Physical Optics and Light Measurements, Volume 26 (Methods in Experimental Physics)

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Methods of Experimental Physics VOLUME 26 PHYSICAL OPTICS AND LIGHT MEASUREMENTS

METHODS OF EXPERIMENTAL PHYSICS Robert Celotta and Judah Levine, Editors-in-Chief

Founding Editors

L. MARTON

C. MARTON

Volume 26

Physical Optics and Light Measurements Edited by

Daniel Malacara Centro de lnvestigaciones en Optice Leon, Gto Mexico

ACADEMIC PRESS, INC. Harcourt Brace Jovanovich, Publishers

Boston

San Diego New York Berkeley London Sydney Tokyo Toronto

Copyright 0 1988 by Academic Press, Inc. All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information qtorage and retrieval system, without permission in writing from the publisher. ACADEMIC PRESS, INC. 1250 Sixth Avenue, San Diego, CA 92101 United Kingdom Edifion publid~edbv ACADEMIC PRESS INC. ( L O N D O N ) LTD. 24-28 Oval Road. London NWI 7DX

Library of- Congress Cbraloging-in-Publication Data

Physical optics and light measurements. (Methods of experimental physics; v. 26) Includes bibliographies and index. 1. Optics, Physical. 2. Optical measurements. 1. Malacara, Daniel, Date11. Series. QC395.2.P49 1988 535’.2 88-978 ISBN 0-12-47597 1-8

Printed in the United States of America 88 89 90 91 9 8 7 6 5 4 3 2 1

CONTENTS LIST OF CONTRIBUTORS . .

. . . . . . . . . . . . . . . . . ix PREFACE. . . . . . . . . . . . . . . . . . . . . . . . . . xi ... LIST OF VOLUMESI N TREATISE . . . . . . . . . . . . . . . xi11

1 . Interference

by D A N I E LMALACARA 1.1 Introduction

. . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . 1.3. Multiple-Beam Interferometers . . . . . . . . . . . . . 1.4. Multiple-Pass Interferometers . . . . . . . . . . . . . 1.2. Two-Beam Interferometers

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.5. Applications of Interferometry References

1

7 25 37 38 45

2 . Diffraction and Scattering by D A N I E LMALACARA

..................... 49 2.2. Fresnel Diffraction . . . . . . . . . . . . . . . . . . 53 2.3. Fraunhofer Diffraction and Fourier Transforms . . . . . 62 2.4. Diffraction Gratings . . . . . . . . . . . . . . . . . 70 2.5. Resolving Power of Optical Instruments . . . . . . . . . 79 2.6. The Abbe Theory of the Microscope . . . . . . . . . . 86 2.7. Scattering . . . . . . . . . . . . . . . . . . . . . . 92 ..................... References 102 2.1. Diffraction

V

vi

CONTENTS

3. Optical Polarization by FREDERIC R . STAUFFER

3.1. Introduction

. . . . . . . . . . . . . . . . . . . . .

107

3.2. Electromagnetic Description of Light

108

3.3. Wave Propagation in Isotropic

. . . . . . . . . . Media . . . . . . . . . . . . . . . . . . . . . . .

113

3.5. Thin Films

125

3.6.

..................... Wave Propagation in Anisotropic Media . . . . . . . .

133

. . . . . . . . . . . . . . . . .

157

3.4. Wave Propagation for Metals

3.7. Slits, Gratings, and Metal Grid Polarizers 3.8. Light Source and Detector Polarizations

3.9. Polarization Determination and Mathematical Description

References

. . . . . . . . . . . . . . . . . . . . .

120

158 159 162

4 . Holography

by R. D. BAHUGUNA A N D D . MALACARA

. . . . . . . . . . . . . . . . . . . . . Holography . . . . . . . . . . . . . . . .

4.1. Introduction

167

4.2. Theory of

168

. . . . . . . . . . . . . Some Applications of Holography . . . . . . . . . . . Experimental Procedures in Holography . . . . . . . . References .. .. .. .. .. .. . . . . . . . . .

4.3. Different Types of Holograms

175

4.4.

191

4.5.

199 206

5 . Photometry and Radiometry by WILLIAM L. WOLFE 5.1. Introduction

.....................

5.2. Symbols, Units. and Nomenclature

5.3. Formulas for Blackbody Radiation

. . . . . . . . . . . . . . . . . . . . . .

213 214 219

vii

CONTENTS

. . . . . . . . . . . . . . Radiometric Temperature Measurements . . . . . . . Radiometric Instruments . . . . . . . . . . . . . . Measurements . . . . . . . . . . . . . . . . . . . . Photometry: Radiometry of Visible Light . . . . . . . References .....................

5.4. Simple Radiative Transfer

5.5. 5.6. 5.7. 5.8.

. . .

223 239 246 263

.

284 287

6. Detectors

by T. 0. POEHLER

........... 6.2. Figures of Merit . . . . . . . . 6.3. Thermal Detectors . . . . . . . 6.4. Photon Detectors . . . . . . . . 6.1. Introduction

6.5. Noise

. . . .

......... 291 . . . . . . . . . . 292 . . . . . . . . . . 293 . . . . . . . . . . 303

. . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .....................

6.6. Optical Window Material References

328 331 332

This Page Intentionally Left Blank

CONTRIBUTORS Numbers in parentheses indicate pages on which the authors’ contributions begin.

RAMENDRADEOBAHUGUNA(167), Centro de Znvestigaciones en Optica, Apartado Postal 948, 37000 Leon, Gto. Mexico DANIELMALACARA ( 1 , 49, 167), Centro de Investigaciones en Optica, Apartado Postal 948, 37000 Leon, Gto. Mexico THEODORE 0 . POEHLER(291), Applied Physics Laboratory, Johns Hopkins Road, Laurel, Maryland 20707 FREDERICR. STAUFFER(107), Sacramento Peak Observatory, Sunspot, New Mexico 88349 WILLIAM L. WOLFE(213), Optical Sciences Center, University of Arizona, Tucson, Arizona 85721

ix


PREFACE Two books covering the field of modern optics have been prepared in this series “Methods of Experimental Physics”, separating the material into two parts, one with the title “Geometrical and Instrumental Optics”, and the other with the title “Physical Optics and Light Measurements”. The purpose of these books is to help the scientist or engineer who is not a specialist in optics to understand the main principles involved in optical instrumentation and experimental optics. Our main intent is to provide the reader with some of the interdisciplinary understanding that is so essential in modern instrument design, development, and manufacture. Coherent optical processing and holography are also considered, since they play a very important role in contemporary optical instrumentation. Radiometry, detectors, and charge coupled imaging devices are also described in these volumes, because of their great practical importance in modern optics. Basic and theoretical optics, like laser physics, nonlinear optics and spectroscopy are not described, however, because they are not normally considered relevant to optical instrumentation. In this volume, “Physical Optics and Light Measurements”, Chapter One describes the theory and applications of interference and interferometers. Chapter Two studies diffraction, its basic theoretical fundamentals, and some practical applications. Polarized light and its uses are considered in Chapter Three. Holography and holographic methods are studied in detail in Chapter Four. The photometric and radiometric principles are covered in Chapter Five. Finally, Chapter Six considers detectors. There might be some overlapping of topics covered in different chapters, but this is desirable, since the points of view of different authors, treating different subjects, may be quite instructive and useful for a better understanding of the material. This book has been the result of the efforts of many people. Professor H. W. Palmer started this project and spent many fruitful hours on it. Unfortunately, he did not have the time to finish his editorial work due to previous important commitments. I would like to express my great appreciation of and thanks to Professor Palmer and all of the authors, without whom this book could never have been finished. I also thank Dr. R. E. Hopkins and many friends and colleagues for their help and encouragement. Finally, I appreciate the great understanding of my family, mainly my wife Isabel, for the many hours taken away from them during the preparation of these books. DANIEL MALACARA Leon, Gto. Mexico.


METH0DS OF EXPERIMENTAL PHYSICS Editors-in-Chief

Robert Celotta and Judah Levine Volume 1. Classical Methods Edited by lmmanuel Estermann Volume 2. Electronic Methods, Second Edition (in two parts) Edited by E. Bleuler and R. 0. Haxby Volume 3. Molecular Physics, Second Edition (in two parts) Edited by Dudley Williams Volume 4. Atomic and Electron Physics-Part A: Atomic Sources and Detectors; Part B: Free Atoms Edited by Vernon W. Hughes and Howard L. Schultz Volume 5. Nuclear Physics (in two parts) Edited by Luke C. L. Yuan and Chien-Shiung W u A: Preparation, Structure, Volume 6. Solid State Physics-Part Mechanical and Thermal Properties; Part B: Electrical, Magnetic, and Optical Properties Edited by K. Lark-Horovitz and Vivian A. Johnson Volume 7. Atomic and Electron Physics-Atomic Parts) Edited by Benjamin Bederson and Wade L. Fite

Interactions (in two

Volume 8. Problems and Solutions for Students Edited by L. Marton and W. F. Hornyak Volume 9. Plasma Physics (in two parts) Edited by Hans R. Griem and Ralph H. Lovberg Volume 10. Physical Principles of Far-Infrared Radiation By L. C. Robinson Volume 11. Solid State Physics Edited by R. V. Coleman

xiv

M E T H O D S OF EXPERIMENTAL PHYSICS

Volume 12. Astrophysics-Part A: Optical and Infrared Astronomy Edited by N. Carleton Part B: Radio Telescopes; Part C: Radio Observations Edited by M. L. Meeks Volume 13. Spectroscopy (in two parts) Edited by Dudley Williams Volume 14. Vacuum Physics and Technology Edited by G. L. Weissler and R. W. Carlson Volume 15. Quantum Electronics (in two parts) Edited by C. L. Tang Volume 16. Polymers-Part A: Molecular Structure and Dynamics; Part B: Crystal Structure and Morphology; Part C: Physical Properties Edited by R. A. Fava Volume 17. Accelerators in Atomic Physics Edited by P. Richard Volume 18. Fluid Dynamics (in two parts) Edited by R. J. Emrich Volume 19. Ultrasonics Edited by Peter D. Edmonds Volume 20. Biophysics Edited by Gerald Ehrenstein and Harold Lecar Volume 21. Solid State: Nuclear Methods Edited by J. N. Mundy, S. J. Rothman, M . J. Fluss, and L. C. Smedskjaer Volume 22. Solid State Physics: Surfaces Edited by Robert L. Park and Max G. Lagally Volume 23. Neutron Scattering (in three parts) Edited by K. Skold and D. L. Price Volume 24. Geophysics-Part A: Laboratory Measurements; Part B: Field Measurements Edited by C. G. Sammis and T. L. Henyey Volume 25. Geometrical and Instrumental Optics Edited by Daniel Malacara Volume 26. Physical Optics and Light Measurements Edited by Daniel Malacara

1. INTERFERENCE Daniel Malacara Centro de lnvestigaciones en Optica A.C. Apdo. Postal 948 37000 Leon, Gto. Mexico.

1 .l.Introduction The luminous phenomenon called interference is a direct consequence of the wave nature of light. Using the interference of light, we can make interferometers, which are instruments that use this phenomenon to measure very accurately many physical parameters. The general subject of interference has been treated extensively in many classical textbooks on optics like those by Born and Wolf,’ Cook,’ Franqoq3 Candler,4 Steel,’ and Tolansky.6 There are also special chapters on the subject of interference in many advanced books like those by Baird,’ Baird and Hanes,’ and Dyson: and others. This chapter describes very briefly the interference phenomenon and some interferometers. Special emphasis is placed on the applications of these useful instruments, which have played a very important role in the development of physics due to their extremely high accuracy. 1.1 .l. Methods to Obtain Interference Fringes

To obtain interference fringes, the phases of the two interfering waves must be synchronized, that is, they must be coherent. Before the advent of lasers, this was possible only if both waves originated from the same light source. In order to produce two waves from a single source, we must have either a division of the wave front or division of its amplitude. Division of Wave Front. This class of interference is produced when the two interfering wave fronts are taken from different portions of the original wave front. The typical examples are Young’s experiment, the Fresnel biprism, and Lloyd’s mirror, but there are many others. Young’s double-slit experiment is performed as shown in Fig. 1. The lenses are not strictly necessary, but their addition makes the theory easier. The light source S must be either a very narrow slit or a point, and lens L collimates the light to obtain an approximately flat wave front. Normally, 1 M E T H O D S OF E X P E R I M E N T A L PHYSICS Val. 26

Copyright 0 1988 by Academic Pres,. Inc. All rights of reproduction in any form reserved ISBN 0-12-475971-8

2

INTERFERENCE

screen

1.0-.8-

(b)

6.4 -

.2-

FIG. 1. Young's experiment. (a) Experimental arrangement and (b) Diffraction pattern.

the interference pattern would be observed at infinity, but lens L serves to bring the pattern closer over the screen. It can be found in almost any good textbook on optics' that the interference pattern is given by 7T-1

I

= I,,,

cos'{

A sin e ]

sin{ ?sin 7TD

e] 9

(1.1)

sin 0

A

where I,,, is the irradiance at the center of the pattern, and 8 is the angular deviation from the optical axis. This radiation pattern also appears with

3

INTRODUCTION

4

----

&l--

screen

s* (a)

SO"

rce

FIG.2. ( a ) Lloyd's mirror and (b) Fresnel biprism.

dipole antennas and radio waves. The total width of the central maximum is given by sin 0 = A l l . The Fresnel biprism and Lloyd's mirror also produce interference fringes by division of the wave front, as shown in Fig. 2. If two waves are to interfere producing fringes with good contrast, the polarization states of both waves must be the same. This condition is always satisfied in the Fresnel biprism. However, in Lloyd's system, only one beam is reflected. Therefore, the reflection coefficients and the phase shifts under reflection must remain nearly constant over the range of incident angles used. This is possible only near grazing incidence.

4

INTERFERENCE

A second reason for using grazing incidence in the Lloyd system is that the spacing between fringes decreases rapidly as the separation between the virtual sources S, and S2 is increased. The sources must be quite close to each other to make fringes visible, and this is possible only near grazing incidence. If the screen in the Lloyd system is placed near the edge of the mirror, we can observe that a dark fringe appears at this edge. This happens because there is a phase shift upon reflection with grazing incidence. Wolfe and Eisen" have studied the coherence requirements in Lloyd's mirrors. Division of Amplitude. This class of interference occurs when both interfering beams are obtained by division of the amplitude of the original wave front by means of a partially reflecting optical surface. Then, both beams travel different paths, and interference occurs when they are recombined. Typical examples are Newton rings and the Michelson interferometer described in the next section. 1.1.2. Classification of Interference Fringes

A classification for interference fringes can be made according to the way they are observed, namely, fringes of equal thickness and fringes of equal inclination. Fringes of Equal Thickness. Each fringe represents the locus of all points in which two optical surfaces (or wave fronts) have a constant separation. This is much better understood by means of the following example. Consider the optical arrangement in Fig. 3 where the flat or convex surface of one lens is placed against the flat surface of another lens. Monochromatic light enters the first lens and impinges upon the second surface from which

observing eye

observed

FIG. 3. Arrangement to observe Newton fringes.

fringes

INTRODUCTION

5

some of the light is reflected. Some of the light that goes through the first lens is reflected from the upper flat surface of the second lens. The two reflected beams interfere constructively when the phase difference is an integral multiple of 27~.We can see that the optical path difference is twice the surface separation. If one of the surfaces is spherical, each fringe is a ring that represents the locus of points with equal surface separation. These circular fringes, also called Newton rings, are a particular case of fringes of equal thickness. Fringes of Equal Inclination. I n this case, each fringe is the locus of points in the field of view with the same angle of the incidence 8 at the interferometer. As an example, let us consider two parallel reflecting surfaces as in Fig. 4. As in the previous example, the two interfering beams are produced by division of amplitude. Fringes of equal thickness cannot be formed because the optical path difference (OPD) is the same for the entire field. One way to change the phase difference and hence to observe fringes is to introduce a range of angles of incidence by using an extended light source. Circular fringes are observed with an angular radius 8 given by mA cos 8 =-) 2d

where rn is an integer smaller than or equal to 2 d / A .

FIG. 4. Arrangement to produce fringes of equal inclination.

6

INTERFERENCE

To directly observe these fringes, the lens and the screen may be replaced by the unaided eye (focused at infinity). Then, the observation must be made almost perpendicularly to the optical surfaces, and the eye must be placed very close to them, because of the small diameter of the pupil of the eye. If the observing distance d between the surfaces is so large that the fringes cannot be resolved with the naked eye, a telescope must be used. 1 .I .3. Light Sources for Interferometers

If two light waves are to interfere, they must have their phase synchronized. Then, the two waves are said to be coherent to each other. When the two interfering waves are taken from the same light source, either by division of the amplitude or by division of the wave front, this condition is satisfied under certain conditions that we will describe. Let us first assume that the light source is perfectly monochromatic, but that the light source is extended. Each point of the light source may produce a complete interference pattern. Good fringe contrast is obtained only if the phase difference for both waves is the same for all points over the extended light source. If the above condition is not satisfied, the light source has to be very small. In that case, it is said that the interferometer has to be illuminated by a spatially coherent beam. If the condition is satisfied, the light source can be extended, and the interferometer is said to be source-size compensated. Let us now assume that the illuminating beam is spatially coherent or that the interferometer is source-size compensated. The light, however, may not be perfectly monochromatic. Then, each wavelength component of the light produces an interference pattern. Good fringe contrast is obtained only if all patterns produced by the interference between both light beams are the same for all wavelengths. In general, this condition can be achieved only if this OPD is zero for all wavelengths, the interferometer is then said to be white-light compensated. Otherwise, the light has to have a narrow bandwidth whose maximum permissible value depends on the particular interferometer. A light beam with a narrow bandwidth is said to have light-temporal coherence. Very few interferometers are white-light compensated, and, therefore, their light sources have, in general, to be quite monochromatic. The monochromaticity of a light beam can be measured either by its bandwidth or by its coherence length, which is the maximum optical path difference that can be introduced before the fringe contrast is completely lost. The coherence length of light sources varies from a fraction of a millimeter to as much as one meter for the KrR6standard lamp (Baird and

TWO-REAM INTERFEROMETERS

7

Howlett”), or even several hundreds of meters for a single-frequency gas laser.

1.2. Two-Beam Interferometers Another possible classification for interferometers takes into account the number of light beams that form the interference pattern. Then, we may have either two-beam o r many-beam interferometers. In this section, the most common two-beam interferometers will be briefly described.” 1.2.1. The Newton Interferometer

We call a Newton interferometer any arrangement of two surfaces in contact with, and illuminated by, a monochromatic source of light, as shown in Fig. 5 . This interferometer has been described in detail by Murty.” monochromatic source of light

4

in contact

FIG.S . Newton interferometer.

8

INTERFERENCE

This instrument forms equal-thickness fringes; and a bright fringe is formed whenever the phase difference is equal to an integral multiple of 2 7 ~If . we take into account the fact that there is a phase shift under reflection for one of the two beams, we can say that a dark fringe occurs when

2nd

= mA,

(1.3)

where m is an integer, d is the thickness of the gap between the two reflecting surfaces, and n is the refractive index of the gap medium. This expression assumes that the fringes are observed normally to the reflecting surfaces, and hence the two following rules should be followed to obtain good accuracy: (a) If the surfaces are flat, a collimating lens is required in order to observe the fringes from the focus ofthis lens. Another acceptable alternative is to use a large extended source of light and then to observe the fringes from a minimum distance of about five times the diameter of the flat surface under test. (b) If the surfaces are spherical, the observation point should effectively be placed at the center of the curvature, as shown in Fig. 6. Figure 7 shows some typical defects that may be observed with this interferometer. 1.2.2. The Fizeau Interferometer

This instrument, shown in Fig. 8, has also been described in detail by Murty.” Equal-thickness fringes are formed by the two reflected beams. The lens is used to collimate the light and to optically place the observing eye at infinity. The optical path difference between the two beams is

OPD = 2nd,

( 1.4)

where d is the thickness of the gap, and n is its refractive index. If the field is free of fringes, we infer that the product nd is constant over the gap. We can conclude that d is a constant only if n is assumed to be constant. If straight and parallel fringes with separation S are observed, the two faces forming the gap are flat and with an angle 8 in radians between them, given by

e=-.

A

2 nS

This makes this interferometer quite useful for measuring small wedges in glass plates.

9

TWO- BEAM INTERFEROMETERS

k

I

surface

observation point

(a 1

under t e s t

observat Ion polnt

k

't2

(b)

urf ace under test

F I G .6. Testing of spherical surfaces with Newton interferometer.

1.2.3. The Michelson Interferometer

Probably the most famous of all interferometers was invented by Michelson for his well-known experiment. Figure 9 shows this interferometer. An extended light source sends the light to the beam splitter P which divides the beam in two, with one going to mirror MI and the other going to mirror Mz. Both mirrors send the beams back to the beams splitter and from there to the observing eye. When the mirrors M , and M2 are exactly perpendicular to their optical axes, the fringes are of the equal-inclination type. In this case, the fringes

10

INTERFERENCE

FIG. 7. Typical interference patterns with Newton interferometer.

are circles with angular radii given by Eq. (1.2), where d is given by

d = -OPD, 2 ’ and OPD, is the optical path differences along the optical axis.

11

TWO-BEAM INTERFEROMETERS

t-

c

light source

FIG.

8. Fizeau interferometer.

The interfering beam coming from mirror MI traverses the beam splitter three times, while the beam from mirror M 2 passes through the plane only once. To equalize the thickness of glass that each beam traverses, a plate P2 of the same thickness and orientation as P, is inserted into the beam to M 2 . This extra plate is called a compensating plate, and the interferometer containing one is said to be compensated. When the interferometry is not compensated, only one beam goes through the inclined beam splitter, and thus the optical path difference for a given value of 8 is different in different planes. In that case, the Haidinger fringes are not circular but elliptical in shape, as described by Guild.I3

light source

4observed pattern

F i c . 9. Michelson interferometer.

12

INTERFERENCE

Fic;. 10. Three kinds of Michelson fringes.

The fringes have good contrast only if the OPD, is smaller than the coherence length. Therefore, the fringes are observed with white light only if the variation of the OPD within the spectral region being observed becomes smaller than about A/4. Since the index of refraction is a function of the wavelength, the former condition cannot be satisfied unless the total amount of the glass traversed by both beams is the same, and then the OPD, is adjusted to be smaller than a few wavelengths. Hence, white-light fringes can be observed only in a compensated interferometer. If the mirrors MI and M 2 are not perpendicular to their optical axes, equal-thickness fringes will be observed instead of equal inclination fringes. However, they are exactly equal-thickness fringes only if the observing eye is very far from the interferometer, or a lens is used as in Fig. 9 to optically place the eye at an infinite distance from the interferometer. If the eye is close to the interferometer and if the mirrors are not perpendicular to their optical axes, intermediate fringe shapes are observed. These fringes are curved, with the convexity directed toward the narrower part of the wedge formed by the two mirror images. The three kinds of Michelson fringes are shown in Fig. 10. The extended light source permits the fringes to be observed only at a well-defined plane near the virtual images of the light source. These fringes are said to be “localized.”

1.2.4. Modifications of t h e Michelson Interferometer

Several modifications have been made of the Michelson interferometer. The most important ones are the Mach-Zehnder, Jamin, and TwymanGreen interferometers.


13

r-i

I

t

1

I 1 sample 1 I I 1

FIG. 11. Jamin interferometer.

The Jamin interferometer is shown in Fig. 11. The light from the light source is split at the surfaces of a thick glass plate. Then, the two beams are recombined at a second identical glass plate. For some applications, the advantage of this arrangement over the Michelson interferometer is that the light passes through the sample only once. The plates have a large thickness so that enough room is left for the sample. This interferometer was used for refractometry, but it has now been replaced by other kinds of interferometers. The Mach-Zehnder interferometer, shown in Fig. 12, can be considered as a Jamin interferometer in which the four reflecting surfaces have been replaced by four plates. This instrument is generally used to detect small changes or space variations in the refractive index of the samples. Both, the Jamin and the Mach-Zehnder interferometers, are compensated if correctly adjusted, so that white-light sources can be used if desired. However, it must be pointed out that this adjustment is particularly difficult in the case of the Mach-Zehnder. The Twyman-Green interferometer14*" is one of the most useful modifications of the Michelson interferometer, used mainly to test the quality of optical components. The system is illuminated with a collimated

14

INTERFERENCE

,attern 2

9 p2

pattern I

2

+

3

\

Q

F I G . 12. Mach-Zehnder interferometer

beam of light, as shown in Fig. 13. The fringes that are observed by placing the eye at the focus of lens L are of equal-thickness. The Twyman-Green interferometer is not generally compensated, and large values of the OPD are frequently obtained, so that the light source

Y F I G . 13. Twyman-Green interferometer.

15


--(C)

FIG. 14. Testing optical components with a Twyman-Green interferometer.

mu t be monochromatic. Many different kinds of optical components ca be tested with this interferometer as described by Malacara.I6 The simplest one to be tested is a plate of glass with plane parallel faces, which is inserted into one of the beams as shown in Fig. 14(a). The optical path difference introduced by the presence of the glass plate may be found considering that in the space with thickness d occupied by the plate, the index of refraction is changed from 1 (without the plate) to n (with the plate). A factor of 2 appears because the light passes twice through the plate. Thus,

O P D = 2 ( n - 1)d.

(1.7)

If the interferometer is adjusted so that no fringes are observed before introducing the plate into the beam, and so that fringes are observed after the plate is inserted, we establish that the quantity ( n - l ) d is a constant throughout the plate. Assuming that the index n does not vary over the plate, we may conclude that d is a constant. The variations in d and in n can be obtained independently only if another measurement is made with a Fizeau interferometer.

16

INTERFERENCE

If the plate faces are flat but not parallel and n is constant, straight and parallel fringes appear. Using the arrangements in Figs. 14(b) and 14(c), the quality of a lens or photographic objective can be tested. It is assumed in this test that the convex or concave mirrors are perfectly spherical. If the lenses are assumed to be perfect, the quality of the spherical mirror is tested. Interference patterns associated with various lens aberrations are shown in Fig. 15. Many

FIG. 15. Some Twyman-Green interferograrns.


17

other types of optical elements, such as prisms and diffraction gratings, can also be tested with this interferometer. 1.2.5. Shearing Interferometers

These are a class of interferometers in which a perfect reference beam is not needed. The reference wave front is the same as the one under test but with a different orientation, position, or lateral extension. Basically, there are four types of shearing interferometers as illustrated in Fig. 16. Lateral shearing interferometers are probably the most popular of the four and have been extensively described by Murty.” Radial, rotational, and reversal shearing interferometers have also been described in detail by Malacara” and Bryngdahl.” The lateral shearing interferometer measures the wave-front deformation by comparing the wave front with a laterally displaced reproduction of itself, thus avoiding the need for a flat reference wave front. Probably, the first interferometer of this kind was designed by Bates.*’ This is basically a Mach-Zehnder interferometer used with a convergent wave front and with the plate P2 tilted to produce two laterally displaced

I ate ra I

rotational shear

radial

reversal shear

FIG.16. Four types of shearing interferometer operations.

18

INTERFERENCE h

front sheared wavef ront s microscope objective

gas LASER

rl

lens

under test

FIG. 17. Two lateral shearing interferometers.

wave fronts as shown in Fig. 17(a). The lens under test is placed between the light source and the interferometer. Murty” has proposed a simpler version of a lateral shearing interferometer in which the large temporal coherence of laser light permits large values for the OPD [see Fig. 17(b)]. Lateral shearing interferometers provide rapid and easy identification of lens aberrations as shown in Fig. 18. Given an interferogram, the wave front that produced it can be computed by using several methods, such as those described by Saunders,” M a l a ~ a r a , ’and ~ Rimmer and W ~ a n t . * ~ Many radial shear interferometers have been designed, but the first one was invented by Brown.25Basically, this is a Jamin interferometer that works in convergent light and has a small meniscus lens in one of the beams. To compensate the interferometer, a small parallel plate is placed in the other beam [see Fig. 19(a)]. Another interesting radial shear interferometer, shown in Fig. 19(b), was designed by M ~ r t y . ~ ~ Radial shear interferometers are sensitive to all types of aberrations, as the Twyman-Green interferometers, but to a lesser degree.

TWO-REAM I N T E R F E R O M E T E R S

19

FIG. 18. Lateral shearing interferometer patterns.

Rotational shear interferometers are not sensitive to rotationally symmetric aberrations like spherical aberration but are quite useful for the detection of nonrotationally symmetric aberration, like coma or astigmatism. These interferometers have been described by Murty and Hagerott2’ (see Fig. 20). Reversal shear interferometers are insensitive to rotationally symmetric aberrations and also to astigmatism if the axis of reversion coincides with the x or y axes. It is quite sensitive, however, to coma. Common examples of this kind of interferometers are the Koester prism interferometers shown in Fig. 21, and described by Gates28 and Saunders.” 1.2.6. Common-Path Interferometers

Where the reference and test beams follow separated paths in an interferometer, they can be differently affected by vibrations, turbulence, and

20

INTERFERENCE

RADIAL L I SHEARED WAVEFRON rs

BROWN'S

INTERFEROMETER

WAVEFRON r

UNDER T E S r I

.

."&

.^CC."

EYlSPHERlCAL

AM

SPLITTER

MURTY'S INTERFEROMETER

FIG. 19. Brown's and Murty's radial shear interferometers.

temperature variations. An interferometer in which both beams follow the same path is called a common-path interferometer. They have been extensively treated by M a l l i ~ k . ~ ' Common-path interferometers have zero optical path difference for all wavelengths and can therefore be used with white light. They are also very stable. Most of these instruments are either of the lateral shearing or radial shearing types. A large radial shearing is obtained when one of the beams is made to traverse a small area of the system under test, while the other beam traverses the whole aperture. The most popular instrument of this kind is the so called scatter-plate or Burch interfer~meter,~' illustrated in Fig. 22. After passing the light through the first scatter plate, there will be two beams: a diffracted one, which we


21

FIG. 20. Murty and Hagerott's rotational shear interferometers.

FIG. 21. Koester's prism interferometers. (a) Gates interferometer and ( b ) Saunders interferometer.

22

INTERFERENCE

FIG. 22. Burch’s scatter plate interferometer.

call DF, and an undiffracted one (zero order), which we call DO. The undiffracted beam DO covers only a small area near the center of the surface under test, while the diffracted beam covers the whole surface. The first scatter plate is imaged onto the second plate, which is a mirror image of the first. The beam DO produces two beams upon passing through the second plate, one again undiffracted (DOO) and one diffracted for the first time (DOF). The beam D F also originates two beams, one undiffracted (DFO) and one diffracted a second time (DFF). The beam DO0 is observed as a small bright spot near the center of the mirror. The beam D F F is very faint and unobserved. The beams DOF and DFO form a fringe pattern. The pattern is sometimes interpreted as a radial shear interferogram with a very large amount of radial shear; however, it is best to think of it as a Twyman-Green pattern. There i s a large variation in common-path interferometers, many of them using birefringent materials, as described by FranGon’ and Mallick.”’ 1.2.7. Other Two-Beam Interferometers

There are many other two-beam interferometer designs as, for example, those that involve division of the wave front rather than division of the amplitude. Two very important instruments of this kind are the Rayleigh and the Michelson stellar interferometers. M i ~ h e l s o nwas ~ ~the first to apply interference phenomena to the measurement of refractive indices. Later, Lord Rayleigh designed an interferometer for this purpose that has been

23


fringes

FIcj. 23. Rayleigh interferometer.

well described by Candler.4 It is very useful for the measurement of refractive indices of gases or liquids (see Fig. 23). The light from a narrow slit is collimated by means of a lens, it then goes through two glass tubes containing two gases whose indices are to be compared. The beams are then recombined by another lens in order to form a set of interference fringes. The working principle is the same as that of Young's double-slit interferometer. When the gases or liquids in the two tubes have different refractive indices, the fringes are laterally displaced. The Rayleigh interferometer (or refractometer) has more accuracy in the measurements of small refractive indices variations than any other instrument. It can detect changes in the refractive index of the order of lo-', or one unit in the sixth decimal place. The Michelson stellar interferometer ( Michelson6") is also based on the Young interferometer, used in order to measure the angular diameter of stars. The measurement is indirectly made by measuring the degree of spatial coherence of the incoming light. This interferometer is illustrated in Fig. 24.

f l a t mirrors

s c r e e n or d

photographic plate --La"

'IrnI" flat mlrrors

F I G . 24. Michelson stellar interferometer.

24

INTERFERENCE

The light received from opposite sides a and b of the star disc has an angular separation 8. The optical path difference from point a assumed to be on the axis is represented by OPD,. Thus, the optical path difference from point b, off-axis, is given by OPD,,

= OPD,

-

ed,

(1.8)

where d is the separation between the two small mirrors receiving the light from the star. The irradiances produced by the two sources at points a and b can be obtained from Eq. (1.1) as I, =21[1 +cos(koOPD,)], and Ih

= 21[ 1

+ COS( koOPDh)].

(1.9)

Since the two point sources are not coherent with respect to each other, the total irradiance on the pattern is 1, = 1,

+Ib

=41

+ ~ I [ c okoOPD,) s( + cos ko(OPD, - Od )].

(1.10)

The fringe visibility versus the slit separation is plotted in Fig. 25. The angular diameter of the star may be calculated from the measurement of the distance d, which produces the first minimum in the visibility. As an example, the star Betelgeuse has an angular diameter of about 0.05 sec of arc, which lets d equal about three meters for green light. A modern version of the Michelson stellar interferometer is the irradiance interferometer in which the light from each slit is not made to interfere. Instead two irradiance detectors are used, and then the irradiance variations in the two signals are electronically correlated. This instrument has a

t"

t"

1.22 X/8

X./28

01 Two point source8

b) C l r c u l a r

rxtrndrd

FIG.25. Fringe visibility in Michelson stellar interferometer.

SOUICI

M ULTIPLE-BEAM I N T E R F E R O M E T E R S

25

quantum theory explanation, after an experiment devised by Brown, Hanbury, and Twiss.3’

1.3. Multiple-Beam Interferometers Some interferometers form their interference patterns with the superposition of more than two wave fronts. These interferometers have many advantages, as we shall see in this section. 1.3.1. The Fabry-Perot Interferometer

Multiple-reflection interferometers have been described in detail by Royc h ~ u d h u r and i ~ ~ Polster er a/.” The most popular of this class of interferometers is the F a b r y - P e r ~ t .The ~ ~ irradiance pattern of a two-beam interferometer is sinusoidal, and therefore the exact position of a fringe maximum cannot visually be located with an accuracy greater than about one tenth of the fringe spacing. The Fabry-Perot interferometer produces extremely narrow equal-inclination fringes, and thus the fringe maxima can be located with an accuracy better than one hundredth of a wavelength. The principle involves the use of multiple reflections, so that interference is produced by an almost infinite number of beams as shown in Fig. 26.

Sl FIG.

s2

26. Multiple reflection in a Fdbry-Perot interferometer.

26

INTERFERENCE

(b) FIG. 27. Fabry-Perot configurations.

The two parallel surfaces may be in two plates as in Fig. 27(a), or in one plate as in Fig. 27(b). Such plates are sometimes called etalons, from the French word for "standard." The transmitted irradiance 1 ( 6 ) as a function of the phase difference, assuming the amplitude transmission and reflection coefficients in Fig. 26, may be given by (1.11)

where it has been assumed that rl and rz are equal, in other words, that the phase shift upon reflection is either 0" or 180". Here, I is the incident irradiance, the asterisk denotes the complex conjugate, and the phase difference is given by

477

6 =-

A

nd cos 0 2 ,

(1.12)

where d is the separation between the reflecting surfaces, n is the refractive index of the material between the reflecting plates, and O2 is the angle of incidence of the rays inside the gap. If the reflectances R , and R2 and the transmittances T, and Tz are always measured in air, not inside of the glass plates as sometimes defined, Eq.

MULTIPLE-BEAM I N T E R F E R O M E T E R S

27

( 1 . 1 1 ) reduces to

I ( 6 ) = I”

T , T2 6’ ( 1 - R,)2+4R, sin22

(1.13)

as shown by Ba~meister,~’where the average reflectance R, is defined as (R~R~)~’~. This expression can be generalized to include the case in which there are phase shifts E , and E~ upon reflection as measured in the medium of the spacer (shown by Baumeister and Jenkins3’) by writing Eq. (1.12) as 47T 6 = - nd cos O2 -$( A

+

E ~ ) .

(1.14)

As pointed out before, the great advantage of multiple-reflection over two-beam interferometers is the increased sharpness of the fringes. The width of the fringes depends very much on the reflectances of the plates as illustrated in Fig. 28. If we define the half-width A6 of the fringe as the width at half the maximum fringe irradiance, it is given by

A6=2-

1-R, R;I2

(1.15) ’

The “finesse” N R is a number used to specify the sharpness of the fringes. It is defined as the ratio of the fringe spacing to the width in Eq. (1.15). Thus, (1.16) As shown in Fig. 28, the finesse can be extremely large even for relatively small values of R,.

FIG. 28. Fringe profile in a Fabry-Perot interferometer.

28

INTERFERENCE

As pointed out by Baird and Hanes,' with a Fabry-Perot interferometer we can observe fringes of equal inclination, fringes of equal thickness, and fringes of equal chromatic order, which will be described in this section. Fringes of equal inclination are formed when the light source is extended or convergent, so that a wide range of angles of incidence are present and the two plates are parallel to each other. The fringes are observed at an infinite plane by means of a lens with a screen at its focus or a small telescope focused at infinity. The fringes are circles located at positions such that

OPD = 2nd cosz 8 = mA,

(1.17)

where m is an integer. The smallest circular fringe corresponds to the largest value of m, given by 2 nd m,,,=-- A

e,

(1.18)

where e is the fractional value of ( 2 n d l A ) . Thus, the value of m for the pth fringe is (1.19) hence the angular radius of this fringe is given by cos

A

e2= 1 --2 nd ( p - 1 + e ) .

( 1.20)

Two rings with different wavelengths may have the same angular diameters, because their value of m may also be different. However, this ambiguity is eliminated by measuring two sets of rings with different values of d. Fringes of equal thickness may also be formed with a Fabry-Perot interferometer by forming a small wedge with the two reflecting surfaces. In this case, an extended source is not needed, and the light is normally collimated by a lens. The instrument may now be called a multiple-reflection Fizeau interferometer. Fringes of equal chromatic order appear when a white light source is used to illuminate the interferometer adjusted with parallel faces, and then the outgoing light is seen through a spectrograph. Narrow bright bands are seen in the spectrum. This effect is called a channeled spectrum, and the fringes are called fringes of equal chromatic order or Edser- Butler fringes. Tomkins and Fred39 have used these fringes to calibrate a spectrograph by using the fact that they are located at positions A N = 2 n d / ( N - 4 / 2 ~ ) ,

MULTIPLE-BEAM INTERFEROMETERS

29

where N is an integer, and C#I is the total phase change on the two internal reflections. T ~ l a n s k y ~ has ~ . ~also * ~used ' these fringes to measure the microtopography of crystal surfaces and films. 1.3.2. The Spherical Fabry-Perot Interferometer

The optical path diff erence in a Fabry-Perot interferometer changes with the angle of incidence. This property is not desirable in photoelectric spectrometry where a large solid angle of acceptance of the light is required together with a small variation in the optical path difference. In this technique, the light spectrum is analyzed not by changing the angle of incidence to form rings but by changing the plates' separation by a small amount, usually of the order of a wavelength or less. An interferometer in which the optical path difference does not change with small variations in the angle of incidence was designed by C ~ n n e s . ~ * It consists of two spherical confocal surfaces as shown in Fig. 29. The two spherical surfaces have their focal planes coincident at some plane between them. If the two mirrors have the same radius of curvature, the center of curvature of each mirror is located on the opposite mirror. An incident ray reaching mirror M , at point A travels to point B, where it is partially reflected. The reflected ray then follows path BDCAB, so that it comes back to the original trajectory AB at A, where it is again partially reflected. The optical path difference between two consecutive rays is

OPD = 4t,

(1.21)

where r is the separation between the vertices of the two mirrors. The lower half of each mirror is totally reflecting, while the upper half is partially reflecting. We can see that within the limits of the paraxial optics (small angles of incidence), the phase difference is independent of the angle of incidence.

/

\

FIG. 29. Connes's confocal interferometer.

30

INTERFERENCE

We should also notice that the emerging rays are all coincident along the incident ray and not the parallel and laterally separated rays as in the Fabry-Perot interferometer. To keep the rays within the limits of the paraxial optics, two small circular diaphragms are frequently used in front of the mirrors. The only adjustment that this interferometer needs is to fix the correct distance between the mirrors with a tolerance of a few microns. Since the mirrors are spherical, there is no need to adjust their parallelism. The main use of this instrument is to examine the spectrum of lasers, as described by Herriott4' and Fork et al.,44by oscillating one of the mirrors a small distance (usually about h / 4 by means of an electromechanical device). 1.3.3. Other Multiple-Reflection Interferometers

In some other interferometers, the principle of multiple reflection is also used in order to increase the accuracy of the instrument. These interferometers are not used much and are mainly of historical interest. An example is the Lummer-Gehrke plate4 shown in Fig. 30(a): It may be

FIG. 30. Other multiple-beam interferometers. ( a ) Lummer-Gehrke plate and ( b ) Echelon.

