Adaptive Optics Engineering Handbook

ADAPTIVE OPTICS ENGINEERING HANDBOOK EDITED IY ROBEIT K. TYSIN The University of North Carolina at Charlotte Charlotte...

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ADAPTIVE OPTICS ENGINEERING HANDBOOK

EDITED IY

ROBEIT K. TYSIN The University of North Carolina at Charlotte Charlotte, North Carolina

Copyright 2000 by Marcel Dekker, Inc. All Rights Reserved. MARCEL

D D E K K E R

MARCEL DEKKER, INC.

NEW YORK • BASEL

ISBN: 0-8247-8275-5 This book is printed on acid-free paper. Headquarters Marcel Dekker, Inc. 270 Madison Avenue, New York, NY 10016 tel: 212-696-9000; fax: 212-685-4540 Eastern Hemisphere Distribution Marcel Dekker AG Hutgasse 4, Postfach 812, CH-4001 Basel, Switzerland tel: 41-61-261-8482; fax: 41-61-261-8896 World Wide Web http:/ /www.dekker.com The publisher offers discounts on this book when ordered in bulk quantities. For more information, write to Special Sales/Professional Marketing at the headquarters address above. Copyright  2000 by Marcel Dekker, Inc. All Rights Reserved. Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, microfilming, and recording, or by any information storage and retrieval system, without permission in writing from the publisher. Current printing (last digit): 10 9 8 7 6 5 4 3 2 1 PRINTED IN THE UNITED STATES OF AMERICA

Copyright 2000 by Marcel Dekker, Inc. All Rights Reserved.

OPTICAL ENGINEERING Founding Editor Brian J. Thompson University of Rochester Rochester, New York

Editorial Board Toshimitsu Asakura Hokkai-Gakuen University Sapporo, Hokkaido, Japan

Nicholas F. Borrelli Corning, Inc. Corning, New York

Chris Dainty Imperial College of Science, Technology, and Medicine London, England

Bahram Javidi University of Connecticut Storrs, Connecticut

Mark Kuzyk Washington State University Pullman, Washington

Hiroshi Murata The Furukawa Electric Co., Ltd. Yokohama, Japan

Edmond J. Murphy JDS/Uniphase Bloomfield, Connecticut

Dennis R. Pape Photonic Systems Inc. Melbourne, Florida

Joseph Shamir Technion–Israel Institute of Technology Hafai, Israel

David S. Weiss Heidelberg Digital L.L.C. Rochester, New York

1. Electron and Ion Microscopy and Microanalysis: Principles and Applications, Lawrence E. Murr 2. Acousto-Optic Signal Processing: Theory and Implementation, edited by Norman J. Berg and John N. Lee 3. Electro-Optic and Acousto-Optic Scanning and Deflection, Milton Gottlieb, Clive L. M. Ireland, and John Martin Ley 4. Single-Mode Fiber Optics: Principles and Applications, Luc B. Jeunhomme 5. Pulse Code Formats for Fiber Optical Data Communication: Basic Principles and Applications, David J. Morris 6. Optical Materials: An Introduction to Selection and Application, Solomon Musikant 7. Infrared Methods for Gaseous Measurements: Theory and Practice, edited by Joda Wormhoudt 8. Laser Beam Scanning: Opto-Mechanical Devices, Systems, and Data Storage Optics, edited by Gerald F. Marshall 9. Opto-Mechanical Systems Design, Paul R. Yoder, Jr. 10. Optical Fiber Splices and Connectors: Theory and Methods, Calvin M. Miller with Stephen C. Mettler and Ian A. White 11. Laser Spectroscopy and Its Applications, edited by Leon J. Radziemski, Richard W. Solarz, and Jeffrey A. Paisner 12. Infrared Optoelectronics: Devices and Applications, William Nunley and J. Scott Bechtel 13. Integrated Optical Circuits and Components: Design and Applications, edited by Lynn D. Hutcheson 14. Handbook of Molecular Lasers, edited by Peter K. Cheo 15. Handbook of Optical Fibers and Cables, Hiroshi Murata 16. Acousto-Optics, Adrian Korpel 17. Procedures in Applied Optics, John Strong 18. Handbook of Solid-State Lasers, edited by Peter K. Cheo 19. Optical Computing: Digital and Symbolic, edited by Raymond Arrathoon 20. Laser Applications in Physical Chemistry, edited by D. K. Evans 21. Laser-Induced Plasmas and Applications, edited by Leon J. Radziemski and David A. Cremers 22. Infrared Technology Fundamentals, Irving J. Spiro and Monroe Schlessinger 23. Single-Mode Fiber Optics: Principles and Applications, Second Edition, Revised and Expanded, Luc B. Jeunhomme 24. Image Analysis Applications, edited by Rangachar Kasturi and Mohan M. Trivedi 25. Photoconductivity: Art, Science, and Technology, N. V. Joshi 26. Principles of Optical Circuit Engineering, Mark A. Mentzer 27. Lens Design, Milton Laikin 28. Optical Components, Systems, and Measurement Techniques, Rajpal S. Sirohi and M. P. Kothiyal 29. Electron and Ion Microscopy and Microanalysis: Principles and Applications, Second Edition, Revised and Expanded, Lawrence E. Murr

30. Handbook of Infrared Optical Materials, edited by Paul Klocek 31. Optical Scanning, edited by Gerald F. Marshall 32. Polymers for Lightwave and Integrated Optics: Technology and Applications, edited by Lawrence A. Hornak 33. Electro-Optical Displays, edited by Mohammad A. Karim 34. Mathematical Morphology in Image Processing, edited by Edward R. Dougherty 35. Opto-Mechanical Systems Design: Second Edition, Revised and Expanded, Paul R. Yoder, Jr. 36. Polarized Light: Fundamentals and Applications, Edward Collett 37. Rare Earth Doped Fiber Lasers and Amplifiers, edited by Michel J. F. Digonnet 38. Speckle Metrology, edited by Rajpal S. Sirohi 39. Organic Photoreceptors for Imaging Systems, Paul M. Borsenberger and David S. Weiss 40. Photonic Switching and Interconnects, edited by Abdellatif Marrakchi 41. Design and Fabrication of Acousto-Optic Devices, edited by Akis P. Goutzoulis and Dennis R. Pape 42. Digital Image Processing Methods, edited by Edward R. Dougherty 43. Visual Science and Engineering: Models and Applications, edited by D. H. Kelly 44. Handbook of Lens Design, Daniel Malacara and Zacarias Malacara 45. Photonic Devices and Systems, edited by Robert G. Hunsperger 46. Infrared Technology Fundamentals: Second Edition, Revised and Expanded, edited by Monroe Schlessinger 47. Spatial Light Modulator Technology: Materials, Devices, and Applications, edited by Uzi Efron 48. Lens Design: Second Edition, Revised and Expanded, Milton Laikin 49. Thin Films for Optical Systems, edited by François R. Flory 50. Tunable Laser Applications, edited by F. J. Duarte 51. Acousto-Optic Signal Processing: Theory and Implementation, Second Edition, edited by Norman J. Berg and John M. Pellegrino 52. Handbook of Nonlinear Optics, Richard L. Sutherland 53. Handbook of Optical Fibers and Cables: Second Edition, Hiroshi Murata 54. Optical Storage and Retrieval: Memory, Neural Networks, and Fractals, edited by Francis T. S. Yu and Suganda Jutamulia 55. Devices for Optoelectronics, Wallace B. Leigh 56. Practical Design and Production of Optical Thin Films, Ronald R. Willey 57. Acousto-Optics: Second Edition, Adrian Korpel 58. Diffraction Gratings and Applications, Erwin G. Loewen and Evgeny Popov 59. Organic Photoreceptors for Xerography, Paul M. Borsenberger and David S. Weiss 60. Characterization Techniques and Tabulations for Organic Nonlinear Optical Materials, edited by Mark Kuzyk and Carl Dirk

61. Interferogram Analysis for Optical Testing, Daniel Malacara, Manuel Servín, and Zacarias Malacara 62. Computational Modeling of Vision: The Role of Combination, William R. Uttal, Ramakrishna Kakarala, Sriram Dayanand, Thomas Shepherd, Jagadeesh Kalki, Charles F. Lunskis, Jr., and Ning Liu 63. Microoptics Technology: Fabrication and Applications of Lens Arrays and Devices, Nicholas F. Borrelli 64. Visual Information Representation, Communication, and Image Processing, Chang Wen Chen and Ya-Qin Zhang 65. Optical Methods of Measurement: Wholefield Techniques, Rajpal S. Sirohi and Fook Siong Chau 66. Integrated Optical Circuits and Components: Design and Applications, edited by Edmond J. Murphy 67. Adaptive Optics Engineering Handbook, edited by Robert K. Tyson 68. Entropy and Information Optics, Francis T. S. Yu 69. Computational Methods for Electromagnetic and Optical Systems, John M. Jarem and Partha P. Banerjee 70. Laser Beam Shaping: Theory and Techniques, edited by Fred M. Dickey and Scott C. Holswade 71. Rare-Earth-Doped Fiber Lasers and Amplifiers: Second Edition, Revised and Expanded, edited by Michel J. F. Digonnet 72. Lens Design: Third Edition, Revised and Expanded, Milton Laikin 73. Handbook of Optical Engineering, edited by Daniel Malacara and Brian J. Thompson 74. Handbook of Imaging Materials, edited by Arthur S. Diamond and David S. Weiss 75. Handbook of Image Quality: Characterization and Prediction, Brian W. Keelan 76. Fiber Optic Sensors, edited by Francis T. S. Yu and Shizhuo Yin 77. Optical Switching/Networking and Computing for Multimedia Systems, edited by Mohsen Guizani and Abdella Battou 78. Image Recognition and Classification: Algorithms, Systems, and Applications, edited by Bahram Javidi 79. Practical Design and Production of Optical Thin Films: Second Edition, Revised and Expanded, Ronald R. Willey 80. Ultrafast Lasers: Technology and Applications, edited by Martin E. Fermann, Almantas Galvanauskas, and Gregg Sucha 81. Light Propagation in Periodic Media: Differential Theory and Design, Michel Nevière and Evgeny Popov 82. Handbook of Nonlinear Optics, Second Edition, Revised and Expanded, Richard L. Sutherland

Additional Volumes in Preparation

Optical Remote Sensing: Science and Technology, Walter Egan

From the Series Editor

The field of adaptive optics as a focused activity is now over 25 years old. The first concepts appeared in the 1950s; however, to see it as a really active field with significant progress, we look to the literature starting in the 1970s. Adaptive optics originated from the desire to correct images degraded by atmosphereic turbulence before detection. The approach contrasts, but in many ways complements, the techniques available for postdetection processing; i.e., both pre- and postdetection processing can be used in conjunction with each other. Clearly the initial problem was with earthbound telescopes; telescopes with large apertures collect more light than smaller-diameter intruments but do not necessarily provide any greater resolution. Adaptive optics methods require that an incoming wavefront be sensed and evaluated in a time interval shorter than the fluctuation rate of the wavefront. Once the distortion of the wavefront has been determined, a real-time correction is needed to compensate for that distortion. It is the fundamental science, engineering, and technology of these methods and their implementation that have allowed adaptive optics to become a reality. Although adaptive optics started with the desire to control and improve image formation in earth-based telescopes, its methods can be applied to a variety of other systems that suffer from time-varying effects, for example, interferometric sensing, speckle systems, radiant energy delivery systems, and holographic systems, among others. The Adaptive Optics Engineering Handbook provides a detailed description and discussion of the methods of adaptive optics, including the key topics of wavefront sensing and wavefront correction, and how they are applied in practice. Brian J. Thompson

Preface

Adaptive Optics Engineering Handbook is a practical guide to the development and implementation of adaptive optics systems and technology. Chapters by experts in the various subfields of adaptive optics have contributed detailed descriptions of system design techniques, the use of wavefront sensors, deformable mirrors and wavefront estimators (reconstructors), and advanced technologies and applications. The Handbook is a unique compendium of work from 16 authors that has never before been published in book form. Chapter 1 provides a survey of the history of adaptive optics and starts the Handbook with a ‘‘how-to’’ guide by referencing many operational systems showing ‘‘how it was done by the best in the business.’’ Chapters 2 and 3 provide a detailed description of system considerations for modeling, analyzing, and designing a conventional adaptive optics system with laser guide stars. Chapters 4, 5, and 6 describe the three major subsystems and components of a system: the wavefront sensor, the deformable mirror, and the control system wavefront estimators. Chapters 7, 8, and 9 describe state-of-the-art technology in high-speed wavefront correction devices. Chapter 10 gives an overview of cutting-edge research in medical imaging using adaptive optics, and Chapter 11 covers innovative techniques for wavefront compensation that overcome field-of-view limits common to many system configurations. The Adaptive Optics Engineering Handbook is intended for scientists, astronomers, engineers, and technicians working with adaptive optics systems, or those developing optical components and associated technologies such as highresolution cameras, high-speed computers, diffractive and micromachined optics, active mirrors, and high-power lasers. Robert K. Tyson

Contents

From the Series Editor Preface Contributors 1. Introduction Robert K. Tyson 2. System Design and Optimization Ronald R. Parenti 3. Guide Star System Considerations Richard J. Sasiela and John D. Shelton 4. Wavefront Sensors Joseph M. Geary 5. Deformable Mirror Wavefront Correctors Ralph E. Aldrich 6. Innovative Wavefront Estimators for Zonal Adaptive Optics Systems Walter J. Wild 7. Micromachined Membrane Deformable Mirrors Gleb Vdovin

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8. Surface Micromachined Deformable Mirrors William D. Cowan and Victor M. Bright 9. Liquid Crystal Adaptive Optics Gordon D. Love 10. Wavefront Sensing and Compensation for the Human Eye David R. Williams, Junzhong Liang, Donald T. Miller, and Austin Roorda 11. Wide Field-of-View Wavefront Sensing Erez N. Ribak

Contents

Contributors

Ralph E. Aldrich, Ph.D. Technology Consultant, Acton, Massachusetts Victor M. Bright, Ph.D. Department of Mechanical Engineering, University of Colorado at Boulder, Boulder, Colorado William D. Cowan Air Force Institute of Technology, Wright-Patterson AFB, Ohio Joseph M. Geary, Ph.D. Center for Applied Optics and Department of Physics, University of Alabama in Huntsville, Huntsville, Alabama Junzhong Liang* University of Rochester, Rochester, New York Gordon D. Love, Ph.D. Department of Physics and School of Engineering, University of Durham, Durham, United Kingdom Donald T. Miller‡ University of Rochester, Rochester, New York Ronald R. Parenti, Ph.D. Optical Communications Technology, Massachusetts Institute of Technology Lincoln Laboratory, Lexington, Massachusetts

Current affiliations * Intel Corporation, Santa Clara, California. ‡ Indiana University, Bloomington, Indiana.

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Contributors

Erez N. Ribak, Ph.D. Department of Physics, Technion–Israel Institute of Technology, Haifa, Israel Austin Roorda, Ph.D.* College of Optometry, University of Rochester, Rochester, New York Richard J. Sasiela, Ph.D. Radar Systems Engineering, Massachusetts Institute of Technology Lincoln Laboratory, Lexington, Massachusetts John D. Shelton, Ph.D. Massachusetts Institute of Technology Lincoln Laboratory, Lexington, Massachusetts Robert K. Tyson, Ph.D. Department of Physics, University of North Carolina at Charlotte, Charlotte, North Carolina Gleb Vdovin, Ph.D. Information Technology and Systems, Delft University of Technology, Delft, The Netherlands Walter J. Wild† University of Chicago, Chicago, Illinois David R. Williams University of Rochester, Rochester, New York

* University of Houston, Houston, Texas. † Deceased.

1 Introduction Robert K. Tyson University of North Carolina at Charlotte, Charlotte, North Carolina

Adaptive optics are used to enhance the capability of optical systems by actively compensating for aberrations. These aberrations, such as atmospheric turbulence, optical fabrication errors, thermally induced distortions, or laser device aberrations, reduce the peak intensity and smear an image or a laser beam propagating to a target. Normally, increasing the aperture size decreases the diffraction angle and makes an image sharper. However, for many optical systems, the beam or image quality is limited, not by the aperture, but by the propagation medium. The twinkling of stars or distorted images across a paved road on a hot summer day is caused by turbulence in the atmosphere. Distortions like these can be corrected by adaptive optics. The result of more than three decades of technology development, adaptive optics systems are being used at observatories around the world. This Handbook is a guide to the implementation of adaptive optics, a collection of analysis tools for system design and development, and an introduction to up-to-date developments in the multidisciplinary adaptive optics field. The principal uses for adaptive optics are improving image quality in optical and infrared astronomical telescopes, imaging and tracking rapidly moving space objects, and compensating for laser beam distortion through the atmosphere (1). Although these missions differ, the techniques used to compensate for the underlying distortions are similar. Adaptive optics are real-time distortion-compensating systems (2). Although many types of adaptive optics systems have been tried in the laboratory or field, the most common adaptive optics system in use today consists of three subsystems. Figure 1 shows the subsystems in an observing application. A wavefront sensor measures the distortion induced by the atmosphere by evaluating the light from a natural source or an artificial beacon placed high above the telescope.


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Figure 1 Basic components of an adaptive optics system.

An active mirror, called a deformable mirror, can rapidly change its surface shape to match the phase distortions measured by the wavefront sensor. A control computer is used to evaluate the wavefront sensor measurements and translate the signals into control signals to drive the actuators of the deformable mirror. Over large apertures, like those used in modern astronomical telescopes, the wavefront tilt is a dominant effect which, as it varies rapidly during the exposure time, further distorts the image. Adaptive optics systems often offload the tilt wavefront measurement to a specialized tilt control mirror to remove the large stroke requirements from the deformable mirror. Because the adaptive optics compensation is performed by macroscopic movement of an optical element, the system is called inertial. Because the compensation is linearly proportional to the disturbance, the system is considered linear. These terms are in contrast to nonlinear phase conjugation techniques which employ atomic or molecular changes in optical materials and exploit their nonlinear phase compensation properties. Nonlinear systems are discussed elsewhere in the literature and will not be a topic of discussion in this practical examination of adaptive optics. While most adaptive optics systems look like Fig. 1, there are innovative variations on the standard design. For example, the 6.5-m Smithsonian Institution–University of Arizona Monolithic Mirror Telescope located at the Steward Observatory in Arizona will put the deformable mirror on the Cassegrain second-


Introduction

3

ary mirror surface instead of using the separate deformable mirror like that shown in Fig. 1. In astronomy, adaptive optics provide the means for increasing the angular resolution in direct imaging, and they provide higher performance for many spectroscopic, interferometric and photometric measurements. For example, if the scientific goal is to make a simple detection of a faint point source such as a star in the presence of a bright sky background, the final detected signal-to-noise ratio is proportional to D/α, where D is the aperture diameter of the telescope’s primary mirror and α is the angular resolution at the time of detection. Large telescopes now have apertures up to 10 meters, today’s practical engineering limit. From the above ratio, decreasing α is just as important as increasing D. Adaptive optics provides the opportunity to decrease α to the theoretical limit.

I. THE EARLY HISTORY There was no one inventor of adaptive optics. Systems and technology have evolved over the past 30 years to become a major subset of modern astronomical telescopes and high energy laser propagation systems. More than 2000 research papers have been published, and hundreds of investigators, engineers, and technicians have provided advances in theory and technology that support the multidisciplinary field of adaptive optics. Because of the recent, rapid rise in interest, usage, and development of adaptive optics (see Fig. 2), we find that technological advances, experimental results, and even breakthroughs are reported almost on a weekly basis. The concept of adaptive optics is not particularly new, and like many excellent ideas it seems to have been discovered more than once. In 1953 Horace Babcock (3), then director of the Mount Wilson and Palomar observatories, was the first to suggest how one might build an astronomical adaptive optics instrument. Independently, in 1957 Vladimir P. Linnik (4) described the same concept in the Soviet journal Optika i Spektroskopiya. Although Babcock was very specific in his design concepts and had the resources of Mount Wilson and Palomar at his fingertips, his adaptive optics system was not built. It was simply beyond the technological capabilities of the 1950s. The first practical developments in adaptive optics technology followed in the late 1960s when American industry, driven by requirements to image satellites and project high energy laser beams into space, created the first operational systems. The first fully operational adaptive optics system was installed on a surveillance telescope at Haleakala Observatory in Maui, Hawaii, where it imaged satellites (5) launched by the Soviet Union. Components and full closed-loop systems were developed during the 1970s and early 1980s under various programs by companies such as Itek, Hughes, Ford Aerospace, United Technologies,


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Figure 2 Adaptive optics-related publications in the open literature are rising at an increasing rate. This sample from two domestic sources, Optical Engineering and Proceedings of the SPIE (both published by SPIE, Bellingham, WA), shows the trend in interest, proliferation, and advancement of adaptive optics throughout the world. The excursions in 1994 and 1998 are due to the quadrennial space- and ground-based astronomy meeting held in each of those years that draws large international participation.

Lockheed, Perkin-Elmer, MIT Lincoln Laboratory, Ball Aerospace, and Adaptive Optics Associates. By the mid-1980s, Thermotrex Corporation, United Technologies Optical Systems, Laserdot, and the European consortium ONERA were building and supplying wavefront sensors, deformable mirrors, and control systems. Through acquisitions and corporate spinoffs, companies like Hughes Danbury Optical Systems (later Raytheon Optical Systems) and Xinetics continue to advance the state of the art in adaptive optics. During the development period from 1965 to the mid-1980s, theoretical developments paralleled the progression of more sophisticated hardware. David Fried developed much of the theory to understand the phenomena of adaptive optics phase conjugation and provided a number of tools for performance prediction and assessment (6). Darryl Greenwood wrote a seminal paper describing the temporal nature of adaptive optics compensation (7). Supporting work by Tyler (8) and Lukin (9) contributed greatly to the understanding and practical analysis methodology of system requirements and performance parameters. Many others


Introduction

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contributed volumes on adaptive optics theory and analysis, primarily in the areas of thermal blooming compensation, wavefront control algorithms, and temporal control theory applied to multichannel adaptive systems. The need for a bright stellar source was always a concern for astronomical applications of adaptive optics. To operate, a wavefront sensor must have sufficient light to overcome photon noise and background noise with enough light left over to form the image. In astronomy, there are few stars of scientific interest that are sufficiently bright. For imaging uncooperative satellites, reflected light is often too dim or nonexistent. In 1985, French astronomers Foy and Labeyrie published work detailing how one might use backscatter from a laser focused to a point in the atmosphere as an artificial beacon (a guide star) for astronomical adaptive optics (10). As work progressed in the astronomy community to build and test a laser powerful enough to have sufficient backscatter for the FoyLabeyrie method, the political changes in eastern Europe began to have an effect upon adaptive optics. Since the beginning of the 1980s, classified U.S. military work was addressing the problems of projecting high energy laser beams from the ground to space for missile defense and secure communications. Suggestions by Adaptive Optics Associates founder Julius Feinleib and alternative concepts suggested by Richard Hutchin (11), with theoretical support from Fried, showed how laser guide stars would be a means to avoid the problems associated with uncooperative targets or point-ahead angles for space relays. The research from 1982 at the U.S. Air Force Starfire Optical Range directed by Robert Fugate advanced the laser guide star concept and produced a wealth of information about laser performance requirements, adaptive optics system operation, atmospheric physics, and closed-loop images of spaceborne objects (12). By 1991, the bulk of the military work on laser guide starts was declassified and made available to astronomers around the world. Low altitude Rayleigh scattering has a serious drawback for guide star applications. All the atmosphere above the guide star, nominally at 20 km altitude, is undetected and still distorts the image. In 1982 Will Happer of Princeton University suggested using resonant backscatter from mesospheric atomic sodium for a high altitude (90 km) laser guide star. This suggestion was experimentally demonstrated at the Starfire Optical Range once a laser with sufficient power was obtained at the 589 nm sodium line. One wants the guide star to be close enough in the sky to the target object so that their light is affected by the same atmospheric path. Otherwise the coherence between the reference wavefront and the light from the target object will be lost. The time scale for readjusting the electro-optics is therefore about r0 /ν, where ν is the wind velocity in the turbulent atmosphere and r0 is the coherence length of the atmosphere. On an average night at an astronomical observatory, r0 /ν is of order 30 ms at visual wavelengths. To monitor and then remove the turbulence,


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an electro-optic control system must operate about 10 times faster than the atmospheric changes. While these restrictions discouraged the development of visual-light astronomical adaptive optics systems before 1980, there was a clever solution for the military system designed for satellite surveillance. At the Haleakala Observatory the adaptive optics system was used primarily in twilight hours. During twilight, sky background contamination is relatively low, yet an Earth-orbiting satellite is still illuminated by sunlight. Bright glints of sunlight reflected off the satellite itself provide the reference wavefront. There are many cases where the object itself, such as a sunlit satellite, is insufficient for wavefront sensing. In astronomy, there are billions of stellar objects too dim for sensing and not near enough in the sky to bright objects. For high energy laser propagation to uncooperative targets or satellite tracking and imaging, an artificial source must be placed above, or high in, the atmosphere to provide photons for the wavefront sensor and subsequent compensation. Lasers actually provide only partial correction, because a natural star still is required for the lowest-order (tip-tilt) correction. The laser light experiences equal and opposite overall tilt upon travelling up into the atmosphere and returning. Chapter 2 has a detailed discussion and explanation of a number of system design tools for laser guide star systems.

