detecting at harmonics of the modulation frequency when small-amplitude field modulation is used. Doppler-broadened line showing the Lamb dip. (a) 30-MHz scan of a partially saturated Dopplerbroadened line showing the Lamb dip; (b) 30-MHz scan with first-derivative detection; (c) 30-MHz scan with second-harmonic, second-derivative detection. The gain is increased by a factor of four from (a); and (d) 30-MHz scan with third-harmonic, third-derivative detection.

If the modulation amplitude is much smaller than the linewidth, detection at the fundamental modulation frequency results in a first derivative lineshape as in analogous electron spin resonance spectra. In order to remove the effects of sloping baselines produced by transitions with a slow Stark effect, it is common to use detection at either the second or third harmonic of the modulation frequency. Second-harmonic detection produces a second-derivative signal resembling a sharpened absorption line but with negative side lobes. Third-harmonic detection produces a third-derivative signal which resembles a sharpened first derivative, but which again possesses side lobes. Theoretical lineshapes are illustrated in Fig. 21. Second- and third-harmonic detection are particularly useful for


31.29

the observation of narrow saturation features free from background effects. The detectors used are quantum-limited liquid nitrogen cooled devices, PbSnTe or CdHgTe in the 10-μm region and InSb or Au doped Ge in the 5-μm region. In the far infrared, Golay cells have been used but in order to achieve a better signal-to-noise ratio it is necessary to use detectors cooled by liquid helium. Just as in atomic spectroscopy one may use atomic beam spectroscopy as an alternative to absorption saturations, so too may one use molecular beam systems in high-resolution studies of the rotational-vibrational spectra of molecules.

Fluorescence Line Narrowing Fluorescence line narrowing (FLN) is a technique complementary to that of OHB. It may also be understood by referring to Fig. 17. A narrow laser line is used to pump within the inhomogeneous linewidth 'inh. The laser interacts only with the subset of levels spanning the bandwidth of the laser 'L. These centers reradiate to some lower lying level, with a fluorescence linewidth much narrower than the inhomogeneous width. The fluorescence linewidth approaches the homogeneous width. In fact, for centers involved in a resonance fluorescence transition, the total FLN lineshape is a convolution of the laser lineshape and twice the homogeneous lineshape (once for the pump bandwidth and once for the fluorescence). The FLN linewidth ' is then usually written as ' 'L 2'h. Experimentally, FLN requires a little more sophistication than does holeburning spectroscopy. Of course, one still requires a stable, high-resolution laser. Care must be used in extracting the true homogeneous linewidth, especially for nonresonant fluorescence. Many of the experimental problems relative to solid samples are discussed in the review by Selzer,13 and numerous examples are given by Yen and Selzer.17 The CW FLN spectrum shown in Fig. 22 is for the Cr3 ion in aluminum oxide.18 The fluorescence lifetime is 3.4 ms at 4.2 K. Hence the homogeneous width is of the order 0.3 kHz. A direct-phonon relaxation process between the two 2E levels, 2A and E, separated in energy by 29 cm 1, broadens the homogeneous width to y130 kHz. In CW measurements, a homogeneous width in excess of 100 MHz was reported.18 The problem is relaxations due to super-hyperfine interactions with neighboring aluminum nuclei. The application of a dc magnetic field of only 40 mT has the effect of inhibiting relaxation due to local fields at the Cr3 ions due to the 17Al nuclear moments. A very considerable narrowing of the Cr3+ FLN spectrum is then achieved.

$!" !#"

!!"!"$

%

%

%

$!""&

FIGURE 22

FLN in the R1 transition of ruby.18

31.30

INSTRUMENTS

31.8

LIGHT SCATTERING Light-scattering experiments are now a routine feature in many optical laboratories. The first observations of light scattering by small particles were reported by Tyndall.19 Subsequently, theoretical work by Lord Rayleigh20 showed both that the scattered intensity varied as the fourth power of the frequency and that the scattering was due to molecules rather than dust particles. Many of the early studies were concerned with the depolarization of the light after being scattered by the dust-free atmosphere. Of course, in the prelaser era, sufficient light intensity could only be achieved by use of strongly condensing lenses to focus light onto the gas cell. Very great care was then necessary to obtain reliable depolarization measurements. Even in the laser era it is still essential to avoid any effects due to parasitic light which often plague light-scattering experiments. A significant early result from scattering of light by gases was that the scattered light intensity varied with the density of the gas being used as the sample. However, Lord Rayleigh discovered that the intensity scattered per molecule decreased by a factor of order 10 on condensation to the liquid phase. There is a somewhat smaller decrease in going from the liquid phase to the solid. Obviously, some scattering experiments become rather difficult in the solid state. The classical experimental geometry for studying Rayleigh scattering is in the 90° orientation for the scattered radiation. This is also the most useful orientation for Raman scattering in solids.21 One important feature of the structure of solids is the periodic disturbance of the crystal structure by the propagation of quantized elastic waves (i.e., phonons). Those elastic waves which travel at the velocity of sound (i.e., sonic waves) are essentially thermal density fluctuations in the elastic medium. Brillouin predicted that such fluctuations should give rise to fine structure in the Rayleigh scattered light when the Bragg coherence condition L1 2L p sin(F /2) is obeyed. Here N1 is the wavelength of light, Np is the wavelength of those phonons responsible for scattering the light, and H is the scattering angle. Because the scattering centers are in motion, the scattered light is frequency shifted by the Doppler effect. It is an easy matter to show that the Doppler shift, $v, is given by $v o v p o 2v1 (v/c) sin (F /2)

(20)

where vp is the frequency of the density fluctuations in the medium and v is the velocity of sound in the medium. For light in the visible region then, that part of the phonon spectrum probed by the Brillouin scattering is in the gigahertz frequency region. In addition, the Brillouin components are completely polarized for 90° scattering. Before the advent of lasers, the study of Brillouin scattering effects in solids was exceedingly difficult. It remains a technique more used in gases than in condensed media. Raman was one of numerous scientists engaged in research into light scattering during the decade 1920 to 1930. Much of his work was carried out using sunlight as a source. However, in experiments using monochromatic light, he observed in the spectrum of light scattered at 90° by liquid samples, new lines at wavelengths not present in the original light.21 The frequency displacement of these new lines from source frequency was found to be independent of the wavelength of the incident light. This was contrary both to fluorescence excitation and Brillouin scattering [Eq. (20)]; hence was born a new scattering phenomenon for which Raman was awarded the Nobel prize and which now bears his name. The frequency shifts in the Raman spectrum of a particular substance are related to but not identical to infrared absorption frequencies. In general, infrared transitions occur when there is a change in the electric dipole moment of a center as a consequence of the local atomic vibrations. The Raman lines occur when a change in polarizability is involved during atomic vibrations. This usually means that infrared transitions occur only between states of opposite parity whereas Raman transitions occur between states of the same parity. Thus the infrared and Raman spectra give complementary information about the vibrational spectra of spectroscopic centers. Raman scattering measurements have found wide application in condensed matter physics. The spectrometer systems have much in common with fluorescence spectrometers, although lasers provide the excitation source almost without exception. Single-frequency lasers (He-Ne, Ar, Ke) and tunable dye lasers and solid-state lasers have all been used. Most lasers provide a polarized output and it is necessary to modify this to allow the excitation polarization to be varied. The scattered


31.31

!

"

FIGURE 23 Raman spectra of F-centers in NaCl for different polarizations of the detected radiation. The shifts are measured relative to the 514.5 nm wavelength of the Ar+ laser.20

light is observed via a monochromator in a direction normal to the laser beam. Again, provision is made for the polarization of the scattered radiation to be analyzed. To permit observation closer to the laser line, double or triple monochromators are used to eliminate all traces of stray light. Furthermore, one must take trouble to separate out the Raman-scattered light from any fluorescence signal. Since the Raman signal is instantaneous, it is comparatively straightforward to recover the desired signal from the decaying fluorescence signal using time-resolution techniques. An example of the application of Raman spectroscopy in color center physics is shown in Fig. 23. The intensity of scattering versus wavelength shift from the Ar laser excitation is shown for F-centers in NaCl22 for which the longitudinal optic frequency is 270 cm 1. The major Raman-shifted spectrum occurs below 200 cm 1, showing that the vibrational interaction is due to ionic displacements close to the defect. These local modes have broad peak centers near Y 175 cm 1. A comparison of the polarized and unpolarized excitation spectra shows that the local mode scattering is supplemented by a lattice vibrational contribution covering much of the 0 to 500 cm 1 frequency shift.

31.9

REFERENCES 1. B. Henderson and G. F. Imbusch, Optical Spectroscopy of Inorganic Solids, Oxford University Press, Oxford, England, 1989. 2. J. R. Lakowicz, Principles of Fluorescence Spectroscopy, Plenum Press, New York, 1983. 3. H. Engstrom and L. F. Mollenauer, Phys. Rev. B 7:1616 (1973). 4. M. O. Henry, J. P. Larkin, and G. F. Imbusch, Phys. Rev. B 13:1893 (1976).

31.32

INSTRUMENTS

5. 6. 7. 8. 9. 10. 11.

12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22.

J. R. Lakowicz and H. Charek, J. Biol. Chem. 256:6348 (1981). L. F. Mollenauer, J. M. Wiesenfeld, and E. P. Ippen, Radiation Effects 72:73 (1983). P. P. Feofilov, The Physical Basis of Polarized Emission, Consultants Bureau, New York, 1961. J. C. Kemp, J. Opt. Soc. Amer. 59:915 (1966). S. Geschwind, R. J. Collins, and A. L. Schawlow, Phys. Rev. Lett. 3:545 (1959). P. Edel, C. Hennies, Y. Merle d’Aubigné, R. Romestain, and Y. Twarowski, Phys. Rev. Lett., 28:1268 (1972). The Stanford group made a number of experimental improvements; see for example T. W. Hänsch, I. S. Shakin, and A. L. Shawlow, Nature (Lond.) 235:63 (1972); T. W. Hänsch, M. H. Nayfeh, S. A. Lee, S. M. Curry, and I. S. Shahin, Phys. Rev. Lett. 32:1336 (1974); and C. E. Wieman and T. W. Hänsch, Phys. Rev. Lett. 36:1170 (1976). A. I. Ferguson and J. M. Tolchard, Contemp. Phys. 28:383 (1987). P. M. Selzer, in W. M. Yen and P. M. Selzer (eds.), Laser Spectroscopy of Solids I, Springer-Verlag, Berlin, 1981, p. 113. W. Yen, in W. M. Yen (ed.), Laser Spectroscopy of Solids II, Springer-Verlag, Berlin (1988). B. Cagnac, G. Grynberg, and F. Biraben, J. Phys. Paris 34:845 (1973). G. Duxbury, International Reviews in Physical Chemistry 4:237 (1985). W. M. Yen and P. M. Selzer, in W. M. Yen and P. M. Selzer (eds.), Laser Spectroscopy of Solids I, SpringerVerlag, Berlin, 1981. P. E. Jessop, T. Muramoto, and A. Szabo, Phys. Rev. B 21:926 (1980). J. W. Tyndall, Notes of a Course of Nine Lectures on Light, Royal Institution of Great Britain, Longmans, London, 1869. Lord Rayleigh (Hon. J. W. Strutt), Phil. Mag. x/i:107 (1871). C. V. Raman, Indian J. Phys. 2:1 (1928). J. M. Worlock and S. P. S. Porto, Phys. Rev. Lett. 15:697 (1965).

32 INTERFEROMETERS Parameswaran Hariharan School of Physics University of Sydney Sydney, Australia

32.1

GLOSSARY A a C d E F FSR I Ji( ) L m N n p R r T v N S P L [ Y

area amplitude ratio of peaks to valleys distance electric field finesse free spectral range intensity Bessel function length integer number of fringes refractive index optical path difference reflectance radius transmittance velocity wavelength angle frequency phase phase difference angular velocity 32.1

32.2

32.2

INSTRUMENTS

INTRODUCTION Optical interferometers have made possible a variety of precision measurements using the interference phenomena produced by light waves.1,2 This chapter presents a brief survey of the basic types of interferometers and discusses some of their applications.

32.3

BASIC TYPES OF INTERFEROMETERS Interferometric measurements require an optical arrangement in which two or more beams, derived from the same source but traveling along separate paths, are made to interfere. Interferometers can be classified as two-beam interferometers or multiple-beam interferometers according to the number of interfering beams; they can also be grouped according to the methods used to obtain these beams.

The Fizeau Interferometer In the Fizeau interferometer, as shown in Fig. 1, interference fringes of equal thickness are formed between two flat surfaces separated by an air gap and illuminated with a collimated beam. If one of the surfaces is a standard reference flat surface, the fringe pattern is a contour map of the errors of the test surface. Absolute measurements of deviations from flatness can be made by an intercomparison of three surfaces. Modified forms of the Fizeau interferometer are also used to test convex and concave surfaces by using a converging or diverging beam.3

The Michelson Interferometer The Michelson interferometer, shown schematically in Fig. 2, uses a beam splitter to divide and recombine the beams. As can be seen, one of the beams traverses the beam splitter three times, while the other traverses it only once. Accordingly, a compensating plate of the same thickness as the beam splitter is introduced in the second beam to equalize the optical paths in glass. With an extended source, the interference pattern is similar to that produced in a layer of air bounded by the mirror M1 and M 2` , the image of the other mirror in the beam splitter. With collimated light, fringes of equal thickness are obtained. The Michelson interferometer modified to use collimated light (the Twyman-Green interferometer) is used extensively in optical testing.4

FIGURE 1 The Fizeau interferometer.

INTERFEROMETERS

32.3

FIGURE 2

The Michelson interferometer.

The Mach-Zender Interferometer The Mach-Zehnder interferometer uses two beam splitters and two mirrors to divide and recombine the beams. As shown in Fig. 3, the fringe spacing and the plane of localization of the fringes obtained with an extended source can be controlled by varying the angle between the beams and their lateral separation when they emerge from the interferometer. The Mach-Zehnder interferometer has been used for studies of gas flows and plasmas.

The Sagnac Interferometer In the Sagnac interferometer, as shown in Fig. 4, the two beams traverse the same closed path in opposite directions. Because of this, the interferometer is extremely stable and easy to align, even with an extended broadband light source.

FIGURE 3

The Mach-Zehnder interferometer.

32.4

INSTRUMENTS

FIGURE 4

The Sagnac interferometer.

The Sagnac interferometer has been used for rotation sensing. When the interferometer is rotated with an angular velocity Y about an axis making an angle S with the normal to the plane of the interferometer, a phase difference L is introduced between the beams given by the relation

J (8PW A cos Q )/Lc

(1)

where A is the area enclosed by the light path, N is the wavelength, and c is the speed of light. Polarization Interferometers Polarization interferometers are used in interference microscopy.5 The Nomarski interferometer, shown schematically in Fig. 5, uses two Wollaston (polarizing) prisms to split and recombine the beams. If the separation of the beams in the object plane (the lateral shear) is small compared to the dimensions of the object, the optical path difference corresponds to the phase gradients in the test object. Grating Interferometers Gratings can be used as beam splitters in the Michelson and Mach-Zender interferometers. Such an arrangement is very stable, since the angle between the beams is affected only to a small extent by the orientation of the gratings. Figure 6 is a schematic of an interferometer that has been used to test fineground surfaces at grazing incidence utilizing two diffraction gratings to split and recombine the beams.6 Shearing Interferometers Shearing interferometers are widely used for optical testing, since they eliminate the need for a reference surface. As shown in Fig. 7, in a lateral shearing interferometer two images of the test wavefront are superimposed with a mutual lateral displacement, while in a radial shearing interferometer one of the images is contracted or expanded with respect to the other.7,8 The Fabry-Perot Interferometer The Fabry-Perot interferometer9 is used widely in high-resolution spectroscopy. It consists of two flat, parallel surfaces with highly reflecting, semitransparent coatings. If the surfaces are separated by a distance d and the medium between them has a refractive index n, the normalized value of the transmitted intensity at a wavelength N for rays traversing the interferometer at an angle S is I T (L ) T 2 /(1 R 2 2 R cos J )

(2)

INTERFEROMETERS

32.5

FIGURE 5

The Nomarski interferometer.

where T and R are, respectively, the transmittance and reflectance of the surfaces and J (4P /L )nd cos Q . With an extended source of monochromatic light, the fringes seen by transmission are narrow, concentric rings. The free spectral range (FSR), which corresponds to the range of wavelengths that can be handled without successive orders overlapping, is given by the relation FSR L L 2 /2nd

(3)

FIGURE 6 Grating interferometer used to test fine-ground surfaces at grazing incidence. (From Ref. 6.)

32.6

INSTRUMENTS

y

x

S (a)

d1

d2

(b) FIGURE 7 Fields of view in (a) lateral and (b) radial shearing interferometers.

while the width of the peaks at half the maximum intensity corresponds to a change in L given by the relation $ J 2(1 R)/R1/2

(4)

The ratio of the free spectral range to the width of the fringes at half maximum intensity is known as the finesse F, and is given by the relation F P R1/2 /(1 R)

(5)

Two useful variants of the Fabry-Perot interferometer are the multiple-passed Fabry-Perot interferometer and the confocal Fabry-Perot interferometer. With the conventional Fabry-Perot interferometer, the ratio of the intensity at the maxima to that at the minima between them is C [(1 R)/(1 R)]2

(6)

INTERFEROMETERS

FIGURE 8 ferometer.

32.7

Ray paths in a confocal Fabry-Perot inter-

and for typical values of reflectance (R y 0.95), the background due to a strong spectral line may mask a neighboring weak satellite. A much higher contrast factor may be obtained by double- or multiple-passing the interferometer.10,11 The confocal Fabry-Perot interferometer uses two spherical mirrors whose spacing is chosen, as shown in Fig. 8, so that their foci coincide. Any ray, after traversing the interferometer four times, then emerges along its original path.12 The confocal Fabry-Perot interferometer has a higher throughput than the plane Fabry-Perot interferometer and produces a uniform output field. It is, therefore, the preferred form for operation in a scanning mode by using piezoelectric spacers to vary the separation of the mirrors.

32.4 THREE-BEAM AND DOUBLE-PASSED TWO-BEAM INTERFEROMETERS Because of the sinusoidal intensity distribution in two-beam interference fringes, it is difficult to estimate their position visually to better than 1/20 of their spacing. However, it is possible to detect much smaller optical path variations using the intensity changes in a system of interference fringes.

Three-Beam Interferometers Zernike’s three-beam interferometer, shown schematically in Fig. 9, uses three beams produced by division of a wavefront at a screen containing three parallel, equidistant slits.13 In this arrangement, the optical paths of all three beams are equal at a point in the back focal plane of the lens L2. The two outer slits provide the reference beams, while the beam from the middle slit, which is twice as broad, is used for measurements. The intensity at any point in the interference pattern is then given by the relation (7)

I I 0[3 2 cos 2Y 4 cos Y cos J ]

where [ is the phase difference between the two outer beams, and L is the phase difference between the middle beam and the two outer beams at the center of the field. The intensities at adjacent maxima are equal only when L is an odd multiple of R/2. Two positions of the plane of observation can

FIGURE 9 Zernike’s three-beam interferometer.

32.8

INSTRUMENTS

be found that satisfy this condition, one inside and the other outside the focus, and any small change in the optical path of the middle beam can be measured from the shift in these positions. Three-beam fringes can also be produced with an optical system similar to that in the Jamin interferometer.14 Settings are made by means of a compensator in the middle beam and can be repeated to N/200 by visual observation, and to better than N/1000 with a photoelectric detector.15 Double-Passed Two-Beam Interferometers Fringes whose intensity is modulated in the same manner as three-beam fringes can be produced by reflecting the beams emerging from a two-beam interferometer back through the interferometer.16 In this case also, the intensity of the adjacent fringes is equal when the phase difference between the single-passed beams is

J (2m 1)P /2

(8)

where m is an integer. Measurements can be made with a precision of N/1000.

32.5

FRINGE-COUNTING INTERFEROMETERS One of the main applications of interferometry has been in accurate measurements of length using the wavelengths of stabilized lasers. Electronic fringe counting has become a practical technique for such measurements.17 The very narrow spectral line widths of lasers make it possible to use a heterodyne system. In one implementation of this technique, a He-Ne laser is forced to oscillate simultaneously at two frequencies, v1 and v2, separated by a constant frequency difference of about 2 MHz, by applying an axial magnetic field.18 These two waves, which are circularly polarized in opposite senses, are converted to orthogonal linear polarizations by a N/4 plate. As shown in Fig. 10, a polarizing beam splitter reflects one beam to a fixed reflector, while the other is transmitted to a movable reflector. A differential counter receives the beat frequencies from the photodetector Ds and a reference photodetector DR. If the two reflectors are stationary, the two beat frequencies are the same, and the net count is zero. However, if one of the reflectors is moved, the change in the optical path is given by the net count.

!

!

FIGURE 10 Heterodyne fringe-counting interferometer. (After Ref. 18 © Copyright Hewlett-Packard Company. Reproduced with permission.)

INTERFEROMETERS

32.9

32.6 TWO-WAVELENGTH INTERFEROMETRY If a length is known within certain limits, the use of a wavelength longer than the separation of these limits permits its exact value to be determined unambiguously by a single interferometric measurement. One way to synthesize such a long wavelength is by illuminating the interferometer simultaneously with two wavelengths N1 and N2. The envelope of the fringes then corresponds to the interference pattern that would be obtained with a synthetic wavelength (9)

Ls L1L2 /| L1 L2|

This technique can be implemented very effectively with a carbon dioxide laser, since it can operate at a number of wavelengths that are known very accurately, yielding a wide range of synthetic wavelengths.19 Two-wavelength interferometry and fringe-counting can be combined to measure lengths up to 100 m by switching the laser rapidly between two wavelengths as one of the mirrors of a TwymanGreen interferometer is moved over the distance to be measured.20

32.7 FREQUENCY-MODULATION INTERFEROMETERS New interferometric techniques are possible with laser diodes which can be tuned electrically over a range of wavelengths.21 One of these is frequency-modulation interferometry. Figure 11, shows a frequency-modulation interferometer that can be used to measure absolute distances, as well as relative displacements, with high accuracy.22 In this arrangement, the signal beam reflected from the movable mirror returns as a circularly polarized beam, since it traverses the N/8 plate twice. The reference beam reflected from the front surface of the N/8 plate interferes with the two orthogonally polarized components of the signal beam at the two detectors to produce outputs that vary in quadrature and can be fed to a counter to determine the magnitude and sign of any displacement of the movable mirror. To make direct measurements of the optical path difference, the frequency of the laser is ramped linearly with time by using a function generator to vary the injection current of the laser. An optical

Laser diode

Function generator

Fixed reflector (/8 plate)

Isolator

Polarizing beam splitter

Movable mirror

Detector

Detector

Fringe counter

FIGURE 11 (From Ref. 22.)

Storage CRO

Frequency-modulation interferometer for measurements of distance.

32.10

INSTRUMENTS

"$%"$"#%"

%#"

$$" $"

$#"

"$%" $"" $$" $"% '#"

"!%' %$"

$ &$$"

%$"

&"$" %$ (

"

"

&$

FIGURE 12 (From Ref. 24.)

Heterodyne interferometer for measurements of thermal expansion.

path difference p introduces a time delay p/c between the two beams, so that they produce a beat signal with a frequency f ( p/c)(dv /dt )

(10)

where dv/dt is the rate at which the laser frequency is varying with time.

32.8

HETERODYNE INTERFEROMETERS In heterodyne interferometers, a frequency difference is introduced between the two beams by means of two acousto-optic modulators operated at slightly different frequencies. The output signal from a square-law detector then contains an ac component at the difference frequency whose phase corresponds to the phase difference between the interfering light waves.23 Heterodyne techniques can also be used for measurements of very small changes in length.24,25 In the setup shown in Fig. 12, the frequency of a laser is locked to a transmission peak of a Fabry-Perot interferometer formed by attaching two mirrors to the ends of the sample. The beam from this slave laser is mixed at a photodetector with the beam from a stable reference laser. Changes in the separation of the mirrors can be evaluated from the changes in the beat frequency. A simple arrangement for measuring small displacements uses two diode lasers with external cavities. A displacement of the reflecting mirror of one cavity results in a change in the beat frequency.26

32.9

PHASE-SHIFTING INTERFEROMETERS In phase-shifting interferometers, the phase difference between the two beams in the interferometer is varied linearly with time and the values of intensity at any point in the interference pattern are integrated over a number of equal segments covering one period of the sinusoidal signal.

INTERFEROMETERS

32.11

FIGURE 13 Schematic of a compact optical system (the Mirau interferometer) used for phase-stepping interference microscopy.

Alternatively the phase difference between the two beams can be changed in a number of equal steps, and the corresponding values of intensity at each data point are measured and stored. In both cases, the values obtained can be represented by a Fourier series, whose coefficients can be evaluated to obtain the original phase difference between the interfering beams at each point. 27,28 Typically, four measurements are made at each point, corresponding to phase intervals of 90°. If I1, I2, I3, and I4 are the values of intensity obtained, the phase difference between the interfering beams is given by the relation tan J (x , y) (I1 I 3 )/(I 2 I 4 )

(11)

Phase-shifting interferometers are used widely in optical testing, since a detector array can be used in conjunction with a microcomputer to make measurements simultaneously at a large number of points covering the interference pattern. Figure 13 is a schematic of a compact optical system (the Mirau interferometer) used for phaseshifting interference microscopy. In this setup, the phase-steps are introduced by mounting the sample on a piezoelectric transducer (PZT) to which an appropriately varying voltage is applied. In a Fizeau interferometer, it is possible to use a laser diode as the light source and vary its output frequency.29 If the initial optical path difference between the beams in the interferometer is p, a frequency shift $v in the output of the laser diode introduces an additional phase difference between the beams $ J (2P p/v ) $ v

(12)

Another way of shifting the phase of a beam of light is by a cyclic change in its state of polarization. Since the resulting phase shift (the Pancharatnam phase) is very nearly achromatic, measurements can be made with white light, so that phase ambiguities at steps are eliminated.30

32.10

PHASE-LOCKED INTERFEROMETERS The output intensity from an interferometer depends on the phase difference between the beams. In phase-locked interferometers, any variation in the output intensity is detected and fed back to a phase modulator in the measurement path so as to hold the output intensity constant. The changes in the optical path can then be estimated from the changes in the drive signal to the phase modulator.31

32.12

INSTRUMENTS

Laser diode Object () Mixer () Amplifier

CCD array FIGURE 14 Schematic of a phase-locked interferometer using a laser diode source. (From Ref. 32.)

Drifts can be eliminated by using an ac amplifier. If the phase of one beam in the interferometer is subjected to a sinusoidal modulation $ J (t ) $ J sin W t

(13)

with an amplitude $L which is perpendicular to it. Now, if E|| is in phase with E>, the net magneto-optic effect will be a pure rotation of the polarization vector. On the other hand, if E|| and E> are 90° out of phase, then the reflected polarization will be elliptical, with no rotation whatsoever. In practice, the phase difference between E|| and E> is somewhere between 0° and 90°, resulting in a reflected beam which has some degree of ellipticity k, with the major axis of the polarization ellipse rotated by an angle Sk (relative to the incident E vector). By inserting a Soleil-Babinet compensator in the reflected beam’s path, one can change the phase relationship between E|| and E> in such a way as to eliminate the beam’s ellipticity; the emerging polarization then will become linear with an enhanced rotation angle. In this chapter, reference to Kerr angle implies the effective angle which includes the above correction for ellipticity.

35.22

INSTRUMENTS

Incident polarization Collimator

Polarizing beam-splitter

Objective

Laser

Substrate

Magnetic layer

Photodiode #2 Polarizing beam-splitter

Photodiode #1 Differential amplifier (a) PBS axis #1

Amplitude along axis #1:

1 ( + ) 2

Amplitude along axis #2:

1 ( – ) 2

45

PBS axis #2 (b)

FIGURE 17 Differential detection scheme utilizes a polarizing beam splitter and two photodetectors in order to convert the rotation of polarization to an electronic signal. E|| and E> are the reflected components of polarization; they are, respectively, parallel and perpendicular to the direction of incident polarization. The diagram in (b) shows the orientation of the PBS axes relative to the polarization vectors.