M ULTIPLE-BEAM INTERFEROMETERS

31

considered as a Fabry-Perot etalon using a very large angle of incidence. The angle of incidence of the light inside the plate is very near the critical angle, so that a large reflectivity is obtained without any kind of coating on the surfaces. Light enters the system through a small prism cemented onto one end of the plate. This system was very much used4’ at the beginning of the century for spectrographic analysis, but is has now been largely replaced by the Fabry-Perot interferometer. The e ~ h e l o n ~ ~of. ~Michelson .~’ [Fig. 30(b)] is another multiple-beam device, formed by a stack of flat parallel plates with constant thickness t. The plates are arranged so that steps with approximate height of 1 cm and width of 0.1 cm are formed. In the transmission echelon, the wave front is divided in M portions with the relative optical path difference as a coarse grating, blazed for a very high order. A complete description of transmission and reflection echelons is found in the book by Candler.4 1.3.4. Interference Filters

A common application of the Fabry-Perot interferometer is to isolate narrow spectral regions. Used in this way (Fig. 31), it is called an interference filter. It should be used only with collimated light. The wavelength transmitted by the filter is given by

A=

2nd cos e m

(1.22)

If a filter is to isolate a line with wavelength A, a very low order of interference, m about 2, is chosen so that there is a large wavelength difference between the different orders. The undesired orders are then eliminated by means of any broad-band absorptive filter. An interference filter can be made by evaporating a thin reflective film, like silver or aluminum, on a piece of glass. Then a transparent film, like magnesium fluoride, is evaporated on the metal. On this layer, another film of metal is evaporated to complete the films. It is interesting to point out that the glass substrate does not have to be extremely flat, since the important thing is only the constant thickness of the evaporated spacer.

-glass m -c-i-elatl

coating dielectric spacer -glass

Fic;. 31. Interference filter.

32

INTERFERENCE

ANGLE OF INCIDENCE

F l c . 32. Wavelength change with orientation in an interference filter.

An interference filter may be slightly tunned by changing the filter orientation with respect to the light beam. Keeping in mind that 8 is the angle inside the spacer with refractive index n, and 4 is the angle of incidence outside the filter, we can see that ( 1.23)

where A, and A. are the wavelengths for an angle of incidence 4 and for normal incidence, respectively. Figure 32 shows a plot of (Ao- & ) / A o versus 4. A consequence of this angle sensitivity is that the narrower the passband, the more accurate the angle of incidence and collimation have to be. Half-widths of interference filters can have a wide range of values, between about 80 A and about 1 A. In order to reduce the passband width (or half-width), interference filters are often made with more than one cavity, that is, with several spacers with reflecting films made of dielectric multilayers instead of metal. 1.3.5. Thin Films A multilayer film consists of thin layers of different dielectric materials deposited by evaporation, one on top of the other. Table I lists a few of the materials used. A good review of this subject was written by Ba~meister,~’ and a more complete description is given in Chapter 13 in this book. Due

33

M ULTI PLE-BEAM I N T E R F E R O M E T E R S

TABLE1.

Some Dielectric Materials for Thin Films

Formula

Name

Refractive index A = 550 nrn

Transparency range in pm

MgF, ZnS Ti,O, SiOz NalAIF,

Magnesium fluoride Zinc sulfide Titanium oxide Silicon dioxide Cryolite

1.38 2.3 2.3 1.46 1.35

0.13-8 0.39-14.5 0.39-12 0.2-9 0.2-14

to the interference produced by multiple reflections at the interfaces between layers, these filters are highly color selective in both transmission and reflection. There are many uses for these films, for example: (a) Antireflection coatings. A lens can be coated with one or more dielectrics for the purpose of minimizing reflected light, thereby enhancing the transmitted light. The flare around the image is reduced by eliminating the unwanted light that would otherwise come from reflections at the lens surfaces. (b) Enhancement of the reflection and/or protection of metal mirrors. (c) Beam splitters and semitransparent mirrors for interferometers and other kinds of optical instruments. For example, partially transmitting mirrors are used in lasers. These mirrors have an advantage over the metallic ones in that they do not absorb energy. (d) Band-pass filters. These filters reflect all the colors that are not transmitted, as opposed to the gelating or glass filters that absorb all the energy that is not transmitted. They are often called dichroic mirrors, but the term dichrom strictly applies to materials that absorb two polarizations differently. Among the uses for these filters are cold mirrors for movie projectors that do not reflect infrared but only visible light; color beam splitters for cold television cameras; and others. Single-layer Coatings. A single-layer coating deposited over a substrate can be used to reduce the reflectance of a dielectric surface or to enhance the reflectance of a metallic surface. Let us consider the case illustrated in Fig. 33. The minimum reflectivity is obtained when the optical thickness of the layer is h / 4 and its refractive index is - nl/2

2-

s

9

(1.24)

where n , is the substrate refractive index. This condition cannot be exactly

34

INTERFERENCE

surface surface 2

+

V/II

I coating Y ' ' H

I

I

I

glass

@

fl

substrate

@

f

N 2 I

,&

d

N,

FIG. 33. Single-layer coating.

satisfied with practical dielectrics. It may, however, be approached very closely, for example, by the magnesium fluoride. The reflectance of the system varies with the optical thickness of the coating as shown in Fig. 34. We can see that the maximum reflectance that can be obtained for any thickness is the reflectance value with no coating. In other words, the reflectance can be reduced, but never increased, when this coating is deposited on a dielectric surface. This result can be shown to be true whenever the coating has a refractive index smaller than that of the substrate. The minima have zero reflectance only when the condition in Eq. (1.24) is exactly satisfied. In order that the wavelength interval between the selected minimum at A,, and its adjacent maxima be as large as possible, the minimum optical thickness that produces a minimum is used, that is, Ao/4. Then, if R is the ,reflectance

uncoated

0

x/4

substrate

A/

2

3x/4

OpllCOl thickness N d

FIG. 34. Change in the reflectance of a single antireflecting layer with the optical thickness nd.

35

MULTIPLE-BEAM I N T E R F E R O M E T E R S

0

’0°

600

wavelength

700nm

FIG. 35. Reflectance for a single antireflecting film.

reflectance of the uncoated substrate, the final reflectance becomes

R

h).

= Ro cos2( .rrA 0

(1.25)

as shown graphically in Fig. 35. If the reflectance is to be increased rather than decreased, the index of refraction of the coating should be as high as possible with respect to that of the substrate. In this case, there is a phase change under reflection at one of the interfaces. Therefore, when increasing the optical thickness, the first maximum of the reflectance occurs at Ao/4 (see Fig. 36). Multilayer Films. Several films with different thickness and alternatively of high and low index of refraction can be superimposed by evaporation. By selecting the appropriate thickness and number of layers, it is possible to obtain almost any desired bandwidth, reflectance, and transmittance of the multilayer. A particularly interesting multilayer is the “quarter-wave stack.” As shown in Fig. 37, this multilayer consists of an even number of layers of alternatively low and high refractive indices. All layers have the same optical thickness Ao/4. Figure 38 shows the reflectance curve of a quarter-wave stack with four periods of high and low index layers.

36

INTERFERENCE

.3c

.2c

u

0

-" -

.I 0

(

4 1

I

I

so0

wavelength

600

FIG.36. Reflectance of a single reflecting layer.

FIG.37. A quarter-wave stack.

I

reflectance

F I G .38. Reflectance curve for a quarter-wave stack.

I

700nrn

37

M ULTI PLE-PASS I N T E R F E R O M E T E R S

-==--+I ,test

l

M

lens

3

FIG. 39. Double-pass Twyman-Green and Fizeau interferometers.

1.4. Multiple-Pass Interferometers These are interferometers in which at least one of the wave fronts traverses the normal trajectories more than once. The instruments, which have been described in detail by H a r i h a ~ - a nand ~ ~ by Langenbe~k,~’ have some advantages, as we shall see in this section. A double-pass interferometer is designed with the purpose of separating the symmetrical and the antisymmetrical parts of the wave aberration and to display them in separate interferograms. Figure 39 shows the double-pass Twyman-Green interferometer described by Hariharan and Sen.” The two beams going out from the interferometer are reflected back to it by a small mirror M 3 . Then, the patterns are observed by means of the beam splitter S 2 , where there are four wave fronts. By moving the light source slightly off-axis, two different patterns are observed. One pattern represents the symmetrical component and the other one the antisymmetrical component of the wavefront. Double-pass Fizeau interferometers have also been designed by Sen and Pu~~tambekar’”’~ and by Puntambekar and Sen” in order to reduce the coherence requirements of the light source. The multiple-pass principle can also be used in order to increase the sensitivity of an interferometer. Two examples are the multiple-pass Twyman-Green and Fizeau4’ interferometers shown in Fig. 40.

38

INTERFERENCE

aux i lio r y be a rn divider

test

surf ace

coated mirror

surface

FIG.40. Multiple-pass (a) Twyman-Green and (b) Fizeau interferometer.

1.5. Applications of Interferometry Interferometric techniques are very powerful tools in many different branches of science for very accurate measurements of a great number of parameters. A complete list of all applications of interferometry would be almost infinite,47~s4*ss~sh but a brief description of the main uses of these techniques is given in this section. 1.5.1. Relativity Measurements

One of the crucial experiments in modern physics was the MichelsonMorley e ~ p e r i m e n t , 'which ~ is now described in many modern textbooks of p h y s i ~ s . ~The ' experiment was performed with the famous Michelson interferometer in order to determine the frame of reference (the aether) with respect to which the light moves. It was assumed that the light had a constant speed with respect to the aether and that the earth moves in space,

APPLICATIONS OF INTERFEROMETRY

39

with the aether being stationary. Hence, if the interference pattern is observed with one arm of the interferometer being parallel to the earth's direction of movement, a small movement of the fringes is to be expected when the interferometer moves with the earth at 90" with respect to the former direction of movement. The negative result of the experiment led to many hypotheses trying to explain it,SXbut the satisfactory explanation came only with the special theory of relativity. Modern versions of this experiment have recently been performed with the same results.

1.5.2. S t a r Diameter Measurements

Michelson used his stellar interferometerh0in order to measure the angular diameter of stars (see Section 1.2.7). Although this instrument has been successful in measuring stellar diameters, it has many stability problems that limit its accuracy.

1.5.3. Refractometry

Interferometry can also be used to measure very small changes in the refractive index of a gas or to compare the refractive indices of two liquids. As mentioned before, Michelson3' applied interference for the first time for this purpose. The most commonly used instrument for this application is the Rayleigh interferometer4 described in Section 1.2.7. A Jamin or a Fabry-Perot interferometer can also be used to measure refractive indices.

1.5.4. Schlieren Techniques

The well-known Foucault or knife-edge test and many variations described under the general name of schlieren techniques can be considered to be interferometric. A good description of this test can be found in many books" and papers6' As shown in Fig. 41, the general idea behind this test is to detect small lateral displacements of rays focused to a point. This test can be applied to measure wave-front or surface deviations in spherical mirrors or lenses. Another application is to study the turbulence in a cell placed between the wave-front generator and the knife-edge. The Ronchi testh3 could be considered as a variation of the classical schlieren techniques.

40

INTERFERENCE

FIG.41. Foucault test.

1.5.5. Wave-front Topography

Almost all interferometers can be applied to the study of the topography of wave fronts or optical surfaces. An extensive bibliography on this subject is found in a paper by Malacara et ~ 1 . "Brief ~ reviews are given by Briers" and by Schulz and Schwider.66 An extensive study is presented in the book edited by M a l a ~ a r a , ~where ' even modern techniques such as fringe-scanning interferometers are described by Bruning."' It would be impossible to describe the numerous testing techniques in this chapter, and so the reader is referred to the specialized publications. 1.5.6. The Transfer Function of Lenses A way to specify the resolving power of a lens is by means of the optical transfer function (OTF). This function gives the contrast modulation of the phase shift for each spatial frequency in lines per unit length. There are a large number of ways to measure the optical transfer function of a lens, as explained by R o s e n h a ~ e r . "Some ~ of these methods are interferometric. The obvious way is to measure the wave-front deviations by means of any interferometer, as for example the Twyman-Green interferometer, and then to compute the transfer function. A more direct way is to use a lateral shearing interferometer and to make use of the fact that the transfer function can be mathematically obtained from a convolution of the wave front, as

APPLICATIONS OF INTERFEROMETRY

41

shown by hop kin^.^' Some lateral shearing interferometers made for this purpose are described by Fran~on.' 1.5.7. Length and Angle Measurements

Very precise length measurements can also be made by means of interferometry. For example, the separation between the two reflecting flats of a Fabry-Perot interferometer can be accurately measured. From Eq. (1.17), the order of interference m is given by m=

2nd cos 8

(1.26)

A

If this order of interference m is somehow determined and the wavelength is known, the distance d can be determined. A shown by F r a n ~ o n ,there ~ are several methods to determine the order of interference. M i ~ h e l s o n , ~at' the Bureau International des Poids et Mesures, was the first one to make an experimental determination of the length of the meter by means of interferometry. He used a Michelson interferometer with the device in Fig. 42 instead of one with flat mirrors, and found that the meter was equal to n, where n = 1,553,165.3 and h o = 6438.472 8, for the red cadmium line. A more recent determination of the length of the meter was made by Baird and H ~ w l e t t , ~after * the adoption in 1960 of the new international meter as 1,650,763.73 vacuum wavelengths of the 6057 8, line of Kr.x6The precision with this method is about lo8. Modern definitions of the meter, however, involve the use of lasers, with the advantage of their great coherence length.5 The frequency of lasers is calibrated against the cesium standard, thus defining the unit of time. The

I

I

FIG. 42.

Michelson etalon.

42

INTERFERENCE

second step was to measure the wavelength of the laser line by interferometric comparison with the Krypton line. Next, the vacuum speed of light was computed from the product of the wavelength and the frequency, and this result, equal to c = 299,792,498 meters/sec. was defined to be exact. In conclusion, the definition of the meter fixes the speed of light, and any source of known frequency may become a reference standard for length. An obvious application of these techniques is the calibration of scales in many instruments, for example in the fabrication of diff raction gratings.73 Interferometry using amplitude-modulated beams can be used to measure longer distances. It is like having a much longer wavelength, but the accuracy of the measurements can be much greater than that obtained by other methods. Instruments using this principle in order to measure long distances of the order of several meters have been made by Dukes and Gordon.74 A good description of these instruments can be found in a paper by Bruning.68 Even the distance from the earth to the moon has been measured with this principle by H a m m ~ n d . ~ ' as well as l a ~ - g e ~angles ~ * ~ ' can also be measured very accurately by interferometric procedures. 1.5.8. Microscopy

The principle of multiple-beam interference in thin films can be used with great advantage to study the microtopography of small crystal or glass surfaces. This subject has been fully described by Tolansky4' and by Krug et al." One of the many applications of this method is the observation of the polishing defects of a metal or glass surface. 1.5.9. Doppler Interferometry If a light source is moving with a velocity u with respect to the observer, the observed wavelength and frequency shift by an amount given by AA A

-

Av - u v

c

(1.27)

These small wavelength and frequency shifts can be measured very accurately by means of interferometry as described by Foreman er a/.''' An interesting application is the measurement of fluid-flow velocities by using small light-scattering particles in colloidal suspension.".'* Another application of the Doppler effect to interferometry is the measurement of slow rotations, as for example the earth rotation, with very high accuracy. This measurement is performed with the so-called Sagnac or ring interferometer," shown in Fig. 43. Two beams travel in the interferometer,

APPLICATIONS O F I N T E R F E R O M E T R Y

43

MONOCROMATIC COLIMATED LIGHT SOURCE OBSE LIGH'

E R OR ETECTOR

FIG. 43. Sagnac interferometer

one moves clockwise, and the other one counterclockwise. The result is that there is a fringe shift proportional in magnitude to the angular rotation w , as follows: 4Aw

AN=-

CA '

( 1.28)

where A is the area of the interferometer ( A = 2R'). M i c h e l ~ o n ~ and ~ , ' ~Michelson and Gale'' were the first to use this interferometer to measure the earth's rotation. Macek and Davisx7 designed a modern version of this instrument, using four laser tubes with a cavity in a ring configuration. With lasers, a stronger signal is obtained, but the two counter-running methods tend to lock together at low speeds, but with special methods this can be avoided, as shown by Aronowitz.xx The cyclic interferometer as well as the ring laser may be used as gyroscopes, as described by Rowland and A g r a ~ a l . ' ~

44

1 NTERFERENCE

1.5.10. Spectroscopy Probably one of the most popular and oldest applications of interferometers is the measurement of the spectral components of light beams, as was already mentioned in Section 1.3. The resolution of interferometric methods is so good that the hyperfine structure46and the wavelength values3' can be measured with high accuracy. Another very important and relatively new spectroscopic method that we want to describe here is called Michelson-Fourier spectroscopy. This method has been treated by P. Connes'" and Vanasse and Strong." It works on the principle that the temporal coherence or wave-train shape determines the frequency spectrum and vice versa. A Twyman-Green interferometer can be used to measure the wave-train shape, and then the spectrum can be computed mathematically. Suppose that we want to determine the spectrum of the light source illuminating the interferometer in Fig. 13. This is done by moving one of the flat mirrors along its optical axis to change the OPD. A light detector then replaces the eye in order to permit the average irradiance of the interference pattern to be recorded. The irradiance measured by the detector is a function of the optical path difference. Defining (1/2)Zl(k)dk as the average irradiance due to either of the two beams on a narrow band dk centered at k, the irradiance d l at the detector will vary sinusoidally with the OPD as follows: d l = 11(k ) [1+ COS( kOPD)] dk.

( 1.29)

Since different wavelength light beams are mutually incoherent, the irradiance I due to all colors acting simultaneously is

I= If we now define

lom lom I,(k)[l+cos(kOPD)] dk.

( 1.30)

I,=

(1.31)

I,(k) dk,

and I(OPD)=

I,'

I , ( k ) c o s ( k . O P D ) dk,

(1.32)

the irradiance at the detector can be written

I = I"+ I ( 0 P D ) .

(1.33)

After I ( 0 P D ) has been experimentally measured, the irradiance spectrum I ( k ) can be obtained from the cosine Fourier transform of I(OPD), which

45

APPLICATIONS O F I N T E R F E R O M E T R Y

gives d(0PD). Z(0PD) C O S ( ~ OPD)

( 1.34)

In practice, the mirror displacements cannot have infinite magnitude, and thus this integral is approximated by

I , ( k ) =-

Z(0PD) cos(k- OPD) d(OPD),

(1.35)

where the value of OPD,,, is twice the maximum mirror displacement. The larger this OPD,,, , the better the spectral resolving power. There are two clear advantages of interference spectroscopy over conventional dispersing spectroscopy. The first advantage is called Jucquinot and Dufour” or etendue advantage. The spectroscopic field of view of an interferometer is circular and much greater than the narrow and long field given by the slit of a spectrometer. For instance, with a Fabry-Perot interferometer, the spectrum of the whole image of the sun can be simultaneously studied, while with a spectroheliograph, this has to be done by dividing the image of the sun into slits. The second advantage is called Fellgett9’ or multiplex advantage. In a spectroscope, different wavelengths are studied either at different times if a photoelectric detector is used or with different photographic regions if a photographic plate is used. In interference spectroscopy, on the other hand, all spectral lines are measured simultaneously with a single photoelectric detector. This is equivalent to coding each optical frequency by an electrical cosine signal with different frequency, and then decoding numerically by Fourier analysis. In this manner, the signal-to-noise ratio is much greater than in conventional spectroscopy. We could describe many other useful applications of interferometry, but it would be impossible to mention all of them in a single chapter. Acknowledgment The author wishes to express his gratitude to Professor W. H. Steel for his many valuable suggestions.

References M . Born and E. Wolf, Principles ofoptics, Pergamon Press, New York, 1959. A. H . Cook, Interference of Electromagnetic Waves, Clarendon Press, Oxford, 1971. M . Fraqon, Optical Interferometry, Academic Press, New York, 1966. C. Candler, Modern Interferometers, Hilger and Watts, London, 1951. 5. W. H. Steel, Inferferomerry, 2nd ed., Cambridge University Press, London, 1983.

1. 2. 3. 4.

46

INTERFERENCE

6. S. Tolansky, An Introduction to Inrerferometry, Longmans Green, London, 1955. 7. K. M. Baird, “Interferometry: Some Modern Techniques,’’ in Advanced Optical Techniques (A.E.S. Van Heel, ed.), p. 123, North-Holland Publ., New York, 1967. 8. K. M. Baird, and G. R. Hanes, “Interferometers,” in Applied Optics and Optical Engineering, Vol. 4, p. 309 (R. Kingslake, ed.), Academic Press, New York, 1967. 9. J. Dyson, “Interferometers,” in Concepts ofClassical Optics (J. Strong, ed.), p. 377, W. H. Freeman, San Francisco, 1958. 10. R. N. Wolfe and F. C. Eisen, “Irradiance Distribution in a Lloyd Mirror Interference Pattern,” J. Opt. SOC.Am. 38, 706 (1948). 1 1 . W. H. Steel, “Two Beam Interferometry,” in Progress in Optics, Vol. 5, Chap. 3 ( E . Wolf, ed.), North-Holland Publ., Amsterdam, 1966. 12. M. V. R. K. Murty, ”Newton, Fizeau and Haidinger Interferometers,” in Optical Shop Testing ( D . Malacara, ed.), Chap. 1, Wiley, New York, 1978. 13. J. Guild, “Fringe Systems in Uncompensated Interferometers.” Proc. Phjjs. Soc. (London) 33, 40 (1920-1921). 14. F. Twyman, “Camera Lens,” British Patent 130224 (1919). 15. F. Twyman, “The Testing of Microscope Objectives and Microscopes by Interferometry,” Trans. Faraday Soc. 16, 208 (1920). 16. D. Malacara, “Twyman-Green Interferometer,” in Optical Shop Testing ( D . Malacara, ed.), Chap. 2, Wiley, New York, 1978. 17. M. V. R. K. Murty, “Lateral Shearing Interferometers,” in Optical Shop Testing (D. Malacara, ed.), Chap. 4, Wiley, New York, 1978. 18. D. Malacara, “Radial, Rotational. and Reversal Shear Interferometers,” in Optical Shop Testing (D. Malacard, ed.), Chap. 5, Wiley, New York, 1978. 19. 0. Bryngdahl, “Applications of Shearing Interferometry,” in Progress in Optics Vol. 4, Chap. 2 (E. Wolf, ed.), North-Holland Publ., Amsterdam, 1965. 20. W. J . Bates, “A Wavefront Shearing Interferometer,” Proc. Phys. Soc. 59, 940 (1947). 21. M. V. R. K. Murty, “The Use of a Single Plane Parallel Plate as a Lateral Shearing Interferometer with a Visible Gas Laser Source,” Appl. Opt. 3, 531 (1964). 22. J . B. Saunders, “Measurement of Wavefronts without a Reference Standard. I: The Wavefront Shearing Interferometer,” J. Res. Nut. Bur. Stand. 65b, 239 (1961). 23. D. Malacara. Testing of Optical Surfaces, Ph. D. Thesis, Institute of Optics, University of Rochester, New York, 1965. 24. M. P. Rimmer and J. Wyant, “Evaluation of Large Aberrations Using a Lateral Shear Interferometer Having Variable Shear,” Appl. Opt. 14, 143 (1975). 25. D. S. Brown, Interferometry N.P.L. Symposium No. I / , p. 253, Her Majesty‘s Stationery Office, London, 1959. 26. M. V. R. K. Murty, “A Compact Radial Shearing Interferometer Based on the Law of Refraction,” Appl. Opt. 3, 853 (1964). 27. M. V. R. K. Murty and E. C. Hagerott, “Rotational Shearing Interferometer,” Appl. Opt. 5, 615 (1966). 28. J. W. Gates, “The Measurement of Comatic Aberrations by Interferometry,” Proc. fhvs. Soc. B 68, 1065 (1955). 29. J . B. Saunders, “Inverting Interferometer,” J. Opt. Soc. Am. 45, 133 (1955). 30. S. Mallick, “Common-Path Interferometers,” in Optical Shop Testing ( D . Malacara, ed.), Chap. 3, Wiley, New York, 1978. 31. J. M. Burch, “Scatter Fringes of Equal Thickness,” Nature 171, 889 (1953). 32. A. A. Michelson, “Interference Phenomena in a New Form of Refractometer,” Phil. Mag. 5, 236 (1882). 33. Hanbury R. Brown, and R. Q. Twiss, “A new Type of Interferometer for Use in Radio Astronomy,” Phil. Mag. 45, 663 (1954).

REFERENCES

47

34. C. Roychoudhuri, “Multiple Beam Interferometers.” in OpficalShopTesfing ( D . Malacara, ed.), Chap. 6, Wiley, New York, 1978. 35. H. D. Polster, R. M. Scott, R. L. Crane, P. H. Langenbeck, R. Pilston, and G. Steinberg, “New Developments in Interferometry,” Appl. O p f .8, 521 (1969). 36. C. Fabry and A. Perot, “Thtorie et Applications d’une Nouvelle Mithode de Spectroscopie Interferentielle,” Ann. Chim. Phys. 7, 115 (1899). 37. P. W. Baumeister, “Interference and Optical Interference Coatings,” in Applied Optics and Opfical Engineering, Vol. 1, p. 285 (R. Kingslake. ed.), Academic Press, New York, 1965. 38. P. W. Baumeister and F. A. Jenkins, “Dispersion of the Phase Change for Dielectric Multilayers. Application to the Interference Filter,” J. Opt. Sac. Am. 47, 57 (1957). 39. F. S. Tomkins and M. Fred, “Wavelength Measurements with a Concave Grating Spectrograph,” Appl. O p f . 2, 715 (1963). 40. S. Tolansky, Multiple-beam Interferomefry of Surfaces and Films, Clarendon Press, Oxford, 1948. 41. S. Tolansky, Surjace Microtopography, Wiley Interscience, New York, 1960. 42. P. Connes, “L‘italon d e Fabry-Perot Sphirique,” J. Phvs. Radium 19, 262 (1958). 43. D. R. Herriott, “Spherical-Mirror Oscillating Interferometer,” Appl. Opt. 2, 865 (1963). 44. R. L. Fork, D. R. Herriott, and H. Kogelnik, “A Scanning Spherical Mirror Interferometer for Spectral Analysis of Laser Radiation,” Appl. Opt. 3, 1471 (1964). 45. K. W. Meissner, “Interference Spectroscopy. Part I,” J. Opt. Soc. Am. 31, 405 (1941). 46. K. W. Meissner, “Interference Spectroscopy. Part 11,” J. O p f . Sac. A m . 32, 185 (1942). 47. W. E. Williams, Applicafions of Inferferomefry,4th ed., Methuen, London, 1950. 48. P. Hariharan, “Multiple Pass Interferometers,” in Optical Shop Tesfing, ( D . Malacara, ed.), p. 217, Wiley, New York, 1978. 49. P. Langenbeck, “Multipass Twyman-Green Interferometer,” Appl. O p f . 6 , 1425 (1967). 50. P. Hariharan and D. Sen, “The Separation of Symmetrical and Asymmetrical Wavefront Aberrations in the Twyman-Green Interferometer,” Proc. Phys. Sac. (London) 77, 328 (1961 ). 51. D. Sen and P. N. Puntambekar, “An Inverting Fizeau Interferometer,” O p f . A c f a 12, 137 (1965). 52. D. Sen and P. N. Puntambekar, “Shearing Interferometers for Testing Corner Cubes and Right Angle Prisms,” Appl. O p f . 5, 1009 (1966). 53. P. N. Puntambekar and D. Sen, “A Simple Inverting Interferometer,” Opt. Acta 18, 719 (1971 ). 54. W. E. Williams, “Applications of Interferometry,” in Concepfs of Classical Opfics (J. Strong, ed.), p. 373, W. H. Freeman, San Francisco, 1958. 55. J. Dyson, Inferferomefry as a Measuring Tool, Machinery Publishing Co., Brighton, 1970. 56. P. Mollet, Opfics in Mefrology, Pergamon Press, Oxford, 1960. 57. A. A. Michelson and M. Morley, Silliman J. 34, 333 (1887). 58. R. B. Leighton, Principles of Modern Physics, Chap. 1, McGraw-Hill, New York, 1959. 59. T. S. Jaseja, A. Javan, J. Murray, and C. H. Townes, Phys. Rev. 133A, 1221 (1964). 60. A. A. Michelson, “Measurement of the Diameter of Orionis with the Interferometer,” Asfrophys. J. 53, 249 (1921). 61. J. Ojeda-Castatieda, “Foucault, Wire and Phase Modulation Tests” in OpticalShop Tesfing, ed. D. Malacara, John Wiley and Sons, New York, 1978. 62. V. Grigull and H. Rottenkolber, “Two-Beam Interferometer Using a Laser,” J. Opf. Soc. Am. 57, 149 (1967). 63. A. Cornejo-Rodriguez, “Ronchi Test’’ in Optical Shop Tesfing (D. Malacara, ed.), Chap. 8, Wiley, New York, 1978. 64. D. Malacara, A. Cornejo, and M. V. R. K. Murty, “Bibliography of Various Optical Testing Methods,” Appl. Opf. 14, 1065 (1975).

48

INTERFERENCE

65. J. D. Briers, “Interferometric Testing of Optical Systems and Components: A Review,” Opt. Laser Technol. 4, 28 (1972). 66. G. Schulz and J. Schwider, “lnterferometric Testing of Smooth Surfaces,” in Progress in Optics, Vol. 13, Chap. 4, (E. Wolf, ed.), North-Holland Publ., Amsterdam, 1976. 67. D. Malacara, ed. Optical Shop Testing, Wiley, New York, 1978. 68. J. H. Bruning, “Fringe Scanning Interferometers,” in Optical Shop Testing (D. Malacara ed.), Chap. 13, Wiley, New York, 1978. 69. K. Rosenhauer, “Measurement of Aberrations and Optical Transfer Functions of Optical Systems,” in Advanced Optical Techniques, Chap. 18, Wiley, New York, 1967. 70. H. H. Hopkins, “Interferometric Methods for the Study of Diffraction Images,” Optica Acta 2, 23 (1955). 71. A. A. Michelson, “Ditermination Expirimentale de la Valeur du Metre en Longeurs d’Ondes Lumineuses,” Trau. Mem. Bur. Int. Poids Mes. 11, 1 (1895). 72. K. M. Baird and L. E. Howlett, “The International Length Standard,’. Appl. Opt. 2, 455 (1963). 73. G. R. Harrison, N. Sturgis, S. P. Davis, and Y. Yamada, “Interferometrically Controlled Ruling of Ten-Inch Diffraction Grating,” J. Opt. SOC.A m . 49, 205 (1959). 74. J. N. Dukes and G. B. Gordon, Hewlett-Packard Journal 21, 2 (1970). 75. A. L. Hammond, Science 170, 1289 (1970). 76. V. Met, “Determination of Small Wedge-Angles Using a Gas Laser,” Appl. Opt. 5, 1242 (1966). 77. D. Malacara and 0. Harris, “Interferometric Measurement of Angles,” Appl. Opt. 9, 1630 (1970). 78. D. T. Tentori and M. Celaya, “Continuous Angle Measurement with a Jamin Interferometer,” Appl. Opt. 25, 215 (1986). 79. W. Krug, J. Rienitz, and G. Schultz, Contributions to Interference Microscopy (English trans.), Hilger and Watts, London, 1964. 80. J. W. Foreman, Jr., E. W. George, and R. D. Lewis, Appl. Phys. Lett. 7 , 77 (1965). 81. C. P. Wang and D. Snyder, “Laser Doppler Velocimetry: Experimental Study,” Appl. Opr. 13, 98 (1974). 82. Y. Yeh and H. Z. Cummins, Appl. Phys. Lett. 4, 176 (1964). 83. G. Sagnac, 1. Phys. Radium 4, 177 (1914). 84. A. A. Michelson, Phil. Mag. 8, 716 (1904). 85. A. A. Michelson, Astrophys. J. 61, 137 (1925). 86. A. A. Michelson and H. G. Gale, Asrrophys. J. 61, 140 (1925). 87. W. M. Macek and D. T. M. Davis, Jr., Appl. Phys. Lett. 2, 67 (1963). 88. F. Aronowitz, Laser Applications, Vol. 1, p. 133, (M. Ross, ed), Academic Press, New York, 1971. 89. J. T. Rowland and Agrawal, Opt. Laser Technol. 13, 239 (1981). 90. P. Connes, “Recherches sur la Spectroscopie par Transformation de Fourier,” Rev. d’Optrque 40, 116 (1961). 91. G. A. Vanasse and J. Strong, “Application of Fourier Transformations in Optics: Interferometric Spectroscopy,” in Concepts of Classical Optics (J. Strong, ed.), p. 419, W. H. Freeman, San Francisco, 1958. 92. P. Jaquinot and C. Dufour, J. Rech. Cent. Natl. Rech. Sci. 2, 91 (1948). 93. P. Fellgett, “A Propos de la Thiorie du Spectromitre Interfirentiel Multiplex,” J. Phys. Radium 19, 187 (1958).

2. DIFFRACTION A N D SCATTERING Daniel Malacara Centro de lnvestigaciones en Optica. A.C Apdo. Postal 948 37000 Leon, Gto. Mexico

2.1. Diffraction Diffraction phenomena have been some of the most interesting and most thoroughly studied manifestations of light in the history of physics. Many good textbooks present this subject in detail like those by Born and Wolf,’ Meyer,2 Marechal and F r a n ~ o n ,and ~ Ra~leigh.~ The , ~ earliest known mention of this phenomenon appears in the works of Leonard0 da Vinci (1452-1519).However, Grimaldi7 is the first one to seriously treat the subject in his book. It is interesting to know that Newton* was aware of Grimaldi’s observations. Huygens’ was the first one to develop a wave theory (see Fig. l), although he also seems to have been aware of Grimaldi’s observations, since he did not attempt to explain the existence of diffraction bands. An attempt to explain diffraction qualitatively was made by Young, 1 0 * ’ 1 - 1 2 who assumed that the wave passes undisturbed through the aperture, except at the edges, where the edge wave is formed. In this case, he thought that the interference between the direct wave and the edge waves formed the diffraction pattern. It was not until 1818 that FresnelI3 used the wave theory together with the principle of interference to explain diffraction quantitatively. He assumed mutual interference between all secondary waves, adding on the screen the amplitudes of each Huygens wavelet, and taking into account any phase differences. The Huygens principle, modified in this way, is called the HuygensFresnel principle; it leads to quantitatively good predictions of the light distribution on the screen. This theory, nevertheless, is not perfect. One of the problems is that there is no explanation as to why the wavelets do not travel backwards to the source, and another is that the resultant phase on the screen, computed by this theory, is 7r/2 less than the experimental value. 49 M E T H O D S O F E X P E R I M E N T A L PHYSICS Vol. 26

Copyright 0 1988 by Academic Press. Inc. All rights of reproduction in any form reserved ISBN 0- 12-475971-8

50

DIFFRACTION A N D SCATTERING

FIG. 1. Huygen's explanation of diffraction.

2.1.1. Kirchhoff's Diffraction Theory

In 1876 Kirchhoff 14.15,'6.17 established a complete mathematical theory of diffraction. He then proved that the Huygens-Fresnel principle could be regarded as an approximate form of his theory. Only a brief description will be given here, since very complete and detailed accounts can be found in several books.' Kirchhoff's theory starts with the so-called HelmholtzKirchhoff theorem derived from the scalar differential wave equation. He considers an arbitrary closed surface and an observation point P inside it. The idea is to evaluate the field at the internal point P, from a knowledge of the field at the closed surface (see Fig. 2). A point light source is placed outside the closed surface, at a distance r much greater than A. Kirchhoff then shows that the amplitude U ( P )at this point is given by:

This expression may now be applied to the solution of diffraction problems by considering the surface S as formed by three parts: portion A, the aperture through which the light passes, portion B, the diffracting flat screen,

51

DIFFRACTION

FIG.2. Helmholtz-KirchoH theorem

and portion C, a spherical shell which intersects the diffracting screen and which has its center at the observation point P. The amplitude over the diffracting screen must be zero. Therefore the integral over portion B must also be zero. The integration over the sphere C also goes to zero when its radius goes to infinity. If the angle between the incident and the diffracted rays is called 8, it may be shown that: cos( n, s) + cos( n, r ) = 1 + cos 8.

(2.2)

Hence, Eq. (2.1) becomes rs

(2.3)

which is the well-known Kirchhoff diffraction integral, where u is the diffracting aperture. We notice here a few differences between the results of this theory and those of Fresnel. The first important difference is the presence of the complex number i in front of the integral. This means that the phase of the observation point P, obtained with the Kirchhoff theory, has a difference 7r/2 with respect to the phase obtained with the Huygens-Fresnel theory. This also means that the phase of the illumination at point P, using a very small diffraction pinhole (only one Huygens wavelet), and the phase of the illumination at the same point with no diffraction, differ by 7r/2.

52


Second, we have the presence of the factor (1 + cos 0)/2. Usually this is called the inclination factor, first derived by Stokes. It implies that the amplitude of each secondary wave decreases when the inclination angle increases. This inclination factor appeared naturally in Kirchhoff’s theory and is only postulated in the Huygens-Fresnel theory. The third difference is the presence of the product l / r s in the integral in Eq. (2.3) which would appear in the Huygens-Fresnel theory only when the decrease of the amplitude, as l / r s , is considered. Experimental verifications of Kirchhoff’s scalar theory of diffraction were made by Silver” in 1962. Even Kirchhoff’s theory is incomplete, not only because of the approximations made, but mainly because the vectorial transverse nature of light waves is ignored. The scalar theory of diffraction produces, in general, very accurate results, but a further refinement is achieved if the vectorial nature of light20.2’.22.23324.2s and the optical properties of the screen or stops are considered by means of the electromagnetic theory. 2.1.2. Other Theories

As pointed out before, Kirchhoff’s theory is not perfect. PoincarC26and S ~ m m e r f e l dshowed ~~ that Kirchhoff’s hypotheses were inconsistent, because the theory does not reproduce the assumed values on the screen and in the plane of the aperture. As described by Bouwkamp28 (including numerous references), many modifications to Kirchhoff’s theory were made. One of the most important modifications of Kirchhoff’s hypotheses was made by K ~ t t l e r , ~ who ~ . ~ ’showed that it is more appropriate to prescribe some discontinuities at the edge of the aperture rather than continuous boundary values. This is called a saltus problem, and it is especially interesting when considering the diffraction at a black, completely absorbing screen. Another interesting theory again concerns the existence of Young’s edge waves. Young postulated these waves when noticing that the edge of the aperture appeared luminous when observed from points within the geometrical shadow. It is interesting to know that Fresnel considered these edge waves but finally discarded them. The first one to seriously consider the edge waves again was M a e ~ , but ~’ he did not go very far. Sommerfeld” used the edge waves to find an exact solution to the problem of diffraction by an infinite half plane. The problem was further studied by R u b i n o ~ i c z ,confirming ~ ~ . ~ ~ the existence of Young’s edge waves for flat or spherical diffracted wave fronts. Miyamoto3’ and Miyamoto and developed even further the Rubinowicz theory showing that Kirchhoff‘s diffraction integral can be

FRESNEL DIFFRACTION

53

FIG.3. Some Fresnel diffraction patterns of recognizable objects.

separated into two waves, one coming from the aperture and another one coming from its edge. In other words, they showed that a diffraction wave appears as arising from the scattering at the edge of the aperture. Descriptions of the Miyamoto-Wolf theory can be found in R ~ b i n o w i c z . ~ ~ . ~ ~ Even a corpuscular quantum approach to the diffraction problem in gratings has been given, namely by Duane.40

2.2. Fresnel Diffraction Depending on the positions of the light source and the observation plane, we may consider two types of diffraction. Fresnel diffraction occurs when either the light source or the observation plane or both are at finite distances from the diffracting aperture or object. Fraunhofer diffraction occurs when both the light source and the observation plane or both are at finite distances from the diffracting aperture. In the following sections, we will consider some selected configurations for Fresnel diffraction. Figure 3 shows some interesting Fresnel diffraction patterns. 2.2.1. Single-Slit Diffraction

Let us consider the configuration shown in Fig. 4, with a point source at 0 and the observing point at P. A direct ray OFP and a diffracted ray O Q P meet at P, with an optical path difference (OPD) equal to the small line segment Q R .

54


-

c

a

c

b

l

\

\ \ \

\

FIG;. 4.

Fresnel diffraction geometry.

Assuming that the distances a and b are very large compared to the height S, the segment QR may be approximated by the sum of the sagittas of the two arcs passing through Q and R, respectively. S 2 S2 a + b OpD=QR=-+-=2a 2b 2ab

s2,

(2.4)

giving a phase difference 6: S=kOPD=

n( a

+b ) S2.

abh

(2.5)

To simplify the mathematics, we consider this as a two-dimensional problem. Defining a nondimensional variable u as:

we can write the phase difference 6: (2.7)

If we divide the wave front into narrow parallel strips of width ds, we may use the Huygens-Fresnel principle and add the contribution of all the strips to the amplitude at point P. This sum is made by considering the

55

FRESNEL DIFFRACTION

FIG.5. Total amplitude at point P.

phase differences between the different strips and by adding them vectorially as shown in Fig. 5. Each vector, representing the contribution of each slit, has a magnitude directly proportional to the width ds and hence to dv. Each vector forms an angle 6 with the x axis. The resultant amplitude at the point P is directly proportional to the resultant vector R. We can mathematically generate a curve, as in Fig. 5, that can be used to find the amplitude at point P for any width and position of the diffracting slit. Using Eq. (2.7), we find: dx = dv cos 6 = cos dy = dv sin 6 = sin

and by integrating, we obtain: x= y=

(q)

dv,

(G) dv,

lo" ($) lo" ($)

(2.9)

cos

dv,

(2.10)

sin

dv.

(2.11)

56

DIFFRACTION A N D SCAITERING

f

q

A2

FIG.6 . Cornu spiral.