II. ADAPTIVE OPTICS SUBSYSTEMS: WAVEFRONT SENSORS The wavefront sensor (located at the bottom of the adaptive optics system shown in Fig. 1) aims to squeeze from a minimal number of photons the maximum amount of wavefront information possible. Dividing the available light into the image and the wavefront measurement instrument is a balancing act. If fewer photons can be used, fainter guide stars can be used, and the amount of light becomes available for interesting targets. Laser guide stars do their best work correcting higher order wavefront irregularities. To achieve this goal it is important that the detector have the highest quantum efficiency and the lowest system induced noise. While some researchers considered avalanche photodiodes to be the detector of choice, newer systems rely on custom-designed charge-coupled devices with quantum efficiencies approaching 80–90%. These CCDs use on-chip amplifiers designed in such a way as to match the CCD readout rates to the time scale of the variations in the wavefront (13). Early military adaptive optics systems used both shearing interferometers and those of the Shack-Hartmann design (see Fig. 3). The most common type


Introduction

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Figure 3 Three principal types of wavefront sensors.

now being used is a Shack-Hartmann sensor which relies on a lenslet array to create a grid of subimages, each of which measures the local slope of the wavefront. The rectangular grid geometry ordinarily used with the Shack-Hartmann sensor is conceptually simple. For measuring the most common atmospheric disturbances, radially symmetric low order aberrations, a curvature sensor can be used. It compares the signal strength from two equally defocused images, one just inside and the other just outside the focal plane (14). The two-dimensional difference between these signals provides information on low order wavefront perturbations. In addition to the imaging system configuration with a separate wavefront sensor, systems can make use of the image signal itself for sharpening the images. When there is no explicit wavefront measurement and the sharpness of the image is measured in the image plane, varying the correction process can converge toward the ‘‘best image.’’ This is analogous to trying on a number of pairs of glasses to find out which one is best. Trial and error, with a ‘‘sharp’’ image as the figure of merit, is the basis of this approach. Other approaches, such as multidither, have been used in laser propagation but are range and sensitivity dependent.


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Details of wavefront sensing techniques and associated electro-optical instrumentation can be found in Chapter 4. The important parameters regarding the state of technology of wavefront sensors are • Number of subapertures (a measure of spatial resolution) • Type of sensor: Hartmann, shearing interferometer, or curvature (determines complexity; enabling technologies, processor requirements) • Sensor geometry (affects spatial resolution and processor requirements). • Detector type and characteristics, quantum efficiency and noise (specifies source brightness and determines bandwidth) • Wavelength (related to atmospheric parameters and detector choice) • Readout rate or frame rate (constrains closed-loop bandwidth) III. ADAPTIVE OPTICS SUBSYSTEMS: DEFORMABLE MIRRORS Once the wavefront is measured and the control signal determined, an optical element must be moved to change the phase of the beam of light. The tilt of a beam can be removed with fast tilting or scanning mirrors. For larger beams, particularly those required for transmission of high energy lasers, specialized designs are used. For correcting higher order aberrations, there are three dominant deformable mirror (DM) types (see Fig. 4). A segmented mirror, with individually controlled tip, tilt, and piston motion on the segments can be used. These mirrors have the advantage of segments being easily replaced, matching closely to a geometrically registered wavefront sensor, and having an unlimited aperture (by being able to just add more segments). Its primary drawback is the gap between segments which can scatter and diffract energy in an undesired and uncontrolled manner. The gaps can be avoided by using a continuous faceplate with an array of actuators behind it. The actuators push and pull on the surface. Since the surface is continuous, there is some mechanical crosstalk from one actuator to the next (called the influence function), but that can be controlled by mechanical design and through software within the control computer. Because most deformable mirrors have regular arrays of actuators, either square or hexagonal geometries, the alternate pushing and pulling of adjacent actuators can impart a patterned surface resembling a waffle. The ‘‘waffle mode’’ can appear in images because of its regular pattern which acts as an unwanted diffraction grating. Computer processing can be employed to avoid or eliminate this drawback. Curvature wavefront sensors measure the second derivative (Laplacian) of the wavefront. A bimorph mirror is made up of PZT material capable of deforming when voltage is applied across a region. The natural shape of the bimorph


Introduction

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Figure 4 Three principal types of deformable mirrors.

is also in the shape of curvature, making it ideal for operation in conjunction with a curvature sensor. The computer reconstructor is very simple because there is a one-for-one relationship between the sensor signals and the mirror commands. Unfortunately, bimorphs are limited in their ability to compensate very high order spatial frequencies. Details of deformable mirrors and their construction can be found in Chapter 5. The important parameters regarding the state of technology of deformable mirrors are • Number of actuators (specifies spatial resolution). • Type: segmented, continuous faceplate, or bimorph • Actuator, or segment, separation (related to spatial resolution and manufacturing complexity). • Faceplate material and cooling requirements (determines application and total power) • Type of actuators (related to drive voltages, speed, heat buildup, total stroke, and hysteresis).


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• Total stroke and hysteresis (limits aperture, total amount of compensation, and bandwidth)

IV. ADAPTIVE OPTICS SUBSYSTEMS: LASER GUIDE STAR PROJECTION A third major technology issue is how to create and project laser guide stars in the atmosphere. The two atmospheric scattering processes that can provide the brightest laser guide star return signal are Rayleigh scattering of photons off molecules in the stratosphere and resonance scattering off sodium atoms in the mesosphere. For Rayleigh scattering, there are two laser systems suited to the task: a 530-nm/550-nm copper-vapor laser and a 351-nm excimer laser working with XeF (15). Both systems are available as reliable commercial products capable of delivering, respectively, 200 W and 50 W output power, sufficient for each to create a tenth magnitude star at 10–20 km altitude. The second laser guide star technique relies on resonant scattering at 589 nm off the neutral sodium atoms present in abundance at an altitude of about 92 km. For this technique, special experimental lasers must be built and tailored to the requirements of the sodium excitation. The copper-vapor and excimer systems operate in the pulsed mode, while sodium-wavelength lasers are either pulsed or continuous wave. Laser guide star systems produce a reference wavefront that emanates from a finite altitude in the Earth’s atmosphere, and this leads to complications that have been given the name focal anisoplanatism. Because the laser guide star is not at infinite focal distance, all of the atmosphere is not sensed and there is an error in the focus component of the wavefront. Sodium laser guide stars have less focal anisoplanatism because they are created 5–10 times higher in the atmosphere than Rayleigh laser guide stars (see Fig. 5). Trades must be performed to properly design a laser guide star. If the raw laser beam has poor beam divergence properties, which is generally the case for commercial excimer and copper-vapor lasers, the telescope’s full primary mirror must be used as a primary element in the laser project system. If the laser emits a near-diffraction-limited output beam, the projection system becomes simpler. A side-mounted projection system can be placed on the telescope structure and boresighted with the main telescope (the current design of Keck and one of two proposals for the NOAO’s Gemini 8.5-m telescope). Alternatively, the laser beam can be sent to the top of the telescope structure, where it can be projected along the telescope’s optical axis from behind the Cassegrain secondary mirror; this setup has been proposed for the monolithic mirror telescope (MMT). In the MMT design, low-altitude scattered light that would otherwise be a contaminant remains in the shadow of the secondary mirror as viewed from the astronomy detectors. Details of laser guide star systems considerations can be found in Chapter 3.


Introduction

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Figure 5 Laser guide stars can be produced with Rayleigh scattering at an altitude of about 20 km or with resonant sodium backscatter at an altitude near 90 km.

The important parameters regarding the state of technology of laser guide stars are • Laser wavelength (specifies altitude, Rayleigh at 20 km or sodium at 90 km). • Laser power (specifies brightness of LGS) • Pulse repetition rate and pulse length (related to brightness, altitude, and sensor bandwidth)

V.

ADAPTIVE OPTICS SUBSYSTEMS: CONTROL COMPUTERS

The control computer, often called a wavefront reconstructor, takes the signals from the wavefront sensor. Images from a curvature system, spot intensity pat-


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terns from a Hartmann sensor, or interferograms from a shearing interferometer are used to compute appropriate drive signals for the deformable mirror. The speed and accuracy of this computation directly affect the closed-loop bandwidth of the system. Early systems used analog resistor matrices to do the calculations and did not limit bandwidth. Current systems, to make better use of changing conditions, variations in the wavefront sensor configuration, high noise conditions, and programmability, use digital signal processors to convert wavefront sensor signals to drive signals. For ‘‘thousand-channel systems,’’ and the like, the multiple control processors act in parallel to keep the overall bandwidth in the range of atmospheric time scales. Most commonly, combinations of Intel i860 chips or Texas Instruments TMS320C40 or TMS320C50 DSP chips are used on VME boards or other interfaces. Processors with hundreds of MFLOPS (millions of floating point operations per second) are required for atmospheric compensation systems. The important parameters regarding the state of technology of control computers are • Type: analog or digital (relevant to system complexity and flexibility). • Processor brand and number of units (related to parallelism and bandwidth) • MFLOPS (relates to system complexity and ultimate processor delay time)

VI. ADAPTIVE OPTICS TECHNOLOGIES Adaptive optics historically developed because of advances in various engineering fields that came together and progressed at the right time. Better visible and infrared sensors and detectors, high speed digital electronic processing, reliable and low-cost electromechanical actuators, and rugged and powerful lasers all advance the adaptive optics systems where they are used. New technologies provide for advances and breakthroughs in adaptive optics beyond mere improvements in existing ground-based imaging systems or laser propagation systems. These technologies will make it possible for a proliferation of miniature, lowcost adaptive optics systems in medicine, communications, surveillance, weather and climate monitoring, astronomy, and tactical military systems. Wavefront sensors are dependent upon the device that converts photons to an electronic signal. Many systems use avalanche photodiodes, but, because of their versatility, speed, wavelength range, and small size and weight, two-dimensional CCD, CID, or CMOS cameras are used as the focal planes for wavefront sensors. Rapid improvements in size (number of pixels), sensitivity, residual noise, and quantum efficiency allow for operation with fainter wavefront beacons.


Introduction

13

Being able to divide the wavefront into smaller and smaller regions for higher and higher resolution links the advances in wavefront sensor technology to the advances in visible and infrared cameras and focal planes. In 1996, a breakthrough in technology made it possible to consider very low cost and even ‘‘throw-away’’ adaptive optics. For decades, advances in sensors and electronic processing have helped to bring down the cost of adaptive optics. Only one subsystem, the deformable mirror, remained at a costly level because of the difficulty in manufacturing multiple units and the large size of the mirrors themselves. A few researchers in the microelectromechanical (MEM) field have recently developed prototype deformable mirrors using conventional CMOS microchip technology. The result has been demonstrations of small (1 cm) mirrors with hundreds of actuators, a few microns of stroke, an optical wavefront surface better than λ/20, actuator bandwidths up to 10 kHz, and projected production cost of about $200 per unit. This reduction in cost (by 3–4 orders of magnitude) and reduction in required power and weight (by 3 orders of magnitude) make possible many new applications of adaptive optics that were once considered prohibitive. Chapter 7 has details about these devices and applications. In recent years, there have been considerable advances in phase modulators such as liquid crystals, which can be used for low-cost adaptive optics. The first devices were only able to extend the on-off characteristics of spatial light modulators to make crude inroads into adaptive optics applications. For example, ferroelectric liquid crystal devices had one wave of throw in 10 µs, but were limited to only bistable operation (on-off). Now, nematic liquid crystals with up to 10 waves of throw with analog (continuous) control are possible. Although the nematic devices are slower (40 ms for one wave, i.e., 25 Hz), they are polarization independent and show high optical quality (λ/15). Devices up to 127 pixels have been reported (16). Technology improvement is expected by changing the viscosity of the crystal, varying its operational thickness, and optimizing the control voltage. See Chapter 7 for the details on recent liquid crystal adaptive optics development and construction. In addition to devices used for wavefront correction, new advances in optics, primarily binary optics and diffractive optics, have application directly in wavefront sensors and beam shaping applications. Diffractive optics can be applied to make very efficient, high fill factor lenslet arrays for the input aperture of Hartmann wavefront sensors. By making these devices accurate and repeatable, wavefront sensors can be integrated with small correction devices (liquid crystals or MEMs) to make complete single-chip adaptive optics systems for a fraction of the cost of labor-intensive discrete systems. Although lasers themselves are nearly 40 years old, and their use as potential weapons has been under study for nearly that long, the applications of new types of lasers and advances in new materials for lasers remain a strong element of adaptive optics technologies. Lasers with internal adaptive optics, either linear


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or nonlinear, remain a very intense area of development. By increasing the wavefront quality of an emerging beam, the extraction efficiency is improved, thereby making the resultant application cheaper and more useful. Specialized lasers for adaptive optics, such as mid- to high-power lasers for Rayleigh beacons and tuned lasers for 589-nm sodium laser beacons, are the subject of much of the work at institutions developing adaptive optics. Faster detectors and faster deformable mirrors will not be the bandwidthlimiting components of adaptive optics. With hundreds of channels of information to process at kilohertz rates, the numerical bottleneck may be the digital signal processing. Two costs are related to driving up the speed of the wavefront control system. Parallel processing requires multiple special purpose electronic boards and processors along with software that can be adaptable and robust to make use of the flexibility of the processors. Although the current state of electronic processing can handle closed-loop adaptive optics bandwidths of 100–200 Hz, high resolution imaging or high speed modulation of communications traffic while tracking fast moving low-Earth-orbit satellites requires bandwidths of 10 times that (⬃2500 Hz). Processing delays, for hundreds of floating point matrix multiplies, must be reduced to less than 100 µs.

VII. A SURVEY OF OPERATIONAL SYSTEMS Rather than presenting the engineering details of existing operational adaptive optics systems in this volume, which would be impractical and outdated within months, we present here a summary of the systems, subsystems, technologies, and developers. Specifics about each system can be obtained in working documents and current literature by each of the investigators. At the present time there are many new adaptive optics programs in the design and construction phase. By far, the most common application of adaptive optics is for imaging. Since 1982, adaptive optics (AO) have been used for imaging through the atmosphere using the basic components that consist of a wavefront sensor (WFS), a corrector like a deformable mirror (DM), a control computer to perform the real-time numerical calculations, and sometimes a laser guide star (LGS). A. U.S. Air Force Starfire Optical Range The Starfire Optical Range (SOR) now consists of a 3.5-m telescope, built by Contraves Brashear Systems (Pittsburgh, PA), with a high speed, multielement adaptive optics system. The deformable mirror (Xinetics, Devon, MA) has 941 lead-magnesium-niobate (PMN) actuators. A 700-channel Hartmann-Shack wavefront sensor provides 1400 slope commands to the wavefront reconstructor


Introduction

15

and signal processor. The wavefront sensor operates at a wavelength between 0.45 and 1.0 µm. The processing electronics, wavefront reconstructor, wavefront slope processor and most other associated electronics were built ‘‘in house’’ at SOR by USAF or contractor employees. The imaging camera is a 2000 ⫻ 2000 pixel CCD. The reported frame rate for the wavefront sensor is 2500 Hz. The reported closed-loop bandwidth is 80 Hz. Because the SOR is used primarily for satellite imaging (SOR reports only 10% usage for astronomy), the pseudowind as the telescope slews across the sky induces very high speed aberrations. The SOR 3.5-m system has operated near diffraction-limited at 0.8 µm wavelength. SOR also has a 1.5-m telescope with an adaptive optics system that is ‘‘mostly decommissioned.’’ The SOR 1.5 m was clearly the state-of-the-art (577 actuators, 500 subaperture wavefront sensor, 1-kHz frame rate) until the completion of the 3.5-m system. The 1.5-m system had a 128 ⫻ 128 pixel CCD (MIT/ Lincoln Lab) with 10 electrons read noise and 90% quantum efficiency. At the SOR site there is a 200-W Cu vapor laser (Oxford Lasers, UK) to produce a Rayleigh guide star. The current 3.5-m telescope has a number of beam paths that can be used to alternately select various lasers for atmospheric probing, guide star generation, or other testing (17).

B.

Canada-France-Hawaii Telescope

The most advanced operational nonmilitary adaptive optics system is installed on the Canada-France-Hawaii 3.6-m telescope (CFHT) on Mauna Kea (18). CFHT is a consortium that includes Canada, France (including French industries such as ONERA), and the University of Hawaii Institute for Astronomy. The adaptive optics system installed on CFHT has evolved over the past six years. The original system was an experimental adaptive optics system with a 13-element bimorph mirror and a curvature sensor. The second generation, called Pueo (Hawaiian for ‘‘owl’’) or the AO Bonnette, had a 19-element bimorph deformable mirror built by CILAS (formerly Laserdot in France). The wavefront sensor was a 19-channel curvature sensor using avalanche photodiodes as the detectors. The computer was also built by Laserdot. About 1996, the system was again upgraded to a 36channel system called Hokupa’a (Hawaiian for ‘‘immovable star’’). The curvature wavefront sensor operates in the band 0.5 to 1.0 µm. The sensor readout rate is 1.2 kHz. The SPARC2 wavefront processor has 0.2-ms delay, resulting in a reported closed-loop bandwidth of 120 Hz. Near-diffraction-limited imaging has been achieved near 1.0 µm with diffraction-limited observations at 2.2 µm. CFHT plans to install a sodium laser guide star within the next few years. The projected brightness from a 5-W Na LGS will be magnitude 11. A Strehl ratio of 0.1–0.3 at 1.2 µm is expected, 0.3–0.5 at 1.6 µm, and 0.4 to 0.8 at 2.2 µm with essentially 100% sky coverage.


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C. European Southern Observatory The European Southern Observatory (ESO) has developed adaptive optics for incorporation on the 3.6-m telescope at La Silla, Chile. The architecture of the telescope-adaptive optics system was developed mostly in Garching, Germany, the headquarters of ESO, in conjunction with Max Planck Institute. Laboratories also are located in Garching. French and Italian collaborators, primarily the Observatoire de Paris, ONERA, CILAS, LEP, and Shakti are responsible for the principal adaptive optics hardware components, the wavefront sensor and the deformable mirror. The WFS has evolved from a series of systems, first called COME-ON (19) (an acronym of the collaborators), to COME-ON⫹ (20), and now to the current generation, ADaptive Optics for Near Infrared Systems (ADONIS). COME-ON had a first generation Laserdot bimorph deformable mirror and an ONERA-built WFS with 20 subapertures (5 ⫻ 5 configuration with corners cut off) and an intensified Reticon array (QE ⫽ 0.1). The processor was a hardwired device with sampling at about 100 Hz, resulting in a closed-loop bandwidth of 10 Hz. The computational delay in the processor was 2–4 ms. The science wavelength was primarily in the infrared at 1–5 µm. ADONIS is being built for the 3.6-m ESO New Technology Telescope (NTT) at La Silla (21) and as a test bed for the ESO 8.2-m Very Large Telescope (VLT) (22), an array of four 8-m telescopes at Paranal, Chile. ADONIS has a 52-actuator CILAS DM with piezoelectric (PZT) actuators. The WFS is a 32subaperture (7 ⫻ 7 array with corners cut off ) system with a Reticon array for wavefront sensing in the band 0.4–0.9 µm. For low flux conditions, the WFS will use an EBCCD array (LEP, France) that has a QE of 0.1 and 0.27 electrons/ pixel noise. The sample rate can be varied from 25 to 200 Hz with a closed-loop bandwidth specified at 33 Hz. The delay in the EBCCD is reported to be less than 1 ms. The wavefront processor uses Texas Instruments C40 processors. The highest Strehl ratio reported for the system is about 0.2 in the K band with a magnitude 10.5 guide star brightness (23). The system for the VLT will be coordinated with the French company SFIM Industries Etablissement d’Asnieres.

D. Telescopio Nazionale Galileo (TNG) The most significant program within Italy is the adaptive optics module for the Telescopio Nazionale Galileo (TNG) (24). The program is named AdOpt@TNG (25). The bulk of the work is performed at Asiago Astrophysical Observatory. Groups in Milano-Merate are developing the tip/tilt system. The wavefront sensor prototype is being tested at Arcetri (Firenze, Italy). The system consists of the following elements (26): The wavefront sensor is either a 4 ⫻ 4 Shack-Hartmann configuration, an 8 ⫻ 8 Shack-Hartmann,


Introduction

17

or a flexible, but slower Shack-Hartmann configuration. The wavefront control computer is a VME-based design with TI C40 boards to measure centroid, estimate slopes, and command the deformable mirror. An ELTEC EUROCOM-7 68040 CPU is the basic processor, and a Loughborough Sound Images (UK) DSP 56001 board is used. The wavefront sensor will use an EEV39 80 ⫻ 80 CCD array with 1-kHz frame rate readout electronics from ElettroMare (La Spezia, Italy). The maximum anticipated read noise is 10 electrons at 500 Hz. The 96actuator deformable mirror will come from Xinetics. In addition to the hardware development at Asiago, a large amount of unique work on determining global tilt from artificial laser guide stars is done by the astronomers at the Astronomical Observatory of Padova (Ragazzoni et al.) and collaborators at Arcetri. E.

ALFA, Calar Alto

In addition to ESO, the Max-Planck Institutes for Astronomy (Heidelberg, Germany) and the Max Planck Institute for Extraterrestrial Physics (Garching, Germany) have installed a laser guide star and adaptive optics system on the 3.5-m telescope at Calar Alto, Spain. The AO system, adaptive optics with a laser for astronomy (ALFA), is one of only three operational sodium laser guide star astronomical telescopes in the world (27). The AO instrumentation was integrated as a turnkey system by Adaptive Optics Associates, Inc. (AOA). The first generation deformable mirror built by Xinetics has 97 PMN actuators and 2-µm maximum stroke. The wavefront sensor, built by AOA, is a 100-subaperture system using an MIT/LL focal plane array with 64 ⫻ 64 pixels with a 1206 frame/s readout rate and 6 electrons/pixel noise. For dim guide stars, this chip can be read out at 60 Hz, but the electron read noise increases to nine electrons because of higher dark current. The processor uses 20 Texas Instruments TMS320C40 boards and Motorola 68060 CPUs. It can calculate 15 aberration modes in 0.7 ms (900 Hz closed loop) from a subset of 20 subapertures. A 350-actuator Xinetics DM is expected to replace the earlier version. ALFA also has a state-of-the-art sodium laser guide star. The LGS is a 3.75-W dye laser from Coherent (Santa Clara, CA) pumped by a 25-W Ar ion laser (28). This produces a magnitude 9–10 LGS. A natural guide star is used for tip/tilt, stabilization in a 5–10 Hz control loop. The tip/tilt camera is a EEV CCD39 (80 ⫻ 160 pixel) camera from AstroCam Ltd., Cambridge, UK (29). F.