Differential Detection Figure 17 shows the differential detection system that is the basis of magneto-optical readout in practically all erasable optical storage systems in use today. The beam splitter (BS) diverts half of the reflected beam away from the laser and into the detection module. The polarizing beam splitter (PBS) splits the beam into two parts, each carrying the projection of the incident polarization along

PRINCIPLES OF OPTICAL DISK DATA STORAGE

35.23

one axis of the PBS, as shown in Fig. 17b. The component of polarization along one of the axes goes straight through, while the component along the other axis splits off to the side. If, upon reflection from the disk, the polarization did not undergo any rotations whatsoever, then the beam entering the PBS would be polarized at 45° to the PBS axes, in which case it would split equally between the two branches. Under this condition, the two detectors generate identical signals and the differential signal $S will be zero. Now, if the beam returns from the disk with its polarization rotated clockwise (rotation angle Sk), then detector 1 will receive more light than detector 2, and the differential signal will be positive. Similarly, a counterclockwise rotated beam entering the PBS will generate a negative $S. The electronic signal $S thus reproduces the pattern of magnetization along the scanned track.

Enhancement of the Signal-to-Noise Ratio by Multilayering The materials suitable for optical recording presently have very small Kerr angles (typically Sk 0.5°), with the result that the signal $S is correspondingly small. Multilayering schemes designed for the enhancement of the MO signal increase the interaction between the light and the magnetic medium by encapsulating a thin film of the MO material in an antireflection-type strucure. By providing a better index match between the MO film and its surroundings, and also by circulating the light through the MO film, multilayered structures manage to trap a large fraction of the incident light within the magnetized medium, and thus increase the Kerr rotation angle. These efforts inevitably result in a reduced reflectivity, but since the important parameter is the magneto-optically generated component of polarization, E> R sin Qk , it turns out that a net gain in the signal-to-noise ratio can be achieved by adopting the multilayering schemes. Reported enhancements of E> have been as large as a factor of 5. The popular quadrilayer structure depicted in Fig. 15 consists of a thin film of the MO material, sandwiched between two transparent dielectric layers, and capped off with a reflecting metallic layer. The entire structure, which is grown on a transparent substrate (through which light must travel to reach the MO film), is protected by a lacquer layer on the top. Numbers shown in Fig. 15 for the various layer thicknesses are representative of currently designed quadrilayers. The advantage of sending the light through the substrate is that the front facet of the disk stays out of focus during operation. In this way, small dust particles, finger prints, and scratches will not block the passage of light, and their deteriorating effects on the quality of the focused spot (which affects the integrity of both writing and readout) will be minimized. Any optical storage medium designed for removability ought to have the kind of protection that illumination through the substrate provides. The note of caution with substrate-side illumination is that, if the objective is simply designed for focusing in the air, then the oblique rays will bend upon entering the substrate and deviate from nominal focus, causing severe aberrations (see Fig. 7c). Therefore, the substrate thickness and refractive index must be taken into account in the objective’s design.

Sources of Noise in Readout The read signal is always accompanied by random noise. The effective noise amplitude (relative to the strength of the signal) ultimately limits the performance of any readout system. Part of the noise is thermal in nature, arising from random fluctuations of charge carriers within the photodiodes, resistors, and amplifiers. In principle, this source of noise can be mitigated by reducing the operating temperature of the device. However, since operating below the normal room temperature is not very practical for data storage systems, one must accept some of the limitations brought about by the thermal noise. Another source of readout noise is shot noise which, in classical language, is due to random arrival of photons at the photodetector(s). This noise is a permanent companion of the read signal and cannot be eliminated, but the system parameters may be adjusted to minimize its effect. One

INSTRUMENTS

10 Carrier (4 MHz)

0 –10 Spectral density (dBm)

35.24

–20 Second harmonic

–30 –40

(c) Erased track

(d) Recorded track

–50 –60 –70 (a) Thermal (b) Disk stopped 0

2

4 6 Frequency (MHz)

8

10

FIGURE 18 Spectra of the various noise components in magneto-optical readout.

property of the shot noise is that its rms amplitude is proportional to the square root of the available optical power Po. Since the signal strength is directly proportional to Po, it is clear that by increasing the read power of the laser one can enhance the ratio of signal-to-shot noise. There is, however, an upper limit on the laser read power, since the rise in the temperature of the medium will force the decline of its magneto-optical response. Other sources of noise in magneto-optical readout include the laser noise, the media noise, and the data noise. Laser noise is caused by amplitude/phase fluctuations of the electromagnetic radiation that comprises the optical beam. Media noise arises from variations in the reflectivity/ magneto-optic activity of the medium across its surface. The presence of grooves with rough and nonuniform edges can be a source of media noise as well. The term data noise refers to the unpredictable variations of the read signal arising from the imperfect shape/position of the recorded marks. Figure 18 shows the various components of noise in a typical MO readout system, as detected by a spectrum analyzer. In (a) the light beam is blocked and the trace on the analyzer screen is solely due to the thermal noise. The trace in (b) where the beam reaches the detectors but the disk is stationary shows the combined effect of thermal, shot, and laser noise. Trace (c) corresponds to reading an erased track on a spinning disk; the noise here includes all of the above plus the media noise. When a single-frequency tone was recorded on the track and the read-back signal was fed to the spectrum analyzer, trace (d) was obtained. The narrow pulse at frequency f0 is the first harmonic of the recorded signal; the corresponding second harmonic appears at 2f0. The noise level in this case is somewhat greater than that from the same track before the data was recorded. This difference is due to “data noise” and arises from jitter and nonuniformity of the recorded marks. A commonly used measure of performance for optical recording media is the carrier-to-noise ratio (CNR). This is the ratio of the signal amplitude at the carrier frequency f0 to the average level of noise. On a logarithmic scale the ratio is simply the difference between the two levels; in Fig. 18 the CNR is 53 decibels (dB).


35.25

35.8 MATERIALS OF MAGNETO-OPTICAL RECORDING Amorphous rare earth transition metal alloys are presently the media of choice for erasable optical data storage applications. The general formula for the composition of the alloy may be written (TbyGd1−y)x(FexCo1−z)1−x where terbium and gadolinium are the rare earth (RE) elements, while iron and cobalt are the transition metals (TM). In practice, the transition metals constitute roughly 80 atomic percent of the alloy (i.e., x 0.2). In the transition metal subnetwork the fraction of cobalt is usually small, typically around 10 percent, and iron is the dominant element (z 0.9). Similarly, in the rare earth subnetwork Tb is the main element (y 0.9) while the Gd content is small or it may even be absent in some cases. Since the rare earth elements are highly reactive, RE-TM films tend to have poor corrosion resistance and, therefore, require protective coatings. In a disk structure such as that shown in Fig. 15, the dielectric layers that enable optimization of the medium for the best optical/thermal behavior also perform the crucial task of protecting the MO layer from the environment. The amorphous nature of the material allows its composition to be continuously varied until a number of desirable properties are achieved. (In other words, the fractions x, y, z of the various elements are not constrained by the rules of stoichiometry.) Disks with large surface areas are coated uniformly with thin films of these media, and, in contrast to polycrystalline films whose grains and grain boundaries scatter the light beam and cause noise, these amorphous films are smooth and substantially noise-free. The films are deposited either by sputtering from an alloy target, or by cosputtering from multiple elemental targets. In the latter case, the substrate moves under the various targets and the fraction of a given element in the alloy film is determined by the time spent under the target as well as the power applied to that target. Substrates are usually kept at a low temperature (by water cooling, for instance) in order to reduce the mobility of deposited atoms and to inhibit crystal growth. Factors that affect the composition and short-range order of the deposited films include the type of the sputtering gas (argon, krypton, xenon, etc.) and its pressure during sputtering, the bias voltage applied to the substrate, deposition rate, nature of the substrate and its pretreatment, temperature of the substrate, etc. Ferrimagnetism The RE-TM alloys of interest in MO recording are ferrimagnetic, in the sense that the magnetization of the TM subnetwork is antiparallel to that of the RE subnetwork. The net magnetic moment exhibited by the material is the vector sum of the two subnetwork magnetizations. Figure 19 shows a typical temperature dependence of RE and TM magnetic moments, as well as the net saturation moment of the material. The exchange coupling between the two magnetic subnetworks is strong enough to give them the same critical temperature Tc. At T 0 K the rare earth moment is stronger than that of the transition metal, giving the material a net moment along the direction of the RE magnetization. As the temperature rises, thermal disorder competes with interatomic exchange forces that tend to align the individual atomic dipole moments. The decrease of MRE with the increasing temperature is faster than that of MTM, and the net moment Ms begins to approach zero. At the compensation point temperature Tcomp, the net moment vanishes. Between Tcomp and Tc the net moment is dominated by the TM subnetwork and the material is said to exhibit TM-rich behavior (as opposed to when T < Tcomp, where it exhibits RE-rich behavior). At the Curie temperature, thermal agitations finally break the hold of the exchange forces on magnetic dipoles, and the magnetic order disappears. Beyond Tc the material is in the paramagnetic state. The composition of the materials of magneto-optical storage is chosen so that Tcomp appears near the ambient temperature of Ta 300 K. Thus, under normal conditions, the net magnetization of the material is close to zero. Figure 20 shows a schematic drawing of the magnetization pattern in the cross section of a recorded track. Note that, although the net magnetization is nearly zero everywhere, the subnetwork moments have opposite orientations in adjacent domains. During readout the light from the GaAs laser interacts mainly with the transition metal subnetwork; thus, the MO Kerr signal is strong even though the net magnetization of the storage layer may be small. The magnetic

35.26

INSTRUMENTS

RE moment

Groove Land

TM moment

Recorded track

Substrate

Magnetic film

FIGURE 19 Temperature dependence of magnetization in amorphous RE-TM films. The moments of RE and TM subnetworks decrease monotonically, until they both vanish at the critical (Curie) temperature Tc. The net magnetization is the difference between the two subnetwork moments, and goes through zero at the compensation point Tcomp.

FIGURE 20 Schematic diagram showing the pattern of magnetization along a recorded track. The rare earth and the transition metal moments couple antiferromagnetically, so that the net magnetization everywhere is small. However, since the read beam interacts mainly with the TM subnetwork, the readout signal is not necessarily small.

electrons of iron and cobalt are in the 3d electronic shell, which forms the outer layer of the ion once the 4s electrons have escaped into the sea of conduction electrons. The magnetic electrons of Tb and Gd, in contrast, are within the 4f shell, concealed by the 5s, 5p, and 5d shells, even after the 6s electrons escape to the conduction band. A red or near-infrared photon is not energetic enough to penetrate the outer shell and interact with the magnetic 4f electrons, but it readily interacts with the exposed 3d electrons that constitute the magnetic moment of the TM subnetwork. It is for this reason that the MO Kerr signal in the visible and in the infrared is a probe of the state of magnetization of the TM subnetwork.

Perpendicular Magnetic Anisotropy An important property of amorphous RE-TM alloy films is that they possess perpendicular magnetic anisotropy. The magnetization in these films favors perpendicular orientation even though there is no discernible crystallinity or microstructure that might obviously be responsible for this behavior. It is generally believed that atomic short-range order, established in the deposition process and aided by the symmetry-breaking at the surface of the film, gives preference to perpendicular orientation. Unequivocal proof of this assertion, however, is not presently available due to a lack of high-resolution observation instruments. The perpendicular magnetization of MO media is in sharp contrast to the in-plane orientation of the magnetization vector in ordinary magnetic recording. In magnetic recording, the neighboring domains are magnetized in head-to-head fashion, which is an energetically unfavorable situation, since the domain walls are charged and highly unstable. The boundary between neighboring domains in fact breaks down into zigzags, vortices, and all manner of jagged, uneven structure in an attempt to reduce the magnetostatic energy. In contrast, adjacent domains in MO media are highly stable, since the pattern of magnetization causes flux closure, which reduces the magnetostatic energy.


35.27

Coercivity and the Hysteresis Loop Typical hysteresis loops of an amorphous RE-TM thin film at various temperatures are shown in Fig. 21a. These loops, obtained with a vibrating sample magnetometer (VSM), show several characteristics of the MO media. (The VSM applies a slowly varying magnetic field to the sample and measures its net magnetic moment as a function of the field.) The horizontal axis in Fig. 21a is the applied field, which varies from −12 to +12 kOe, while the vertical axis is the measured magnetic moment per unit volume (in CGS units of emu/cm3). The high degree of squareness of the loops signifies the following: 1. The remanent magnetization Mr is the same as the saturation magnetization Ms. Thus, once the sample is saturated with the help of an applied field, removing that field does not cause a reduction of the magnetic moment. 2. Transitions of the magnetization from up to down (or from down to up) are very sharp. The reverse field does not affect the magnetization until the critical value of Hc, the coercive field, is reached. At the coercive field the magnetization suddenly reverses direction, and saturation in the opposite direction is almost immediate.

300 Mr

Ms (emu/cc)

180

Ms 200 K

60

300 K 425 K

–60

Hc

–180 –300 –20

–10

0 H (KOe) (a)

10

20

275 375 Temperature (K) (b)

475

20

Hc (KOe)

15 10

Tcomp

5 0 75

175

FIGURE 21 (a) Hysteresis loops of an amorphous Tb27(FeCo)73 film, measured by VSM at three different temperatures. The saturation moment Ms, the remanent moment Mr, and the coercive field Hc are identified for the loop measured at T 200 K. (b) Coercivity as function of temperature for the above sample. At the compensation temperature, Tcomp 400 K, the coercivity is infinite; it drops to zero at the Curie point Tc 450 K.

35.28

INSTRUMENTS

The fact that Mr is very nearly equal to Ms in MO media is significant, since it means that the recorded domains remain fully saturated and exhibit maximum signal during readout. The coercivity Hc, in addition to being responsible for the stability of recorded domains, plays an important role in the processes of thermomagnetic recording and erasure. The coercivity at room temperature, being of the order of several thousand oersteds, prevents fields weaker than Hc from destroying (or disturbing) any recorded data. With the increasing temperature, the coercivity decreases and drops to zero at the Curie point, Tc. Figure 21b is the plot of Hc versus T for the same sample as in (a). Note that at the compensation point the coercivity goes to infinity, simply because the magnetization vanishes, and the external field does not see any magnetic moments to interact with. Above Tcomp the coercive field decreases monotonically, which explains the process of magnetization reversal during thermomagnetic recording: M switches sign once the coercivity drops below the level of the applied field.

35.9

CONCLUDING REMARKS In this chapter we have reviewed the basic characteristics of optical disk data storage systems, with emphasis on magneto-optical recording. The goal has been to convey the important concepts without getting distracted by secondary issues and less significant details. As a result, we have glossed over several interesting developments that have played a role in the technological evolution of optical data storage. In this final section some of these developments are briefly described.

Multiple-Track Read-Write with Diode Laser Arrays It is possible in an optical disk system to use an array of lasers instead of just one, focus all the lasers simultaneously through the same objective lens, and perform parallel read/write/erase operations on multiple tracks. Since the individual lasers of an array can be modulated independently, the parallel channels thus obtained are totally independent of each other. In going from a single-channel drive to a multiple-channel one, the optics of the system (i.e., lenses, beam splitters, polarization-sensitive elements, focus and track servos, etc.) remain essentially the same; only the lasers and detectors proliferate in number. Parallel track operations boost the sustainable data rates in proportion to the number of channels used.

Diffractive Optics The use of holographic optical elements (HOEs) to replace individual refractive optics is a promising new development. Diffractive optical elements are relatively easy to manufacture, they are lightweight and inexpensive, and can combine the functions of several elements on a single plate. These devices are therefore expected to help reduce the cost, size, and weight of optical heads, making optical drives more competitive in terms of price and performance. An example of the application of HOEs in MO systems is shown in Fig. 22, which shows a reflection-type element consisting of four separate holograms. The light incident on the HOE at an angle of 60° has a p component which is the original polarization of the laser beam, and an s component (parallel to the hologram’s grooves) which is the magneto-optically generated polarization at the disk. Nearly 90 percent of the s and 70 percent of the p polarization in this case are reflected from the HOE without suffering any diffraction (i.e., in the zero-order beam); they are captured in the differential detection module and yield the MO read signal. The four holograms deflect 20 percent of the incident p-polarized light in the form of first-order diffracted beams and bring them to focus at four different spots on a multielement detector. The two small holograms in the middle, H3 and H4, focus their first-order beams on detectors P3 and P4, to generate the push-pull


35.29

"& $&"! #$%

"$(!#$% !%

$$"$

!%

"'%!!%

%

"$(! #$%

FIGURE 22 Application of holographic optical element (HOE) in optical recording. (a) Configuration of MO head using a polarizationsensitive reflection HOE. (b) Geometrical relation between holograms and detectors. (After A. Ohba et al., SPIE Proceedings, Vol. 1078, 1989.)

tracking-error signal. The other two holograms, H1 and H2, send the diffracted beams to a fourelement detector in order to generate a focus-error signal based on the double knife-edge scheme. This HOE, therefore, combines the functions of beam splitting, masking, and focusing all in one compact unit.

Alternative Storage Media The GaAlAs lasers of the present optical disk technology will likely be replaced in the future by light sources that emit in the blue end of the spectrum. Shorter wavelengths allow smaller marks to be recorded, and also enable the resolving of those marks in readout. Aside from the imposition of tighter tolerances on focusing and tracking servos, operation in the blue will require storage materials

35.30

INSTRUMENTS

FIGURE 23 Direct overwrite in exchangecoupled magnetic multilayers involves the formation of domains that do not extend through the entire thickness of the magnetic medium.

that are sensitive to the short wavelengths. The current favorites for erasable optical recording media, the amorphous RE-TM alloys, may not be suitable for readout in the blue, since their magneto-optic Kerr signal drops at short wavelengths. A new class of magnetic materials which holds promise for future-generation device applications is the class of TM/TM superlattice-type media. The best-known material in this class is the Co/Pt-layered structure which consists of very thin layers of cobalt (typically one or two atomic layers), separated by several atomic layers of platinum. These polycrystalline films which have very small crystallites (grain diameter 200 Å) are prepared either by electron beam evaporation or by sputtering. Co/Pt films have large perpendicular magnetic anisotropy, good signalto-noise ratios in the blue, sufficient sensitivity for write/erase operations, and are environmentally more stable than the RE-TM media.

Direct Overwrite in Magneto-Optical Systems The problem of direct overwrite (DOW) on MO media has been the subject of extensive research in recent years. Some of the most promising solutions have been based on exchange-coupled magnetic multilayered structures. The basic idea of recording on exchange-coupled bilayers (or trilayers) is simple and involves the writing of reverse domains that do not extend through the entire film thickness, such as those shown schematically in Fig. 23. Such domains are under pressure from their excessive wall energies to collapse and can readily be erased with a moderate-power laser pulse. DOW on exchange-coupled media is thus achieved by writing (i.e., creating reverse domains) with a high-power pulse, and erasing (i.e., eliminating domains) with a moderate-power pulse. An external magnetic field is usually required for writing on such media, but neither the direction nor the magnitude of this field needs to change during erasure. Optical recording is an evolving technology, which will undoubtedly see many innovations and improvements in the coming years. Some of the ideas and concepts described here will hopefully remain useful for some time to come, while others may have a shorter lifetime and limited usefulness. It is the author’s hope, however, that they all serve as a stepping-stone to profound new ideas.

35.10

FURTHER INFORMATION Proceedings of the Optical Data Storage Conference are published annually by SPIE, the International Society for Optical Engineering. These proceedings document each year the latest developments in the field of optical recording. Two other conferences in this field are the International Symposium on Optical Memory (ISOM) whose proceedings are published as a special issue of the Japanese Journal of Applied Physics, and the Magneto-Optical Recording International Symposium (MORIS) whose proceedings appear in a special issue of the Journal of the Magnetics Society of Japan.


35.11

35.31

BIBLIOGRAPHY 1. A. B. Marchant, Optical Recording, Addison-Wesley, Boston, Massachusetts, 1990. 2. P. Hansen and H. Heitmann, “Media for Erasable Magneto-Optic Recording,” IEEE Trans. Mag. 25:4390– 4404 (1989). 3. M. H. Kryder, “Data-Storage Technologies for Advanced Computing,” Scientific American 257:116–125 (1987). 4. M. H. Kryder, “Data Storage in 2000—Trends in Data Storage Technologies,” IEEE Trans. Mag. 25:4358–4363 (1989). 5. A. E. Bell, “Optical Data Storage Technology: Status and Prospects,” Computer Design, pp. 133–146 (1983). 6. D. S. Bloomberg and G. A. N. Connell, “Magneto-Optical Recording,” Magnetic Recording, vol. II, C. D. Mee and E. D. Daniel (eds.), McGraw-Hill, New York, 1988, chap. 6. 7. S. Miyaoka, “Digital Audio is Compact and Rugged,” IEEE Spectrum, pp. 35–39 (1984). 8. G. Bouwhuis and J. J. M. Braat, “Recording and Reading of Information on Optical Disks,” Applied Optics and Applied Engineering, vol. IX, R. R. Shannon and J. C. Wyant (eds.), Academic Press, New York, 1983, chap. 3. 9. G. Bouwhuis, J. Braat, A. Huijser, J. Pasman, G. Van Rosmalen, and K. S. Immink, Principles of Optical Disk Systems, Adam Hilger Ltd., Bristol and Boston, 1985, chaps. 2 and 3. 10. H. H. Hopkins, “Diffraction Theory of Laser Read-Out Systems for Optical Video Discs,” J. Opt. Soc. Am. 69:4–24 (1979). 11. Special issue of Applied Optics on video disks, July 1, 1978. 12. M. Mansuripur, “Certain Computational Aspects of Vector Diffraction Problems,” J. Opt. Soc. Am. A6:786– 805 (1989). 13. P. Sheng, “Theoretical Considerations of Optical Diffraction from RCA Video Disk Signals,” RCA Review 39:513–555 (1978). 14. D. O. Smith, “Magneto-Optical Scattering from Multilayer Magnetic and Dielectric Films,” Opt. Acta 12:13 (1985). 15. P. S. Pershan, “Magneto-Optic Effects,” J. Appl. Phys. 38:1482–1490 (1967). 16. K. Balasubramanian, A. S. Marathay, and H. A. Macleod, “Modeling Magneto-Optical Thin Film Media for Optical Data Storage,” Thin Solid Films 164:341–403 (1988). 17. Y. Tomita and T. Yoshini, “Optimum Design of Multilayer-Medium Structures in a Magneto-Optical Readout System,” J. Opt. Soc. Am. Al:809–817 (1984). 18. K. Egashira and T. Yamada, “Kerr Effect Enhancement and Improvement of Readout Characteristics in MnBi Film Memory,” J. Appl. Phys. 45:3643–3648 (1974). 19. G. A. N. Connell, “Interference Enhanced Kerr Spectroscopy for Very Thin Absorbing Films,” Appl. Phys. Lett. 40:212 (1982). 20. K. Y. Ahn and G. J. Fan, “Kerr Effect Enhancement in Ferromagnetic Films,” IEEE Magnet. 2:678 (1966). 21. H. Wieder and R. A. Burn, “Direct Comparison of Thermal and Magnetic Profiles in Curie Point Writing on MnGaGe Films,” J. Appl. Phys. 44:1774 (1973). 22. P. Kivits, R. deBont, and P. Zalm, “Superheating of Thin Films for Optical Recording,” Appl. Phys. 24:273–278 (1981). 23. M. Mansuripur and G. A. N. Connell, “Laser-Induced Local Heating of Moving Multilayer Media,” Appl. Opt. 22:666 (1983). 24. J. P. J. Heemskerk, “Noise in a Video Disk System: Experiments with an (AlGa) As Laser,” Appl. Opt. 17:2007 (1978). 25. G. A. Acket, D. Lenstra, A. J. DenBoef, and B. H. Verbeek, “Influence of Feedback Intensity on Longitudinal Mode Properties and Optical Noise in Index-Guided Semiconductor Lasers,” IEEE J. Quant. Electron. QE-20:1163 (1984). 26. A. Arimoto, M. Ojima, N. Chinone, A. Oishi, T. Gotoh, and N. Ohnuki, “Optimum Conditions for the High Frequency Noise Reduction Method in Optical Video Disk Players,” Appl. Opt. 25:1398 (1986). 27. M. Ojima, A. Arimoto, N. Chinone, T. Gotoh, and K. Aiki, “Diode Laser Noise at Video Frequencies in Optical Video Disk Players,” Appl. Opt. 25:1404 (1986).

35.32

INSTRUMENTS

28. 29. 30. 31. 32. 33. 34. 35. 36. 37.

J. W. Beck, “Noise Consideration of Optical Beam Recording,” Appl. Opt. 9:2559 (1970). D. Treves and D. S. Bloomberg, “Signal, Noise, and Codes in Optical Memories,” Opt. Eng. 25:881 (1986). D. Treves, “Magneto-Optic Detection of High-Density Recordings,” J. Appl. Phys. 38:1192 (1967). B. R. Brown, “Readout Performance Analysis of a Cryogenic Magneto-Optical Data Storage System,” IBM J. Res. Dev. pp. 19–26 (1972). R. L. Aagard, “Signal to Noise Ratio for Magneto-Optical Readout from MnBi Films,” IEEE Trans. Mag. 9:705 (1973). M. Mansuripur, G. A. N. Connell, and J. W. Goodman, “Signal and Noise in Magneto-Optical Readout,” J. Appl. Phys. 53:4485 (1982). B. G. Huth, “Calculation of Stable Domain Radii Produced by Thermomagnetic Writing,” IBM J. Res. Dev., pp. 100–109 (1974). D. H. Howe, “Signal-to-Noise Ratio for Reliable Data Recording,” Proc. SPIE 695:255–261 (1986). K. A. S. Immink, “Coding Methods for High-Density Optical Recording,” Philips J. Res. 41:410–430 (1986). R. Hasegawa (ed.), Glassy Metals: Magnetic, Chemical and Structural Properties, CRC Press, Boca Raton, Florida, 1983.