Equations (2.10) and (2.1 1 ) are the Fresnel integrals, whose numerical values are given in table form in many book^.^'.^^ The curve produced with these integrals is the Cornu spiral shown in Fig. 6. Strictly speaking, we should have used the inclination factor (1 + cos 8 ) / 2 here, since the amplitude of each secondary wavelet decreases when the angle 8 increases. We assumed, however, that 8 remains very small everywhere, and therefore the inclination factor is always very close to 1. The Corm spiral allows us to find the amplitude of the illumination at the point P on the axis. If the diffracting slit is centered on the optical axis and has a width S, we first compute u by using Eq. (2.6). Then we choose two symmetrical points A , to A, on the Cornu spiral, such that their separation measured along the spiral be equal to u. The distance from A , and A * , measured along a straight line, represents the amplitude of the illumination at P. If the diffracting slit is decentered off the optical axis, we again use Eq. (6.6) to compute the decentering and the slit width in terms of u. Then two

57

FRESNEL DIFFRACTION

points A; and A2 are chosen such that their positions as measured along the curve corresponding to the slit width and to the decentering. Again, the separation between these two points, measured along a straight line, represents the amplitude at P. The whole diffraction pattern for a diffracting slit, with a width corresponding to a certain value of u, is found by moving two points, A; and A;, along the curve in such a way that the distance between these two points along the curve remains constant and equal to u. The distance between the two points in a straight line will change continuously to give the amplitude of the whole diffraction pattern. The irradiance which is the square of the amplitude is shown in Fig. 7(a). In order to relate the resultant irradiance to the value of the unobstructed irradiance, the squares of amplitudes obtained from Fig. 6 must be divided by 2. If we have only straight edges, we may consider that we have a decentered slit, with one edge on the optical axis and the other edge at infinity. In this case, we have one fixed point 2, and a moving point A; on the Curnu spiral. We then obtain the diffraction pattern in Fig. 7(b). If there is no INTENSITY

I

I

I

r

GEOMETRICAL

LIMITS

(A)

INTENSITY

G

GEOMETRICAL EDGE

(B) FIG.7. Diffraction light distribution in (a) a wide slit, (b) a straight edge.

58


diffraction obstacle, the intensity everywhere on the screen is represented by the distance along a straight line from 2, to 2,. In this case, we can see that the phase at the point P, with no diffraction, is ~ r / 4(45") ahead of the phase at the same point P, produced with the infinitely narrow diffracting slit, and is 7r/4 ahead of the phase at P produced with an infinitely small diffracting pinhole. 2.2.2. Circular Aperture Diffraction

Fresnel diffraction by a circular aperture can be easily studied for a point on the optical axis. In doing so, we can obtain very interesting results, as we shall see. Using the same geometry as in Fig. 2, we can find the amplitude at the point P by dividing the wavefront into annular zones concentric with the point F, where each zone has a radius S, and then by adding the contributions of each zone to the final amplitude at P. From Eq. (2.5), the phase difference between the light on a ring with radius S and the light through the center is 6 = KS2,

(2.12)

where K=

v(a

+b )

abh

.

(2.13)

The contribution of each zone is directly proportional to its area, and this area in turn is directly proportional to its width ds. Taking this into account, we can find a graph analogous to the Cornu spiral, which can be easily shown to be given by:

[

x + y--

;I*=[&l'~

(2.14)

This is a circle with its center on the y axis, tangent to the x axis, and with radius A/2K. This means that, if the radius S of the diffracting circle is increased continuously, the amplitude on the axis at the point P oscillates, as shown in Fig. 8(a). This is physically unacceptable, because these oscillations in the amplitude do not occur when the diffracting aperture is very large. The problem disappears when taking into account the inclination factor (1 +cos 8)/2, because the radii of the circles will become smaller when the aperture grows, increasing the angle 8. The curve is now a spiral, as in Fig. 9. When the aperture becomes very large, the amplitude reaches the value A/2K at the point P on the axis. The color at the center of the diffraction pattern produced by a circular aperture and illuminated with white light has been studied by Bergsten and H ~ b e r t y . ~ ~

59

F R E S N E L DIFFRACTION AMPLITUDE AT POINT P

A 2K

S

AMPLITUDE AT POINT P

FIG.8. Amplitude variation on axis with a diffracting circle, (a) without obliquity factor, ( b ) with obliquity factor.

2.2.3.Fresnel Zone Plate Diffraction The amplitude on the axis of a diffracting circular aperture oscillates because some annular zones contribute constructively to the interference, while others contribute destructively. If we mask all the zones that contribute destructively, the irradiance at the point P is drastically enhanced. Then the diffracting hole becomes a Fresnel zone plate, which is illustrated in Fig. 10.

60


FIG.9. Spiral for a circular aperture.

F I G .10. Fresnel zone plate

FRESNEL DIFFRACTION

61

Since the irradiance is greatly enhanced at the point P, it must be greatly reduced at all other points by the energy conservation principle. This means that a Fresnel zone plate acts like a lens and can even form images. The outer edge of a dark ring occurs for 6 = nm, while the inner edge of the same ring occurs for 6 = (n - 1/2)7r. Thus, from Eqs. (2.12) and (2.13), it may be shown that the radii S of the limits of a dark ring n are given by (2.15)

and ab s . =[ h ( n - 1/2) a+b

(2.16)

n'n

Defining So= SIin a n d f = 2Si/h (for a zone plate with a transparent center zone), we may show that (2.17)

which is the equivalent of the thin-lens formula for the positions of the object and the image, at a and b, respectively. In spite of the close similarity between a thin lens and zone plate, there are some important differences. The first difference for a lens is that the focus is not unique, but there are an infinite number of them with focal lengths fm, given by 2s; mh '

fm =-

m =integer,

(2.18)

where rn is any positive or negative odd integer. It should also be noticed that for every convergent focus there is a corresponding diverging focus. In other words, a zone plate acts simultaneously as a convergent and a divergent lens. A very useful application of zone plates is for alignment, as described by Van A pinhole camera uses a small pinhole, instead of a lens, to form the image. A consequence of Eq. (2.17) is that in order to optimize the resolution and light-collecting power, the pinhole must have a radius S, and the photographic-plate-to-pinholedistance should be f: A Gabor plate is quite similar to a zone plate, with the difference being that its transmittance does not change in steps but in a quasi-sinusoidal manner. It is normally produced by the interference between a flat and a spherical wave front.

62

D I F F R A C T I O N A N D SCATTERING

I/ I

/-

-I+-POINT COLLIMATOR

I

\‘I

DIFFRACTING APERTURE

FIG.1 I . Arrangement to observe Fraunhofer diffraction.

2.3. Fraunhofer Diffraction and Fourier Transforms When the light source and the observing screen are both at infinite distances from the diffracting aperture, we say that we are observing Fruunhofer difruction. The light source may really be at infinity as in the case of a star, but more frequently the light source is placed “optically at infinity” by means of a collimating lens. In this case the light source is at the focus of a convergent lens, as shown in Fig. 1 1 . The observing screen cannot be really placed at infinity, but it is again done optically by placing it in the focal plane of a convergent lens. The fact that the observing screen is at infinity may be interpreted by saying that what we observe on the screen in Fraunhofer diffraction is the angular distribution of the light after the diffracting aperture. Figure 12 shows the slow transition from Fresnel to Fraunhofer diffraction, with a point source at infinity, by increasing the distance b of the observing screen from the diffracting aperture. We say that we have reached the far field or Fraunhofer diffraction region when b >> D‘/A, where D is the diameter of the diffracting object. The inclination factor is not important in the case of Fraunhofer diffrdction, and if we also take the case of normal incidence ( r = constant), the

F R A U N H O F E R DIFFRACTION A N D FOURIER T R A N S F O R M S

0 44

63

64


diffraction integral of Eq. (2.3) becomes

U(P)=i{

{

AcdS, S

(2.19)

where A is a constant whose magnitude is directly proportional to the amplitude on the diffracting aperture plane. If the aperture is on the x, y plane, and the amplitude on this plane is a function of x and y , we consider A also to be a function of x and y. The distance s is variable enough to produce significant changes in the phase ks, but constant enough to include l / s within the value of A. Thus, U (P ) =

J J ,~ ( xy ,) eiksdx dy.

(2.20)

We may show that s = x s i n O , + y s i n O,.;

(2.21)

and by defining

k,

=k

sin 0,

and

k,, = k sin O v ,

(2.22)

we can write Eq. (2.20) in the form

U(k,, k,) = i

{

A(x, y ) e’(k\r+k”’) dx dy.

(2.23)

On the other hand, from Fourier theory, the function A(x, y ) can be given by: (2.24) The functions A(x, y ) and U (k,, k,) are two-dimensional Fourier transforms of each other. Thus, except for unimportant constants, we can say that the angular diffraction pattern is the Fourier transform of the amplitude function on the diffracting aperture. In a similar way, for diffraction apertures in one dimension, we may write the one-dimensional Fourier transforms as (2.25)

FRAUNHOFER DIFFRACTION A N D FOURIER TRANSFORMS

65

and U ( k , ) e - l k \ ' dk,.

(2.26)

Fraunhofer diffraction has been widely studied by many authors, among whom we must mention S c h ~ e r d , ~whose ' excellent work includes many fine handmade color diffraction pictures. This work was reviewed by Hoover and Harris.46Harris4' has computed diffraction patterns for many different apertures. The colors observed when the aperture is illuminated with white light have been studied by Bergsten and P ~ p e l k a . ~ '

2.3.1. Single-Slit and Rectangular Aperture Diffraction

Let us take a slit on the x, y plane centered on the y axis. If the slit width is 2 a , we can write Eq. ( 2 . 2 5 ) in the form:

U ( k , )= A

['

J

(2.27)

elk\' dx,

-0

assuming that the amplitude A ( x ) remains constant inside the slit. Integrating, and using Eqs. ( 2 . 2 1 ) and ( 2 . 2 2 ) , we obtain:

U (k ) = U,

I

sin( k a sin 6) k a sin 6 '

(2.28)

where U, is a constant; Fig. 13 shows this amplitude and its corresponding irradiance versus k a sin 6. The first dark zone in the irradiance pattern can be shown to be at an angle 6 given by A

sin 6 =-. 2a

(2.29)

Therefore, the diffraction pattern becomes narrower when the slit width 2 a increases. For a rectangular aperture, with width 2 a and length 2b, we can use Eq. ( 2 . 2 3 ) to obtain U(O,, 4.) = U"

sin( k a sin 0,) k a sin Ox

sin( k b sin Or) k b sin 6,

][

3.

(2.30)

66


k o sin

(B)

6

FIG. 13. Fraunhofer diffraction pattern for single slit. ( a ) Amplitude distribution. ( h ) lrradiance distribution.

2.3.2.Circular Aperture Diffraction The diffraction pattern of a circular aperture may be found by integrating Eq. (2.23) inside the aperture, thus obtaining sin 0 ) u(e)=2u0 J,(ka ka sin 0

[

1’

(2.31)

where a is the radius of the aperture and J , ( x )is a first-order Bessel function


-

AIRY DISC

--

67

ka sin 8

(6) F l c . 14. Fraunhofer diffraction pattern for a circular aperture. ( a ) Amplitude aperture. ( b ) Irradiance distribution.

of x. The function U ( 0 ) and its corresponding irradiance U 2 (0 ) are plotted in Fig. 14. The first dark ring, which outlines the central spot or Airy disc, studied by Airy in 1835,49 can be seen to have a direction given by h

sin 0 = 1.22 -= 2a

where D is the diameter of the aperture.

A 1.22 D’

(2.32)

68


If the pattern is formed at the focal plane of a lens of clear aperture D with focal length A the radius r of the Airy disc is:

f

r = 1.22h -. D

(2.33)

If a collimated beam of light illuminates a well-designed convergent lens, the image is not a point but a Fraunhofer diffraction pattern of the free aperture. This is the reason why the resolving power of any perfect optical instrument (an instrument with no geometrical aberrations) is not infinity, but is limited by the size of the diffraction image. Sletten and Blacksmithso have shown that a paraboloidal microwave antenna has also an image at the focus, limited in size by Fraunhofer diffraction. It should be pointed out, however, that in a paraboloidal antenna, the f / D ratio is between 0.2 and 0.5, and thus the image structure differs from the Airy pattern because of the large angle of convergence, as described by Minnett and Thomas.’’ Figure 15 shows the Fraunhofer diffraction patterns produced by four different aperture shapes.

2.3.3.Babinet Principle Two pupil functions, A l ( x , y ) and A 2 ( x , y ) are complementary to each other if A,(x,y)+A,(x,y)=constant.

(2.34)

If the amplitude is constant over the whole x, y plane, then we say that two diffracting apertures are complementary if the transparent parts in one are the opaque parts in the other, and vice versa. For example, the complementary aperture of a circular aperture is the whole x, y plane, excluding only the opaque disc of the same size as the circular aperture. Let us assume that we have two arbitrary complementary apertures, giving diffraction pattern U , ( P ) and U , ( P ) . If U ( P ) is the amplitude of the illumination produced without any diffraction apertures, the Babinet principle’’ says that (2.35)

which can be very easily shown to be true. Applying this principle to the Fraunhofer diffraction patterns produced by a circular aperture and an opaque disc, we can see that U ( P ) is a very high and narrow peak at the origin, because all the light will travel in a single direction if there is no diffracting aperture.


69

FIG. 15. Fraunhofer diffraction pattern for four different apertures.

Thus, except for the point at the origin, the Fraunhofer diffraction patterns of complementary apertures are identical in shape but with amplitudes of opposite sign. 2.3.4. Parseval‘s Theorem As in any other physical process, energy is preserved, but it is probably more evident in Fraunhofer diffraction, where all computations are made by means of Fourier transforms as given by Eqs. (2.23) and (2.24). The following result we quote here can be mathematically proved from the definition of the Fourier transform.

(2.36)

70

D I F F R A C T I O N A N D SCATTERING

This equation is known as Parseval’s or the energy theorem, because it may be used to show that energy is always conserved in Fraunhofer diffraction experiments.

2.4. Diffraction Gratings A diffraction grating is just a diffraction screen formed by many parallel and equidistant slits. The principle of the diffraction grating was discovered by Rittenhouse” and later rediscovered by F r a ~ n h o f e r A. ~good ~ modern review of diffraction gratings is found in the book by Hutleyss and in the papers by Richardson” and Welford.” We could study the diffraction gratings by applying Kirchhoff’s integral directly to obtain all their properties. However, by studying them using more elementary principles, we gain better physical insight. 2.4.1. Angular Distribution of Light As shown in Fig. 16, the maxima of irradiance occur at angles such that the interference between the different slits is constructive. The interference is assumed to take place at infinity, since we are considering Fraunhofer diffraction. In Fig. 16, we have represented only two consecutive slits. It is easy to see that the maxima of the intensity are at an angle such that the optical path difference between two consecutive slits is given by

OPD = CA - BD = mA,

FIG. 16. Maxima of irradiance in a diffraction grating.

(2.37)

DIFFRACTION GRATl NGS

71

where m is any integer including zero, commonly called the order of interference. If d is the separation between the centers of the slits, and 8 , and Or are the angles of incidence and diffraction, we can see that d(sin

-sin 8,)

= mh.

(2.38)

The angles 8 , and O2 are both positive, as shown in Fig. 16, because they are each measured counterclockwise from the normal to the grating. The complete angular distribution of the light may be obtained by adding the amplitude contributions of each of the slits at a given angle 0,. Assuming that the grating has N slits and that 6 is the phase difference between two consecutive slits, the total amplitude at an angle 02 is:

E

= A[1

+ elS+ e2"+.

. . + e ( N p l ) 'I,S

(2.39)

which can be transformed to give (2.40)

Irradiance is then given by (2.41)

which can be shown to be equal to

(2.42)

i n order to visualize the form of this function, we have plotted the numerator, the denominator, and their quotient for a grating with five slits ( N = 5) in Fig. 17. The high peaks of the irradiance occur at values of 6, such that

_6 -- mr,

m =integer,

2

since at those points,

[

sin'(

lim S/2-mrr

y) (I")

and hence the peaks have a maximum irradiance A 2 N 2 .

(2.43)

(2.44)


I

I

I

I

I

I

I

I

I

I

m=O

m=l

I

A

A

m=2

m =3

FIG.17. Irradiance in a grating with five slits.

The phase difference 6 is given by:

S=kOPD=

2~rd(sinO1 -sin 0,)

A

(2.45)

Therefore, from Eq. (2.43) we may obtain Eq. (2.38) as expected. There are ( N- 2) secondary peaks with much lower irradiance between any two of the main peaks. If N is increased, the number of secondary maxima increases, but their height decreases very rapidly. Figure 18 shows the diffraction patterns produced by four gratings with a different number of slits. 2.4.2. Chromatic Dispersion and Resolving Power

If light of several wavelengths (several colors) impinges upon a diffraction grating at a given angle of incidence O,, there will be a different angle of diffraction O2 for each color. This property is used to make spectrographs and spectrometers using diffraction gratings instead of prisms. From Eq.

DIFFRACTION GRATINGS

73

FIG. 18. Diffraction patterns for four different arrays of slits.

(2.38), the chromatic dispersive power can be given by:

m A AA - d cos 02'

02 --

(2.46)

The chromatic dispersive power increases linearly with the order of interference. In a spectrograph using gratings, the angle O2 does not change much within a given order; therefore, cos O2 may be considered to be a constant. Thus A 0 J h A is approximately constant. In other words, the chromatic dispersion is approximately linear with A, as opposed to that for a dispersing prism. The chromatic resolving power of a grating may be computed by means of the Rayleigh criterion illustrated in Fig. 19. This criterion may be stated by saying that two spectral lines with slightly different wavelengths are just resolved when the maximum of one of the lines coincides with the first minimum of the other. We can see that the phase difference between the peak of the line and the first minimum is given by: (2.47) but from Eq. (2.45), this difference in phase corresponds to a difference

74

DIFFRACTION A N D SCATTERING SUM OF

TWO LINES

FIG. 19. Two lines just resolved by a diffraction grating.

he2 in the angle 0 2 ,given by 2 .rrd A~=--cos A

8,AO2.

(2.48)

From these two equations, the angular difference between the line maximum and its first minimum is

A = - Nd cos B2’

(2.49)

If two spectral lines with wavelength difference AA are separated by just this angle, from Eqs. (2.49) and (2.46), we obtain:

A _ - mN. AA

(2.50)

This is the well-known expression for the resolving power of a grating. If we define the effective width We, of a grating as

we,,= w cos 82 = Nd cos 8 2

(2.51)

and use Eq. (2.49) we find a n alternative expression showing that the resolving power of a grating is limited only by its effective width and the wavelength, as shown by Michelson,5s (2.52)


15

We may also write: (2.53) which shows that the chromatic resolving power of a grating is directly proportional to its effective diameter times its dispersive power.

2.4.3. Energy Distribution Among Different Orders

Until now, we have been considering the grating formed by infinitely thin slits. If the width of each slit is 2a, we cannot assume that the light diffracted by each slit travels in all directions with equal irradiance but according to the diffraction pattern for a single slit [Eq. (2.28)]. Thus, the irradiance in a direction O will be the product of Eq. (2.42) times the square of Eq. (2.28): sin'( sin

y) (Ifi)

sin ka(sin O1 -sin 0,) [ka(sin O1 -sin O,)] '

(2.54)

We have plotted this irradiance in Fig. 20(a) for a grating with five slits of widths 2a = d/3. A line with some order of interference does not appear when its direction coincides with the direction of a zero of the second factor in Eq. (2.54). The zeros of the second factor may be shown to occur at angles such that 2a(sin O1 -sin Oz) = nh,

n = a n y integer excluding zero.

(2.55)

The line positions are given by Eq. (2.38). Thus, the missing order numbers are: d 2a

m = n--,

(2.56)

where n is any integer, excluding zero, such that m is also an integer. The influence of the slit width is illustrated in Fig. 21, which shows the diffraction patterns for two slits with different widths but with the same separation. 2.4.4. Ghost Lines in Diffraction Gratings

Often spurious spectrum lines are seen in grating spectra, due to nonconstant separation in the slits. There are two main types of these so-called ghost lines.

76

DIFFRACTION A N D SCATrERING

-6

-4

-2

I

0

2

4

6

m

(B) FIG.20. lrradiance distribution in a grating with slits with finite width. ( a ) Finite-size grating. ( b ) Infinite-size grating.

Rowland ghosts,59 which have been studied by Michelson"" and Anderson,6' are due to periodic errors in the spacing of the slits because of imperfections in the ruling engine. These spurious lines appear as two faint lines symmetrically located with respect to the main or parent line. Lyman ghosts have also been studied, among others, by Anderson6' and by Runge.62They are due to nonperiodic errors in the line spacing. These lines are usually observed at large distances from the parent lines.


0

5 LI1

m

rn

5

5 0

77

78


A ) T R A N S M I S S I O N TYPE

B ) REFLECTION TYPE

FIG. 22. Basic types of diffraction gratings.

There are some other types of spurious lines and ghosts that may appear in a diffraction grating spectrum, but all of them are due to ruling imperfections. 2.4.5. Different Types of Diffraction Gratings

There are reflecting as well as transmission diffraction gratings, as shown in Fig. 22. Both of these gratings usually have the groove faces made at an angle (called the blaze angle) from the plane of the grating. The purpose of the blaze angle is to shift the maximum of the light distribution among the different orders, from the zero order to some other desired order, as first described by W00d63.h4and by Babco~k.'~ To avoid absorption and other undesirable effects due to the light transmission through glass, the grating type used most is the reyedon grating. The blaze angle a in this case is given by 6 ,- 6 2 2 '

(2.57)

y = o1 62.

+

(2.58)

mh - 2 cos Y sin a. _ d 2

(2.59)

a=-

and the total deflection angle is

Then, from Eq. (2.38), we find:

A case of special interest arises, y

= 0;

( a = el = &), reducing Eq. (2.59) to

mh = 2 d sin a = 2 h ,

(2.60)

RESOLVING POWER O F OPTICAL I N S T R U M E N T S

79

GROOVE

\

.

B L A Z E D GRATING

I / ECHELLE GRATING

FIG.23. Blazed reflection grating and echelle grating.

where h is the step height as shown in Fig. 23. The blaze angle is often specified as the wavelength that fulfills this condition in the first order. WoodG6pointed out that very efficient gratings for the infrared could be obtained by using coarse-blazed gratings with this condition ( y = 0). Since the light is reflected on the grooves' faces in a mirror like manner, he called them echellettes, Harrison6736x considered that since resolving power depends only on the total grating width, the echellette could be made with spacings as large as about 100 grooves/mm. The typical blaze angle for these gratings, named echelles by Harrison, is 63"26'. Echelles are very useful in order to achieve high resolving powers in the infrared. Gratings can also be concave and can be used in many configurations that will not be described here. The reader is referred to the book by Sawyer69 for a more complete description. The art of grating manufacture is a very complicated and interesting one. The reader can consult the excellent papers by Harrison,67768S t r ~ n g , ~ and " Stroke7' for details. In closing the subject of diffraction gratings, it should be mentioned that there are also two- and three-dimensional gratings. Figure 24 shows the diffraction patterns for two two-dimensional arrays of apertures.

2.5. Resolving Power of Optical Instruments Because of diffraction effects, the resolving powers of all optical instruments, even if they have no geometrical aberrations, are limited by the sizes of their apertures.

80


FIG.24. Diffraction of a two-dimensional array of holes. (a) A regular array and (b) its diffraction pattern. (c) An irregular array, and (d) its diffraction pattern.

If a lens is free of aberrations of any kind, its resolving power is determined by the size of the Fraunhofer diffraction pattern produced by the lens source in the Airy disc. We can see that the angular resolution of a lens in a telescope is limited only by the diameter D of the lens. If we define the image distance ratio ( L ' /T )as (L'/ D ) , which becomes equal to the focal ratiof/T when L ' = J we may write the linear radius of the Airy disc on the image plane as: r = 1.22A

(b).

(2.61)

Let us consider two very close stars, with their Airy discs overlapping in the image plane. The light of the two stars is mutually incoherent, so that there is no fixed phase relation between the two images. The resultant combined image is then obtained by adding the intensities of the two images.

RESOLVING POWER O F OPTICAL INSTRUMENTS

SUM

81

OF TWO IMAGES

-

RADIUS OF AIRY DISC

FIG. 25. Two point images just resolved.

Rayleigh established that under these conditions of incoherent illumination, the two images are just resolved when they are separated by a distance equal to the radius of the Airy disc, as illustrated in Figs. 25 and 26. Until now, we have considered images produced by lenses with no geometrical aberrations that produce perfectly spherical wave fronts. When the wave fronts have deformations smaller than A / 4 (Rayleigh criterion), the image is not much different from that produced by a perfect lens, as shown by Marechal.’* If the wave-front deformations are large compared with the wavelength, the image is that predicted by geometrical optics. The intermediate case, however, is extremely complicated, as described by Marechal and F r a n ~ o n Figure .~ 27 shows the appearance of an image of a point source in the presence of aberrations. The angular resolution of a telescope increases, that is, it resolves a smaller angle when the diameter is increased. However, a telescope with a resolution greater than that permitted by the atmospheric “seeing” or turbulence is unnecessarily precise, because in this case the resolution limiter will be the seeing and not the telescope. Depending on the geographic place, the seeing is of the order of a few tenths of a second of arc, which may be matched by a telescope with a diameter of about 40cm. To take full advantage of the resolution of a telescope with diameters much larger than 40 cm, we have to use it outside of the terrestrial atmosphere. However, very large telescopes are made because the light-gathering power increases with the square of the diameter.

82


F i c . 2 6 . Two point images with equal irradiance. ( a ) Unresolved. ( b ) Just resolved. (c) Completely resolved.

2.5.1. Optical Transfer Function

The different points in the image formed by a photographic lens are also mutually incoherent. Therefore, we say that the object is incoherently illuminated. The quality of the image in a photographic lens is seldom completely determined by the maximum resolving power of the lens. For example, two lenses might be able to resolve the same fine detail, but with very different contrast. As in electronic amplifiers, it is not enough to know


83

ill

0)

C

.-.g D 0

L

d

P N

! i

84


the high-frequency cutoff of an audio amplifier, but the complete frequency response curve must also be known. The optical equivalent for lenses of the frequency response curve for electronic amplifiers may be found by defining the spatial frequency content of an image. To illustrate this concept, we may take any picture and take a measurement of the irradiance along a straight line. The graph of the irradiance, with respect to x, may be considered, by means of Fourier analysis, as the superposition of many sine waves as shown in Fig. 28, with different wavelengths L. Then, we define the spatial frequency w as: w=-

2T L’

(2.62)

The frequency response curve of a lens described first by O’Nei1173 in 1954 is called the optical transferfunction (OTF) and is defined as the ratio of the harmonic components in the image to those in the object. It has two

m

-wK 3 0 Lb

FIG. 28. Spatial frequency content of an image.

85


parts. The plot of the amplitude ratio versus spatial frequency is often called the modulation transferfunction (MTF). The second part, the plot of the phase ratio, is less frequently used. The best way to measure the MTF is to form the image of an object that contains all spatial frequencies in equal amounts. The plot of the amplitudes of the spatial frequency components in the image thus formed would then represent the frequency response of the lens. An object that contains all spatial frequencies in equal amounts is a point source, as may be proved without much difficulty. The image of this point source is called point spreadfunction. The point spread function is the same as the Fraunhofer diffraction pattern of the lens aperture only if the lens is perfectly free of geometrical aberrations. We shall assume a point source at infinity. The point spread function in the focal plane of a lens with focal length F, with coordinate x F and y F can be obtained from Eq. (2.23) if we first obtain the amplitude as follows: A(XF, yF) =

II-:

T ( ~y ), e ~ ( k / F ) ( r ~ f i + i ,)i ldx dy,

(2.63)

where T ( x ,y ) represents the pupil function, including any deformations of the wave front due to aberrations of the lens. The point spread function S ( x F ,y F ) is then given by:

and therefore, from Eq. (2.63):

.

[I

1

T * ( x ,y ) e - l ( k I ~ ) ( X X f i + Y v F ) d x d y . -‘m

(2.65)

The variables x and y in these integrals do not remain after the integration and, therefore, may be replaced by the symbols 2 and J in the second integral, so that we may move the entire expression under the same integral signs.

(2.66)

Once the point spread function is computed from this expression or becomes known by some other means, the optical transfer function F ( w , , w , , ) may be computed by using the Fourier transform of S ( x F ,y F )

86


as follows: F(w,,w,) =

I:I

s ( x F y, F )e - “ W \ X F + ’ > ” ,

)

dXF dYF.

(2.67)

There is a simpler way to compute the OTF when T ( x , y ) instead of S ( X , , y F )is known, as will be described here. If the function F ( w , , w , ) is known, the point spread function can be computed by taking the Fourier transform of F ( w , , w ) ) , as follows:

Then, except for an unimportant constant, Eq. (2.66) becomes identical if we define: w,

k =-(x-Z) F

(2.69)

and

k F

w , =- ( Y

-?I,

(2.70)

obtaining thus:

If a lens is perfectly free of aberrations, we can set T ( x , 1 ) = T*(x,y ) = 1 inside the clear parts of the aperture. Then F ( q , w.,.) is the common area of two sheared images of the aperture, as in Fig. 29. Summarizing, the optical transfer function is the two-dimensional Fourier transform of the point spread function and also the autocorrelation of the pupil function. In Fig. 30, we have the optical transfer function of a perfect circular lens for various defocusing magnitudes. The cutoff frequency in focus is given by: kD F

w, =-.

(2.72)

2.6. The Abbe Theory of the Microscope When we defined the resolution of a lens, we assumed that the two neighboring points to be resolved were perfectly incoherent to each other.

87

T H E A B B E T H E O R Y OF T H E MICROSCOPE

COMMON AREA: F(wx,wy)

FIG.29. Computation of the optical transfer function of an aberration-free lens.

\ \ \

INFOCUS-,

\

h I

0

-.2

0

I

0.2

1

I

0.4

I

I

I

0.6

dr

0.0

(i&P FIG.30. Some optical transfer functions.

88


FIG.

31. Abbe theory of the microscope.

The appearance of the image is quite different if the two points are coherent. This is, for example, the case if the object is illuminated by a single point source or a laser. Ernest Abbe74 developed a theory, usually called the Abbe theory of the microscope, which allows the computation of the resolving power of a lens when the object is illuminated with coherent light. His main assumption was that the spatial frequency structure of the object to be imaged acts like a diffraction grating when illuminated by a coherent light source, as shown in Fig. 31. If the object acts as a diffraction grating, the light will be diffracted into many beams with different orders of diffraction. In this case the image may be regarded as the interference pattern arising from the superposition of all diffracted beams. If the lens is so small that only the zero order of diffraction passes through the lens, no interference takes place at the image plane, and therefore no image is formed. The object structure is first detected at the image plane when the lens is large enough to let the two first orders of diffraction pass through the lens. Under these conditions, the image is not an exact reproduction of the object, because the orders of diffraction with magnitude greater than 1 are not reaching the image plane. The image, in this case, has a sinusoidal profile with a spatial frequency equal to that of the object (see Fig. 32). When the lens is larger, so that the order of diffraction + 2 and -2 passes through the lens, the image will more closely resemble the object, as shown in Fig. 32.

89

T H E A B B E T H E O R Y O F T H E MICROSCOPE

FIRST ORDER IMAGE

OBJECT

(A)

THIRD ORDER IMAGE

(B)

(C)

FIG.32. Explanation of Abbe theory.

The resolving power limit may be easily found by using the diffraction grating equation, (2.38) for the first order rn and normal incidence:

d=- A sin 8’

(2.73)

where A, is the wavelength in the object medium, d is the minimum object diameter, and 8 is the angular radius of the lens, as seen from the object. Alternatively, we can write this diameter d as (2.74) where A. is the wavelength in vacuum, N is the refractive index in the object medium, and NA is the numerical aperture of the lens. We can see that the resolution is greater when the object is illuminated with incoherent light. The resolution with coherent illumination is almost half of that obtained with incoherent illumination. Another difference between illumination by coherent and incoherent light is that the optical transfer function in the coherent case is constant for all spatial frequencies, dropping suddenly to zero at the cutoff frequency, while in the incoherent case, it decreases continuously as the frequency increases. It must be pointed out, however, that the differences between the two types of illumination are so complex that a superficial examination must be avoided. Comparison should be limited only to certain particular aspects. The resolution in the case of partial coherence (illumination with a mixture of coherent and incoherent light) is very interesting and has been studied by many authors, like

90


Ar2

A6 2

A) NORMAL MICROSCOPE

B ) P H A S E C O N T R A S T MICROSCOPE

FIG.33. Amplitudes in a phase-contrast microscope.

2.6.1. The Phase-Contrast Microscope

A logical application of the Abbe theory of the microscope is the phasecontrast microscope developed by Zer~~ike,’~.’’ (he received the 1953 Nobel Prize in Physics for this development). This microscope permits the observation of transparent objects without staining. It is only required that the object’s index of refraction be different from that of the medium in which it is immersed. If the object has a refractive index different from that of the surroundings, a wave front passing through it becomes deformed because of the different optical paths. Thus the light becomes diffracted as in any other kind of object. No image is visible, however, because the interference at the image plane produces a constant amplitude everywhere. To understand this, we might consider (see in Fig. 33) that the amplitude A, at a given point on the image is the sum of the nondiffracted amplitude A,, (zero order) and the diffracted amplitude Ad (nonzero order). The diffracted amplitude changes from point to point on the image, because it contains the information about the object structure. The nondiffracted amplitude is constant in phase and amplitude everywhere on the image plane. The resultant amplitude A, has the same magnitude for all image points, even though Ad and the phase difference A 6 are variables. The object is transparent, but by shifting the phase of the nondiffracted wave by an angle 7r/2, as we see in Fig. 33, the resultant amplitude A, will be a function of the phase shift As. The object then becomes visible, especially at the edges, where the light is diffracted. As shown in Fig. 34, the phase shift in the zero order of diffraction is obtained by means of a “phase plate” placed at the back focus of the microscope objective. The phase plate is a thin glass plate, on which a ring

91

THE ABBE THEORY O F THE MICROSCOPE

CON DENSER

ANULAR DIAPHRAGM

OBJECTIVE

OBJECT

PHASE PLATE

IMAGE PLANE

FIG. 34. Optical arrangement in a phase-contrast microscope.

of some dielectric material has been evaporated to produce the phase shift. The illuminating system has a diaphragm with an annulus aperture and a condenser. This illuminating system is designed so that all the nondiffracted light passes through the ring on the phase plate. The diffracted light tends to pass out of the ring on the phase plate. Figure 35 shows the objected observed in a normal and in a phase-contrast microscope. 2.6.2. Spatial Filtering of Images

This is a technique based on the Abbe theory of the microscope. We have seen before that any irradiance distribution may be considered as the

(A)

( B)

FIG. 35. Pictures taken using a microscope. (a) Normal. (b) Phase contrast.

92


FRONT FOCAL PLANE OF FOURIER L E N S

BACK FOCAL PLANE OF FOURIER L E N S

FIG.36. Spatial filtering of images.

superposition of the light diffracted by the many spatial frequencies contained in the object. Alternatively, it can be said that each order of diffraction contributes to the formation of the image with a different spatial frequency. From the point of view of Fraunhofer diffraction, it is important to remember that the angular distribution of the light diffracted by an aperture is the two-dimensional Fourier transform of the amplitude distribution over the plane of the diffracting screen. Because of this property, if a convergent lens is illuminated with a collimated beam of light, the light distribution on the back focal plane is the two-dimensional Fourier transform of the amplitude distribution over the lens plane, or pupil function, except for a phase factor. The phase factor can be eliminated in order to obtain the exact Fourier transform if the diffracting object is not in contact with the lens, but in front of it on the front focal plane, as shown in Fig. 31. The spatial filtering of images is used to analyze the spatial frequency content of any image and even to remove undesired frequencies by means of appropriate stops at the back focal plane of the Fourier lens. This is usually done with the optical arrangement in Fig. 36. The object to be filtered is at the front focal plane of the lens and the two-dimensional Fourier transform is at the back focal plane, where the filtering stops are placed. Examples of spatial filtering are shown in Figs. 37 and 38. There have been a great number of applications for the use of this technique in the analysis of many graphic results. For example, in seismic studies where a Fourier transform of the picture is desired. It is often used to remove the noise from a picture if the spatial frequency of the noise is known.

2.7. Scattering Most of the light that our eyes receive comes not directly from the light source but through scattering. Looking at almost any object, what we see

SCATTERING

93

FIG.37. Spatial filtering of a two-dimensional grid.

is the light scattered from its surface. If we look at the blue sky or at clouds, we also see scattered light. These examples give us an idea of the great importance of scattering phenomena. Furthermore, the subject is very complicated, and many good books have been written covering it, for example, those by Van de Hulst” and Kerker.79 Scattering phenomena may be classified according to the size or shape of the scattering object or to the wavelength. In general, scattering is an anisotropic phenomenon, depending on the shape and orientation of the

94


FIG.38. Spatial filtering of a picture with TV scanning lines

object and on the orientation of the plane of polarization. The shapes of scattering objects which are most easily analyzed mathematically are spheres, ellipsoids, and cylinders. Miex" developed a very complete theory for scattering by spheres of arbitrary size, using electromagnetic theory. 2.7.1. Scattering by Large Objects

Depending on the size of the object, several models could be used to explain scattering. If the object is very small compared with the wavelength, the electric field of the wave forms electric dipoles with each particle. This theory will be considered in Section 2.7.2. If the size of the object is of the order of the wavelength or larger, the main mechanism producing scattering is diffraction. As shown in Fig. 24, if a scattering cloud is formed of many spherical particles of the same diameter, their size can be found by examination of the diffraction pattern. Using this idea, Young designed an instrument called eriometer to measure the size of blood corpuscles.

SCAnERING

95

Young's eriometer does not work for particles of varied sizes, because the Fraunhofer diffraction pattern is formed from the irradiance sum of all the individual patterns. Parrent and Thompson" and Silverman et aLX2 pointed out this problem and proposed to observe the pattern in the far field (s >> n 2 / A ) of the individual particles but close enough to be in the near field of the whole sample. Under those conditions, the diffraction patterns of each particle will be in different places instead of all being superimposed. It is interesting that the diffraction patterns are not Airy patterns, because they may be interpreted as Fresnel patterns, or alternatively, as Fraunhofer patterns with a coherent background. If the object is very large compared with the wavelength, the mechanisms producing scattering are diffraction on the edge of the object and refractions and reflections inside it. These particles may be water drops, such as in fog, rain, mist, and haze. The rainbow is one example of scattering by water drops. This phenomenon has been considered by poets as well as scientists. The scientific description is often assumed to be an exercise in geometrical optics, but it is much more complicated than that, as shown by Khare and Nussenzveigs3 and Van de H ~ l s t . ' ~ The best approximate theory to explain the rainbow was formulated by Airy. An exact but complicated theory is obtained applying the general Mie theory. The glory is another beautiful atmospheric phenomenon due to scattering by water drops, as described in the book by Mir~naert.'~ It is observed as a rainbow surrounding the sun or moon, covered by a thin veil of clouds. More often, persons standing on a high point, observe their shadows projected onto low clouds, showing a colored ring around each of their heads. The glory can be seen by each observer, with the ring only around his or her own head. 2.7.2. Scattering by Small Particles

When an electromagnetic wave passes through matter or a cloud of particles much smaller than the wavelength, its electric field acts on each molecule or particle, displacing the positive charges in the opposite direction to the negative charges. The resulting deformed molecule or particle may thus be considered as an electrical dipole, formed by two charges of opposite sign joined by an elastic force. An electrical dipole is the system formed by two charges to a -9 with masses M and m, respectively, and joined by an elastic force ky, where k is a constant, and y is the separation of each charge from the equilibrium position. We shall assume that M >> m.Thus, if the electric charges are free

96


of any friction or driving force, their movement equation would be m -d=2Y- k dt2

(2.75)

’

with the solution y

= A e-‘wo‘

(2.76)

which, substituted into Eq. (2.75), gives the characteristic vibration frequency w =27ruo=

&

(2.77)

where vo is called the resonance frequency. Let us assume that an electromagnetic wave with the electric field

acts as a driving force for the dipoles. With this force, the movement equation would become (2.79)

The choice of sign for the driving force is made by taking into account that the electron feels the force in the direction opposite to E, because of its negative charge -9. However, this equation is incomplete, because we know that any accelerated charge radiates, and therefore the dipole charges will emit an electromagnetic wave. Due to this radiation, the electrons experience some kind of reaction force that opposes their movement. This reaction force has been shown by Lorentz to be d3y F , = - . -2q2 3~x10’ dt3’

(2.80)

Thus, the complete movement equation is d3y m -d- 2Y - = - k y + - 2q2 --qEo dt2 3cx lo7 dt3

e-””.

(2.81)

The effect of the magnetic field of the driving wave is negligible compared with that of the electric field and thus it is ignored. By proposing again a solution of the type Y = A e-‘”‘

(2.82)

97

SCATTERING

and by substituting the value for the amplitude, A is found to be equal to

A=

-9Eo

m[wi-w2- iyw3/oi]'

(2.83)

where y is given by (2.84)

Therefore, we see that the dipole oscillates with a frequency equal to that of the driving wave. In order to study the amplitude and phase of the dipole oscillation, we shall consider three different cases. We assume that y w 3 / w i is very small compared with ( w 2 - w ' ) . (1) If w > w o , as in the case of free electrons or most materials at X-ray frequencies, A is positive. Thus, the dipole vibration is 180" out-of-phase with respect to that of the driving wave, as shown in Fig. 39. DRIVING ELECTRIC FIELD

W 80°).2y~'0 Also, the slope for the reflectance versus the incidence angle for the dielectric is very steep, thus making the angular acceptance very sensitive. A gold substrate with its lower reflectance can be used with a high-index film to produce a reasonable P or S extinguishing polarizer which has high extinction, smaller incidence angle, and better angular sensitivity. Another solution is to add some absorption to the film (this cannot be used in high-energy beams due to the absorption) which produces multiple incidence angles (75'-85') at which the P or S component is extinguished and the sensitivity to angular acceptance is d e ~ r e a s e d . There ~' is a trade-off of throughput for angular insensitivity and the unwanted component's intensity. These are simple, single-film, wavelength-dependent reflection polarizers which are useful when used with low-power densities. There also exist multilayer dielectric films on a transparent substrate which are used as reflection/transmission polarizers for higher-power densities.