Lick Observatory

Lawrence Livermore National Laboratory (LLNL) supported the Lick observatory in finding usage for technologies such as the AVLIS laser used in laser isotope separation. The adaptive optics system (30) consists of a 127-actuator (61-active) deformable mirror built by LLNL. The DM was reported to have a 4-


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µm stroke. Adaptive Optics Associates built the 37-subaperture Shack-Hartmann wavefront sensor using a 64 ⫻ 64 pixel CCD array built by MIT/LL. The array was reported to have seven electrons read noise at 1200 fps. The controller used a 160 MFLOP Mercury VME board with four Intel i860 chips. The closed-loop bandwidth was reported to be 30 Hz. The Lick adaptive optics system was the first commercial/university observatory to use a sodium guide star (31). The LLNL laser guide star is a frequency doubled Nd:YAG operating on the Na line (589 nm), with 18W power, 100-ns pulses, 11-kHz rep rate, and projected out of a 30-cm projection telescope. An updated LGS is based on the Oxford Lasers (UK) Cu vapor pumped dye laser. This system has a pulse repetition frequency of 26 kHz with 32-ns pulse length and a peak power of 1100 W. The LGS was supported by a natural guide star tilt loop operating at 120 Hz. G. Mt. Palomar The Mt. Palomar adaptive optics system was retrofitted to the 50-year-old 200inch (5 m) Hale telescope (32). The system consists of a Xinetics 349-actuator DM with PMN actuators and 4-µm stroke. The 241-subaperture wavefront sensor was built by AOA. The WFS contains a MIT/LL 64 ⫻ 64 pixel CCID with six electrons read noise and 600 fps readout rate. The system was integrated at the NASA Jet Propulsion Laboratory and placed on the telescope in 1997. H. Mt. Wilson The University of Illinois Seeing Improvement System (UnISIS) adaptive optics system (33) on the Mt. Wilson 2.5-m telescope contains a 177-actuator Xinetics DM and a 351-nm Rayleigh guide star. For natural guide stars a 64 ⫻ 64 pixel MIT/LL sensor is used. The sensor has 6.4 electrons read noise and approximately 90% peak QE at 700 nm. For the 351-nm LGS the detector is an EEV model 39A 80 ⫻ 80 pixel CCD (34). I. Monolithic Mirror Telescope Retrofit of the 6.5-m multiple mirror telescope (MMT) on Mt. Hopkins, AZ, with a monolithic 6.5-m primary mirror becomes the monolithic mirror telescope (MMT) (35). The adaptive optics system is a unique implementation. To keep the number of reflective surfaces at a minimum, the staff has designed a 300actuator deformable secondary mirror. (Most other adaptive optics implementations use existing primary and secondary mirrors, reimage the pupil to a position within the optical train either in the Coude´ path or along the Cassegrain focus path, and place a nominally flat DM in that location.) The curved DM is being built by Thermo Trex Corp. (San Diego, CA). The wavefront sensor is a Shack-


Introduction

19

Hartmann design with an EEV CCD39A 80 ⫻ 80 pixel chip with 3.6 electrons read noise operating at 390 kHz. The wavefront control computer uses Texas Instruments TMS320C40 DSPs for slope processors and matrix multipliers. The LGS is a Lite Cycles (Tucson, AZ) doubled Nd:YAG Raman shifted with CaWO4 to achieve 10-W output at 589 nm at 108 Hz and 0.73-ns pulse length. The laser will be upgraded to a doubled Raman-shifted fiber laser to operate CW at 10 W. J. National Solar Observatory The National Solar Observatory, Sacramento Peak, NM, experimented with adaptive optics on the 76-cm vacuum tower telescope (VTT) in 1991. The early system used a Lockheed segmented deformable mirror and a Shack-Hartmann WFS based on an array of quad-cell detectors. The system that is being designed now will use a 97-actuator Xinetics deformable mirror and a ‘‘correlating Shack-Hartmann’’ wavefront sensor that uses arbitrary scenes like solar granulation as targets for wavefront sensing (36). The control computer uses off-the-shelf digital signal processor components. The unique advance of this system is the ability to use a non-point-like source as the wavefront beacon that is imaged on a focal plane array. Each subaperture is a 12 ⫻ 12 pixel array with a 6-arcsec square field-of-view. The individual images are cross correlated to extract the wavefront modes for driving the deformable mirror. K.

W. M. Keck Telescope

The two 10-m telescopes on Mauna Kea will be fitted with adaptive optics (37). Keck II will be fitted first in 1998–1999, followed by Keck I in 1999–2000. The Keck AO system allows the science beam to reach the detector suite with the same F/# with or without the adaptive optics in place. The laser guide star system, built by Lawrence Livermore National Laboratory, was delivered in February 1998. The adaptive optics system has a 349-actuator Xinetics DM. The wavefront sensor, built by Georgia Tech Research Inst. (Atlanta, GA) and AOA, uses a 64 ⫻ 64 pixel array CCD, built by MIT/LL, with 11 electrons read noise at 2-kHz frame rate. The reconstructor is based on 16 i860 Mercury boards. The tilt sensor uses an EG&G avalanche photodiode. The sodium LGS is a Nd:YAG-pumped three-stage LLNL dye laser. It has a 30-kHz PRF, 100-ns pulses, 20 W to create a tenth magnitude star from a 50-cm projection telescope. L.

Apache Point Observatory—Chicago Adaptive Optics System

Adaptive optics has been installed on the Apache Point 3.5-m ARC telescope (38). It is primarily operated at 0.85 µm using a MIT Lincoln Laboratory 64 ⫻ 64


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element CCD array and an Adaptive Optics Associates lenslet array for wavefront sensing. The University of Chicago built the deformable mirrors with 201 actuators and 4-µm stroke. M. Alpha LAMP Integration The Alpha LAMP Integration (ALI) program is the United States’ technology demonstration system for space-based high energy laser (HEL) weapons. The program, under the direction of the U.S. Ballistic Missile Defense Organization (BMDO) and the U. S. Air Force, has resulted in a number of demonstrations of technology, including the successful firing of a high energy hydrogen fluoride laser in a vacuum, wavefront control of the HEL using adaptive optics, beam pointing and jitter control, and development, manufacture, and operation of a 4-m segmented, active HEL primary mirror. The adaptive optics make use of the unique characteristics available to a space-based device. Without the atmosphere to be compensated, the disturbances of the laser and optics are the principal problems. The wavefront is sampled from the final optical element of the beam train, the LAMP primary mirror (PM). LAMP has 404 holographically etched gratings on it that transmit a part of the outgoing beam (a small fraction of 1%) through a hole in the center of the secondary mirror. Behind the hole is the outgoing wavefront sensor (OWS) with a series of transfer optics and two orthogonal focal plane arrays. A total of 512 centroids are measured in a Hartmann configuration. In addition to the outgoing wavefront measured at the OWS, 8 centroids from the autoalignment system are measured for boresighting the tracker to the HEL pointing optics, and 100 null centroids are measured for calibration. The centroid processor then transmits the information to the control computer, which multiplies the centroid signals by an estimation matrix and produces 241 deformable mirror commands and offloads the global tilt commands to a fast steering mirror. System focus is an open-loop command to the secondary mirror based on target range or, in the case of the ALI experiment, the focal point of the diagnostics. The system is designed to work at near 500 Hz. Other active elements of the system include the actuators on the segments of the LAMP primary mirror, a series of beam walk mirrors to keep the HEL beam on the clear aperture throughout the optical train, and the jitter control between the laser resonator and the beam control system. N. U. S. Air Force Airborne Laser The Airborne Laser (ABL) is a demonstration USAF high energy laser weapon system designed for endoatmospheric destruction of missiles in their boost phase. To compensate for atmospheric distortion in the long slant paths between the ABL and the target, the system has three adaptive optics systems which are


Introduction

21

interconnected. The beam path where compensation takes place is 30 cm in diameter. There are three deformable mirrors. The atmospheric compensation deformable mirror (ACDM) and the beam cleanup deformable mirror (BCDM) are identical. Both are 256-actuator PMN designs with 1.67 cm interactuator spacing and a silicon or ULE facesheet. There is an 18 ⫻ 18 array where 196 actuators are controlled and 60 are slave actuators. The stroke is specified at ⫾4 µm with a 1-kHz bandwidth at 0.8-µm wavelength. The third DM is the focus offload DM (FODM), which is designed to compensate for just the focus aberration to offload that correction from the ACDM and BCDM. The FODM will have 15-µm stroke, interactuator spacing of 3.34 cm, and a 10-Hz bandwidth. The PMN actuators of the FODM are in a 9 ⫻ 9 array with 69 actuators (44 active, 25 slave). The baseline wavefront sensor detector is an EBCCD with 20% QE and 128 ⫻ 128 pixels. Two illuminators are used in the ABL. The beacon illuminator is a 7.5-kHz, 1.9-cm beam, ⬍100-ns pulse laser. The power level is classified. The beacon is split into eight smaller beacons to avoid scintillation effects on the target that reduce the wavefront compensation performance by causing dropout of the wavefront signals. The Tracker illuminator is to operate at 5 kHz with 1.75cm beam and ⬍100-ns pulse. O.

Gemini

Gemini, an astronomy consortium of the United States, Canada, United Kingdom, Chile, Brazil, and Argentina, is building two identical 8.1-m telescopes, one on Mauna Kea (G-MK) the other at Cerro Pachon, Chile (G-CP) (39). The adaptive optics system for Gemini-Mauna Kea is being designed by the Herzberg Institute of Astrophysics, Dominion Astrophysical Observatory, Canada. The system consists of a 8 ⫻ 8 subaperture Shack-Hartmann wavefront sensor and continuous faceplate DM. The sensor uses a 64 ⫻ 64 pixel CCD with up to 200-Hz readout for tilt. One unique characteristic of the G-MK adaptive optics system is the configuration where the pupil of the DM is imaged to the strongest atmospheric turbulence layer at 6.5-km altitude (40). This was done to maximize the isoplanatic angle and, therefore, sky coverage. All other astronomical imaging systems and laser projection systems image the DM pupil to the primary mirror for simplicity of design and flexibility with a changing atmosphere. P.

Subaru

The Subaru telescope, the Japanese National Astronomical Observatory effort on Mauna Kea, is an 8.3-m telescope with an adaptive optics system consisting of a French CILAS bimorph deformable mirror with 185 actuators (41). The system will use a wavefront sensor with 144 subapertures for bright guide stars and


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detuned to 36 subapertures for dim guide stars. Both a visible wavefront sensor (500-Hz frame rate) and an IR sensor (200-Hz frame rate) will be used. The system is expected to be operational by the year 2000. Q. United Kingdom Adaptive Optics Program The UK Adaptive Optics Programme (42) is the coordinating entity for a number of university astronomy efforts throughout the United Kingdom. The most developed is the ELECTRA deformable mirror system, used with the Nasmyth adaptive optics for multi-purpose instrumentation (NAOMI), which is installed on the 4.2-m William Herschel telescope at La Palma, Canary Islands. The adaptive optics system consists of a 76-element segmented DM with each segment having tip-tilt-piston motion. The result is a 228-degree-of-freedom system. The mirror is driven by a Shack-Hartmann wavefront sensor with 8 ⫻ 8 subapertures operating in the 0.5–0.8 µm band. The system can be detuned to a 4 ⫻ 4 subaperture array for dim guide stars. The wavefront sensor camera is an EEV Ltd. (UK) device. The system uses natural guide stars and a 2.2-µm science camera (43). The principal investigators are at the University of Durham, the Royal Observatory Edinburgh, the Royal Greenwich Observatory, the University of Kent, and Imperial College London. R. Mexican IR-Optical New Technology Telescope (TIM) A collaboration between the Instituto de Astronomia, Universidad Nacional Autonoma de Mexico (UNAM) in Ciudad Universitaria D.F. Mexico, the Departement d’Astrophysique de l’Universite de Nice, France, the European Southern Observatory, Santiago, Chile, and the Institute for Astronomy, University of Hawaii, Honolulu, HI, has resulted in a design for an operational adaptive optics system for the 6.5-m segmented Mexican IR-optical new technology telescope (TIM) (44). The proof-of-principle AO system is composed of a 19-element bimorph with curvature sensing tested on the 2.1-m San Pedro martir (SPM) telescope. The system, named GUIELOA (Zapotec language for ‘‘our eyes’’), is very similar to the CFHT Pueo system. The Mexican AO program will use EG & G active quenching APDs for sensing and will be controlled by two Force SPARC5 computers and a SPARC workstation. Another less ambitious system is being built for the Observatoro Astronomico Nacional in Tonantzintla, Puebla, Mexico. The tip-tilt only system is built at the Instituto de Astronomia (National Autonomous University of Mexico), Coyoacan, Mexico. The system contains an EG & G C30927E-03 chip for a quadrant detector with Physik Instrumente (Germany) E-809.00 piezodrivers on a 2-cm tilt mirror (45).


Introduction

S.

23

Gran Telescopio Canarias (GTC)

The Instituto de Astrofisica de Canarias is designing an adaptive optics system for the Gran Telescopio Canarias (GTC) to be installed within the next few years (46). The telescope is a 10-m aperture with 36 segments, similar to the Keck telescope. There will be either 10 ⫻ 10 or 16 ⫻ 16 subapertures for the wavefront sensor and 11 ⫻ 11 or 17 ⫻ 17 actuators. The wavefront sensing band is 0.4– 0.8 µm with a variable sampling rate between 10 and 1000 Hz. The wavefront error from optical aberrations has been minimized to 122 nm. The system is being designed for diffraction limited performance (0.94 Strehl ratio) at 2.2 µm using a 280-actuator DM.

T.

Yunnan Observatory

A second-generation adaptive optics system is operating at the 1.2-m Yunnan telescope. The wavefront sensor is a 838-Hz Shack-Hartmann sensor. It contains a Chinese-built lenslet array and wavefront processor. The image intensifier for the WFS is a Hamamatsu C2166-1. The focal plane array for the WFS is a Dalsa (Waterloo, Ont., Canada) CCD CA-0128A. The wavefront processor uses 12 Texas Instruments TMS320C50 DSP chips. The closed-loop bandwidth is 50 Hz (47). The deformable mirror is built from PZT stacks, with an 8.8-cm clear aperture; the actuators are on 1.2-cm spacing, with ⫾2.2-µm stroke and 3% hysteresis. The Rayleigh laser guide star is still under development. The laser will be a frequency doubled Nd:YAG, 100-mJ/pulse, device operating at 838 Hz to match the WFS. The first-generation adaptive optics system was demonstrated previously on the Yunnan 1.2 m telescope. The 21-actuator system operated for stellar imaging in visible wavelengths. This system had a 300-Hz open-loop bandwidth, with 700-V DM and separate tilt mirror. The WFS was the 32-subaperture shearing interferometer, with λ/59 wavefront detection error, and a digital 65-Hz control system. This same system operated on the Yunnan 0.375-m telescope at Phoenix Hill and Xinglong. The wavefront sensor used 0.4–0.9 µm photon counting PMTs with 95 photons/subaperture sensitivity (48). To support the observatory work, the Institute of Optics and Electronics, Chengdu, has developed a number of prototype deformable mirrors and demonstrated closed-loop performance. A 21-actuator DM and tilt mirror with a 32-subaperture shearing interferometer and an analog 300-Hz control system was used to demonstrate atmospheric turbulence compensation in a horizontal path experiment (49). Other technology included a 55-actuator, 10-cm 400V DM that used the inner 37 actuator (50) and a larger 69-actuator, 12-cm DM (51).


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U. Beijing Observatory On the Beijing Observatory 2.16-m telescope, stellar imaging in the visible and IR was reported using the Chengdu-built 21-actuator DM and tilt mirror, 32subaperture shearing interferometer, operating at 2.2 µm with a digital 25-Hz control system (52). In 1988, on this same telescope, a 19-actuator Chengdubuilt DM (1.5 µm/600V) with a 37-subaperture 380 frames/s, 8-bit wavefront sensor was used to demonstrate atmospheric compensation. The WFS had 16 ⫻ 14 pixels/subaperture, an EG & G Reticon 128 ⫻ 128 camera, operating at 1.315 µm. The control computer used five Texas Instruments TMS320C25 processor chips. There have been supporting adaptive optics technology experiments at the National University of Defense Technology, Changsha (53). A deformable mirror with 63 actuators was integrated with a 52-subaperture WFS containing a CCD camera. The camera readout is 1 ms. The 104 slopes from the WFS are processed with eight TMS320C50 chips by a control algorithm with 500 computations/frame, 250 µs/frame, using a computer with two TMS320C50 chips.

V. U.S. Air Force Advanced Electro-optic System The advanced electro-optic system (AEOS) is a U.S. Air Force telescope placed on Mt. Haleakala, HI, to replace the AMOS compensated imaging system. AEOS will use a Xinetics 941-actuator DM, identical to that in use at SOR, with an advanced control system to allow variations in the reconstructor matrix during observations to optimize the bandwidth, constrained by guide star brightness and site conditions.

W. Anglo-Australian Telescope Because Australia has no high altitude sites suitable for high resolution astronomy, most of the science is accomplished through sky mapping and other widefield efforts. The Anglo-Australian Telescope will employ a basic adaptive optics system (54) consisting of a 19-element bimorph, 7.5-cm (3.5-cm active) diameter with a 22-µm stroke for a 500-V drive signal. The WFS is a 19-element curvature sensor with 100-Hz closed-loop bandwidth. The system is designed to maximize sky coverage by conjugating the phase at different altitude layers. However, since the data on the atmospheric layers is sparse, the system must be flexible to adjust the conjugation distance. This process requires intensive computer processing, but it has been determined that an off-the-shelf SPRC workstation is capable of tip-tilt computations.


Introduction

25

REFERENCES 1. M. C. Roggemann, B. Welsh, Imaging through Turbulence, CRC Press, Boca Raton, FL, 1996. 2. R. K. Tyson, Principles of Adaptive Optics, 2nd ed., Academic Press, Boston. 1997. 3. H. W. Babcock, Publ. Astron. Soc. Pac. 65, 229 (1953). 4. V. P. Linnik, Opt. Spektrosk. (USSR) 3, 401 (1957); English translation in F. Merkle, ed., Active and Adaptive Optics, European Southern Observatory Conf. and Proc. 48, European Southern Observatory, Garching bei Munchen, Germany (1994), p. 535. 5. J. W. Hardy, Sci. Am., June (1994), p. 60 6. D. L. Fried, J. Opt. Soc. Am. 56, 1380 (1966). 7. D. P. Greenwood, J. Opt. Soc. Am. 67, 390 (1977). 8. G. A. Tyler, J. Opt. Soc. Am. A, 1, 251 (1984). 9. V. P. Lukin, Atmospheric Adaptive Optics, SPIE Opt. Engr. Press, Bellingham, WA, 1995. 10. R. Foy, A. Labeyrie, Astron. Astrophys. 152, L29 (1985). 11. R. A. Hutchin, OSA Annual Mtg., San Jose, CA, Paper FH2, 1991. 12. R. Q. Fugate, D. L. Fried, G. A. Ameer, B. R. Boeke, S. L. Browne, P. H. Roberts, R. E. Ruane, L. M. Wopat, Nature (London) 353, 144 (1991). 13. B. M. Levine, J. R. Janesick, J. C. Shelton, Proc. SPIE 2201, 596 (1994) C. D. McKay, J. E. Baldwin, J. Rogers, G. Cox, Proc. SPIE 2201, 613 (1994). 14. F. Roddier, Appl. Opt. 27, 1223 (1988). 15. R. Q. Fugate, Top. Mtg. on Adap. Opt. Soc. Am. Tech. Dig. Series 13, 90 (1996); L. A. Thompson, R. M. Castle, Opt. Lett. 17, 1485 (1992). 16. S. R. Restaino, D. Payne, M. Anderson, J. T. Baker, S. A. Serati, G. C. Loos, Proc. SPIE 3353, 776 (1998). 17. J. M. Spinhirne, J. G. Allen, G. A. Ameer, J. M. Brown II, J. C. Christou, T. S. Duncan, R. J. Eager, M. A. Ealey, B. L. Ellerbroek, R. Q. Fugate, G. W. Jones, R. M. Kuhns, D. J. Lee, W. H. Lowrey, M. D. Oliker, R. E. Ruane, D. W. Swindle, J. K. Voas, W. J. Wild, K. B. Wilson, J. L. Wynia, Proc. SPIE 3353, 22 (1998). 18. R. Arsenault, D. Salmon, F. Roddier, G. Monnet, J. Kerr, J. Sovka, ‘‘The CanadaFrance-Hawaii Telescope Adaptive Optics Instrument Adaptor,’’ Proc. SPIE 1920, 364 (1993); J. E. Graves, M. J. Northcott, F. J. Roddier, C. A. Roddier, L. M. Close, Proc. SPIE 3353, 34 (1998). 19. F. Rigaut, ESO Conf. Prog. Telesc. Instrum. Tech., 479, 1992. 20. G. Rousset, J. L. Beuzit, N. Hubin, E. Gendron, P.-Y. Madec, C. Boyer, J.-P. Gaffard, J.-C. Richard, M. Vittot, P. Gigan, P. J. Lena, Proc. SPIE 2201, 1088 (1994). 21. J. L. Beuzit et al., Proc. OSA Top. Mtg. Adap. Opt., ESO, p. 57, 1996. 22. D. Bonaccini, F. J. Rigault, A. Glindemann, G. Dudziak, J.-M. Mariotti, F. Paresce, Proc. SPIE 3353, 224 (1998). 23. D. Bonaccini, E. Prieto, P. Corporon, J. C. Christou, D. le Mignan, P. D. Drado, R. Gredel, N. N. Hubin., Proc. SPIE 3126, 589 (1997).


26

Tyson

24.

R. Ragazzoni, D. Bonaccini, ‘‘The Adaptive Optics System for the Telescopio Nazionale Galileo,’’ Proc. Top. Mtg. on Adaptive Optics, ESO Conf. and Workshop Proc. 54, 17 (1996). R. Ragazzoni, A. Baruffolo, J. Farinato, A. Ghedina, S. Mallucci, E. Marchetti, T. Niero, Proc. SPIE 3353, 132 (1998). R. Ragazzoni, ed., AdOpt@TNG Yearly Status Report, Dec. 1997. Some summaries are also available in Proc. SPIE 3126 (1997) and Proc. SPIE 3353 (1998). S. Hippler, A. Glindemann, M. Kasper P. Kalas, R.-R. Rohloff, K. Wagner, D. P. Looze, W. K. Hackenberg, Proc. SPIE 3353, 44 (1998). A. Quirrenbach, Proc. SPIE 3126, 35 (1997). A. Glindemann et al., Publ. Astron. Soc. Pac. 109, 688 (1997). H. D. Bissinger, S. S. Olivier, C. E. Max, ‘‘Conceptual Design for a User-Friendly Adaptive Optics System at Lick Observatory,’’ Top. Mtg. on Adap. Opt., Opt. Soc. Am. Tech. Dig. Series 13, 37 (1996). D. T. Gavel, H. W. Friedman, Proc. SPIE 3353, 254 (1998). R. G. Dekany, ‘‘The Palomar Adaptive Optics System,’’ Top. Mtg. on Adap. Opt., Opt. Soc. Am. Tech. Dig. Series 13, 40 (1996); R. G. Dekany, G. Brack, D. Palmer, B. R. Oppenheimer, T. L. Hayward, Proc. SPIE 3353, 56 (1998). C. Shelton, S. Baliunas, Proc. SPIE 1920, 371 (1993). L. A. Thompson, R. M. Castle, S. W. Teare, P. R. McCullough, S. L. Crawford, Proc. SPIE 3353, 282 (1998). M. Lloyd-Hart, J. R. P. Angel, D. G. Sandler, T. K. Barrett, P. C. McGuire, T. A. Rhoadarmer, D. G. Bruns, S. M. Miller, D. W. McGarthy, M. Cheselka, Proc. SPIE 3353, 82 (1998). T. R. Rimmele, R. R. Radick., Proc. SPIE 3353, 72 (1998). D. S. Acton, P. L. Wizinowich, P. J. Stomski, J. C. Shelton, O. Lai, J. M. Brase, Proc. SPIE 3353, 125 (1998). E. J. Kibblewhite, M. R. Chun, J. E. Larkin, V. Scor, F. Shi, M. F. Smutko, W. J. Wild, Proc. SPIE 3353, 60 (1998). L. K. Saddlemyer, G. Herriot, J.-P. Veran, J. M. Fletcher, Proc. SPIE 3353, 150 (1998). G. Herriot, S. Morris, S. Roberts, J. M. Fletcher, L. K. Saddlemyer, G. Singh, J.P. Veran, E. H. Richardson, Proc. SPIE 3353, 488 (1998). H. Takami, N. Takato, M. Otsubo, T. Kanzawa, Y. Kamata, K. Nakashima, M. Iye, Proc. SPIE 3353, 500 (1998). R. Myers, A. Longmore, R. Humphreys, G. Gilmore, B. Gentiles, M. Wells, R. Wilson, ‘‘The UK Adaptive Optics Programme,’’ Proc. SPIE 2534, 48 (1995). A. J. Longmore, M. Wells, M. Strachan, C. Dixon, T. Peacocke, R. M. Myers, R. A. Humphreys, A. B. Gentles, S. P Worswick, A. J. Weise, Proc. SPIE 3126, 18 (1997). S. Cuevas, P. Sotelo, F. Garfias, A. Iriarte, L. A. Martinez, V. G. Orlov, V. V. Voitsekhovich, O. Chapa, S. J. Tinoco, J. Vernin, R. Avila, F. Marchis, J. E. Graves, M. J. Northcutt, F. J. Roddier, C. A. Roddier, Proc. SPIE 3353, 531 (1998). P. Sotelo, R. Flores, F. Garfias, S. Cuevas, Proc. SPIE 3353, 1202 (1998). N. Devaney, C. D. Bello, J. Keen, L. Jochum, Proc. SPIE 3353, 561 (1998).