INDEX

Index note: The f after a page number refers to a figure, the n to a note, and the t to a table. Abbe numbers, 29.36 and axial chromatic aberrations, 17.22 of binary optics, 23.6, 23.6f in gradient index optics, 24.3, 24.7 of molded microlenses, 22.12, 22.12t of reflective and catadioptric objectives, 29.17 Abbe sine condition, 29.36 and reflective and catadioptric objectives, 29.34 of stigmatic conditioning, 1.30–1.31, 17.10 Abbe-Porter experiments, 11.1 Abbe’s prisms, 19.3t, 19.7, 19.7f–19.8f ABC model, of surface finish, 8.14–8.15 Abel transform, 8.13 Abelès method, 12.11–12.12 Aberrated wavefronts, 2.12, 2.13 Aberration(s): in binary optics, 23.4–23.7, 23.5f, 23.6f chromatic, 1.91–1.92 defined, 1.28 defocus as, 1.85–1.86 and general aspheres, 29.3 higher-order, 29.37 in instrumental optics, 1.85 of point images, 1.85–1.92, 1.88f polarization, 15.35–15.37, 15.35f–15.37f pupil, 1.76 ray, 1.87–1.88, 1.88f Seidel, 29.38 spherical, 1.90, 29.7, 29.8, 29.15, 29.21, 29.37, 29.38 and stop position, 1.92 and stop size, 1.92 in systems with rotational symmetry, 1.89–1.90 third-order, 1.90–1.91, 29.38 transverse primary chromatic, 17.22 wavefront, 1.86–1.88, 1.86f Aberration theory, in gradient index optics, 24.3 Ablation, microscopes, 28.54

Absolute instruments, 1.29 Absorption cross section, 7.5 Absorption cross section, 31.3 Absorption index, 12.6 Acceptance (étendue), 1.22, 1.81, 13.7 Access time, for optical disk data, 35.6 Accessories, for cameras, 25.16–25.17, 25.18f Achromatic doublets (lenses), 17.22–17.25, 17.23f–17.25f, 17.24t Achromatic retardation plates, 13.48–13.52, 13.50f, 13.53t Acousto-optic cells, 11.8–11.9, 11.9f Acousto-optic correlators, 11.10–11.12, 11.11f Acousto-optic scanners, 30.44–30.45 Active autofocus systems, for cameras, 25.11–25.12, 25.12f Active devices: fabrication of, 21.14 for integrated optics, 21.25–21.31, 21.26f–21.31f Active layer removal, in PIC manufacturing, 21.19 Active pixel sensors, 26.2, 26.8–26.9, 26.8f Active scanning, 30.4 Active-passive transitions, in PICs, 21.19–21.20 Addition, as analog operation, 11.2 Aerial cameras, 25.20 Aerial images, 1.26 Aerosol Polarimeter Sensor, 15.38 Affine transformations, 1.57 Afocal Cassegrain-Mersenne telescope objective, 29.9 Afocal Gregorian-Mersenne telescope objective, 29.12 Afocal lenses, 18.1–18.22 for binoculars, 18.13–18.14, 18.14f catadioptric, 18.21–18.22, 18.22f Galilean, 18.15–18.17, 18.15f, 18.16f Gaussian, 1.45, 1.46f, 1.53–1.54, 1.53f, 1.54f

The combined index for all five volumes can be downloaded from www.HandbookofOpticsOnline.com. I.1

I.2

INDEX

Afocal lenses (Cont.): Gaussian analysis, 18.4–18.6 and focusing lenses, 18.2–18.4, 18.3f, 18.5f and optical invariant, 18.7 subjective aspects of, 18.6, 18.7f Keplerian, 18.7–18.14 and eye relief manipulation, 18.8–18.10, 18.9f, 18.10f field-of-view limitations in, 18.11 finite conjugate afocal relays, 18.11–18.12, 18.12f thin-lens model of, 18.7–18.8, 18.8f paraxial matrix methods, 1.70 for periscopes, 18.19, 18.19f reflecting, 18.19–18.21, 18.20f, 18.21f in relay trains, 18.17–18.19, 18.17f, 18.18f for scanners, 18.13, 18.13f for telescopes, 18.10–18.11, 18.11f Afocal magnification, 18.5–18.6 Afocal objectives, 29.9, 29.12, 29.29–29.30 Agile beam steering, 30.51–30.63 with decentered-lens and mircolens arrays, 30.57–30.60, 30.58f–30.60f, 30.62–30.63 with digital micromirror devices, 30.60–30.61 with gimbal-less two-axis scanning micromirrors, 30.61–30.62, 30.62f phased-array, 30.52–30.57, 30.53f, 30.62–30.63 Ahrens method of spar cutting, 13.12 Ahrens Nicol prisms, 13.16f, 13.17 Ahrens prisms, 13.14 Aircraft, synthetic aperture radar for, 11.6–11.7, 11.7f Airy disks: in confocal microscopy, 28.50 defined, 3.26 of DIC microscopes, 28.39 of microscopes, 28.17–28.19, 28.18f, 28.19f of solid-state cameras, 26.15 and vector diffraction, 3.33 Airy equation, 12.10 Airy pattern, 17.38 Airy-Drude formula, 16.5 Altenhof objectives, 29.32–29.33 Alternative Paul three-mirror objective, 29.28 Alvarez plates, 22.16 Alvarez-Humphrey plates, 22.37 Amici lenses, 17.10, 17.10f Amici prisms, 19.3t, 19.11, 19.12f, 20.6f Amplitude, of waves, 2.4, 2.5, 12.5

Amplitude division, interference by, 2.14, 2.19–2.28 and extended sources, 2.20 and Fizeau interferometers, 2.24–2.26, 2.25f and fringes of equal inclination, 2.20–2.22, 2.21f, 2.22f and fringes of equal thickness, 2.22–2.24, 2.23f and Michelson interferometers, 2.26–2.28, 2.26f–2.27f plane-parallel plate, 2.19, 2.20f, 2.30–2.33, 2.30f, 2.32f, 2.33f thin films, 2.24 Amplitude penetration depth, 12.5 Amplitude reflection coefficients, Fresnel, 12.7–12.8, 12.10 Amplitude scattering matrix, 7.10, 7.13 (See also Jones matrix) Amplitude transmission coefficients, Fresnel, 12.8 Amplitude-shift-keyed (ASK) transmission, 21.30 Analog optical signal and image processing, 11.1–11.20 Fourier transforms in, 11.3–11.5, 11.3f, 11.5f and fundamental analog operations, 11.2–11.3 incoherent processing, 11.17–11.20, 11.18f, 11.19f and spatial filtering, 11.5–11.6, 11.6f of synthetic aperture radar data, 11.6–11.8, 11.7f–11.8f of temporal signals, 11.8–11.12, 11.9f–11.11f of two-dimensional images, 11.12–11.17, 11.13f Analog transmission, 21.32–21.34, 21.33f, 21.34f Analytical signal representation, in coherence theory, 5.2–5.3 Analyzed states, of polarizers, 15.19 Analyzer vectors, 15.11 Anamorphic afocal attachments (anamorphosers), 18.16–18.17, 18.16f Anamorphic error control, 30.49, 30.50 Anastigmatic (term), 29.36 Anastigmatic objectives, 29.12–29.13 Angle characteristic function, 1.14–1.15, 1.15f, 1.17 Angle of incidence, 1.23, 1.39, 13.48 Angle-point characteristic function, 1.16, 1.17 Angular change, of optical beams, 30.5 Angular correction function, 5.6, 5.7f Angular magnification, 1.52, 1.78, 18.4 Angular scan, 30.28–30.29 Angular spectrum representation, 5.14–5.15, 5.14f

INDEX

Anisotropy, perpendicular magnetic, 35.26 Annealed proton exchange (APE) process: for fiber optic gyroscopes, 21.35, 21.36, 21.36t for LiNbO3 waveguides, 21.16–21.17 Anomalous diffraction, 7.5, 7.6f Antinodal points, of lens systems, 17.7 Antiprincipal points, of lens systems, 17.7 Aperture(s), 1.74–1.76, 1.75f circular diffraction of light from, 3.6–3.7, 3.6f, 3.7f, 3.9–3.11 Fraunhofer patterns for, 3.25, 3.26, 3.27f double-slit, 3.26–3.28, 3.27f, 3.28f image space numerical, 1.79 linear, 30.54 rectangular, 3.19–3.20 Fraunhofer patterns for, 3.25, 3.26 Fresnel diffraction from, 3.19–3.20, 3.20f uniformly illuminated, 30.9, 30.10, 30.10f Aperture, numerical, 1.79, 17.9 Aperture stops, 1.74, 1.75f, 17.8, 29.5, 29.36 Aperture-scanning microscopy, 28.53–28.54 Aplanatic (term), 29.36 Aplanatic lenses, 17.5, 17.11–17.12, 17.11f Aplanatic objectives, 29.11–29.13 Arbitrary phase profiles, 23.8 Arbitrary systems, paraxial matrix methods for, 1.67 Area-solid-angle-product, 1.22 (See also Étendue) Arrayed waveguide gratings (AWGs), 21.24 Aspherical surfaces: and axial gradients, 24.3 and reflective/catadioptric objectives, 29.3 in systems of revolution, 1.35 Astigmatic difference (term), 1.43 Astigmatism, 1.90, 1.91, 29.34, 29.37 Astronomical telescopes, 18.10 Astronomy, radio, 5.23 Atmospheric particles, scattering by, 7.2 Atoms: electronic structure of, 10.12–10.16, 10.13f–10.15f multielectron, 10.10–10.11 one-electron, 10.7–10.9, 10.8f, 10.9f spectra of, 10.3 Augmented resolution, 30.12–30.14, 30.13f Autofocus, of cameras, 25.11–25.15, 25.12f–25.15f Autofocus SLRs, 25.12–25.14, 25.13f

I.3

Automatic focusing, optical disks and, 35.12–35.14, 35.13f Automatic tracking, on optical disks, 35.14–35.17, 35.14f–35.16f Average degree of polarization (average DoP), 14.32–14.33 Axial astigmatism, 29.34, 29.37 Axial color, 1.91, 29.9, 29.37 Axial gradient lenses, 24.3–24.5, 24.4f Axial image point, 1.27 Axial (longitudinal) magnification, 1.28, 1.52, 17.5 Axicons, 11.7, 11.7f Axis wander, of prisms, 13.15 AxoScan Mueller matrix polarimeters, 15.33 Azimuth, 15.10, 16.16 Babinet compensators, 13.53–13.55, 13.54f Babinet principle, 3.9–3.11, 3.10f, 3.11f, 3.13 Babinet-Soleil compensators, 13.55–13.56, 13.55f Back focal length (BFL), of camera lenses, 27.2, 27.3f–27.5f, 27.7f–27.16f, 27.18f–27.22f, 27.25 Backscatter and backscattering: coherent, 9.14–9.15, 9.14f enhanced, 6.5–6.7, 6.5f Baker relays, 18.18, 18.18f Baker super-Schmidt objective, 29.21 Baker-Nunn objective, 29.22 Balmer A-spectra, 10.7–10.8, 10.8f Balmer B-transition, 10.9 Barrel distortion, 1.91 Barrel length (BRL), of camera lenses, 27.3f–27.5f, 27.7f–27.16f, 27.18f–27.22f, 27.25 Beam deviation and displacement, 19.2 Beam diffusers, 23.13 Beam propagation method (BPM), 21.8 Beam shapers and shaping: in binary optics, 23.13 for optical disks, 35.8, 35.9f, 35.10f Beam splitters, polarizing, 13.41–13.42 Beam steering (see Agile beam steering) Beam-splitter gratings, 23.11, 23.12 Dammann approach, for binary gratings, 23.12 Beam-splitter prisms, 13.18–13.22 Foster, 13.7, 13.18f, 13.21–13.22 Glan-Thompson, 13.18f, 13.22 Rochon, 13.7, 13.18–13.21, 13.18f, 13.24 Sénarmont, 13.7, 13.18, 13.18f, 13.21 Wollaston, 13.7, 13.18, 13.18f, 13.21, 13.24

I.4

INDEX

Becke line, 28.27 Beilby-layer polarizers, 13.28 Benes architecture, for switches, 21.34–21.35 Berremann calculus, 15.32 Bertrand lenses, 28.8, 28.44f Bertrand-type Feussner prisms, 13.23 Bessel functions, 7.12, 7.14–7.15, 17.38, 28.17 Bidirectional reflectance distribution function (BRDF), 8.4, 15.38–15.39, 15.38f–15.40f Bifocal lenses, 23.12, 23.12f Billet’s split lens, 2.16, 2.17f Binary optics, 23.1–23.17 fabrication of mask layout, 23.14–23.16, 23.15f, 23.15t micromachining techniques, 23.16, 23.16f, 23.17 and geometrical optics, 23.2–23.9 aberration correction, 23.4–23.7, 23.5f, 23.6f analytical models, 23.2–23.4, 23.6 micro-optics, 23.7–23.8, 23.7f–23.8f optical performance, 23.8–23.9, 23.9f, 23.10t and scalar diffraction theory, 23.10–23.13, 23.11t, 23.12f, 23.13f and vector diffraction theory, 23.13–23.14, 23.14f Binoculars, 18.13–18.14, 18.14f Binomial vectors, of space curves, 1.19 Biotar lenses, 17.28 Biplates, 13.56 Birefringence, 15.6, 15.41 Bistatic radar cross-section (RCS), in surface scattering, 8.3, 8.4 Blocks (optical disk data), 35.6–35.7 Bohr frequency condition, 10.4 Bohr’s theory of hydrogen, 10.3 Born series, 9.4 Born-Oppenheimer approximation, 10.19, 10.20, 10.22 Bound modes, of optical waveguides, 21.3 Brace half-shade plates, 13.57 Bragg cell spectrum analyzers, 11.9–11.10, 11.10f Bragg cells, 11.11–11.12, 11.19 Bragg diffraction, 11.9, 11.9f Bragg reflection filters, 21.30 Bragg reflectors, 21.8, 21.30 Bragg regime, 30.39–30.41, 30.41f, 30.42f Brashear-Hastings prisms, 19.3t, 19.25, 19.25f Bravais biplates, 13.56

Brewster angle: and Airy equation, 12.10 defined, 12.12 and extinction ratio, 12.18, 12.22, 12.22f, 12.23f, 12.24 and polarization, 12.15 Brewster angle prisms, 13.13 Brewster angle reflection polarizers, 12.16–12.18, 12.16f, 12.17f, 13.34–13.37, 13.34t–13.36t Brewster angle transmission polarizers, 12.18–12.24, 12.19t–12.20t, 12.21f, 13.37–13.39, 13.38t–13.39t Bright field microscopy, 28.25, 28.27–28.28, 28.27f Brillouin scattering, 31.30 Broadening, of lineshapes, 10.7 Brownian fractals, 8.9, 8.17 Brownian movement, 28.28 Buckbee Mears wire-grid polarizers, 13.31 Bunsen-Kirchhoff spectrometers, 20.5f Butterfly scanners, 30.50f, 30.51 Cable television (CATV), 21.2, 21.32–21.34 Caged compounds, in microscopy, 28.55 Calcite: double refraction in, 13.2–13.6, 13.2f–13.3f, 13.4t–13.5t Feussner prisms of, 13.23 Rochon prisms of, 13.20 Camera formula, for solid-state cameras, 26.13 Camera lenses: classification system for, 27.17, 27.23t, 27.24 design limitations of, 27.1–27.2 fish-eye, 27.6 inverted telephoto, 27.2, 27.6, 27.7f–27.14f extreme wide-angle, 27.6, 27.13f, 27.14f highly complex extreme speed, 27.6, 27.9f–27.12f very compact moderate speed, 27.6, 27.7f–27.8f SLR normal lenses, 27.2, 27.3f–27.4f telephoto lenses, 27.6, 27.13, 27.15f–27.16f wide-angle lenses, 27.2, 27.5f zoom lenses, 27.17, 27.20f–27.22f Cameras, 25.3–25.26 accessories for, 25.16, 25.17, 25.17f aerial, 25.20 and autoexposure, 25.10–25.11 and autofocus, 25.11–25.15, 25.12f–25.15f characteristics of, 25.3–25.4 clandestine, 25.21

INDEX

Cameras (Cont.): critical features of, 25.8, 25.9f display types, 25.6–25.7 endoscopic, 25.21, 25.21f features of, 25.8, 25.18, 25.19f film for, 25.5, 25.6 and flash, 25.16, 25.17f formats for, 25.18 high-speed, 25.21–25.22, 25.22f and image, 25.5 images from, 25.7 and instant pictures, 25.8 large-format, 25.18–25.20 and red eye, 25.16 and resolution of fine detail, 25.5–25.6, 25.6f sewer, 25.22–25.23 solid-state (see Solid-state cameras) stereo, 25.23–25.24, 25.23f streak, 25.24, 25.24f thermal imaging, 25.25 and time lag, 25.8–25.9, 25.9f underwater, 25.25 video, 25.7–25.8 view, 25.18–25.20, 25.19f for wide-angle photography, 25.25, 25.26f Canon EOS A2E camera, 25.14f, 25.15, 25.16, 25.15f Cantor sets, 8.9 Cardinal points, of lenses, 1.44, 17.7 Carl Zeiss prism system, 19.3t, 19.16, 19.16f Carlisle objectives, 29.8 Carlson, Chester, 34.1 Carrier effects, in integrated optics, 21.10–21.12 Carrier-to-noise ratio (CNR), 35.24 Cartesian coordinates, 1.20, 1.21 Cassegrain objectives, 29.6, 29.7 afocal Cassegrain-Mersenne telescope, 29.9 dual magnification, 29.9–29.10 with field corrector and spherical secondary, 29.8–29.9 Houghton-Cassegrain, 29.22–29.23 Mangin-Cassegrain with correctors, 29.24 reflective Schmidt-Cassegrain, 29.17 Schmidt-Cassegrain, 29.16–29.17 Schmidt-meniscus Cassegrain, 29.21 with Schwarzschild relay, 29.32 solid Makutsov-Cassegrain, 29.19 spherical-primary, with reflective field corrector, 29.9 three-mirror, 29.30

I.5

Cassegrainian telescopes, 18.21 Catadioptric (term), 29.37 Catadioptric Herschelian objective, 29.27 Catadioptric lenses: afocal, 18.21–18.22, 18.22f systems of, 1.9 Catadioptric objectives (see Reflective and catadioptric objectives) Cathode ray tube (CRT) monitors, 25.6–25.7 Cathode-ray tubes (CRTs), 30.4, 30.25–30.26 Catoptric systems, 1.9 Caustics, ray densities and, 1.88 Center, of afocal lens, 1.54 Channels, for wave propagation, 9.16 Characteristic functions (geometrical optics), 1.13–1.18 angle characteristic function, 1.14–1.15, 1.15f, 1.17 angle-point characteristic function, 1.16, 1.17 and expansions about rays, 1.16 and expansions about the axis, 1.16–1.17 ideal, 1.17–1.18 mixed, 1.13 paraxial forms of, 1.17 and paraxial matrices, 1.74 point characteristic function, 1.14 point eikonal, 1.14, 1.17 point-angle characteristic function, 1.15–1.17 Charge injection devices (CIDs), 26.6–26.7, 26.6f–26.8f Charge spectrographs, 34.8, 34.8f Charge-coupled devices (CCDs), 25.7, 26.3–26.5, 26.4f–26.6f Charged area development (CAD), in xerographic systems, 34.4 Chebyshev polynomials, 7.15 Chemical beam epitaxy (CBE), 21.17, 21.18 Chief rays, 1.75, 17.8, 29.20, 29.37 Chiolite, 13.42 Chiral particles, scattering by, 7.2 Cholesky decomposition, 14.41, 14.42 Chromatic aberration correction, 23.5–23.6, 23.6f Chromatic aberrations, 1.91–1.92 Chromatism, of axial gradients, 24.3–24.6 Circle (Zernike) polynomials, 1.90, 23.3 Circuits, in integrated optics, 21.21–21.31 for active devices, 21.25–21.31, 21.26f–21.31f for passive devices, 21.21–21.25, 21.22f–21.25f Circular analyzers, of polarized light, 15.18, 15.19

I.6

INDEX

Circular apertures: diffraction of light from, 3.6–3.7, 3.6f, 3.7f, 3.9–3.11 Fraunhofer patterns for, 3.25, 3.26, 3.27f Circular polarizers, 15.17–15.19 Circular scan, 30.16, 30.18f Clandestine cameras, 25.21 Clausius-Mosotti theory, 7.16 Cleaning, in xerographic systems, 34.10 Clebsch-Gordon coefficients, 31.17 Coated spheres, scattering by, 7.14 Coddington’s equations, 1.44 Coefficient of finesse (interference), 2.31 Coercivity, of optical disk data, 35.17n, 35.27, 35.27f, 35.28 Cogging, of streak cameras, 25.24 Coherence, 5.1–5.23, 6.2–6.13 analytical signal representation, 5.2–5.3 applications of, 5.22–5.23 in binary optics, 23.7, 23.8f classical, 5.1–5.2 coherence area, 5.3 and coherence functions, 5.4–5.9 angular correction function, 5.6, 5.7f complex degree of coherence, 5.4 complex degree of spectral coherence, 5.5 cross-spectral density function, 5.5 efficient sampling of, 6.10–6.12, 6.11f higher-order functions, 5.8–5.9 intensity, 5.7 mutual coherence function, 5.4 radiance, 5.8 radiant emittance, 5.7–5.8 radiant intensity, 5.8 spectrum and normalized spectrum, 5.5–5.6 coherence time, 5.3 complex degree of, 2.37, 5.4 and enhanced backscatter, 6.5–6.7, 6.5f and general linear systems, 6.3–6.4 and image formation, 6.9–6.10, 6.9f and interference, 2.13, 2.36–2.42 laser sources, 2.41–2.42, 2.42f Michelson stellar interferometers, 2.40–2.41, 2.40f mutual coherence function, 2.36–2.38, 2.36f spatial coherence, 2.38–2.40, 2.38f–2.39f temporal coherence, 2.41 and Koehler-illumination, 6.12–6.13, 6.12f and laser modes, 5.23 and Lau effect, 6.7–6.8, 6.8f of light sources, 5.9–5.13

Coherence (Cont.): and Lukosz-type super-resolving systems, 6.9–6.10, 6.9f measurements of coherence, 5.3–5.4 and noncosmological red shift, 5.23 and optical image enhancement, 11.14–11.17 partial, 2.38 and polarization effects, 5.22 propagation in, 5.13–5.19, 5.14f–5.16f, 6.4 and radio astronomy, 5.23 scalar field amplitude, 5.3 spatial coherence, 5.3 and speckle, 5.22 and spectral representation, 5.22 and spectrum of light, 5.19–5.22, 5.20f and statistical radiometry, 5.22 temporal, 2.41, 5.3 time averages in, 6.4–6.5 Coherence area, 5.3 Coherence length, 2.19 Coherence time, 5.3 Coherence volume, 5.3 Coherency matrix, 12.29–12.30, 14.41 Coherent arrays, scattering by, 7.2–7.3 Coherent backscattering, 9.14–9.15, 9.14f Coherent mode representation (spectrum of light), 5.20–5.21 Coherent optical image enhancement, 11.14–11.17 Coherent radiation, 30.2, 30.25–30.26 Coherent scattering, 7.3, 9.2, 9.3, 9.5–9.7, 9.6f Collimation, 35.8, 35.9f, 35.10f Collineation, 1.56–1.63 of conjugate lines, 1.59 of conjugate planes, 1.58–1.59 coordinate systems and degrees of freedom for, 1.57 equations of, 1.57–1.58 general properties of, 1.62–1.63 matrix representation of, 1.59–1.60 of rotationally symmetric lenses, 1.60–1.62, 1.62f Colocalization, single molecule high-resolution, 28.23 Color: axial, 1.91, 29.9, 29.37 in human visual system, 26.18 lateral, 1.91–1.92 in xerographic systems, 34.11–34.12, 34.11f–34.13f

INDEX

Color filter arrays (CFAs), in solid-state cameras, 26.18 Coma, 1.31, 24.7, 29.37 Coming’s glass molding process, 22.9, 22.9f Compact disks (CDs), 35.1–35.2, 35.5n Compensators, 13.53–13.56, 13.54f, 13.55f, 28.38 Complementary aperture screens, 3.9–3.11 Complementary metal-oxide semiconductors (CMOSs), 26.8–26.9 Complex amplitude, 2.4, 2.5, 11.2 Complex degree of coherence, 2.37, 5.4 Complex degree of spectral coherence, 5.5 Complex refractive index, 7.12–7.13, 12.5, 12.6 Composite retardation plates, 13.52, 13.53 Compound microscopes, 17.10 Compound mirror optics configurations, 30.15–30.16, 30.15f Conductance channels, in speckle patterns, 9.16 Conductive magnetic brush (CMB), 34.6 Conductivity, diffraction and, 3.32–3.33 Configurational coordinate model, of lineshapes, 10.22, 10.23f Confocal Cassegrainians (telescopes), 18.21 Confocal microscopy, 28.49–28.51, 28.49f, 28.51f Confocal parabolas, 18.20, 18.20f, 18.21 Conic constant (term), 1.34, 29.37 Conic mirrors, 29.3f, 29.4f Conical surfaces, in systems of revolution, 1.34–1.35 Conjugate lines, collineation of, 1.59 Conjugate matrices, 1.68–1.71, 1.73 Conjugate planes: collineation of, 1.58–1.59 in microscopes, 28.4–28.5 Conscopic imaging, 28.8–28.9 Conservation of étendue, law of, 1.22 Constructive interference, 2.7 Continuous wave (CW) dye lasers, 10.8 Contrast, in microscopy: bright field microscopy, 28.25, 28.27–28.28, 28.27f dark field microscopy, 28.28 Hoffman modulation contrast, 28.29 interference microscopy, 28.33–28.44, 28.35f, 28.37f, 28.38f, 28.40f, 28.42f, 28.43f and modulation transfer function, 28.24–28.25, 28.25f, 28.26f phase contrast, 28.28–28.29, 28.29f SSEE microscopy, 28.29, 28.30, 28.30f–28.33f, 28.33

I.7

Contrast transfer function (CTF), 4.7–4.8, 4.8f, 28.24 Cook objectives, 29.31 Coordinate systems: for aberrations of point images, 1.86 for collineation, 1.57 for Fresnel equations, 12.6–12.7, 12.7f left-handed, 12.6 for Mueller matrices, 14.19–14.20 Cornu’s spiral, 3.16–3.19, 3.18f Corotron, in xerographic systems, 34.2, 34.3f Correctors, for reflective and catadioptric objectives: aplanatic, anastigmatic Schwarzschild with aspheric corrector plate, 29.13 Cassegrain with spherical secondary and field corrector, 29.8–29.9 Mangin-Cassegrain with correctors, 29.24 Ritchey-Chretien telescope with two-lens corrector, 29.8 spherical-primary Cassegrain with reflective field corrector, 29.9 three-lens prime focus corrector, 29.10 Correlated double sampling (CDS), 26.11 Correlators, acousto-optic, 11.10–11.12, 11.11f Cosine condition, of stigmatic imaging, 1.30, 1.30f Cosine-to-the-fourth approximation, 1.81 Cotton polarizers, 13.21 Couder objective, 29.12 Coulomb repulsion, 10.10, 10.12, 10.16 Coupled-dipole method, 7.15 Couple-mode theory, in integrated optics, 21.8 Coverslip correction, 28.10–28.11, 28.12f, 28.13 Critical illumination, 28.7 Cross-spectral density function, 5.5, 5.9, 5.10, 5.16 Crystalline-quartz retardation plates, 13.46–13.48 Curie temperature, 35.25 Curvature: of space curves, 1.18–1.19 vertex (paraxial), 1.32–1.33 Curved surfaces, radial gradients with, 24.7 Cylinders, scattering by, 7.14 Cylindrical lenses, 22.45, 22.46f Cylindrical wavefronts, 3.13–3.21, 3.14f and Cornu’s spiral, 3.16–3.19, 3.18f, 3.19t and opaque strip construction, 3.20–3.21 from rectangular apertures, 3.19–3.20 from straight edges, 3.14–3.16

I.8

INDEX

Czerny-Turner monochromators, 31.6 Czerny-Turner mounts, 20.8f, 20.14t Dall-Kirkham objective, 29.8 Dammann approach, for binary gratings, 23.12 Dark decay, of xerographic photoreceptors, 34.3 Dark field microscopy, 28.28 Data noise, 35.24 Data rates, 30.6–30.8 Data-reduction equations, polarimetric, 15.14–15.15 Decentered lens arrays, for agile beam steering, 30.57–30.60, 30.58f–30.60f, 30.62–30.63 Decomposition: of conjugate matrices, 1.71–1.72 of Mueller matrices, 14.33 Cholesky decomposition, 14.41, 14.42 polar decomposition, 14.39–14.40 SVD, 15.25–15.27 Weyl’s plane-wave, 3.23 Deep reactive ion etching (DRIE), 22.7, 22.23, 30.62 Deep x-ray lithography (DXRL), 22.7 Defocus, 1.82, 1.83, 1.83f, 1.85–1.86, 1.90, 1.91 Degenerate integrated structures, 21.12 Degree of circular polarization (DoCP), 15.10 Degree of coherence, 2.37 Degree of linear polarization (DoLP), 15.10 Degree of polarization (DoP), 12.14–12.15, 15.9 Degree of polarization (DoP) surfaces and maps, 14.31–14.32, 14.32f Density (coherency) matrix, 12.29–12.30, 14.41 Depolarization: defined, 15.7 and diagonal depolarizers, 14.30 Mueller matrices for, 14.30–14.39 depolarization index, 14.32 generators of, 14.33–14.39, 14.36f, 14.37f and volume scattering, 9.16–9.17, 9.17f Depolarization index, 14.32 Depth of field, 1.84, 17.37, 28.22–28.23 Depth of focus, 1.84, 17.37, 17.37f, 28.22 Derivative matrices, 1.73 Derotation, of polygon scanners, 30.35–30.36 Destructive interference, 2.7 Development, in xerographic systems, 34.5–34.10, 34.5f–34.9f