3.5.2. Multilayer Films 3.5.2.1.Polarization Effects. Another way to make thin-film polarizers is to use periodic multilayer dielectric films of high and low in dice^.^*-^^

THIN FILMS

129

These are suitable for beams with high-energy densities. The time dependence of the wave equation, wt, is assumed fixed, and the equations are expressed as second-order differential equations in space for the magnetic and electric wave propagation. The multilayer’s effect on the incident beam’s magnetic and electric field is expressed as a 2 x 2 unimodular transformation matrix. The new beam produced from the first layer is applied to the second layer, producing a modified beam for the third layer, etc. The total thin-film stack can thus be expressed as a single unimodular transformation matrix. To see the effect of the dielectric multilayer on the S and P polarization, first consider the mth thickness z, at normal incidence. The matrix transformation of the electric and magnetic fields is

The diagonal and cross-diagonal terms are pure real cosine and pure imaginary sine terms with arguments which are the phase thickness,

A,=-.

2rn,z,

(3.28)

A0

To apply this at oblique incidence angles, the effective phase thickness and effective index (a, p m )are used to describe the uniform plane wave at an oblique incidence as a nonuniform plane wave at normal incidence, via 6, = A, cos 4,

and

where 4, is the refracted angle through the layer. The last two relations were introduced in Fresnel’s equations earlier and are substituted into normal incidence Fresnel equations to produce nonnormal incidence equations. Periodic multilayer dielectric thin films have a transformation matrix, also known as a Herpin matrix, which is equivalent to a two-film combination for a particular wavelength. A symmetric three-layer film is a special case which simplifies to an equivalent single-film matrix for all wavelengths. The single film is described by an equivalent index, N, and an equivalent phase thickness, r.

(3.29)

130

OPTICAL POLARIZATION

A symmetric three-film combination with an effective phase thickness (Y + /3 can be made by sandwiching one layer between the halves of the other layer. Three films with index layers n,,, n p , n, and phase thicknesses a / 2 , p / 2 , a / 2 result. Equating the three-film transformation matrix to the singlefilm equivalent matrix gives an equation for the equivalent phase thickness,

cosh

r = cos

(Y

cos /3

(3.30)

For values of lcosh rl> 1, the effective index N is complex in order to preserve the pure imaginary cross diagonal terms for the equivalent single film. This means that there is attenuation of the beam propagating through the medium (a stop band), while for lcosh I'l< 1, N is real and the beam propagates through the medium (pass band). The number of periodic layers increases the attenuation and improves performance. Since the effective index for the S and P components are functionally different in 4, an incidence angle and phase thickness can be found for a given wavelength which will give lcosh rl> 1 for the S component and lcosh rl< 1 for the P component, thus making an S-reflecting, P-transmitting polarizer. If it is possible, N is adjusted to the medium surrounding the film, or else an antireflection coating is needed. In practice, there is more to multilayer polarizer design, such as antireflection coatings to index match the multilayer to the air and/or glass interface, or to design the multilayer to take into account the electric field distribution through the stack to reduce laser damage.37-41 3.5.2.2. Linear Polarizers and Phase Retarders. Mahlein has published articles on quarter-wave multilayer dielectric stacks for laser mirror His graphs represent various thin-film combinations of n, versus np ranging from 1.3 to 3.0 with various incidence angles for P or S extinguishing reflection/transmission polarizers. He also shows possible designs for nonpolarizing beam splitters and polarizing interference filters. As for reflection quarter-wave retarders, Southwell has designed a dielectric multilayer on a silver substrate with a reflectance equal to 99'/0.~~His design starts with a quarter-wave-thick periodic multilayer stack, and the film thicknesses is adjusted according to a merit function which maximizes the desired characteristics-90" retardance *3' over a 5% range about the design wavelength and a reflectivity of 99.8%. These coatings can be used to reduce polarization effects in corner cube reflectors. 3.5.2.3. MacNeille Polarizing Prisms. Another application of thin films as a polarizer is in beam-splitting glass cubes first described by M a ~ N e i l l e . ~A' *glass ~ ~ cube is cut along a diagonal, and a multilayer stack is deposited such that the Brewster angle condition is met at all the diagonal interfaces, as drawn in Fig. 17. If the effective indices for the high- and

THIN FILMS

131

FIG. 17. A P-polarizing MacNeille prism using multiple reflections at Brewster's angle &.

low-index films are set equal, the parallel Fresnel-reflection coefficient is zero. This condition with the Brewster angle condition produces an equation in terms of the refractive indices and the incident angle in the prism. nG sin

cPG = n H sin 4H= nL sin 4L, (3.31)

Commercially-made, broad-spectral-band, thin-film-polarizing cubes (4000 A to 7000 A) are available with an extinction ratio of 100 to 1, and narrow-band cubes are available with an extinction ratio of 1000 to 1. Optically coupling the prism halves for high-power densities is difficult, and residual birefringence exists in the glass. Liquid prisms with immersed interference-coated substrates have been demonstrated as practical to overcome these problems and can have a wide spectral range from the UV through the ~ i s i b l e . ~ ' 3.5.2.4. Total Internal Reflection Devices. Total internal reflection phase-retarding devices, such as the Fresnel rhomb mentioned earlier, control the phase difference introduced on reflection to make quarter- and half-wave retarder^.^"'^ The phase difference between the P and S components, A, for most glasses in air does not occur at the maximum, A,,,, in the plot A versus the incidence angle (Fig. 6). Usually the larger incidence angle is used to make the rhomb, since its angular sensitivity is not as large as the smaller incidence angle. For the least angular sensitivity, Amax would equal the desired phase retardation. Increasing the refractive index increases A,,,, so a film with a smaller refractive index is deposited, and its thickness is adjusted to give A,, = 45". For a *3" incidence angle variation over the

132

O P T I C A L POLARIZATION

visible spectrum, the retardation variation is 0.4" for a coated rhomb and 5.0" for an uncoated version. Imperfections in total internal reflection devices also are due to strain birefringence which is inversely proportional to wavelength, and for applications in the ultraviolet it becomes critical, especially if the retarder is to be achromatic. Also, it is reported that polishing the faces that are not used reduces strain 10-15 mm inside the glass by a factor of two over that for a fine-ground face. Attention should be given to mounting the retarder, and silicone rubber sealant as an adhesive does not introduce any measurable strain. Another error source is the refractive index of the glass and thin film. For the glass, the surface is slightly different in phase retardation, about 0.5" more than the theoretical values, because surface shaping and polishing changes the refractive index. Also, generally for most materials, the refractive index increases with a decrease in wavelength, except for polyvinyl alcohol. Retardation error sources can be corrected by adjusting the reflected beam's phase difference experimentally with film thickness. The film can increase achromatism, because the increase in refractive index with decrease in wavelength can be balanced by the thin film's increased phase thickness due to p. King has made Fresnel rhombs with either 144A magnesium fluoride films on each reflecting surface or a 268A film on one surface leaving the other uncoated, which produce a retardation of 90" f 0.2" from 3440 A to 6680 A compared to 90"*2.5" for an uncoated rhomb. In Fig. 18, various other retardation devices are shown. They have about the same chromaticity but various other advantages and disadvantages with regard to beam displacement, physical size, and angular acceptance. A Fresnel rhomb displaces the beam; and, if not optimized by thin films, it has a 9" variation in retardation with a *7" incidence angle. Achromatic device one, ADl, is self-compensating in incidence angle, because the first two reflections on the minus side of the optimum angle will have two reflections on the positive side, therefore almost canceling the off -angle effects. It is a very long prism, 175 mm long for a 10-mm beam, and it would be hard to meet theoretical predictions due to strain birefringence. It has a 0.7" retardance variation for a *7" incidence-angle variation. The Mooney rhomb is made of two dense flint 60" prisms and is self-compensating in incidence angle with a performance similar to AD1. Also, its physical size is small. Achromatic device two, AD2, is physically small and does not deviate or displace the beam, but its incidence angle sensitivity is terrible with a 13.2" retardation variation for a *3" incidence-angle variation.'' Half-wave or full-wave rhombs have also been designed to deviate the light beam 90" without changing the polarization. A simple design is the dove prism.

WAVE PROPAGATION I N A N I S O T R O P I C M E D I A

133

FIG. 18. Total internal reflection devices. (a) Fresnel rhomb, ( b ) Mooney rhomb, (c) achromatic device 1 ( A D l ) , and (d) achromatic device 2 (AD2).

3.6. Wave Propagation in Anisotropic Media 3.6.1. Birefringence in Crystals 3.6.1.1. Dielectric Tensor and Energy Distribution. Another way to separate orthogonal polarizations is by dielectrics with anisotropic optical constants either naturally occurring in crystals or induced by an electromagnetic field or stress and compression. This produces different velocities for orthogonal polarizations depending on the travel direction in the crystal.” The medium can be described in terms of its dielectric constants as a nine-element tensor.

(3.32)

where D is now not necessarily parallel to E as for an isotropic media, and the angle difference is between the wave front’s normal, s, and Poynting’s energy propagation vector, S. This means that there is a wave-front phase velocity and an energy ray velocity as shown in Fig. 19. The phase velocity is a projection of the ray velocity onto the wave front’s normal. Maxwell’s equations and the material equations with the dielectric tensor give an

134


equation relating the phase velocity to the wave normal direction and the material constants. Considering the functional form of the energy density, the dielectric tensor is seen to be symmetric, and at most six independent values need to be determined. A coordinate system, termed the principal axes, can be fixed to the tensor’s diagonal. In this coordinate system, the material equations simplify to the diagonal terms. Eyy+Ey,

E,,+Ex,

EZZ+EZ,

i = x, y , x.

Assume the media is homogeneous, nonconductive, nonmagnetic, and electrically anisotropic. A plane wave, Eq. (3.4) simplifies the derivatives in Maxwell’s equations.

V x H =j w D = -jks x H V x E = -jpwH

= -jks x E

(3.33)

:);(

= phase

velocity = V,

(kY2

=velocity in i direction = (3.34)

where s is the unit vector for propagation direction and k defines equal phase planes A apart. D and H are perpendicular to s, but E is not. E is perpendicular to the Poynting energy flow vector, S.

WAVE PROPAGATION I N ANISOTROPIC M E D I A

135

S

E FIG. 19. The orientation of the electromagnetic vectors H, D, and E, the wave normal s, and the energy propagation vector S in an anisotropic medium.

The Ei can be eliminated, leaving an equation of the material constants, the propagation direction, and the phase velocity.

(3.35) The result is known as Fresnel’s equation of wave normals. V ; has two values and implies with Eq. (3.34) that the medium has two linearly polarized beams with different phase velocities. It can also be shown that the two electric displacement vectors are perpendicular. 3.6.1.2. Crystal Types by Optical Properties. The dielectric tensor is a second-order symmetric tensor, and therefore it needs six parameters to describe it. These parameters can be interpreted as the principal axes’ lengths and angular orientations with respect to the crystallographic axes. The number of parameters to describe the tensor is dependent upon their optical anisotropies which are subdivided according to crystallographic structure.” (1) Optically isotropic crystals are called cubic crystals. Three crystallographically, mutually orthogonal axes can be found, and the diagonal terms of the dielectric tensor are equal. Only one parameter, the refractive index, is needed to describe the dielectric tensor. (2) Uniaxial crystals have a plane with equal optic constants and a direction normal to the plane with a different optic constant called the optic axis. Two parameters, the lengths of the two principal axes, are needed to describe the dielectric tensor. A set of crystallographically equivalent directions are in the plane about an axis of symmetry aligned with the optic axis. Crystals can have three, four, or six crystallographically equivalent directions in a plane or three-, four-, or sixfold symmetry about the optic axis. Trigonal, tetragonal, and hexagonal crystals are in this group. Calcite is an example. It is a rhombohedra1 crystal, and the optic axis is a threefold

136

0 PTlC AL POLAR1 ZATIO N

symmetry axis directed into the rhomb’s corner. A surface cut perpendicular to the optic axis on the corner accurately determines the crystal’s principal axes.’ Unpolarized light incident on a uniaxial crystal produces two linearly polarized beams determined by the optic constants for the plane and the optic axis. For all field orientations with respect to the optic axis, an electric vector parallel to the plane exists and behaves as for an isotropic medium. This is termed the ordinary beam, 0, and is described by a spherical velocity surface. The wave perpendicular to the 0 beam depends on the incident beam’s orientation and varies from the optic constants for the plane to the optic constants perpendicular to the plane. This is the extraordinary beam, E, and its phase velocity as a function of propagation direction is a spheroidal surface. The phase velocity surfaces for a uniaxial crystal in terms of the principal axes are found by applying Fresnel’s equation of wave normals. The z axis is made parallel to the optic axis, and the angle from the wave normal to z is 9.

v, = v, = v,, s: + s: = sin2 9, ( V ; - vf)[(V ; - ~ f sin’ ) 9

v, = v,, s t = cos2 8,

+ ( V : - v;)cos2 e] = 0,

(3.36)

v;,= vf, v;*= vf sin2 e + v’,cos2 e. Figure 20 is a plot of the two phase velocity surfaces and an arbitrary wave normal, s, with its two phase velocities. When s is in the optic axis direction, the phase velocities are equal, and when s is perpendicular, the difference between phase velocities is maximum.

z O.A.

z O.A.

FIG. 20. Cross sections of phase-velocity surfaces for a negative and a positive uniaxial crystal predicted by the Fresnel equation for wave normals. The phase velocities are determined by the wave normal’s orientation.

WAVE PROPAGATION IN ANISOTROPIC MEDIA

137

Birefringence is the difference between the 0 and E refractive indexes. birefringence = nE - no

(3.37)

The sign means that the ordinary velocity surface is ahead of or behind the extraordinary surface. (3) Crystals with principal axes as a function of three dielectric constants and angle orientation with the crystal axes are called biaxial crystals. TE waves are refracted at an angle depending on orientation with respect to the crystal's axes and wavelength. Both waves perpendicular to s act as extraordinary waves, and the velocity surfaces are general ellipsoids. Orthorhombic crystals have their principal axes aligned with the crystallographic axes, and so only three parameters, namely, the lengths of the three principal axes, are needed. Monoclinic crystals have a fixed principal axis parallel to a crystal axis and need four parameters for the dielectric tensor. Triclinic crystals need all six parameters to describe the dielectric tensor. Applying the Fresnel equation of wave normals to biaxial crystals has three dimensions that can vary. Three cross-sections, each about a different principal axis, simplify the problem. Figure 21 shows three cross sections for a biaxial crystal. One cross section has the curves touching at four points, and the two lines connecting the origin with the points are the directions of the optic axes. With the wave normal in these directions, the two phase velocities 'are equal. Quartz and calcite are the most commonly used crystals for crystalretarding plates and prism polarizers and beamsplitters. Quartz is a biaxial crystal. A beam propagating along the optic axis separates into L and R circular polarizations due to a weak circular birefringence. for

nL- n R = *0.000071

A = 5892 A.

Separating L and R polarization in phase is equivalent to rotating the PoincarC sphere about the L and R poles, and the resulting elliptical 2

Y

Z

vx

>

vy

>

vz

FIG. 21. Three cross sections of phase-velocity surfaces described by the Fresnel wave normal equation for a biaxial crystal.

138


polarization azimuth is rotated according to the phase shift. This phenomenon is termed optical activity. The beam’s ellipticity changes quickly to linear polarization for angles not far from the optic axis, and quartz behaves almost as a uniaxial crystal. Quartz’s weak linear birefringence makes it useful for retardation plates. nE- no = 0.0091

for

A = 5892 A.

Calcite is a negative uniaxial crystal. Prism polarizers and beamsplitters are usually made from calcite, because the transmission and birefringence are large, from 2000 to 12,000 A. nE = 1.486, no

= 1.658

for

A = 5892 A.

The literature has references with material constants given for the more commonly used crystals and wavelength ranges.

3.6.2.Crystal Polarizers 3.6.2.1. Prism Beamsplitters. Beam-splitting polarizers, such as the Rochon, SCnarmont, and Wollaston are biprisms which separate the 0 and E beams by introducing a discontinuity in the refractive index for either the E beam (Rochon and Sknarmont) or for both the E and 0 beams (Wollaston). Figure 22 shows the E and 0 rays for a Rochon and a Wollaston. The Rochon and SCnarmont are basically the same with their first prisms’ optic axis perpendicular to the entrance face. The Rochon gives a slightly larger E-beam deviation than the SCnarmont, because the beam sees the maximum birefringence. These prisms can be used backwards, but the E-beam deviation will be slightly less. Snell’s law at the prism discontinuity, now from nE to no, reverses the incident beam’s refracted angle from greater to smaller deviation. The beam deviation and

FIG.22. ( a ) Rochon and ( b ) Wollaston polarization beam splitters. The optic axis direction is oriented parallel I or perpendicular 0 to the page.

WAVE PROPAGATION IN ANISOTROPIC MEDIA

139

accepted field angle are a p p r ~ x i m a t e l y ~ ~

(3.38)

A Wollaston polarizing beamsplitter has the optic axis of its first prism parallel to the face to produce an E and an 0 beam which sees the second prism's optic axis rotated 90" about the beam axis to produce a maximum birefringence and deviation for both beams. Wollaston prisms can produce angular separations up to 20" with a slight asymmetry, and triple-element Wollastons produce larger angular separations. At large angles through crystal polarizers, image distortion occurs due to nE varying in angle with respect to the optic axis. Since the birefringence of Rochon's first prism is not used, it can be substituted with strain-free glass of either index, nE or no. Glass eliminates image distortion due to the effects of the birefringent crystal and is easy to manufacture cheaply with a good tolerance. The residual birefringence in glass needs to be c o n ~ i d e r e d . ~ ~ 3.6.2.2.Prism Polarizers. Polarizing prisms are birefringent crystals cut into two pieces with an interface of an index which totally internally reflects the higher-index beam, for calcite the ordinary beam.' A Glan-Thompson prism is illustrated in Fig. 23. A Nicol prism was the first polarizing prism design, and it is essentially a calcite rhomb cut along a diagonal and cemented together. Glan-type prism polarizers arrange the optic axis to give

i"

i i i I

i

i

,-I

_._.-.- 0 - - - - - - -. E

FIG. 23. Clan-Thompson prism drawn with normal and extreme rays. The optic axis is perpendicular to the page, 0.

140


maximum use of birefringence, thus increasing acceptance angle. Beam distortion and polarization uniformity are better controlled for Glan prisms. The prism’s angular acceptance is limited by either the critical angle of the larger-index O-rays, or by the lower-index transmitted E-ray being refracted along the prism’s cut angle. The refractive index discontinuity, n,, at an angle S in Fig. 23 has its normal rotated ad angle ( 9 0 ” - S ) with respect to the entrance normal. Applying Snell’s law to the 0 beam at surface 1-2 gives an equation relating the maximum incidence angle for total internal reflection to the angle S and the cement’s refractive index, n,, n, sin di= ( n i - n:)”, cos s - n, sin S.

(3.39)

For the E beam’s limit, the total internal reflection case n E > n, can be described with the 0 beam’s equation and the appropriate substitutions, n , sin

di= n2 sin s - ( n i- n:)”, cos S.

( 3.40)

To make the maximum usable acceptance angle for both 0 and E beams, the two incidence angle equations are made equal in magnitude by adjusting the cut angle and cement. This gives, tan S =

( n h- n:)”,

(3.41)

2n2

The acceptance angle reaches a maximum for n2 = n,. For n E < n,, the maximum accepted incident angle for the transmitted beam is refracted along the cut. n, sin d ,,

= n2 sin

S.

By equating the maximum incident angle for the transmitted beam to the totally reflected beam gives an equation relating the cut angle and the (3.42)

Considering the totally reflected beam, the maximum incident angle to be reflected increases with the smallest cut angle which is achieved with n, = n E . The tan S equations reduce to the same form for n2 = nE. Glan-type prisms are sold with the length-to-aperture ratio assuming a square aperture, and this gives an idea of its acceptance angle. L A

1 tans’

(3.43)

Since optical cements have absorption, air-spaced calcite prism polarizers are made for use up to calcite’s ultraviolet limit and for high-energy densities.

WAVE PROPAGATION I N ANISOTROPIC MED IA

141

The usable maximum (symmetric about the prism's normal) incident angle or semifield angle is about 8" in the ultraviolet, decreasing for longer wavelengths, and is limited by the transmitted E beam being totally internally reflected at the cut. Large prism apertures can be made, since most of the crystal is used. An air-spaced Glan-Thompson is termed a Clan- Foucault The optic axis is parallel to the cut surface, so that the transmitted beam has its electromagnetic vibration oriented parallel to the cut surface, which has the higher Fresnel reflection coefficient, r,. If the optic axis is rotated 90" about the beam axis, the reflection coefficient is now rp,which is significantly reduced. The transmission is increased in the ultraviolet by 10% for this design, which is called a Clan- Taylor. The optic axis orientation decreases the useful birefringence which reduces the semifield angle. The cement n, is unity, and the angular acceptance primarily depends on the cut angle and the crystal's wavelength-dependent birefringence. The L / A ratio is about equal to one. To increase the semifield angle or to decrease the L / A ratio for a given maximum incident angle, double prisms are constructed.' Examples are the Ahrens prism (two Glan-Thompson prisms next to each other), the Grosse prism (an air-spaced Ahrens), and the Marple-Hess prism (a double GlanTaylor back to back). Since the number of components increases in double prisms, the manufacturing difficulty is increased, and extinction performance is about an order of magnitude worse than for single polarizing prisms. The Ahrens and Grosse prisms have half the L / A ratio of their Glan single-prism counterparts. For example, an Ahrens prism with an L / A of 1.8 has been made with a 26" semifield angle. The Grosse prism has an L / A about equal to one-half and is useful for tight physical limitations. MarpleHess prisms have definite advantages due to the cut angles S and -S. 0-rays below the axis will be totally reflected by the first surface, and 0-rays above the axis will be totally reflected by the second surface. The axial 0-ray and its total internal reflection due to n , ( A ) decreasing in A is the design parameter for the ordinary beam. The prism's limiting field angle is determined by the E-ray's total reflection, and the limiting wavelength depends on the axial 0-ray's transmission. These prisms have a lower transmission in the ultraviolet because of their increased L / A ratio. Glass-calcite Glan-type prism polarizers are possible, because the second prism is used to transmit the E-ray, and its birefringence is not used. The second crystal prism can be replaced with an index n, glass prism. Or, the first prism can be replaced with an no index glass, and an n, cement used to assemble the prism. The E-ray is totally reflected in this case. These prisms are usually not as good as all-crystal polarizers, because the greater L/ A ratio and optical cement enhance the resulting residual strain.s3

142


As with prism beamsplitters, image distortion in prism polarizers occurs for off-axis beams. Other contributions to degradation in extinction ratio is scattered light and the matter of eliminating the unwanted ray. One solution is to blacken the crystal’s sides. Another method is to cut and polish the prism side to make the unwanted beam exit normal to the exit face. The unwanted beam is saved. Prism performance studies show an order of magnitude increase in extinction ratio when the detector is moved farther from the prism, and entrance and exit apertures are used.s4 The crystal grades, optical finish, strain birefringence in contacting the two prisms, and optical alignment affect the prism’s p e r f ~ r m a n c e . ~ ~

3.6.3. Retarding Waveplates 3.6.3.1. Crystal Retardation Plates. A quartz or calcite plate with its optic axis parallel to the entrance surface introduces a phase shift between linear orthogonal polarizations, d 6 = 274 n E - n,) -. A

(3.44)

For n E > no, linearly polarized light travels faster parallel to the optic axis, and for nE < no, the fast direction is perpendicular to the optic axis. A fast (F) and a slow (S) coordinate system oriented with the optic axis describes the retarder’s orientation. The PoincarC sphere is useful to describe retardation effects. Incident and resultant elliptical polarizations are points on the sphere which are transformed due to phase shifts. A retarder’s phase shift between F and S rotates the incident point about the point marking F’s orientation. A positive phase shift is a clockwise rotation about F equal to the phase shift. For example, a quarter-wave retarder with its fast axis at +45” will pass H as R and V as L polarizations. The spectrum viewed with a retarding plate between crossed polarizers is fringed with the polarizations’ cyclical wavelength retardation. The wavelength interval between equal polarization states decreases for the thicker plates and can be used to filter or select a wavelength by selecting the corresponding polarization ellipticity. Lyot birefringent filters use this method to make typically a 1/4-A pass band isolating the fringe from its neighbors by cascading plates with double the fringe spacing in the spectrum or half the previous element’s thickness until the final element’s fringe spacing is 128 or 256 A. Retardation plates are usually made of quartz, mica, or plastic for the near ultraviolet to the near The spectral range is expanded by using other birefringent crystals. Calcite’s birefringence is very high, thus


143

enhancing its dependence on thickness and temperature. A quarter-wave plate for a particular wavelength would be more difficult to obtain and maintain than for quartz. The thickness of a retardation plate can be a single-order retardation period (which can be achieved with split mica or quartz in the infrared) or multiple orders thick. d

= mh.

(3.45)

Thick multiple-order plates are sensitive to changes in the retardation equation. Wavelength, birefringence, refracted angle, and temperature variations are factors in determining the resulting retardation. The sensitivity can be decreased by using thick plates by orienting one plate’s optic axis perpendicular to the other plate and by adjusting their thickness difference to a single-order thick. This combination acts as a single-order plate and is called a jirst-order plate. 3.6.3.2.Compensating Retardation Plates. All simple retardation plates still depend on wavelength. The retardation’s dependence on wavelength can be corrected for either with variable thickness plates called compensators, or with composite plates to produce the desired retardation characteristics. Compensators change the net thickness experiencing a birefringence which directly changes the phase difference between the two orthogonal polarizations, as demonstrated by Poincare’s sphere. The Babinet compensator is a fixed and a movable quartz wedge with the optic axis parallel to the entrance and exit surfaces, but crossed to each other. The field’s center is marked with a reference line on the fixed wedge, and the relation between the movable-wedge displacement and retardation can be empirically determined by using crossed polarizers as a polarizer and an analyzer. White light viewed with a Babinet between crossed polarizers produces colored bands and extinction indicating changing retardation orders. Zero phase difference at the center will produce a dark line with the reference line. Introducing a retarding plate will shift the fringes, and a correction opposite the retarding plate’s phase shift is needed to extinguish the center again. The distance moved by the wedge determines the introduced phase shift. To increase the useful field, a Soleil compensator is used. It is a fixed retarding plate and a variable-thickness plate with their optic axis crossed. The variable thickness plate introduces a uniform variable phase shift across the viewing aperture. If white light is viewed with a Soleil between crossed polarizers, the field is tinted with phase-shifted wavelengths. Zero net thickness between the plates results in extinction. A retarding plate introduced between the polarizers will transmit a particular wavelength, and a

144


phase correction made by the variable-thickness plate will again extinguish that wavelength. The Senarmont compensator is an elliptical polarization analyzer consisting of a quarter-wave plate and a polarizer. Elliptical polarization is incident with its major axis aligned with the wave plate's fast and slow axes. The wave plate will retard either the major or minor axis by a quarter wave, resulting in a linear polarization with an azimuth determined by the ellipticity. An elliptical polarization is selected with the azimuth of a following linear polarization analyzer with respect to the wave plate's axes. If the elliptical polarization analyzer is turned around with light now incident on the linear polarizer, then a known elliptical polarization with its axes aligned with the wave plate is produced. For example, consider how circular polarization is produced. The linear polarizer oriented parallel to the wave plate's axes produces linear polarization. The linear polarizer oriented 45" to the wave plate will produce either L or R circular polarization. 3.6.3.3.Achromatic Wave Plates. Composite achromatic wave plates can be divided into composites of the same or into composites of positive and negative birefringent crystals similar to focus correction for lenses.60 Possible positive and negative birefringent wave-plate composites for visible applications are magnesium fluoride and ammonium deuterium phosphate or magnesium fluoride and potassium deuterium phosphate designs with a 0.5% deviation; a calcite, quartz, and magnesium fluoride triplet with a 0.5% deviation; a magnesium fluoride and quartz doublet with a 5% deviation over the spectral region. Composite wave plates with the same material can be made with zero first-order variations in retardation and wave plate axis. This is demonstrated using Jones' matrix techniques. Three- and nine-plate combinations act as pure wave plates, and four- and ten-plate combinations behave as a wave plate plus a rotator. A three-element achromatic wave plate made of the same material with the first and third elements identical leads to the consequence that they are parallel. Various achromatic combinations of three-element wave plates for a desired retardation can be made using these assumptions, but the retardation axis may vary to first order in wavelength. Constraining the retardation axis leads to the middle plate being a half-wave plate, which is rotated by an angle 0 with respect to the outer elements. The total retardation is 26. COS~O=-T~~

3

cos 6 +-

(

= dlsin(26,),

where *A6 is the tolerance as a function of wavelength.

(3.46)


145

For three-plate combinations, some chromaticity is left, due to a quadratic term. A central-design wavelength needs to be chosen, and the allowable deviation from the desired retardation gives the useful spectral range. The correct retardation occurs at a wavelength above and below the central wavelength for all but the half-wave combination which varies cubicly. Combinations for the nine-element retarder (a threefold three-element retarder) are possible to make a retarder's retardation vary about 1% from 3514 to 10,000 A. For example, the three-element quarter-wave plate consists of two outer plates with a 115.5" retardation and a half-wave plate. A half-wave plate is made of three half-wave plates. A small additional rotation of the central half-wave plate introduces a first-order term which cancels the third-order term, improving the retarder's performance. If a three-element half-wave plate is split in the middle and the halves are then rotated, retardation is equal to twice the complement of the angle rotated. Also, the polarization is rotated by the complement of the separation angle. This combination of quarter-wave and half-wave plates is a fourelement variable-retardation plate which rotates the polarization axis. Nineelement half-wave combinations split in the middle give a ten-element variable wave plate with a performance similar to the nine-element one and in addition, it rotates the polarization as the four-element wave plate does. Two-element composite retarding plates have been mentioned in the literature, but upon considering the Jones' matrix calculation for two wave plates, no degree of achromaticity can be obtained, and the plates need to be tuned for the wavelength. Wave plates are usually made of quartz, mica, or stretched polyvinyl. alcohol. Since the birefringence of mica and polyvinyl alcohol is variable, wave plates need to be selected for the application. Some of the plastic laminates have a curl to them, and it is best to leave the curl alone. Mica sheets, usually muscovite, can be split into large apertures to the desired retardation by trial and error to account for the birefringence differences for different crystal samples.6' This can be accomplished with adhesive tape, which can remove layers that are 5-micron-thick or less and can remove high spots in large apertures. Since the wave plate is thin, little beam deviation and wave-front distortion occurs, but multiple coherent reflections can occur which can be eliminated by cementing thin films into an optically thick medium. Ultraviolet light curing cement is suitable for this purpose, taking care that the illumination is uniform across the aperture to minimize stress. 3.6.3.4. Measuring Wave Plates. To determine the optic axis for a wave plate, it is placed between crossed polarizers and rotated for maximum intensity. The entrance polarizer and wave plate act as an elliptical polarizer

146


with the resultant polarization ellipse axes aligned 45" to the wave plate's extraordinary and ordinary axes. The analyzer is then oriented 45" to the ellipse axes. Rotating the wave plate about the optic axis direction has no effect on the birefringence. The light-path length in the crystal and retardation will change, but the analyzer will transmit. Rotating the wave-plate surface perpendicular to the optic axis changes the birefringence from a maximum to a zero value for a 90" rotation, and so extinction occurs.62 The wave plate's retardation is related to the intensity of the elliptical polarization's major and minor axes, as seen earlier. It is a matter of measuring the maximum intensity of the above crossed polarizers and wave-plate arrangement and rotating the analyzer 90" to measure the minimum or minor axis intensity, (3.47)

The error in measuring the wave plate's retardation is determined by the intensity's error measurement and the crossed polarizer's extinction ratio. A 0.001YO intensity uncertainty produces less than 0.1" retardation measurement error.63 3.6.4. Induced Birefringence and Phase Modulation

Anisotropies can also be induced in a material's dielectric tensor by elastic stresses, electric fields, and magnetic fields, divided into elasto-optic, electrooptic, and magneto-optic effect^.^^'^^ Linear electro-optic and elasto-optic effects are due to nonlinearities between the material polarization vector P and the electric field. The material effects are described in terms of the impermeability tensor, B = E - ' . Small changes in the tensor due to linear changes in the electric field ( E ) and strain (S) is A B,

= rzkEk

+ pfklsk[= A B z -k A B: .

(3.48)

where rllk is the electro-optic coefficient, and P Y k l is the elasto-optic coefficient. The superscript implies holding that variable constant while determining the other variable. Summation is implied for indices which appear in both terms of a product of matrix elements. For example, the term r,,kEk means to sum the elements varying k to account for all k contributions. Also, symmetric tensors can be expressed in a compressed matrix notation to simplify the matrix element numbering from row and column to an ordering as follows:

r I 3= r3, = r s ,

r,* = rz, = r,.

WAVE PROPAGATION IN ANISOTROPIC M E D I A

147

Crystals with a linear electro-optic effect are also piezoelectric. Strained piezoelectric crystals produce an electric displacement related to the strain, so that the electro-optic and elasto-optic coefficients are defined while holding the other effect constant. The variable that is held constant, either E or S, is denoted by a superscript. The effect on a strained piezoelectric crystal is,

(3.50) where C:k/ is the elastic tensor element, ekliis the piezoelectric tensor element, and Tj is the strain tensor element. Conversely, an applied electric field in a properly cut piezoelectric crystal can generate shear or longitudinal waves which are used for transducers. The change in refractive index can be found by considering how the impermeability matrix changes for either effect. Under the assumption that the index change is small to the material’s initial index, the index change is

nips Ans = -2 ’ (3.51)

where p and r are chosen for a particular crystal orientation or application, and p S and rE are typically less than lop4. 3.6.4.1. Electro-optic Effect. Electro-optic modulators use either the linear Pockels electro-optic effect or the quadratic Kerr electro-optic effect, and the light beam travels either parallel to (longitudinal mode) or perpendicular to (transverse mode) the applied electric field. The retardation across an electro-optic modulator is limited by the beam’s maximum incident angle, electric field uniformity, scattered light, and refractive index variation. These modulators work best in collimated light, and less than 5% retardation variation across the aperture can be attained. The aperture should be checked between crossed polarizers in the on and off state. Extinction ratios between the on and off state are determined by the crossed polarizers extinction and the modulator’s retardation uniformity; extinction values range from 100 : 1 to better than 1000: 1. Some useful materials for Pockels cells are ammonium dihydrogen phosphate (ADP), potassium dihydrogen phosphate (KDP), potassium dideuterium phosphate (KD*P), lithium tantalate, lithium niobate, hexamine, and lead lanthanum zirconate titanate (PLZT) ceramic.66-68Spectral transmission, chemical stability, optical constants, and modulation frequency determine the material’s application. The natural birefringence

148


can be either offset with a bias voltage or can be eliminated by using crossed crystal pairs to cancel each other’s birefringence (this also reduces temperature-birefringence effects). The voltage applied to produce half-wave retardation is in the kilovolts range. The drive voltages for modulators depend also on the path length through the crystal and on electrode configuration. With many passes through a crystal or with cascaded crystal pairs, the drive voltage is reduced. High-frequency modulation is limited by capacitance, mechanical stress, and heating; gigahertz frequencies can be achieved. Low-frequency modulation on the order of a second or longer has the problem of current leaking across the crystal. The charge needs to be replaced. This is accomplished with a slightly conducting grease for contacting the electrodes to the crystal.69 Longitudinal Pockels cells have either transparent electrodes or ring electrodes on the entrance and exit surfaces. The phase shift is

(3.52) where I is the distance through the crystal, and E is the electric field. A specification for longitudinal . modulators is the voltage for half-wave retardation, VAI2, which is needed to produce maximum transmission between crossed polarizers. Large apertures can be made. Transverse Pockels cells have electrodes on the crystal’s sides, and the light beam passes perpendicular to the applied field. Retardation increases with an increased path length through the crystal and with a decreased entrance aperture or electrode gap. Capacitance increases with increased crystal length so that the frequency bandwidth decreases. The retardation is

(3.53)

where I is the crystal length, a is the electrode gap, and is the half-wave field distance product defined for a particular crystal, its orientation, wavelength, and temperature. The quadratic Kerr effect aligns the material’s molecular polarization vectors by an applied electric field. The time to orient and induce birefringence with a high voltage and to disorient it again determines the operating


149

frequency for an electro-optic Kerr cell. The retardation is S = 2.rrKE21,

K = Kerr constant

(3.54)

I = length Electro-optic Kerr cells use a variety of gases, or liquids, or PLZT. For liquids and gases, the drive voltages are typically tens to hundreds of kilovolts, and the switching times are nanoseconds. For PLZT, the drive voltages are hundreds of volts and the switching speeds are microseconds. Also, liquid and gas Kerr cells do not have any residual natural birefringence. Subpicosecond laser pulses and high-power densities can be used to induce birefringence in Kerr materials, This is the optical Kerr efect. The optical Kerr cell can be either transverse or longitudinal; the type determines the modulated beam’s direction with respect to the gating laser pulse. The transverse cell has a long interaction path with the modulated laser beam, and self-focusing problems can be induced for crystal lengths greater than one or two centimeters thick. Usually, the optical Kerr cell is a longitudinal cell with the gating laser pulse colinear with the light beam to be modulated. The limiting switching speed is the time to randomize the molecular polarization after a gating pulse has passed. For nitrobenzene, the relaxation time is 32 ps and for carbon disulfide about 2 ps. The induced birefringence due to the electric field E ( t ) from the laser is 6nll- Sn, = n28

I:,

E 2 ( t ) exp[-(t - r’)]

dt’

-, T

(3.55) where n is the refractive index, n28 is the ac Kerr coefficient, Sn, and Snll are the refractive index changes perpendicular and parallel to the E field, T is the relaxation time, t is the gating pulse length, I?( t ) is the time average over one period, and P ( t ) is the power density. For t > T, the birefringence ~ n ,-,~ n = , n,,E2( t ) .

(3.56)

The sample beam’s retardation is 6 = 27rn,,P( t )

I ~

(Acne,) ‘

(3.57)

The on/off extinction ratio is typically about 1000: 1, determined mostly by the extinction ratio of the entrance and exit polarizer, and the aperture is the gating beam diameter.

150


3.6.4.2. Acousto-optic Effect. There are several ways to use the elastooptic effect depending on incident beam diameter and frequency modulat i ~ nA. uniaxial ~ ~ static strain deforms the isotropic media’s spherical phase velocity surface into a spheroidal phase velocity surface with an optic axis oriented along the strain direction with a birefringence,

Anli= - n o3- PSSC. 2 C = stress-optic constant

(3.58)

A useful wave plate can be made with a stressed plate and light traveling perpendicular to the stress. The equation for phase induced by stress is,

2 IT A nl

6=---

A

-

2 TSCl A

’

(3.59)

where 1 is the distance through the crystal. For quartz, the stress needed to produce quarter-wave retardation at 5500 A is an order of magnitude less than needed to destroy the plate. If the stress is a sound wave with a wavelength, A, much larger than the beam diameter, the phase modulation varies sinusoidally with the stress.

s = s,,, for

e-i(kr-RO

-

s,,,

COS(Rf)

(3.60)

A > than the beam diameter

6 = cos(Rr),

where R is the angular frequency, and k is the wave number for sound. If the sound wavelength is comparable to the beam diameter, the phase modulation or birefringence will vary across the aperture, and the intensity modulation contrast between crossed polarizers will not be as large. This limits the maximum frequency for resonating elasto-optic modulators to less than 1 MHz. Standing-wave photo-elastic modulators can drive in a self-resonating mode and can be made with zinc selenide, calcium flouride, KRS-5 infrared glass, germanium, and fused quartz. There are two types of construction, either the bar type73or an isometric-contour element.74 The bar type is the first design which is made of either one or two transducers connected to either side of the optical element to be modulated. The transducers and the optical element all resonate in a standing-wave mode with nodes at the joints. The elements can be cemented together with silicone cement to reduce strain birefringence. Bar-type modulators have two error sources which are eliminated by the isometric-contour-type modulator. The errors are, ( 1 ) a


151

nonlinear, anharmonic dc birefringence proportional to the driving amplitude squared (an error of about l % ) , and (2) the reflectance at the modulator’s surfaces is modulated by the modulated refractive index adding a small variation on the order of 0.001%. The isometric-contour-type modulator is a rectangular optical element supported at its corners along the length and modulated across the width. The light beam travels the long direction through the bar’s center. The usable entrance aperture is a little larger for these types. For photo-elastic modulators in general, their entrance apertures can be large, and the incidence angle can be as large as 50”. Modulators can be driven by transducers at sound frequencies with many sound wavelengths across the beam diameter.72s75 The compressions moving at sound’s velocity will appear to the light beam as an almost stationary phase grating with a grating spacing equal to the sound’s wavelength. If light’s interaction with the sound waves is small, it is diffracted into orders, N, with angle, + N , according to the thin phase grating equation, sin+N-sin+,,=-,

NA A

N=0,*1,*2

,....