25. 26. 27. 28. 29. 30.

31. 32.

33. 34. 35.

36. 37. 38. 39. 40. 41. 42. 43.

44.

45. 46.


Introduction

27

47. W. Jiang, N. Ling, G. Tang, M. Li, F. Shen, C. Rao, Y. Zhu, B. Xu, Proc. SPIE 3353, 696 (1998). 48. W. Jiang et al., Proc. SPIE 1920, 381 (1993). 49. W. Jiang et al., Sci. Engr. Frontiers for 8–10 m Telesc., Tokyo (1994). 50. Z. Zhige et al., High Pwr. Laser & Part. Beams, 8, 88, 1996. 51. N. Ling, Proc. SPIE 2828, 472 (1996). 52. R. Changhui, High Pwr. Laser Part. Beams, 8, 469 (1996). 53. J. Yongmei et al., J. Nat. Univ. Def. Tech., 18, 90 (1996). 54. J. J. Bryant, J. W. O’Byrne, R. A. Minard, P. W. Fekete, ‘‘Low Order Adaptive Optics at the Anglo-Australian Telescope,’’ Proc. Top. Mtg. on Adaptive Optics, ESO Conf. and Workshop Proc. 54, 23 (1996).


2 System Design and Optimization Ronald R. Parenti Massachusetts Institute of Technology Lincoln Laboratory, Lexington, Massachusetts

An adaptive-optics system for real-time phase conjugation can be characterized as a highly parallel servo device capable of sensing and correcting the incoming wavefront at the pupil plane of an imaging sensor. Performance close to the diffraction limit of the input aperture can be achieved in the limit that the angular separation between the turbulence probe and the target object is small (⬍θ 0 ), the spacing between control elements on the active optical element is well matched to the turbulence coherence length (⬍r0 ), and a sufficiently high update rate is maintained (⬎1/τ 0 ). However in most practical implementations of this concept, performance compromises are necessary to reduce component costs and improve the sensitivity of the wavefront sensor. The four principal elements of a conventional adaptive-optics system are shown in Fig. 1. Light originating from an exoatmospheric source is corrupted in both amplitude and phase as a result of random fluctuations in the refractive index of the intervening air. The quality of the image formed at the focus of a ground-based telescope is largely driven by phase distortions, which can be corrected through the insertion of an optical surface having the conjugate optical path difference (OPD). Most compensation systems apply this conjugate phase with a pair of active elements consisting of a high-speed tilt mirror and a deformable mirror that removes the figure (tilt-removed) component of the distortion. Error signals are generated by the phase sensor, which actually measures the first derivative of the phase of the incoming wavefront. A phase-computation device, referred to as a reconstructor, transforms the output of the wavefront sensor into a set of drive signals that control the active optical elements. Each of the components just described has a unique set of error mechanisms that together establish the overall effectiveness of the adaptive-optics system. In


30

Parenti

Figure 1 Essential components of an adaptive-optics phase compensation system. Error signals developed from the phase difference between the incoming wavefront and the deformable mirror are measured by the phase sensor and subsequently applied to the two active optical elements. The overall performance of the system is primarily a function of the correction bandwidth, mirror actuator density, and measurement noise.

the following sections the first-order effects of these errors are quantified; the most important effects are then combined in a servo loop model that provides a concise description of the dynamic performance characteristics. I. ATMOSPHERIC MODELS To motivate this discussion it will be useful to refer to a specific set of atmospheric models so that numerical results can be generated. During the past two decades, several dozen models have been developed in an attempt to describe refractive-index measurements made at a variety of locations throughout the world. For daytime conditions at inland sites, the SLC-Day (1) (standardized under the Submarine-Laser Communications program) and the Hufnagel-Valley (2) turbulence models are frequently employed for systems analysis. For nighttime viewing at good astronomical sites a more benign model is probably appropriate; the following formulation is based on the Hufnagel-Valley profile: C 2n (h) ⫽ 8.16 ⫻ 10⫺54 h 10 exp ⫹ 1.90 ⫻ 10⫺15 exp

冢冣冢冣

冢冣

⫺h ⫺h ⫹ 3.02 ⫻ 10⫺17 exp 1000 1500

⫺h 100

where the units of the altitude, h, is meters.


(m⫺2/3)

(1)

System Design and Optimization

31

Temporal fluctuations in the cumulative phase distortion are somewhat less critical to this analysis; therefore, the standard wind-velocity profile proposed by Bufton (3) is probably appropriate for most calculations:

冦冤

h ⫺ 9400 v(h) ⫽ 5 ⫹ 30 exp ⫺ 4800

冥冧 2

(m/sec)

(2)

The turbulence and wind models are plotted as a function of altitude in Fig. 2 and 3, respectively. The physical properties of turbulence and its effect on optical systems can usually be described in terms of the turbulence and wind-velocity moments, defined by the integrals µ n ⬅ ∫ C 2n (h) h n dh

and

v n ⬅ ∫ C 2n (h) v n (h) dh

The zeroth-order turbulence moment provides a measure of the spatial coherence length (4): r0 ⫽ {0.423 k 2 sec(ζ) µ 0 }⫺3/5

(m)

(3)

and the atmospheric time constant is derived from the 5/3 velocity moment (5): τ 0 ⫽ {2.91 k 2 sec(ζ)v 5/3 }⫺3/5

(sec)

(4)

Figure 2 Comparison of the Hufnagel-Valley model that is typically used for calculations of daytime turbulence, and the modified HV model that was developed by the authors to represent nighttime turbulence conditions at good seeing locations.


32

Parenti

Figure 3 The Bufton wind velocity profile.

where k ⫽ 2π/λ is the optical wavenumber and ζ is the zenith angle. Astronomers prefer to characterize turbulence strength by the seeing angle, which is inversely related to r0 : θ seeing ⫽ 1.22

λ r0

(rad)

(5)

Angular coherence effects can be estimated from the 5/3 turbulence moment: θ 0 ⫽ {2.91 k 2 sec 8/3 (ζ)µ 5/3 }⫺3/5

(rad)

(6)

where the parameter θ 0 is known as the isoplanatic patch. A numerical tabulation of these quantities for the three turbulence models introduced earlier in this section is provided in Table 1. The performance of closed-loop servo systems is somewhat dependent on the power spectrum of the input disturbance, and it is useful, therefore, to specify nominal spectra for the figure (tilt-removed) and tilt components of turbulence. The figure component of turbulence is developed from a paper by Greenwood and Fried (6). This spectrum, which is plotted in Fig. 4, can be approximated by a function having two segments: Ffig ( f ) ⫽

⫺7/5 4/3 0.132 sec(ζ)k 2 D 4 µ 12/5 , 0 v 5/3 f

冦0.0326 sec(ζ)k v 2

5/3

f ⫺8/3,


f ⱕ 0.705 D⫺1 µ ⫺3/5 v 3/5 0 5/3 f ⱖ 0.705 D⫺1 µ ⫺3/5 v 3/5 0 5/3

(7)


33

Table 1 Summary of Turbulence Parameters for Imaging Systems Operating at 0.55 µm Model SLC-Day HV-21 Modified HV

Zenith

r0 (cm)

θ seeing (arc-s)

θ0 (µrad)

τ0 (ms)

0° 45° 0° 45° 0° 45°

5.7 4.6 5.6 4.5 20.0 16.2

2.4 3.0 2.5 3.1 0.69 0.85

13.8 8.0 7.8 4.5 20.0 11.5

2.7 2.2 2.1 1.7 6.3 5.1

The units are rad 2 of phase error per Hz. A concise treatment of full-aperture tilt was recently published by Tyler (7). This spectrum also has a simple form in its low- and high-frequency limits that can be joined at their intersection to give Ftilt ( f ) ⫽

冦0.028 sec(ζ) D

1.60 sec(ζ)v ⫺1/3 f ⫺2/3, ⫺5

v 14/3 f ⫺17/3,

1/5 f ⱕ 0.445 D⫺1 v ⫺1/5 ⫺1/3 v 14/3 1/5 f ⱖ 0.445 D⫺1 v ⫺1/5 ⫺1/3 v 14/3

(8)

Figure 4 Power spectrum for turbulence-induced figure error. This model incorporates atmospheric moments developed from the modified Hufnagel-Valley turbulence profile and the Bufton wind profile.


34

Parenti

Figure 5 Power spectrum for turbulence-induced tilt error, given in units of rad 2 of single-axis tilt per Hz. This model incorporates atmospheric moments developed from the modified Hufnagel-Valley turbulence profile and the Bufton wind profile.

where the units here are rad 2 of single-axis tilt per Hz and normalized for integration over positive values of f. A plot of this function for a 4-m aperture is given in Fig. 5.

II. PHASE SENSOR AND TRACKING SENSOR MEASUREMENT NOISE Phase sensors and tracking systems have a number of similarities, and a good first-order analysis of both can be made by investigating the properties of a simple quadrant detector. The results given in this discussion are equivalent to those derived elsewhere (8), except for minor differences in the multiplicative constants. In the one-dimensional view of a motion detector shown in Fig. 6a, light from a lens of diameter d s is focused on a pair of detectors that meet in the null position of the focal spot. The off-null position is determined by measuring the difference between the signals generated by the two detectors, i.e., x⫽C

⫺n L ⫹ n R nL ⫹ nR



35

Figure 6 One-dimensional view of tracking sensor (a) and a Hartmann phase sensor (b) show that they are functionally equivalent in that they convert aperture-averaged tilts into spot displacements at the focus of the lens. The wavefront sensor measures local gradients of the phase of the incoming radiation with a spatial resolution that is comparable to the turbulence coherence length.

where n L and n R represent the number of photoelectrons collected by the left and right detector elements, respectively, and C is a constant determined by the geometry of the optical system. If the focal spot is assumed to be approximately square and to have a diffraction-limited width of f λ/d s , then C⫽

fλ 2d s

and the angular displacement, θ, of a point source with respect to the boresight of the sensor is equal to θ⫽

冤

x λ ⫺n L ⫹ n R ⫽ f 2d s n L ⫹ n R

冥

For most tracking scenarios the size of the focal spot will not be diffractionlimited, but will instead be dominated by an effective collection aperture established by the turbulence coherence diameter, r0 , or determined by the angular extent of the object, θ obj . To first order, the increase in beam diameter due to


36

Parenti

these two effects can be incorporated into the displacement equation in the following manner: θ⫽

1 2

√冢冣冢冣 2

λ ds

λ r0

⫹

2

⫹ (θ obj ) 2

冤⫺nn ⫹⫹nn 冥 (radians of tilt) L

R

L

(9)

R

A Hartmann phase sensor incorporates an array of quadrant trackers, each of which has a collection diameter comparable to r0 so that the local phase gradient, g, can be accurately measured; a single subaperture element is illustrated in Fig. 6b. In this case the object does not physically move with respect to the system boresight, but only appears to be displaced by an angle, θ, as a result of a phase shift across the subaperture of magnitude 2πθd s /λ. Under some conditions an artificial beacon of wavelength λ b may be necessary to provide an adequate light source for the wavefront sensor, whereas image compensation is optimized for the object wavelength, λ; for these systems an appropriate scaling factor must be included in the gradient equation. The final expression for the phase-gradient measurement is

冢冣√ 冢冣冢冣冤 λb

ds

2

ds

2

⫺n L ⫹ n R

冥

(radians of phase) (10) λ r0 λb nL ⫹ nR where r0 is computed for the beacon wavelength. The calculation of the measurement errors associated with Eqs. (9) and (10) is straightforward as long as the signal fluctuations are random and independent. When the focal spot of the motion detector is near its null position, on average the same number of electrons will be collected by each detector so that n L ⫽ n R ⫽ n. The variance of the difference calculation, n R ⫺ n L , is twice the individual measurement variance, σ 2n , thereby yielding the following descriptions of the tilt and phase noise:

g⫽π

1⫹

⫹

θ obj

冦冢冣冢冣 ⫹ (θ ) 冧 (radians of tilt) π σ λ d d σ ⬇ 1⫹ ⫹ θ 冦冢冣冢冣冢冣冧 (radians of phase) 2n λ r λ σ 2θ ⬇

σ 2n

8n 2

2 n

2

2 g

λ ds

2

2

⫹

λ r0

2

obj

2

b

2

2

2

(11)

2

s

s

0

b

obj

2

(12)

At this point all of the important parameters have been defined, with the exception of σ 2n . Although optical sensors are subject to a wide range of noise mechanisms, the most important of these effects can be characterized by a noise model containing two terms. For an ideal detector, photon noise, which results from the random arrival of photons, determines the sensor’s performance. Photon noise is governed by Poisson statistics, in which case the variance in the number of photoelectrons is equal to the average number collected (9). The readout process for nonideal detectors generates an additive noise contribution that is usually



37

described in terms of equivalent rms electrons, n rms . This noise source is independent of the input light level, but may be a function of operational factors such as the readout frame rate. The relative impact of additive noise on the output signal can be significantly reduced if an optical intensifier having a photoelectron gain of G e is placed in front of the detector, although this optical configuration can adversely affect the detector’s quantum efficiency, η. For a detector exposed to N photons from which n ⫽ ηN photoelectrons are generated, its measurement variance will be σ 2n ⫽ ηN ⫹

n 2rms

(13)

G 2e

In order to correctly apply Eqs. (11) through (13) to the behavior of a twodimensional quadrant detector, it is necessary to specify the manner in which all of the photons captured by the sensor, represented by Np , are distributed among the individual detectors and subsequently processed. In the case of the tracking sensor, the object’s motion along one dimension is determined by first summing the detector outputs along the orthogonal direction and then forming their difference, as shown in Fig. 7a. Each detector receives Np /4 photons, and the summation process doubles the noise variance from the left and right outputs, so the final error expression is σ 2θ ⬇

G 2e ηNp ⫹ 4n 2rms 4(G e ηNp ) 2

冦冢冣冢冣 ⫹ (θ ) 冧 λ ds

2

⫹

λ r0

2

obj

2

(radians 2 of single-axis tilt)

(14)

Figure 7 Comparison of the data processing for a quadrant motion detector (a) and a Hartmann phase sensor (b). The motion detector can provide signals for the orthogonal direction by interchanging the processing of the four elements. Hartmann sensors typically incorporate a second camera to compute phase gradients along the orthogonal axis.


38

Parenti

Hartmann sensors can be constructed in a variety of ways. In one of the more common configurations, separate cameras are used to obtain tilts in the two orthogonal directions as a means of reducing the net readout bandwidth. On-chip summation techniques are used, which effectively provide elements having a 2to-1 aspect ratio, as indicated in Fig. 7b. As a result of the initial division of light between two cameras, each of the detector elements within a single subaperture collects Np /4 photons; therefore, σ 2g ⬇

冢冣冦冢冣冢冣冧

2π 2 (G 2e ηNp ⫹ 4n 2rms ) λ b (G e ηNp ) 2

2

λ

1⫹

ds r0

2

⫹

ds

λb

2

θ obj

(radians 2 of phase) (15) Although Eqs. (14) and (15) are only first-order approximations of a very complex noise process, they accurately describe the more important parametric relationships.

III. PHASE RECONSTRUCTOR ERROR PROPAGATOR As the servo model described in Fig. 1 indicates, the error signals that are input to the deformable-mirror servo system are the result of phase estimates generated by the phase reconstructor. This component introduces an estimation error that is proportional to the gradient variance described by Eq. (15). To compute the constant of proportionality (also known as the error propagator, ε p ), the form of the reconstructor algorithm must be specified. In the limit of infinite spatial resolution, the relationship between the phase, φ, and the phase gradient, g, is described by the Poisson equation with Neumann boundary conditions, i.e., ∇ 2 φ ⫽ div g

(16)

For a discrete-measurement system, this relationship can be represented by the matrix product g ⫽ A␾

(17)

in which g and ␾ are the gradient and phase vectors, respectively; A is a twodimensional matrix that uniquely associates each gradient element with two or more phase points. In the Hudgin sensor geometry (10), shown in Fig. 8a, the incoming radiation is sampled by a pair of wavefront sensors that separately measure one-dimensional x and y gradients. The sampling regions of the two sensors are displaced such that each measurement is centered at the midpoint of a horizontal or vertical line drawn between two adjacent phase points. The Fried geometry (11), illustrated in Fig. 8b, can be realized by a single camera that



39

Figure 8 Separate one-dimensional x and y gradient measurements may be made between two adjacent phase points, as specified by the Hudgin sampling geometry (a), or two-dimensional gradient samples can be obtained within a region centered at the midpoint of four phase points according to the Fried geometry (b).

makes collocated two-dimensional gradient measurements at the center of a region defined by four nearest-neighbor phase points. Fried’s approach requires an array of quadrant detectors, similar to the tracking sensor shown in Fig. 7a. For reasons relating primarily to sensor readout bandwidth, the Hudgin scheme is the more popular of the two sensor implementations. For either sampling geometry, the least-square solution to Eq. (17) is equivalent to the generalized inverse of the A matrix, ␾ ⫽ {(At A)⫺1 At } g ⫽ Bg

(18)

where A⫺1 and At represent the inverse and transpose, respectively. Since B is invariant for a given sensor geometry, this operation can be precomputed and permanently programmed into the reconstruction device. The structure of the B matrix provides a complete description of the noise characteristics of the phase reconstructor. Herrmann (12) has shown that the error propagator for a compensation system having Na phase elements is εp ⫽

1 Trace[(At A)⫺1 ] Na

(19)

Numerical calculations performed by Herrmann indicate that when Na is larger than 50, εp ⬇

冦0.24 ln(N ),

0.12 ln(Na ), a

Hudgin geometry Fried geometry


(20)

40

Parenti

Although the error propagator for the Hudgin reconstructor is a factor of 2 smaller than that for Fried’s reconstructor, the net phase error associated with the two computational techniques is essentially identical. This is due to the fact that the former collects fewer photons due to the need to divide the light between two cameras, whereas the latter has a higher readout noise since each subarray incorporates twice as many detectors. Thus, the phase error, σ 2φ , for either geometry is approximately equal to σ ⬇ 0.24 ln(Na ) 2 φ

冢冣

2π 2 (G 2e ηNp ⫹ 4n 2rms ) λ b (G e ηNp ) 2

λ

冦冢 r 冣 ⫹ 冢λ θ 冣冧

⫻ 1⫹

ds

2

2

ds

2

(21)

obj

0

b

The error propagator is approximately equal to unity for deformable-mirror systems having about 100 actuators. IV. THE DEFORMABLE MIRROR The active component of the figure-control servo system is the deformable mirror, which imposes both spatial and temporal limitations on the operation of the compensation system. A one-dimensional cross section of a typical structure is shown in Fig. 9. A thin facesheet is bonded to an array of actuators, each of which is capable of moving a distance equivalent to several optical wavelengths. Many types of actuators have been successfully employed in these structures; some of the more recent designs have included stacks of an electrostrictive material, which can be displaced by several microns by applying a voltage of a few hundred volts. The time required to achieve full-stroke displacement is typically a few

Figure 9 Cross section of a typical deformable mirror employing a flexible facesheet and a two-dimensional array of actuator stacks. The actuators are cemented to the facesheet so that the surface can be deformed by either extending or contracting the actuator dimension.



41

hundred microseconds. Mirrors have been constructed with the actuator elements arranged in rectangular or hexagonal patterns (zonal control) (13,14) as well as in nonuniform distributions designed to allow independent control of the Zernike modes (modal control) (15). The choice of actuator density is one of the principal drivers in the cost of an adaptive-optics system. Light-collection efficiency is also an important tradeoff when imaging dim sources, since the phase-sensor noise decreases as the subaperture collection area is increased. When the subaperture dimensions become too large, however, the high-frequency components of the turbulence-induced phase distortion cannot be accurately compensated. The resulting spatial distortions in the corrected wavefront are known as fitting error. The uncorrected phase error due to the figure (tilt-removed) component of turbulence is a function of the ratio of the aperture diameter and the turbulence coherence length (16): σ 2fig ⫽ 0.134

冢冣 D r0

5/3

(22)

which corresponds to an rms phase distortion of several wavelengths for a nominal astronomical installation. The correction achievable by an adaptive-optics system having a specified actuator density has been investigated extensively. Some of the earliest work in this field was conducted by Greenwood (17), who derived the following expression to relate the fitting error to the number of actuators, Na: 2 fig fitting

(σ )

冢冣

5/3

D ⬇ 0.274 r0

N ⫺5/6 a

which is perhaps more conveniently expressed in terms of the subaperture spacing, d s : (σ 2fig ) fitting ⬇ 0.34

冢冣

5/3

ds r0

Subsequent experiments have shown that the mechanical constraints applied at the mirror boundaries can significantly increase the value of the leading constant. For this reason, a somewhat more conservative estimate is probably more appropriate for system optimization studies: 2 fig fitting

(σ )

⬇ 0.5

冢冣 ds

5/3

r0


(23)

42

Parenti

V. SERVO BANDWIDTH LIMITATIONS In the temporal domain, the selection of the correction bandwidth involves cost and light-collection trade-offs that are very similar to those encountered in the determination of actuator density. High bandwidths provide better phase correction, but require higher pixel readout rates that necessitate more expensive sensor designs. Shorter integration times also reduce the number of photons collected per integration cycle, which increases the phase-sensor noise. The complex interaction of these effects is most easily investigated through the use of a simplified servo model. The servo model illustrated schematically in Fig. 10 incorporates the essential components of a CW control loop. The input wavefront is reflected from the deformable mirror and a fraction of the output is sensed by the wavefront sensor. In addition to the residual phase error, a noise component is also generated by the wavefront sensor, and the composite output is averaged for a dwell time, τ d .

Figure 10 Servo loop diagram for a digital control system employing a boxcar averaging device and a digital accumulator. The error signal at the wavefront sensor is averaged for a period of τ d . The accumulator provides the deformable mirror with an impulse drive signal at the end of each integration interval, which is subsequently held until the next drive signal.