Diagonal depolarizers, 14.30 Dialytes (lenses), 17.25, 17.25f Diamagnification, 1.23 Diamond turning, 22.15–22.18, 22.16t, 22.17f, 22.18f Diattenuation and diattenuators, 14.6, 15.7 linear, 14.8, 14.17 Mueller matrices for, 14.16–14.19, 14.16f Dichroic polarizers, 13.24–13.33, 13.26f, 13.27f coatings as, 13.28 measuring polarization of, 13.33 pyrolytic-graphite polarizers, 13.28–13.29, 13.29f sheet polarizers, 13.25–13.28 Dichroism, 15.19, 15.41, 31.17, 31.20, 31.21 Dielectric impermeability, 21.9 Differential detection, of optical disk data, 35.22–35.23, 35.22f Differential geometry, of rays, 1.19–1.21 Differential scattering cross-sections (DSCs), 7.8, 8.3, 8.4 Differential-interference contrast (DIC) microscopy, 28.27, 28.39–28.41, 28.40f Differential-phase-shift-keyed (DPSK) transmission, 21.30, 21.32 Diffraction, 3.2–3.3, 3.3f anomalous, 7.5, 7.6f Bragg, 11.9, 11.9f from circular apertures, 3.6–3.7, 3.6f, 3.7f and Cornu’s spiral, 3.16–3.17 of cylindrical wavefronts, 3.13–3.21, 3.14f Cornu’s spiral, 3.16–3.19, 3.18f, 3.19t opaque strip construction, 3.20–3.21, 3.21f from rectangular apertures, 3.19–3.20, 3.20f from straight edge, 3.14–3.16, 3.14f, 3.15f definition of, 3.6 from disks, 3.7–3.8 Fraunhofer, 3.24–3.28, 3.24f–3.26f Airy diffraction as, 28.17 conducting screens for, 3.33f and gratings, 20.3 Fresnel-Kirchhoff formula, 3.21, 3.22, 3.32 Green’s function, 3.22–3.23 Huygens-Fresnel construction, 3.4–3.13 Babinet principle, 3.9–3.11, 3.10f, 3.11f circular apertures and disks, light from, 3.6–3.9, 3.6f–3.9f Fresnel zones, 3.4–3.6 zone plates, 3.11–3.13

INDEX

Diffraction (Cont.): mathematical theory of, 3.21–3.29 diffraction grating, 3.28–3.29, 3.29f, 3.30f Fraunhofer diffraction, 3.24–3.28 Fresnel and Fraunhofer approximations, 3.23–3.24 Rayleigh-Sommerfeld, 3.9, 3.10, 3.23, 3.29 and resolution of microscopes, 28.19–28.22, 28.20f, 28.21f scalar diffraction theory, 23.10–23.13, 23.11t, 23.12f, 23.13f by spheres, 7.4 stationary phase approximation for, 3.29, 3.31–3.32 vector, 3.32–3.37, 3.32f–3.37f, 23.13–23.14, 23.14f Diffraction efficiency, 23.9, 23.9f, 23.10t Diffraction gratings, 3.28–3.29, 3.29f, 3.30f, 20.4 arrayed waveguide, 21.24 beam-splitter, 23.11, 23.12 binary, 23.13 dispersive gratings vs., 20.3–20.4 interference, 33.14 multiple beam, 2.28–2.29, 2.29f, 2.30f for PICs, 21.19 as polarizers, 13.30–13.33 Diffraction patterns, three-dimensional, 28.19–28.22 Diffraction-limited depth of focus, 28.22 Diffraction-limited lenses, 17.37–17.39, 17.41f–17.42f Diffraction-type polarizers, 13.30–13.33 Diffractive optics, 33.13, 35.28, 35.28f, 35.29 Diffusers, 6.5, 6.5f, 6.7, 23.13 Diffusion approximation, of radiative transfer, 9.11–9.12 Digital holographic microscopy (DHM), 28.42, 28.43 Digital light processing (DLP), 30.3, 30.60 Digital micromirror devices (DMDs), 22.23, 30.60–30.61 Digital transmission, in integrated optics, 21.31–21.32 Digitized Green’s function, 7.15, 7.16 Dioptric systems, 1.9 Dipole approximation, discrete, 7.16 Dipole model of light, 3.33–3.36, 3.37f Dirac delta function, 5.9, 5.12, 6.8 Dirac equation, for one-electron atom, 10.10 Dirac series, 30.8

I.9

Direct overwrite (DOW), of optical disks, 35.30, 35.30f Direct-vision prisms, 19.2 Discharged area development (DAD), 34.4 Discrete dipole approximation, 7.16 Discrete signals, incoherent processing of, 11.17–11.20, 11.18f, 11.19f Disk rotation speed, of optical disks, 35.5–35.6 Disks, as aperture screens, 3.7–3.11, 3.8f Dispersion, 20.1 Dispersive prisms and gratings, 20.1–20.15 configurations of, 20.4–20.15 diffraction gratings vs., 20.3–20.4 Eagle configuration, 20.7, 20.11f Ebert-Fastie configuration, 20.8, 20.12f Littrow configuration, 20.7f, 20.10 Paschen-Runge configuration, 20.7, 20.11f Pfund configuration, 20.8f, 20.10, 20.13f in spectrometers, 20.2–20.3, 20.3f in spectroradiometers, 20.1, 20.2f, 20.14t Wadsworth configuration, 20.5f, 20.8, 20.12f Displays, of cameras, 25.6–25.7 Distances: in Gaussian lenses, 1.51–1.53 hyperfocal, 1.85 Distortion, 29.37 barrel, 1.91 nonrectilinear, 27.6 of objectives, 29.6 pincushion, 1.91 pupil, 1.78 rectilinear, 27.6, 27.13f, 27.14f Distortion-free focusing lenses, 18.4 Distributed Bragg reflector (DBR) lasers, 21.25, 21.30, 21.32, 21.37, 21.37f Distributed feedback (DFB) lasers, 21.25, 21.29, 21.30, 21.32, 21.38, 21.42 Distributed index of refraction, 24.1 [See also Gradient index (GRIN) optics] Distributed-index planar microlenses, 22.26–22.31, 22.27f–22.30f, 22.27t, 22.31t Disturbance of wavefront, 3.5–3.6, 3.14–3.15, 3.14f Division-of-amplitude photopolarimeter (DOAP), 16.15, 16.15f, 16.16 Division-of-amplitude polarimeters, 15.5–15.6 Division-of-aperture polarimeters, 15.5 Division-of-wavefront photopolarimeter (DOWP), 16.14, 16.14f Doppler broadening, 10.2, 10.7, 31.23

I.10

INDEX

Doppler shift, 2.13, 5.23, 11.6, 31.30 Double Dove prisms, 19.3t, 19.10, 19.10f Double refraction, in calcite, 13.2–13.6, 13.4t–13.5t Double-beam spectrophotometers, 31.4–31.5, 31.5f Double-Gauss lenses, 17.27–17.28, 17.28f, 27.2 Double-pass monochromators, 20.9f Double-pass objective optics, 30.32–30.33, 30.32f Double-passed two-beam interferometers, 32.8 Double-reflection error control, 30.50–30.51, 30.50f, 30.51f Double-slit apertures, 3.26–3.28, 3.27f, 3.28f Doublets, achromatic (lenses), 17.22–17.25, 17.23f–17.25f, 17.24t Dove prisms, 19.3t, 19.9, 19.9f, 19.10, 19.10f Drude model, 21.10 Dual magnification Cassegrain objective, 29.9–29.10 Dual rotating retarder polarimeters, 15.16, 15.16f Dummond, D. G., 13.47 Duty cycle, for input/output scanning, 30.14 Dynamic range, of solid-state cameras, 26.11, 26.14 Dynamic scattering, 9.7–9.8, 9.7f Dyson interference microscopes, 28.41, 28.42, 28.42f Dyson lenses, 18.21, 18.22, 18.22f Eagle configuration, of dispersive prisms, 20.7, 20.11f Ebert-Fastie configuration, 20.14t Ebert-Fastie configuration, of dispersive prisms, 20.8, 20.12f Eccentric field arrangement, of lenses, 18.22 Eccentricity, 1.34, 15.10 Edge response, OTF and, 4.7 Effective focal length (EFL): of camera lenses, 27.3f–27.5f, 27.7f–27.16f, 27.18f–27.22f, 27.25 of Gaussian lenses, 1.48 of microlenses, 22.10 Effective index method, 12.11, 12.12, 21.4 Effective medium theories (EMTs), 16.4, 16.9 Effective-medium representation, of volume scattering, 9.8 Efficiency factors, for scattering by particles, 7.5, 7.6f E-folding distance, 7.13 Eigenpolarization, 15.7

Eikonals, 1.12–1.14 Einstein coefficients: for spontaneous emission, 31.3 for stimulated absorption, 10.6 Eisenburg and Pearson two-mirror, three reflection objective, 29.25 Elastic scattering, 7.3 Electric fields, 2.3–2.4, 3.2, 3.3 Electromagnetic dipole model of light, 3.33–3.36, 3.37f Electromagnetic theory, 12.4 Electronic holography, 33.9–33.14, 33.11f–33.13f Electronic structure, of atoms, 10.12–10.16, 10.13f–10.15f Electronic-speckle pattern interferometry (ESPI), 33.9–33.13, 33.11f–33.13f Electrons: lifetimes of, 10.6 in multielectron atoms, 10.10–10.11 in one-electron atoms, 10.7–10.9, 10.8f, 10.9f and optical spectra, 10.12–10.16, 10.13f–10.15f Electro-optic holography (EOH), 33.11–33.13, 33.12f, 33.13f Electro-optic (gradient) scanners, 30.45–30.48, 30.46f–30.48f Electro-optic tensors, 21.10 Electro-optical modulators, 15.23 Ellipsometers, 16.10–16.18, 16.10f–16.12f for azimuth measurements, 16.16 four-detector photopolarimeters, 16.14–16.16, 16.14f–16.16f interferometric arrangements of, 16.18 normal-incidence rotating-sample, 16.18 null, 16.11, 16.12 perpendicular-incidence, 16.17–16.18, 16.18f photometric, 16.12–16.14, 16.13f, 16.14f return-path, 16.16–16.17, 16.17f rotating-analyzer, 16.13, 16.13f, 16.14 rotating-detector, 16.14, 16.14f Ellipsometric angles, 16.3 Ellipsometry, 16.1–16.21 about, 16.2–16.3, 16.2f applications, 16.21 conventions, 16.3–16.4, 16.3f defined, 15.7 generalized, 16.19 instrumentation for (see Ellipsometers) Jones-matrix generalized, 16.19 modeling and inversion, 16.4–16.9, 16.6f–16.8f, 16.10f

INDEX

Ellipsometry (Cont.): Mueller-matrix generalized, 16.19–16.21, 16.20f, 16.20t, 16.21f multiple-angle-of-incidence, 16.3 and polarimetry, 15.30–15.32, 15.31f, 15.32f spectroscopic, 16.3 transmission, 16.10 variable-angle spectroscopic, 16.3 Elliptical polarizers, 14.10, 15.17–15.18 Elliptical retarders, Mueller matrices for, 14.14 Ellipticity, of polarization elements, 15.10 Empty magnification limit, 28.17 Endoscopic cameras, 25.21, 25.21f Enhanced backscatter (EBS), 6.5–6.7, 6.5f Entrance pupil (lens), 1.76, 17.8, 18.4–18.6, 29.37 Entrance pupil distance (ENP) of camera lenses, 27.3f–27.5f, 27.7f–27.16f, 27.18f–27.22f, 27.25 Entrance window (lens), 17.9 Epi-illumination, in microscopes, 28.7–28.9, 28.7f, 28.8f Epitaxy, 21.17–21.20 Equivalent lenses, 17.20 Equivalent particles, in volume scattering, 9.6 Erasing, in xerographic systems, 34.10 Etching, for PICs, 21.18–21.19 Étendue, 1.22, 1.81, 13.7 Euler equations, 1.20 Excitonic effects, in integrated optics, 21.11 Exit pupil (lens), 1.76, 17.8, 18.6, 29.37 Exit pupil distance (EXP), of camera lenses, 27.3f–27.5f, 27.7f–27.16f, 27.18f–27.22f, 27.25 Exit window (lens), 17.9 Exiting beam, of polarimeters, 15.4 Exposure, in xerographic systems, 34.3 Extended boundary condition method (EBCM), 7.15 Extended objects, images of, 1.27 Extended sources, interference by, 2.20 Extinction cross section, 7.5, 7.8 Extinction paradox, 7.8 Extinction ratio: about, 12.14–12.15, 12.17 of polarizers, 12.21–12.24, 12.22f, 12.23f, 14.6 Extreme wide-angle lenses, 27.6, 27.13f, 27.14f Eye loupes, 17.9–17.10 Eye relief (ER), 18.8–18.10, 18.9f, 18.10f Eye space, of afocal lenses, 18.4 Eye space pupil diameter, 18.6

I.11

Eye tracking, in cameras, 25.14–25.15, 25.14f, 25.15f Eyepieces, in afocal systems, 18.7 Fabry-Perot etalon (cavity), 2.33, 2.33f Fabry-Perot interferometers, 32.4–32.7, 32.7f, 32.14 in dynamic wave meters, 32.17 in gravitational wave interferometers, 32.21, 32.21f as heterodyne interferometers, 32.10 and multiple beam interference, 2.33–2.36, 2.34f, 2.35f and wire-grid polarizers, 13.31 False polarization, 15.38 Faraday cages, 34.8, 34.8f Faraday rotators, 28.46 Faraday shutters, 25.22 Fast axis, 12.25, 15.7 Fax machines, 24.6 Feedthrough, of optical disk data, 35.14 Fermat’s principle, 1.11–1.13, 1.24 Ferrimagnetism, 35.25, 35.26, 35.26f Feussner prisms, 13.6, 13.7, 13.22–13.23, 13.22f Fiber interferometers, 32.14–32.16, 32.15f Fiber optic gyroscopes (FOG), 21.2, 21.35–21.37, 21.36f, 21.36t Fiber squeezers, 15.24 Fiber-to-fiber excess loss, 21.13–21.14 Fick’s law, 9.12 Field (lens), 1.74 Fields, of rays, 1.13 Field angles: of apertures, 1.75, 1.75f, 1.76 of Glan-Thompson type prisms, 13.12 Field curvature, 1.91, 29.7, 29.37 (See also Petzval curvature) Field flatness (aberration), 1.91 Field flattener lenses, 17.28 Field intensities, of waves, 2.5–2.6 Field lenses, 1.82, 1.82f, 17.10 Field of view (FOV), 1.74 in Keplerian afocal lenses, 18.11 for reflective and catadioptric objectives, 29.34–29.35, 29.35f, 29.36f in telescopes, 18.15–18.16, 18.15f Field size (lens), 28.13 Field stop, 1.74, 17.9, 29.5, 29.37 Field-effect-transistors (FETs), 21.38 Field-flattened Schmidt objective, 29.14–29.15

I.12

INDEX

Fill factor, of binary optics, 23.8 Film (camera), 25.5, 25.6 Filters: for coherent optical image enhancement, 11.14–11.17 of Mach-Zehnder interference, 21.23, 21.24f, 21.25 narrowband, 3.3 for pattern recognition, 11.12–11.14 spatial, 11.5–11.6, 11.6f Finesse: coefficient of, 2.31 of Fabry-Perot etalon, 2.34, 2.35 of interferometers, 32.6 Finish models, for surface scattering, 8.14–8.15 Finite conjugate afocal relays, 18.11–18.12, 18.12f Finite rays, 1.35 Finite-difference time-domain (FDTD) technique, 7.16–7.17 First-order optics, 1.29, 1.37 First-order retardation plates, 13.46 Fish-eye lenses, 27.6 Fixed-pattern noise (FPN), 26.11 Fizeau interferometers, 2.24–2.26, 2.25f, 32.2, 32.2f, 32.17 Flame hydrolysis (FHD), 21.13–21.14 Flash (camera), 25.16, 25.17f Flat-field objective optics, 30.30–30.33 Flat-medial-field objectives, 29.11 Fluorescence imaging with one nanometer accuracy (FIONA), 28.23 Fluorescence line narrowing (FLN), 10.18, 31.29, 31.29f Fluorescent microscopy, 28.48–28.49 Flux: of polarization elements, 15.9 and radiative transfer, 9.12–9.13, 9.13f F-number, 1.79, 17.9 Focal Gaussian lenses, 1.45–1.53, 1.47f conjugate equations of, 1.49–1.50, 1.49f magnifications and distances in, 1.50–1.53 nodal points of, 1.48, 1.48f, 1.49 principal focal points of, 1.47 principal planes of, 1.47, 1.47f, 1.48 reduced coordinates of, 1.53 Focal length: effective of camera lenses, 27.3f–27.5f, 27.7f–27.16f, 27.18f–27.22f, 27.25 of Gaussian lenses, 1.48 of microlenses, 22.10

Focal length (Cont.): in gradient index optics, 24.5–24.6 primary, 3.12 of surfaces, 1.39–1.40 in systems of revolution, 1.40 Focal lines, 1.58 Focal planes, 1.57, 1.70 Focal plane-to-focal plane conjugate matrices, 1.69 Focal points: front and rear, 1.40, 1.47 of lens systems, 17.7 principal, 1.47, 1.58 Focal ratio, 29.5, 29.37 Focal-plane-to-focal-plane geometry, 11.3–11.4, 11.3f Focus, range of, 1.85 Focus error signal (FES), 35.12–35.14, 35.13f Focusing, of optical disks, 35.9–35.12, 35.11f, 35.12f Form birefringence (term), 15.41 Form dichroism (term), 15.41 Förster resonance energy transfer (FRET), 28.23 45° half-wave linear retarders, Mueller matrices for, 14.12t 45° linear polarizers, 14.10t 45° quarter-wave linear retarders, Mueller matrices for, 14.12t Forward-looking infrared (FLIR) systems, 30.22, 30.23, 30.51 Foster prisms, 13.7, 13.18f, 13.21–13.22 Foucault prisms, 13.7, 13.17 Four-detector photopolarimeters (FDPs), 16.14–16.16, 16.14f–16.16f Fourier analysis, for radiative transfer, 9.11 Fourier domain filters, 11.13 Fourier transform lenses, 18.12 Fourier transform plane, 33.16–33.19, 33.18f, 33.20f Fourier transforms: in analog optical and image processing, 11.3–11.5, 11.5f and coherence theory, 5.20 and diffraction, 3.2, 3.24 for focal plane-to-focal plane matrices, 1.69 and transfer functions, 4.2 of uniformly illuminated linear aperture, 30.54 Four-powered-mirror lenses, 18.21, 18.21f Fractal model of surface finish, 8.14

INDEX

Fractals: brownian, 8.9, 8.17 Fresnel-Kirchhoff approximation for, 8.8–8.9 Frankford Arsenal prisms, 19.3t, 19.18–19.24, 19.18f–19.24f Frank-Ritter-type prisms, 13.6, 13.6f, 13.13–13.14 Franz-Keldysh effect, 21.11, 21.11f Fraunhofer approximation, 5.14, 5.16 Fraunhofer diffraction, 3.24–3.28, 3.24f–3.26f Airy diffraction as, 28.17 conducting screens for, 3.33f and gratings, 20.3 Fraunhofer theory, 7.11 Fredericksz cells, 14.31, 15.21 Free spectral range (FSR), of interferometers, 2.34, 2.35, 2.35f, 32.5 Frenet equation, 1.19 Frequency domain, 7.17 Frequency-modulation interferometers, 32.9, 32.9f, 32.10 Fresnel amplitude reflection coefficients, 12.7–12.8 Fresnel amplitude transmission coefficients, 12.8 Fresnel diffraction, 3.24 Fresnel equations, 12.6–12.13, 12.15 for absorbing materials, 12.10–12.13, 12.13f coordinate system for, 12.6–12.7, 12.7f for nonabsorbing materials, 12.8–12.10, 12.9f Fresnel intensity reflectivities, 8.6 Fresnel lenses, micro-, 22.31–22.37, 22.31f–22.33f, 22.35f–22.37f Fresnel losses, 22.10 Fresnel number, 3.25 Fresnel propagation kernels, 6.6 Fresnel reflection coefficients, 8.10 Fresnel rhombs, 13.45, 13.50f Fresnel Risley prisms, 19.27f Fresnel zones, 3.4–3.6, 3.4f for cylindrical wavefronts, 3.6f, 3.7f, 3.13–3.14, 3.14f and Fraunhofer diffraction, 3.25 opaque strip obstruction, 3.20–3.21, 3.21f Fresnel-Kirchhoff approximation, 8.5–8.9 Fresnel-Kirchhoff diffraction formula, 3.21, 3.22, 3.32 Fresnel’s biprism, 2.16, 2.16f Fresnel’s mirror, 2.16, 2.16f Fringe localization, 2.14 Fringe visibility (contrast), 2.7–2.8

I.13

Fringe-counting interferometers, 32.8, 32.8f Fringes of equal inclination, 2.20–2.22, 2.21f, 2.22f Fringes of equal thickness, 2.22–2.24, 2.23f Front focal lengths, 1.40 Front focal points, 1.40, 1.47 Front principal planes, 1.48 Front vertex distance (FVD), camera lens performance and, 27.3f–27.5f, 27.7f–27.16f, 27.18f–27.22f, 27.25 Full width at half maximum (FWHM) points, 30.9 Full-frame arrays, of CCDs, 26.3–26.4, 26.4f Fusing, in xerographic systems, 34.10 Gabor objective, 29.20 Galilean lenses, 18.7, 18.15–18.17, 18.15f, 18.16f Galilean telescopes, 18.15 Galvanometer scanners, 30.41–30.44, 30.43f, 30.44f Gauss points, of lenses, 1.44 Gaussian analyses, of lenses: afocal, 18.4–18.7, 18.7f focusing, 18.2–18.4, 18.3f, 18.5f Gaussian lenses, 1.44–1.55 afocal, 1.45, 1.46f, 1.53–1.54, 1.53f, 1.54f focal, 1.45–1.53, 1.47f conjugate equations of, 1.49–1.50, 1.49f magnifications and distances in, 1.50–1.53 nodal points of, 1.48, 1.48f, 1.49 principal focal points of, 1.47 principal planes of, 1.47, 1.47f, 1.48 reduced coordinates of, 1.53 notation for, 1.45t properties of, 1.54 systems of, 1.54–1.55 Gaussian optics, 1.29, 1.44 GDx Nerve Fiber Analyzer, 15.41 Gédamine, 13.9, 13.10, 13.11f, 13.20 General deviation prisms, 19.3t, 19.28, 19.28f–19.29f General Photonics, 15.24 Generalized ellipsometry (GE), 16.19 Generalized Lagrange invariant (étendue), 1.22, 1.81, 13.7 Geometrical optics: for aberrations of point images, 1.86–1.92, 1.86f and binary optics, 23.5–23.9, 23.5f–23.9f, 23.10t characteristic functions of, 1.13–1.18, 1.15f

I.14

INDEX

Geometrical optics (Cont.): for collineation, 1.56–1.65, 1.65f and conservation of étendue, 1.22 defined, 1.8 for Gaussian lenses, 1.44–1.55 afocal, 1.45, 1.46f, 1.53–1.54, 1.53f, 1.54f focal, 1.45–1.55, 1.47f–1.49f notation for, 1.45t properties of, 1.54 systems of, 1.54–1.55 and images about known rays, 1.43–1.44, 1.44f for imaging, 1.26–1.31, 1.30f at interfaces of homogeneous media, 1.23–1.26 lens sizes and fields in, 1.74–1.85 apertures, 1.74–1.77, 1.75f cosine-to-the-fourth approximation, 1.81 field lenses, 1.82, 1.82f fields, 1.74, 1.77, 1.84 F-number, 1.79 focus and defocus, 1.82–1.85, 1.83f irradiance, 1.80 power per pixel, 1.80 pupils, 1.76–1.79, 1.76f, 1.78f telecentricity, 1.83–1.84 total lens étendue, 1.81 vignetting, 1.81, 1.81f, 1.82 paraxial matrix methods, 1.65–1.74 and rays, 1.8–1.13 in heterogeneous media, 1.18–1.22 paths of, 1.10–1.13 and skew invariant, 1.23 of systems of revolution, 1.32–1.43 paraxial optics of, 1.37–1.43 ray tracing in, 1.35–1.37, 1.36f surfaces, 1.32–1.35 unfolded reflections, 1.32 Geometrical path length, 1.11 Geometrical wavefronts, 1.12–1.13 Gimbal-less two-axis scanning-micromirror devices (GSMDs), 30.61–30.62, 30.62f Glan-Foucault prisms, 13.7, 13.9, 13.11f, 13.12–13.14 Glan-Taylor prisms, 13.7, 13.9n, 13.10–13.14, 13.10f, 13.11f Glan-Thompson prisms, 13.6, 13.6f, 13.9–13.12, 13.10f, 13.18f, 13.22 field angle of, 13.12 and optical spectrometers, 31.7 sheet polarizers vs., 13.27 transmission by, 13.9–13.10, 13.11f

Glan-type prisms, 13.6, 13.6f, 13.8–13.15 defects and testing of, 13.14–13.15 Frank-Ritter, 13.6, 13.6f, 13.13–13.14 Glan-Foucault, 13.7, 13.9, 13.11f, 13.12–13.14 Glan-Taylor, 13.7, 13.9n, 13.10–13.14, 13.10f, 13.11f Glan-Thompson, 13.6f, 13.9–13.12, 13.10f, 13.11f, 13.27 Lippich, 13.6, 13.6f, 13.12–13.13 Marple-Hess, 13.12, 13.13 precautions with, 13.14 Glass: for microlenses, 22.9–22.10, 22.10t, 22.11f, 22.12t, 22.13t, 22.14f for objectives, 29.2, 29.2t sol-gel formed, 24.8 Glass-calcite Rochon prisms, 13.20 Glazebrook prisms, 13.6, 13.9 (See also Glan-Thompson prisms) Goerz prism system, 19.3t, 19.17, 19.17f Goos-Hanchen shift, 21.4 Graded index profile, of optical waveguides, 21.4 Graded-index (GRIN) films, 16.9 Gradient dispersion, 24.3 Gradient index (GRIN) optics, 24.1–24.8 analytic solutions of, 24.2 axial gradient lenses, 24.3–24.5, 24.4f materials, 24.8 mathematical representations of, 24.2–24.3 radial gradients, 24.5–24.8, 24.5f, 24.7f Grating equation, 23.4 Grating interferometers, 32.4, 32.5f Grating multiplexers, 23.11, 23.12 Grating polarizers, 13.33 Gratings (see Diffraction gratings; Dispersive prisms and gratings) Gravitational-wave interferometers, 32.21, 32.21f Gray code, on optical disks, 35.16 Gray-scale masks, 22.23–22.25, 22.23f–22.25f Green’s function, 3.22–3.23, 3.22f in approximations of multiple scattering, 9.10 digitized, 7.15, 7.16 in scattering, 9.3 Gregorian objective, 29.10 Grooved regions, on optical disks, 35.15, 35.15f, 35.16 Gunn diodes, 31.21

INDEX

Haidinger fringes, 2.22, 2.22f, 2.27 Half-period zones, 3.5 Half-shade devices, 13.56–13.57 Half-wave linear retarders, Mueller matrices for, 14.12t, 14.13 Half-wave plates, 12.25, 12.27 Half-wave retarders, Mueller matrices for, 14.14 Halle prisms, 13.16f, 13.17 Hamiltonian optics, 1.13, 1.43 Hamilton’s equations for rays, 1.21 Harting-Dove (Dove) prisms, 19.3t, 19.9, 19.9f, 19.10, 19.10f Hartman testing, 22.26 Hartnack-Prazmowski prisms, 13.16f, 13.17 Hartree-Fock variational approach, 10.11 Hastings (Brashear-Hastings) prisms, 19.3t, 19.25, 19.25f Hausdorff-Besicovitch dimension, 8.8 Header information, on optical disks, 35.6 Heisenberg uncertainty principle, 21.11 Heisenberg’s matrix mechanics, 10.3 Helmholtz equation, 3.2, 5.9, 5.10, 33.3 Helmholtz invariant, 1.77 (See also Two-ray paraxial invariant) Hermitian matrices, 14.19, 14.41 Herschel condition, 1.31, 29.34, 29.37 Herschelian objectives, 29.6, 29.27 Heterodyne interferometers, 32.10, 32.10f, 32.20, 32.20f Heterogeneous media, rays in, 1.9, 1.18–1.22 High-dry objectives, 28.11, 28.12f Higher-order aberrations, 29.37 Highlight color, in xerographic systems, 34.12, 34.13f High-performance miniature systems, 22.5–22.8 High-speed cameras, 25.21–25.22, 25.22f Hilbert space formulation, 5.2 Hoffman modulation contrast microscopy, 28.29 Holographic inspection, 33.16–33.19, 33.17f–33.18f, 33.20f–33.22f Holographic memory, 33.24–33.25 Holographic microscopes, 28.42, 28.43, 28.43f Holographic optical elements (HOEs), 33.13–33.16, 33.15f, 35.28–35.29, 35.29f Holographic optics, 33.13 Holographic scanners, 30.38–30.41, 30.40f–30.42f