(3.61)

The amplitude of the various orders is given by Bessel functions and the maximum phase shifts, 6, in the medium due to the sound wave. (3.62) where I,, is the incident intensity. Changing the sound frequency changes the orders’ angles, and changing the sound amplitude modulates the intensity in other orders. These effects can be used in a feedback loop as an active element for beam steering and modulating. Since the phase grating is traveling, the diffracted light in the N t h order will be frequency-shifted by an amount, (3.63) where V, is the material’s sound velocity, and & is the light frequency. For high sound frequencies or for a long interaction path with light, the behavior changes from a thin phase grating operating in the Raman- Nath regime to the thick phase grating behavior similar to X-ray diffraction called the Bragg regime. The light is diffracted into two orders, because the sound wave is a sinusoidally varying density function needing one frequency term to describe it, while for X-ray diffraction in particle distributions, higher spatial harmonics are needed to describe the particles’ positions producing a range of diffracted angles.

152


Light incident at the angle OB determined by the Bragg conditions will be diffracted into a zero-order beam and either the plus or minus first-order beam, depending on the incident angle. A

sin Be = 2n0A’

(3.64)

where no is the material’s refractive index. Modulators are usually Bragg-type modulators, and the on/off contrast in the first-order beams are limited by scattering. Extinction ratios of 1000 : 1 are possible in the first-order beams. But, the modulator is not able to diffract all of the energy out of the zero-order beam, because rediffraction of higher orders and the sound wave’s refractive effects add light to the zero-order beam. Contrast ratios of 10: 1 are typical. To improve the on/off contrast, spatial or polarization filtering is used to separate the zero-order beam. Both filtering methods use two modulators. Spatial filtering uses aperture stops. The first modulator has an aperture placed far behind it to isolate the zero order; the beam is then passed through a second modulator and aperture combination like the first. Contrast ratios of about 100: 1 are produced. Polarization filtering uses a shear-driven modulator which acts as a half-wave retardation for the first-order beams. The zero-order beam is selected by an entrance polarizer parallel to the transducer surface and a parallel exit polarizer. To improve contrast, the beam is passed through a second combination or reflected through the first combination. The contrast ratio is again 100 : 1 .76 The efficiency to drive an acousto-optic modulator is a measure of a material’s usefulness as a modulator and its frequency b a n d ~ i d t h . ~The ’ stress retardation for an elasto-optic material can be rewritten by using a relation between stress, velocity v, and mass density p ; this gives an expression for the strain now replaced with an expression from the sound power density P, of energy propagating at sound velocity under strain.

’= -[

‘

2 Ps1n:p2 (pv3hA2)]

(3.65)

where M is the acousto-optic figure of merit and is related to the power needed to produce a given retardation, h is the height perpendicular to the light and sound waves. Other figures of merit take into account refinements in the theory such as the bandwidth.


153

Acousto-optic modulators recently have been used for spectral filters tuned by sound frequency and observing the first-order output as the sound frequency is swept. Rapid scans, of usually less than a second, can cover large spectral regions, and applications in the infrared are common.78 Another use for acousto-optic modulators is to impress light-phase information on electrical signals controlling a modulator in a phase-locked loop to measure without ambiguity light-phase shifts. Other uses are beam steering, light-frequency shifting, and m o d ~ l a t o r s . ~ The ~*~ useful ~ bandwidth is usually hundreds of MHz, and the maximum useful frequency modulation is limited by the number of sound waves across the beam diameter, usually tens of MHz. 3.6.4.3. Magneto-optic Effect. Magneto-optic effects are the Faraday, Kerr, and Cotton-Mouton effect; the Faraday effect is the most prominent effect and most often used. A helical electric current coiling along a material demonstrating the Faraday effect will rotate the plane of polarization along the current direction due to the phase retardation between left and right circular polarization. The retardation introduced is S

=

VHI.

(3.66)

V = Verdet constant H =field strength 1 = light path length

This also means that multiple reflections will show an accumulation of multiple retardations and rotations. Faraday rotating glasses are available, too. 3.6.4.4. Depolarizers and Polarization Scramblers. Polarization can be scrambled such that the detection system sees a net zero polarization created by either a time average, a spectral average, or a spatial average across the beam, or by all three effects. Incoherent multiple scattering randomizes the polarization to produce an unpolarized beam.7 The Lambertian sphere, an opal glass diffusing plate, powder-liquid mixtures, hot-pressed polycrystalline aluminum oxide or thorium oxide, cold-pressed magnesium oxide, and cloudy fused quartz are examples. The powder-liquid mixtures can effectivelydepolarize the light for very thin films. But, scattering depolarizers destroy the image and usually strongly attenuate the beam. The preservation of the image and intensity suggests using birefringence to phase average incident polarization either in wavelength, across the aperture, or in time.'' A birefringent plate with its optic axis oriented 45" to the azimuth of the incident polarization produces a cyclically varying retardation in wavelength increasing in frequency with plate thickness. A

154

OPT1C A L PO LA R I ZATIO N

thick plate with many retardation cycles in a small wavelength range produces a fairly unpolarized beam for a system with poor spectral resolution. To allow for a varying azimuth incident on a thick-plate pseudodepolarizing element, a Lyot pseudodepolarizer is made of two plates with a 2 : 1 thickness ratio and a 45" angle between their optic axes.82 The 45" angle is most critical to the depolarizer's performance, and the thickness ratio only needs to be different from unity. Lyot depolarizers also limit spectral resolution. Birefringent crystal wedges can be made to introduce a variable phase shift in one direction across the beam diameter for a particular wavelength and is a monochromatic depolarizer if the optical system's final image does not spatially resolve the phase variation. Two wedges with the optic axes oriented 45" to e8ch other are independent of the azimuth of the incident polarization. Double images and beam deviation occur for these depolarizer types. A varying driving function with a single optic modulator aligned with its axes horizontal and vertical produces a polarization intensity modulation expressed in four intensities, for example total intensity, horizontal linear, +45" linear, and right circular polarization intensities modulated by the driving function. driving function

(3.67)

Sb = so

total intensity polarized + unpolarized

S : = S, cos S - S3 sin 6

horizontal linear polarization

s; = s,

+45" linear polarization

S ; = S, sin 6 + S3 cos S

circular polarization,

where S' and S are the transformed and incident polarization vectors. The phase-modulated beam will always have one linear component unmodulated. If the modulator is rotated 45", then the modulated exiting linear polarization changes from horizontal to 45" linear.

s; = s,, S ; = S, cos S - S, sin 6.

If the driving function is a sinusoidally varying function, then the intensity phase modulation can be expressed with Bessel function expansions for the sine and cosine intensity modulation.

155


S = S,,, sin wt, u)

sin 6 =sin(S,,,

sin w t ) = 2

C

(3.68)

J2n+l(Smax) sin[(2n + l)ot],

n=O

m

C

cos 6 =cos(&,,, sin wt)=Jo(6,,,)+2

JZn(Smax) sin(2nwt).

fl=l

The time averages for the sine terms are zero, and the remaining term is J,( S,,,) = 0 for S,,, = 2.405, the first zero of J, . This means zero polarization can be produced in either horizontal linear and circular polarizations, or in 45" linear and circular polarizations, and can be used to calibrate instrumental polarization to an accuracy of 0.01 to o . o o ~ ~ ~ . ~ ~ If an optic modulator were driven by a sawtooth function such that the phase modulation varied linearly from 0 to 2rr, the time-averaged output light would have only a linear polarized component. If a second modulator is added with its optic axis 45" to the first modulator and driven at an integral multiple of the first frequency nw, then the resulting polarization is zero. Another method to produce depolarized light makes use of the timeaverage effects for rotating retardation plates.83

s=

plate retardation

+ unpolarized

st,= s,

total intensity polarized

S' -I - 2( 1 + cos 6)Sl

horizontal linear polarization

S ; = $(1 + cos 6)S2

+45" linear polarization

s; = s3 cos 6

circular polarization

A rotating achromatic quarter-wave plate will pass only unpolarized and linearly but not circularly polarized light. A rotating achromatic half-wave plate will pass only unpolarized and circularly but not linearly polarized light, and the circular polarization's handedness is reversed. A combination of two wave plates rotating in opposite directions produces an element passing only unpolarized light.

3.6.5. Liquid Crystals

Liquid crystals (LCD) are organic molecules in a state between liquid and solid and show electromagnetic anisotropies. They are structured in three basic geometries which are easily disrupted by electric fields, magnetic fields, or temperature, depending on the design. The liquid crystals can be

156


chosen such that the net molecular polarization lines up parallel or perpendicular to the crystal's long axis; this corresponds to positive or negative birefringence, A n 0.2. For example, tunable birefringence can be produced by treating the container's walls such that negative nematic crystals line up parallel to the light beam in the off state. An electric field applied to the walls will flip the liquid crystals perpendicular to the electric field and light beam and produce a retardation proportional to the voltage up to a threshold voltage. Or, the cell's interior wall can be treated to align the positive nematic crystal structure perpendicular to the beam; this is done by rubbing the glass to form parallel microgrooves and then treating the surface with a high-energy surfactant. Then, a 90" twist is introduced between the entrance and exit surface, rotating the beam 90". A voltage of 5-10 volts will completely disrupt the rotation and extinguish the beam between crossed polarizers. Liquid crystals are low-current, low-voltage, high-contrast, thindisplay p o l a r i z e r ~ . ~ ~

-

3.6.6. Dichroic Polarizers

Linear dichroic molecules and crystals absorb polarization with different absorption coefficients K, and K y . To reduce scattering effects, molecules of small dimensions are used, usually needle-shaped iodine molecules. To align the molecules, they are either applied to a burnished glass surface with parallel microgrooves and allowed to settle, or are placed on a polyvinyl alcohol or nitrocellulose base and stretched, which also introduces unwanted birefringence. Better, controllable performance is obtained with the plasticsheet p ~ l a r i z e r . ~ ~ Extinction ratios and transmission vary with the concentration of dichroic molecules; and concentration of molecules, standard densities, and extinction ratios are commercially available. Heavily dyed polarizers have a parallel transmission of 12-14% and a crossed transmission of 0.001%. As the dye is decreased, the short-wavelength extinction or optical density decreases, thus decreasing the spectral region with high uniform optic densities for crossed polarizers. Their parallel transmission is 30% or more. To extend the plastic's usefulness into the infrared, the polyvinyl alcohol sheet is catalytically dehydrated to polyvinylene. It is useful in the infrared from about 8000 A-20,OOO A, with a large ttansmission of the unwanted component at 9200 A. For near ultraviolet use, the polyvinyl alcohol/iodine polarizer can be purified. Its range is down to 2000A. The thin Polaroid sheets are usually cemented in glass or plastic laminates. King has reported on dichroic polarizers and crystal prism polarizers. The dichroic polarizer type has its place according to degree of p e r f o r m a n ~ e . ~ ~

SLITS, GRATINGS, A N D METAL GRID WLARIZERS

157

3.7. Slits, Gratings, and Metal Grid Polarizers An ideal slit is an infinite conductor and is very thin. Approximate examples are scratches on thin silver-coated mirrors. The light transmitted by narrow slits of width d satisfying A / d > 2 are completely S polarized and perpendicular to the slit. This is the electromagnetic theory region, and the effect described is Hertzian polarization. Either finite conductivity or a thick slit will decrease the slit’s transmission. When A / d < O S , then the polarization effect for P or S polarizations is negligible. This is called the scalar theory region described by infinite conductivity. For the region 2 > h / d > 0.5, the transmitted polarization is predominantly P polarization due to the slit’s finite conductivity and thicknesss6 Wire grids with a spacing of A / d > 0.5 are made to S polarize light. Successful wire grids are either gratings with their ridges metal-coated at a high-incidence angle or metal whiskers grown by a metal beam directed at a substrate at a high-incidence angle.87Grid spacings as small as visible-light wavelengths can be made for polarizers for wavelengths greater than 8000 A. Extinction ratios are greater than 100: 1 and improve with wavelength. Also, wire-grid polarizers can have a large acceptance angle and aperture.’ Diffraction gratings polarize the light according to the gratings material and overcoat, the groove shape and spacing d, the wavelength A, and the order N. As for slits, the P polarization is predicted fairly well with a scalar theory down into the ultraviolet.** The S polarization perpendicular to the slit is not as well behaved, and polarization anomalies in different orders and wavelengths are more pronounced than for P polarizations. Energy can be either diffracted into or out of the observed order, so the anomalies can appear either bright or dark. The influence of a dielectric coating is to broaden the anomaly and to shift it to longer wavelengths. For h / d < 0.2, the polarization effects become negligible, and scalar theory applies to both the P and S components. The grating’s efficiency in wavelength and diffracted angle is strongly influenced by anomalous behavior. The grating anomalies are either Wood’s anomalies or resonance a n ~ m a l i e s . *Wood’s ~ * ~ ~ anomalies occur in a grating order at a wavelength which is at the same time being diffracted at *90° in another order, passing onto or off the grating. Wood’s anomalies occur at Rayleigh’s wavelengths given by the grating equation nh = a(sin i * I ) ,

(3.69)

where n is the passing-off order.“ Resonance anomalies are leaky surface waves supported by the grating and depending on the grating surface structure. A grating with shallow grooves and a slightly modulated surface has little reactance and little

158

0 PTIC A L PO LA RI ZATlO N

anomalous effect. The resonance and Wood's anomalies are close together for this type of surface. For a strongly modulated surface, the reactance increases and the anomalies separate. Blazed gratings in the Littrow configuration (the incident and. diffracted beams have the same angle), with right triangle-shaped grooves and the 90" corner at the apex, can be described by the blaze angle only. First-order diffraction is most strongly influenced by anomalies which decrease for higher orders because A / d decreases. Higher-order anomalies and diffracted angle deviations can be deduced in terms of the first-order Littrow configuratiom9* First-order gratings with a blaze angle of less than 5" behave close to scalar theory and show few anomalies. The P and S polarizations have a peak in their efficiency at the blaze angle equal to 100%. For gratings blazed between 5" to 18", Wood's anomalies at the Rayleigh wavelength for the -1 and + 2 order at A/d = 2 / 3 , about 19.5", decrease in strength with the blaze angle. The S grating efficiency is 100% at the blaze angle, and the P efficiency peak is decreasing. For gratings blazed at 18" to 22", the anomalies are suppressed because the Rayleigh pass-off anomaly occurs in the same direction as the blaze angle. The S efficiency becomes more uniformly high for diffraction angles greater than about 19". The P efficiency is a minimum of about 80%. For blaze angles greater than 22", the S efficiency for diffraction angles greater than 19" is high up to 60" and is 100% at the blaze angle. The P efficiency peak is 100% at the Rayleigh wavelength A / d = 2 / 3 . Gratings with an angular deviation between the incident and reflected beams have lower P and S efficiencies, and the anomalies and peak efficiencies shift in wavelength. Gratings with known P and S efficiencies can be used as a polarizing element in an instrument.

3.8. Light Source and Detector Polarizations Light sources can be polarized for a number of reasons. Lasers are usually linearly polarized coherent emission sources. Incandescent and fluorescent emissions are polarized because the emitted radiation is refracted at the surface; and radiation refracted at a grazing angle to the surface is strongly P polarized.93 Metal filaments, ribbons, and Nernst glowers have demonstrated strong polarization effects for radiation from the highly angled cylindrical edges. Also, deuterium lamps have been studied, which demonstrate strong polarization effects for old lamps with a thin metal film deposited on the glass envelope.94 New lamps are not as strongly polarized across the light spot, but the polarization is still nonuniform away from the central light spot, which is possibly due to polarized reflections from the

POLARIZATION DETERMINATION A N D MATHEMATICAL DESCRIPTION

159

envelope. The envelope’s polarization effects have been demonstrated for tungsten lamps. Limiting light to the light spot reduces polarization effect^.^^'^^ Depolarizers can be used to remove the polarization effects if the effects are found to be significant enough. Detectors are also polarization sensitive for various reasons which sometimes cannot be neglected. For example, photomultipliers or photocathodes are sensitive to polarization due to their intrinsic characteristics. Also detectors and arrays can have windows with varying static birefringence. The polarization effects need to be considered for the particular application and can be removed, if necessary, by either calibration or a depolarizer.

3.9. Pola rizat ion Deter mi nation and Mat hemat i ca I Descri ption Polarization is either coherent and totally polarized or incoherent with an unpolarized and a polarized component. In either case, the PoincarC spherical surface description is applicable. An elliptical polarization is marked on the sphere and transformed to a new polarization after passing through an optical element. Coherent polarized light is known in terms of phase shift between orthogonal polarizations and their intensities, for example linear or circular polarizations. Incoherent polarized light is described by Cartesian coordinates as the amount of linear polarization in the x and +45” axes, and the amount of L or R polarization. This coordinate system plus intensity is known as Stokes’ parameters. Two- or four-dimensional transformation matrices describe the polarization vector and the optical element’s effects; and the cumulative effect is a matrix multiplication of all the effects in their order, resulting in a single transformation matrix describing the system. Coherent light can be written as a two-dimensional vector multiplied by a complex phase term and transformed with 2 x 2 matrices known as Jones’ matrices. Incoherent radiation can be described in real numbered Stokes’ parameters and transformed with 4 x 4 matrices known as Mueller’s matrices. Light represented in terms of Jones’ matrices is the vector sum of two linear orthogonal amplitudes E , and E,, and a phase shift 6. The fact that they have the same frequency can be implied and removed from the calculations. The modified form for an incident beam E and output beam E’ is,

E = E x e x+ EyeY=

(3 ,

(3.70)

160


Clear isotropic materials, linear polarizers, retarders, reflections, and coordinate rotations transform E to E’. E‘=M*E

transparent isotropic medium in x - y M = ( kx 0

0

k,

),

kx 1

Ex

k, E,

Eb

)

partial linear polarizer in x - y

6 phase retardation in x - y

reflection in x - y -sina

Ex cos a + E, sin a -Ex sin (Y + E, cos a

cosa

rotation of an angle a counterclockwise looking into the beam cos2 a

1 0

cos a sin a

general orientation of linear polarizer M

=R(7Y)(

‘6“

cos2a e i r r I 4 j ( 2 ) ” 2 cos a sin a e P * , 4 ) R ( a ) = ( j ( 2 ) ’ / 2cos a sin a sin2a ePJrrI4

general orientation of quarter-wave plate

(3.71)

Light represented in terms of Stokes’ vectors and Mueller’s matrices are related to the Jones’ matrices. The Stokes’ vector has several labeling schemes,

(I Q

u

V)

(I M C S) (So

s,

s 2

SJ

POLARIZATION DETERMINATION A N D MATHEMATICAL DESCRIPTION

161

The following relations hold between the Stokes' vectors and the timeaveraged electromagnetic vectors,

I:=(E:)+(E:)+I~, 2

Q 2 + U 2 + V2,

0 = ( E 3 -W;L U = 2( E,E,,) cos S

= Re( I?$,,),

V = 2( ExE,,)sin 6 = Im( ExEy).

(3.72)

The Stokes' vectors can also be written in terms of the light's polarization azimuth a,ellipticity e, and the percent polarization P.

P=

( Q 2 + U 2 + V2)'/2 3

IT

a = 0.5 arctan(

:),

e = 0.5 arcsin Q = ITP cos(2e) sin(2a),

U = ITP cos(2e) cos(2a), V = ITP sin(2e).

(3.73)

The Mueller transformation matrices are similar to the Jones' matrices and are determined by how they transform the incident light to output light. 1 1 0 0

0" linear polarization -1

0 0

0 0 0 90" linear polarization

162


1 0 1 0

45” linear polarization

right circular polarization 1 0 0 - 1

-1

0 0

left circular polarization 0

0

0 rotation of an angle

cy

counterclockwise looking into the beam.

The 4 x 4 transformation matrices can also be determined experimentally to take out an instrument’s polarization. A fairly complete list of matrices is given by Shurcliff,2’ and recent articles go into details for error analysis with Jones’ and Mueller’s matrices for polarization measurement^.^'-^^' References J. Strong, Concepts of Classical Optics, p. 110, W. H. Freeman, San Francisco, 1958. M. Born and E. Wolf, Principles ofOprics, p. xxi, Pergamon Press, New York, 1964. R. H. Muller, “Definitions and Conventions in Ellipsometry,” Surface Sci. 16, 14 (1969). J. D. Jackson, Classical Electrodynamics, p. 269, Wiley, ,New York, 1975. Ramachendran and Ramaseshan, Encyclopedia of Physics, Vol. 2 5 , Part 1, S, p. 1 (Flugge, ed.), Springer-Verlag, 1961. 6. R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light, p. 270, NorthHolland Publ. Amsterdam, 1977. 7. J. M. Bennett and H. E. Bennett, “Polarization,” in Handbook ofoptics, Chap. 10, ( W . G. Driscoll and W. Vaughan, eds.), McGraw-Hill, New York, 1972.

1. 2. 3. 4. 5.

REFERENCES

163

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164


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61. S. Chu, R. Conti, P. Bucksbaum, and E. Commins, “Making Mica Retardation Plates: A Simple Technique,” Appl. Opt. 18, 1138 (1979). 62. J. Strong, Procedures in Experimental Physics, p. 388, Prentice-Hall, New York, 1952. 63. W. C. Davis, “Simplified Wave Plate Measurements,” Appl. Opt. 20, 2879 (1981). 64. E. Hartfield and B. J. Thompson, “Optical Modulators,” in Handbook ofOptics, Chap. 17 (W. G. Driscoll and W. Voughan eds.), McGraw-Hill, New York, 1972. 65. E. K. Sittig, “Elastooptic Light Modulation and Deflection,” in Progress in Optics, Vol. 10, Chap. 6, p. 231 (E. Wolf, ed.), North-Holland Publ., Amsterdam, 1972. 66. C. Forno and 0. C. Jones, “Hexamine Electro-Optic Light Modulators,” J. Phys. E 7, 101 (1971). 67. B. P. Blumich and R. Germer, “Electro-Optic Shutters with PLZT Ceramics for Infrared Applications,” J. Phys. E 12, 770 (1979). 68. E. R. Kocher and R. P. Novak, “Fast Electro-Optical Shutter with Programable Transmission Control,” 1. Phys. E 13, 542 (1980). 69. T. G. Baur, D. E. Elmore, R. H. Lee, C. W. Querfeld, and S. R. Rogers, “Stokes 11. A New Polarimeter for Solar Observation,” Solar Physics 70, 395 (1981). 70. J. Etchepare, G. Grillon, R. Muller, and A. Orszag, “Kinetics of Optical Kerr Effect Induced by Picosecond Laser Pulses,” Opt. Comm. 34, 269 (1980). 71. M. A. Duguay, “The Ultrafast Optical Kerr Shutter,” in Progress in Optics, Vol. 14, Chap. 4, p. 163 (E. Wolf, ed.), North-Holland Publ., Amsterdam, 1976. 72. A. Korpel, Applied Optics and Optical Engineering, Vol. 6, p. 89 (R. Kingslake and B. J. Thompson, eds.), Academic Press, New York, 1980. 73. J. C. Cheng, L. A. Nafie, S. D. Allen, and A. I. Braunstein, “Photoelastic Modulator for the 0.55-13 pm Range,” Appl. Opt. 15, 1960 (1976). 74. J. C. Kemp, “Photoelastic-Modulator Polarimeters in Astronomy,” in Polarizers and Applications (G. B. Trapani, ed.), Proc. Soc. Phot. Opt. Inst. Eng. 307, 80 (1981). 75. A. Korpel, Electro Optics-Principles and Applications (B. Thompson and J. B. DeVelis, eds.), Proc. SOC.Phof. Opt. Inst. Eng. 38, 3 (1973). 76. G. B. Brandt, M. Gottlieb, and J. J. Conroy, “Improved Contrast for Acousto-Optical Modulators Using Spatial Filtering,” J. Opt. Soc. A m . 67, 1269 (1977). 77. A. Korpel, “Acousto-Optics,” in Applied Oprics and Optical Engineering, Vol. 6, Chap. 4 (R. Kingslake and B. J. Thompson, eds.), Academic Press, New York, 1980. 78. 1. C. Chang, “Acousto-Optic Tunable Filters,” Opt. Eng. 20, 824 (1981). , 79. J. B. Houston, Jr., M. Gottlieb, Shi-Kay Yao, I. C. Chang, J. Tracy, L. M. Smithline, and G. J. Wolga, “The Potential for Acousto-Optics in Instrumentation: An Overview for the 1980s.” Opt. Eng. 20,712 (1981). 80. M. Gottlieb and G. W. Roland, “Infrared Acousto-Optic Materials: Applications, Requirements and Crystal Development,” Opt. Eng. 19,901 (1980). 81. Shan-Ling Lu and A. P. Loeber, “Depolarization of White Light b y a Birefringent Crystal,” J. Opt. SOC.A m . 65, 248 (1975). 82. A. P. Loeber, “Depolarization of White Light by a Birefringent Crystal. 11. The Lyot Depolarizer,” J. Opt. SOC.A m . 72,650 (1982). 83. B. H . Billings, “Monochromatic Depolarizer,” J. Opt. SOC.A m . 41,966 (1951). 84. R. A. Saref, in Electro Optics-Principles and Applications (B. J. Thompson and J. B. DeVelis, eds.), Proc. SOC. Phof. Opt. Inst. Eng. 38,23 (1973). 85. E. H. Land, “Some Aspects of the Development of Sheet Polarizers,” J. Opt. SOC.A m . 41, 957 (1951). 86. R. V. Jones and J. C. S. Richards, “The Polarization of Light by Narrow Slits,” Proc. Roy. SOC.(London), A225, 122 (1954). 87. R. E. Slocum, “Evaporative Thin Metal Films as Polarizers,” in Polarizers and Applicafions (G. B. Trapani ed.), Proc. SOC.Phot. Opt. Inst. Eng. 307,25 (1981).

166


88. A. J . Caruso, G. H. Mount, and B. E. Woodgate, “Absolute S- and P-Plane Polarization Efficiencies for High Frequency Holographic Gratings in the VUV,” Appl. Opt. 20, 1764 (1981 ). 89. C. H . Palmer, “Parallel Diffraction Grating Anomalies,” J. Opt. SOC.Am. 42, 269 (1952). 90. C. H. Palmer and F. W. Phelps, Jr., “Grating Anomalies as a Local Phenomenon,” J. Opt. SOC. Am. 58, 1184 (1968). 91. A. Hessel and A. A. Oliner, “A New Theory of Wood’s Anomalies on Optical Gratings,” Appl. Opt. 4, 1275 (1965). 92. E. G. Loewen, M. Neviere, and D. Maystre, “Grating Efficiency Theory as it Applies to Blazed and Holographic Gratings,” Appl. Opt. 16, 271 1 (1977). 93. 0. Sandus, “A Review of Emission Polarization,” Appl. Opt. 4, 1634 (1965). 94. H . P. Lengkeek and P. Winsemus, “Polarization of Light Sources: a Photographic Method,” Appl. Opt. 14, 2766 (1975). 95. R. K. Kostuk, “Polarization Characteristics of DXW-Type Filament Lamp,” Appl. Opt. 19, 2274 (1980). 96. D. L. Spooner, “Polarization Pattern of a High Intensity Incandescent Lamp,” Appl. Opt. 11, 2984 (1972). 97. M. A. Bouchiat and L. Pottier, “A High-Purity Circular Polarization Modulator: Application to Birefringence and Circular Dichroism Measurements on Multidielectric Mirrors,” Opt. Comm. 37, 229 (1981). 98. H. G. Berry, G . Gabrielse, and A. E. Livingston, “Measurement of the Stokes Parameters of Light,’’ Appl. Opt. 16, 3200 (1977). 99. R. M. A. Azzam, “Division of Amplitude Photopolarimeter (DOAP) for the Simultaneous Measurement of all Four Stokes Parameters of Light,” Opt. Acta 29, 685 (1982). 100. V. M. Bermudez and V. H. Ritz, “Wavelength-Scanning Polarization Modulation Ellipsometry: Some Practical Considerations,” Appl. Opt. 17, 542 (1978). 101. P. S. Hauge, ”Techniques of Measurement of the Polarization-Altering Properties of Linear Optical Systems,” in Optical Polarimetry (R. M. A. Azzam and D. L. Coffeen, eds.), Roc. SOC.Phot. Opr. Inst. Eng. 112, 2 (1977).

4. HOLOGRAPHY

R. D. Bahuguna* Rose-Hulman Institute of Technology 5500 Wabash Avenue Terre Haute, Indiana 47803

D. Malacara Centro de lnvestigaciones en Optica. A X Apartado Postal No. 948 37000 Leon, Gto. Mexico

4.1. Introduction The invention of the laser has had a great impact on the science of optics. In particular, the high coherence of the laser light has completely revitalized those methods of optics which use interference as the basic phenomenon. Among these is the method of wave-front reconstruction. Also called holography, this method was invented by Denis Gabor in 1948'*2.3 to improve the resolution in the image obtained with the electron microscope. Although the initial goal did not prove successful, holography in itself became an important subject of investigation. However, due to the lack of a coherent light source, it remained in obscurity for more than a decade. In the 1960s, with the availability of the laser, a major advance was made at the University of Michigan Institute of Science and Technology. Leith and Upatnieks in 1962495*6 applied the laser light to holography producing excellent three-dimensional images which astonished all those who saw them. In the same year, Denisyuk' made a remarkable contribution to holography by combining it with the volume-recording process, invented by Lippmann' in 1891, to make color photographs. Using a thick emulsion to record the interference fringes, a method was produced for color holography in which reconstruction could be carried out with a point source of white light. His hologram is, in effect, a hologram and a narrow-band spectral filter, combined into a single structure. *Work was done while the author was at Centro de lnvestigaciones en Optica, A.C., Apartado Postal No. 948, 37000 Leon, Gto. Mexico. 167 METHODS OF t X P E R I M t N T A L PHYSICS Vol ?b

Copyright 01988 by Academic Press. Inc. All rights of reproduction in an) form reserred. I S B N 0-12-475971-8

168

HOLOGRAPHY

Stephen Bentong of the Polaroid Corporation made another significant contribution in display holography. The method, known as rainbow holography or Benton holography, makes it possible to view the hologram with an extended white light source. Cross” combined the rainbow holograms with a technique known as composite or multiplex holography to produce cylindrical holograms. Holography, although applicable to all waves, i.e., optical waves, microwaves, X-rays, electon waves, seismic waves, and acoustic waves, has progressed exceptionally fast in the first category. This is essentially because of the availability of a highly coherent source: the laser. Research continued throughout the sixties and seventies. To date, hundreds of scientific papers and many books like those by Stroke,’’ Smith,” Collier et al.” Francon,14 Caulfield,” Abramson,I6 and Hariharan” have been published about holography and its applications to various other areas. Holography is a relatively new process that is similar to photography in some respects, but is nonetheless fundamentally different. Some of the basic differences are the following: (a) Photography records the image of an object formed by the lens of a camera. The image is recorded as two-dimensional and contains information about the object as viewed from a single particular angle. In holography, the object wave front is reconstructed, giving a three-dimensional view. It is as if one is viewing the object itself, possessing depth and parallax properties. (b) The photograph is in itself a direct record of the scene and can be viewed by focussing one’s eyes onto its flat surface. The hologram is the interference record of the object wave with a reference phase-related wave. To reconstruct the image, the record is illuminated by the reference beam. The image is reconstructed at the same position as the object (or its image) during recording, and thus, not necessarily on the holographic plate itself. (c) Photography can be done in ordinary light, whereas holography requires some amount of coherence. (d) In photography the intensity is a quantity averaged over all the phases of the light wave, and so the phase information about the object is lost. On the other hand, in holography the amplitude and phase are encoded with the help of the reference wave and can be decoded in the reconstruction step.

4.2. Theory of Holography The of-axis holography4 is the best-known form of holography and is, conceptually, the simplest to treat from the theoretical point of view. In this section, we describe the theory of this method for plane holograms and

THEORY OF HOLOGRAPHY

169

later see the effect of thick emulsions. Finally, we discuss the nonlinear effects and aberrations involved in the process. 4.2.1. Basic Theory of Plane Holograms

By a plane or thin hologram we mean one in which the recording material is thin compared to the spacing of the highest spatial frequency of the exposure variations, i.e., the recording medium is essentially twodimensional. Consider the arrangement shown in Fig. 1 , where 0 is the object wavefront and R is the reference field. They both interfere on the photographic emulsion H. The total field on H is O + R. The irradiance I ( x ) on the photographic plate is given by:

I ( x ) = ( O + R ) ( O +R ) * =

I 0l2+ IR 1' + OR * + O* R.

(4.1)

Assuming that we are in the linear region of the amplitude transmittance f ( x ) versus the exposure E ( x ) curve, we have:

t(x)=to+BE(x),

(4.2)

where to is a constant background transmittance and B is a parameter determined by the emulsion and the processing conditions. Since E ( x ) for a given exposure time is directly proportional to Z(x), we may write using eqns. (4.1) and (4.2): t(x)=

t,+B'(1012+lR12+OR*+O*R)

where B'= B T ; t is the exposure time.

FIG.

1. Basic arrangement when recording a hologram.

(4.3)

170

HOLOGRAPHY

The developed plate which is now a hologram is illuminated by the reference wave R . The transmitted field X ( x ) at the hologram plane is then given by: X ( x )= R ( x ) t ( x ) =

R~,,+B’(R~O~*+RIR~~+

I R ~*o+R*o*) .

(4.4)

Now, if the reference beam is sufficiently uniform so that IRI2 can be treated as constant, then the fourth term of Eq. (4.4)is merely a constant times the object wave, showing that the object is reconstructed as such. The hologram is then like a window through which a virtual image of the object is seen. The last term represents the conjugate of the object wave times R’.

(C

1

(d)

FIG. 2. Method to determine the conjugate image: Join the reference point source R and the object point 0, by a straight line. Draw a perpendicular R P to the plate H through R. Measure the angle RPO = a say. Draw a line passing through P at an angle ( - a ) and let it 0;is then the conjugate image. ( a ) , (b) and (c) are different possible meet O , R at 0;. recording geometries; (d) is similar to ( a ) except that one is viewing the image in reflection. Reflection images of 0, and 0:are seen as 0,and 0;respectively.

T H E O R Y OF HOLOGRAPHY

171

The conjugate image might be a real or a virtual image, depending on the particular geometry used to form the hologram. When the image is real, it is spatially located in front of the hologram. A simple method to determine the type of the conjugate image and its position is described in the book by AbramsonI6 and is shown in Fig. 2. The paths of the light in a hologram are reversible. Thus, if the reconstructing beam reaches the hologram from the opposite side and in an antiparallel direction to the original reference beam, the two images are formed at the same places. The only difference is then that a virtual image becomes real, and vice versa. If, when making a hologram, the object is real on the same side as the collimated reference beam, we have the following property. The real images produced by holograms, unlike those produced by lenses, are pseudoscopic, that is, they are “inside out.” Concave surfaces appear as convex, and vice versa. However, an orthoscopic image may be obtained if the pseudoscopic real image of another hologram is used as the object when taking the hologram. Figure 3 shows two alternative ways to reproduce these holographic images.

FIG. 3. Basic arrangement to reproduce the images of a hologram: (a) recording set up; (b) reconstruction with a beam identical to the reference beam; (c) reconstruction with a beam antiparallel to the reference beam.

172

HOLOGRAPHY

4.2.2. Nonlinear Effects and Aberrations

The effect of film nonlinearities on holograms has been studied by many authors, like Friezem and Zelenka,'' Goodman and Knight," Bryngdahl and Lohmann,20 and Kozma." When the visibility of the fringes is high, the variations on the exposure E ( x ) are large, and hence the linear approximation in Eq. (4.2) might not be valid. Then, in general: t ( x ) = t o + B l E ( x ) + B 2 E 2 ( x ) + B 3 E 3 ( ... x) .

(4.5)

The third- and higher-order terms produce higher-order images. The contrast of the object is reduced and some ghost images appear. If the object is a set of discrete points, false images may be produced. This analysis explains nonlinear effects in flat transmission holograms, but not in volume or phase holograms. Besides film nonlinearities, intrinsicphase nonlinear effects also appear in phase holograms. The images of a hologram may be reproduced by using any of the two basic configurations described in the former section. However, the images may also be obtained after changing one or more of the following parameters: ( a ) position of the reconstructing light source, (b) wavelength, ( c ) size of the hologram, etc. The changing of these parameters in general affects the position of the images, the type of these images, and their size. However, it is very important to point out that these changes also affect the quality of the images, because some aberrations are also introduced, as pointed out by Meier.22 4.2.3. Effects of Thickness: The Bragg Effect

If the photographic emulsion is somewhat thicker than the period of the interference fringes, the hologram is called a volume hologram. This means that the interference pattern extends not only across the surface of the emulsion but, since the emulsion is quite transparent, throughout its depth as well. A volume hologram has a number of properties not shared by the plane hologram. Volume holograms can be made to produce an image either by light transmitted through them (transmission holograms) or b y light reflected from their surfaces (reflection holograms). We can schematically represent the formation of a volume-transmission hologram, as in Fig. 4, where R and 0 are the reference and object plane waves. (The results of plane waves can be generalized for any object, as the object wave can always be decomposed into plane waves). The waves R and 0 interfere with an angle 4 between them and 4 / 2 with respect to the normal to produce a set of parallel fringes perpendicular to the emulsion.

T H E O R Y O F HOLOGRAPHY

173

R

3

Emulsion

FIG. 4. Arrangement to form a volume-transmission hologram.

The spacing is easily seen to be given by: d=

A (2 sin(4/2))'

(4.6)

Let us now illuminate the hologram at an angle 8, which is, in general, not equal to 4/2 at recording. In order to obtain an image reconstructed from the plane object wave, it is necessary that all the rays add in phase after reflection from the semireflecting planes. The following relation must then be satisfied:

+A

sin 0 =-, 2d

(4.7)

which is the familar Bragg condition. On comparing Eqs. (4.6) and (4.7), we have the following four solutions:

and

e = * ( n - 4/21. We will consider each case separately as follows: (a) When the illuminating beam is identical to the reference beam ( 0 =

4/2), the object wave is reconstructed, and the virtual image formed. No conjugate image is observed. (b) When the illuminating beam is in the opposite direction to the initial reference wave 6 = - ( n - 4 / 2 ) , a real image is reconstructed. This real image is pseudoscopic, and no virtual image is formed.

174

HOLOGRAPHY

(c) When the illuminating beam is in the direction of the object beam ( 0 = - 4 / 2 ) , only an imperfect real image, with aberrations, is formed. The

complete object cannot be seen, becasue the Bragg condition is not satisfied for all waves from the object. (d) When the illuminating beam is in an opposite direction to the object wave ( 0 = T - 4 / 2 ) , a partial, imperfect virtual image, with aberrations, is reconstructed. The complete image is not observed for the same reason given before. Summarizing, we can say that in thick holograms only one image is reconstructed, at,the same position where the object was when the hologram was formed. If the reconstructing beam is parallel to the reference beam, the image is orthoscopic. However, if the reconstructing beam is antiparallel to the reference beam, the image is pseudoscopic. So far, we have discussed the transmission volume holograms. The reflection type can be made as shown in Fig. 5 . The object wave 0 and the reference wave R come from opposite directions. The interference fringes thus recorded are approximately planes parallel to the emulsion. The distance between the planes is h / 2 . If such a hologram is illuminated with a beam identical to the reference beam, it will reconstruct the virtual image of the object. The reconstruction is viewed in reflection rather than transmission. The major feature of the refection holograms is that a single-color wave front may be reconstructed with white-light illumination. The Bragg planes act as an interference filter and have a high reflectivity only for those wavelengths that produced them. However, in practice, such a hologram recorded at one wavelength may reconstruct in a slightly shorter wavelength.

Object

i'

wave 0

Reference wave R

F I G . 5. Arrangement to form a volume-reflection

hologram.

DIFFERENT TYPES O F HOLOGRAMS

175

For example, the recording in red color may reconstruct in green. This is because of emulsion shrinkage during photographic processing. The Bragg planes pull together so that the final spacing is smaller than the one at recording, leading to reconstruction at a different wavelength. It is worth pointing out that white-light illumination results in a loss of image resolution, since the hologram acts as a spectral filter with a bandwidth of about 100 A, which is too large for high-resolution imagery.

4.3. Different Types of Holograms There are a number of hologram forming configurations depending upon the position and form of the object and the reference beams. The simplest one, invented by Gobor, is the in-line hologram. In this method, the object is a transparency illuminated by a reference beam. The scattered light contains information about the object, and the unscattered light acts as a reference wave. The source is the green line of the mercury arc lamp, Although successful in reconstructing the transparency, this hologram suffered from a serious drawback: the twin-image problem. The primary image and the conjugate image are in the same direction and so are inseparable. The problem was overcome successfully by Leith and Upatnieks4 using their off-axis configuration, and later by Bryngdahl and Lohmann” and by Meier24using single-side band holography. In the following section, we describe Fresnel, Fraunhofer, Fourier transform, image-plane, color, composite holographic stereogram, rainbow, and computer (synthetic) holograms.

4.3.1. Fresnel Holograms

When the recording plate is placed in the Fresnel diffraction region of the object and at an arbitrary distance from the reference source, we obtain a Fresnel hologram. A practical arrangement is shown in Fig. 6 . It is the simplest and most convenient configuration, requires no lenses either in formation or in reconstruction, and produces images of three-dimensional objects as shown in Fig. 7.

4.3.2. Fraunhofer Holograms

A Fraunhofer hologram can be defined as one which records interference of the far-field diffraction pattern of the object, with a reference light source not coplanar with the object.

176

HOLOGRAPHY

L

S

FIG.6. Recording a Fresnel Hologram.

FIG.7. Reconstructed image from a Fresnel Hologram.

177


\

d

-

. $

-4

H 0 FIG.8. Recording of a Fraunhofer hologram with a free space configuration.