43

The averaged error signal is input to an accumulator, which has an adjustable gain parameter, g l , that is used to vary the loop bandwidth. The mirror response is modeled as having an exponential time constant, τ m . The overall performance of the phase-conjugation system can be characterized by its ability to simultaneously suppress turbulence-induced phase distortions and wavefront sensor noise. The properties of individual components of the system are most conveniently described by a frequency-domain transfer function, which is applied to the power spectrum of the input disturbance; these transfer functions are described in Fig. 10. The net transfer function of the loop also depends on the point at which the disturbance is applied (18). The transfer function with respect to an optical disturbance inserted at the deformable mirror is Tt ( f ) ⫽

冷

1 1 ⫹ H av ( f ) H acc ( f )Hm ( f )

冷

2

(24)

whereas the transfer function for noise generated by the wavefront sensor has the form Tn ( f ) ⫽

冷1 ⫹ H ( f ) H

H av ( f ) H acc ( f ) H m ( f ) av

acc

( f ) Hm( f )

冷

2

(25)

The functions H av , H acc and Hm represent the averager, accumulator, and mirror component transfer functions, respectively. The power spectrum of the residual error is the sum of the turbulence and noise outputs: Ferror ( f ) ⫽ Tt ( f ) Ffig ( f ) ⫹ Tn ( f ) Fnoise ( f )

(26)

where F fig corresponds to the function defined by Eq. (7). This analysis can also be applied to the tilt correction system by replacing the turbulence power spectrum with F tilt and applying the appropriate dwell-time and mirror-response parameters in the component transfer functions. To a limited extent, the response of the servo system can be tuned by adjusting the loop gain setting, although care must be taken to avoid an instability condition. As the accumulator gain is increased, rejection of low-frequency turbulence improves but high-frequency components are amplified; this behavior is illustrated in Fig. 11. The transmission function for noise (examples of which are shown in Fig. 12) is essentially a low-pass filter, which has a cutoff frequency that increases with increasing gain. Parametric investigations similar to those shown in Figs. 11 and 12 indicate that a good compromise between turbulence suppression and noise transmission is achieved when the gain is [g l ] optimal ⬇

1 1 ⫹ 2(τ m /τ d )


(27)

44

Parenti

Figure 11 Turbulence transfer function for closed-loop compensation as a function of loop gain. In this example, the mirror-response time constant is assumed to be negligible.

Figure 12 Noise transfer function for closed-loop compensation as a function of loop gain. In this example, the mirror-response time constant is assumed to be negligible.



45

When the servo gain is set to this value, the effective servo time constant, τ s , is τ s ⬇ τ d ⫹ 2τ m

(28)

Since servo performance is commonly described in terms of an equivalent compensation bandwidth, it is useful to relate the turbulence and noise transfer functions given in Eqs. (24) and (25) to a characteristic bandwidth parameter. To first order

冦冢冣冧 1⫹冦冢53冣 2π f τ 冧

{2π f τ s} ⫹ 2

Tt( f ) ⬇

3 2π f τ s 5 4

4

(29)

s

and 1

Tn( f ) ⬇ 1⫹

冦冢冣

冧

3 2π f τ s 5

4

(30)

where the temporal frequency, f, is in Hz. The low-frequency behavior of the turbulence transfer function approximates that of a two-pole high-pass filter of bandwidth 1/(2 πτ s ), whereas the noise transfer function is equivalent to a fourpole low-pass filter of bandwidth 5/(6πτ s ). When the integrals of Eqs. (29) and (30) are compared to those for ideal binary filter functions, the following approximate relationships are obtained: Bt ⬇

1 10 τ s

(31)

Bn ⬇

1 3 τs

(32)

and

It is perhaps surprising that the noise bandwidth is a factor of 3 larger than the turbulence-correction bandwidth; however, both of these expressions are in good agreement with the usual design guidelines for a discrete servo system operating at a sample rate of 1/τ s . The transfer functions just derived can now be combined with the power spectra discussed in Sec. I and the noise expressions derived in Secs. II and III to develop a composite description of the servo output. The system’s affect on noise generated within the sensing device is most easily understood by recognizing that the relationships given in Eqs. (14) and (15) assume a simple photon integrator having a dwell time of τ d . It is easy to show that the equivalent band-


46

Parenti

width of this type of integrator is approximately 1/(2 τ d ); therefore, the bandwidth indicated in Eq. (32) represents a noise-throughput reduction corresponding to a multiplicative factor of 2τ d /3τ s . The figure component of turbulence derives from the integral σ 2fig ⫽

冮

∞

Tt ( f )Ffig ( f ) df

0

⬇ 0.0326 sec(ζ) k 2 v 5/3

冮

∞

0

冦冢冣冧 f 1⫹冦冢53冣 2π f τ 冧

{2π f τ s}2 ⫹

3 2π f τ s 5 4

4

⫺8/3

df

s

where the assumption has been made that the low-frequency turbulence component given in Eq. (7) does not contribute significantly to the residual error. The result of this integration yields the approximate expression σ

2 fig

⬇

冢冣 τs

5/3

(33)

τ0

which incorporates the turbulence time constant, τ 0 , defined in Eq. (4). An approximate expression for the tilt component of turbulence is obtained by ignoring the high-frequency component of the tilt power spectrum given in Eq. (5): σ 2tilt ⫽

冮

∞

Tt ( f ) Ftilt ( f ) df

0

⬇ 1.60 sec(ζ) v⫺1/3

冮

1/5 0.445 D⫺1 v ⫺1/5 ⫺1/3 v 14/3

(2π f τ st )2 f ⫺2/3 df

0

where τ st ⫽ τ dt ⫹ 2τ mt represents the effective tilt-servo time constant for a measurement dwell time of τ dt and a mirror time constant of τ mt . The result of this integration is σ 2tilt ⬇ 0.20

冢冣冢冣 λ D

2

τ st

2

τ 0t

(34)

where the characteristic tilt time constant is defined to be 7/15 ⫺1/3 ⫺1/2 τ 0t ⬅ {0.512 k 2 sec(ζ)v 8/15 } ⫺1/3 v 14/3 D

(35)

The leading constant in Eq. (35) is chosen such that the beam area due to tracking jitter is approximately twice the diffraction-limited value when τ st /τ 0 t ⫽ 1. The subject of tilt jitter will be treated in more detail in Sec. VI. The analytical development presented in the last four sections can be con-



47

cisely summarized by a pair of relationships that provide an estimate of the uncorrected servo error. The figure component of the residual error is 2 fig total

(σ )

冢冣冢冣冢冣

1 ds ⬇ 2 r0

5/3

⫹

fitting error

τs

5/3

τ0

2τ d

⫹

servo lag

3τ s

σ 2φ (rad 2 of phrase)

(36)

sensor noise

and the tilt component is 2 tilt total

(σ )

冢冣冢冣冢冣

1 λ ⬇ 5 D

2

τ st

τ 0t

servo lag

2

⫹

2τ dt 3τ st

σ 2θ (rad 2 of single-axis tilt)

(37)

sensor noise

These two expressions represent the first step in the optimization of an adaptiveoptics system. To complete the process, a relationship must be developed between correction error and overall performance.

VI. HEURISTIC MEASURES OF SYSTEM RESOLUTION The ability of a phase compensation system to correct for the effects of atmospheric distortion can be characterized by a variety of metrics, including the Strehl ratio, beam area, point spread function (PSF), and modulation transfer function (MTF). The Strehl, which is defined as the ratio of the measured onaxis beam intensity to the diffraction-limited intensity, is a convenient and easily defined quantity that is frequently employed by the high-energy laser community. For imaging systems, the PSF and MTF are more relevant measures of performance, but for the purpose of design optimization it would be preferable to define a single scalar quantity to represent focal-plane resolution. The image of a point source viewed through the atmosphere evolves from a simple Airy shape with elevated sidelobes under weak turbulence conditions to a complex speckle pattern as the turbulence becomes more intense; these two limiting conditions are illustrated in Fig. 13. Both pictures are representative of short-exposure data, which means that the sensor integration time is small compared to the atmospheric time constant. When many short-exposure images are averaged, the random speckle patterns merge to form a beam profile that contains a diffraction-limited core having an angular width of approximately λ/D and an extended skirt of dimensions λ/r0 , see Fig. 14. Unfortunately, there is no simple and unambiguous definition of resolution that can be assigned a point spread function of this complex shape. The form indicated schematically in Fig. 14 suggests a description of resolution that evolves from the main-lobe width to the skirt diameter as the strength


48

Parenti

Figure 13 Comparison of typical far-field beam shapes for (a) weak turbulence and (b) strong turbulence conditions. The main-lobe beamwidth for the weak-turbulence example is approximately λ/D and the Strehl is 0.37. The Strehl for the strong-turbulence irradiance distribution is 0.02 and the linear extent of the beam is roughly λ/r0 .

of turbulence is increased. To investigate the nature of this transition it is useful to begin with the extended Marećhal approximation, which provides a relationship between the figure error and the Strehl under weak-turbulence conditions Strehl ⬇ exp(⫺σ 2fig )

weak-turbulence conditions

(38)

When the turbulence becomes sufficiently strong, nearly all of the energy is contained in the skirt, in which case the Strehl is better described by a form suggested by Yura (19):

Figure 14 Point-spread function obtained by averaging many short-exposure images corrupted by atmospheric turbulence. The fraction of the energy contained in the outer skirt is a function of the ratio D/r0 .



Strehl ⬇

49

1 strong-turbulence conditions 1 ⫹ (D/r0 ) 2

(39)

A composite short-exposure Strehl can be developed by assuming that all of the energy lost from the central core of the beam is scattered over a region of width of λ/r0 : [Strehl] SE ⬇ exp(⫺σ 2fig ) ⫹

1 ⫺ exp(⫺σ 2fig ) 1 ⫹ (D/r0 ) 2

(40)

The net resolution afforded by a beam of this description will be greatly dependent upon the nature of the object being imaged. Although it may be possible to accurately determine the separation of two closely spaced point sources when only a small fraction of the total energy is contained in the main lobe of the beam, for more complex scenes the resolution will be limited by sidelobe smearing effects. For these more general conditions, an energy-weighted average of the main-lobe and sidelobe contributions may provide a reasonable working definition of resolution, which yields the following result: [Resolution] SE ⬇ 1.22

冢冣

λ 1 D √ [Strehl] SE

(41)

As indicated in Fig. 15, this representation predicts a transition from the diffraction-limited Rayleigh criterion to a turbulence-dominated resolution that occurs at a figure variance of approximately 2.5 rad 2. This transition value corresponds to an rms figure error of about 1/4 wavelength, which may be regarded as an upper limit on the tolerable error for a compensated imaging system. For most viewing applications it is necessary to employ a camera dwell time that is much longer than the atmospheric time constant in order to collect sufficient light to form a high signal-to-noise image. Under these conditions, the residual tilt jitter of the compensation system must be incorporated into the Strehl and resolution calculations to derive the appropriate long-exposure quantities. When the residual jitter is small, its effect can be modeled as the convolution of a Gaussian profile with the main lobe of the short-exposure beam. The width of the Gaussian distribution is defined by the single-axis tilt jitter, σ tilt , and it is observed that its shape approximates the core of the Airy function when σ tilt ⫽ 0.45(λ/D). Therefore, the addition of tilt jitter to the short-exposure Strehl can be modeled in the following manner: [Strehl] LE ⬇

exp(⫺σ 2fig ) 1 ⫹ 5(D/λ) 2 σ 2tilt

⫹

1 ⫺ exp(⫺σ 2fig )


1 ⫹ (D/r0 ) 2

(42)

50

Parenti

Figure 15 Composite short-exposure resolution for uncorrected turbulence as a function of figure error. The transition from diffraction-limited resolution to turbulence-dominated resolution occurs in the range of 3–6 rad 2.

and the long-exposure resolution is developed according to the same logic used for the short-exposure case: [Resolution] LE ⬇ 1.22

冢冣

λ 1 D √ [Strehl] LE

(43)

Using Eq. (42) or (43), a system designer can specify a tolerable degradation factor that can subsequently be used to establish an appropriate error budget. A baseline procedure for design optimization is outlined in the next section.

VII. DESIGN GUIDELINES FOR ADAPTIVE-OPTICS SYSTEMS Design strategies for adaptive-optics systems are strongly dependent on the operational environment. High-energy laser beam propagation systems for military applications require near-unity Strehl ratios to achieve good performance; therefore, all sources of compensation error must be kept to a minimum. For astronomical imaging applications, the goals are usually much more modest, and budgetary



51

constraints are a primary consideration. The design problem for astronomers is further complicated by a requirement to image extremely dim objects, so care must be taken to make optimal use of all available light. Since the design criteria for military systems are more easily developed, this problem will be addressed first. When developing component specifications, the usual approach is to begin with a performance goal and then work backwards through the list of degradation factors to assign a tolerance factor to each. The flowchart shown in Fig. 16 embodies the design relationships assembled earlier in this chapter. Although the formulation of the matrix that minimizes technical risk and monetary cost typically requires extensive iteration, the process can be initiated by making all of the contributions roughly equivalent. As a further refinement of this process, it is observed that when appropriate approximations are applied, the error variance can usually be rewritten as σ 2 (x, y) ⫽ ax m ⫹ by n ⫹ cx⫺r y⫺s

(44)

where x and y are the independent design parameters. The third term in this expression is the noise contribution, which may be represented by the variable z. Equation (44) is minimized when x⫽

冢冣 rcy⫺s ma

1/(m⫹r)

and

y⫽

冢冣 scx⫺r nb

1/(n⫹s)

(45)

Figure 16 Flowchart showing the interaction of component performance factors on the net operation of an adaptive-optics system.


52

Parenti

When these conditions are simultaneously satisfied for a specified performance goal, σ 2goal , the following design guidelines are developed: xⱕ

冤冢

冣冥

zⱕ

冢

冣

σ 2goal nr nr ⫹ ms ⫹ mn a

1/m

yⱕ

,

冤冢

冣冥

σ 2goal ms nr ⫹ ms ⫹ mn b

1/n

(46)

and mn σ 2goal nr ⫹ ms ⫹ mn

(47)

Consider, for example, a military system in which the Strehl, S, must be nearly unity. In this case the second term of Eq. (42) may be safely ignored and an equal division of the figure and tracking errors yields the following pair of equations: 2 fig total

(σ )

1 ⱕ ⫺ ln(S) 2

2 tilt total

(σ )

and

冢冣冤 2

1 λ ⱕ 5 D

1

√S

冥

⫺1

(48)

By incorporating judicious assumptions regarding the dominant resolution and noise effects for high-Strehl operation, one obtains the following forms for the figure and tilt errors: (σ 2fig ) total ⬇

冢冣冢冣冦冢冣冧

1 ds 2 r0

5/3

⫹

τs

5/3

τ0

2

π D ⫹ 0.24 ln 4 ds

(49)

16π 2 n 2rms 3[(λ/hc)G e ηγτ s d 2s I p ] 2

and (σ 2tilt ) total ⬇

冢冣冦冢冣 λ D

2

1 τ st 5 τ0 t

2

⫹

2 1 3π (λ/hc)ηγ τ r 20 τ st I p

冧

(50)

where I p is the photon irradiance and the γ parameters represent throughput factors for the individual subsystems. When Eq. (46) is applied one obtains d s ⱕ 0.68 [⫺ln(S)] 3/5 r0 ,

τ s ⱕ 0.29 [⫺ln(S)] 3/5 τ 0 , and


τ st ⱕ 0.58

√

1

√S

⫺ 1 τ 0 t (51)


53

To satisfy the wavefront sensor noise requirement for sensor-noise-limited operation, the number of photons collected per subaperture per dwell time must exceed Np ⱖ

18n rms η

√

ln(D/d s ) ⫺ln(S)

(52)

Thus, the beacon requirement is critically dependent upon the sensor noise and quantum efficiency. For a system incorporating a high-efficiency CCD array having a 10-electron readout fluctuation, the achievement of near-unity Strehl requires about 500 photons per subaperture per dwell time. Methods for achieving adequate target illumination are one of the principal concerns for the designers of directed-energy systems. The design of adaptive-optics systems for astronomical telescopes is strongly driven by the brightness of the more interesting target objects, and economic factors may constrain the affordable actuator density and bandwidth. The optimization process for this case is somewhat more complex and is likely to involve an adaptive adjustment of operational characteristics in order to achieve best performance under all conditions. When necessary, the effective servo bandwidth of a high-performance system can be reduced by introducing analog or digital filters, and the spatial resolution of the applied compensation can be degraded through an appropriate modification of the phase-reconstruction matrix. To illustrate the optimization process for a low-brightness object, Figs. 17 and 18 trace the characteristics and performance of a compensation system for an 8-m-class telescope, for which the following parametric values are assumed: D λ r0 τ0 τ0 t τm

⫽ ⫽ ⫽ ⫽ ⫽ ⫽

8m 0.55 µm 16 cm 5 ms 12 ms 1 ms

τ mt η n rms γ t /γ γ ⫹ γt

⫽ ⫽ ⫽ ⫽ ⫽

10 ms 0.8 10 electrons 0.1 0.5

The last of these assignment represents an allocation of 50% of the available light for the science camera. Source intensity has been converted to stellar magnitude, m ν , by way of the relationship I p ⫽ 10 ⫺(m ν⫹21.2)/2.5

(53)

Figure 18 suggests that, for high-quality visible imaging, one needs to locate or generate an eighth magnitude source to serve as the reference beacon for the wavefront sensor. The likelihood of finding a star of the requisite brightness in the vicinity of the target object is the subject of the next section.


54

Parenti

Figure 17 Spatial and temporal design parameters for an optimized adaptive-optics system as a function of target-object intensity. Beyond 12th magnitude, no significant improvement of the high-spatial-frequency (figure) component of turbulence is achieved for this system.

Figure 18 Resolution and Strehl as a function of target-object intensity for an optimized adaptive-optics system. Loss of signal necessitates the use of larger subapertures and a lower correction bandwidth, which results in a rapid degradation of resolution for objects dimmer than eighth magnitude.



55

VIII. OPERATIONAL UTILITY OF NATURAL GUIDE-STAR SYSTEMS Adaptive-optics systems are inherently high-light-level devices that cannot function usefully below some minimum signal threshold. When the target object fails to exceed this brightness requirement, a nearby star may provide a suitable reference. The maximum allowable displacement between the target and reference objects is established by the isoplanatic angle. The effect of a reference offset on the quality of phase compensation is most easily treated as an additional figure error, which can be appended to the expression given in Eq. (36). The magnitude of the figure error due to a referencesource offset of θ is 2 fig reference offset

(σ )

冢冣

θ ⬇ θ0

5/3

(54)

where θ 0 is the isoplanatic angle defined in Eq. (6). Since it has been demonstrated that the total phase error can be no more than a few rad 2 in order to maintain a moderate level of correction, it is reasonable to assume that the reference star can be displaced from the target object by no more than θ 0 . This criterion is strongly wavelength dependent, because the turbulence coherence length, time constant, and isoplanatic angle are all proportional to λ 6/5. Estimates of star densities may be extracted from the Infrared Handbook (20), which tabulates average population values as well as numbers for the galactic pole and equator; this information is summarized in Fig. 19. For first-order

Figure 19 Comparison of stellar densities for the galactic pole, the galactic equator, and the average density as a function of visual magnitude.


56

Parenti

numerical calculations it is convenient to use the following approximation for the average stellar density: Ψ(m ν ) ⬇ 1.45 exp(0.96 m ν ) (stars/rad 2 )

(55)

which can be combined with Eq. (53) to rewrite the density as a function of the received visible-wavelength irradiance: Ψ(I p ) ⬇ 2.02 ⫻ 10 ⫺9 I p⫺1.04 (stars/rad 2 )

(56)

The final step in the operational utility calculation involves the combination of the density expressions and the isoplanatic angle. To first order, the probability of finding a star of brightness I p within an angular displacement of θ is P(θ, I p ) ⬇ πθ 2 Ψ(I p )

(57)

By applying the optimization process discussed in the previous section, this probability function can be plotted as a function of the short-exposure Strehl enhancement and the wavelength of the viewing system, as shown in Fig. 20. In this comparison, the phase error due to beacon-offset anisoplanatism is permitted to be comparable to the total phase distortion computed from Eq. (49). The operational utility analysis summarized in the previous figure emphasizes one of the most stressing issues facing the designers of adaptive-compensa-

Figure 20 Operational utility as a function of the achievable enhancement of the shortexposure Strehl. The operational utility defines the likelihood that a suitable reference star can be found within a small angular separation from the target object. In this analysis, the reference offset error was allowed to be comparable to the error due to the compensation system.



57

tion systems for astronomical applications. Although all of the essential hardware and data processing techniques required to achieve substantial improvements in resolution have been developed, the illumination requirements severely limit its practical utility at short wavelengths. This restriction can be eliminated, however, if provision is made to generate a synthetic beacon of sufficient intensity. Issues associated with synthetic beacon production and utilization are addressed in Chapter 3.

REFERENCES 1. M. G. Miller and P. L. Zieske, ‘‘Turbulence Environmental Characterization,’’ RADC-TR-79-131, Rome Air Development Center, Griffiss Air Force Base, NY, 1979. 2. R. E. Hufnagel, ‘‘Variations of Atmospheric Turbulence,’’ in Digest of Technical Papers, Topical Meeting on Optical Propagation Through Turbulence, Optical Society of America, Washington, DC, July 9–11, 1974. 3. J. L. Bufton, ‘‘Comparison of Vertical Profile Turbulence Structure with Stellar Observations,’’ Appl. Opt. 12, 1785–1793 (1973). 4. D. L. Fried, ‘‘Limiting Resolution Looking down through the Atmosphere,’’ J. Opt. Soc. Am. 56, 1380–1384 (1966). 5. D. P. Greenwood, ‘‘Bandwidth Specification for Adaptive Optics Systems,’’ J. Opt. Soc. Am. 67, 390–393 (1977). 6. D. P. Greenwood and D. L. Fried, ‘‘Power Spectra Requirements for Wave-FrontCompensation Systems,’’ J. Opt. Soc. Am. 66, 193–206 (1976). 7. G. A. Tyler, ‘‘Bandwidth Considerations for Tracking through Turbulence,’’ The Optical Sciences Company, Report No. TR-887 (March 1988). 8. G. A. Tyler and D. L. Fried, ‘‘Image Position Error Associated with a Quadrant Detector,’’ J. Opt. Soc. Am. 72 (1982). 9. R. H. Kingston, Detection of Optical and Infrared Radiation, Springer-Verlag, 1978. 10. R. H. Hudgin, ‘‘Wave-front Reconstruction for Compensated Imaging,’’ J. Opt. Soc. Am. 67, 374–378 (1977). 11. D. L. Fried, ‘‘Least-Squares Fitting a Wave-front Distortion Estimate to an Array of Phase-Difference Measurements,’’ J. Opt. Soc. Am. 67, 370–375 (1977). 12. J. Herrmann, ‘‘Least-Squares Wave Front Errors of Minimum Norm,’’ J. Opt. Soc. Am. 70, 28–35 (1980). 13. J. H. Everson, ‘‘New Developments in Deformable Mirror Surface Devices,’’ Proc. SPIE 141, Adaptive Optical Components (1978). 14. J. E. Harvey and G. M. Gallahan, ‘‘Wavefront Error Compensation Capabilities of Multi-actuator Deformable Mirrors,’’ Proc. SPIE 141, 50–57 (1978). 15. F. Roddier, ‘‘Curvature Sensing and Compensation: A New Concept in Adaptive Optics,’’ Appl. Opt. 27, 1223–1225 (1988). 16. R. J. Sasiela, Electromagnetic Wave Propagation in Turbulence, Springer-Verlag, 1994.


58 17.

Parenti

D. P. Greenwood, ‘‘Mutual Coherence Function of a Wavefront Corrected by Zonal Adaptive Optics,’’ J. Opt. Soc. Am. 69, 549–553 (1979). 18. A. D. Poularikas and S. Seely, Signals and Systems, PWS, 1985. 19. H. T. Yura, ‘‘Short-Term Average Optical-Beam Spread in a Turbulent Medium,’’ J. Opt. Soc. Am. 63, 567–572 (1973). 20. W. L. Wolfe and G. J. Zissis, Eds., The Infrared Handbook, Environmental Research Institute of Michigan, 1985.