I.15

Holography, 33.1–33.25 electronic, 33.9–33.14, 33.11f–33.13f and holographic inspection, 33.16–33.19, 33.17f–33.18f, 33.20f–33.22f and holographic memory, 33.24–33.25 and interferometry, 33.4–33.9, 33.8f and lithography, 33.22–33.24, 33.22f–33.24f optical elements for, 33.13–33.16, 33.15f principles of, 33.2–33.4, 33.3f Holtronic Technologies holographic system, 33.23 Homogeneous broadening, of lineshapes, 10.7 Homogeneous coordinates, of collineation matrix, 1.59 Homogeneous media, 1.9, 1.23–1.26 Homogeneous polarization elements: defined, 15.7 in Mueller matrices, 14.25–14.26 Homogeneous sources of light, 5.11–5.12, 5.19 Hopkin’s formula, 5.13 Horizontal half-wave linear retarders, Mueller matrices for, 14.12t Horizontal linear polarizers, 14.9, 14.10t Horizontal quarter-wave linear retarders, Mueller matrices for, 14.12t Houghton objectives, 29.22–29.23 Huang-Rhys parameter, 10.22 Hubble space telescope, 29.8 Human visual system (HVS), color in, 26.18 Hund’s rule, 10.12 Hurst dimension, 8.8 Huygens wavelets, 5.16 Huygens-Fresnel construction (diffraction), 3.4–3.13 Babinet principle, 3.9–3.11 for double refraction in calcite, 13.5 Fresnel zones in, 3.4–3.6 and light from circular apertures and disks, 3.6–3.9 zone plates in, 3.11–3.13, 3.12f Hydrogen, Bohr’s theory of, 10.3 Hyperfocal distance, 1.85 Hysteresis loops, of optical disks, 35.27, 35.27f, 35.28 Ideal imaging (term), 1.28, 1.38 Identity matrix, 14.8 Illumination: critical, 28.7 epi-, 28.7–28.9, 28.7f, 28.8f for input/output scanning, 30.14 trans-, 28.5–28.7, 28.6f

I.16

INDEX

Illumination area, surface scattering and, 8.12 Image(s), 1.26 about known rays, 1.43–1.44, 1.44f aerial, 1.26 from cameras, 25.5, 25.7 of extended objects, 1.27 in focal Gaussian lenses, 1.51 formation of, 6.9–6.10, 6.9f, 17.5–17.8, 17.5f, 17.6f latent, 34.1–34.4, 34.2f–34.3f medial, 29.11, 29.37 point, 1.85–1.92, 1.88f of points, 1.27 received, 1.26 recorded, 1.26 in systems of revolution, 1.42 two-dimensional, 11.12–11.17, 11.13f virtual, 29.38 Image derotation, 30.36 Image distance, 18.3 Image distribution, for cameras, 25.7 Image enhancement, coherent optical, 11.14–11.17 Image erectors, 18.10 Image height, 1.27, 18.4 Image inversion, 19.2 Image plane, 1.27 Image point, 1.27 Image processing, analog (see Analog optical signal and image processing) Image receptors, in xerographic systems, 34.1 Image reversion, 19.2 Image rotation, 30.35–30.36 Image space, 1.26, 1.83–1.84, 1.83f Image space numerical aperture, 1.79 Image-forming cone (bundle), 1.74 Image-forming rays, 1.74 Imaging: conscopic vs. orthoscopic, 28.8–28.9 fluorescence, 28.23 in geometrical optics, 1.26–1.31 ideal, 1.17, 1.28–1.29, 1.38 in microscopes, 28.44–28.54 aperture-scanning microscopy, 28.53–28.54 confocal microscopy, 28.49–28.51, 28.49f, 28.51f fluorescent microscopy, 28.48–28.49 light field microscopy, 28.53 polarizing microscopes, 28.44–28.48, 28.44f, 28.46f, 28.47f structured illumination, 28.52

Imaging (Cont.): Newtonian equation for, 17.8 in polarimeters, 15.6 pupil, 1.76 stigmatic, 1.29–1.31, 1.30f Image-space scanners, 30.18–30.23 Immersion lenses, 17.11 Imperfect polarizers, 13.33 Impermeability, dielectric, 21.9 Inclination factor, 3.5 Incoherent arrays, scattering by, 7.2–7.3 Incoherent light sources, 5.12, 5.18–5.19 Incoherent processing, of discrete signals, 11.17–11.20, 11.18f, 11.19f Incoherent radiation, 30.2, 30.26, 30.27 Incoherent scattering, 9.2–9.5 Incomplete polarimeters, 15.4 Incomplete sample-measuring polarimeters, 15.16–15.17 Index ellipsoid equation, 21.9–21.10 Index of refraction (refractive index), 1.9 for Brewster angle transmission polarizers, 12.21–12.22 complex, 7.12–7.13, 12.5, 12.6 distributed, 24.1 in gradient index optics, 24.2–24.3 in integrated optics, 21.8–21.9 of polarizers, 12.16, 12.18 for rays in heterogeneous media, 1.21–1.22 of shallow radical gradients, 24.7–24.8 Infrared emitting diodes (IREDs), 25.14, 25.14f Infrared radiation, single-order plates and, 13.47 Inhomogeneous (heterogeneous) media, rays in, 1.9, 1.18–1.22 Inhomogeneous optics, 24.1 (See also Gradient index (GRIN) optics) Inhomogeneous polarization elements, 14.25–14.26, 14.26f, 15.7, 15.20 Input planes, translations of, 1.68 Input/output scanning, 30.2, 30.4–30.6, 30.25–30.34 objective, preobjective, and postobjective, 30.28–30.29, 30.29f, 30.30f objective optics, 30.30–30.33, 30.32f–30.33f power density and power transfer of, 30.25–30.28, 30.27f, 30.28f resolution of, 30.8–30.14, 30.10f, 30.10t, 30.11t, 30.12f–30.13f Inspection, holographic, 33.16–33.19, 33.17f–33.18f, 33.20f–33.22f Instant pictures, 25.8

INDEX

Instrumental polarization, 12.15 Insulating magnetic brush (IMB), 34.6 Integrated circuits: photonic of III-V materials, 21.17–21.20 in integrated optics, 21.2 in WDM systems, 21.37, 21.38 polarizers for, 13.57 Integrated optic circuits (IOCs), 21.2 Integrated optics (IO), 21.1–21.41 applications of, 21.31–21.39 analog transmission, 21.32–21.34, 21.33f, 21.34f digital transmission, 21.31–21.32 fiber optic gyroscopes, 21.35–21.37, 21.36f, 21.36t silicon photonics transmission, 21.38, 21.39 switching, 21.34–21.35, 21.34f WDM systems, 21.37–21.38, 21.37f–21.39f circuit elements of, 21.21–21.31 active devices, 21.25–21.31, 21.26f–21.31f passive devices, 21.21–21.25, 21.22f–21.25f future trends in, 21.39–21.41 advanced integration, 21.40–21.41 shift from R&D to manufacturing, 21.39–21.40 materials and fabrication techniques for, 21.13–21.21 ion-exchanged glass waveguides, 21.13 LiNbO3 and LiTaO3, 21.16–21.17, 21.28, 21.33 PICs of III-V materials, 21.17–21.20 silicon photonics, 21.14–21.16, 21.15f thin film oxides, 21.13–21.14 physics of, 21.3–21.12 carrier effects, 21.10–21.12 index of refraction, 21.8–21.9 linear electro-optical effect, 21.9–21.10 nonlinear effects, 21.12 optical waveguides, 21.3–21.8, 21.3f–21.5f, 21.7f thermal effects, 21.12 Intensified CCD (ICCD), 26.3 Intensity, 5.7, 11.2 Intensity interferometers, 32.19 Intensity reflection coefficients, 12.8–12.12 Intensity transmission coefficients, 12.9–12.10

I.17

Interference, 2.3–2.42 by amplitude division, 2.19–2.28 extended source, 2.20 Fizeau interferometers, 2.24–2.26, 2.25f fringes of equal inclination, 2.20–2.22, 2.21f, 2.22f fringes of equal thickness, 2.22–2.24, 2.23f Michelson interferometer, 2.26–2.28, 2.26f–2.27f plane-parallel plate, 2.19, 2.20f thin films, 2.24 applications of, 2.42 and coherence, 2.36–2.42, 2.36f, 2.38f–2.40f, 2.42f constructive and destructive, 2.7 effects of, 2.5–2.14 aberrated wavefronts, 2.12, 2.13 coherence, 2.13 interference fringes, 2.6–2.8, 2.6f, 2.7t plane wave and spherical wave, 2.9–2.11, 2.10f temporal beats, 2.13 two plane waves, 2.8–2.9, 2.9f two spherical waves, 2.11–2.12, 2.12f, 2.13f multiple beam, 2.28–2.36 diffraction gratings, 2.28–2.29, 2.29f, 2.30f Fabry-Perot interferometer, 2.33–2.36, 2.34f, 2.35f plane-parallel plates, 2.30–2.33, 2.30f, 2.32f, 2.33f order of, 2.7 by wavefront division, 2.14–2.19, 2.15f–2.18f and wavefronts, 2.4–2.5, 2.5f and waves, 2.3–2.4, 2.5f Interference filters, Mach-Zehnder, 21.23, 21.24f, 21.25 Interference fringes, 2.6–2.8, 2.6f, 2.7t Interference gratings, 33.14 Interference microscopy, 28.33–28.44 differential-interference contrast microscopes, 28.39–28.41, 28.40f Dyson microscopes, 28.41, 28.42, 28.42f holographic, 28.42, 28.43, 28.43f Jamin-Lebedev microscopes, 28.38–28.39, 28.38f Linnik microscopes, 28.36–28.38, 28.37f Mach-Zehnder microscopes, 28.36, 28.37f Mirau microscopes, 28.41, 28.42, 28.42f and optical coherence tomography, 28.43–28.44 optical path difference (OPD) in, 28.33–28.34, 28.35f, 28.36

I.18

INDEX

Interference polarizers, 13.39–13.41, 13.40f Interferometers (interferometry), 32.1–32.21 double-passed two-beam, 32.8 Fabry-Perot, 32.4–32.7, 32.7f, 32.14 in dynamic wave meters, 32.17 in gravitational wave interferometers, 32.21, 32.21f as heterodyne interferometers, 32.10 and multiple beam interference, 2.33–2.36, 2.34f, 2.35f and wire-grid polarizers, 13.31 fiber, 32.14–32.16, 32.15f finesse of, 32.6 Fizeau, 2.24–2.26, 2.25f, 32.2, 32.2f, 32.17 free spectral range of, 2.34, 2.35, 2.35f, 32.5 frequency-modulation, 32.9, 32.9f, 32.10 fringe-counting, 32.8, 32.8f grating, 32.4, 32.5f gravitational-wave, 32.21, 32.21f heterodyne, 32.10, 32.10f, 32.20, 32.20f and holography, 33.4–33.9, 33.8f intensity, 32.19 and interferometric optical switches, 32.19 and interferometric wave meters, 32.16–32.17, 32.16f, 32.17f laser-Doppler, 32.12–32.13, 32.13f laser-feedback, 32.13–32.14, 32.14f Mach-Zehnder, 21.10, 21.12, 21.14, 21.16, 21.32, 21.40, 32.3, 32.3f, 33.5 Michelson, 2.26–2.28, 2.26f–2.27f, 32.2, 32.3f, 32.21, 33.4 Michelson stellar, 2.40–2.41, 2.40f, 32.19, 32.19f Newton, 2.25 Nomarski, 32.4, 32.5f nulling, 32.20–32.21 phase-conjugate, 32.17, 32.18, 32.18f phase-locked, 32.11–32.12, 32.12f phase-shifting, 32.10–32.11, 32.11f polarization, 32.4, 32.5f Sagnac, 21.35, 21.36, 21.36f, 32.3–32.4, 32.4f second-harmonic, 32.17–32.18, 32.18f shearing, 32.4, 32.6f stellar, 32.19–32.21, 32.19f, 32.20f three-beam, 32.7–32.8, 32.7f and two-wavelength interferometry, 32.9 Twyman-Green, 2.28, 32.2, 32.9, 33.5 Young’s two pinhole, 6.3 Interferometric arrays, 32.20–32.21 Interferometric ellipsometry, 16.18

Interferometric optical switches, 32.19 Interferometric wave meters, 32.16–32.17, 32.16f, 32.17f Interline transfer, of CCDs, 26.4–26.5, 26.5f, 26.6f Intersection points, ray tracing for, 1.36, 1.36f Invariance properties, of rays, 1.10 Invariants, optical, 18.7 Inverse filters, for coherent optical image enhancement, 11.14–11.17 Inverse Galilean telescopes, 18.15, 18.16, 18.16f Inverse systems, conjugate matrices for, 1.71 Inverted telephoto lenses, 27.2, 27.6, 27.7f–27.14f highly complex extreme speed, 27.6, 27.9f–27.12f with nonrectilinear distortion, 27.6 with rectilinear distortion correction, 27.6, 27.13f, 27.14f very compact moderate speed, 27.6, 27.7f–27.8f Ion exchange process, 24.8 Ion-exchanged glass waveguides, 21.13 Ions, tri-positive rare earth, 10.16–10.18, 10.16t, 10.17f Iris (lens), 17.8 Irradiance, 3.3 of circular patterns and disks, 3.7–3.8 of complementary aperture screen patterns, 3.10–3.11 and diffraction gratings, 3.28, 3.29 of double-slit patterns, 3.27, 3.27f of lambertian objects, 1.80 of straight-edge patterns, 3.15–3.17, 3.18f as vector, 3.33, 3.37f and zone plates, 3.11, 3.12f Irradiation, microbeam, 28.54 Isotropic homogenous spheres, scattering by, 7.11–7.14 Jamin-Lebedev microscopes, 28.38–28.39, 28.38f Jellett-Cornu prisms, 13.56 J-K model, of surface finish, 8.13, 8.15 Jones calculus, 12.29–12.30 Jones matrix, 7.10 in ellipsometry, 15.30, 16.19 and Mueller matrices, 14.3, 14.22–14.24, 14.27, 14.33 tensor product of, 14.23

INDEX

Keplerian afocal lenses, 18.7–18.14 in binoculars, 18.13–18.14, 18.14f and eye relief manipulation, 18.8–18.10, 18.9f, 18.10f field-of-view limitations in, 18.11 finite conjugate afocal relays, 18.11–18.12, 18.12f for scanners, 18.13, 18.13f in terrestrial telescopes, 18.10–18.11, 18.11f thin-lens model, 18.7–18.8, 18.8f Keplerian telescopes, 18.10 Kerr cell shutters, 25.22 Kerr cells, 31.9 Kerr effect, 15.23, 35.21, 35.23, 35.25 Kinematic lens mounts, 22.7, 22.8f Klein bottle, 14.15 Kodak Cameo Motordrive camera, 25.16, 25.16f Kodak Cobra flash, 25.16, 25.17f Kodak DCS 200 camera, 25.7 Kodak Fun Saver Portait 35, 25.16, 25.17f Kodak glass molding process, 22.9, 22.10 Koehler (Köhler) illumination: and coherence theory, 6.12–6.13, 6.12f in microscopes, 28.4–28.5, 28.5f, 28.7, 28.9 Korsch objectives, 29.30–29.32, 29.34 Kramers states, 31.19, 31.19f, 31.20 Kramers-Kronig relation, 21.11 Kronecker delta function, 5.20 Kubelka-Munk theory, 9.13 Lagrange invariant, 1.22, 1.41, 1.77, 1.81, 13.7, 17.5, 30.11 Lamb shift, 10.4 Lambertian objects, irradiance for, 1.80 Lambert’s law, 5.12 Lamipol structures, 13.57 Landé interval formula, 10.17 Landolt fringe, 13.14, 13.17, 13.18 Lands, of optical disks, 35.3, 35.5f Landscape lenses, 17.17–17.20, 17.17f–17.20f, 17.19t Laporte selection rule, 10.10 Large-format cameras, 25.18–25.20 Large-format film, 25.6 Laser direct write (LDW) fabrication, 22.19–22.23, 22.20f–22.22f Laser modes, coherence theory and, 5.23 Laser noise, 35.12, 35.24 Laser power modulation (LPM), 35.17–35.19 Laser radar (LIDAR) systems, 30.51

I.19

Laser speckle, 9.14 Laser Stark spectroscopy, 31.27–31.29, 31.27f, 31.28f Laser-assisted chemical etching (LACE), 22.45 Laser-Doppler interferometers, 32.12–32.13, 32.13f Laser-feedback interferometers, 32.13–32.14, 32.14f Lasers: continuous wave (CW) dye, 10.8 distributed Bragg reflector, 21.25, 21.30, 21.32, 21.37, 21.37f distributed feedback, 21.25, 21.29, 21.30, 21.32, 21.38, 21.40 for interferometry, 2.41–2.42 Latent images, in xerographic systems, 34.1–34.4, 34.2f–34.3f Lateral color, 1.91–1.92, 29.14, 29.37 Lateral magnification, 17.5 Lateral resolution, of microscopes, 28.17–28.19 Lau effect, 6.7–6.8, 6.8f Laurent half shades, 13.56 Leaky waveguides, 21.3 Left circular polarizers, Mueller matrices for, 14.10t Left half-wave circular retarders, Mueller matrices for, 14.12t Left-circularly polarized light, 12.27 Left-handed coordinate systems, 12.6 Legendre transformations, 1.13, 1.15, 1.16 Leica Summitar lenses, 17.28, 17.28f Leman prisms, 19.3t, 19.13, 19.13f Length-to-aperture (L/A) ratio, for prisms, 13.7 Lens axis, 1.32 (See also Optical axis) Lens law, 17.8 Lenses, 17.3–17.39 axial separations in, 1.52 cardinal points of, 1.44, 17.7 conjugate matrices for, 1.71 defined, 1.9 effective focal length of, 1.48, 22.10, 27.3f–27.5f, 27.7f–27.16f, 27.18f–27.22f, 27.25 entrance pupil of, 1.76, 17.8, 18.4–18.6, 29.37 entrance window of, 17.9 equivalent, 17.20 exit pupil of, 1.76, 17.8, 18.6, 29.37 exit window of, 17.9 field size of, 28.13 fields of, 1.74

I.20

INDEX

Lenses (Cont.): F-number and numerical aperture of, 17.9 Gauss points of, 1.44 Gaussian analyses of, 18.2–18.7, 18.3f, 18.5f, 18.7f geometrical optics for, 1.74–1.85 apertures, 1.74–1.77, 1.75f cosine-to-the-fourth approximation, 1.81 field lenses, 1.82, 1.82f fields, 1.74, 1.77, 1.84 F-number, 1.79 focus and defocus, 1.82–1.85, 1.83f irradiance, 1.80 power per pixel, 1.80 pupils, 1.76–1.79, 1.76f, 1.78f telecentricity, 1.83–1.84 total lens étendue, 1.81 vignetting, 1.81, 1.81f, 1.82 image formation in, 17.5–17.8, 17.5f, 17.6f inverses of, 1.71 for magnifiers, 17.9–17.10 for microscopes, 28.9–28.17 (See also Reflective and catadioptic lenses) compound, 17.10 objective lenses, 28.9–28.15, 28.10t, 28.11t, 28.12f, 28.13t, 28.14f–28.16f oculars, 28.16–28.17 natural stop shift of, 22.3 nodal points of, 1.48, 1.49 optical center point of, 17.16, 17.17 performance of, 17.29–17.36, 17.30f–17.35f for periscopes, 18.19, 18.19f pupils of, 17.8–17.9 rays in, 1.35 for scanners, 18.13, 30.57–30.60, 30.58f–30.60f shape factor of, 17.12–17.13, 17.13f single element, 17.12–17.17, 17.13f–17.16f, 17.17t stops of, 17.8–17.9 systems of, 17.7, 17.20–17.22, 17.21f–17.22f See also specific types of lenses Lenslets, monolithic, 22.25–22.26, 22.25f Lifetimes, of electrons, 10.6 Light-emitting diodes (LEDs): in integrated optics, 21.2 and parallel matrix-vector multipliers, 11.18 and serial incoherent matrix-vector multipliers, 11.17–11.18 transmission by, 21.32

Light field microscopy, 28.53 Light grasp (étendue), 1.22, 1.81, 13.7 Light sources, 5.6, 5.9–5.13 Light-gathering power (étendue), 1.22, 1.81, 13.7 Light-measuring polarimeters, 15.3–15.4, 15.11–15.13 Linear diattenuation and diattenuators, 14.8, 14.17 Linear electro-optic effect, 21.9–21.10 Linear magnification, 18.4 Linear polarization sensitivity, 14.17 Linear polarizers, 14.9, 14.10t, 15.7 Linear systems, coherence theory for, 6.3–6.4 Linearity: in paraxial matrix methods, 1.66 of systems of revolution, 1.41 and transfer functions, 4.2 Linnik interference microscopes, 28.36–28.38, 28.37f Liouville’s theorem, 1.22 Lippich-type prisms, 13.6, 13.12–13.13 Glan-Taylor, 13.7, 13.9n, 13.10, 13.10f, 13.11f, 13.12–13.14 half-shade, 13.12n, 13.56 Marple-Hess, 13.12, 13.13 Liquid crystal (LC) cells, 15.32f, 15.33–15.35, 15.33f, 15.34t Liquid crystal displays (LCDs), 25.7 Liquid crystal (LC) lenses, 22.40–22.41, 22.42t Liquid crystal on silicon (LCOS) panels, 15.28 Liquid crystal retarders, 15.21–15.23, 15.22f Liquid crystal variable retarders (LCVRs), 15.21–15.23, 15.22f Liquid immersion development, in xerographic systems, 34.9, 34.10 Liquid lenses, 22.37–22.41, 22.38f–22.41f, 22.42t Liquid-phase epitaxial (LPE) growth, 21.17 Lissajous-type figures, from Risley prisms, 19.25, 19.26f Lithium niobate (LiNbO3), 21.16–21.17, 21.28, 21.33, 21.39 Lithium tantalate (LiTaO3), 21.17 Lithography: deep x-ray, 22.7 and holography, 33.22–33.24, 33.22f–33.24f and miniature and micro-optics, 22.18–22.25 with gray-scale masks, 22.23–22.25, 22.23f–22.25f laser direct write fabrication, 22.19–22.23, 22.20f–22.22f

INDEX

Littrow configuration, of dispersive prisms, 20.7f, 20.10 Littrow mirrors, 20.4 Lloyd’s mirror, 2.16, 2.17f, 2.18 Localization, in volume scattering, 9.13–9.17, 9.14f Longitudinal aberrations, 1.87 Longitudinal magnification, 1.28, 1.52, 17.5 Lorentzian distribution of frequencies, 10.7 Loupes, eye, 17.9–17.10 Low inertia scanners, 30.43 Low-light-level television (LLTV) systems, 26.3 Low-order flux models, of radiative transfer, 9.12–9.13, 9.13f Lu-Chipman polar decomposition, 14.39–14.40 Lukosz-type super-resolving systems, 6.9–6.10, 6.9f Luminescence, 31.17, 31.19–31.21, 31.19f Luminescence excitation spectrometers, 31.11–31.12, 31.11f Luminescence spectrometers, 31.5–31.12, 31.8f, 31.11f Luminosity (étendue), 1.22, 1.81, 13.7 Luneburg lenses, 1.21, 22.26 (See also Distributed-index planar microlenses) Lyot coronagraphs, 29.5 Lyot stop, 29.5, 29.5f, 29.37 Mach-Zehnder (MZ) interference filters, 21.23, 21.24f, 21.25 Mach-Zehnder (MZ) interference microscopes, 28.36, 28.37f, 28.39 Mach-Zehnder (MZ) interferometers, 21.10, 21.12, 21.14, 21.16, 21.32, 21.40, 32.3, 32.3f, 33.5 Mach-Zehnder (MZ) modulators, 21.26, 21.27f, 21.28, 21.32, 21.34 Magnetic brush development, in xerographic systems, 34.5–34.7, 34.5f–34.7f Magnetic circular dichroism (MCD), 31.20, 31.21 Magnetic circular polarization (MCP), 31.21 Magnetic field modulation (MFM), 35.19, 35.19f, 35.20 Magnetic resonance, optically detected, 31.21–31.23, 31.22f, 31.23f Magneto-optical (MO) disks, 35.2 Magneto-optical (MO) modulators, 15.23 Magneto-optical (MO) readout, 35.21–35.24, 35.22f, 35.24f

I.21

Magneto-optical (MO) recording, 35.25–35.28, 35.26f, 35.27f (See also Optical disk data storage) Magnification, 1.28 afocal, 18.5–18.6 angular, 1.52, 1.78, 18.4 axial (longitudinal), 1.28, 1.52, 17.5 dia-, 1.23 by focal Gaussian lenses, 1.52–1.53 lateral, 17.5 linear, 18.4 longitudinal, 1.28 and magnifiers, 17.9–17.10 in microscopes, 28.3–28.4 pupil, 1.76 pupil angular, 1.78 scan, 30.5, 30.12–30.14 secondary, 29.12, 29.38 in systems of revolution, 1.42 transverse, 1.28, 1.50–1.51 visual, 1.28 Maksutov objectives, 29.19, 29.20 Malus-Dupin principle, 1.12 Mangin elements, in objective design, 29.6 Mangin objectives, 29.7, 29.24 Mapping, of object and image space, 1.27 Marginal rays, 1.75, 17.8, 29.37 Marple-Hess prisms, 13.12, 13.13 Mask layout, for binary optics, 23.14–23.16, 23.15f, 23.15t Maskless lithography tool (MLT), 22.23 Mass-transport process, for miniature and micro-optics, 22.45, 22.46f Master groups, of zoom lenses, 27.17 Master-oscillator/power-amplifiers (MOPAs), 21.30, 21.30f Matched filters, for pattern recognition, 11.12–11.13 Matrices: amplitude scattering, 7.10 for collineation, 1.59–1.60 computing polarization with, 12.27–12.30 density (coherency), 12.29–12.30, 14.41 identity, 14.8 Jones, 7.10 in ellipsometry, 15.30, 16.19 and Mueller matrices, 14.3, 14.22–14.24, 14.27, 14.33 tensor product of, 14.23 paraxial, and geometrical optics, 1.65–1.74