The following are the two geometries with which to record such a hologram. (1) This configuration is a free-space geometry (see Fig. 8). The subject (a small transparency) is illuminated by a broad plane-parallel beam. The recording plate is kept in the far field of the subject. The light scattered from the subject interferes with the uniform background to form the hologram. This geometry was utilized by Thompson et aLZ5for measuring the size and shape of dynamic aerosol particles. The twin-image problem in this geometry is apparently removed. (2) In this configuration (see Fig. 9), the subject is placed close to the lens and is illuminated by a plane-parallel beam. The recording plate is

FIG. 9. Recording of a Fraunhofer hologram with a lens.

178

HOLOGRAPHY

kept at the back focal plane of the lens, and the reference wave is incident obliquely from the side. 4.3.3. Fourier Transform Holograms

A Fourier transform hologram records the interference pattern of the Fourier transform fields of both the object and the reference source. A practical arrangement is shown in Fig. 10. Such holograms are useful as spatial filters for pattern recognition, where the properties of the Fourier transform provide the basis for the recognition process, as described by Vander Lugt.26 It is worth noting that when the Fourier transform hologram is formed with a plane-reference wave, the generated image remains stationary with respect to the translation of the hologram. For example, a moving reel of such holograms would project stationary images. By a variation of the above geometry, one obtains the so-called quasiFourier transform h01ogram.l~ In this case, the subject is adjacent to the Fourier transforming lens with a coplanar reference source. As is well known, parallel illumination of the transparency produces its Fourier transform at the back focal plane together with a phase factor, as shown by G ~ o d m a n . ~The ’ coplanar reference source removes the phase factor, and thus we obtain the same properties as obtained with the Fourier transform hologram. Another variation of the geometry leads to the lensless Fourier transform hologram by virtue of the absence of a lens.13 Here the Fresnel diffraction field of the transparency is recorded together with the field from a coplanar reference source. The phase factor is again removed, and the hologram

L,

H

FIG. 10. Recording of a Fourier transform hologram.


179

formed with this arrangement has a transmittance resembling that of the Fourier transform hologram. 4.3.4. Image-Plane Holograms

When the image of an object is formed on the plane of the recording plate and a hologram is formed with the help of a reference wave, the hologram is known as an image-plane hologram. A practical arrangement is shown in Fig. 11. The image reconstructed with the reference beam in the original direction is similar to that obtained with a Fresnel hologram, except that the field of view is limited by the aperture of the lens. The field of view can be increased significantly by using the conjugate illumination. Then, the illuminating beam is on the opposite side and in opposite direction to the original reference beam, as described by Brandt28and R o ~ e n s . * ~ Since the conjugate illumination is preferred, the image in the reconstruction is pseudoscopic, or “inside out.” The image becomes orthoscopic if a conventional hologram is used instead of the lens to form a real image over the holographic plate. In image-plane holograms, the requirements for the illuminating source are less stringent. The effect of the extension of the source on the resolution or image blur As of the reconstructed image can be seen from the following relations. As =

(2)

Ar,

(4.8)

\ I

I

1 p,

L

Q2

FIG. 1 1 . Arrangement to form an image plane hologram.

180

HOLOGRAPHY

where Ar is the extent of the source, Z, its distance from the hologram, and Z1the distance of a point on the object from the hologram. Further, the equation describing the effect of the finite bandwidth Ah on the resolution Au of the reconstructed image is given by the following relation: AU =

z1sin 0,

(y) ,

(4.9)

where Or is the angle of the reference beam with the normal to the hologram plane (the subject and image are considered to be on-axis). Equations (4.8) and (4.9) reveal that for small 2,,extension of the source and finite bandwidth have little effect. Thus reconstructions from imageplane holograms are sharp and achromatic even with white-light-extended sources. However, for image points which are far outside the plane of the hologram, the reconstruction shows color dispersion and blur. In conclusion, image-plane holography is particularly suitable whenever display of holograms, double-exposure interferometric holograms, or vibration-analysis holograms are desired in white light.

4.3.5.Color Holograms Multicolor holography was first suggested by Leith and Upatnieks6 The basic idea of color holography is to use three-color object and reference beams to record a hologram on a single photographic plate. If the object and the reference beams contain the three primary colors, the hologram would, when illuminated with a beam identical to the reference beam, reconstruct the object wave in full color. There are basically two main techniques to record color holograms ( a ) holograms recorded as planar diffraction gratings and (b) holograms recorded as volume diffraction gratings. The difference in response of the two grating types is made evident by considering the grating equations: d(sin i + sin 0) = A 2d sin i

=A

(planar grating),

(4.10)

(volume grating).

(4.11)

The planar grating equation reveals that for a given A, diffracted light would always be observed for any incidence. Thus the response is that broad-band and special techniques are required to separate or eliminate the spurious images which result from light of wavelength A l , diffracting from the component hologram recorded with the wavelength A2. The volume grating equation, on the other hand, states that the incident and diffraction angles are equal, and thus only wavelengths satisfying Eq.


181

(4.11) would be significantly diffracted. Thus the spurious images are suppressed by the filtering action of the volume grating. Further, there is no loss of resolution, contrast, or spatial frequency bandwidth. We now describe the various techniques used to record color holograms. 4.3.5.1. Plane Hologram Techniques. There are several basic techniques to make color plane holograms, as will be described next. ~ In the three-reference-beam method due to Leith and U p a t n i e k ~ ,the object beam is a mixture of three primary colors. The reference beam is a combination of three separate beams in the three different primary colors. Each reference beam is inclined at a different angle to the recording plate. In this way, the spurious or the cross-talk images resulting from diffraction of light of wavelength A , and from the component hologram recorded at wavelength A 2 are spatially separated. This method, however, is not very practical because the field of view is very much restricted. Furthermore, during reconstruction, the illuminating beam must be accurately aligned so as to be identical to the reference beam. The spatial multiplexing method is essentially due to Collier and Penningt01-1.~'Here, we spatially multiplex, in a nonoverlapping manner, the several holograms corresponding to the several colors used, i.e., the various colored interference patterns are not allowed to overlap on the recording medium. During reconstruction, the colored illuminating light is incident in such a way that each component hologram is illuminated with only that color light used to form it. There are several experimental arrangements which could be used to form such a hologram. One possible scheme is shown in Fig. 12. A mask F consisting of strips of red, green, and blue filters is imaged onto the hologram plane by a lens with the multiwavelength reference beam. The image, which consists of discrete areas of single-wavelength light, acts as

FIG.

12. Recording a color hologram by spatial multiplexing.

182

HOLOGRAPHY

the reference wave front. The light in each strip then interferes with the light of the same color coming from the object, giving nonoverlapping multiplicity of single-color holograms. The reconstruction step consists of placing the hologram in register and illuminating through the same filter mask. Some resolution is sacrificed in this method, since each component hologram will be smaller than the total recording. Also, this scheme results in the production of some noise, because in a A ,-component hologram there will also be light present of A2 and A 3 from the object, thereby decreasing the diffraction efficiency of the hologram ( A , , A 2 , A 3 are the three primary colors). The coded-reference-beam method, also due to Collier and Pennington,30 involves coding of the reference beam. The amplitude and phase of the reference wave are made to vary across the hologram plane in a significantly different manner for each wavelength. During reconstruction the developed plate has to be replaced precisely in the same position that it occupied during the formation of the hologram. A possible arrangement for recording such a hologram is shown in Fig. 13. The ground-glass screen D serves as the reference-beam coding plate. Each colored component of the reference beam, when passing through the diffuser, produces a complicated speckle pattern on the hologram plane. The component speckle patterns are randomly different from each other with respect to phase and intensity at a point, meaning that each colored reference wave is uniquely coded. Collier and Pennington have noted that when only light of wavelength 0.514 km was used, an excellent image was obtained. However, when the wavelength was changed by only 0.0128 km and the coding plate was illuminated in exactly the same way, only a uniform noise was observed.

FIG.

13. Recording a color hologram using a coded reference beam.

183


Successful holograms have been recorded using this method, although the noise problem still remains. The geometry of the code plate plays a major part in determining the degree to which the noise affects the reconstruction. The noise, which is principally due to the interaction of the wavelengths A , and A 2 , with the hologram formed with A 3 , is more uniformly distributed over the image area by increasing the solid angle subtended by the diffuse screen at the hologram. 4.3.5.2. Volume-Hologram Techniques. There are two kinds of volume hologram techniques, namely, transmission and reflection volume holograms, as will be described in this section. The transmission volume hologram technique was used to record the first mulitcolor hologram by Pennington and Lin3' in Kodak 649F photographic emulsion. Light of wavelength 6328 8, from a He-Ne laser was mixed with light of wavelength 4880 8, from an argon laser, using a beam splitter. Of the two doubly colored beams so produced, one acts as a reference and the other illuminates a color transparency. The reference and the subject beams then combine together at an angle of 90" with respect to each other at the hologram plane. As noted before, the interference is throughout the depth of the thick emulsion. An arrangement to record a three-color hologram devised by Friesem and Fedorowicz3* is shown in Fig. 14. The two three-colored wave fronts are generated in a similar manner as in the earlier r e f e r e n ~ e . ~A' better color fidelity is, of course, obtained in the reconstruction than with two-color wave fronts. Another technique uses the properties of the reflection holograms already discussed in Section 4.2.3. The hologram can be reconstructed with a multicolor wave front identical to the reference, or with white light as noted earlier.

L2

0

FIG. 14. Recording a color volume-transmission hologram.

184

HOLOGRAPHY

4.3.6. Composite Holographic Stereograms

An interesting application of holography is the three-dimensional display of an object by the use of two-dimensional pictures or projections. This field, known as composite holography, started in the late sixties with Redman," and De B i t e t t ~ . ~ ~ McCrikerd and George,33 George et Composite holography consists of producing a multiple hologram formed by many vertical strip holograms, each with a different paralax. The observer then obtains the stereo effect by looking at a collection of very thin vertical holographic windows. A different way to synthetize three-dimensional holograms from two-dimensional pictures has been described by Sopori and Chang3' and Haig.38 They multiplexed many images in a single plate by changing the angle of incidence of the object or the reference beams when making the hologram. , ~ ~and Koch? and Grossman4' applied composite Kasahara et ~ l . Groh holography to the three-dimensional display of internal human organs from ordinary X-ray projections. 4.3.7. Rainbow Holograms

White-light illumination of display holograms has proven to be so much more practical than laser or arc-lamp illumination, that most of the holograms that we see today are of some white-light type. One of these is the rainbow or Benton hologram. Rainbow holography was first reported by Benton' as a two-step process, but recent developments have simplified the procedure to one step with some limitations. Another development is the cylindrical holographic stereogram derived from the rainbow holograms by Cross." Below, we describe these three methods one by one. 4.3.7.1. Two-step Method. A conventional hologram H , is first recorded as shown in Fig. 15(a). The real image is then reconstructed by a beam that is antiparallel to the original reference beam. However, during reconstruction, instead of illuminating the whole plate, only a horizontal slot about 10mm wide is illuminated. A second hologram plate H 2 is placed near the real image, and a new hologram is recorded with a reference beam coming from the same side as the rays producing the real image [see Fig. 15(a)l. After development, the second hologram H2is illuminated by a white-light beam in a direction antiparallel to the reference beam. The object rays are retraced in the direction they came from, back to the position of the slit. In other words, the light is diffracted only towards the real image of the thin horizontal slot of the master hologram. Further, because of dispersion, we obtain many colored slit images one over the other (see Fig. 1%).


185

FIG. 15. Recording a rainbow hologram by the two-step method.

The presence of the slit makes a significant reduction in the information content in the second hologram, allowing a relaxation in the coherence requirements for the reconstructing light source. When the eye is placed at one of the real images of the slit, one sees the complete virtual image of the object in one color, depending on through which colored slit image one is watching. The holographic image of the object is almost monochromatic, and there is almost no blurring effect due to dispersion. Instead of a point light source, an incoherent vertical line source may be used to produce a continuum of vertically displaced horizontal strip images. Another advantage is that the reconstruction efficiency is very high, since most of the light is diffracted towards the observer’s eyes. A disadvantage is that the vertical parallax is sacrificed. This is not a serious drawback, as most of the time the horizontal parallax is sufficient, and it is the one that is necessary for stereoscopic vision. As noted earlier, in a Lippman-Denisyuk hologram, the colors are separated by an interference filter effect, and one single color is selected by the illumination angle. On the other hand, in rainbow holography, the colors are separated by diffraction, and only one single color is selected by the observation angle.

186

HOLOGRAPHY

4.3.7.2. One-Step Method. The one-step method was first introduced by Chen and Yu4* and was further developed by Chen.4'.44 Review papers on this subject have been presented by Chen et al.45and by Benton et al.46 The orthoscopic recording scheme is shown in Fig. 16(a). A horizontal slit A is placed on the lens which images the object near the recording plate P2. The reference-wave point source SLis located in the plane of the lens and is centered above the slit. The finished hologram, when reconstructed with the same reference wave, produces an orthoscopic image. But, the observer can only view a part of this image directly in front of the slit image; the rest is vignetted out and one must move about to sample it. Illumination with a white light source S,, however, produces a dispersed image of the slit, allowing the observer to see more of the image but in a spectrum of colors (Fig. 16(b)]. Chen and Yu42 have shown that by locating the slit between the object and the

FIG. 16. One-step orthoscopic recording scheme of a rainbow hologram ( a ) Recording; (b) Observation.


187

front focal plane of the lens, a monochromatic image can be obtained. This is because the image of the slit is formed on the same side of the plate as the observer, thus allowing the observer to place his eyes directly at the position of the slit image. But the size of the image becomes limited due to vignetting by the lens aperture. The pseudoscopic configuration is shown in Fig. 17(a). The geometry is almost similar to the one used for orthoscopic configuration, with the only difference that the reference wave is convergent instead of divergent. In the reconstruction step [see Fig. 17(b)], the conjugate illumination with white

.

\

\

'\

\ \

\

FIG. 17. One-step pseudoscopic recording scheme of a rainbow hologram. ( a ) Recording; (b) Observation.

188

HOLOGRAPHY

light is used to give real images of the slit through which the observer can see the image of the object. Unfortunately, the real image that is obtained positions the front of the object closest to the lens, farthest from the viewer. Thus the viewer is presented with the familiar “inside-out” or reversed-depth experience of a pseudoscopic image. However, there is an advantage that the image is bright and sharp and is undistorted and unvignetted by the lens aperture. Therefore, for applications where the depth perception is not important, the pseudoscopic configuration would be more desirable. The astigmatic recording scheme is yet another method for producing rainbow holograms in one step and produces a good orthoscopic image with large field of view in the horizontal direction. The recording is illustrated in Fig. 18. The cylindrical lens L, images the object 0 in the vertical direction near the hologram plate H. The slit S is also imaged but beyond the recording plate. A divergent beam is used as the reference wave front. The image of the object can be reconstructed with a white-light point source in the same position as the reference source. Since in the vertical direction the hologram is an image plane hologram, the image in this direction is sharp (see Section 4.3.4). It is worth pointing out that in the one-step rainbow process using conventional spherical lenses, the field of view is narrow in the horizontal direction, which is a major drawback. This is primarily because of the difflculty in getting a large aberration-free spherical lens. The astigmatic recording has removed this obstacle, since long convex cylindrical lenses are easily available. The lensless method of Bahuguna and Santoyo4’ is particularly suitable for transparencies. In this method, a master hologram of a diffuse slit is recorded. By using conjugate illumination a wave converging to a slit is reconstructed. This reconstructed wave is utilized to illuminate the object transparency for creating a rainbow hologram. Since the master hologram can be used for making a rainbow hologram of an unlimited number of transparencies, this method is essentially a one step method. A variation of the above method has been described by B a h ~ g u n a ~ ~ .

FIG. 18. Astigmatic recording scheme of a rainbow hologram.

DIFFERENT TYPES OF HOLOGRAMS

189

4.3.7.3. Rainbow Cylindrical Holograms. Lloyd Cross” invented the cylindrical holographic stereograms, utilizing the earlier work in rainbow holography invented by Benton,q and composite holography. Composite holography consists of making a motion picture film of a subject, placed on a slowly rotating table, with a cine camera. The individual frames of the film are hologramed on strips of about one millimeter wide and twenty cemtimeters long. The holographic film is then processed and bent into a cylindrical shape. When the observer views the film, each eye sees a photograph of the object taken from a different angle (just as with a stereo viewer), and consequently a three-dimensional image is perceived at the center of the cylinder as the observer walks around the hologram. An ordinary tungsten lamp with a transparent glass envelope is used as a light source. A machine to make this kind of hologram using anamorphic optics (plastic and holographic cylindrical lenses) was described by Fuesk and Huff .49 4.3.8. Computer Holograms A computer-generated hologram (CGH), also called synthetic hologram, is produced with a graphical output from a digital computer. The hologram is calculated from the knowledge of a wave front or the object to be represented. The actual hologram is made by photoreducing the computer graphical output on a photographic film or glass plate. The first CGHs were developed by Brown and Lohmann” with the purpose of making spatial filters for optical-data processors. Since then, the number of applications of CGHs have grown, and they may now be divided into the following five’areas:

a. three-dimensional image display b. optical-data processing c. interferometry d. optical memories e. laser-beam scanning All of these applications have been described in a very complete review by Lee.s’ Computer-generated holograms may be of three kinds: 1. Binary holograms. Most computer plotters may produce only highcontrast prints without gray tones. When the print is copied and minified by means of a high-contrast film, the resulting hologram is like a black screen with many small holes and is called a binary hologram, as described by Lohmann and Paris.” Efficient holograms have a total transparent area of about 50%.

190

HOLOGRAPHY

Binary holograms are analogous to normal holograms recorded on a very-high-contrast and nonlinear photographic film. Then, high-order images should be expected, but they do not overlap if the sampling theorem is obeyed. There are many applications for binary holograms, but their most interesting applications are probably in the field of optical testing, as shown by W ~ a n and t ~ loo ~ mi^.'^ Computer-generated holograms for this application are used to produce reference aspheric wave fronts in order to test an aspheric optical surface or lens. This kind of hologram looks like an interferogram, as shown in Fig. 19. 2. Half-tone holograms. The computer hologram may also be obtained by displaying the results on a suitable half-tone printout or on a highresolution television screen. This kind of representation approaches a real hologram. 3. Phase holograms (kinoforms). A kinoform, as described by Lesem et a ~ , is ~ ’a phase hologram obtained by the following method. A variabledensity display is calculated in such a way that after bleaching, the profile of the emulsion gives a phase hologram which reconstructs only one image.

FIG. 19. Binary computed holograms to test aspherical surfaces (from Ref. 53).

S O M E APPLICATIONS OF HOLOGRAPHY

191

Then, the picture is taken and bleached. Such a hologram is of interest, because the entire flux is concentrated in this image.

4.4. Some Applications of Holography There are numerous applications for holography. Most of them utilize and exploit the fundamental difference between holography and photography, namely that holography records the whole wave front, while photography is a recording of the irradiance distribution in the image. In this section, we deal with the more important applications. 4.4.1. Holographic Interferometry

Interferometry is a laboratory technique by which extremely small movements of objects are detected and measured by exploiting the interference properties of light. It can detect changes as small as a fraction of the wavelength of light. Conventional interferometry is mainly useful for making measurements on highly polished surfaces of relatively simple shape. Holographic interferometry extends the range to include three-dimensional surfaces of arbitray shape and surface condition. It is concerned with the formation and interpretation of the fringe pattern which appears when a wave generated at some earlier time and stored in a hologram is later reconstructed and made to interfere with a comparison wave. Holographic interferometry has been applied with success to test and measure aspheric optical surfaces, for example, by Broder and M a l a ~ a r a , ’ ~ FouCrC and M a la ~a ra ,~’ Malacara and M a l l i ~ k The . ~ ~ time-delay aspect makes the holographic method a very useful tool in many fields. For example, a machine part can be tested before and after stress. Further, due to the basic three-dimensional structure of the holographic reconstruction, a complex object can be examined interferometrically from many different perspectives: a single interferometric hologram is equivalent to many observations with a conventional optical interferometer. We shall discuss below the three basic forms of holographic interferometry. 4.4.1.l. Real-Time Holographic Interferometry. The hologram of the test object is first recorded in the usual way. The hologram plate is processed and put back in exact register on the plate holder, with the object and the reference beam undisturbed. The illumination can be adjusted so as to equalize the absolute values of the original and reconstructed object-wave amplitudes. Under the ideal conditions, the two wave fronts must exactly cancel, and the object should not be visible. In practice, at least one broad

192

HOLOGRAPHY

bright fringe is usually observed, because the processing of the hologram inevitably distorts the emulsion to some small degree. In the next step, the object may be placed under stress and caused to deform, or it may be released from stress and allowed to flow and creep or expand. As a result, the actual object wave differs at several points with the reconstructed wave, and the field of view is now crossed with fringes. Mathematically we can write the object wave in the initial and final state as Oo(x)exp(i+o(x)) and Oo(x) exp(+;(x)) respectively. The location of the fringes would then be defined by the locus of all points for which &(x) - 40(x)= const.

(4.12)

A dark fringe is produced whenever 4 & ( ~ ) - 4 ~ ( ~ ) = 2 ( m + l ) .m r r=;O , 1,2,....

(4.13)

Many authors, like Aleksandrov and B o n ~ h - B r u e v i c hEnnos,60 ,~~ Froehly et a1.,6' Tsuruta et S0llid,6~Molin and Stetson: and Hecht et ~ 1 1 . 6 ~ have worked on the problem of relating the observed fringes to the actual object deformation. According to Abramson,""-" the interpretation of the fringes is simple in most of the cases. His holodiagram allows measurement of deformation even for relatively complex and large objects. This technique and other methods, for example the method of Hecht et al.,6' have made holographic interferometry a very useful metrological tool for routine measurements. A spherical form of the single-exposure interferometer is shown and described in Fig. 20. First a hologram of a diffuser is taken [Fig. 2O(a)]. The hologram is then put back in the plate holder in exact register so that both the actual and the reconstructed waves superpose. If a test flat T is inserted in the object beam [Fig. 20(b)], interference fringes appear which D T

D

P

H (a)

P

H (b)

FIG. 20. Instrument to perform single-exposure holographic interferometry

SOME APPLICATIONS OF HOLOGRAPHY

193

are related to the optical inhomogeneities, wedge and thickness variations present in the test plate. 4.4.1.2. Double-Exposure Holographic Interferometry. When it is sufficient to form a permanent record of the relative surface displacement occurring after a fixed interval of time, a simpler technique, the doubleexposure method is employed. In this method, two exposures are taken, one for the initial state and the other for the final state of the object, on the same hologram plate. The interferogram so formed, yields the necessary fringes on illumination by the reference beam. Figure 21 shows an interferogram of an aluminum block tilted between the exposures. This method is similar to the single-exposure method in most respects. Unlike the previous method, exact manual registration of the reconstructed wave with the original one is not required. Further, distortion due to emulsion shrinkage is identical for both reconstructed waves and therefore does not affect the spacing of the fringes formed by the interference of the two waves.

FIG. 21. Holographic interferogram of an aluminum block, tilted between two exposures.

194

HOLOGRAPHY

4.4.13. Time-Averaged Holographic Interferometry. The multiple exposure interferometry can be extended to the limiting case of continuum of exposures. The resulting technique, known as time-averaged holographic interferometry, is used to study how surfaces vibrate. It was first reported by Powell and Stetson.” Suppose the object is vibrating in one of its normal modes. The timeaveraged hologram of such an object gives what is effectively a double exposure of the vibrating surface seen at either extreme of its oscillation. Thus, the object would appear crossed with fringes. Figure 22 shows some of the time-average interferograms of a 35-mm film can at its second resonance. Here the line across the middle of the can is a node of vibration of the can, and the contours to either side are loci of constant amplitude of vibration. Each bright area represents an amplitude of vibration of approximately a multiple of a half wavelength. 4.4.1.4. Sandwich Holography. The fringes in interferometric holography appear because the object either moved or changed its shape between the first and the second exposure. Many times, however, it is not easy to eliminate the effect of the movement of the object as a rigid body and to determine the object deformations due to loads or stresses. Once the second exposure is taken, the effect of the movement of the object as a rigid body can only be compensated if the two holograms were recorded with two different and separated reference waves. However, an elegant solution is the sandwich hologram invented by Abram~on.’*-’~ In sandwich holography, a pair of holographic plates are simultaneously exposed by placing them in the same holder, with their emulsions towards the object and with the antihalation backing removed from the front plate. Two sandwiches at least must be exposed, one without any stress in the object and the second after the stress is introduced, as shown in Fig. 23. Upon reconstruction, both sandwich holograms will reconstruct the object

FIG. 22. Time-average interferograms of a 35-mm film can at its second resonance (from Ref. 12).

195

SOME APPLICATIONS OF HOLOGRAPHY

A , A2

OBJECT

OBJECT

FIG. 23. Sandwich holography.

without any fringes. However, if the sandwiches are now formed by the pairs of plates A l with B2 or by BI with A * , the reconstructed image of the object will show the fringes. The tilting of a hologram laterally moves the image of the object when reconstructing the image of an object. However, if the sandwich pair is tilted, the images of both holograms move almost together, producing only a very small tilt of one image with respect to the other. This property of sandwich holograms permits compensation or simulation of small rigid movements of the body by means of small tiltings of the sandwich. In this manner, the fringe pattern due to the object stresses is easily isolated from any possible object rigid motion fringes. 4.4.2. Holographic Microscopy

The application of holography to microscopy dates back to the origin of holography itself. Gabor had intended to change the wavelength between formation of the hologram and reconstruction of the same in order to obtain a magnification in the ratio of the wavelengths. The magnification can also be obtained by scaling down the hologram, or by decreasing the curvature

196

HOLOGRAPHY

of the reconstructing wave as compared to the reference wave. The general formula for magnification is given by

M=

mL m2Zo I*--PZC

z,'

(4.14)

Z R

where the upper sign is for the primary image and the lower sign is for the conjugate image. Here m is the linear magnification of the hologram, is the ratio of the two wavelengths, and Z,, Z,, and 2, are the z-coordinates of the object-point, the reference-point source and the illuminating point source respectively. The above principles have been effectively utilized to form holographic microscopes. For details of the theory, construction, and working, the reader is referred to the literature, specially the papers by E l - S t ~ m El-Sum ,~~ and Kirkpatri~kB , ~ a~ e ~ Van , ~ ~Ligten and O ~ t e r b e r g and , ~ ~ Thompson et al.25 4.4.3. Imagery through Diffusing Media

The recovery of the image of an object obscured by a distorting medium, for example the distorting atmosphere, has been a subject of investigation for several years. One possible use for holography is to solve such problems. Let the object be obscured by a diffuser. To image the object faithfully, we proceed as described by Leith and UpatnieksBOA hologram of the object wave through the diffuser is taken with the help of a reference wave. The processed plate is put back in register, with the diffuser still in place. The hologram is now illuminated by a conjugate wave. This reconstructs the complex conjugate of the signal plus diffuser which, after passing through the diffuser, produces the conjugate real image, with the noisy part due to the diffuser having been filtered out. The explanation of the phenomena is as follows. The diffuser can be mathematically represented by the function exp[ i e ( x , y )]. During reconstruction, the conjugate real image of the diffuser, given by exp[-iO(x, y ) ] , is superposed on the diffuser, mutually cancelling the phase factors. It is as if the diffuser was not there, and we observe the real image of the object. In case the disturbance is time varying, the above method does not help. The possibility of a solution exists if the reference and the object wave are made to traverse essentially through the same part of the intervening medium, as explained by Goodman." Then, at a first approximation, the phase difference between the two waves at the hologram will be largely independent of the intervening medium, implying that the hologram is practically unaffected by the perturbing medium.

SOME APPLICATIONS O F HOLOGRAPHY

197

4.4.4. Matched Filtering and Character Recognition

An optical matched filter is a photographic recording of the Fourier transform of the amplitude distribution of the object. Since the amplitude as well as the phase are recorded, this is essentially a Fourier transform hologram of the object. The primary use of such a filter is character recognition, as shown by Vander LugtZ6and Vander Lugt er a/.'* The procedure is as follows. The filter, corresponding to the character to be read, is placed in the filter plane of a 4f-filtering system. The filtering process performs a cross-correlation between the object and hologram filter. In the image plane, we get a bright spot corresponding to the detected character. In case a set of characters are to be stored, the hologram filter consists of a number of holograms, corresponding to these characters, stored in a single master hologram.

4.4.5. Holographic Optical Elements and Aberration Corrections

Conventional optical elements like lenses and prisms may be replaced by holographic optical elements in many applications, with a lower cost, as described by Close." Computer-generated holograms may be produced to simulate and substitute aspheric lenses that produce complicated wave fronts. A holographic optical element may even be analyzed by conventional lens methods, like ray tracing. It has been shown by Sweattx4that the analogy is so close that a holographic lens may be considered as a normal lens with an infinite index of refraction. A hologram may also be used as an extra optical element in an optical system in order to correct some aberrations, as shown by Upatnieks et a/.,*' and to improve its performance. An obvious problem, however, is that the light efficiency of a hologram is, in general, quite low. A second problem is that monochromatic light must be used due to the color dispersion of the hologram.

4.4.6. Acoustical Holography

Most of the principles pertaining to optical holography are also applicable to acoustical holography. However, the recording and the readout techniques so far applied have been rather complex and difficult, and the results have been of a poor quality. But, due to the large number of interesting applications, it is hoped that new and simpler techniques will evolve in the near future.

198

HOLOGRAPHY

There are principally four methods for recording an acoustic hologram: a. recording by the surface of a liquid

b. recording by a matrix of detectors c. hologram is obtained by scanning, pointwise

d. hologram is obtained by direct interaction of light and acoustic waves In this section, we only deal with the first method. For details and other methods, the reader is referred to the literature like the papers by Mueller and Sheridon,86 Young and Wolf,87 and Preston.88 Figure 24 illustrates the geometry for recording and reading such a hologram. S , and S2 are the two coherent ultrasonic sources. S , produces the reference wave and S2 illuminates the object A. The reference wave and the scattered wave from the object interfere on the surface of the liquid. The information about the object is in the form of ripples on the surface of the liquid. In the reconstruction step, the distorted surface is illuminated by a parallel beam of monochromatic light, and the scattered light wave is collected by the lens, as shown in Fig. 24. The image of the light source is blocked by a small stop. The two reconstructed images A' and A" appear very near the focal plane of the lens. The large differencebetween the recording ultrasonic wavelength and the illuminating light has two deteriorating effects: ( a ) the images are formed far from the surface, ( b ) the images are greatly aberrated. In order to

s,

s,

FIG. 24. Recording a reconstruction of an acoustic hologram

EXPERIMENTAL PROCEDURES I N HOLOGRAPHY

199

eliminate these effects, the photographic record of the distorted liquid surface is taken with a lens of sufficiently small aperture and later reduced by the ratio of the two wavelength. This photograph can now serve as a hologram to obtain an optical reconstruction of the acoustically illuminated object.’“ Acoustical holography is finding several applications in various areas as described by Green,’g and by Hildebrand and Brenden.” Since many objects apaque to light waves are transparent to ultrasonic waves, they can be optically examined through ultrasonic holography. Because of this feature, it is also applicable to nondestructive testing, as shown by Byron and Collins,” and to medical imaging, as shown by Malacara.’’

4.5. Experimental Procedures in Holography The basic requirement in holography is to record a stable fringe pattern due to the interference between two light beams. In order to achieve this, certain procedures are to be followed in general. In what follows, we discuss these together with the necessary equipment required to produce a good hologram. 4.5.1. Light Sources for Holography

The requirements placed on the light sources to be used in hologram formation depend on the subject and the experimental setup. Specifically, the light disturbances reaching any point on the hologram plane (via reflections from the subject surface or reference mirrors) must be mutually coherent. In other words, the maximum optical path difference for light rays between the source and the hologram via the above-mentioned path must be less than the coherence length. Besides, the object and the reference waves must be spatially coherent. Poor coherence would effect the fringe contrast and thus the diffraction efficiency. Fortunately, with the availability of the laser, the above requirements are easily achieved, and thus making a hologram has become a routine task in the laboratory. It is, however, worth noting that Leith and Upatnieks,’3 by using an achromatic fringe system, have produced holograms with a high-pressure mercury arc, although of a lower quality than those made with a laser source. During reconstruction, the restrictions on the source depend on the type of the hologram. For example, an image-plane hologram, a rainbow hologram, or a volume hologram can be reconstructed with a n extended white-light source. For best results, however, the reconstructing wave must be identical to the one used during formation of the hologram. In general,

200

HOLOGRAPHY

deviation from this condition and poor coherence of the source affect the resolution in the image. Normally, for a small object of a few centimeters in size, a 2-mW He-Ne laser is quite adequate in terms of the required power. For larger objects, one can use a higher-power He-Ne laser. If still higher power is needed, one may want to use an argon laser which gives output in the range of a few watts. To study transient phenomena, Brooks er have shown that a pulsed laser is a must.

4.5.2. Mechanical Stability

There are essentially two stability conditions that must be met to record a fringe pattern. We discuss them in the following two sections. 4.5.2.1. Stability of the Recording Plate. The recording plate must be stable relative to the fringe pattern in order to record the pattern faithfully. Its stability requirements depend on the highest frequency content in the fringe pattern. In most holograms, the minimum fringe spacing is approximately equal to the wavelength of the forming light. So the recording medium must not move more than a small fraction of a wavelength during exposure. 4.5.2.2. Stability of the Fringe Pattern. Special efforts have to be made to ensure the stability of the fringe pattern. The whole optical setup must be isolated from external vibrations. An inexpensive method is to mount the optical setup on a massive slab of iron or concrete (weighting about 0.5 tons) supported on partially inflated motorcycle tubes. The resonance frequency must be much lower (about 1 Hz) than the frequency of the building vibrations. Sophisticated ready-made vibration-free tables are available from the Newport Corporation. Acoustic and thermal disturbances are more troublesome. The best way is to eliminate the sources of such disturbances themselves, for example switching off air-conditioning units, fans, and heaters, etc. Reducing the exposure time is also advantageous, but at the expense of having a more powerful laser. One must also reduce, as far as possible, the optical path lengths, especially those between the beam splitter and the recording medium.

4.5.3. Beam Expansion

The laser-output cross-section is typically 1 mm in diameter to illuminate the subject. While making a hologram, one needs a larger cross-section, and hence the need of expanding the beam arises. The beam can be expanded by one or more lenses or spherical mirrors.

201

EXPERIMENTAL PROCEDURES IN HOLOGRAPHY

Because of the diffraction effects at the walls of the laser cavity, the TEMoo mode (the usual mode of operation) may be modulated, giving a speckled appearance to the expanded beam. This is especially the case when the laser resonant cavity is formed with two large-radius mirrors. One can get rid of this modulation by focusing the laser output through a microscopic objective and putting a spatial filter (a fine pinhole) of diameter between 10 to 40 pm, depending on the magnification of the microscope objective being used, at the focus of the beam. The output is a clean expanded beam. 4.5.4. Holographic Materials

The subject of holographic materials is so important that complete books have been written about it, for example, by Smith.95Good chapters in books covering this topic have been written by Hariharan” and by Gladden and L e i g h t ~ Additional .~~ general references are given in this last chapter, an interesting reference being that of Bartolini et aL9’ 4.5.4.1. Photographic Emulsion. Silver halide photographic emulsions are by far the most popular of all recording media. They are available in film or glass plates, mainly from Kodak or from Agfa-Gevaert. Their spectral sensitivities are of different types, as shown in Fig. 25 and Fig. 26. With

Y

a

5

a;

1000

e m

-

Moo-

swo

4000

4MK)

1100

eooo

0100

Too0

7100

WAVELENGTH OF LIGHT

F i c . 25. Spectral sensitivity curves for some Kodak holographic silver halide emulsions.

202

HOLOGRAPHY

7

WAVELENGTH OF LIGHT

FIG. 26. Spectral sensitivity curves for some Agfa-Gevaert holographic silver halide emulsions.

them it is possible to match most laser wavelengths. Table I summarizes some of the most important characteristics of these photographic materials. For ordinary practical purposes, the optical density versus log E (exposure) is the characteristic normally specified. For holographic purposes, the transmittance t versus the exposure E was originally suggested; however, from the point of view of diffraction efficiency, it is more important to plot the transmittance t versus the logarithm of the exposure. These curves for some holographic photographic emulsions are given in Fig. 27. Glass photographic plates are available with and without antihalation backing. If desired, the antihalation backing may be removed with alcohol. 4.5.4.2. Dichromated Gelatin. The most important feature of hardened gelatin is that it is not dissolved in water, but it only swells three to four times its original thickness. If a gelatin film contains a small amount of dichromate((NH.&CrzO,), when exposed to blue light, the Crh' ion is reduced to Cr3+,making the gelatin even harder, as described by ChangY8and Curran and Shankoff." If the plate is adequately processed, as shown for example by Hariharan," a modulation of the thickness as well as of the refractive index is obtained, producing in this manner a highly efficient phase hologram.

203

EX PER1 M E N T A L P R O C E D U R E S I N H O L O G R A P H Y

TABLE1.

Silver Halide Films and Plates for Holography

TYPe Plates

Films

Spectral Sensitivity

Resolving Power in I/mm.

Emulsion Thickness in pm.

649 F SO-I73 SO-424 SO-453

Panchromatic Blue and Red Blue-Green Blue-Green

2000+ 2000+ 1250 1250

17/16 6 7/3 9

10E56 10E75 8E56 H D 8E75 H D

Blue-Green Panchromatic Blue-Green Blue and Red

1500f 1500+ 2500+ 2500+

7 7 7 7

Kodak 649 F 120-01/02 125-01/02 131-01/02 Agfd-Gevaert

10E56 10E75 8E56 H D 8E75 H D

EXPOSURE

IN

ERGS/cm'

FIG. 27. Curves of amplitude transmittance versus exposure, in a logarithmic scale, for some holographic silver halide emulsions, using a wavelength of 6328 A.

204

H 0 LOG RAPH Y

4.5.4.3. Photoresists. The photoresists are resins that, like dichromated gelatin, are hardened by the action of the light. Their use in holography has been described in several articles by Bartolini100-'02and by Norman and Singh.Io3 Blue or green light produces chemical changes in the resin that make it more difficult to dissolve with a solvent. Phase-diffraction gratings or holograms may be produced with photoresists, but unlike those produced with dichromated gelatin, no modulation of the refractive index is obtained, but only of the thickness. The developer is a suitable solvent that dissolves the unexposed region faster than the exposed ones, or vice versa, depending upon whether the photoresist is negative or positive, respectively. A positive photoresist should be used for holographic purposes. Shipley AZ-1350 has been the most widely used photoresist, but recently it has been replaced by Shipley Microposit 1350 J photoresist. Since the sensitivity of this photoresist is greater for shorter wavelengths, blue light for a He-Cd laser or green light from an argon laser should be used. If linearity is desired, the appropriate development procedure should be performed, using a developer different from that used for the normal uses of the photoresist, as described by Norman and Singh."' 4.5.4.4. Other Materials. Besides the materials just described, there are various other recording materials, for example: photochromics,'04~'09 photopolymers,' ' O J ' ~ thermoplastics,' l 4 electro-optic crystaIs,"5-'20 magnetooptic materials,'2"'22 metal films,123and ferroelectric m a t e ~ i a 1 s . I ~ ~

4.5.5. Recording, Processing, and Bleaching

These three basic steps in hologram-making using photographic emulsions have been described in detail by Benton'" and will now be considered. 4.5.5.1. Recording. There is no universal recording material that is suitable for all of the varied applications for holography. However, the photographic emulsion is still the principal recording medium, as it comes close to having ideal properties for a wide range of requirements. The most essential characteristic of the recording medium is that it should be capable of recording the generally high frequencies encountered in holography. For example, for an angle of 30" between the object and the reference beams, the frequency is about 1000 cycles/mm, which is beyond the resolution limit of most of the recording materials. Besides this, the medium must be reasonably sensitive to the particular radiation involved. 4.5.5.2. Processing. The following procedure, not very different from processing of the high-resolution photographic image, is recommended for the exposed holographic plate:

EXPERIMENTAL P R O C E D U R E S IN HOLOGRAPHY

205

1. Development: 5 minutes continuous agitation in Kodak D-19 developer at 20°C, followed immediately by a 1-minute rinse in water. 2. Fixation: 5 minutes continuous agitation in Kodak rapid fixer, followed by a 1-minute rinse in water. 3. Removal of residual fixer: I f minutes in Kodak hypoclearing agent. 4. Wash: 5 minutes in flowing water. 5 . Removal of dye-sensitizer: 5 minutes in methanol, followed by a l-minute rinse in water. 6. Drying: Emulsion shrinkage is a real problem in holography. A onewavelength shrinkage can cause a 180O-phasereversal of the reconstructed wave front, a disadvantage for interferometric applications. Also, there is a color shift in reflection holograms when reconstructed in white light (see Section 11.2.2). Some special measures are to be taken in order to avoid emulsion ~ h r i n k a g e . ' ~ . ' ~ 4.5.5.3. Bleaching. An absorption hologram recorded in a photographic emulsion is converted into a phase hologram by bleaching out the silver image, thus obtaining higher diffraction efficiencies. There are essentially two methods, as described in detail by Benton'" which are: ( a ) Direct bleaching and (6) Reversal bleaching. In the direct bleaching method, the film or plate is developed and fixed as usual. It is then put in a bleach bath which converts the metallic silver into a transparent insoluble salt having a refractive index significantly higher than that of the gelatin. One of such agents is potassium ferricyanide, as described by Burckhardt and Doherty.'" The hologram efficiency of this and other similar processes is analyzed in a paper by Upatnieks and Leonard,'27 showing that it is between 25% and 50%. Higher efficiencies, up to 70%, may be obtained with a wet bromine process described by Lehmann er al.,'" or with a dry process using bromine vapor, as shown by G r a ~ b e .Unfortunately, '~~ holograms bleached by these methods are associated with flare light due to the enhancement of the low spatial frequencies. This is due to the fact that the resulting relief images augment the index images. The noise can be substantially reduced by substituting Kodak D-76 for the usual D-19 developer. It can be further reduced by immersing the hologram in an index-matching liquid gate or by coating the hologram surface with an index-matching film. The reversal bleach method described by Smith'30 overcomes the noise problem associated with the previous method by producing relief images which tend to cancel the index variation at low spatial frequencies. Here, the plate is developed but is not fixed. It is then put in the R-9 bleach bath, which converts the metallic silver to a soluble salt and removes it entirely from the emulsion. The result is that the remaining silver halide appears

206

HOLOGRAPHY

reversed with respect to the case of the direct bleach method, the relief however remaining at regions where development took place. Thus the relief image tends to cancel the phase variations due to the silver halide, and consequently the speckle noise is reduced. 4.5.6. Embossing of Holograms

Phase photoresist holograms may be replicated by a process similar to the one used to make photographic records, as described by Bartolini ef C I ~ . , ’ ~ ’ Clay , ” ~ and Gore,’33and by Burns.’34 The process consists first of coating the photoresist hologram with a thin layer of nickel or silver either by vacuum evaporation or by an electroless method in order to make it conductive. The second step is to deposit on top of this metallic film a thicker layer of nickel by electrolysis to form a metal stamper master. The next step is to emboss this master into a transparent flexible thermoplastic like PVC or polycarbonate that softens at a temperature far from that of thermal decomposition, as shown by Clay,’35to produce the desired replicas. Thousands of holograms can be produced with a single nickel master, at a very low cost for each copy. Acknowledgments The authors want to thank Prof. N. Abramson for his very valuable comments, Mr. Fernando Mendoza for his help with the laboratory work, and Mr. Krishna Morales for the drawings in this chapter.