3 Guide Star System Considerations Richard J. Sasiela and John D. Shelton Massachusetts Institute of Technology Lincoln Laboratory, Lexington, Massachusetts

I. INTRODUCTION Adaptive-optics system design must take into account many factors. The beacon that provides a measure of the atmospheric turbulence may be in a different direction from the one that needs the correction. In this case the performance depends on the isoplanatic patch size, which is the region over which the turbulence is fairly constant. This patch size can be different for different Zernike modes. The rate of change of turbulence with time affects the acceptable servo bandwidth. The target must be tracked through turbulence and the effect of tilt on system performance is very important. Apertures tend to smooth out some effects, and this affects system design. The system performance with artificial beacon is affected by the beacon height and the number of beacons. Formulas to determine all these effects to enable one to design a system are developed in this chapter. The solution of most turbulence problems of practical interest involving wave propagation rely on applying the theory in the region in which the Rytov approximation applies. Generally, these are cases in which the scintillation caused by turbulence is small. It is also assumed that the turbulence can be represented in the spatial Fourier domain as an isotropic spectrum. If one also assumes that the turbulent cells do not change rapidly, then observed changes in the turbulence are caused by the transport of the turbulent eddies by the local winds. This is called the frozen turbulence assumption. With this assumption one can readily transform the spatial spectrum into a temporal spectrum. Adaptive-optics systems are generally used under conditions in which these approximations apply. Adaptive-optics performance depends on the characteristics of turbulence that affects the tilt jitter, other Zernike modes, and how these


60

Sasiela and Shelton

modes change with direction, displacement, and time. Formulas that enable one to calculate these effects are given in this chapter. In adaptive-optics system design one is interested in the effect that turbulence has on system performance. The turbulence spectra is modified by the aperture and the subsequent processing to obtain the tilt or other Zernike modes or some function of the difference between the turbulence seen in different directions. All these processes are linear and can be represented in the Fourier domain as a filtering operation on the turbulence spectrum. The aperture operations decrease the observed turbulence strength for some spatial frequencies and increase it at others. Often one is interested in the resulting phase variance or correlation, which is the sum of the contributions at individual frequencies. Since any aperture operation can be viewed as a filtering operation, it is not surprising that there is a general way to solve these problems. One can set up a general problem to find the variance or correlation function of the aperture amplitude and phase functions. The expression contains a sixfold integration. With reasonable assumptions three of these integrals can be evaluated in general. The reduced expression has just three integrals—two over the transform of the coordinates perpendicular to the direction of propagation, and the third along the spatial coordinate in the direction of propagation. A specific problem is solved by inserting the appropriate filter function in the general formula. To complete the solution, the triple integral must be evaluated. The double integral in the transverse Fourier coordinate can be expressed as an integral over angle and magnitude. Either because the problem has circular symmetry or because of a relatively simple angular dependence, for most problems, the angular integration is easily evaluated. Surprisingly, for all the common filter functions the integral over the magnitude of the spectrum can be evaluated and the answer expressed in terms of analytic functions; thus, one need not resort to numerical integration. The Mellin transform is the vehicle for performing these integrals. For the simplest problems in which there is only one parameter that varies, the solution is found by a table look-up. Pole-residue integration using the Cauchy residue theorem is used to find the solution when there are two parameters that vary. The evaluation of the integral in this case can also be obtained with computer algebra programs such as Mathematica or Maple. The results are expressed as a sum of generalized hypergeometric functions. Luckily, computer algebra programs can readily plot these functions. The programs also calculate power series for these solutions so that one can calculate results easily for small parameter values and observe their dependence. For large values of the parameter, the Mellin transform technique can be used to obtain asymptotic series, which are very useful in determining trends and performing system calculations. For even more complicated problems where there are more than two parameters one can use the Mellin transform technique to obtain the solution as multiple infinite series. Asymptotic solutions can also be obtained in these cases.


Guide Star System Considerations

61

After evaluating the integral in the Fourier spectral domain, the last remaining integration along the propagation path can often be evaluated so that the solution is expressed as a sum of one or more moments of the turbulence distribution along the propagation path.

II. GENERAL SOLUTION OF TURBULENCE PROBLEMS Most turbulence problems of interest are concerned with the turbulence effects on a single beam or the difference in effects between two beams. The singlebeam case is a special case of the double-beam case if different amplitudes are allowed for the two beams. In this section the general solution to the turbulence problem in which the phase variance between two beams that can be of different size, different direction, and have different aperture functions applied is given. Diffraction effects are included. Either beam can be collimated or focused at the aperture or any point in space. The background and detailed derivation of the general solution are given in Sasiela and Shelton [1993a,b] and Sasiela [1994a]. Tatarski [1961, 1971] uses the Rytov approximation to obtain formulas for phase and log-amplitude variances. His method is generalized here to obtain an expression that can be applied to many problems and particularly those associated with adaptiveoptics system design. In order to cover the cases of collimated and spherical waves, the derivation for the expression for variance of the log-amplitude and phase is done on a Gaussian wave using the approach of Ishimaru [1969]. The results given here apply to the case in which the beam radius is allowed to go to infinity. These results can be generalized to the finite radius case. Beam-wave theory is developed by using the scalar wave equation, the paraxial approximation (beam-propagation vector is close to the z-axis), and the Rytov approximation to find the Green’s function for the problem. From the Green’s function one can write an expression for the phase, φ(rជ, L), or logamplitude, χ(rជ, L), of an infinite Gaussian beam propagating between z ⫽ 0 and z ⫽ L as φ(rជ, L)

冤χ(rជ, L) 冥

⫽ k0

冮

L

0

dz′ ∫ dv(κជ, z′)

冤sin[P(γ, κ, z′] 冥 exp[i γκជ ⋅ rជ] cos[P(γ, κ, z′]

where k0 λ κ dν(κជ , z)

⫽ ⫽ ⫽ ⫽

2π/λ wavelength of the propagating wave spatial wavenumber transverse to the z-direction turbulence-induced refractive index variations


(1)

62

Sasiela and Shelton

γ

⫽ propagation parameter that distinguishes between collimated and focused beams P(γ, κ, z) ⫽ diffraction parameter defined below For a source at z ⫽ 0 centered on the origin, one finds γ ⫽ 1 for the plane wave case, and γ ⫽ z/L for the spherical wave case. For a source at z ⫽ L centered on the origin, γ ⫽ 1 for the plane wave case, and γ ⫽ (L ⫺ z)/L for the spherical wave case. For Gaussian beams, in general, γ is complex. Here γ is taken to be real. For propagation from z ⫽ 0 to z ⫽ L, the diffraction parameter is P(γ, κ, z) ⫽

γ κ 2 (L ⫺ z) 2k 0

(2)

and for propagation from z ⫽ L to z ⫽ 0, it is P(γ, κ, z) ⫽

γ κ2 z 2k 0

(3)

For many problems the weighted average of phase or log-amplitude over a receiving aperture is wanted. If g(rជ ) is a normalized weighting function that multiplies the above expression, the integral over the aperture of diameter D is recognized as the Fourier transform of the aperture function and is given by G(γ κជ) ⫽ ∫ d rជ ′ g(rជ ′) exp[i γκជ ⋅ rជ ] (4) where G(γ κជ) is called the complex aperture filter function. We choose the normalization such that 4 ∫ d rជ ′ g 2 (rជ ′) ⫽ 1 (5) πD 2 Using these results, the phase or log-amplitude related difference between two waves with different propagation and diffraction parameters and aperture weightings, and where the second wave has a relative amplitude A(κជ , z) with respect to the first as illustrated in Fig. 1, is

Figure 1 Geometry of focused and collimated beams of different diameters propagating in different directions with a separation that depends on the axial coordinate.



φR(rជ, L)

冤χR(rជ, L) 冥

⫽ k0

冮

63

L

dz′ ∫ dv(κជ, z′)

0

冦

⫻ G 1 (γ 1 κជ)

冤 sin[P (γ , κ, z′)] 冥 cos[P1 (γ 1 , κ, z′)] 1

(6)

1

⫺ A(κជ, z′)G 2 (γ 2 κជ)

冤 sin[P (γ , κ, z′)] 冥冧 cos[P2 (γ 2 , κ, z′)] 2

2

If the two beams have the same amplitude, but one is displaced from the other by a distance dជ, which can vary with axial distance, then A(κជ, z) ⫽ exp[iκជ ⋅ dជ ]

(7)

Care must be exercised in using this relation with focused beams. For a point source at z ⫽ L, one has γ ⫽ 1 ⫺ z/L. A displacement of dជ means that the waves propagate from the point at z ⫽ 0 and rជ ⫽ 0 to two receiving apertures separated by dជ. In this case A(κជ, z) ⫽ exp[i(1 ⫺ z/L)κជ ⋅ dជ ]. If one wanted the difference between two focused beams in which the beam focus points and receiving apertures were each separated by dជ, then A(κជ , z) ⫽ exp[iκជ ⋅ dជ ]. To find the variance, this expression is multiplied by its complex conjugate and the expected value is found. The techniques that have been used in the past [Strohbehn, 1978] to perform three of the six integrations are used to obtain the final expression: σ 2φR

冤σ 冥 ⫽ 0.2073 k 冮 dz C (z) ∫ dκជ f (κ) cos[P (γ , κ, z)] ⫻ 冦G (γ κជ) 冤 sin[P (γ , κ, z)] 冥 cos[P (γ , κ, z)] ⫺ A(κជ, z)G (γ κជ) 冤 sin[P (γ , κ, z)] 冥冧 L

2 0

2 χR

2 n

0

1

2

冦

1

1

1

1

1

⫻ G*1 (γ 1 κជ)

2

2

2

2

2

(8)

冤 sin[P (γ , κ, z)] 冥 cos[P1 (γ 1 , κ, z)] 1

1

⫺ A*(κជ, z)G*2 (γ 2 κជ)

冤 sin[P (γ , κ, z)] 冥冧 cos[P2 (γ 2 , κ, z)] 2

2

where f(κ) is the normalized two-dimensional turbulence spectrum and C 2n (z) is turbulence strength along the propagation path.


64

Sasiela and Shelton

This expression assumes that the wave is infinite. For a finite size beam, for which γ is complex, a similar expression can be found. This expression can be evaluated by Mellin transform techniques for specific problems. The beamwave case is discussed in detail in Sasiela [1994a]. If the inner and outer scales are negligible, then the spectrum is the normalized Kolmogorov spectrum given by f (κ) ⫽ κ⫺11/3

(9)

The coefficient 0.033 that normally multiplies this spectrum has been absorbed into the coefficient of the variance expression. This spectrum requires infinite energy because the turbulence becomes unbounded as the wavelength gets large. The behavior at large wavelengths or outer scale depends on the generation mechanism of the turbulence, and it can have a different dependence in different circumstances. Spectrums have been proposed by von Ka´rmańn and Greenwood [1974]. The von Ka´rmańn turbulence spectrum is given by f (κ) ⫽ [κ 2 ⫹ κ 20 ]⫺11/6

(10)

and the Greenwood spectrum is f (κ) ⫽ (κ 2 ⫹ κκ 0 )⫺11/6

(11)

where κ 0 ⫽ 2π/L 0 , and L 0 is called the outer scale of turbulence; it can depend on the axial coordinate. There are also difficulties with large wavenumbers. From the hydrodynamic equations it can be shown that the above spectra require infinite energy dissipation due to the rapid increase in dissipation with decreasing scale size. Physically, below a certain scale size dissipative effects force the velocity to be laminar, leading to a rapid decrease in the spectrum with decreasing scale size. Several ad hoc models have been given for the spectrum in this inner scale region; the Gaussian form was proposed by Tatarski. A spectrum with both inner scale and von Ka´rmańn outer scale—the modified von Ka´rmańn spectrum—is Ψ(k) ⫽ 0.033 C 2n (k 2 ⫹ κ 20 )⫺11/6 exp[⫺k 2 /κ 2i ]

(12)

where κ i ⫽ 2π/L i ⫽ 5.91/L in , and L in is the commonly used inner scale. The spectrum is shown in Fig. 2. Greenwood and Tarazaro [1974] and Gurvich et al. [1974] noted that the turbulence does not decay as rapidly as that predicted by the Kolmogorov spectrum in the region just before it starts its rapid decay due to inner scale effects. Hill and Clifford [1978] did a hydrodynamic analysis to derive a spectrum in the region where inner scale is significant, and found a slight peak before an exponen-



65

Figure 2 Von Ka´rmań spectrum with inner and outer scale present. The value of κ i is equal to 2π/0.1, and the value of κ o is equal to 2π/10. The presence of the outer scale causes the spectrum to approach a constant at low wavenumbers. The inner scale causes the spectrum to decay faster than Kolmogorov turbulence at high wavenumbers. The spectrum has an ⫺11/3 power law behavior in the inertial subrange at intermediate wavenumbers.

tial decay. Frehlich [1992] has modeled the spectrum as Ψ(k) ⫽ 0.033 C 2n (k 2 ⫹ κ 20 )⫺11/6 g(k)

(13)

where a four-term approximation for g(k) is 4

g(k) ⫽ exp(⫺δ | k| L in )

冱 a (|k| L ) n

in

n

(14)

n⫽0

The constants are δ ⫽ 1.1090, a 0 ⫽ 1, a 1 ⫽ 0.70937, a 2 ⫽ 2.8235, a 3 ⫽ ⫺0.28086, and a 4 ⫽ ⫺0.08277. This function is plotted versus the Tatarski function exp[⫺(kL in /5.92) 2 ] in Fig. 3. Scintillation measurements agree more closely with the Hill spectrum.

Figure 3 Factor that multiplies the Kolmogorov turbulence spectrum in the Tatarski and Hill models of inner scale.


66

Sasiela and Shelton

Inner scale affects the scintillation but has little effect on the Zernike modes. For that reason its effects are normally neglected in adaptive-optics system design. For specific cases Eq. (8) can be simplified. If there is only one beam, the weighting function A(κជ , z) is zero, and the formula reduces to σ 2φR

冤冥 σ 2χR σ

冮

⫽ 0.2073 k 20

冤

dz C 2n (z) ∫ dκជ f(κ)

0

2 φχR

⫻

L

cos 2 [P(γ, κ, z)]

sin 2 [P(γ, κ, z)] 0.5 sin[2P(γ, κ, z)]

冥

(15) F(γκជ)

where the aperture filter function is defined as F(γ κជ) ⫽ G(γ κជ) G*(γ κជ)

(16)

To find the phase or log-amplitude variance at a point, F(γ κជ) should be set to unity. In this equation σ 2φχR is the cross variance of phase times log-amplitude related quantities. A series of formulas based on the general equation that apply in different situations will now be given. If there are two beams with weighting on the phase or log-amplitude differing by a real constant, i.e., A(κជ, z) is real, and whose values of γ are the same, and of which the same aperture-averaged component is wanted, then

冤σ 冥 σ 2φR 2 χR

⫽ 0.2073 k 20 ⫻

冮

L


0

{cos[P1 (γ, κ, z)] ⫺ A(κជ, z) cos[P2 (γ, κ, z)]} 2

冤{sin[P (γ, κ, z)] ⫺ A(κជ, z) sin[P (γ, κ, z)]} 冥 F(γκជ) 1

(17)

2

2

If there are two beams with equal weighting on the phase and log-amplitude whose values of γ are equal, but where different aperture modes are wanted, then


⫽ 0.2073 k 20 ⫻

冮

L


0

冤sin [P(γ, κ, z)] 冥 | G (γ κជ) ⫺ G (γ κជ) | cos 2 [P(γ, κ, z)] 2

1

2

(18) 2

If there are two beams with equal weighting on the phase and log-amplitude



67

whose values of γ differ, but where different aperture modes are wanted, and diffraction can be neglected, then σ 2φR ⫽ 0.2073 k 20

冮

L

dz C 2n (z) ∫ dκជ f (κ) | G 1 (γ 1 κជ) ⫺ G 2 (γ 2 κជ) | 2

0

(19)

If there are two beams displaced from each other by a distance dជ that have the same propagation constant, and for which the same aperture mode is wanted, one obtains D φR (dជ )

冤σ 冥冤D (dជ ) 冥 σ 2φR

⫽

2 χR

⫽ 0.2073 k 20

χR

冮

L


0

冤sin [P(γ, κ, z)] 冥 F(γ κជ) 2[1 ⫺ cos(κជ ⋅ dជ )] cos 2 [P(γ, κ, z)]

⫻

(20)

2

D φ (dជ ) and D χ (dជ ) are the phase and log-amplitude structure functions when F (γ κជ) ⫽ 1. Equation (17) can be used to find the phase and log-amplitude of a wave (called a beacon) that has propagated down through the atmosphere. If an adaptive-optics system is used in which a collimated beam has the conjugate of this phase applied at the origin, then the variances along the propagation path that is displaced by a distance dជ (the anisoplanatic offset) from the beacon path can be found from


⫻

⫽ 0.2073 k 20

冮

L

dz C 2n (z) ∫ dκជ f (κ)

0

冷冤冥冤冥冤冥冷冷 sin 冤κ2kL冥 cos 冤κ2kz冥 ⫺ exp[i κជ ⋅ dជ ] sin 冤κ (L2k⫺ z)冥冷

冤

cos

κ2L κ2z κ 2 (L ⫺ z) cos ⫺ exp[i κជ ⋅ dជ ]cos 2k 0 2k 0 2k 0 2

2

0

2

2

0

0

2

冥

F(κជ)

(21)

If the anisoplanatic offset is zero, the expression is


⫽ 0.2073 k 20

⫻

冮

L

dz C 2n (z) ∫ dκជ f (κ)

0

冷 sin 冤冥冤冥冷冷 cos 冤κ2kL冥 sin 冤κ2kz冥冷

冤

2

κ2L κ2z sin 2 2k 0 2k 0 2

2

2

2

0


0

2

2

冥

(22) F(κជ)

68

Sasiela and Shelton

To solve a specific problem, the relevant aperture filter function and turbulence spectrum must be inserted into the above equations. The next section contains filter functions relating to Zernike modes and gradient tilt on an unobscured and an annular aperture, and those due to a finite aperture or source distribution. The general formula for the power spectral density of phase, S φ (ω), or logamplitude, S χ (ω), in the frozen turbulence approximation can be found from the above expressions. Power spectral density, S(ω), is related to variance, σ 2, by


⫽

1 π

冮

∞

dω

冤S (ω)冥 S φ (ω)

(23)

χ

0

The integral over κ can be converted to one over frequency by making the substitutions κx ⫽

ω v(z)

(24)

and c2 ⫽

κ 2y v 2 (z) ω2

⫹1

(25)

where v(z) is the velocity profile along the propagation path. As an example, the case of a single wave is illustrated. It is assumed the spectrum is reduced by a system for which Fservo (ω) is the servo response. Recognizing that everything multiplying dω in the resulting integral over ω is the power spectral density, one finds

冤S (ω)冥 S φ (ω) χ

⫽ 2.606 k 20 ωFservo (ω)

冮

C 2n (z)

L

dz

v 2 (z)

0

冮

∞

c dc U(c ⫺ 1) √c 2 ⫺ 1

0

冤冥 γωc 冤冥 sin γω c (L ⫺ z) F 冤v(z) 冥冤 2v (z) k 冥

ωc ⫻f v(z)

冤

cos

2

γω 2 c 2 (L ⫺ z) 2v 2 (z) k 0 2 2

2

2

0

冥

(26)

The Heaviside unit step function is defined by U(x) ⫽ 1

for x ⬎ 0

U(x) ⫽ 0

for x ⬍ 0

(27)

From the Huygens-Fresnel approximation the Strehl ratio for a circular



69

aperture [Tyler, 1984] can be found by evaluating the beam profile at the origin to obtain

冤 1 D(αជ) ⫽ ∫ d αជ K(α) exp 冤⫺ 2π 2 冥

SR ⫽

1 αជ D(αជ D) ∫ d αជ K(α) exp ik 0 Dρជ ⋅ ⫺ 2π z 2

冥冷

ρជ⫽0

(28)

The integral is over a circular aperture of unit radius, D(αជ) is the structure function, and K(α) is the optical transfer function modified by a numerical factor such that 16 [cos⫺1 (α) ⫺ α(1 ⫺ α 2 )1/2 ] U(1 ⫺ α) π

K(α) ⫽

(29)

When the structure function is isotropic, the integration over angle in the aperture can be performed to yield SR ⫽

冮

∞

0

冤

D(α) α dα K(α) exp ⫺ 2

冥

(30)

III. FILTER FUNCTIONS The solution for the phase variance in the last section contained a filter function. This function depends on the particular problem that is being solved. For standard problems the filter function can be chosen from those given below. Filter functions to determine the Zernike modes on a full or annular aperture and the gradient tilt are listed below. The filter functions to calculate second moment quantities with a finite receive aperture or a finite size source are listed.

A.

Zernike Modes

The general formula for the filter function to find a Zernike component on a circular aperture of diameter D was given by Noll [1976] as Fm, n (κជ) x Fm, n (κជ) y F0, n (κជ)

冧

⫽ (n ⫹ 1)

(κD/2) 冤2 J κD/2 冥 n⫹1


2

冦

2 cos 2 (mϕ)

2

sin 2 (mϕ)

1

(m ⫽ 0)

(31)

70

Sasiela and Shelton

The formula for specific components is found from the general result. The piston, Zernike mode (0, 0), phase-variance filter function is F(κជ ) ⫽

冤

2 J 1 (κD/2) κD/2

冥

2

(32)

This filter function extracts the phase variance due to piston. To obtain the meansquared piston displacement in physical as opposed to phase space, the filter function must be multiplied by (1/k 0 ) 2. The filter function to determine the x, y, and total phase variance from the aperture tilt, Zernike mode (1, 1), is Fx (κជ) Fy (κជ ) F(κជ)

冧

⫽

(κD/2) 冤4 JκD/2 冥 2

2

冦

cos 2 (ϕ)

sin 2 (ϕ)

(33)

1

The filter functions to determine the x and y components and total tilt angle is obtained from the above by multiplying by (4/k 0 D) 2 where the local diameter must be inserted for an uncollimated beam. In some problems one requires the variance with some Zernike modes removed. This is easily done by subtracting the phase variance due to these components from the total variance. For instance, the filter function to remove piston and tilt variance from a single wave is F(κជ) ⫽ 1 ⫺

冤

2 J 1 (κD/2) κD/2

冥冤 2

⫺

4 J 2 (κD/2) κD/2

冥

2

(34)

B. Gradient Tilt The gradient or G-tilt, in a geometric sense, is equal to the average ray direction that is obtained by finding the average x and y components and the total phase gradient over the aperture. The filter function of G-tilt variance is Fx (κជ ) Fy (κជ) F(κជ)

冧

⫽ J 21 (κD/2)

冦

cos 2 (ϕ)

sin 2 (ϕ)

(35)

1

The filter function to determine the tilt angle is obtained from the above by multiplying by (4/k 0 D) 2.



C.

71

Tilt on an Annulus

Tilt filter functions for the Zernike and gradient tilts on an annulus can be derived from the expressions given in Yura and Tavis [1985]. The Zernike tilt filter function on an annulus with inner diameter D i is Fx (κជ) Fy (κជ) F(κជ)

冧

(κD/2) D J (κD /2) ⫺ 冤k D[1 ⫺16(D /D) ]冥冤J κD/2 冢 D 冣 κD /2 冥 2

⫽

3

2

0

i

2

i

4

i

2

i

冦

cos 2 (φ)

sin 2 (φ)

(36)

1

The gradient tilt on an annulus is Fx (κជ) Fy (κជ) F(κជ)

D.