I.22

INDEX

Matrices (Cont.): Pauli spin, 14.24, 14.41 point spread, 15.36, 15.36f power, 1.67 for radiative transfer, 9.11 for single scattering, 9.16, 9.17 Stokes, 14.4 T-matrix method, 7.15 See also Mueller matrices Matrix generators, 14.27 Maxwell equations, 7.3 Maxwell fisheye (lens), 1.21 Maxwell-Boltzmann velocity distribution, 31.24 Maxwell-Garnett mixing formula, 9.8 Maxwellian ideal imaging, 1.17, 1.28, 1.38 Maxwell’s electromagnetic theory, 10.3 Maxwell’s equations: and binary gratings, 23.13 and coherence theory, 5.2, 5.3 and diffraction, 3.1, 3.2, 3.4 and laws of reflection and refraction, 1.24 and optical waveguides, 21.3, 21.5 and surface scattering, 8.4 McCarthy objective, 29.32 McKinley relays, 18.18, 18.18f, 18.19 Mean-field approximation, 7.16 Mechanical distances, in focal Gaussian lenses, 1.53 Medial images, 29.11, 29.37 Medium-format film, 25.6 Meinel-Shack objective, 29.28 Mellin transforms, 11.14 Melted-resin arrays, 22.42–22.45, 22.42f–22.44f Memory, holographic, 33.24–33.25 Meridians (meridional planes), 1.27, 1.32 Meridional rays, 1.35, 1.37 Mersenne objectives, 29.9, 29.12 Mersenne telescopes, 18.20 Metal-insulator semiconductor (MIS) capacitors, 26.2 Metal-organic molecular beam epitaxy (MOMBE), 21.17, 21.18 Metal-organic vapor-phase epitaxy (MOVPE), 21.17–21.18 Metal-oxide semiconductor (MOS) capacitors, 26.2, 26.6 Mica retardation plates, 13.45–13.46 Michel Lévy Color Chart, 28.35f Michelson interferometers, 2.26–2.28, 2.26f–2.27f, 32.2, 32.3f, 32.21, 33.4

Michelson stellar interferometers, 2.40–2.41, 2.40f, 32.19, 32.19f Microbeam irradiation, 28.54 Microelectromechanical systems (MEMS), 30.3 Micro-Fresnel lenses (MFLs), 22.31–22.37, 22.31f–22.33f, 22.35f–22.37f Microlenses: distributed-index planar, 22.26–22.31, 22.27f–22.30f, 22.27t, 22.31t micro-Fresnel lenses (MFLs), 22.31–22.37, 22.31f–22.33f, 22.35f–22.37f molded glass, 22.9–22.10, 22.10t, 22.11f, 22.12t, 22.13t, 22.14f molded plastic, 22.10, 22.12–22.15 Micromachining techniques, for binary optics, 23.16, 23.16f, 23.17 Micromirrors, 22.23, 30.61–30.62, 30.62f Micro-optical table (MOT) techniques, 22.6, 22.7f Micro-optics (see Miniature and micro-optics) Microscopes, 28.1–28.56 aperture-scanning microscopy, 28.53–28.54 bright field microscopy, 28.25, 28.27–28.28, 28.27f compound, 17.10 confocal microscopy, 28.49–28.51, 28.49f, 28.51f contrast in bright field microscopy, 28.25, 28.27–28.28, 28.27f dark field microscopy, 28.28 Hoffman modulation contrast, 28.29 interference microscopy, 28.33–28.44, 28.35f, 28.37f, 28.38f, 28.40f, 28.42f, 28.43f modulation transfer function, 28.24–28.25, 28.25f, 28.26f phase contrast, 28.28–28.29, 28.29f SSEE microscopy, 28.29–28.33, 28.30f–28.33f dark field microscopy, 28.28 differential-interference contrast, 28.39–28.41, 28.40f Dyson, 28.41, 28.42, 28.42f fluorescent microscopy, 28.48–28.49 history of, 28.1–28.3 holographic, 28.42, 28.43, 28.43f imaging modes of, 28.44–28.54, 28.44f, 28.46f, 28.47f, 28.49f, 28.51f interference microscopy, 28.35f, 28.37f, 28.38f, 28.42f, 28.43f

INDEX

Microscopes (Cont.): Jamin-Lebedev, 28.38–28.39, 28.38f lenses in, 28.9–28.17, 28.10t, 28.11t, 28.12f–28.16f light field microscopy, 28.53 Linnik, 28.36–28.38, 28.37f Mach-Zehnder, 28.36, 28.37f Mirau, 28.41, 28.42, 28.42f optical arrangements in, 28.3–28.9, 28.4f–28.8f optical path difference (OPD) in, 28.33–28.34, 28.35f, 28.36 resolution, 28.17–28.24, 28.18f–28.21f specimen manipulation for, 28.54–28.55 SSEE microscopy, 28.29–28.33, 28.30f–28.33f Mie scattering, 7.11, 7.12, 9.17 Miller algorithm, 7.15 Miniature and micro-optics, 22.1–22.46, 22.46f and binary optics, 23.7–23.8, 23.7f–23.8f design considerations, 22.2–22.8 diamond turning, 22.15–22.18, 22.16t, 22.17f, 22.18f distributed-index planar microlenses, 22.26–22.31, 22.27f–22.30f, 22.27t, 22.31t drawn preform cylindrical lenses, 22.45, 22.46f high-performance miniature systems, 22.5–22.8 laser-assisted chemical etching (LACE), 22.45 liquid lenses, 22.37–22.41, 22.38f–22.41f, 22.42t and lithography, 22.18–22.25, 22.20f–22.25f mass-transport process, 22.45, 22.46f melted-resin arrays, 22.42–22.45, 22.42f–22.44f micro-Fresnel lenses, 22.31–22.37, 22.31f–22.33f, 22.35f–22.37f molded microlenses, 22.8–22.15 molded glass, 22.9–22.10, 22.10t, 22.11f, 22.12t, 22.13t, 22.14f molded plastic, 22.10, 22.12–22.15 monolithic lenslet modules, 22.25–22.26, 22.25f Minimum signal, for solid-state cameras, 26.13–26.14 Minox camera, 25.21 Mirages, 24.1, 24.2f Mirau interference microscopes, 28.41, 28.42, 28.42f Mircolens arrays, for agile beam steering, 30.57–30.60 Mirror-image effect, 12.6

I.23

Mirrors: compound, 30.15–30.16, 30.15f conic, 29.3f, 29.4f Fresnel’s, 2.16, 2.16f Littrow, 20.4 Lloyd’s, 2.16, 2.17f, 2.18 micromirrors, 22.23, 30.61–30.62, 30.62f plane, 1.25 in reflecting afocal lenses, 18.20, 18.20f, 18.21, 18.21f reflection from, 1.25 for scanners, 30.14–30.16, 30.15f, 30.60–30.62, 30.62f as thin lenses, 1.55 See also Reflective and catadioptric objectives Misfocus, 1.82 Mixed characteristic functions, 1.13 Mobius strip, 14.15 Modulation transfer function (MTF), 4.3 calculations, 4.3–4.6, 4.4f, 4.5f and camera lens performance, 27.3f–27.5f, 27.7f–27.16f, 27.18f–27.22f, 27.24, 27.25 and characteristics of objective detectors, 28.16 and contrast of microscopes, 28.24–28.25, 28.25f, 28.26f and development in xerographic systems, 34.7 diffraction-limited, 4.4–4.5, 4.4f, 4.5f measurements of, 4.6–4.8, 4.8f for microscopes, 28.24, 28.25f for scanners, 4.6 for solid-state cameras, 26.14 at specific wavelengths, 17.38, 17.39 of uniformly illuminated apertures, 30.9, 30.10, 30.10f Modulators: electro-optical modulators, 15.23 Mach-Zehnder, 21.26–21.28, 21.27f, 21.32, 21.34 magneto-optical modulators, 15.23 photo-elastic, 15.21, 16.13 polarization (retardance), 15.20–15.24 traveling wave, 21.26 Molecular beam epitaxy (MBE), 21.17, 21.18 Molecular Expressions (website), 28.3 Molecular scattering, 7.11 Monocentric Schmidt-Cassegrain objectives, 29.16 Monocentric systems, 29.37 Monochromatic aberration correction, 23.6–23.7

I.24

INDEX

Monochromatic sources of light, 5.11 Monochromators: Czerny-Turner, 31.6 double-pass, 20.9f Perkin-Elmer Model 99, 20.9f Unicam double-, 20.10, 20.13, 20.15f Monogon scanners, 30.34–30.36 Monolithic lenslet modules (MLMs), 22.25–22.26, 22.25f Moving electron beam exposure system (MEBES), 23.15, 23.16, 23.16f Mueller calculus, 12.28–12.30 Mueller matrices, 7.10, 14.1–14.42 about, 14.3–14.4 coordinate system, 14.19–14.20 and depolarization, 14.30–14.31 depolarization index, 14.32 generators of, 14.33–14.39, 14.36f, 14.37f nondepolarizing matrices, 14.24–14.25, 14.27–14.30 diattenuators/diattenuation, 14.16–14.19, 14.16f in ellipsometry, 15.30, 16.19–16.21, 16.20f, 16.20t, 16.21f and Jones matrices, 14.22–14.24 normalization of, 14.19 physically realizable, 14.40–14.42 polar decomposition of matrices, 14.39–14.40 and polarimetry, 15.8–15.9, 15.11 elements of, 15.13–15.14 in error analysis, 15.28 singular value decomposition, 15.25–15.27 polarizance, 14.18 and polarization, 14.7, 14.8, 14.25–14.27, 14.33 average degree of polarization, 14.32–14.33 degree of polarization surfaces and maps, 14.31–14.32, 14.32f ideal polarizers, 14.8–14.10, 14.10t nonpolarizing, 14.8 for radiative transfer, 9.11 for refraction and reflection, 14.20–14.22 retarder, 14.11, 14.12t, 14.13–14.15, 14.15f for single scattering, 9.16, 9.17 and Stokes parameters, 14.4–14.6 transmittance, 14.16–14.17 Mueller matrix bidirectional reflectance distribution function (MMBRDF), 15.39, 15.39f, 15.40f Mueller matrix polarimeters, 15.26–15.27

Mueller polarimeters, 15.4 Mueller vectors, 15.15 Mueller-Jones matrices, 14.24, 14.27–14.29 (See also Nondepolarizing Mueller matrices) Muller convention, 12.6 Multielectron atoms, 10.10–10.11 Multifocal lenses, 23.12, 23.12f, 23.13f Multiple beam interference, 2.28–2.36 diffraction gratings, 2.28–2.29, 2.29f, 2.30f Fabry-Perot interferometers, 2.33–2.36, 2.34f, 2.35f plane-parallel plates, 2.30–2.33, 2.30f, 2.32f, 2.33f Multiple (volume) scattering, 9.2, 9.3, 9.8–9.17 analytical theory of, 9.9–9.10, 9.9f depolarization, 9.16–9.17, 9.17f effective-medium representation, 9.8 radiative transfer, 9.10–9.13, 9.11f, 9.13f speckle patterns, 9.15–9.16, 9.15f weak localization, 9.13–9.17, 9.14f Multiple-angle-of-incidence ellipsometry (MAIE), 16.3 Multiple-order retardation plates, 13.47–13.48 Multiple-track read-write with diode laser arrays, 35.28 Multiplexed image scanning, 30.23, 30.24f Multiplexed sensors, for fiber interferometers, 32.16 Multiplexing, spatial, 23.8 Multiplication, 11.3 Multipliers, serial incoherent matrix-vector, 11.17–11.18, 11.18f Mutual coherence function, 2.36–2.38, 2.36f, 5.4, 5.10, 6.2–6.3 Mutual intensity, 5.11, 6.3–6.4, 6.7, 6.8 Nakamura biplates, 13.56 Narrowband filters, 3.3 Natural broadening, of lineshapes, 10.7 Natural stop shift, of lenses, 22.3 Net complex amplitude, 2.5 Neutron scattering, 9.6 Newton interferometers, 2.25 Newton ring method, 29.22 Newtonian form, of Gaussian equations, 18.4 Newtonian imaging equation, 17.8 Newtonian objectives, 29.6 Newton’s equation, for Gaussian focal lenses, 1.49 Newton’s ring pattern, 2.10, 2.19, 2.25–2.26 Nicol curtate prisms, 13.16f, 13.17

INDEX

Nicol-type polarizers, 13.6, 13.6f, 13.10f, 13.15–13.18 conventional, 13.6f, 13.15–13.16 Glan-type vs., 13.8–13.9 trimmed, 13.16–13.18, 13.16f Nikon N8008s camera, 25.7 Nikon Plan Apochromat, 28.13, 28.14f Nipkow disks, 28.50, 28.51f Nippon Sheet Glass, 24.2, 24.6, 24.7 Nodal planes, of Gaussian lenses, 1.48–1.49 Nodal plane-to-nodal plane conjugate matrices, 1.69 Nodal points, of lenses, 1.48, 1.48f, 1.49, 17.7 Nodal rays, 17.16 Noether’s theorem, 1.21 Noise: data, 35.24 fixed-pattern, 26.11 laser, 35.12, 35.24 and optical disk data, 35.12, 35.23–35.24, 35.24f pattern, 26.11–26.12, 26.12f shot, 26.11 in solid-state cameras, 26.11–26.12, 26.12f Noise equivalent exposure (NEE), 26.10 Nomarski interferometers, 32.4, 32.5f Nomarski prisms, 28.40, 28.41 Nonafocal lenses, 1.46 (See also Focal lenses) Noncalcite prisms, 13.23–13.24 Noncosmological red shift, 5.23 Nondepolarizing Mueller matrices, 14.24–14.25, 14.27–14.30 Nondispersive prisms, 19.1–19.29 Abbe’s, 19.3t, 19.7, 19.7f–19.8f Amici (roof), 19.3t, 19.11, 19.12f and beam deviation, 19.2 and beam displacement, 19.2 Brashear-Hastings, 19.3t, 19.25, 19.25f Carl Zeiss, 19.3t, 19.16, 19.16f Dove, 19.3t, 19.9, 19.9f, 19.10, 19.10f Frankford Arsenal, 19.3t, 19.18–19.24, 19.18f–19.24f general deviation, 19.3t, 19.28, 19.28f–19.29f Goerz, 19.3t, 19.17, 19.17f and image inversion/reversion, 19.2 Leman, 19.3t, 19.13, 19.13f Pechan, 19.3t, 19.11, 19.11f penta, 19.3t, 19.13, 19.14f Porro, 19.3, 19.3t, 19.5f, 19.6, 19.6f retroreflectors, 19.3t, 19.28, 19.28f

I.25

Nondispersive prisms (Cont.): reversion, 19.3t, 19.14, 19.14f rhomboidal, 19.3t, 19.25, 19.25f right-angle, 19.3, 19.3t, 19.4f Risley, 19.3t, 19.25, 19.25f–19.27f, 19.27 Schmidt, 19.3t, 19.12, 19.12f Wollaston, 19.3t, 19.15, 19.15f Nonhomogeneous polarization elements, 14.25–14.26, 14.26f, 15.7, 15.20 Nonhomogeneous polarization elements (Mueller matrices), 14.25–14.26, 14.26f Nonlinear effects, in integrated optics, 21.12 Nonnormal-incidence reflection: in Brewster angle reflection polarizers, 13.34–13.37, 13.34t–13.36t in pile-of-plates polarizers, 12.15–12.18, 12.16f, 12.17f in polarizing beam splitters, 13.41–13.42 Nonnormal-incidence transmission: in Brewster angle transmission polarizers, 13.37–13.39, 13.38t–13.39t in interference polarizers, 13.39–13.41, 13.40f in pile-of plates polarizers, 12.18–12.24, 12.19t–12.20t, 12.21f in polarizing beam splitters, 13.41–13.42 Nonpolarizing elements, 15.7 Nonpolarizing Mueller matrices, 14.8 Nonradiating sources of light, 5.18 Nonrectilinear distortion, 27.6 Non-return to zero inverted (NRZI) scheme, 35.17 Non-return to zero (NRZ) scheme, 35.17 Nonrotationally symmetric systems, 1.74 Nonspherical particles, scattering by, 7.15–7.17 Normal congruence rays, 1.10 Normal vectors, 1.18, 1.19 Normal-incidence rotating-sample ellipsometers (NIRSE), 16.18 Normalization, of Mueller matrices, 14.19 Normalized spectrum, coherence functions for, 5.5–5.6 Null ellipsometers, 16.11, 16.12 Nulling interferometers, 32.20–32.21 Numerical aperture, 1.78, 1.79, 17.9 Nyquist frequency, 26.16–26.20, 26.17f Object relief distance, 18.11–18.12, 18.12f Object space, 1.26, 1.83 Object space numerical aperture, 1.78 Object space pupil diameter, 18.6

I.26

INDEX

Object transparencies, 11.4–11.5, 11.5f Objective optics, 30.30–30.33, 30.32f–30.33f Objective scanning, 30.5, 30.28 Objective speckle, 33.9 Objectives: for microscopes, 28.9–28.15, 28.10t, 28.11t in afocal systems, 18.7 corrections for tube length, 28.13, 28.13t coverslip correction, 28.10–28.11, 28.12f, 28.13 design of, 28.13–28.15, 28.14f–28.16f field size, 28.13 working distance, 28.13 reflective and catadioptric (see Reflective and catadioptric objectives) for telescopes afocal Cassegrain-Mersenne, 29.9 afocal Gregorian-Mersenne, 29.12 Ritchey-Chretien, 29.8 three-mirror afocal, 29.29–29.30 Object-space scanners, 30.18–30.23, 30.19f–30.23f Object-to-image distance, 1.51–1.52 Oblique spherical aberrations, 1.90, 29.15, 29.21, 29.37 Obliquity factor, 3.5 Obscurations, of reflective and catadioptric objectives, 29.4–29.5, 29.4f Oculars, for microscopes, 28.16–28.17 Off-axis, eccentric-pupil Paul-Gregorian objective, 29.28–29.29 Off-axis double-pass grating spectrograph, 20.10, 20.13f Offner relay, 29.33 Offset quantum wells, 21.19 On-axis objective optics, 30.30 One-electron atoms, 10.7–10.9, 10.8f, 10.9f 135° linear polarizers, Mueller matrices for, 14.10t 135° half-wave linear retarders, Mueller matrices for, 14.12t 135° quarter-wave linear retarders, Mueller matrices for, 14.12t Ophthalmic polarimetry, 15.39, 15.41 Optic axis, of calcite crystals, 13.2, 13.2f, 13.3f, 13.3n Optical absorption spectrometers, 31.2–31.5, 31.5f Optical axes, 1.32, 14.8, 18.2, 29.5, 29.37 Optical center point, of lenses, 17.16, 17.17

Optical coherence microscopy (OCM), 28.44 Optical coherence tomography (OCT), 22.2, 22.39, 22.40f, 28.43–28.44 Optical constants, 7.12, 12.4–12.6, 16.5 Optical disk data storage, 35.1–35.30 alternative storage media, 35.29, 35.30 automatic focusing, 35.12–35.14, 35.13f automatic tracking, 35.14–35.17, 35.14f–35.16f data format and layout for, 35.2–35.7, 35.3f–35.5f developments in, 35.28–35.30 diffractive optics, 35.28, 35.28f, 35.29 direct overwrite, 35.30, 35.30f materials for recording, 35.25–35.28, 35.26f, 35.27f multiple-track read-write with diode laser arrays, 35.28 and optical path, 35.7–35.12, 35.7f, 35.9f–35.11f readout, 35.21–35.24, 35.22f, 35.24f thermomagnetic recording process, 35.17–35.20, 35.18f–35.20f Optical extent (étendue), 1.22, 1.81, 13.7 Optical fibers, polarizers for, 13.57 Optical hole burning (OHB), 10.18 Optical holeburning (OHB) spectroscopy, 31.24–31.26, 31.24f–31.26f Optical invariants, 18.7 Optical matched filtering, for pattern recognition, 11.12–11.14, 11.13f Optical metrology, 15.35 Optical path, of optical disks, 35.7–35.12, 35.7f, 35.9f–35.12f Optical path difference (OPD), 2.7, 28.33–28.34, 28.35f, 28.36 Optical path length (OPL), 1.11, 2.5 Optical processing systems, for synthetic aperture radar data, 11.7–11.8 Optical sine theorem, 17.5 Optical spectrometers (see Spectrometers) Optical spectroscopy (see Spectroscopy) Optical theorem, 7.8 Optical train, in microscopes, 28.3–28.5, 28.5f Optical transfer function (OTF): calculations of, 4.3, 4.5 and camera lens performance, 27.3f–27.5f, 27.7f–27.16f, 27.18f–27.22f, 27.24 measurements of, 4.6–4.7 Optical tube length, 17.10

INDEX

Optical tweezers, 28.55 Optically detected magnetic resonance (ODMR), 31.21–31.23, 31.22f, 31.23f Optoelectronic integrated circuit (OEIC), 21.2 Orthogonal matrices, 14.11 Orthoscopic imaging, 28.8 Orthotomic systems, 1.10, 1.12 Oscillations, wavelength interval between, 12.10 Osculating planes, of space curves, 1.18, 1.19 Outer product processors, 11.19 Output planes, conjugate matrices for, 1.68 Overillumination, 30.14 Oxides, thin film, 21.13–21.14 Pancharatnam phase, 32.11 Panoramic cameras, 25.25, 25.26f Parabasal optics, 1.43 Paraboloid objective, 29.6 Parallax stereogram, 25.24 Parallel matrix-vector multipliers, 11.18–11.19, 11.18f Parallel-beam scanners, 30.23–30.25, 30.24f–30.25f Paraxial chief rays, 1.75 Paraxial curvature, 1.32–1.33 Paraxial invariant, 1.41 Paraxial invariants, 1.77 Paraxial limit, of systems of revolution, 1.38 Paraxial matrices, for geometrical optics, 1.65–1.74 angle instead of reduced angle, 1.72 arbitrary systems, 1.67 and characteristic functions, 1.74 conjugate matrices, 1.68–1.71, 1.73 linearity, 1.66 nonrotationally symmetric systems, 1.74 operation on two rays, 1.68 possible zeros, 1.68 power matrix, 1.67 skew rays, 1.73 transfer matrices, 1.66 two-ray specification, 1.72 unit determinants, 1.67 Paraxial optics, 1.29, 1.37 Paraxial optics, of systems of revolution, 1.37–1.43 angle of incidence at a surface, 1.39 axial object and image locations, 1.40 image location and magnification, 1.42 linearity of, 1.41

I.27

Paraxial optics, of systems of revolution (Cont.): paraxial limit, 1.38 principal focal lengths of surfaces, 1.39–1.40 ray tracing, 1.40 reflection and refraction, 1.38 switching axial objects and viewing positions, 1.43 three-ray rule, 1.42 transfer, 1.38 two-ray paraxial invariant, 1.41 Paraxial pupils, 1.77 Paraxial rays, 1.35 Partially polarized light, 15.7 Particles, scattering by, 7.1–7.17 coherent vs. incoherent arrays, 7.2–7.3 concepts of, 7.4–7.5, 7.6f–7.7f, 7.8–7.10, 7.9f, 7.10f isotropic homogenous spheres, 7.11–7.14 Mie, 7.11, 7.12 nonspherical particles, 7.15–7.17 regular particles, 7.14–7.15 single particles, 7.2–7.3 theories of, 7.3–7.4 Paschen-Runge configuration, 20.7, 20.11f, 20.14t Passive autofocus systems, for cameras, 25.12 Passive devices, for integrated optics, 21.21–21.25, 21.22f–21.25f Pattern noise, 26.11–26.12, 26.12f Pattern recognition, optical matched filtering for, 11.12–11.14, 11.13f Paul objectives, 29.28–29.29 Pauli spin matrices, 14.24, 14.41 Pechan prisms, 19.3t, 19.11, 19.11f Pellin-Broca prisms, 20.6f Penta prisms, 19.3t, 19.13, 19.14f Percus-Yevick approximation, 9.5 Perfectly reflecting (PEC) surfaces, 8.10 Periphery cameras, 25.22 Periscopes, lenses in, 18.19, 18.19f Periscopic lenses, 17.27, 17.27f Perpendicular magnetic anisotropy, 35.26 Perpendicular-incidence ellipsometers (PIEs), 16.17–16.18, 16.18f Petzval (field) curvature: in gradient index optics, 24.4, 24.6, 24.7 of reflective and catadioptric objectives, 29.7, 29.11, 29.14, 29.15, 29.32 as wavefront aberration, 1.91 Petzval lenses, 17.10, 17.28, 17.28f, 17.35f Petzval sum, 28.15, 29.37

I.28

INDEX

Pfund configuration, of dispersive prisms, 20.8f, 20.10, 20.13f Pfund objectives, 29.6 Phase contrast microscopy, 28.28–28.29, 28.29f Phase plates, 28.28 Phase retardation, 12.24 Phase transfer functions (PTF), 4.3, 4.7 Phase-conjugate interferometers, 32.17, 32.18, 32.18f Phased-arrays, for agile beam steering, 30.52–30.57, 30.53f, 30.62–30.63 Phase-locked interferometers, 32.11–32.12, 32.12f Phase-matching, in integrated optics, 21.12 Phase-sensitive detection (PSD), 31.8–31.11, 31.8f Phase-shifting interferometers, 32.10–32.11, 32.11f Phasors, 5.2 Photo-activated localization microscopy (PALM), 28.23 Photocopolymerization, 24.8 Photocopying, 24.6 Photo-elastic modulators (PEMs), 15.21, 16.13 Photographic plates, 3.3 Photography, wide-angle, 25.25 Photoluminescence decay time, 31.12–31.15, 31.13f, 31.14f Photometric ellipsometers, 16.12–16.14, 16.13f, 16.14f Photon correlation spectroscopy (PCS), 9.8 Photon migration approach, to radiative transfer, 9.12 Photon transfer, 26.12 Photonic integrated circuits (PICs): of III-V materials, 21.17–21.20 in integrated optics, 21.2 in WDM systems, 21.37, 21.38 Photopolarimeters, 16.13–16.16, 16.14f–16.16f Photoreceptors, in xerographic systems, 34.1, 34.2–34.4, 34.2f, 34.3f Photoresponse nonuniformity (PRNU), 26.11–26.12, 26.12f Photosensitive compounds, in microscopy, 28.55 Physically realizable Mueller matrices, 14.40–14.42 Picosecond and sub-picosecond relaxation, 31.14, 31.14f Pictures, instant, 25.8 Piezoelectric transducer (PZT), 15.21, 15.24

Pile-of-plates polarizers: nonnormal-incidence reflection, 12.15–12.18, 12.16f, 12.17f nonnormal-incidence transmission, 12.18–12.24 Pincushion distortion, 1.91 Piston error, 1.91 Pixels, 30.8 Planar lenses, 17.28 Planar objects, transmissive, 6.3 Planar secondary source of light, 5.9 Plane mirrors, 1.25 Plane of incidence, 1.23, 12.6 Plane of polarization, 12.6n Plane waves, 2.4, 2.5f, 3.3, 3.17 decomposition of, 3.23 interference of, 2.8–2.9, 2.9f and spherical waves, 2.9–2.11, 2.10f Plane-parallel plates, 2.19, 2.20f, 2.30–2.33, 2.30f, 2.32f, 2.33f Planes of vibration, 12.6 Pockels cells, 28.45, 31.9 Pockels effect, 15.23, 21.9 (See also Linear electro-optic effect) Poincaré spheres, 12.27–12.29, 14.4–14.6, 14.5f, 14.26f, 28.45 Point characteristic function, 1.11, 1.14 Point eikonal, 1.14, 1.17 Point images, aberrations of, 1.85–1.92, 1.86f Point objects, image planes of, 28.19 Point spread matrix, 15.36, 15.36f Point-angle characteristic function, 1.15–1.17 Points, images of, 1.27 Polacoat dichroic polarizers, 13.25, 13.26, 13.28 Polanret system, 28.29 Polar decomposition, of Mueller matrices, 14.39–14.40 Polarimeters, 15.3–15.6 AxoScan Mueller matrix, 15.33 classes of, 15.5 complete and incomplete, 15.4 defined, 15.7 design metrics for, 15.24–15.25 division-of-amplitude, 15.5–15.6 division-of-aperture, 15.5 dual rotating retarder, 15.16 dual rotating retarder polarimeters, 15.16, 15.16f imaging, 15.6 light-measuring, 15.3–15.4, 15.11–15.13