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Suppression of Photochromic Effects in Bleached Silver Halide Holography,” Appl. Opt. 9, 1948 (1970). 129. A. Graube, “Advances in Bleaching Methods for Photographically Recorded Holograms,” Appl. Opt. 13, 2942 (1974). 130. H. M. Smith, “Photographic Relief Images,” J. Opt. SOC.Am. 58, 533 (1968). 131. R. A. Bartolini, W. J. Hannan, D. Karlsons, and M. J. Lurie, “Embossed Hologram Motion Pictures for Television Playback,” Appl. Opt. 9, 2283 (1970). 132. R. A. Bartolini, N. Feldstein, and R. J. Ryan, “Replication of Relief-Phase Holograms for Prerecorded Video,” J. Elecrrochem. SOC.120, 1408 (1973). 133. B. R. Clay and D. A. Gore, “Holographic Moving Map Display,” in Coherenr Oprics in Mapping, Proc. SOC.Phot. Opt. Instr. Eng. 45, 149 (1974). 134. J. R. Burns, “Large Format Embossed Holograms,” in Applications of Holography, Proc. SOC.Phot. Opt. Instr. Eng. 523, 7 (1985). 135. 8 . E. Clay, “Two-Dimensional Displays,” in Handbook of Optical Holography (H. J. Caulfield, ed.), Academic Press, New York, 1979.


5. PHOTOMETRY AND RADIOMETRY William L. Wolfe Optical Sciences Center University of Arizona Tucson, Arizona 85721

5.1. Introduction The practice of radiometry deals with the calculation of radiation from objects of interest, in optical systems, and on detectors, and with the measurement of various radiometric quantities on both a relative and absolute basis. The calculation of radiative transfer alone has been the subject of several monographs (e.g., Chandrasekhar’), and the measurement of radiation has likewise been the subject of entire books (e.g., Bauer,’ and Smith et d 3 )This . can only serve as an introduction to the field in general and descriptions of several topics of special interest. All equilibrium radiation problems can begin with a consideration of the famous blackbody distribution first described by P l a n ~ k .The ~ angular, spatial, and areal characteristics of real bodies can then be expressed in terms of an efficiency factor, the emissivity, that multiplies a blackbody function to obtain the true flux characteristics. Several important laws of radiation distribution and transfer apply to the blackbody idealization, and most experimental situations can be modeled with only minor variations from these laws. Accordingly, the first portion of the chapter is devoted to blackbody laws, partly because many of the variations and forms of this law are not widely known. An area of radiation transfer that is becoming increasingly important is transfer in partially transparent media. The new techniques of sounding the atmosphere from satellites to infer temperature and of probing media like molten glass depend upon this theory. It also gives some insight concerning the nature of Lambertian radiators. Temperature measurements made with multispectral radiometers have received new emphasis from the satellite techniques of reconnaissance with “crude spectrometers.” The techniques have application in areas other than those of assessing crops and drainage patterns and may prove to be a most powerful and practical radiometric technique. The section on multicolor 213 METHODS OF EXPERIMENTAL PHYSICS Vol. 26

Copyright 01988 by Academic Press. Inc. All rights of reproduction in any form reserved ISBN 0-12-475971-8

214

PHOTOMETRY A N D RADIOMETRY

techniques provides some new results or perhaps an enhanced understanding of old techniques applied to new problems. Last and maybe least is the question of nomenclature. There has been confusion in the field of radiometry. Some of it is due to the many different fields in which radiometric measurements are made and discussed, and some of it is apparently due to the fact that photometry preceded radiometry just as the awareness of light preceded that of infrared and ultraviolet radiations (Herschel,’ and Ritter6).

5.2. Symbols, Units, and Nomenclature Table I presents a list of all the symbols used in this chapter along with their definitions and units associated with them. The units are given not just to define the dimensionality, but they are intended to help clarify the meaning. SI units are generally used, although frequently used units like centimeters and micrometers are indicated rather than meters where it is appropriate. The symbols are listed alphabetically, but the ensuing discussion treats them topically. Perhaps the simplest of the variables to be discussed is the independent spectral variable. Most of us think in terms of the optical wavelength A of the radiation to be measured. It is also useful to think in terms of the frequency which can be expressed in several ways. The wave number i j is just the reciprocal of the wavelength. The frequency v is the speed of light times that ( v = c/A), and the radian frequency w is 27rv. Similarly, k, the radian wavenumber is 27rl or 2 7 r l A . The nondimensional frequency x is c2/AT (where c2 is the second radiation constant defined below, and T is the temperature), a convenient independent variable for many calculations. The spectral distribution of blackbody radiation can be given in terms of any of these variables. In order to treat this somewhat generally, the symbol y is used to represent any of these spectral variables. Radiation transfer is ultimately expressed in terms of a quantity like power, the time rate of energy. An equivalent concept is the time rate of the transfer of a number of photons or of the amount of power which evokes a visual response. Each of these is a flux and is denoted by the symbol P. To distinguish between them, the subscripts u, q, and u respectively, are recommended by the International Illumination Society.’ I have not used a subscript when the symbol can refer to any of these or when it refers specifically to energy transfer. I believe, no confusion will result. The rest of the symbols which are used to describe radiative transfer are L, M , E, I, and S. These are, respectively, the radiance or flux per unit normal area and solid angle; the irradiance or incidance or incident flux per unit area;

S Y M B O L S , UNITS, A N D NOMENCLATURE

215

the emittance or exitance (emitted flux per unit area); the intensity or flux emitted per solid angle-generally from a subresolution source; and the radiance per unit path length in a medium which scatters or is a volume emitter. This is also called the volume radiance. In several sections I have dealt with irradiance and electric field strength in the same equation and have used U for the electric field. These equations have also involved m U U * for which I have used 9.I would prefer using I and E for these, and J and H for intensity and irradiance, but that would be contrary to the recommendations of the International Commission on Illumination (CIE). Several writers have proposed nomenclature schemes to make these definitions logical, consistent, and easy. Table I1 is a summary of these. The fluometry scheme of Jones has as its main feature the unifying concept that these flux terms are really all geometric concepts. He therefore used sterance, incidance, exitance, intensity, and sterisent as generic terms to be modified by “radiant,” “photon,” and ‘‘luminous.’’ Nicodemus,’ largely in an effort to overcome the ambiguity of “intensity” (it is variously used to mean E, M , I, and L by different scientists), has coined the geometric terms of areance, and pointance for E or M and I, respectively. Worthing,” years ago proposed that “-ancy” be used to mean “per unit area.” He went further and proposed that the transmission, absorption, and reflection properties be specified with “-ance” endings when a sample is specified, and “-ivity” endings when a substance is specified.” This, of course, makes emittance an ambiguous term if one uses the “conventional” terms. The nomenclature system has not yet been proposed. The concept of path radiance may be somewhat foreign to some readers. In a scattering medium or volume radiator the radiance is not constant. It can increase with path length, because each element of path contributes more radiation to the total. It thus has units of W cm-* sr-’ cm-I, where the final cm-’ is the (reciprocal of the) unit of path length. There is no recommended symbol for this. I have chosen S for this quantity because it agrees with “sterisent,” and many other appropriate letters have been preempted. I have also replaced with P (to stand for phlux) for ease of typing. Finally, to indicate any of these radiometric quantities, I have used the symbol R. The atomic constants are listed in a conventional way: cI and c2 are the first and second radiation constants, equal to 27rc2h and hc/ k g , respectively. The speed of light in vacuo is c; Planck’s constant, h ; the Stefan-Boltzmann constant a ; that for photon distributions, aq;and the Boltzmann constant k g (because k is the radian wave number). Some of the common symbols are x, y, and z for general distances. These do not get confused with the spectral variables. Angles are denoted by 6 and 4 ; solid angles by R; areas by A; time by 1 ; temperature by T ; integers by m, P, 4.

216

PHOTOMETRY A N D R A D I O M E T R Y

TABLE1. Symbols, Units and Nomenclature Symbol a, b, c

A

B BRDF b C C CI c2

D D*

E F

f g

h

I K

Kt,

k k, L I

M m. P, 9

N NEFD NEAL n

P R 9 r

S SNR T I

U U

V V(A) u W X

Y I

Definition constants area effective bandwidth Bidirectional Reflectance Distribution Function base of a prism constant speed of light in a vacuum first radiation constant second radiation constant diameter specific detectivity irradiance or incidance focal ratio focal length o r area factor constant ( 2 or 4 ) Planck's constant or height intensity luminous efficacy radiation interaction factor radian wavenumber Boltzmann's constant radiance length radiant emittance or exitance integer constants number of photons noise equivalent flux density noise equivalent change in radiance refractive index power or flux general radiometric variable or range responsivity radius volume radiance signal-to-noise ratio temperature time electromagnetic field quantity ( E o r H) normalized nondimensional frequency volume or voltage or voltage ratio visibility curve velocity width nondimensional frequency or coordinate length general spectral variable or coordinate length coordinate length

Units

mZ Hz sr ' m

m s-'

w cm2 s-I cm K m cm Hz'/'W w cm-:

I

-

m or

-

-

w s2 W sr-' ImW

J K-' -

'

Wcm 'sr-' m Wcm W cm-' W cm-'sr-'

'

-

W various VW-' or AW-' m W cm-'sr-' -

K S

v m-'

or

v

-

m' or

v or -

-

m

SCI

m

- or m various or m m

217

SYMBOLS, UNITS, A N D NOMENCLATURE

TABLEI. (coni.) Symbol a

Y E

1)

1)"

8 A

v v P U

u(A) T

@

R w

Definition absorptivity or absorptance absorption coefficient emissivity quantum efficiency luminous efficiency angle wavelength frequency wavenumber reflectance or reflectivity Stefan Boltzmann constant slit function transmittance or transmissivity angle solid angle radian frequency

Subscripts a, b

a C

d e h I

q S

U V

1.2.i W

indicating different spatial points atmosphere collector or optical element detector emitted hemispherical incident photon source energy visible first, second ith of a group infinite (number of reflection)

Superscripts BB h m 0

P -

blackbody hemispherical maximum, molecular optics projected time rate weighted average

Units

TABLE11. Radiometric Nomenclature ~~

Quantity Flux Density [Wm-*] Incident Exiting Flux per solid angle [w sr-'] Flux density per solid angle [w sr-' m-*] Volume Radiance

~

~

Jones

Nicodemus

Worthing

Incidance Exitance Intensity

Areance Incident Areance Exitent Areance Pointance

Radiancy Irradiancy

Sterance Sterisent

Sterance

-

Field Theorists

Meteorologists

Intensity

Flux Density

-

-

-

-

-

-

-

-

> z 0

Specific Intensity

-

rn -I P

5 mW/cm2), several methods are available for direct visualization of the radiation pattern in spectral regions where photographic film is not sensitive. Liquid crystals are cholesteric compounds whose structure varies with temperature, often in a small temperature range. Absorption of radiant energy by a thin liquid-crystal layer will raise its temperature and cause a change in the wavelength of light reflected from the crystal surface. Observation of this change provides a means of monitoring the spatial and temporal behavior of a radiant source, often an infrared laser. Liquid-crystal displays have been used to monitor laser mode patterns and infrared interferograms at wavelengths where conventional photographic plates fail to respond.28 Another widely used method of observing the spatial variation of infrared radiation is through thermal quenching of visible l u m i n e ~ c e n c eIf. ~a~ZnCdS phosphor is excited by an ultraviolet lamp, visible yellow fluorescence is

303

PHOTON DETECTORS

TABLE11.

Thermal Detectors

Operating Temperature Detector

( O K )

NEP(Wj

R" (V/W)

D* (cm Hz'"/W)

T

(secj ~

Thermocouple BoI ome t e r Carbon Bolometer Ge Bolometer Si Bolometer TGS Pyroelectric SBN Pyroelectric Golay Cell

3 00 3 00

2.5~ lo-"'

X I

1 I x 10' 2 x 104 1 x 10' 1 x 10' 1x10~

5x10-I'

2x10'

L

Ix

2

5 x lo-'' 5 x 10-I)

2 3 00 3 00 3 00

10-l'

3x10' 2 x IOU 4.5 x io10 8 x 10"

3x 1x 1x 1.ox 2x 1 x loy 1x 5x10" 1x 1.6~10~ 1x

lo-' lo-' lo-) lo-'

induced which can in turn be quenched wherever the phosphor is locally heated by radiant energy. Thin plastic films can be coated by a lacquer bearing the phosphor, after which the film is mounted on an aluminum heat sink to provide units with sensitivities of approximately 100 mW/cm* and a response time of 0.1 sec.3"

6.3.6.Summary The performance of a number of different thermal detectors is summarized by the characteristics shown in Table 11. The room-temperature thermal detectors, such as thermocouples and bolometers, are relatively slow and exhibit moderately good detection sensitivity as measured by NEP and D*. The cooled bolometers are significantly more sensitive, have better response times, but are vastly more difficut to use. The pyroelectric detectors exhibit a sensitivity similar to the other uncooled thermal detectors and are becoming the standard room-temperature sensor for energy and power measurements.

6.4. Photon Detectors Optical radiation in the visible portion of the spectrum is most frequently detected by a photon detector where the incident photons directly interact with electrons in the detector medium to produce an electrical signal proportional to the intensity of the radiation. There are three basic processes which may be used to categorize the majority of practical photon detectorsthe photoemissive effect, the photovoltaic effect, and the photoconductive effect. Since these processes directly convert radiation into electrical signals by a quantum (rather than thermal) process, detectors of this class are

304

DETECTORS

typically characterized by high sensitivity, short response times, and strong spectral variations in the response to radiation. 6.4.1. Photomultiplier

One of the most important direct photon effects used to detect radiation is the photoemissive effect, also known sometimes as the external photoeffect. Photoemissive devices are particularly well suited to detection of low-intensity radiation and offer major advantages in detection of fast low-level signals. In a photoemissive detector, the action' of incident radiation is to induce emission of an electron from the surface of a photocathode into space, where it is subsequently collected by an anode. There are a number of devices in which the photoemissive effect is applied for sensing in a simple photomultiplier or for imaging radiation in image intensifiers or image tubes, primarily in the visible spectrum. The most commonly employed device is the photomultiplier, where the photoemitted electrons impinge upon a sequence of electrodes called dynodes, which act as secondary electron Each electron which is incident on a dynode induces emission of one or more additional electrons, so that an amplification or gain mechanism exists for the production of large numbers of electrons. The spectral properties of the photomultiplier are primarily determined by the photocathode. Photocathodes are generally thought to be of two types, conventional photocathode^^^ and so-called negative electron affinity cathodes.33734The two types of cathodes differ from each other in the magnitude of the photoelectric work function associated with the energy required to remove an electron from its equilibrium thermal level inside of a solid to the energy level above the potential barrier at the solid surface which allows the electron to escape from the solid. The minimum energy which a photon must possess to induce photoemission equals the work function, 4, of the metal [Fig. 6(a)]. Within the category of conventional photocathodes, both metals and semiconductors are used as electrode materials. The energy band diagrams associated with the photoemissive properties in these types of materials are illustrated in Fig. 6(a) and (b), respectively. In the conventional semiconductor electrode, the minimum energy required for a photon to induce photoemission is still that quantity which will raise the electron to a level higher than the potential barrier at the surface of the material [Fig. 6(b)]. Semiconductor photocathodes with a positive electron affinity generally require less energy than metallic photocathodes and hence have spectral responses which extend to somewhat longer wavelengths than conventional metallic photocathodes. In either case, these photocathodes can respond to radiation of wavelengths extend-

305

PHOTON DETECTORS

Metal

Vacuum

(a) Semiconductor

Vacuum

-! \

Photoexcitation Conduction band Fermi level Valence band

FIG. 6. Energy band diagrams associated with the photoemissive properties of photocathodes. ( a ) For metals, and (b) for semiconductors.

ing only from the visible spectrum into the near infrared. These material limitations, which have been encountered over most of the history of the use of photomultiplier tubes, have permitted only very marginal response at wavelengths as long as 1 p. In the high-energy ultraviolet part of the spectrum, the tubes often do not respond to wavelengths shorter than 0.2 p as a result of the transmission spectra of typical window materials used in their construction. There have been some extensions of the response to near 0.1 p by using rather delicate materials, such as lithium fluoride, and even further extension of their response into the X-ray spectrum by using very thin wafers and material such as gallium arsenide. An alternative procedure that is used to produce some ultraviolet response in photomultiplier tubes is to coat the entrance window of the tube with an ultraviolet-sensitive material which is capable of fluorescing in a region where the tube does respond sensitively. Materials such as sodium silicate have been used to produce relatively short wavelength response in conventional tubes by using this approach. In recent years, many advances have been made in increasing the sensitivity in the long-wavelength portion of the spectrum and extending that response to the wavelengths as long as 1.5 p.33-36 Prior to this work, photo-

306 Semiconductor

Vacuum

Conduction band Photoexcitation + Fermi leve Valence band

FIG. 7. Energy-band diagram associated with the photoemissive properties of photocathodes with sensitivity in the infrared. (After K r ~ s e . ~ ’ )

emissive detectors were not available for this portion of the spectrum. The extension of spectral response of photomultiplier tubes into the infrared has resulted from the discovery of the negative electronic affinity photocathode. It was discovered that by overcoating the surface of certain p-type semiconductors with evaporated layers of low-work-function materials, the resulting structure has a negative electronic affinity. This is illustrated in Fig. 7. Here, in contrast to the conventional photocathodes previously described, the incident photon must have an energy which only equals or exceeds the energy gap, E,, of a semiconductor to allow photoemission. Hence, by judicious choice of the semiconductor and associated coating, it is feasible to construct photocathodes where the spectral response does extend well into the near infrared. Even here, however, the quantum efficiency does diminish at increasing wavelengths, so that there is yet a relatively severe limitation on use of photomultiplier systems at wavelengths exceeding 1.2 p. The conventional photomultiplier tubes are usually based on a photoemissive structure where a thin evaporated layer composed of a compound including some alkali metal (with the predominate example being that of cesium), metallic elements from Group V of the periodic table, such as antimony and other species, are combined in a ternary or similar alloy. Examples of the spectral response of representative conventional photocathode?’ are shown in Fig. 8. These show some of the conventional materials designated by the typical nomenclature referring to the photocathodes as S1, S17, and S20. For example, the S1, which is silver (Ag) on cesium oxide (CsO),has some response to wavelengths as long as 1.1 p. The conventional photocathode with the most sensitivity in the red portion of the spectrum is the so-called S 2 0 , composed of cesium-sodiumpotassium-antimony [ (Cs)Na2KSb].This system is able to respond, at least weakly, to almost 0.8 p. Some extended response versions of the S 2 0

307

PHOTON DETECTORS

s

1 0.4

1

1

1

0.6

I 0.8

Wavelength

1

I 1

(rm)

FIG. 8. Spectral response of representative conventional photocathodes. (After K r ~ s e . ~ ’ )

photocathode are able to respond sensitively to wavelengths as long as 0.9 p and are therefore useful in sensing red radiation from sources such as the helium-neon and ruby lasers and even the gallium arsenide semiconductor laser at 0.83 micron. The so-called negative electron a f f i n i t ~class ~ ~ ,of ~ ~photoemissive devices utilizes photoconductive semiconductors, such as gallium arsenide, with a very thin surface coating that is most commonly cesium oxide. The production of this surface layer on the semiconductor causes bending of the energy bands of the semiconductor at its surface, as was shown in Fig. 7. The effect of this band-bending is to produce a condition where the energy level of the electron at the conduction-band minimum in the bulk of the semiconductor crystal is higher than the energy level of an electron in the vacuum. Therefore, an electron in the conduction band in the bulk of the material will energetically fall out of the photocathode semiconductor into the vacuum if it is able to reach the activated surface region without recombining. Thus the conduction-band electrons require no energy above that of

3 08

DETECTORS I

0.4

I

0.6

1

I

I

0.8 Wavelength (pm)

FIG. 9. Spectral response of a typical GaAs and InAs,

I

1

1

P,-, photocathode.

E, to escape from the photocathode, while in conventional photocathodes substantially higher energy is required for the electrons to escape into the vacuum [see Fig. 6(b)]. The existence of a region in the negative electron affinity photocathode, where the energy at the bottom of the conduction band exceeds the vacuum potential, is peculiar to this class of photocathodes and does not occur in the conventional photoemitters. The spectral response of a typical GaAs and an InAs,P,-, photocathode is illustrated in Fig. 9. While the photocathode is the most crucial part of the photomultiplier, since it converts the incident radiation to electronic current and hence determines the spectral characteristics of the detector and the ultimate sensitivity, the remainder of the structure is important in achieving the high amplification and noise figure as well as the fast time response associated with the photomultiplier. A simple schematic of a photomultiplier-type structure is illustrated in Fig. 10. The electrons that are emitted from the photocathode are electrostatically focused and accelerated toward a

309

PHOTON DETECTORS

Dynodes

II

Focusing electrode

I I

I

Photocathode (semitransparent)

FIG. 10. Schematic structure of a photomultiplier.

sequence of electrodes termed dynodes. The electrons from the photocathode arrive at the first dynode with a kinetic energy of from 1 to 400 eV. Secondary emission from the surface of the dynode causes a multiplication of the initial current. Using dynode materials which are chosen primarily for high secondary emission ratios, the secondary emission ratio can be from 5 to 10 per electrode, depending on the primary electron energy. Conventional materials such as beryllium oxide (BeO), magnesium oxide (MgO), and cesium antimonide (Cs,Sb) are used as dynode materials in most photomultiplier wavelength systems. The gallium arsenide or gallium phosphide negative electronic affinity secondary emitters are also used in multipliers, since they have high secondary emission ratios. Secondary emission ratios of 5 to 50 are obtained over a primary electron energy range of from 100 to 600 eV. The process of secondary emission from a succession of dynode surfaces causes a large multiplication of the initial current. The process is repeated at a sufficient number of dynodes, so that the initial current emitted from the photocathode will be multiplied by a large factor. For a secondary emission ratio of 5 at each dynode, the multiplication between cathode and anode is approximately lo7 for a series of 10 dynodes. The last electrode in this chain is a collector anode without gain. The structure of the electron multiplier chain can take on a variety of forms, each with special characteristics as illustrated in Fig. 11. A circular arrangement provides a structure which is both simple and compact with A variety of relatively high-bandwidth and electron-collection effi~iency.~’ other ladder or venetian-blind structures have been constructed which are compact, exhibit stable gain characteristics, and allow a relatively large number of stages, but with rather poor time response. The relationship between response and electron-collection efficiency is an inverse one, since the time response is largely determined by a spread in the time of flight of

310

DETECTORS

FIG. 11. Circular arrangement of an electron-multiplier chain. (After Sommer.”)

high-velocity and low-velocity electrons. The avoidance of the large spread in time of flight implies a focusing structure where electrons with unequal paths are either forced into compensating long and short pathlengths, respectively, or rejected as they progress between electrodes. High collection efficiency requires that all electrons be collected independent of their velocity or path. The necessity of minimizing time-of-flight variations between the photoemissive surface and successive electrodes has given rise to photomultiplier designs such as the crossed-field photomultiplier. Here the magnetic field is used together with an electrostatic field to produce varying pathlengths depending on the electron energy, that is, high-energy electrons are forced into long pathlengths while low-energy electrons are induced into correspondingly short paths. This design reduces time-of-flight variations and produces rise times at the anode of about 0.1 ns as compared to several nanoseconds for various other photomultiplier designs. Another rather different photomultiplier is the so-called channel p h o t ~ m u l t i p l i e r .In ~ ~this device, large numbers of parallel channels, each channel being a small diameter cylinder of material exhibiting electron gain, are used to form continuous dynode surfaces. Voltage is applied along the channel length, and multiple reflections of electrons traveling down the channel produce extremely high gain. Devices of this type, where channels with density as high as a million channels per square inch exist in parallel, are used in imaging, and with a high channel density, they can retain a high degree of image resolution.

PHOTON DETECTORS

311

Because of the high current amplification and the noise, the photomultiplier is one of the most sensitive instruments for detection of visible and near infrared radiation. The photomultipliers have been used to detect optical power levels as low as lo-'' W. As a detector of an optical signal modulated at some low frequency, the limiting factor in sensitivity is the so-called dark current of the photomultiplier. The random fluctuations observed in a photomultiplier output are due to either cathode or dynode shot noise, Johnson and 1/ f noise. The cathode shot noise is composed of a fluctuating current emitted by the photocathode due to incident signal power and another component observed in the absence of radiation, which is the so-called dark current arising from random thermal excitation of electrons. All electrons in the cathode which are thermally excited to energies greater than the work function or electron affinity are emitted, and each emitted electron is then effectively replaced by another electron from the cathode. The minimum detectable optical power is determined by the magnitude of this thermally generated dark current and typically leads to a minimum detectable power on the order of W for simple video detection. An obvious means of reducing the thermal dark current is to cool the photo~athode.~'*~" If the temperature of a conventional photoemitter is lowered to at least -2O"C, the thermal dark current becomes negligible. For systems with lower thermal activation energy, such as the negative electron affinity emitters, correspondingly lower temperatures are required. Cooling of this type has traditionally been achieved by using coolants such as dry ice to reduce the temperatures to a point where thermal carrier generation is reduced to negligible levels. More recently, thermoelectric cooling has been found to be more convenient and able to achieve temperatures which are precisely those required for optimum cooling, without reducing the tube temperature to a point where factors such as moisture and condensation become troublesome. 6.4.2. Photoconductive Detectors

For radiation with wavelength in both the visible4' and near infrared42 spectrum, photoconductive and photovoltaic semiconductor detectors are generally employed. A semiconductor photodetector is a simple device with high detectivity. Incident photons excite charge carriers from bound states into free states, increasing the number of mobile charge carriers, and these photoexcited carriers change the electrical properties of the material. The physical realization of this effect depends on the form and properties of the material. In a photoconductive sample, radiation is detected through an increase in electrical conductivity, while in a photovoltaic device it is detected as a photon-generated voltage. Excitation of carriers in a semicon-

312

D E r ECrO RS

hu 2

Ed

\G

hv

Donor level

> Eg

FIG. 12. Energy-band diagram for a photoconductive detector.

ductor may take place from states near the top of the valence band across the energy gap into states near the bottom of the conduction band, producing excess electron-hole pairs, or alternately from impurity levels creating either excess electrons or holes (Fig. 12). The former are designated intrinsic photoconductors, while the latter, in which the photoexcitation is from impurity levels, are termed extrinsic. Extrinsic detectors may incorporate either donor or acceptor impurities, leading to sensors where the radiatively induced carriers are either electrons or holes, respectively. Extremely long wavelength detection may be obtained by electron intraband excitations in materials such as InSb, yielding response at wavelengths in the 100-km to 1-mm range. Intrinsic photoconductors respond to photons of sufficient energy to cause an interband transition on being absorbed, thus creating excess electronhole pairs in the semiconductor. The semiconductor will absorb only photons with energy greater than the energy gap, E B ,so that the maximum wavelength A which can be detected is (6.11) where h is the Planck’s constant and c is the speed of light. The photodetector can respond to all radiation with wavelengths shorter than A,,,, with maximum sensitivity for photon energies slightly greater than the gap energy. The photoresponse of several common infrared photoconductors is shown in Fig. 13. The variation in spectral sensitivity for photon energies greater

PHOTON DETECTORS

313

FIG. 13. Spectral response of several common infrared photoconductors.

than the gap arises from the strong absorption of the radiation which produces excitation only near the surface. The location of the maximum in the spectral response is dependent on the thickness of the crystal and the carrier recombination parameters. The spectral regions in which intrinsic semiconductor photoconductors are capable of responding depend on the energy band gaps in elemental and compound semiconductors which have been developed. As shown in Table 111, the range from the visible spectrum to approximately 8 pm is well covered by a number of common semiconductors. While the points shown are for a fixed temperature, the band gaps of most semiconductors vary strongly with temperature, so that the maximum sensitivity for a given material will change if the detector is cooled. For example, when InSb is cooled from 300°K to 77"K, the energy gap increases from 0.18 eV to 0.23 eV causing a reduction in A,, from 7 pm to 5.5 pm. Materials such as PbS

314

DETECTORS

TABLEI 1 I .

Energy Band Gaps for Photoconductive Semiconductors Band Gap (ev)

Cutoff Wavelength

Material

Si Ge lnAs InSb PbS PbSe PbTe GaAs CdSe CdS Hg,-,Cd,Te( x = 0.2) Pb,-,Sn .Te(x = 0.2)

1.10 0.68 0.33 0.18 0.4 0.25 0.31 1.4 1.74 2.4 0.09 0.083

1.12 1.82 3.75 6.88 3.1 4.95 4.0 0.88 0.71 0.52 14 15

(wm)

~

and PbSe have a positive temperature coefficient describing the band-gap variation and respond to longer wavelengths at lower temperatures. The spectral variation in sensitivity of PbS and PbSe detectors as a function of temperature is illustrated in Fig. 13. Cooling is usually required in infrared photodetectors to reduce competition between the carriers generated by photons and thermally generated carriers. Beyond about 8 pm, there are no simple intrinsic elemental or compound detectors available. The range of intrinsic detectors has been extended beyond this point by alloying two compounds of different bandgaps to yield materials of a desired energy gap. When a compound such as CdTe with a band gap of 1.5 eV (300°K) is combined with HgTe, which is a semimetal in which the valence band and conduction band effectively overlap by 0.15 eV, the resultant band gap is dependent on the fractional composition of the compounds (Fig. 14). In the alloy Hg,-,Cd,Te, the gap varies nearly linearly with x between the two extreme values, so that it goes through zero at x = 0.10 at 300°K.42*43 For x = 0.2 at 77”K, the gap is approximately 0.1 eV, yielding an intrinsic detector in the 8- to 14-pm range where there is an important “window” in atmospheric transmission. Development efforts in this material have centered on this region, although narrow-gap Hg, -,Cd,Te detectors have been made with response beyond 40 pm.44 The binary, semiconductors Pb,_,Sn,Te and Pb, -,Sn,Se also have composition-dependent energy gaps which can be made small corresponding to detection at wavelengths beyond 10 p.m.45.46 Development of the Pb, - .Sn,Te alloy system has received significant emphasis in recent years particularly as photovoltaic detectors. The limiting wavelength response of

315

PHOTON DETECTORS I 1.6

-

1.4

-

1.2

-

m

I

I

I

-

-0.4

0

HgTe

0.2

0.4

0.6

0.8

1.0 CdTe

FIG. 14. Dependence of the resulting band gap on the fractional composition of the compounds in Hg,-,Cd,Te.

a number of the elemental, compound, and alloy semiconductor photoconductors is listed in Table 111. It illustrates that the intrinsic semiconductors cover the visible and infrared spectrum to about 15 pm with a readily available set of photodetectors. In practical use, the PbS and PbSe detectors are widely used in the infrared from 1 to 5 pm, with a reasonable sensitivity at room temperature, although an increase in responsivity and decrease in noise is obtained by operating at low temperature. The intrinsic photoconductors InSb and InAs can be operated at room temperature, but the large number of thermally excited carriers at room temperature limit their response, so that they are customarily cooled. The narrow-gap alloys Hg, -,Cd,Te and Pb, -,Sn,Te, used even further in the infrared, must always be cooled to about 77°K to adequately reduce the number of thermally generated carriers. As shown in Fig. 13, a number of detectors have enhanced sensitivity and extended cutoff wavelengths when operated below 300°K. In addition, detectors used in the wavelength range greater than 10 pm must be cooled to prevent a large number of thermally excited carriers from competing with the photoexcited carriers. These factors have led to the widespread use of cooled photon detectors. When a detector is to be cooled, it is frequently mounted in an insulated dewar flask, illustrated in Fig. 15. The walls of the dewar can be made of glass, Pyrex, or metal and may either be permanently sealed on manufacture

316

DETECTORS

Pump out PO rt

Electrical leads FIG. 15. Cooling of a detector mounted in an insulated dewar flask.

or provide for periodic evacuation. Most infrared detector systems are usually cooled to liquid-nitrogen temperature (-77°K). The semiconductor sensor is usually cemented to the wall of the coolant container and connected by small-diameter wires to an external electrical connector. A window admitting radiation to the detector is selected for a spectral band pass matching the useful range of the detector and may be either removable or permanently mounted to the dewar. Several other methods are also used to cool detectors, in addition to the common cryogenic fluids such as nitrogen. Several open- and closed-cycle refrigeration systems have been used, including Joule-Thomson and Sterling refrigerators. These systems are generally capable of achieving operating temperatures of 70-80°K, with a refrigeration capacity of 10 to 20 W. Units of this type have been used to cool detectors in remote-sensing satellite systems where other methods were not feasible. Finally, there has been a resurgence of interest in solid-state refrigerators using the thermoelectric effect. Following the discovery of the thermoelectric effect by Seebeck, Peltier discovered that when current flows in a circuit of two dissimilar metals, heat is absorbed at one junction and released at the other. The practical use of materials for thermoelectric devices can be evaluated by the thermoelectric figures of merit: (6.12)

where a is the thermoelectric coefficient (V/"K), p the resistivity ( a - c m ) , and k is the thermal conductivity (W/"K cm). Semiconductors generally have the highest thermoelectric figure of merit, with the commonly used bismuth telluride having 2 = 3 x OK-'. With a value of Z such as

PHOTON DETECTORS

317

described, a temperature differential of 60-70°K can be obtained with a semiconductor detector as a heat load.

6.3.3.Extrinsic Photoconductors The requirement for quantum detectors beyond wavelengths of approximately 10 pm frequently makes it necessary to use extrinsic semiconductors. Impurity or defect states in a semiconductor create discrete energy levels in the band gap of a semiconductor with relatively low activation energies. Carriers can be photoexcited from these levels near the conduction or valence band edges, creating a decrease in the detector’s electrical resistance which can then be measured, as was previously shown in Fig. 12. Impurity levels near the conduction band edge are called donor levels, since photoexcitation of carriers will cause extra electrons to be added to the number already in the band at thermal equilibrium. Levels adjacent to the valence band edge are called acceptor levels, since excitation of carriers into these levels will leave excess holes in the band. The nature of the impurity level depends on the valence of the impurity relative to the host semiconductor. Since these are states with very low activation energies, detectors based on photoexcitation from these levels must be cooled to prevent thermal excitation from masking the photoeffect. The technique is limited to semiconductors of a high degree of purity and crystal perfection where the level of incidental impurities and defects is small. In materials which d o meet these requirements, the ionization energy of isolated impurities can be approximately treated by a description similar to an isolated hydrogen atom. The ionization energy in a semiconductor with a dielectric constant K and an effective carrier mass m* is given by

m*

E

= 13.6 m K 2 eV9

(6.13)

where m is the free electron mass. The ionization energy of an impurity state is much smaller than that of a hydrogen atom, due to the difference in the dielectric constant and the effective carrier mass in the solid. For example, the donor ionization energy in GaAs with K = 13.5 and m* = 0.072 m is approximately, 0.005 eV, equivalent to an excitation wavelength of 200 pm. This far-infrared transition is observed in pure GaAs epitaxial films and is used as the basis for a far-infrared photoconductive detector. In materials such as the elemental Group IV semiconductors, Ge and Si, with complicated energy band structure, application of the simple expression for ionization energy yields only an approximate result. Accurate calcula-

318 Conduction band

Au

cu

-

0.33

Egap = 0.75 eV Zn -

Ga

0 2 0.02 025

0.04

-

0.03

Valence band FIG. 16. Typical acceptor impurity levels in germanium.

tions using correct anisotropic masses predict donor ionization energies of about 0.01 eV for germanium and 0.03 eV for silicon. The ionization energies predicted are valid only for elements removed by the valence group of the host element by 1. For example, in Ge, elements of Group Ill are impurities with a single acceptor level, and elements of Group V are impurities with a single donor level with an ionization energy of approximately 0.01 eV. The typical acceptor impurity levels in germanium are illustrated in Fig. 16. Elements further removed from Group IV have more than a single level with substantially different ionization energies, as shown in Fig. 16.47*48*49 Impurity ionization energies in Ge that lead to extrinsic detectors, which are of practical importance, yield photoconductive response as shown in Fig. 17. The small values of most donor and acceptor ionization energies are comparable with thermal energy (0.025 eV) at room temperatures, so that the extrinsic photoconductors must be cooled to reduce thermal ionization. For wavelengths shorter than the maximum, the radiation is strongly absorbed and produces excitation only near the surface. Since it is common that the surface imperfections are characterized by rapid nonradiative recombination, the excitation of photoconductivity and luminescence is less when limited to the surface. The location of the maximum in the spectral response is dependent on the thickness of the crystal and the recombination parameters. For wavelengths longer than the maximum, the excitation decreases simply because the absorption producing free carriers is decreasing.

319

PHOTON DETECTORS

\

Solay cell

I

1.o

I

1

1

1

1

l

Ge:Au I

1

I

I

I

10

I 1

Ill

I

I

1

1

100

1

1

1

1

1000

Wavelength ( r m )

FIG. 17. Spectral response of some extrinsic photoconductors.

6.4.4. Photoconductive Response A photoconductor is operated using a simple electrical circuit consisting of the detector, an external bias supply, and a load resistance. The photosignal generated when radiation is incident on the detector is usually measured across the load resistance, as shown in Fig. 18. When a photoconductor with a bias voltage V, responds to radiation by a change in conductivity us,a signal voltage V, is produced, where

Vs=(:)V",

(6.14)

with (T being the total detector conductivity. For an absorbed power per unit volume P,/Ax, the signal photoconductivity induced by the radiation

320

DETECTORS

Photo voltage Bias SUPPlY

Load resistor Radiation

/

-/

1

1

1

/

/

Semiconductor

/-

/

contacts FIG. 18. Measurement of the photosignal generated when radiation is incident on a photoconductive detector.

is (6.15)

where q is the quantum efficiency, A is the wavelength, and p is the carrier mobility. The detector is assumed to have a thickness x and an illuminated area A. The spectral responsivity RA will then be given by the expression

RA=-qA"0'

hcnxA

(volts/watt),

(6.16)

where n is the carrier concentration. The response of a photoconductive detector depends on having a material with a low thermal equilibrium carrier concentration, small volume, high quantum efficiency, and a long carrier lifetime. Since the excess carrier lifetime is the parameter which determines the response time of the detector, the requirement for a long lifetime for high responsivity conflicts with a short photoconductive response time. Typical response times in intrinsic photoconductors range from lO-'sec to sec, making these detectors very superior in this respect to thermal detectors with equivalent sensitivity. The responsivity also has a linear bias field dependence which is, of course, limited in practice by either Joule heating or other high field effects. Operation of these detectors at low temperatures will typically reduce the equilibrium carrier concentration and

PHOTON DETECTORS

321

increase the excess carrier lifetime, leading to a significant increase in R,. The spectral detectivity 0:can be written as (6.17) where Af is the noise bandwidth and vT is the total noise voltage. Reducing the detector temperature will reduce the thermal noise component, so that the noise will be dominated by generation-recombination noise. The total effect of cooling is a significant increase in detectivity, as was shown for several intrinsic photoconductors in Fig. 13. In addition to specification of responsivity and detectivity, the performance of either extrinsic or intrinsic photoconductors under steady-state incident radiation can be described in terms of the short-circuit photocurrent or the open-circuit photovoltage modes. The short-circuit photocurrent Z,, is typically given by Z,, = TqNg, where 7 is the quantum efficiency, q is the electronic charge, N is the number of photons absorbed in the sample per unit time, and g is the gain of the photodetector. If the photoconductive gain and photon-absorption rate are expressed in terms of fundamental materials parameters, the short-circuit photocurrent can then be written as (6.18) Here P is the absorbed power at a particular wavelength, A, V, is the bias voltage applied to the sample, 1 is the distance between electrodes on the sample, p is the majority carrier mobility, and T is the majority carrier lifetime. From this expression it can be seen that the low-frequency photocurrent is directly proportional to the absorbed power and also depends directly on the wavelength for values of the wavelength below the long-wave cutoff limit associated with each individual semiconductor. An important observation is that the photoconductivity depends on the lifetime and mobility of the majority carriers, since photoconductivity is in fact a majority-carrier phenomenon. In a photoconductive device, minority carriers, which typically have lifetimes significantly shorter than the majority carriers, contribute little to the change in photocurrent and hence can be ignored to a first approximation. If a detector is inserted into an electronic circuit with some effective load resistance, the photocurrent in the total circuit is modified. The photovoltage which appears across such a load resistance can be simply expressed in terms of the product of the short-circuit current and the equivalent load resistance. If the load resistance, R , , is large with respect to the detector resistance, Rdr over its range of operation, the photovoltage is essentially

322

DETECTORS

the open-circuit photovoltage, Voc.When this is the case, the photovoltage V,, can then be written as follows (6.19)

Here the detector resistance was expressed in terms of its majority carrier concentration n, mobility p, and respective length, width, and thickness 1, w, and x. As in the case of the short-circuit current, the open-circuit voltage is then dependent on the incident power, the wavelength, and bias and is inversely proportional to the majority-carrier concentration in the device. These expressions for short-circuit current and open-circuit voltage, are low-frequency or steady-state descriptions, and do not express information about the response time of the device which is crucial in many applications. Since a photoconductor is, in fact, a majority-carrier device, the frequency response of the open-circuit voltage V, is then given by the expression

where w is the angular frequenccy of the modulation of wavelength of radiation of wavelength A. At low frequencies where the product W T is much less than 1, the device response is essentially independent of frequency, while at high frequencies where OT becomes much larger than 1, the response is inversely proportional to frequency and depends explicitly on the majority-carrier lifetime, which is thus the primary limit on high-frequency performance of photoconductive devices.