冧

冤k D[1 ⫺4(D /D) ]冥冤J (κD/2) ⫺ DD J (κD /2冥 2

⫽

2

i

0

1

2

i

1

i

冦

cos 2 (φ)

sin 2 (φ)

(37)

1

Filter Functions for Finite Size Apertures and Distributed Sources

The coherent average of a quantity from a point source at a range L over a receive aperture is equal to the piston of the quantity, which is given by

冤

F(κជ) ⫽ 2

J 1 (κD(1 ⫺ z/L)/2) κD(1 ⫺ z/L)/2

冥

2

(38)

If the point source is at a very long range, this reduces to

冤

F(κជ) ⫽ 2

冥

J 1 (κD/2) κD/2

2

(39)

The filter function for an incoherent source of diameter D s is

冤

F(κជ) ⫽ 2

J 1 (κD s z/2L) κD s z/2L

冥

2

(40)

The filter function of an incoherent source of diameter D s received coherently by a receive aperture of diameter D is the product of the above two filter functions, which is

冤

F(κជ) ⫽ 2

J 1 (κD(1 ⫺ z/L)/2) κD(1 ⫺ z/L)/2

冥冤 2


2

J 1 (κD s z/2L) κD s z/2L

冥

2

(41)

72

Sasiela and Shelton

IV. TURBULENCE MOMENTS As mentioned above, the solution of the last integration along the propagation path is often expressed in terms of turbulence moments. The definitions of the full and partial turbulence moments are given. The zenith dependence is contained in the moments. The full moments are equal to µm ⫽

冮

∞

0

C 2n (z) z m dz ⫽ sec m⫹1 (ξ)

冮

∞

C 2n (h) h m dh

(42)

0

where ξ is the zenith angle. Looking directly up, ξ ⫽ 0. Consider propagation to a distance L, which is at a height H, such that L ⫽ sec(ξ)H. Define the partial moments as µ ⫹m (L) ⫽

冮

∞

冮

L

µ ⫺m (L) ⫽

L


冮


冮

∞

C 2n (h) h m dh

(43)

C 2n (h) h m dh

(44)

H

and

0

H

0

Often turbulence results are expressed in terms of the coherence diameter and isoplanatic angle. The coherence diameter for a plane wave, r 0 , is defined as ⫽ 0.423 k 20 µ 0 r ⫺5/3 0

(45)

and the isoplanatic angle, θ 0 , is defined as ⫽ 2.91 k 20 µ 5/3 θ ⫺5/3 0

(46)

V. MELLIN TRANSFORMS As pointed out in the section where a general formula for the variance was given, if a specific filter function is inserted into one of the equations for phase variance or correlation, then the variance is expressed as a triple integral. The integration over angle in the κជ-plane can usually be performed. The remaining expression contains integrations over spatial wavenumber and propagation direction. The techniques to be presented in this section apply to performing the kappa (spatial wavenumber) integration. The variables are first normalized, and the resulting integration has zero, one, or more parameters that are independent of the integration variables. If no parameters are present, the integration is performed simply by table look-up. If one or more parameters are present, one evaluates the integral



73

by using a computer algebra program or by transforming the integral using the Mellin convolution integral into an integral in one or more complex planes. The highlights of the technique for evaluating the integral by pole-residue methods are discussed in this section. Tatarski considered the evaluation of integrals with one parameter in which inner scale was finite. The form of the inner scale was an exponential that decayed with decreasing scale size. This allowed him to expand the function multiplying the decaying exponential in a Taylor series, and integrate term by term. Since the integral over each term of the power series converged absolutely, the integration could be performed term by term. The resulting power series was then expressed as a hypergeometric function. This approach does not work with zero inner scale or with more than one parameter, thereby restricting the problems that can be solved. This limitation does not apply to the technique discussed below, which applies to every problem of finding second moments or spectral densities using the filter functions given above. The technique is also applicable to finding analytic solutions for the Strehl ratio given in Eq. (30). The Mellin transform pair is given by h(x) → H(s) ⬅ M[h(x)] ⬅

冮

∞

0

dx h(x)x s x

(47)

and h(x) ⫽

1 2πi

冮 ds H(s)x

⫺s

(48)

C

The integral in Eq. (47) will only converge when the value of s is within certain limits. The integration path in the inverse transform integral goes from ⫺i∞ to ⫹i∞, and the value of the real part of s along the integration path is determined by the convergence properties of the function being transformed. The Mellin transform of a function that has the first N terms of its power series subtracted from it is the same as the Mellin transform of the original function with the integration path moved past N poles of the function. This is useful for anisoplanatic problems. The gamma function notation used by Slater [1966] and Marichev [1983] means Γ

] Γ[α ] ⋅ ⋅ ⋅ Γ[α ] 冤 β , . . . , β 冥 ⫽ Γ[α Γ[β ] Γ[β ] ⋅ ⋅ ⋅ Γ[β ] α1, . . . , αm 1

n

1

2

m

1

2

n

(49)

Mellin transforms useful for turbulence problems are given in Table 1. There are many additional Mellin transforms that are tabulated. Mellin


74

Sasiela and Shelton

Table 1 Mellin Transforms Useful for Turbulence Problems a

冮

h(x) → M[h(x)] ⬅ H(s) ⫽

∞

dx h(x)x s x

0

(T1)

exp(⫺x) → Γ[s], Re s ⬎ 0

(T2)

sin(x) → 2 s⫺1 √ π Γ

冤 1 ⫺ s/2 冥 ,

|Re s | ⬍ 1

(T3)

cos(x) → 2 s⫺1 √ π Γ

冤1/2 ⫺ s/2冥 ,

0 ⬍ Re s ⬍ 1

(T4)

1/2 ⫹ s/2

s/2

冤

冥

s/2 √π Γ , ⫺4 ⬍ Re s ⬍ 0 sin 2 (x 2 ) → ⫺ 1/2 ⫺ s/4 8 Jν (x) → 2 s⫺1 Γ J 2ν (x) →

1 2 √π

s/2 ⫹ ν/2

冤ν/2 ⫹ 1 ⫺ s/2冥 , Γ

Jν (x) J ν⫹1 (x) → (1 ⫹ x)⫺p →

⫺Re ν ⬍ Re s ⬍ 3/2

s/2 ⫹ ν, 1/2 ⫺ s/2

冤ν ⫹ 1 ⫺ s/2, 1 ⫺ s/2冥 , 1

2 √π

Γ

⫺2 Re ν ⬍ Re s ⬍ 1

s/2 ⫹ ν ⫹ 1/2, 1 ⫺ s/2

冤ν ⫹ 3/2 ⫺ s/2, 3/2 ⫺ s/2冥 ,

Γ[s, p ⫺ s]

(T5)

⫺1 ⫺ 2 Re ν ⬍ Re s ⬍ 2

(T6)

(T7)

(T8)

, 0 ⬍ Re s ⬍ Re p

(T9)

(1 ⫺ x) a⫺1 U(1 ⫺ x) → Γ[a]Γ

冤s ⫹ a 冥 ,

Re a ⬎ 0, Re s ⬎ 0

(T10)

(x ⫺ 1) a⫺1 U(x ⫺ 1) → Γ[a]Γ

冤

冥,

Re a ⬎ 0, Re(a ⫹ s) ⬍ 1

(T11)

冥

(T12)

Γ[p]

s

1⫺a⫺s 1⫺s

冤

s/2 ⫹ 1/2, ⫺s/2 √π , cos⫺1 (x) U(1 ⫺ x) → ⫺ Γ s/2 ⫹ 1, 1 ⫺ s/2 4

Re s ⬎ 0

U(1 ⫺ x) → Γ

冤s ⫹ 1冥 ,

Re s ⬎ 0

(T13)

U(x ⫺ 1) → Γ

冤1 ⫺ s冥 ,

Re s ⬍ 0

(T14)

s

⫺s



75

Table 1 Continued

Jµ (x) Jλ (x) →

1

Γ

2 √π

冤

µ⫹λ 1 s s s ⫹ , ⫺ ,1⫺ 2 2 2 2 2

1⫺

µ⫹λ µ⫺λ λ⫺µ s s s ⫹ ,1⫺ ⫹ ,1⫺ ⫹ 2 2 2 2 2 2

冥

,

⫺Re(µ ⫹ λ) ⬍ Re s ⬍ 1 (T15) K(x) → a

冤

冥

s/2 ⫹ 1/2 8 1 Γ , Re s ⬎ 0 s/2 ⫹ 2 √π s

(T16)

U(x) is the Heaviside unit step function, and K(x) is a constant times the optical transfer function on a circular aperture.

transforms exist for most of the functions that one normally encounters. A table of Mellin transforms may be augmented by using their following properties: h(ax) → a ⫺s H(s),

a⬎0

(50)

x b h(x) → H(s ⫹ b)

(51)

and h(x p ) →

H(s/p) |p|

p≠0

,

(52)

The parallel to the Fourier convolution integral is the Mellin convolution integral: h(x) ⫽

冮

∞

0

冢冣

dt x h 0 (t) h 1 → M[h(x)] ⬅ H(s) ⫽ H 0 (s) H 1 (s) t t

(53)

The original integral is equal to h(x) ⫽

1 2πi

冮 ds H (s) H (s) x 0

1

⫺s

C

This may be generalized to N complex planes as


(54)

76

Sasiela and Shelton

冮

h(x 1 , . . . , x N ) ⫽

∞

N

dy

0

y

h 0 ( y)

兿冢冣 hj

j⫽1

xj y

→ M[h(x 1 , . . . , x N )] N

⫽ H 0 (s 1 ⫹ s 2 ⫹ ⋅ ⋅ ⋅ ⫹ s N )

兿

H j (s j )

(55)

j⫽1

⫽ H(s 1 , s 2 , . . . , s N ) The original integral is equal to an integration in N complex planes given by h(x 1 , . . . , x N ) ⫽

1 (2πi) N

冮

⋅⋅⋅

C1

冮

1 ⋅ ⋅ ⋅ x ⫺s N ds 1 ⋅ ⋅ ⋅ ds N H(s 1 , s 2 , . . . , s N ) x ⫺s 1 N

(56)

CN

For the problems being considered here, the function H(s 1 , s 2 , . . . , s N ) is the product of ratios of Gamma functions. This fact allows these integrals to be readily evaluated. The analytic solution of all integrals that contain functions for which the Mellin transform is a ratio of Gamma functions can be found. The integrands for finding second moments, power spectral densities, or Strehl ratios using the filter functions given above contain the following functions: sin(x), cos(x), sin 2 (x), cos 2 (x), J ν (x), J ν (x)J µ (x), J 2ν , exp(x), (1 ⫹ x)⫺p, (x ⫺ 1)⫺p U(x ⫺ 1), and K(x). The Mellin transforms of every one of these functions are given in Table 1, thus making these problems amenable to an analytic solution. To evaluate the resulting integrals in κ-space, one uses Cauchy’s poleresidue integral theorem: f (z) ⫽

1 2πi

冮 ds s ⫺ z f(s)

(57)

C

where the contour encloses the pole at s ⫽ z, and the function f (s) is holomorphic (has no singularities) in the integration region. This can be generalized to N complex planes [Range 1986] as f (z 1 , z 2 , . . . , z N ) ⫽

1 (2πi) N

冮

C1

⋅⋅⋅

冮

ds 1 ⋅ ⋅ ⋅ ds N

CN

f (s 1 , s 2 , . . . , s N ) (s 1 ⫺ z 1 ) ⋅ ⋅ ⋅ (s N ⫺ z N )

(58)

where the contour encloses all poles, and the function f(s 1 , s 2 , . . . , s N ) is holomorphic in the integration region. These integrals are easy to evaluate for the problems considered here because the integrand is a ratio of Gamma functions. The only singularities are poles of the numerator Gamma functions. Each Gamma function has an infinite number of poles, one occurring at every negative integers (⫺n) where it has a



77

residue equal to (⫺1) n /n!. To evaluate the integral in one complex plane, simply write two sets of power series corresponding to the poles on each side of the path of integration. The power series that converges for the parameter value of interest is the solution to the problem. If the series resulting from the poles on one side of the integration path is an asymptotic series, then the other solution is valid for all values of the parameter. That power series will converge slowly for large values of the parameter, and in that case, one can use the asymptotic series. A steepest-descent contribution sometimes has to be added to the asymptotic series. A similar method is used to find the solution when the integration is over multiple complex planes. The details are described in Sasiela and Shelton [1993a] and in Sasiela [1994a]. VI. LIST OF RESULTS A.

Tilt

In this section formulas that apply to the calculation of the performance of an adaptive-optics system are given. Most of the results in this section are taken from Sasiela and Shelton [1993b]. First, consider the calculation of Zernike tilt, which is the dominant aberration of the atmosphere. Since this is a single wave problem, Eq. (15) applies. It is assumed that diffraction effects are not important, thereby allowing the cosine term to be replaced by unity. Neglecting inner- and outer-scale effects, the Kolmogorov spectrum given in Eq. (9) applies. Using the filter function to find the two-axis Zernike tilt given in Eq. (33), one obtains for the two-axis tilt variance T ⫽ 0.2073 k 2 Z

2 0

冮

L

dz C (z) ∫ d κជ κ⫺11/3 2 n

0

冢冣冤 16 k0 D

2

J 2 (γκD/2) γκD/2

冥

2

(59)

The integrand in the second integral does not depend on the angle in kappa space. With the substitution x ⫽ γκD/2 one obtains T 2Z ⫽

105.1 D 1/3

冮

L

dz C 2n (z) γ 5/3

0

冮

∞

0

dx ⫺11/3 2 x J 2 (x) x

(60)

Using the Mellin transform in Eq. (T7), one finds the two-axis tilt variance: T 2Z ⫽

6.08 µ 0 D 1/3

⫽ 0.3641

冢冣冢冣 D r0

5/3

λ D

2

(61)

where the coherence diameter, r 0 , is defined in Eq. (45), and the turbulence moments are defined in Sec. IV. Because the variance is proportional to the zeroth moment of turbulence, it varies as the secant of the zenith angle, sec(ξ). For a


78

Sasiela and Shelton

0.6-m-diameter aperture and HV-21 turbulence, the rms tilt is 4 µrad. This is four times the diffraction-limited resolution of the aperture in the visible. For larger apertures the tilt is a larger multiple of the diffraction-limited angle. The tilt phase variance can be converted into a tilt variance by dividing the phase variance by (4/k 0 D) 2 to obtain σ 2Tilt ⫽ 0.38µ0 k 20 D 5/3

(62)

The variance of any Zernike mode is given in the section on Zernike anisoplanatism. In a similar fashion the gradient tilt variance can be found by using the filter function in Eq. (35) and is given by T 2G ⫽ 0.2073 k 20

冮

冢冣冢冣 γκD 4 J 21 k0 D 2 2

L

dz C 2n (z) ∫ d κជ κ⫺11/3

0

(63)

This is easily evaluated using the Mellin transform given in Eq. (T7) to give T ⫽ 2 G

5.675 µ 0 D 1/3

冢冣冢冣

D ⫽ 0.3399 r0

5/3

λ D

2

(64)

The variance of the difference between the Zernike and gradient tilts was studied by Yura and Tavis [1985]. It can be found by inserting the aperture filter functions in Eq. (33) and (35) into Eq. (19) to obtain T 2G-Z ⫽ 0.2073 k 20 ⫻

冤冢

冮

L

dz C 2n (z) ∫ d κជ κ⫺11/3

0

4 J 2 (κD/2) κD/2

冣

冥

冢冣 4 k0 D

2

2

(65)

⫺ J 1 (κD/2)

Using the Mellin transform given in Eqs. (T7) and (T8), one finds T

2 G-Z

⫽

0.102 µ 0 D 1/3

冢冣冢冣

D ⫽ 0.0061 r0

5/3

λ D

2

(66)

If the turbulence is constant along the path of length L, the two-axis Zernike tilt variance is T 2Z ⫽

6.08 C n2 L D 1/3

(67)

For a point source propagating from the ground to space, the value of γ is z/L, and the Zernike tilt variance is



T 2Z ⫽

6.08 µ 5/3 1/3

⫽

5/3

D L

79

0.169 λ 2 D 1/3 L5/3 θ 05/3

(68)

Therefore, a tilt sensor in space observing a point source on the ground is sensitive to the isoplanatic angle. The variance varies as sec 8/3 (ξ). The significant effect of outer scale on tilt jitter was pointed out by Valley [1979]. To set up this problem, the general expression in Eq. (15) is used with the filter function for Zernike tilt in Eq. (33) for a plane wave for which γ ⫽ 1. The von Ka´rmańn turbulence spectrum given in Eq. (10) with diffraction neglected gives for the total tilt variance T ⫽ 0.2073 k 2 0

冮

2 0

L

0

dz C (z) ∫ d κជ [κ ⫹ κ ] 2 n

2 ⫺11/6 0

2

冢冣冤 16 k0 D

2

J 2 (κD/2) κD/2

冥

2

(69)

It is assumed that outer scale does not depend on z. The integrations over angle and z are evaluated. The κ integral can be evaluated using Mellin transform techniques to give T 20 ⫽

6.08 µ 0 D 1/3

冦冤 2

F3

11

/6 , 7/3 ; 5/6 , 29/6 , 17/6 ;

冢冣冤 1/3

D ⫺ 1.4234 L0

冢冣冥 2

πD L0

冢冣冥冧

πD 2 F3 2, /2 ; /6 , 5, 3; L0 5

7

2

(70)

These generalized hypergeometric functions can be evaluated and the results plotted by computer algebra programs. The lowest order terms of the summations in Eq. (70) are T 20 ⬇

6.08 µ 0 D 1/3

冦

冢冣冢冣

1 ⫺ 1.42

D ⫺4.01 L0

D L0

7/3

1/3

冢冣冢冣冢冣冧

⫹ 3.70

D ⫹ 4.21 L0

D L0 4

2

D ⫺ 4.00 L0

13/3

(71)

The tilt jitter is affected by outer scale, and the fractional decrease from the value with infinite outer scale is shown in Fig. 4. If outer scale is 100 times the aperture diameter, the rms tilt jitter is still decreased by 15%. This decrease occurs because the tilt jitter is caused mainly by the long wavelength turbulence. A finite outer scale decreases the turbulence at the long wavelengths and, thereby, decreases the tilt variance. The first two terms in braces give a very good approximation to the tilt variance if the outer scale is larger than several times the aperture diameter. This is an example of how to write the answer as a few terms of a power series. If


80

Sasiela and Shelton

Figure 4 Tilt variance for the von Ka´rmań turbulence spectrum with outer scale, normalized to tilt variance with infinite outer scale. Notice that there is a significant reduction of tilt even if outer scale is 100 times the diameter. This curve is independent of the turbulence distribution along the propagation path.

the outer-scale size were much smaller than the diameter, the series would converge slowly, and an asymptotic series would be appropriate. The Greenwood outer-scale model gives a similar tilt variation. The lowest order terms for this tilt variance are T ⬇ 2 0

6.08 µ 0 D

1/3

冦

冢冣冢冣冢冣冢冣冢冣冧

D 1 ⫺ 1.85 L0 D ⫺5.24 L0

2

1/3

⫹ 1.76

D ⫹ 6.70 L0

D L0

7/3

D ⫺ 3.77 L0

3

(72)

Winker [1991] considered the effect of outer scale on all the Zernike modes. B. Zernike Anisoplanatism and Its Effect on Adaptive-Optics Systems Zernike anisoplanatism refers to the difference in the level of a Zernike mode seen from either two sources or viewed in two apertures. Because the two waves arriving at the aperture or apertures have traveled through different turbulence,



81

there will be a difference in the level of each Zernike mode measured at any time. This difference will increase on average as the angle or displacement of the two apertures increases until the amplitude of the Zernike modes is uncorrelated for the two paths. In that case, the variance of the difference saturates at a level that is two times the variance of the mode measured in a single aperture. The variation of tilt jitter with separation of apertures has been applied in many applications. For instance, if one is sending a laser beam to a target that is being tracked, then the laser beam must lead the target that is being tracked to account for the finite speed of light. The arriving beam will jitter on the target due to tilt anisoplanatism. Anisoplanatism for higher order aberrations is important in applications such as adaptive-optics systems that correct for Zernike modes (modal) rather than local regions of the aperture (zonal). It is important to determine over what angle a correction applies and what error remains after the correction. It is shown that the angle over which any correction is possible decreases as the mode order increases and the aperture diameter decreases. This angle can be quite small. In one limit, if the angle over which the correction applies is less than the resolution of the system, then the correction is worthless. Even for larger angles there can be a severe degradation in performance in some applications. It is shown for some aperture separations that the phase variance for some Zernike modes more than triples when a correction is applied. Applying such a correction is counterproductive. 1. Derivation of Formulas for Zernike Anisoplanatism The calculation of the variance of Zernike anisoplanatism is a direct application of the technique described above. To find the total value of tilt anisoplanatism and both the components parallel and perpendicular to the displacement direction that is taken to be along the x-axis, Eq. (20) with diffraction neglected is used with the filter function for a Zernike mode given in Eq. (31) to obtain for the phase variance σ 2x

冦冧 σ 2y σ

2

⫽ 0.2073 k 20

冮

L

C 2n (z) dz ∫ d κជ

0

n, m

⫻ (n ⫹ 1)

冢

2 J n⫹1 (κD/2) κD/2

冣

冦

冧

cos 2 (mϕ) sin 2 (mϕ) 1

κ ⫺11/3

2

(73)

4[1 ⫺ cos{κd cos(ϕ)}]

where σ 2 is the total variance and is equal to the sum of the x and y variances. The bottom term is also the variance when m is equal to zero, in which cases the top two terms do not apply. There are the requirements that m ⱕ n and n ⫺


82

Sasiela and Shelton

m is even. Common aberrations expressed in terms of (m, n) are (0, 0) for piston, (1, 1) for tilt, (0, 2) for focus, (2, 2) for astigmatism, and (1, 3) for coma. The displacement is considered to be along the x axis and the top term is the x component of the aberration. Therefore, in a more general sense the top term is the phase variance due to a movement parallel to that component of the aberration, and the middle term is due to a perpendicular movement. This expression gives the phase variance for a particular Zernike mode. One can use this result to obtain other information; for instance, if the expression for tilt anisoplanatic phase variance is multiplied by (4k 0 /D) 2 then one obtains the tilt variance, which is the variance of the angular jitter of a signal that arrived with that displacement. The above integral, which has Kolmogorov turbulence, does not converge for piston; therefore the results are not valid for n ⫽ 0. If a turbulence model with finite outer scale is inserted into the integral, the results would be valid for piston. The angular integration can be written as

冮

Ia ⫽ 2

⫽

冮

2π

冦冦

dϕ

0

cos 2 (mϕ) sin 2 (mϕ) 1

冧

[1 ⫺ cos{κ d cos(ϕ)}]

1 ⫹ cos(2mϕ)

2π

dϕ

0

1 ⫺ cos(2mϕ) 2

冧

(74) [1 ⫺ cos{κ d cos(ϕ)}]

The integral in Gradshteyn and Ryzhik Sect 3.715 # 19 can be extended over the full circle to give

冮

2π

0

cos[β cos(x)] cos(2mx)dx ⫽ (⫺1) m 2π J2m (β)

(75)

Use this to obtain σ 2x

冦冧 σ 2y σ2

⫽ 10.42 k 20 (n ⫹ 1)

冮

L

dz C 2n (z)

0

冮

∞

κ ⫺8/3 dκ

0

n, m

⫻

冦

1 ⫺ J0 (κ d) ⫺ (⫺1) m J2m (κ d )

1 ⫺ J0 (κd ) ⫹ (⫺1) J2m (κ d) 2[1 ⫺ J0 (κd )]


m

冧

冤

Jn⫹1 (κD/2) κD/2

冥

2

(76)


83

This can be solved by Mellin transform techniques to obtain for displacements when d ⬍ D σ 2x

冦σ 冧 2 y

⫽ 0.9263 k 20 D 5/3 (n ⫹ 1)

冮

L

dz C 2n (z)