INDEX

Polarimeters (Cont.): Mueller, 15.4 Mueller matrix, 15.26–15.27 polarization modulation, 15.5 sample-measuring, 15.4 incomplete, 15.16–15.17 for Mueller matrix elements, 15.13–15.14, 15.13f spectropolarimeters, 15.6 Stokes, 15.4, 15.5, 15.25 time-sequential, 15.5 See also Photopolarimeters Polarimetric data-reduction equations, 15.14–15.15 Polarimetric measurement equation, 15.12, 15.14–15.15 Polarimetry, 15.3–15.41 applications of, 15.29–15.41 ellipsometry, 15.30–15.32, 15.31f, 15.32f liquid crystal cell and system testing, 15.32–15.35, 15.32f, 15.33f, 15.34t ophthalmic polarimetry, 15.39, 15.41 polarization aberrations, 15.35–15.37, 15.35f–15.37f polarization light scattering, 15.38–15.39, 15.38f–15.40f remote sensing, 15.37–15.38 error analysis in, 15.27–15.29, 15.29f instruments for (see Polarimeters) Mueller matrices in, 15.8–15.9, 15.11 elements of, 15.13–15.14 in error analysis, 15.28 singular value decomposition, 15.25–15.27 polarimetric data-reduction equations, 15.14–15.15 polarimetric measurement equation, 15.14–15.15 and polarization elements, 15.17, 15.19–15.20 polarization generators and analyzers, 15.4–15.5 polarization (retardance) modulators, 15.20–15.24, 15.22f Stokes vectors in, 15.8–15.10 terms in, 15.6–15.7 Polariscopes, Sénarmont, 12.30 Polarizance, 12.14n, 14.18 Polarization, 12.3–12.30 average degree of, 14.32–14.33 and coherence theory, 5.22 concepts and conventions, 12.4–12.6

I.29

Polarization (Cont.): defined, 15.8 degree of, 12.14–12.15 of dichroic polarizers, 13.33 false, 15.38 Fresnel equations for, 12.6–12.13 for absorbing materials, 12.10–12.13, 12.13f coordinate system for, 12.6–12.7, 12.7f for nonabsorbing materials, 12.8–12.10, 12.9f generators and analyzers of, 15.4–15.5 instrumental, 12.15 magnetic circular, 31.21 matrix methods for computing, 12.27–12.30 and Mueller matrices, 14.7, 14.8, 14.25–14.27, 14.33 pile-of-plates polarizers, 12.15–12.24 nonnormal-incidence reflection, 12.15–12.18, 12.16f, 12.17f nonnormal-incidence transmission, 12.18–12.24, 12.19t–12.20t, 12.21f plane of, 12.6n relations for polarizers, 12.14–12.15 retardation plates, 12.24–12.27, 12.25f, 12.26f See also related topics Polarization aberration function (PAF), 15.35 Polarization aberrations, 15.35–15.37, 15.35f–15.37f Polarization analyzer, 15.11 Polarization and Directionality of Earth’s Reflectances (POLDER) instrument, 15.37 Polarization artifacts, 15.38 Polarization coupling, 15.8 Polarization critical region, 15.28 Polarization instruments, 12.29 Polarization interferometers, 32.4, 32.5f Polarization light scattering, 15.38–15.39, 15.38f–15.40f Polarization modulation polarimeters, 15.5 Polarization (retardance) modulators, 15.20–15.24, 15.22f Polarization spectrometers, 31.15–31.23 optically detected magnetic resonance, 31.21–31.23, 31.22f, 31.23f polarized absorption by, 31.15–31.17, 31.15f, 31.18f polarized absorption/luminescence techniques, 31.17, 31.19–31.21, 31.19f principles of, 31.15, 31.15f Polarization state detectors (PSDs), 16.10

I.30

INDEX

Polarization state generators (PSGs), 16.10 Polarization-dependent loss (PDL), 14.17 Polarized absorption, 31.15–31.21, 31.15f, 31.18f, 31.19f Polarized light, 15.8 Polarizer-compensator-sample analyzer (PCSA) ellipsometer arrangement, 16.10–16.14, 16.11f Polarizers, 13.1–13.57 beam-splitter prisms as, 13.6, 13.18–13.22 Foster, 13.7, 13.18f, 13.21–13.22 Glan-Thompson, 13.18f, 13.22 Rochon, 13.7, 13.18–13.21, 13.18f, 13.19f, 13.24 Sénarmont, 13.7, 13.18, 13.18f, 13.21 Wollaston, 13.7, 13.18, 13.18f, 13.21, 13.24 circular, 15.17–15.19 compensators, 13.53–13.56, 13.54f, 13.55f Cotton, 13.21 defined, 15.8 dichroic and diffraction-type, 13.24–13.33, 13.26f, 13.27f dichroic polarizing coatings, 13.28 measuring polarization of, 13.33 pyrolytic-graphite polarizers, 13.28–13.29, 13.29f sheet polarizers, 13.25–13.28 wire-grid and grating polarizers, 13.30–13.33, 13.31f, 13.32t elliptical, 15.17–15.18 Feussner prisms, 13.6, 13.7, 13.22–13.23, 13.22f Glan-Foucault prisms, 13.7, 13.9, 13.11f, 13.12–13.14 Glan-type prisms, 13.6, 13.6f, 13.8–13.15 Frank-Ritter-type, 13.6, 13.6f, 13.13–13.14 Glan-Foucault, 13.7, 13.9, 13.11f, 13.12–13.14 Glan-Thompson type, 13.6f, 13.9–13.12, 13.10f, 13.11f, 13.27 Lippich-type, 13.6, 13.6f, 13.7, 13.9n, 13.10–13.14, 13.10f, 13.11f half-shade devices, 13.56–13.57 ideal, 14.8–14.10, 14.10t, 15.7 imperfect, 13.33 miniature, 13.57 Nicol-type, 13.6, 13.6f, 13.10f, 13.15–13.18 conventional, 13.6f, 13.15–13.16 Glan-type vs., 13.8–13.9 trimmed, 13.6f, 13.7, 13.16–13.18, 13.16f

Polarizers (Cont.): noncalcite prisms as, 13.23–13.24 nonnormal-incidence reflection by Brewster angle reflection polarizers, 13.34–13.37, 13.34t–13.36t pile-of-plates polarizers, 12.15–12.18, 12.16f, 12.17f polarizing beam splitters, 13.41–13.42 nonnormal-incidence transmission by Brewster angle transmission polarizers, 13.37–13.39, 13.38t–13.39t interference polarizers, 13.39–13.41, 13.40f pile-of plates polarizers, 12.18–12.24, 12.19t–12.20t, 12.21f polarizing beam splitters, 13.41–13.42 prism, 13.2–13.8, 13.2f–13.3f, 13.4t–13.5t, 13.6f relations for, 12.14–12.15 retardation plates as, 13.43–13.53, 13.43t–13.44t, 13.50f, 13.53t Polarizing angle, 12.12, 12.15 Polarizing beam splitter (PBS) prisms, 13.6, 13.41–13.42 Polarizing beam splitters (PBSs), 13.41–13.42, 35.22–35.23, 35.22f Polarizing coatings, dichroic, 13.28 Polaroid dichroic polarizers, 13.25–13.28, 13.26f Polygon scanners, 30.34–30.38, 30.34f, 30.35f Porro prisms, 19.3, 19.3t, 19.5f, 19.6, 19.6f Postobjective scanning, 30.5, 30.29, 30.29f, 30.30f Powder cloud development, in xerographic systems, 34.9, 34.9f Power (Gaussian lenses), 1.46–1.47, 1.47f Power density, 30.25 Power exponential (PEX) model, of surface finish, 8.15 Power matrix, 1.67 Power per pixel, 1.80 Power spectra, for surface scattering, 8.12–8.13, 8.13f Power transfer, 30.25–30.28, 30.27f, 30.28f Poynting vectors, 1.8, 3.3 Preflash, of cameras, 25.16 Preobjective scanning, 30.5, 30.28, 30.29f Primary focal length (term), 3.12 Principal plane-to-principal plane conjugate matrices, 1.69 Principal rays (term), 1.75, 17.8 Principal transmittance (term), 12.14–12.16 Prism polarizers, 13.2–13.8, 13.2f–13.3f, 13.4t–13.5t, 13.8f (See also specific types)

INDEX

Prism spectrometers, 20.2–20.3, 20.3f Prismatic facets, 30.34–30.35 Prisms (See also Dispersive prisms and gratings; Nondispersive prisms) axis wander of, 13.15 beam-splitter, 13.7, 13.18–13.22, 13.18f Bertrand-type Feussner, 13.23 Brewster angle, 13.13 of calcite, 13.20, 13.23 Feussner, 13.6, 13.7, 13.22–13.23, 13.22f Foster, 13.7, 13.18f, 13.21–13.22 Foucault, 13.7, 13.17 Frank-Ritter-type, 13.6, 13.6f, 13.13–13.14 Fresnel’s biprism, 2.16, 2.16f Glan-Foucault, 13.7, 13.9, 13.11f, 13.12–13.14 Glan-Taylor, 13.7, 13.9n, 13.10–13.14, 13.10f, 13.11f Glan-Thompson, 13.6, 13.6f, 13.9–13.12, 13.10f, 13.18f, 13.22 field angle of, 13.12 sheet polarizers vs., 13.27 transmission by, 13.9–13.10, 13.11f Glan-type, 13.6, 13.6f, 13.8–13.15 Glazebrook, 13.6, 13.9 Halle, 13.16f, 13.17 Hartnack-Prazmowski, 13.16f, 13.17 Jellett-Cornu, 13.56 length-to-aperture (L/A) ratio, 13.7 Lippich-type, 13.6, 13.9n, 13.10f, 13.11f, 13.12–13.14, 13.12n, 13.56 Marple-Hess, 13.12, 13.13 Nicol curtate, 13.16f, 13.17 Nomarski, 28.40, 28.41 noncalcite, 13.23–13.24 polarizing beam splitter, 13.6, 13.41–13.42 Rochon, 13.7, 13.18–13.21, 13.18f, 13.19–13.20, 13.19f, 13.24 semifield angle of, 13.7 Sénarmont, 13.7, 13.18, 13.18f, 13.21 Steeg and Reuter Nicol, 13.17 Wollaston, 13.7, 13.18, 13.18f, 13.21, 13.24, 28.39, 28.40, 32.4 Projective transformation, 1.56 (See also Collineation) Propagation of light, coherence theory and, 5.13–5.19, 5.14f–5.16f Propagation of mutual intensity, 6.4 Pseudo-Brewster angle, 12.13 Pulse width modulation (PWM), in thermomagnetic recording, 35.18–35.19

I.31

Pupil aberrations, 1.76 Pupil angular magnification, 1.78 Pupil distortion, 1.78 Pupil imaging, 1.76 Pupil magnification, 1.76 Pupils, of lenses, 1.76–1.79, 1.76f, 1.78f, 17.8–17.9 Purcell, M., 7.16 Purcell-Pennypacker method, 7.15 Pure diattenuators, 15.8 Pure retarders, 15.8 Pushbroom scan, 30.18 Pyramidal facets, 30.34–30.35, 30.34f, 30.35f Pyrolytic-graphite polarizers, 13.28–13.29, 13.29f Quantitative phase microscopy, 28.27 Quantum coherence theory, 5.2 Quantum electrodynamic (QED) shifts, 10.4 Quantum-confined Stark effect (QCSE), 21.11–21.12, 21.32 Quantum-well intermixing, 21.19, 21.20 Quarter pitch length, of the rod, 24.6 Quarter-wave circular retarders, Mueller matrices for, 14.12t Quarter-wave linear retarders, Mueller matrices for, 14.11, 14.12t Quarter-wave plates, 12.25–12.27, 12.26f Quartz retardation plates, 13.46–13.48 Quasi-homogeneous sources of light, 5.11–5.12, 5.19 Quasi-monochromatic sources of light, 5.11 Racah parameters, for ion energy levels, 10.12 Radar, synthetic aperture, 11.6–11.8, 11.7f–11.8f Radial gradients, 24.5–24.8, 24.5f, 24.7f Radial symmetry, of scanners, 30.5 Radiance, 5.8 Radiant emittance, 5.7–5.8 Radiant intensity, 5.8 Radiation: coherent, 30.2, 30.25–30.26 incoherent, 30.2, 30.26, 30.27 infrared, 13.47 Radiation fields, coherence theory and, 5.15–5.16, 5.15f, 5.16f Radiation modes, of optical waveguides, 21.4 Radiative lifetime, 31.12–31.13, 31.13f Radiative transfer, in volume scattering, 9.10–9.13, 9.11f, 9.13f

I.32

INDEX

Radio astronomy, 5.23 Radiometry, statistical, 5.22 Radius of torsion, for space curves, 1.19 Radius of torsion, of space curves, 1.19 Ramachandran, G. N., 12.28, 13.53 Raman scattering, 31.30–31.31, 31.31f Raman-Nath diffraction, 11.9, 11.9f Ramsden disk, 28.8, 28.16 Range of focus, 1.85 Rapid Rectilinear lenses, 17.27 Rare earth ions, tri-positive, 10.16–10.18, 10.16t, 10.17f Raster output scanning (ROS) systems, 34.4 Ray aberrations, 1.87–1.88 Ray densities, 1.88 Ray equation, 1.20 Ray fans, 1.35 Ray intercept diagrams, 1.87 Ray optics, 1.8 Ray paths, 1.10–1.13, 24.2 Ray tracing: for binary optics, 23.4, 23.6 in systems of revolution, 1.35–1.37, 1.36f, 1.40 Rayleigh criterion, 17.37, 28.6, 28.18 Rayleigh criterion of resolving power, 3.26 Rayleigh index, 8.6 Rayleigh radius value, 30.10 Rayleigh range of origin, 5.14, 5.16 Rayleigh resolution, 30.56, 33.17 Rayleigh scattering, 7.11, 9.17, 31.30 Rayleigh-Gans approximation, 7.9, 7.9f Rayleigh-Rice (RR) approximation, 8.4, 8.9–8.12 Rayleigh’s diffraction integral, 5.13 Rayleigh-Sommerfeld diffraction, 3.9, 3.10, 3.23, 3.29 Rays, 1.8–1.13 chief, 1.75, 17.8, 29.20, 29.37 for collineation, 1.61 defined, 1.8–1.9 differential geometry of, 1.19–1.21 direction of, 1.10 expansions about, 1.16 fields of, 1.13 finite, 1.35 groups of, 1.10 Hamilton’s equations for, 1.21 in heterogeneous media, 1.9, 1.18–1.22 image-forming, 1.74 images about known rays, 1.43–1.44, 1.44f

Rays (Cont.): invariance properties of, 1.10 in lenses, 1.35 marginal, 1.75 meridianal, 1.37 meridional, 1.35 nodal, 17.16 normal congruence, 1.10 paraxial, 1.35, 1.75 paraxial matrices for, 1.68, 1.73 paths of, 1.10–1.13 principal, 1.75, 17.8 principal index of, 13.3 real and virtual, 1.10, 1.35 reversibility of, 1.9 skew, 1.35, 1.73 variational integral of, 1.19 Reactive ion etching (RIE), 21.18–21.19 Readout, from optical disk data storage, 35.21–35.24 Real pupils, 1.76 Real rays, 1.10, 1.35 Rear focal lengths, 1.40 Rear focal points, 1.40, 1.47 Rear principal plane, of Gaussian focal lenses, 1.48 Received images, 1.26 Receiving surfaces, in imaging, 1.26 Reciprocity theorem, coherence theory and, 5.17–5.18 Recorded images, 1.26 Recording, of optical disk data, 35.25–35.28, 35.26f, 35.27f Rectangular apertures, 3.19–3.20, 3.20f, 3.25, 3.26 Rectilinear distortion correction, 27.6, 27.13f, 27.14f Red eye, cameras and, 25.16 Red shift, noncosmological, 5.23 Rediagonalization, of index ellipsoid equation, 21.10 Reference spheres, for wavefronts, 1.86 Reflectance, 12.17 Reflecting afocal lenses, 18.19–18.21, 18.20f Reflection(s): in Gaussian lens systems, 1.55 in homogeneous media, 1.25 Mueller matrices for, 14.21–14.22 nonnormal-incidence, 12.15–12.18 and phase changes, 12.12–12.13 and ray tracing, 1.37

INDEX

Reflection(s) (Cont.): of systems of revolution, 1.38, 1.39 total internal, 13.20, 21.3 unfolded, 1.32 volumetric, 7.7f Reflective and catadioptric objectives, 29.1–29.38 afocal telescope designs Cassegrain-Mersenne, 29.9 Gregorian-Mersenne, 29.12 three-mirror, 29.29–29.30 Altenhof, 29.32–29.33 anastigmatic designs, 29.12–29.13 aplanatic designs, 29.11–29.13 Baker-Nunn, 29.22 Cassegrain designs, 29.6, 29.7 afocal Cassegrain-Mersenne telescope, 29.9 dual magnification, 29.9–29.10 with field corrector and spherical secondary, 29.8–29.9 Houghton-Cassegrain, 29.22–29.23 Mangin-Cassegrain with correctors, 29.24 reflective Schmidt-Cassegrain, 29.17 Schmidt-Cassegrain, 29.16–29.17 Schmidt-meniscus Cassegrain, 29.21 with Schwarzschild relay, 29.32 solid Makutsov-Cassegrain, 29.19 spherical-primary, with reflective field corrector, 29.9 three-mirror, 29.30 Cook three-mirror, 29.31 correctors, in designs aplanatic, anastigmatic Schwarzschild with aspheric corrector plate, 29.13 Cassegrain with spherical secondary and field corrector, 29.8–29.9 Mangin-Cassegrain with correctors, 29.24 Ritchey-Chretien telescope with two-lens corrector, 29.8 spherical-primary Cassegrain with reflective field corrector, 29.9 three-lens prime focus corrector, 29.10 Couder, 29.12 Dall-Kirkham, 29.8 Eisenburg and Pearson two-mirror, three reflection, 29.25 features of, 29.2–29.5, 29.3f–29.5f field-of-view plots, 29.34–29.35, 29.35f, 29.36f flat-medial-field designs, 29.11 Gabor, 29.20 glass varieties for, 29.2, 29.2t

I.33

Reflective and catadioptric objectives (Cont.): Herschelian catadioptric, 29.27 Houghton designs, 29.22–29.23 Korsch designs, 29.30–29.32, 29.34 Maksutov designs, 29.19, 29.20 Mangin designs, 29.7, 29.24 Mersenne designs, 29.9, 29.12 Paul designs, 29.28–29.29 Ritchey-Chretien with two-lens corrector, 29.8 Schiefspiegler, 29.26–29.27 Schmidt designs, 29.14 Baker super-Schmidt, 29.21 field-flattened, 29.14–29.15 reflective, 29.15 reflective Schmidt-Cassegrain, 29.17 Schmidt-Cassegrain, 29.16–29.17 Schmidt-meniscus Cassegrain, 29.21 Shafer-relayed-virtual, 29.17–29.18, 29.18f solid, 29.16 Schwarzschild designs, 29.12–29.13, 29.32 SEAL, 29.27–29.28 Shafer designs five mirror unobscured, 29.33–29.34 four mirror unobscured, 29.33 Shafer relayed virtual Schmidt, 29.17–29.18, 29.18f two-mirror three reflection, 29.25 Shenker, 29.23 spherical primaries in designs, 29.9, 29.11, 29.18, 29.30 for telescopes afocal Cassegrain-Mersenne, 29.9 afocal Gregorian-Mersenne, 29.12 Ritchey-Chretien, 29.8 three-mirror afocal, 29.29–29.30 terminology, 29.36–29.38 three-mirror designs, 29.28–29.32, 29.34 Wetherell and Womble three-mirror, 29.31 Wright, 29.15 Yolo, 29.26 Reflective Schmidt objective, 29.15 Reflective Schmidt-Cassegrain objective, 29.17 Reflective systems, 1.9 Refraction: in calcite, 13.2–13.6, 13.4t–13.5t double, 13.2–13.6, 13.2f–13.3f, 13.4t–13.5t in Gaussian lens systems, 1.54 in homogeneous media, 1.24–1.25 Mueller matrices for, 14.20–14.21 ray tracing, 1.37 in systems of revolution, 1.38, 1.39

I.34

INDEX

Refraction gradients, index of, 24.1 (See also Gradient index (GRIN) optics) Refractive index (index of refraction), 1.9 for Brewster angle transmission polarizers, 12.21–12.22 complex, 7.12–7.13, 12.5, 12.6 distributed, 24.1 in gradient index optics, 24.2–24.3 in integrated optics, 21.8–21.9 of polarizers, 12.16, 12.18 for rays in heterogeneous media, 1.21–1.22 of shallow radical gradients, 24.7–24.8 Refractive optics, 23.7, 23.8 Refractive systems, 1.9 Region of sag (axial gradients), 24.4f Relativity, of conjugate matrices, 1.70 Relay lenses, 17.10 Relay trains, in afocal lenses, 18.17–18.19, 18.17f, 18.18f Remote sensing, polarimetry and, 15.37–15.38 Remote sensing scanners, 30.2–30.4, 30.14–30.25 circular scan, 30.16, 30.18f compound mirror optics configurations for, 30.15–30.16, 30.15f multiplexed image scanning by, 30.23, 30.24f object- and image-space, 30.18–30.23 parallel-beam, 30.23–30.25, 30.24f–30.25f pushbroom scan, 30.18 resolution of, 30.6–30.8, 30.7f rotating wedge, 30.16, 30.17f single-mirror, 30.14, 30.15f two-dimensional, 30.18, 30.19f Resolution: of cameras, 25.5–25.6, 25.6f of microscopes, 28.17–28.24 Airy disk and lateral resolution, 28.17–28.19, 28.18f, 28.19f depth of field, 28.22–28.23 depth of focus, 28.22 three-dimensional diffraction pattern, 28.19–28.22, 28.20f, 28.21f and objective optics, 30.33 Rayleigh, 30.56, 33.17 of scanners, 30.6–30.14 data rates and remote sensing, 30.6–30.8, 30.7f input/output scanning, 30.8–30.14, 30.10f, 30.10t, 30.11t, 30.12f–30.13f of solid-state cameras, 26.15–26.16, 26.16f

Resolution limit, 1.80 Resonant scanners, 30.41–30.44, 30.43f, 30.44f Responsivity, of solid-state cameras, 26.9–26.10 Retardance, 14.6, 15.8 Retardance modulators, 15.20 Retardance space, 14.6 Retardation plates, 12.24–12.27, 12.25f, 12.26f, 13.43–13.53, 13.43t–13.44t achromatic, 13.48–13.52, 13.50f, 13.53t composite, 13.52, 13.53 crystalline-quartz, 13.46–13.48 defined, 15.8 mica, 13.45–13.46 quarter-wave and half-wave, 12.24–12.27, 12.25f, 12.26f rhomb-type, 13.52, 13.53t variable, 13.53 Retarder space, 14.14–14.15, 14.15f Retarders: defined, 15.8 Mueller matrices for, 14.11–14.15, 14.12t, 14.15f Retrofocus lenses, 27.2 (See also Inverted telephoto camera lenses) Retro-reflection testing and correction, 15.28–15.29, 15.29f Retroreflectors, 19.3t, 19.28, 19.28f Return-path ellipsometers, 16.16–16.17, 16.17f Return-path ellipsometers (RPEs), 16.16–16.17, 16.17f Reverse telephoto lenses, 17.29, 17.34f Reversibility, of rays, 1.9 Reversion prisms, 19.3t, 19.14, 19.14f Revolution, systems of, 1.32–1.43 paraxial optics of, 1.37–1.43 ray tracing in, 1.35–1.37 surfaces, 1.32–1.35 unfolded reflections, 1.32 Rhomboidal prisms, 19.3t, 19.25, 19.25f Rhomb-type retardation plates, 13.52, 13.53t Riccati-Bessel functions, 7.12 Right circular polarizers, Mueller matrices for, 14.10t Right half-wave circular retarders, Mueller matrices for, 14.12t Right-angle prisms, 19.3, 19.3t, 19.4f Right-circularly polarized light, 12.27, 12.28n Ring field lens design, 18.22 Risley prisms, 19.3t, 19.25–19.27, 19.25f–19.27f Ritchey-Chretien objectives, 29.7–29.8

INDEX

Ritchey-Chretien primaries, 29.10 Rochon prisms, 13.7, 13.18–13.21, 13.18f, 13.19f, 13.24 Roof prisms, 19.11, 19.12f (See also Amici prisms) Rotating retarders, 15.20 Rotating wedge scanners, 30.16, 30.17f Rotating-analyzer ellipsometer (RAE), 16.13, 16.13f, 16.14 Rotating-compensator fixed analyzer (RCFA) photopolarimeter, 16.14 Rotating-detector ellipsometer (RODE), 16.14, 16.14f Rotating-element photopolarimeters (REPs), 16.13 Rotation sensors, for fiber interferometers, 32.14–32.15, 32.15f Rotational spectra, 10.20–10.22 Rotationally symmetric lenses, 1.27, 1.60–1.62, 1.62f Rotationally symmetric systems, 1.17, 1.89–1.90 Routers, waveguide grating, 21.24 Rowland circle, 20.5, 20.7, 20.8, 20.10f Rydberg constant, 10.3 Rytov’s series of exponential approximations, 9.4 Sag, of surfaces, 1.32, 1.33f Sagittal fans and foci, 1.35 Sagnac interferometers, 21.35, 21.36, 21.36f, 32.3–32.4, 32.4f Sampled tracking, on optical disks, 35.16, 35.16f Sample-measuring polarimeters, 15.4, 15.13–15.14, 15.13f, 15.16–15.17 Sampling, with solid-state cameras, 26.16–26.19, 26.17f–26.19f SAOBIC processor, 11.20 Saturated absorption spectroscopy, 31.24–31.26, 31.24f–31.26f Saturation equivalent exposure (SEE), 26.10–26.11 Savart plates, 13.56 Scalar diffraction theory, for binary optics, 23.10–23.13, 23.11t, 23.12f, 23.13f Scalar field amplitude, 5.3 Scaling law, of spectrum of light, 5.21 Scan error reduction, 30.48–30.51, 30.49t, 30.50f, 30.51f Scan magnification, 30.5, 30.12–30.14

I.35

Scanners, 30.1–30.63 acousto-optic, 30.44–30.45 agile beam steering, 30.51–30.63 decentered lens and mircolens arrays, 30.57–30.60, 30.58f–30.60f, 30.62–30.63 digital micromirror devices, 30.60–30.61 gimbal-less two-axis scanning micromirrors, 30.61–30.62, 30.62f phased-array, 30.52–30.57, 30.53f, 30.62–30.63 electro-optic (gradient), 30.45–30.48, 30.46f–30.48f error reduction in, 30.48–30.51, 30.49t, 30.50f, 30.51f galvanometer and resonant, 30.41–30.44, 30.43f, 30.44f holographic, 30.38–30.41, 30.40f–30.42f input/output scanning, 30.2, 30.4–30.6, 30.4t, 30.25–30.34 objective, preobjective, and postobjective, 30.28–30.29, 30.29f, 30.30f objective optics, 30.30–30.33, 30.32f–30.33f power density and power transfer of, 30.25–30.28, 30.27f, 30.28f resolution of, 30.8–30.14, 30.10f, 30.10t, 30.11t, 30.12f–30.13f Keplerian afocal lenses for, 18.13, 18.13f modulation transfer function (MTF) for, 4.6 monogon and polygon, 30.34–30.38, 30.34f, 30.35f remote sensing, 30.2–30.4, 30.14–30.25 circular scan, 30.16, 30.18f compound mirror optics configurations, 30.15–30.16, 30.15f multiplexed image scanning, 30.23, 30.24f object- and image-space, 30.18–30.23, 30.19f–30.23f parallel-beam, 30.23–30.25, 30.24f–30.25f pushbroom scan, 30.18 rotating wedge, 30.16, 30.17f single-mirror, 30.14, 30.15f two-dimensional, 30.18, 30.19f resolution of, 30.6–30.14 data rates and remote sensing, 30.6–30.8 input/output scanning, 30.8–30.14, 30.10f, 30.10t, 30.11t, 30.12f–30.13f Scanning, active, 30.4