6.4.5. Photovoltaic Devices

A mode of detection of great practical significance is that using the photovoltaic eff e ~ tWhile . ~ this ~ effect relies on direct conversion of photons to carriers in the material, as does photoconductivity, it places additional requirements on the nature of the detector in that some form of potential barrier with an internal electric field is required to cause a separation of the photoexcited electron-hole pair. Photovoltaic detectors are constructed from materials where the intrinsic photovoltaic effect arising from band-toband transitions generates electron-hole pairs which are subsequently separated by an internal field and separately collected at opposite electrodes of the device. The photovoltaic effect is essentially limited to materials where intrinsic photodetection occurs, although there have been a few limited reports of extrinsic photovoltaic detectors. The simplest and classical structure for separation of the photoexcited electron-hole pairs occurs in the

323

PHOTON DETECTORS

simple p-n junction; however, a number of other structures have recently been vigorously investigated. By far the most common form of photovoltaic effect is that which occurs in a p-n junction fabricated in a common semiconductor, such as silicon, by doping or formation of epitaxial layers at one surface. A simple p-n junction photodiode is illustrated in Figure 19, where two electrical contacts are formed and a thin region at the surface has been prepared to allow the incident radiation to be absorbed and to generate electron-hole pairs. The photoexcited electrons and holes are separated by the high field in the space Incident radiation Top surface

Top contact

n-type

P-n junction

1

I

Contact

N-region

P-region Conduction band

Valence band

FIG. 19. (a) A simple p-n junction photodiode. (b) Associated energy-band diagram.

324

DETECTORS

Current

Saturation current

Ill

current Dark \

Voltage Photovoltage (open circuit)

Photocurrent

Photocurrent (short circuit)

IT-$

+~

Photodiode

Bias source+ '6

I+

Radiation

RL

I

Load

resistor

Photovoltage

FIG. 20. (a) Observation of photovoltage and photocurrent in a photovoltaic device. (b) Circuit to observe the photovoltage.

charge region between the p and n regions. This charge separation creates an open-circuit voltage across the junction and leads to a voltage which can be detected that is proportional to the absorbed photon density. The direct observation of this photovoltage is the simplest and most straightforward mode of operation of a photovoltaic detector [Fig. 20(a)]. Junction detectors are often operated as well under some form of reverse bias voltage which leads to observation of a photocurrent [Fig. 20(a)] rather than a photovoltage and are said to be operated in a photoconductive mode, despite the fact that it is not equivalent to the photoconductive mode previously described. The short-circuit current associated with the photovoltaic detector is identical in form to that for a photoconductive detector, with the exception

PHOTON DETECTORS

325

that the gain is unity, that is, the short-circuit current is given by

I,=-,

nqPA hc

(6.20)

The open-circuit voltage, which is the commonly observed photovoltaic signal, is obtained as the product of the short-circuit current with the dynamic resistance of the junction in the absence of bias. Hence the open-circuit photovoltage V,, for a photovoltaic detector is approximately (6.21) Here I,,, is the saturation current of the junction in the absence of a photo signal, k is Boltzmann’s constant, and T is the absolute temperature. Just as in the case of the photoconductive detector, the photovoltaic signal depends on the incident power and wavelength, but it does not depend on the majority carrier lifetime. The photovoltaic effect, in fact, is essentially independent of the majority-carrier lifetime, since in creating an electronhole pair, the determining factor is the minority-carrier lifetime, which is significantly shorter. Since the frequency response is determined by the product of the modulation frequency of the radiation w and the minoritycarrier lifetime 7, the response is essentially independent of frequency for values of W T < 1. The point at which W T becomes unity is at a significantly higher frequency for a photovoltaic device; therefore, photovoltaic detectors generally display a higher frequency response or a faster time response than photoconductive detectors of exactly the same material. The most widely used photovoltaic detectors for the visible spectrum rely on the common elemental semiconductors Ge and Si with cutoff wavelengths of 1.5 pm and 1.1 pm, r e s p e c t i ~ e l yFor . ~ ~applications at longer wavelengths, photovoltaic structures have been developed using 111-V compounds or the alloy systems Pb, -,Sn,Te and Hg, -,Cd,Te. For optical communications systems, detectors utilizing Ge”,52 or 111-V alloys, such as InxGal-xAsyP1-y,53-55 have been emphasized with peak response in the 1.1- 1.7-pm range. For longer-wavelength infrared extending to wavelengths as long as 14 pm, the Hg,-,Cd,Te and Pb,-,Sn,Te alloy systems have been vigorously de~eloped.’~-’~ While the discussion of photovoltaic detection has centered on the classical p-n junction, there are a number of structures which have been employed for photovoltaic detection. The most direct extension of the simple p-n junction is the p-i-n photodiode which incorporates an intrinsic region between the p and n portions of the traditional junction. If either the p or n side of the junction is made very thin compared to the optical absorption

length in the medium, the incident optical radiation will readily penetrate into the intrinsic region. In the intrinsic region, the absorption will produce a large number of electron-hole pairs. As these electron-hole pairs are produced in a region of high electric field, the carriers will rapidly be drifted out of that region into the adjacent p and n sides of the junction. This rapid carrier separation leads to an improved frequency response and efficiency in the p-i-n diode as compared to that of a pn diode of a comparable material. In addition to being used to enhance detection, p-i-n configurations have also been employed in so-called avalanche photodiodes.""."' An avalanche photodiode is a structure with an internal gain achieved through the so-called avalanche breakdown effect, which occurs in the high field regions of a p-n or p-i-n junction under high reverse bias. The carriers generated by photoexcitation are accelerated to velocities which are sufficiently high so that upon collision with atoms in the lattice, additional free electrons are generated by the impact ionization process. These free electrons are also accelerated and undergo more collisions and hence will also lead to the generation of more free electrons. The so-called avalanche effect, which occurs within a high field region in a reverse bias p-n or p-i-n junction, can lead to a significant internal gain."2 Hence, a larger photosignal is produced in a material with the same incident radiation power than would be produced under conventional diode operation. While the photogenerated current for a p-i-n diode is as previously described by Eq. (6.20), for conventional photovoltaic devices the value of current, I,", in an avalanche diode is I,,

nqPA hc

= -M ,

(6.22)

where M is the avalanche gain. The gain depends on device design and bias conditions and may be as high as 100. The avalanche process is a statistical one, where M fluctuates for each chain of events. Hence the amplification process in these devices is subject to an additional noisegeneration process. In many semiconductors, particularly those of significance for infrared detection, production of both n- and p-type materials of high quality is not always possible within the current state of the art. A method for forming photovoltaic structures in materials where p-n junctions cannot be formed is through creation of a Schottky barrier p h ~ t o d i o d e . " ' .A ~ ~Schottky barrier is a junction formed at the interface of a metal semiconductor structure. In a manner analogous to that described for the p-n junction, a potential barrier is created at the metal-semiconductor interface, which will effectively separate electron-hole pairs generated by incident optical radiation. A diagram of a typical Schottky barrier junction is shown in Fig. 21. In most

PHOTON DETECTORS

EF

321

-

Metal Semiconductor

FIG. 21. Diagram of a typical Schottky barrier function.

cases, the Schottky barrier is formed by producing a thin film of metal on the surface of the semiconductor, where the thickness of the metal is sufficiently small so that it is partially transparent to incident radiation at the wavelengths of interest. Incident photons can generate electron-hole pairs within the semiconductor or in the vicinity of the potential barrier at the metal-semiconductor interface. Depending on the specific structure, the Schottky barrier photodiode may be illuminated either from the metal side of the junction or it may, in fact, be illuminated through the semiconductor if it is made sufficiently thin so that carriers can reach the interfacial region. 6.4.6. Photon-Drag Detector

While most semiconductor quantum detectors respond to radiation by a photo-induced change in free carrier concentration, the photon-drag detector operates through a change in carrier momentum as photon energy is absorbed.65 The simplest description of photon drag attributes the phenomenon to radiation pressure which transfers momentum to the mobile carriers in the semiconductor. As the radiation propagates through the semiconductor, it acts upon mobile carriers, affecting carrier transport in the direction of light propagation. A voltage called the “photon-drag voltage” is generated between a contact at the face of the sample where the radiation enters and another contact where it leaves the sample. The photondrag effect has been exploited for use as a high-speed room-temperature infrared detector, with wide use for high-power CO, lasers at 10.6 pm. While the simple description of the effect is accurate for C 0 2lasers operating

328

DETECTORS

around 10.6 pm with the photon-drag detector at room temperature, observations of the frequency and temperature dependence indicate that for wider use, this description is inadequate. Gibson ef al.66.67have shown that in germanium, a more accurate representation must take into account the differences in energies associated with interband transitions of electrons originating in quantum states of the same energy but different wavevectors when the transitions also involve momentum transfer from the photon filed to the electrons. Based on this model using an accurate band structure for p-type germanium, the frequency and temperature dependence of the effect can be explained. While photon drag has been observed in compounds such as InSb and GaAs. Ge has been found to be the most efficient detector material and is used in commercially produced detectors. The responsivities in typical Ge detectors are on the order of 1 x lop6V/W. While having a very low responsivity, the photon-drag detector does have a linear response at extremely high incident intensities and a very high frequency response. Most detectors are linear up to incident power density of greater than 10 MW/cm2 with response times of less than 1 nsec. It has been proposed that the response time is limited only by the transit time of photons through the detector.

6.5. Noise A fundamental limit exists in the minimum radiation power that can be detected by all forms of detectors, arising from the noise that is present in addition to the Noise may arise from within the detector, may be associated with the source of radiation, or may appear in the electronic system used to enhance the electrical output signal of the detector. The latter element can be largely eliminated through attention to the proper electronic design and may, in fact, include certain low-noise amplifiers that can reduce electronic contribution to the noise to a very small level. The noise which actually arises due to mechanisms internal to the detector generally falls into several distinct categories. Many sources of noise are common to or appear in most forms of radiation detectors. In particular, almost all forms of noise are present, to at least some degree, in the solid state detectors used to detect most forms of visible and infrared radiation. In the absence of electrical bias, there is a low-level source of noise usually described as Johnson noise. This type of noise is generated by the random motion of charge carriers in any resistive medium. The Johnson noise is present at a power level which is determined by the temperature of the material and the bandwidth of the detection system. In addition to these factors which determine the noise power for Johnson

329

NOISE

noise, the noise voltage and current also depend on the resistance value of the detector. The mean square Johnson noise power, which then appears even in the absence of electrical bias as a fluctuating current or voltage, is given by P, = kTB, where k is the Boltzmann's constant, T is the absolute temperature of the detector, and B is the measurement of bandwidth. The open-circuit noise voltage V, is then given by V, = (4kTRB)'/2,

(6.23)

where R is now the resistance of the detector. In most solid state photodetectors, the Johnson noise is present at all frequencies, but it is not the dominant noise, except at high frequencies. At lower frequencies, several other sources of noise dominate. This is illustrated in Figure 22, which illustrates an ideal noise spectrum for a typical solid-state photodetector in the absence of radiation. At low frequencies, another noise source, called l/f noise, dominates, while at intermediate frequencies generation-recombination noise dominates. The specific ranges over which each of these noise sources is predominant varies according to the details of the composition and structure of the detector, but typically below 1 kHz is the range for l/f; which is the principal noise source. Between 1 kHz and as high as 1 MHz, generationrecombination noise would be the dominant noise, while above that frequency, the Johnson noise would be the predominant noise source. At the lowest frequencies, the source of noise found in most solid-state detectors is l/f noise characterized by a spectrum where the noise power I

I

I

I

1

1

I

I

1

I

-

-

Generation-recombination noise

-

-

Johnson noise

I

I 2

1

I 4

I

I 6

I

I 8

I

I 10

FIG. 22. Different types of noise of a solid state photodetector.

330

DETECTORS

varies approximately inversely with frequency. Most solid-state detectors usually exhibit l/f noise at low frequencies. Although at higher frequencies, the amplitude drops below that one or below the other types of frequencyindependent noise. The mechanism for l/f noise is not well understood. It has been attributed to the presence of potential barriers at contacts or at the surface of the detector. The understanding and reduction of l/f noise is then a process which depends on the procedures used in preparation of the contacts and surfaces of the detector and is very much an art. The noise current associated with l/f noise is often expressed as I,=

( c1;-

,

(6.24)

where C is a constant, Ib is the bias current, B is the bandwidth, and f is the frequency. There are often exponents associated with the frequency given here by a, where a is approximately, although not always, precisely equal to unity. A third and very important form of noise found in most solid-state detectors is the so-called generation-recombination noise. Generationrecombination noise is caused by fluctuation in the rate of thermal generation and recombination of charge carriers in a solid, giving rise to a fluctuation in the average charge-carrier density. Because of this variation in carrier density, the resistivity will also fluctuate, and this will, of course, give rise to a fluctuating voltage at the terminals of the detector when a fixed bias current is flowing through the solid. In many solid-state detectors, the fluctuations in charge-carrier concentration will arise either from intrinsic carrier generation, that is, the creation of free electron-hole pairs by the thermal energy of the lattice, or by extrinsic carrier generation where fluctuation in the number of carriers can arise from the generation and recombination rates associated with an impurity level. For a simple intrinsic photoconductor with a single type of thermally generated carrier, usually electrons, the noise current will be given as I

=(%)([*]”” +

1 w2r2

(6.25)

where p is the total number of free holes in the sample, r is the free-carrier lifetime, B is the measurement bandwidth, and n is the total number of electrons in the material. It is clear that when the frequency becomes large with respect to the free-carrier lifetime, the noise power would diminish, which will limit the upper frequency limit over which generation and recombination noise is an important factor.

33 1

OPTICAL WINDOW MATERlAL

A final form of noise which is found in photo-emissive devices is that of shot noise. The shot noise in a photoemissive detector is equivalent to the generation-recombination noise in a semiconductor in that it arises from randomness in the emission of electrons. This noise is generated even at low temperature where thermal carrier generation can be neglected. The shot noise in a photomultiplier includes both the cathode shot noise due to random current fluctuations in the average current emitted by the photocathode due to the incident signal power, and the dynode shot noise due to random secondary emission at the dynodes. Since the current originating at a dynode does not see the full gain contribution of the tube, the dynode shot noise is a small fraction of the cathode shot noise.

6.6. Optical Window Material One of the limitations in the spectral response in many detector systems is associated with the transmission spectrum of optical window material used to protect the detector element. There are approximately one hundred optical materials with a useful transmission band for application as windows. A variety of physical properties can be used to select a fraction of these as technologically significant. The most significant physical properties in window selections are spectral transmittance and the index of refraction of the material. In addition, a number of mechanical and thermal properties are often of importance, including: hardness, thermal conductivity and TABLEIV. Optical Window Materials

Material Borosilicate Glass Fused Silica Sapphire Magnesium Fluoride (IRTRAN-1) Magnesium Oxide (IRTRAN-5) Calcium Fluoride (IRTRAN-3) Zinc Sulfide ( I RTRAN-2) Sodium Chloride Zinc Selenide ( I RTRAN-4) Cadmium Telluride (IRTRAN-6) Silicon*t Germaniumt KRS-5

* Transmission of

Cutoff Wavelengths (pm)* Lower Upper

0.3 0.18 0.15 0.45 0.40 0.15 0.5 0.2 0.5 0.9 1.2 1.8 0.5

2.8 4.3 6.5 9.5 9.5 12. 14.5 2.0 22. 30. 55. 55.

55.

10% through 2-mm specimen thickness.

t Lattice bands cause discontinuous transmission spectrum.

Index of Refraction 1.49 1.44 1.73 1.38 1.68

1.40 2.23 1.51 2.41 2.65 3.46 4.01 2.35

332

DETECTORS

expansion coefficients, specific heat, elastic moduli, and variations of all of these with temperature. There are about 10 to 12 materials that experienced optical-system designers incorporate in most practical systems. These are described in Table IV, which indicates the useful transmission range of each material and its index at the center of this range. Those materials useful as windows in the near ultraviolet and visible part of the spectrum include borosilicate glass, fused silica, sapphire, and calcium fluoride. In the infrared region, there are a number of elemental and compound semiconductors with useful passbands including Ge, Si, CdTe, ZnSe, ZnS, and MgO. As can be seen from Table IV, most of these semiconductors have a high index of refraction so that antireflection coatings are necessary for efficient transmission. A number of the latter group are available commercially prepared by a powder hot pressing process under the trade name IRTRAN.

References 1. R. C. Jones, “Noise in Radiation Detectors,” Proc. Inst. Rad. Eng. 47, 1495 (1959). 2. R. A. Smith, F. E. Jones, and R. P. Chasmar, The Defection and Measurement of Infrared Radiation, Oxford Univ. Press, London, 1968. 3. E. S. Barr, “The Infrared Pioneers, JI; Macedonio Melloni,” Infrared Phys. 2, 67 (1962). 4. A. H. Wilson, The Theory of Metals, Cambridge Univ. Press, London, 1954. Also H. P. R. Frederikse, V. A. Johnson, and W. W. Scanlon, in Methods of Experimental Physics, Vol. 6B, Chap. 7 (L. Marton, ed.), Academic Press, New York, 1959. 5. G. W. Day, 0. L. Caddy, and R. J. Iversen, “Detection of Fast Infrared Laser Pulses with Thin Film Thermocouples,” Appl. PHys. Lett 13, 289 (1968). 6. D. A. Jennings and E. D. West, “A Laser Power Meter for Large Beams,” Rev. Sci. Insfr. 41 565 (1970). 7. J. W. Stearn, “A Radiometer for Continuous Wave Laser Radiation,” J. Sci. Instr. 44, 218 (1967). 8. R. De Waard and E. M. Wormser, “Description and Properties of Various Thermal Detectors,” Proc. Insf. Rad. Eng. 47, 1508 (1959). 9. G. Koren, Y. Yacoby, H. Lotem, M. Kosower, and G. Greenwald, “Thin Film Calorimeter for Low-Energy Laser Pulse Measurements,” Appl. Phys. Left. 23, 73 (1973). 10. A. J. Schmidt and R. C. Greenhow, “A Practical Laser Output Meter,” J. Phys. E 2, 438 (1969). 11. B. Contreras and 0. L. Caddy, “Nanosecond Response Time Room-Temperature Infrared Detection with Thin Film Bolometers,” Appl. Phys. Left. 17, 450 (1970). 12. J. Stricker and J. Rom, “A Fast Response Thin Film Wattmeter for Measurements of CO, Laser Power,” Rev. Sci. Instr. 43, 1168 (1972). 13. W. S. Boyle and K. F. Rodgers, “Performance Characteristics of a New Low-Temperature Bolometer,” J. Opr. SOC.Am. 49, 66 (1959). 14. F. J. Low, “Low Temperature Germanium Bolometer,” J. Opt. SOC.Am. 51, 1300 (1961). 15. R. Bachmann, H. C. Kirsch, and T. H. Geballe, “Low Temperature Silicon Termometer and Bolometer,” Rev. Sci. Insrr. 41, 549 (1970).

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16. D. Bloor, T. J. Dean, J. 0.Jones, D. H. Martin, P. A. Mauer, and C. R. Perry, “Spectroscopy of Extreme Infrared Wavelengths,” Proc. Roy. Sac. A (GB), 260, 510 (1961). 17. M. A. Kinch, “Compensated Silicon-Impurities Conduction Bolometer,” J. Appl. Phys. 42, 5861 (1971). 18. M. J. E. Golay, “A Pneumatic Infrared Detector,” Rev. Sci. Insrr. 18, 357 (1947). 19. J. R. Hickey and D. B. Daniels, “Modified Optical System for the Golay Detector,” Rev. Sci. Insrr. 40 732 (1969). 20. E. H. Putley, in Semiconductors and Sernimerals, Vol. 12, p. 441, (R. K. Willardson and A. C. Beer, eds.), Academic Press, New York, 1977. 21. E. H. Putley, in Semiconductors and Semimerals, Vol. 5, p. 259 (R. K. Willardson and A. C. Beer, eds.), Academic Press, New York, 1970. 22. S. T. Liu and D. Long, “Pyroelectric Detector and Materials,” Proc. IEEE 66, 14 (1978). 23. A. M. Glass, “Investigation of the Electrical Properties of Sr,-,Ba,Nb20, with Special References to Pyroelectric Detection,” J. Appl. Phys. 40, 4699 (1969). 24. A. M. Glass, J. H. McFee, and J. G. Bergman, Jr., “Pyroelectric Properties of Polyvinylidene Fluoride and its Use for Infrared Detection,” J. Appl. Phys. 42, 5219 (1971). 25. M. G. Broadhurst, W. P. Harris, F. 1. Mopsik, and C. C. Malinberg, Polymer Preprints, 14, 820 (1973). 26. A. Hadni, Y. Henninger, R. Thomas, P. Vergnat, and B. Wynche, J. Phys. 26, 345 (1965). 27. A. M. Glass, “Dielectric Thermal and Pyroelectric Properties of Ferroelectric LiTaO,,” Phys. Rev. 172, 564 (1968). 28. F. Keilmann, “Infrared Interferometry with a C 0 2 Laser Source and Liquid Crystal Detection,” Appl. Opr. 9, 1319 (1970). 29. G. A. Condas and D. W. Brown, “Variable Sensitivity Vision Displayer for Infrared Laser Beams,” Rev. Sci. Instr. 41, 888 (1970). 30. T. J. Bridges and E. G. Burkhardt, “Observation of the Output of a C 0 2 Laser by High-Resolution Thermographic Screen,” IEEE J. Quanr. Elecrron. QE-3, 168 (1967). 31. A. H. Sommer, Photoemissive Materials, p. 222, Wiley, New York, 1968. 32. RCA Phoromulriplier Manual, RCA, Electronic Components, p. 3, Hamson, New Jersey, 1970. 33. R. U. Martinelli and D. G. Fisher,. Proc. IEEE, 62, 1339 (1974). 34. R. L. Bell, Negarive Electron Aflniry Devices, Clarendon Press, Oxford, 1973. 35. D. G. Fisher, R. E. Enstrom, J. S. Escher, and B. F. Williams, J. Appl. Phys. 43, 3815 (1972). 36. A. H. Sommer, R C A Rev. 34,95 (1973). Also published in A. H. Sommer, Gallium Arsenide and Related Compounds, p. 143, in 1972 Proceedings of rhe Fourth Inrernarional Conference, Con$ Ser. No. 17, Institute of Physics, London, 1973. 37. P. KNse in Topics in Applied Physics, Vol. 19, p. 5, Oprical and Infrared Detecrors (R. J. Keyes, ed.), Springer-Verlag, New York, 1980. 38. G. W. Goodrich and W. C. Wiley, Rev. Sci. Insrr. 33, 761 (1962). 39. A. H. Sommer, Photoemissive Materials, p. 68, p. 229, Wiley, New York, 1968. 40. R. B. Murray and J. J. Manning, IRE Trans. Nuc. Sci. NS-7, 80 (1960). 41. D. H. Seib and L. W. Aukerman, Advances in Electronics and Elecrron Physics 34, 95-221, (L. Marton ed.) Academic Press, New York, 1973. 42. D. Long, in Topics in Applied Physics, Vol. 19, p. 101, Optical and Infrared Derecrors (R. J. Keyes, ed.), Springer-Verlag, New York, 1980. 43. D. Long and J. L. Schmit, Semiconductors and Semimetals 5, pp. 175-255, (R. K. Willardson and A. C. Beer, eds.) Academic Press, New York, 1970. 44. D. Long, “Properties of Semiconductors Useful for Sensors,” IEEE Trans. Elecrron Devices ED-16, 836 (1969).

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45. T. C. Harman and 1. Melngailis, Applied Solid State Science, p. 1, 4th ed. (R. Wolfe, ed.), Academic Press, New York, 1974. Also: I. Melngailis, J. Luminescence, 7, 505 (1973). 46. 1. Melngailis and T. C. Harman, in Semiconductors and Semimetals, p. 11 1, 5th ed. (R. K. Willardson and A. C. Beer, eds.), Academic Press, New York, 1970. 47. R. B. Emmons, “The Frequency Response of Extrinsic Photoconductors with Reduced Background,” Infrared Phys. 10, 63 (1970). 48. P. R. Bratt, in Semiconductors and Semimetals, Vol. 12, p. 39 (R. K. Willardson and A. C. Beer, eds.), Academic Press, New York, 1977. 49. H. Levinstein and J. Mudar, “Infrared Detectors in Remote Sensing,’’ Proc. IEEE 63, 6 (1975). 50. R. G. Smith, “Photodetectors for Fiber Transmission Systems,” Proc. IEEE 68,1247 (1980). 51. J. Conradi, “Planar Germanium Photodiodes,” Appl. Opt. 14, 1948 (1945). 52. H. Melchior and W. T. Lynch, “Signal and Noise Response of the High Speed Germanium Avalanche Photodiodes,” IEEE Trans. Electron Devices, ED-13, 829 (1966). 53. M. Feng, J. D. Oberstar, T. H. Windhorn, L. W. Cook, G. E. Stillman, and G. E. Streetman, “Be-Implanted 1.3 pm InGaAsP Avalanche Photodetectors,” Appl. Phys. Lett. 34, 591 (1979). 54. T. P. Lee, C. A. Burrus, Jr., and A. G. Dentai, “lnGaAsP/InP Photodiodes: MicroplasmaLimited Avalanche Multiplication at 1-1.3 k m Wavelength,” IEEE J. Quantum Electron, QE-15, 30 (1979). 55. M. A. Washington, R. E. Nahory, M. A. Pollack, and E. D. Beebe, “High Efficiency In,~.,Ga,As,.P,_,/lnP Photodetectors with Selective Wavelength Response Between 0.9 and 1.7 km,” Appl. Phys. Lett. 33, 854 (1978). 56. W. H. Rolls and D. V. Eddolls, “High Detectivity Pb,Sn,-,Te Photovoltaic Diodes,” Infrared Phys. 13, 143 (1973). 57. C. C. Wang and S. R. Hampton, “Lead Teluride-Lead Tin Teluride Heterojunction Diode Array,” Solid State Electron. 18, 121 (1975). 58. A. M. Andrews, J. T. Longo, J. E. Clarke, and E. R. Gertner, “Backside-Illuminated Pb, - ,Sn,Te Heterojunction Photodiode,” Appl. Phys. Lett. 26, 439 (1975). 59. D. Eger, M. Oron, A. Zussman, and Z. Zemel, “The Spectral Response of PbTe/ Pb, -,Sn,Te Heterostructure Diodes at Low Temperatures,” Infrared Phys. 23, 69 (1983). 60. G. E. Stillman and C. M. Wolfe, in Semiconductors and Semimetals, Vol. 12 (R. K. Willardson and A. C. Beer, eds.), Academic Press, New York, 1977. 61. P. P. Webb, R. J. Mclntyre, and J. Conradi, “Properties of Avalanche Photodiodes,” R C A Reu. 35, 234 (1974). 62. A. G. Chynoweth, in Semiconductors and Semimetals, Vol. 4 (R. K. Willardson and A. C. Beer, eds.), Academic Press, New York, 1983. 63. S. M. Sze, Physics ofSemiconductor Deoices, Wiley, New York, 1983. 64. E. M. Logothetis, H. Holloway, A. J. Varga, and E. Wilkes, “Infrared Detection by Schottky Barriers in Epitaxial PbTe,” Appl. Phys. Left. 19, 318 (1971). 65. A. F. Gibson, M. F. Kimmitt, and A. C. Walker, “Photon Drag in Germanium,” Appl. Phys. Lett 17, 75 (1970). 66. A. F. Gibson, C. A. Rosito, C. A. Raffo, and M. F. Kimmit, “An Optical Bridge for the Assessment of Modern Locked C 0 2 Laser,” J. Phys. D 5, 1800 (1972). 67. P. J. Bishop and A. F. Gibson, “Absorption Coefficient of Germanium at 10.6 pm,” Appl. Opt. 12, 2549 (1973). 68. W. B. Davenport and W. L. Root, A n Introduction to the Theory of Random Signals and Noise, McGraw-Hill, New York, 1958.

A

Brewster angle, 116, 121, 122 brightness temperature, 241 Burch interferometer, 20

Abbe theory of the microscope, 86 aberration correction by holography, 197 aberrations, 83 in holography, 172 Achromatic devices 1 and 2, 133 wave plates, 144 acoustical holography, 197 acousto-optic effect, 150 Airy disc, 67, 80 angle measurements, 41 angular resolution of telescopes, 81 anisotropic media, 133 antireflection coatings, 33 astronomical radiometers, 249 atmospheric sounding techniques, 236

C

channeled spectrum, 28 character recognition, 197 chromatic dispersion in gratings, 72 circular aperture, diffracting, 58, 66 polarization, 110 polarizers, 124 coded reference beam for color holograms, 182 coherence length, 6, 41 color holograms, 180 common-path interferometers, 19 compensated interferometer, 11 compensating plate, 11 retardation plate, 143 compensators, 143 complex plane representation of polarization, 118 composite holography, 184 computer hologram, 189 conjugate image in holography, 170 Cornu spiral, 56 Cotton-Mouton effect, 153 crystal polarizers, 138 retardation plates, 142 types, 135 cubic crystals, 135

B

Babinet compensator, 143 principle, 68 band-pass filters, 33 beam expansion in holography, 200 beam splitters, 33 Bennett-Koehler reflectometer, 274 Bessel function, first order, 66 binary hologram, 189 birefringence in crystals, 133 induced, 146 blackbody curves, 223 radiation, 219 blaze angle in gratings, 78 blazed gratings, 78, 158 bleaching of holograms, 204 bolometers, 295 Bragg effect in holograms, 172 regime, 151

0

depolarizers, 153 detection, synchronous, 279 detector polarization, 158 335

336

INDE X

detector Golay cell, 298 photon-drag, 327 pyroelectric, 299 quantum, 303 detectors, 291 figures of merit, 292 liquid crystal, 302 luminescent, 302 noise, 328 photoconductive, 3 I 1 photon, 303 thermal, 293, 303 dichroic mirrors, 33 polarizers, 156 dichroism, 33 dichromated gelatin for holography, 202 dielectric tensor and energy distribution, 133 diffracting circular aperture, 58, 66 rectangular aperture, 65 slit, 57, 65 straight edge, 57 two-dimensional array of holes, 80 diffraction, 49 images with aberrations, 83 dipole radiation pattern, 98 directional reflectance, measurement of, 274 division of amplitude, 4 of wave front, 1 Doppler effect, 42 interferometry, 42 double exposure holographic interferometry, 193 double-pass interferometer, 37 E

echelles, 79 echellettes, 79 echelon, 31 edge waves, 52 Edser-Butler fringes, 28 electro-optic effect, 147 modulators, 147

electromagnetic description of light, 108 elliptical polarization, 113 embossing of holograms, 206 energy distribution in gratings, 75 theorem, 70 equal-thickness fringes, 8 etalon, 41 experimental procedures in holography, 199 extraordinary beam, 136 extrinsic photoconductors, 3 17 F

Fabry-Perot etalon, 31 interferometer, 25 Faraday effect, 153 field equations, 108 measurements, 266 figures of merit for detectors, 292 filter radiometers, 257 finesse of interference fringes, 27 first order plate, 143 Fizeau interferometer, 8 four mirror polarizer, 124 Fourier transform, 62, 69 holograms, 178 Fraunhofer diffraction, 62 hologram, 175 Fresnel biprism, 3 diffraction, 53 equations, I13 hologram, 175 integrals, 56 rhomb, 133 zone plate, 59 fringes of equal chromatic order, 28 of equal inclination, 5 of equal thickness, 4 G Gabor plate, 61 ghost lines in gratings, 75 Clan-Foucault prism, 141

INDEX

Glan-Taylor prism, 141 Clan-Thompson prism, 139 Golay cell, 297 grating angular distribution of light, 70 disperser radiometer, 261 polarizers, 157 gratings blazed, 158 chromatic dispersion, 72 resolving power, 72 gyroscope, optical, 43 H

Haidinger fringes, 11 half-tone hologram, 190 Helmholtz-Kirchhoff theorem, 50 Herpin matrix, 129 hertzian polarization, 157 hologram embossing, 206 hologram, Fourier transform, 178 Fraunhofer, 175 Fresnel, 175 Lippman-Denisyuk, 185 binary, 189 color, 180 computer, 189 half tone, 190 image plane, 179 phase, 190 rainbow cylindrical, 189 volume reflection, 174 holograms, effects of thickness in, 172 rainbow, 184 reflection, 172 transmission, 172 volume, 172 holographic bleaching, 204 interferometry, 191 materials, 201 microscopy, 195 optical elements, 197 processing, 204 recording, 204 stereograms, 184 holography, 167 applications, 191 composite, 184

337

Huygens wavelet, 49 Huygens- Fresnel principle, 49, 54 theory, 5 1 I

image-plane hologram, 179 imagery through diffusing media, 196 imaging radiometers, 252 impermeability tensor, 146 inclination factor, 52, 56 induced birefringence, 146 integrating spheres, 280 interference, 1 filters, 31 fringes, 1 interferometric radiometer, 261 interferometry applications, 38 irradiance interferometer, 24 isotropic crystals, 135 media, 113 J Jamin interferometer, 12 Jones’ matrices, 159 K

Kerr effect, 147, 153 kinoforms, 190 Kirchhoffs diffraction theory, 50, 5 1 knife-edge test, 39 L lambertian sphere, 153 laser power, measurement of, 277 lateral shearing interferometer, 17 length measurements, 41 light source polarization, 158 light sources for holography, 199 for interferometers, 6 linear polarization, 110 linear polarizers, 126, 130 Lippman-Denisyuk hologram, 185 liquid crystals, 155 liquid-crystal detectors, 302

338

INDEX

Littrow configuration, 158 Lloyd’s mirror, 1 localized fringes, I2 luminescent detectors, 302 Lummer-Gehrke plate, 30 Lyman ghosts, 76

N

Newton interferometer, 7 rings, 4, 5 Nicol prism, 139 noise in detectors, 328 nonlinear effect in holography, I72

M

MacNeille polarizing prisms, 130 Mach-Zehnder interferometer, 12, 17 magneto-optic effect, 153 matched filtering. 197 materials for optical windows, 331 Maxwell equations, 108, 134 measurement of directional reflectance, 274 of emissivity, 269 of laser power, 277 of reflectivity, 272 measuring wave plates, 145 mechanical stability in holography, 200 metal grid polarizers, 157 metal mirrors protection, 33 metals, wave propagation for, 120 meter definition, 41 Michelson fringes, 12 interferometer, 4, 9 stellar interferometer, 22 Michelson- Morley experiment, 38 microscopy, 42 microtopography. 29 Miyamoto- Wolf theory, 53 modulation transfer function, 85 Mooney rhomb, 133 Mueller’s matrices, 159 multilayer films, 35, 128 multiple reflection interferometer, 30 multiple reflections in a thin film, 126 multiple-beam interferometer, 25 multiple-pass Fizeau interferometer, 37 Twyman-Green interferometer, 37 interferometers, 37 multiple-reflection Fizeau interferometer, 28 interferometers, 25

0 off-axis holography, 168 one-step method for rainbow holograms. 186 optic axis of a crystal, 135 optical activity, 138 elements, holographic, 197 polarization, 107 transfer function, 82, 84 window material, 331 ordinary beam, 136 orthoscopic recording in holography, 186

P

P-polarization, 114 Parseval’s theorem, 69 partial coherence, 89 partially transparent materials, 231 path radiance, 236 phase contrast microscope, 90 difference in gratings, 72 holograms, 190 modulation, 146 plate, 91 retarders, 126, 130 shift upon reflection, 115 velocities in a crystal, 136 photocathodes, spectral response of, 307 photoconductive detectors, 3 1 1 response, 3 19 photoconductors, extrinsic, 317 photographic emulsions for holography, 201 photometers, 246

lNDEX photometric and radiometric nomenclature, 214 symbols, 214 units, 214 measurements, 263 standards, 264 units, 286 photometry, 213 photomultiplier, 304, 309 photon detectors, 303 photon-drag detector, 327 photoresists for holography, 204 photovoltaic devices, 322 pinhole camera, 61 plane hologram techniques, 181 plane-holograms, 169 Pockels electro-optic effect, 147 PoincarC sphere, 112, 142 point spread function, 85 polarization circular, 110 determination, 159 effects, 128 in metals, 120 in thin films, 125 elliptical, 113 linear, 110 mathematical description, 159 optical, 107 scramblers, 153 polarizer made of four mirrors, 124 made of three mirrors, 123 polarizers, grating, 157 linear, 126, 130 metal grid, 157 slit, 157 polarizing prisms, MacNeille, 130 principal angle, 122 prism beam splitters, 138 disperser radiometer, 260 polarizers, 139 processing of holograms, 204 pseudoscopic configuration in holography, 187 pupil function, 68

339

pyroelectric detector, 299 materials, 301 pyroheliometers, 246 Q

quantum detector, 303 quarter-wave stack, 35 R

radial shear interferometer, 18 radiance temperature, 241 radiation temperature, 239 radiative transfer, 223 in a vacuum, 228 in transparent media, 231 radiometer, interferometric, 261 prism disperser, 260, 261 radiometers, 246 astronomical, 249 filter, 257 imaging, 252 scanning, 252 space defense, 250 standard, 246 radiometric instruments, 246 measurements, 263 standards, 264 temperature measurements, 239 radiometry, 213 of visible light, 284 rainbow cylindrical hologram, 189 holograms, 184 Raman-Nath regime, 151 ratio temperature, 241 difference, 243 Rayleigh interferometer, 22 scattering, 99 real-time holographic interferometry, 191 recording of holograms, 204 rectangular aperture, 65 reference beam in holography, 170 reflectance, directional measurement of. 274

340

INDEX

reflection circular polarizer, 124 diffraction gratings, 78 holograms, 172 polarizers, 122 reflectivity, measurement of, 272 reflectometer, Bennett-Koehler, 274 refractometry, 39 relativity measurements, 38 resolving power in gratings, 72 of optical instruments, 79 resonance anomalies, 157 retarding waveplates, 142 reversal shear interferometers, 19 ring interferometer, 42 Rochon prism, 138 Ronchi-test, 39 rotational shear interferometers, 19 Rowland ghosts, 76 S

S-polarization, 114 Sagnac interferometer, 42 saltus problem, 52 sandwich holography, 194 scalar theory region, 157 scanning radiometers, 252 scatter-plate interferometer, 20 scattering, 49, 92 by large objects, 94 by small particles, 95 cross section, 100 schlieren techniques, 39 second Brewster angle, 121 senarmount compensator, 144 shearing interferometers, 17 signal-to-noise ratio, 247 single layer films, 125 single-layer coatings, 33 single-slit diffraction, 53 slit polarizers, 157 slit, diffracting, 57, 65 solei1 compensator, 143 space defense radiometers, 250 spatial filtering of images, 91 frequencies, 84 multiplexing method for color holograms, 181

spectral response of photocathodes, 307 spectroradiometers, 246, 256 spectroscopy, 43 spherical Fabry-Perot interferometer, 29 standard lamp, 6 radiometers, 246 standards, radiometric and photometric, 264 star diameter measurements, 39 stereograms, holographic, 184 Stokes parameters, 159 straight edge, diffracting, 57 synchronous detection, 279 T

temperature measurements, 213 radiometric, 239 temporal coherence, 6 thermal detectors, 293, 303 thermocouple calorimeters, 293 thermography, medical, 243 thin films, 32, 125 matrix, 129 polarization effects, 125 Thompson scattering, 101 three mirror polarizer, 123 three-reference beam method for color holograms, 181 time averaged holographic interferometry, 194 total internal reflection devices, 131 transfer function of lenses, 40 optical, 82, 84 transmission diffraction gratings, 78 holograms, 172 polarizers, 122 volume color holograms, 183 two dimensional array of holes, diffracting, 80

two-beam interferometers, 7, 22 two-step method for rainbow holograms, 184 Twyman-Green interferometers, 12 U

uniaxial crystals, 135

34 1

INDE X

V

visible light, radiometry of, 284 volume hologram, 172 techniques for color holograms, 183 volume-reflection hologram, 174

wave-front topography, 40 white-light compensation, 6 Wien displacement law, 220 Wollaston prism, 138 Wood anomalies, 157

W wave equations, 108 plates achromatic, 144 measuring, 145 propagation for metals, 120 in anisotropic media, 133 in isotropic media, 113

Y

Young's experiment, 1

2

zone plate, Fresnel, 59


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