I.36

INDEX

Scattering: and backscattering, 6.5–6.7, 6.5f, 9.14–9.15, 9.14f Brillouin, 31.30 by coated spheres, 7.14 coherent and incoherent, 9.2, 9.3 by cylinders, 7.14 Mie, 7.11, 7.12, 9.17 molecular, 7.11 neutron, 9.6 in optical spectrometers, 31.30–31.31, 31.31f and polarization, 15.38–15.39, 15.38f–15.40f Raman, 31.30–31.31, 31.31f Rayleigh, 7.11, 9.17, 31.30 theory of, 9.3–9.4, 9.4f x-ray, 9.6 See also related topics, e.g.: Volume (multiple) scattering Scattering, by particles, 7.1–7.17 in coherent vs. incoherent arrays, 7.2–7.3 isotropic homogenous spheres, 7.11–7.14 Mie scattering, 7.11, 7.12 nonspherical particles, 7.15–7.17 regular particles, 7.14–7.15 single particles, 7.2–7.3 theories of, 7.3–7.4 volume scattering vs., 9.2–9.3 Scattering cross section, 7.4 Scattering length, 9.6 Scattering matrices, 7.10, 16.8–16.9 (See also Mueller matrices) Scattering planes, 7.9 Scattering potentials, 9.6 Scheimpflug condition, 1.61, 17.6f, 17.7, 18.4, 25.18, 25.19f Scheimpflug rule, 18.4, 18.8 Schell model sources (of light), 5.11 Schiefspiegler objectives, 29.26–29.27 Schmidt objectives, 29.14 Baker super-Schmidt, 29.21 field-flattened, 29.14–29.15 reflective, 29.15 reflective Schmidt-Cassegrain, 29.17 Schmidt-Cassegrain, 29.16–29.17 Schmidt-meniscus Cassegrain, 29.21 Shafer-relayed-virtual, 29.17–29.18, 29.18f solid, 29.16 Schmidt prisms, 19.3t, 19.12, 19.12f Schrödinger equation, 10.4 Schwarzschild arrangement, for McCarthy objective, 29.33

Schwarzschild objectives, 29.12–29.13, 29.32 Scophony TV projection system, 30.45 Scorotron, in xerographic systems, 34.2, 34.3f SEAL objective, 29.27–29.28 Second Brewster angle, 12.13 Secondary magnification, 29.12, 29.38 Secondary sources of light, 5.9–5.10 Secondary spectrum, 29.7, 29.38 Second-harmonic interferometers, 32.17–32.18, 32.18f Sectors (optical disk data), 35.4, 35.6–35.7 Seek operation, on optical disks, 35.17 Seidel, Philipp Ludwig von, 17.27 Seidel aberrations, 1.90, 29.38 Selective area epitaxy, 21.19–21.20 Self-centering lens springs, 22.8, 22.8f Self-coherence function, 2.41 Selfoc lenses, 24.2, 24.7f Semiconductors, complementary metal-oxide, 26.8–26.9 Semifield angles, of prisms, 13.7 Sénarmont compensators, 13.53, 28.38 Sénarmont polariscopes, 12.30 Sénarmont prisms, 13.7, 13.18, 13.18f, 13.21 Sensors: active pixel, 26.2, 26.8–26.9, 26.8f generalized, 32.15–32.16, 32.15f multiplexed, 32.16 rotation, 32.14–32.15, 32.15f Serial incoherent matrix-vector multipliers, 11.17–11.18, 11.18f Sewer cameras, 25.22–25.23 Shafer objectives: five mirror unobscured, 29.33–29.34 four mirror unobscured, 29.33 Shafer relayed virtual Schmidt, 29.17–29.18, 29.18f two-mirror three reflection, 29.25 Shallow radial gradient index (SRGRIN), 24.7 Shape factor, of lenses, 17.12–17.13, 17.13f Shearing interferometers, 32.4, 32.6f Sheet polarizers, 13.25–13.28 Shenker objective, 29.23 Shenker objectives, 29.23 Shift invariance, 4.2 Shot noise, of solid-state cameras, 26.11 Siemens star, 28.30f, 28.47f Sierpinski Gasket, 8.9 Signal, for solid-state cameras, 26.9–26.11, 26.13–26.14

INDEX

Signal-to-noise ratio (SNR): of optical disk data, 35.23 of solid-state cameras, 26.13–26.14 Silicon photonics transmission, 21.14–21.16, 21.15f, 21.38–21.40 Silicon-on-insulator (SOI) technology, 21.14–21.15, 21.15f Sine condition, for stigmatic imaging, 1.30–1.31 Singham, Shermila Brito, 7.16 Single element lenses, 17.12–17.17, 17.13f–17.16f, 17.17t Single lens reflex (SLR) cameras: autofocus, 25.12–25.14, 25.13f features of, 25.8, 25.9f formats of, 25.18 lenses for, 27.1–27.2, 27.3f–27.4f normal lenses for, 27.2, 27.3f–27.4f and time lag, 25.8–25.9 Single molecule high-resolution colocalization (SHREC), 28.23 Single scattering, 7.2–7.3 coherent and incoherent, 9.4–9.7, 9.6f dynamic, 9.7–9.8, 9.7f and volume scattering, 9.2–9.8 Single sideband edge enhancement (SSEE) microscopy, 28.29–28.33, 28.30f–28.33f Single speckle, 8.17 Single-component development, in xerographic systems, 34.9, 34.9f Single-mirror scanners, 30.14, 30.15f Single-mode waveguides, 21.4 Single-order plates, 13.47 Singlet lenses, 17.37–17.38 Singular value decomposition (SVD), 15.25–15.27 Sinusoidal ray paths, 24.2 Skew invariant, 1.21, 1.23 Skew rays, 1.35, 1.73 Skewness, 1.23 Skin depth, 7.13 Slater parameters, 10.12 Slow axis, 12.25, 15.8 Small-perturbation approximation, for surface scattering, 8.9–8.12 SMARTCUT technique, 21.14, 21.15 Smith invariant, 1.77 (See also Two-ray paraxial invariant) Smith reflectors, 28.44 Snell’s law, 1.24, 1.38, 12.16, 13.3, 13.5, 13.19 Solano objectives, 29.26 Soleil compensators, 13.55–13.56, 13.55f

I.37

Soleil-Babinet compensators, 35.21n Sol-gel formed glass, 24.8 Solid Makutsov-Cassegrain objective, 29.19 Solid Schmidt objective, 29.16 Solid state spectroscopy, 10.22–10.26, 10.23f–10.27f Solid-state cameras, 26.1–26.20 applications, 26.3 array performance in, 26.9–26.12, 26.12f and charge injection devices, 26.6–26.7, 26.6f–26.8f and charge-coupled devices, 26.3–26.5, 26.4f–26.6f complementary metal-oxide semiconductor (CMOS), 26.8–26.9 modulation transfer function (MTF) for, 26.14 performance metrics for, 26.12–26.16, 26.16f sampling with, 26.16–26.19, 26.17f–26.19f Space curves, differential geometry of, 1.18–1.19 Space-bandwidth product, 6.9 Space-integrating correlator, 11.11 Spar cutting, Ahrens method of, 13.12 Sparrow criterion, 28.18, 28.19 Spatial coherence, 2.38–2.40, 2.38f–2.39f, 5.3, 5.5 Spatial filtering, 11.5–11.6, 11.6f Spatial multiplexing, 23.8 Spatial-frequency content, 4.8 Specimen, for microscopes, 28.22, 28.54–28.55 Speckle: and coherence theory, 5.22 laser, 9.14 objective and subjective, 33.9 single, 8.17 in volume scattering, 9.15–9.16, 9.15f Spectral coherence, 5.5 Spectral transitions, 10.6–10.7 Spectrographs, charge, 34.8 Spectrometers, 31.1–31.31 Bunsen-Kirchhoff, 20.5f dispersive prisms and gratings for, 20.2–20.3, 20.3f high-resolution techniques of, 31.23–31.29 fluorescence line narrowing, 31.29, 31.29f laser Stark spectroscopy of molecules, 31.27–31.29, 31.27f, 31.28f polarized absorption spectrometers, 31.26 saturated absorption, 31.24–31.26, 31.24f–31.26f

I.38

INDEX

Spectrometers (Cont.): and light scattering, 31.30–31.31, 31.31f luminescence, 31.5–31.12, 31.8f, 31.11f optical absorption, 31.2–31.5, 31.5f and photoluminescence decay time, 31.12–31.15, 31.13f, 31.14f polarization, 31.15–31.23 and optically detected magnetic resonance, 31.21–31.23, 31.22f, 31.23f polarized absorption by, 31.15–31.17, 31.15f, 31.18f polarized absorption/luminescence techniques for, 31.17, 31.19–31.21, 31.19f prism, 20.2–20.3, 20.3f Unicam double-monochromator, 20.10, 20.13, 20.15f Spectrophotometric measurements, 12.15 Spectropolarimetry and spectropolarimeters, 15.6, 15.8 Spectroradiometers, 20.1, 20.2f, 20.14t Spectroscopic ellipsometry (SE), 16.3 Spectroscopic lineshapes, 10.6–10.7, 10.22–10.27 in solid state spectroscopy, 10.22–10.26, 10.23f–10.27f of spectral transitions, 10.6–10.7 Spectroscopic transition, rates of, 10.4–10.6 Spectroscopy, 3.29, 10.1–10.22 of multielectron atoms, 10.10–10.11 of one-electron atoms, 10.7–10.9, 10.8f, 10.9f and outer electronic structure, 10.12–10.16, 10.13f–10.15f photon correlation, 9.8 rates of spectroscopic transition, 10.4–10.6 solid state, 10.22–10.26 theoretical basis, 10.3–10.4 of tri-positive rare earth ions, 10.16–10.18, 10.16t, 10.17f and vibrational and rotational spectra, 10.18–10.22, 10.19f, 10.21f Spectrum: power, 8.12–8.13 of primary light source, 5.6 rotational, 10.20–10.22 secondary, 29.7, 29.38 vibrational, 10.18–10.20 and wavefront division, 2.17–2.18, 2.18f Spectrum of light, 5.19–5.22 coherence functions for, 5.5–5.6 coherent mode representation of, 5.20–5.21

Spectrum of light (Cont.): limitations, 5.19, 5.20f for primary sources, 5.6 scaling law, 5.21 Wolf shift, 5.21 Spheres, scattering by, 7.11–7.14 Spherical aberrations, 1.90, 29.7, 29.38 oblique, 29.15, 29.21, 29.37 zonal, 29.8, 29.38 Spherical lenses, 6.3 Spherical primaries, in objective designs, 29.9, 29.11, 29.18, 29.30 Spherical surfaces, in systems of revolution, 1.34 Spherical waves, 2.4, 2.5f, 3.2–3.3 interference from, 2.11–2.12, 2.12f, 2.13f and plane waves, 2.9–2.11, 2.10f Spherochromatism, 24.3–24.6, 29.14 Split-aperture scanners, 30.15–30.16, 30.15f Spontaneous decay rate, 10.6 Square-ended Nicol prisms, 13.16f, 13.17 Squirm, 13.14 Stark effect, 21.11–21.12 Stark spectroscopy of molecules, laser, 31.27–31.29 Stationary phase approximation, in diffraction, 3.29, 3.31–3.32 Stationary surfaces, Fresnel-Kirchhoff approximation for, 8.6–8.8 Statistical radiometry, 5.22 Steeg and Reuter Nicol prisms, 13.17 Stellar interferometers, 32.19–32.21, 32.19f, 32.20f Stereo cameras, 25.23–25.24, 25.23f Stigmatic imaging, 1.29–1.31, 1.30f Stimulated emission depletion (STED), 28.24 Stochastic optical reconstruction microscopy (STORM), 28.23 Stokes matrix, 14.4 (See also Mueller matrix) Stokes parameters: and Mueller matrices, 14.4–14.6 and Poincaré sphere, 14.4–14.6, 14.5f, 14.26f, 14.33 Stokes polarimeters, 15.4, 15.5, 15.25 Stokes vectors, 12.14n, 12.28 and Mueller matrices, 14.15–14.17, 14.19–14.21 for nonhomogeneous polarization elements, 14.26f in polarimetry, 15.3, 15.8–15.13 for radiative transfer, 9.11 for speckle patterns, 9.17

INDEX

Stop shift, 1.92 Stops: aperture, 1.74, 1.75f, 17.8, 29.5, 29.36 field, 1.74, 17.9, 29.5, 29.37 of lenses, 17.8–17.9 S-trace formula, 1.44 Straight edges, cylindrical wavefronts and, 3.14–3.16, 3.14f, 3.15f Stratified-medium model (SMM), in ellipsometry, 16.4 Stray light suppression, 29.5, 29.5f Streak cameras, 25.24, 25.24f Strehl index, 8.6 Sub-Doppler absorption spectroscopy, 31.26–31.27 Subjective speckle, 33.9 Superzone construction, for micro-Fresnel lenses, 22.36 Surface acoustic wave (SAW) devices, 21.16 Surface scattering, 8.1–8.17 finish models for, 8.14–8.15 and finite illumination area, 8.12 Fresnel-Kirchhoff approximation for, 8.5–8.9 fractal surfaces, 8.8–8.9 statistically stationary surfaces, 8.6–8.8 notation for, 8.2–8.4 and power spectra, 8.12–8.13, 8.13f Rayleigh-Rice approximation, 8.9–8.12 second-order statistical functions for, 8.12–8.13 statistics for, 8.12–8.15 and surface finish specifications, 8.15–8.17 Surfaces, 1.32 aspherical, 1.35 conical, 1.34–1.35 spherical, 1.34 Suspension systems, of resonant scanners, 30.43, 30.43f, 30.44 Sweatt lenses, 23.4 Sweatt model, 23.4, 23.6 Switches and switching: in integrated optics, 21.34–21.35, 21.34f interferometric optical switches, 32.19 Symmetrical lenses, 1.71, 17.26–17.27, 17.27f Synthetic aperture radar data, 11.6–11.8, 11.7f–11.8f System-response (SR) function, 8.12 Tangent vectors, of space curves, 1.18, 1.19 Tangential fans and foci, 1.35 Telecentric lenses, 18.12

I.39

Telecentric stop, 17.9 Telecentricity, 1.83–1.84, 30.31, 30.31f, 30.32 Telephoto lenses, 17.29, 27.2, 27.6, 27.7f–27.16f, 27.13 Telescopes: astronomical, 18.10 Cassegrainian, 18.21 field of view in, 18.15–18.16, 18.15f Galilean, 18.15 Keplerian, 18.10 Mersenne, 18.19 objectives for, 29.8, 29.9, 29.12, 29.29–29.30 terrestrial, 18.10–18.11, 18.11f Telescopic lenses, 1.46 (See also Afocal lenses) Telescopic transformations, 1.57 Television: integrated optics and cable, 21.2, 21.32–21.34 Scophony TV projection system, 30.45 Temperature: and crystalline-quartz retardation plates, 13.48 Curie, 35.25 and integrated optics, 21.12 Temporal beats, interference and, 2.13 Temporal coherence, 2.41, 5.3 Temporal signals, analog processing of, 11.8–11.12, 11.9f–11.11f Tensor, electro-optic, 21.10 Tensor product, of Jones matrices, 14.23 Terrestrial telescopes, 18.10–18.11, 18.11f Tessar lenses, 17.26, 17.26f Tewarson, 12.30 Thermal imaging cameras, 25.25 Thermal optimization, of media, 35.20, 35.20f Thermal (Lambertian) sources of light, 5.12–5.13 Thermomagnetic recording process, 35.17–35.20, 35.18f–35.20f Thick lens systems, 1.55 Thin film oxides, 21.13–21.14 Thin films, interference and, 2.24 Thin lens systems, 1.55 Thin-lens model: of Galilean afocal lenses, 18.15, 18.15f of Keplerian afocal lenses, 18.7–18.8, 18.8f Third-order aberrations, 1.90–1.91, 29.38 Thompson reversed Nicol prisms, 13.16f, 13.17 Thomson CSF, 21.35 Three-beam interferometers, 32.7–32.8, 32.7f Three-dimensional diffraction patterns, 28.19–28.22, 28.20f, 28.21f

I.40

INDEX

Three-lens prime focus corrector objective, 29.10 Three-mirror objectives, 29.28–29.32, 29.34 Three-phase model (ellipsometry), 16.5–16.8, 16.6f–16.8f Three-powered-mirror lenses, 18.20, 18.20f, 18.21 Three-ray rule, 1.42 Throughput, 1.22, 13.7 (See also Étendue) Tilted planes, for collineation, 1.61, 1.62, 1.62f Tilted-plane processors, 11.8, 11.8f Time, coherence, 5.3 Time averages, in coherence theory, 6.2n, 6.4–6.5 Time domain, 7.17 Time lag, of cameras, 25.8–25.9, 25.9f Time-integrating correlators, 11.11–11.12, 11.11f Time-sequential polarimeters, 15.5 T-matrix method, 7.15 T-number, 1.79 Toner, in xerographic systems, 34.1, 34.8–34.9, 34.8f Topothesy, of fractals, 8.8 Torsion, radius of, 1.19 Total internal reflection: of optical waveguides, 21.3 of Rochon prisms, 13.20 Track-error signal (TES), 35.14–35.15, 35.15f, 35.17 Tracks, on optical disks, 35.2–35.5, 35.3f–35.5f Transfer, in xerographic systems, 34.10 Transfer functions (see specific functions, e.g.: Modulation transfer function (MTF)) Transfer matrices, 1.66 Transillumination, in microscopes, 28.5–28.7, 28.6f Translation, by scanners, 30.5 Transmission: amplitude-shift-keyed, 21.30 analog, 21.32–21.34, 21.33f, 21.34f differential-phase-shift-keyed, 21.30, 21.32 digital, 21.31–21.32 by Glan-Thompson-type prisms, 13.9–13.10, 13.10f, 13.11f by LEDs, 21.32 in multilayer systems, 16.8–16.9 nonnormal-incidence, 12.18–12.24 silicon photonics, 21.14–21.16, 21.15f by silicon photonics, 21.38, 21.39 Transmission ellipsometry, 16.10 Transmissive planar objects, 6.3

Transmittance: of Brewster angle transmission polarizers, 12.22 of Mueller matrices, 14.16–14.17 of pile-of-plates polarizers, 12.15–12.17 principal, 12.14–12.16 of spherical lenses, 6.3 Transmitted state, of polarizers, 15.19 Transverse electric (TE) modes, of optical waveguides, 21.6, 21.7 Transverse magnetic (TM) modes, of optical waveguides, 21.6, 21.7 Transverse magnification, 1.28, 1.50–1.51 Transverse primary chromatic aberration (TPAC), 17.22 Transverse ray aberrations, 1.87 Transverse translational scan, 30.28 Traveling wave modulators, 21.26 Trigger detection, with ODMR, 31.21 Tri-Level highlight color process, 34.12, 34.13f Trim retarders, for LC panels, 15.33 Trimmed Nicol-type polarizers, 13.16–13.18, 13.16f Triplet lenses, 17.26, 17.26f Tri-positive rare earth ions, 10.16–10.18, 10.16t, 10.17f Trischiefspiegler objective, 29.27 T-trace formula, 1.44 Tube length (objective lenses), 28.13, 28.13t Tutton’s test, 13.45 Two-component magnetic brush development, in xerographic systems, 34.5–34.7, 34.5f–34.7f Two-dimensional images, analog processing for, 11.12–11.17, 11.13f Two-dimensional scanners, 30.18, 30.19f Two-lens systems, 17.20–17.22, 17.21f–17.22f Two-mirror, three reflection objective, 29.26 Two-phase model (ellipsometry), 16.5, 16.6f Two-powered-mirror lenses, 18.20, 18.20f Two-ray paraxial invariant, 1.41 Two-wavelength interferometry, 32.9 Twyman-Green interferometers, 2.28, 32.2, 32.9, 33.5 Unblazed gratings, 20.3–20.4 Underillumination, for input/output scanning, 30.14 Underwater cameras, 25.25 Unfolded reflections, in systems of revolution, 1.32

INDEX

Unicam double-monochromator spectrometer, 20.10, 20.13, 20.15f Uniformity, of binary optics, 23.7 Unit determinants, paraxial matrix methods for, 1.67 Upatnieks, J., 33.2 Van Cittert-Zernike theorem, 2.38, 2.39, 5.13–5.14, 5.17–5.19, 6.4, 6.8 Vander Lugt filters, 11.13–11.14 Vapor-phase epitaxy (VPE), 21.17, 21.18 Variable retardation plates, 13.53 Variable-angle spectroscopic ellipsometry (VASE), 16.3 Variational integral, of rays, 1.19 Vector aberration theory, 29.5 Vector diffraction, 3.32–3.37, 3.32f–3.37f, 23.13–23.14, 23.14f Vector Huygens secondary source (unit), 3.33–3.36, 3.37f Vectors: electric field, 2.3–2.4 of space curves, 1.18, 1.19 Vector-scattering amplitude, 7.8 VeriMask, 33.19 Vertex, for figure of revolution, 1.32 Vertex curvature, 1.32–1.33 Vertical half-wave linear retarders, Mueller matrices for, 14.12t Vertical linear polarizers, 14.10t Vertical quarter-wave linear retarders, Mueller matrices for, 14.12t Vibration, planes of, 12.6 Vibrational relaxation, 31.14, 31.14f Vibrational sidebands, 10.24 Vibrational spectra, 10.18–10.20 Video cameras, 25.7–25.8 Videocassette recorders (VCRs), 35.1 Video-enhanced differential-interference contrast (VE-DIC), 28.41 View cameras, 25.18–25.20, 25.19f Vignetting, 1.81, 1.81f, 1.82, 17.8, 29.6, 29.38 Virtual images, 29.38 Virtual pupils, 1.76 Virtual rays, 1.10 Visual magnification, 1.28 VLSI-CMOS technology, silicon photonics for, 21.15, 21.16 Volume imaging ideal, 1.29

I.41

Volume scattering, 9.1–9.17 multiple scattering, 9.8–9.17 analytical theory of, 9.9–9.10, 9.9f depolarization, 9.16–9.17, 9.17f effective-medium representation, 9.8 radiative transfer, 9.10–9.13, 9.11f, 9.13f speckle patterns, 9.15–9.16, 9.15f weak localization, 9.13–9.17, 9.14f single particle scattering vs., 9.2–9.3 and single scattering, 9.4–9.8, 9.6f, 9.7f theory of, 9.3–9.4, 9.4f Volumetric reflection, 7.7f Volumetric scattering cross section, 7.7f Wadsworth configuration, 20.5f, 20.8, 20.12f, 20.14t WALRUS objective, 29.28 Wave equation, 12.4–12.6 Wave normals, 13.5 Wavefront aberration coefficients, 1.90 Wavefront aberrations, 1.86–1.88 Wavefront division, 2.14 Wavefront division, interference by, 2.14–2.19, 2.15f–2.18f Wavefront multiplexers, 23.11, 23.11t, 23.12 Wavefront quality, of binary optics, 23.8 Wavefronts, 3.4 aberrated, 2.12, 2.13 cylindrical, 3.13–3.21, 3.14f Cornu’s spiral, 3.16–3.19 opaque strip construction, 3.20–3.21 from rectangular apertures, 3.19–3.20 from straight edge, 3.14–3.16 disturbance of, 3.5–3.6 for cylindrical wavefronts, 3.13–3.14, 3.14f and straight edges, 3.14–3.15 geometrical, 1.12–1.13 and interference, 2.4–2.5 Waveguide grating routers (WGRs), 21.24 Waveguides: integrated optics, 21.3–21.8, 21.3f–21.5f, 21.7f leaky, 21.3 Wavelength: and modulation transfer function, 17.38, 17.39 of plane waves, 2.4 Wavelength division multiplexes (WDM) systems: fabrication of, 21.14 filters for, 21.23 in integrated optics, 21.37–21.38, 21.37f–21.39f

I.42

INDEX

Wavelength interval (between oscillations), 12.10 Waveplates, 15.8 Waves, 2.3–2.6 amplitudes of, 2.4, 2.5, 12.5 diffraction of, 3.2–3.3, 3.3f interference of, 2.5–2.6 plane, 2.4, 2.5f, 3.3, 3.17 decomposition of, 3.23 interference of, 2.8–2.9, 2.9f and spherical waves, 2.9–2.11, 2.10f spherical, 2.4, 2.5f, 3.2–3.3 interference of, 2.11–2.12, 2.12f, 2.13f and plane waves, 2.9–2.11, 2.10f Weak polarization elements (Mueller matrices), 14.26–14.27 Wernicke prisms, 20.6f Wetherell and Womble objectives, 29.31 Weyl’s integral, 5.17 Weyl’s plane-wave decomposition, 3.23 Wide-angle photography: cameras for, 25.25, 25.26f lenses for, 27.2, 27.5f with nonrectilinear distortion, 27.6 with rectilinear distortion correction, 27.6, 27.13f, 27.14f Wide-field objective with Maksutov correction, 29.20 Wieman-Hansch experiment, 31.26 Wiener filter, 11.15–11.17 Wiener-Khintchine theorem, 5.5, 5.21 Wien’s wavelength displacement law, 31.4 Wigner distribution function, 5.8 Winchester heads, 35.6 Wire grids, 13.30n Wire-grid polarizers, 13.30–13.33, 13.31f, 13.32t Wiscombe, W., 7.12, 7.15 Wizinowich, P. L., 12.15, 13.52 Wolf shift, 5.21 Wollaston, W. H., 17.17, 17.18

Wollaston prisms, 13.7, 13.18, 13.18f, 13.21, 13.24 for DIC microscopes, 28.39, 28.40 in Nomarski interferometers, 32.4 and nondispersive prisms, 19.3t, 19.15, 19.15f Working distance, of objective lenses, 28.13 Wright objective, 29.15 Wright objectives, 29.15 Write-once-read-many (WORM) technology, 35.2 Xerographic systems, 34.1–34.13, 34.2f cleaning and erasing in, 34.10 color in, 34.11–34.12, 34.11f–34.13f control of, 34.11 development in, 34.5–34.10, 34.5f–34.9f fusing in, 34.10 and latent image, 34.1–34.4, 34.2f–34.3f transfer in, 34.10 X-ray scattering, 9.6 Yolo objectives, 29.26 Young’s astigmatic formulae, 1.44 Young’s double slit experiment, 2.14–2.15, 2.15f Young’s fringes, 13.45 Young’s modulus, for molded microlenses, 22.12t Young’s two pinhole interferometer, 6.3 Young-Thollon half prisms, 20.7f Zeeman effect, 31.17, 31.18f, 31.19–31.21 Zeiss Infinity Color-Corrected Systems Plan Apo, 28.15, 28.15f Zeiss prism system, 19.3t, 19.16, 19.16f Zeiss sheet polarizers, 13.26 Zenger prisms, 20.6f Zernike polynomials, 1.90, 23.3 Zero-phonon transitions, 10.24–10.26, 10.24f Zeros, in paraxial matrices, 1.68 Zonal spherical aberrations, 29.8, 29.38 Zone plates, 3.11–3.13, 3.12f Zoom lenses, 27.17, 27.20f–27.22f

COLOR PLATES

FIGURE 15.2 A sample-measuring polarimeter consists of a source, polarization state generator (PSG), the sample, a polarization state analyzer (PSA), and the detector.

FIGURE 15.3 The dual rotating retarder polarimeter consists of a source, a fixed linear polarizer, a retarder which rotates in steps, the sample, a second retarder which rotates in steps, a fixed linear polarizer, and the detector.

FIGURE 15.6 Imaging polarimeter configured for retro reflection testing using a non polarizing beam splitter and beam dump.

FIGURE 28.28 Michel Lévy Color Chart of interference colors. The horizontal sequence of colors is associated with the interference of two beams of white light whose mutual path difference increases from 0 (left, black for destructive interference) to more than 1700 nm (right, pale green). By comparing an experimentally observed color with the colors of this chart, one can estimate the path difference caused, for example, by the partial reflection off a thin dielectric film (e.g., soap bubble) or transmission through a thin birefringent sheet (e.g., mica) sandwiched between crossed polarizers. For birefringent materials, the path difference is the product of the birefringence (indicated along the upper and right edge) and the thickness (left edge) of the material. The diagonal lines assist in estimating one of the quantities (birefringence or thickness) from the observed color, if the other quantity is known. This brightness/color sequence is specific to interference phenomena that result in destructive interference for zero path difference (e.g., birefringent sheet between crossed polarizers.) A complementary color sequence applies to interference phenomena that result in constructive interference at zero path difference (e.g., birefringent sheet between parallel polarizers.)

FIGURE 32.26

Gravitational-wave detector using two Fabry-Perot interferometers.

Handbook of Optics

Handbook of Optics

Handbook of Optics

Handbook of Neutron Optics