0

冦冤n ⫹ / , / 冥 [ F [n ⫺ / , ⫺n ⫺ / , ⫺ / ; 1, ⫺ / ; ( / ) ] ⫺ 1] ⫺/ (/ ) F [n ⫹ / , ⫺n ⫺ / , / ; / , / ; ( / ) ] ⫺ Γ冤 /, /冥

⫻ ⫺Γ

n ⫺ 5/6 , 7/3 23

7

6

3

17

3

d

10

3

1

⫿ (⫺1) m Γ

5

2

17

6

11

6

4

6

d

3

D

2

6

D

14/3

3

3

2

1

2

2

1

10

2

3

10

d

3

D

2

2

冤2m ⫹ 1, n ⫺ m ⫹

m ⫹ n ⫺ 5/6 , ⫺m ⫹ 7/3 /6 , 17/6 ⫺ m

23

冥( / ) d

2m

D

(77)

⫻ 3 F2 [m ⫹ n ⫺ 5/6 , m ⫺ n ⫺ 17/6 , m ⫺ 11/6 ; 2m ⫹ 1, m ⫺ 4/3 ; ( d/D )2 ] ⫿ (⫺1) m Γ

冤m ⫹

m ⫺ 7/3 , n ⫹ 3/2 /3 , n ⫹ 3/2 , 1/2

10

冥( / )

d D 14/3

冧

⫻ 3 F2 [n ⫹ 3/2 , ⫺n ⫺ 1/2 , 1/2 ; m ⫹ 10/3 , ⫺m ⫹ 10/3 ; ( d/D )2 ] ,

d⬍D

For displacements when d ⬎ D, the result is σ 2x

冦σ 冧 ⫽ 0.9263 k D 2 0

2 y

(n ⫹ 1)

冮

L

dz C 2n (z)

0

n ⫺ 5/6 , 7/3

冦冤n ⫹

⫻ Γ

5/3

23

/6 ,

n ⫺ 5/6 , n ⫹ 3/2

⫺ Γ冤 /冥 ⫺n ⫹

17

6

冥( / ) D

/6 , 2n ⫹ 3, n ⫹ 2

11

d

2n⫺5/3

⫻ 3 F2 [n ⫺ 5/6 , n ⫹ 3/2 , n ⫺ 5/6 ; 2n ⫹ 3, n ⫹ 2; ( D/d )2 ]

⫿ (⫺1) m Γ

n ⫹ m ⫺ 5/6 , n ⫹ 3/2

冤m ⫺ n ⫹

/6 , 2n ⫹ 3, n ⫹ 2

11

冥( / ) D

d

(78)

2n⫺5/3

冧

⫻ 3 F2 [m ⫹ n ⫺ 5/6 , n ⫹ 3/2 , n ⫺ m ⫺ 5/6 ; 2n ⫹ 3, n ⫹ 2; ( D/d )2 ] , d ⬎ D

The total variance and the variance when m ⫽ 0 are given by twice the x or the y variance with the ⫾-terms deleted. Note that the sum of the variances of the parallel and perpendicular components does not depend on m. For very


84

Sasiela and Shelton

large separations the x and y variances are equal, each being one-half the total variance (sum of the x and y components) of that Zernike component. Note that the expression for large separations diverges when n ⫽ 0; the only value of m that is possible for n ⫽ 0 is 0, since m ⱕ n; the (0, 0) mode is the piston. We know that the piston variance is infinite when outer-scale effects are not considered, so this result is expected. Even though this result looks like a formidable expression, it is easily plotted for any Zernike mode by a computer algebra program that evaluates generalized hypergeometric functions. The two-axis variance of a Zernike component for a single wave is given by the first term in Eq. (78) as σ 2m, n ⫽ 0.9263 D 5/3 k 20 (n ⫹ 1) ⫽ 0.6395 µ k D Γ 2 0 0

5/3

冮

L

dz C 2n (z) Γ

0

冤n ⫹

n ⫺ 5/6 , 7/3 23

/6 , 17/6

冥

(79)

冤n ⫹ / 冥 (n ⫹ 1) n ⫺ 5/6 23

6

C. Effect of an Angular Offset between the Waves The tilt-anisoplanatism problem with an infinite outer scale considered in the last section is generalized to allow a finite outer turbulence scale. We will see that it does not have as significant an effect as outer scale did on tilt. One uses Eq. (73) where von Ka´rmańn outer-scale effects are included and m ⫽ 1. To find the tilt variance the equation is multiplied by (4/k 0 D) 2 to give σ 2x

冦冧 σ 2y

⫽ 0.4146

σ2

冢冣冮 16 D

2

dz C 2n (z) ∫ d κជ

0

⫻ (κ 2 ⫹ κ 20 )⫺11/6

冦冧 cos 2 (ϕ)

L

sin 2 (ϕ) 1

冢

J 2 (κD/2) κD/2

冣

(80)

2

[1 ⫺ cos{κ d cos(ϕ)}]

This can be evaluated using Mellin transform techniques in two complex planes. The result is the sum of three double power series. The lowest order terms are σ 2x

冦σ 冧 2 y

⫽

冢冣冦冧冤

2.67 µ 0 d D 1/3 D

2

3

1

1 ⫺ 20.6

冢冣 D L0

2

⫹ 27.4

冢冣 D L0

7/3

冥

⫹ ⋅⋅⋅

(81)

The first term is the same as that in Eq. (77) in which the outer scale was neglected. In Sec. VI where tilt with outer scale was considered, the outer scale



85

had a significant effect on the tilt because the leading term of the series with the ratio of the diameter to the outer scale was raised to the inverse one-third power. Here, the effect is small if the outer-scale size is significantly greater than the diameter because the leading term is the inverse second power of the ratio. The physical reason for this behavior is that for small relative aperture displacements the two beams see the same tilt from long-wavelength turbulence, which cancels when the tilts are subtracted from each other. Mathematically, this subtraction had the result of eliminating one term in the summations. The term that was eliminated has the one-third power-law dependence on outer scale. If the displacement between the two beams is an angle θ, then the expression for Zernike anisoplanatism is obtained by making the substitution d ⫽ θz into Eqs. (77) and (78). The solution for this important case is the sum of a lower and upper contribution expressed as σ 2x

σ 2x

σ 2x

冦σ 冧冦σ 冧冦σ 冧 2 y

⫽

2 y

⫹

2 y

L

(82) U

The lower contribution is found by integrating the expression for small displacements from 0 to H c , where H c ⫽ D cos(ξ)/θ, and ξ is the angle measured from zenith. The upper contribution is obtained by integrating the expression for large displacement from H c to L. The lower contribution is σ 2x

冦σ 冧 2 y

⫽ 0.9263 k 20 D 5/3 (n ⫹ 1) L

冮

Hc

dz C 2n (z)

0

冦冤n ⫹ / , / 冥 [ F [n ⫺ / , ⫺n ⫺ / , ⫺ / ; 1, ⫺ / ; ( / ) ] ⫺ 1] ⫺/ ( / ) F [n ⫹ / , ⫺n ⫺ / , / ; / , / ; ( / ) ] ⫺ Γ冤 /, /冥

⫻ ⫺Γ

n ⫺ 5/6 , 7/3 23

7

10

3

6

3

1

⫿ (⫺1) m Γ

17

3

5

2

17

6

11

6

4

6

θz

3

D

2

6

θz

14/3

D

3

2

3

1

2

2

1

2

10

3

10

3

θz

D

2

2

冤2m ⫹ 1, n ⫺ m ⫹

m ⫹ n ⫺ 5/6 , ⫺m ⫹ 7/3 /6 , 17/6 ⫺ m

23

冥( / ) θz

D

2m

(83)

⫻ 3 F2 [m ⫹ n ⫺ 5/6 , m ⫺ n ⫺ 17/6 , m ⫺ 11/6 ; 2m ⫹ 1, m ⫺ 4/3 ; ( θz/D )2 ] ⫿ (⫺1) m Γ

冤m ⫹

m ⫺ 7/3 , n ⫹ 3/2 /3 , n ⫹ 3/2 , 1/2

10

冥( / ) θz

D

14/3

⫻ 3 F2 [n ⫹ 3/2 , ⫺n ⫺ 1/2 , 1/2 ; m ⫹ 10/3 , ⫺m ⫹ 10/3 ; ( θz/D )2 ]


冧

86

Sasiela and Shelton

The upper contribution is σ 2x

冦σ 冧 2 y

⫽ 0.9263 k 20 D 5/3 (n ⫹ 1) H

冦冤n ⫹

⫻ Γ

n ⫺ 5/6 , 7/3 23

/6 , 17/6

冮

L

dz C 2n (z)

0

冥冤⫺n ⫹ ⫺Γ

n ⫺ 5/6 , n ⫹ 3/2 /6 , 2n ⫹ 3, n ⫹ 2

11

冥( / ) D

θz

2n⫺5/3

⫻ 3 F2 [n ⫺ 5/6 , n ⫹ 3/2 , n ⫺ 5/6 ; 2n ⫹ 3, n ⫹ 2; ( D/θz )2 ] ⫿ (⫺1) m Γ

冤m ⫺ n ⫹

n ⫹ m ⫺ 5/6 , n ⫹ 3/2 /6 , 2n ⫹ 3, n ⫹ 2

11

冥( / ) D

θz

(84)

2n⫺5/3

⫻ 3 F2 [m ⫹ n ⫺ 5/6 , n ⫹ 3/2 , n ⫺ m ⫺ 5/6 ; 2n ⫹ 3, n ⫹ 2; ( D/θz )2 ]

冧

For small displacement angles such that H c is higher than the uppermost turbulence, the anisoplanatic Zernike variance is given solely by the lower contribution. For the parallel component of tilt anisoplanatism the small angle approximation is found by expanding the generalized hypergeometric function in a power series and integrating term by term to obtain

冢

Parallel Tilt Variance x ⫽

θ D

冣

⬇ 0.38 k 20 D 5/3 {1.31733 µ ⫺2 x 2 ⫺ 3.02397 µ ⫺4 x 4 ⫺ ⫹ 2.1965 µ 14/3 x 14/3 ⫹ 0.35035 µ 6⫺ x 6 ⫺ 0.38575 µ ⫺20/3 x 20/3 ⫺ 0.04160 µ ⫺8 x 8 ⫹ x 26/3 ⫺ 0.00096 µ ⫹10 x 10 ⫹ 0.027622 µ 26/3 ⫹ ⫹ 0.0018 µ 32/3 x 32/3 },

(85)

θ HU ⬍ D

H U is the minimum of either of upper height of the turbulence distribution or L. The upper and lower partial turbulence moments, µ m⫹ and µ ⫺m , were defined previously. The perpendicular component is



冢

Perp Tilt Variance x ⫽

θ D

87

冣

⬇ 0.38 k 20 D 5/3 {0.43911 µ 2⫺ x 2 ⫺ 0.60479 µ 4⫺ x 4 ⫺ ⫹ 0.38761 µ 14/3 x 14/3 ⫹ 0.05005 µ ⫺6 x 6 ⫺ ⫺ 0.050315 µ 20/3 x 20/3 ⫺ 0.00462 µ ⫺8 x 8

(86)

⫹ 0.002857 µ ⫹26/3 x 26/3 ⫺ 0.000087 µ ⫹10 x 10 ⫹ ⫹ 0.0001586 µ 32/3 x 32/3 }

θ HU ⬍ D

Notice that for very small angular offsets the variance increases as the square of the angular offset. This is true for all Zernike modes. The upper contribution is found from Eq. (84) to be

冢

Parallel Tilt Variance x ⫽

θ D

冣

⬇ 0.38 k 20 D 5/3 {µ 0⫹ ⫺ 0.53234 µ ⫹⫺1/3 x ⫺1/3 ⫺ 0.024645 µ ⫹⫺7/3 x ⫺7/3 ⫺ 0.000874 µ ⫹⫺13/3 x ⫺13/3 ⫹ x ⫺19/3} ⫺ 0.00015 µ ⫺19/3

(87)

θH U ⬎ D

The perpendicular tilt for large angular separations is

冢

Perp Tilt Variance x ⫽

θ D

冣

⬇ 0.38 k 20 D 5/3 {µ 0⫹ ⫺ 0.79851 µ ⫹⫺1/3 x ⫺1/3 ⫹ 0.018484 µ ⫹⫺7/3 x ⫺7/3 ⫹ 0.000262 µ ⫹⫺13/3 x ⫺13/3 ⫹ 0.000028 µ ⫹⫺19/3 x ⫺19/3} θH U ⬎ D

(88)

Similar series expressions can be written for each of the Zernike modes. For large diameters the first series term for small displacements gives an accurate approximation. For total tilt variance the approximation is Tilt ⬇ 0.667 k 20 D ⫺1/3 µ 2 θ 2

(89)

For total astigmatism the one-term approximation is Astigmatism ⬇ 0.305 k 20 D ⫺1/3 µ 2 θ 2

(90)

For total coma the one-term approximation is Coma ⬇ 0.182 k 20 D ⫺1/3 µ 2 θ 2 The first term for any n is


(91)

88

Sasiela and Shelton

Total Variance ⬇ 1.759 (n ⫹ 1)Γ

冤n ⫹ / 冥 k D n ⫹ 1/6 17

6

2 0

⫺1/3

µ2 θ2

(92)

One is interested in the angular extent over which the correction is useful. There are many ways to define a characteristic angle. First we will define it so that it can be used to gauge when it is useful to apply a correction to an adaptiveoptics system. The angle will be defined as that which would produce the same variance as an aperture with no correction. If the anisoplanatic error is equal to or greater than the value with no correction, then it does no good to correct for that mode. With this definition, the characteristic Zernike anisoplanatic angle, θ nC , is found by setting to unity the ratio of the most significant term of the total Zernike anisoplanatism for small offsets given in Eq. (77) and the Zernike variance given in Eq. (94) to obtain θnC ⫽ 0.603 D

冢

µ0

冣

1/2

µ 2 (n ⫺ 5/6)(n ⫹ 17/6)

(93)

Notice that this is independent of the wavelength; therefore, for shorter wavelengths there are a larger number of pixels that are corrected. For a HV-21 turbulence model for which µ 0 ⫽ 2.23 ⫻ 10⫺12 and µ 2 ⫽ 1.91 ⫻ 10⫺5 this formula reduces to θ nC ⫽

206 D √ (n ⫺ 5/6)(n ⫹ 17/6)

µrad

(94)

For a 1-m-diameter aperture at 0.5-µm operating wavelength, the characteristic tilt anisoplanatic angle is 258 µrad, which is considerably larger than the 21 µrad one gets when the criteria is that the anisoplanatic tilt jitter is one-half a beamwidth. When n ⫽ 2 the angle is 87 µrad, and for n ⫽ 3 the angle is 58 µrad. For large n the angle is 206D/n; the angle gets very small at high Zernike numbers. If the diameter decreases to 0.5 m, then the angles are reduced by a factor of 2. Conversely, a larger diameter increases the angle. The angles above are the largest angles for which correcting that Zernike aberration does any good at improving the image. For the lowest order aberrations a much smaller angle is required if one wants to approach diffraction-limited performance. For a 1-m-diameter aperture operating at a wavelength of 0.5 µm that looks through HV-21 turbulence the variances of a wavefront for the first 14 values of n are 133.7, 6.92, 1.84, 0.73, 0.35, 0.20, 0.12, 0.075, 0.050. 0.035, 0.025, 0.019, 0.014, and 0.011 rad 2. Notice that the tilt variance far exceeds the other variances, which is the reason why tilt must be very well corrected before a correction for other Zernike modes is noticed. The other aberrations can also contribute a significant variance, and since the variance for any Zernike aberration



89

varies as the square of the angular offset, one can easily determine how much the angles given above must be reduced in order to have the variance of a particular aberration drop below a certain level. For a well-corrected system the phase variances for each mode must be below a certain value. One can calculate the error for which the phase variance is under 1 by using the first series term of Eq. (77) to get θ(1 rad 2 ) ⬇ 0.17

λ D 1/6 µ 21/2 √ n ⫹ 1

冢冤 Γ

n ⫹ 17/6 n ⫹ 1/6

冥冣

1/2

(95)

For a 1-m-diameter aperture at 0.5-µm operating wavelength, the angle for n ⫽ 1 is 31.5 µrad; for n ⫽ 2 the angle is 46.6 µrad, and for n ⫽ 3 the angle is 60.3 µrad. For a requirement that the error in any mode be less than 0.1 rad 2, the angles must be reduced by a factor of 3.16. A 0.1-rad 2 requirement results in very small correction angles. These angles increase as the operating wavelength increases. This formula does not apply for all n since, as shown above, the variance never exceeds 0.1 rad 2 when n ⬎ 7. D.

Beam Movement at a Target

Turbulence causes the beam boresight to move at a target. This movement is found here for a source on the ground projected into space, and for a source in space projected to the ground. Tilt causes a beam to change position on a target as shown in Fig. 5. In this problem the filter function is found by considering the beam movement on target for a given tilt realization. It is assumed that scintillations are small, and the beam is not broken up. If there are significant scintillations, these will increase the movement calculated here. It is also assumed that diffraction is unimportant, thereby allowing a ray-

Figure 5 Beam movement at a target board


90

Sasiela and Shelton

optics calculation. With these assumptions, the amount of beam jitter at a target at L is equal to the tilt from turbulence at z times the distance over which it acts, L ⫺ z. The local diameter of the collimated beam must be used. Therefore, the tilt filter function for this problem is F(κជ) ⫽ (L ⫺ z)

冢冣冢 16 k 0 γD

2

2


冣

2

(96)

The variance of beam movement using the formula for one wave in Eq. (15) with Kolmogorov turbulence is X 2 ⫽ 0.2073 k 20

冮

L

0

dz C 2n (z) ∫ d κជ κ⫺11/3 (L ⫺ z) 2

冢冣冢

16 ⫻ k 0 γD

2


冣

2

(97)

This expression can be evaluated for the collimated case, for which case γ ⫽ 1, to give X2 ⫽

6.08 2 [L µ 0 ⫺ 2 Lµ 1 ⫹ µ 2 ] D 1/3

(98)

For distances at which the target is well above the turbulence, the first term in brackets is dominant, giving the physically reasonable result that rms movement is rms tilt times distance. For a 1-m aperture with HV-21 turbulence for which µ 0 ⫽ 2.23 ⫻ 10⫺12 the rms beam jitter is 37 cm at 100 km. If turbulence is constant along the path, one finds X2 ⫽

2.03 L 3 C 2n

(99)

D 1/3

For the beam focused at the target board, for which case γ ⫽ (L ⫺ z)/L, the movement variance is X2 ⫽

6.08 D 1/3

冮

L

dz C 2n (z)

0

(L ⫺ z)5/3 L 5/3

(100)

For constant turbulence along the path this is X2 ⫽

2.28 L 3 C 2n D 1/3

(101)

The movement variance is 112% that of a collimated beam, and the rms movement is 106% that of a collimated beam.



91

For a source in space, the same argument as above gives a filter function F(κជ ) ⫽ z 2

冢冣冢 2

16 k 0 γD


冣

2

(102)

Variance of the beam movement for a collimated beam propagated from space to ground is X2 ⫽

6.08 µ 2

(103)

D 1/3

This movement is equal to 1 cm for the HV-21 model with a 1-m aperture. Thus, it is much easier to hit a target on the ground from space with a laser beam than vice versa.

E.

Scintillation for Collimated and Focused Beams

Expressions that are derived for scintillation are only valid when log-amplitude variance is not saturated, which occurs for values below about 0.5. The variance of log-intensity is four times the variance of log-amplitude. To calculate scintillation of a wave that propagates from 0 to L with inner and outer scales neglected, again use Eq. (15) with the propagation parameter given in Eq. (2), to obtain σ 2χ ⫽ 0.2073 k 20

冮

L

dz C 2n (z) ∫ d κជ κ⫺11/3 sin2

0

冤

γ κ 2 (L ⫺ z) 2k 0

冥

(104)

Using the Mellin transform in Eq. (T5) evaluated at s ⫽ ⫺5/3, one obtains σ 2χ ⫽ 0.5631 k 7/6 0

冮

L

0

dz C 2n (z)(L ⫺ z) 5/6 γ 5/6

(105)

This must be integrated numerically in general. For a receiver well above the turbulence, the approximate value of σ 2χ for a plane wave, for which γ ⫽ 1, is 5/6 σ 2χ ⬇ 0.5631 k 7/6 0 µ0 L

(106)

This varies as sec(ξ). The log-amplitude variance is greater than unity for ranges larger than 2.5 km for a propagation wavelength of 0.5 µm and for a typical turbulence strength of 10⫺14 m⫺2/3. Therefore, scintillation of a beam projected a modest distance along the ground or to space will be saturated.


92

Sasiela and Shelton

For a point source on the ground, for which γ ⫽ z/L, the scintillation observed in space is approximately equal to σ 2χ ⬇ 0.5631 k 07/6 µ 5/6

(107)

11/6

This varies as sec (ξ). If propagation were from L to 0, then L ⫺ z should be replaced by z in Eq. (104). For a plane wave, for which γ ⫽ 1, one obtains σ 2χ ⫽ 0.5631 k 07/6 µ 5/6

(108)

This is the scintillation level seen in space from a point source on the ground. It is also the scintillation observed on the ground from a star and is equal to 0.059 for the HV-21 model at 0.5-µm wavelength and zero zenith angle. Even though this value is small, it is significant. The variance of log-intensity is four times that of the log-amplitude, giving a variance of 0.236. Therefore, the threesigma intensity is twice the mean. Similarly, for a point source in space propagating toward the ground, one obtains σ 2χ ⫽ 0.5631 k 07/6

冮

L

dz C 2n (z)

0

(L ⫺ z) 5/6 z 5/6 L 5/6

(109)

For a point source well above the turbulence, the scintillation is the same as that for a plane wave propagating from space, as would be expected. This result is given in Eq. 108. F. Phase Variance with Finite Servo Bandwidth Greenwood [1977] has derived the phase variance of an adaptive-optics system having finite temporal servo bandwidth with either a one-pole filter or an infinitely sharp filter. All other errors are neglected. Residual rms-phase error from a finite frequency response is calculated here using the same servo filter treated by Greenwood. Consider the following filter:

冷

1 1 ⫹ i(ω/ω3dB ) n

Fservo (ω) ⫽ 1 ⫺

冷

2

(110)

The single-pole filter case corresponds to n ⫽ 1, and the sharp cutoff case corresponds to n ⫽ ∞. Use this filter function in Eq. (26) with diffraction neglected and insert the result into Equation (23). If one considers propagation from L 1 to L 2 , one obtains σ 2φ ⫽ 0.8295 k 20 ⫻

冮

∞

0

冮

L2

dz C 2n (z)

L1

dx x 2n⫺5/3 x 1 ⫹ x 2n

冮

∞

0

v 5/3(z) 5/3 ω 3dB

dc U(c ⫺ 1)c ⫺5/3 c √c2 ⫺ 1


(111)


93

where x⫽

f ω ⫽ ω 3dB f 3dB

(112)

The Mellin transform is used to evaluate both integrals, and the phase variance is σ ⫽ 2 φ

冢冣 fG

f 3dB

5/3

⫽

0.051 k 20 v 5/3

(113)

5/3 f 3dB n sin(5π/6n)

where the velocity moment is defined by vm ⫽

冮

L2

L1

dz C 2n (z) v m (z) ⫽ sec(ξ)

冮

H2

dh C 2n (h) v m (h)

(114)

H1

For propagation between any two points, the integration is over the propagation path, and the velocity is the component transverse to the path. For a single-pole filter (n ⫽ 1) one finds that the characteristic frequency, sometimes referred to as the Greenwood frequency, is f G (n ⫽ 1) ⫽ 0.254 k 06/5 v 3/5 5/3

(115)

For a sharp cutoff filter, the limit of the phase variance as n gets very large is found from L’Hoˆpital’s rule. This gives the same form for the Greenwood frequency; however, the coefficient is smaller. In this case f G (n ⫽ ∞) ⫽ 0.0945 k 06/5 v 3/5 5/3

G.

(116)

Variances for Beams Corrected by Adaptive Optics

If an adaptive-optics system applied the conjugate in both amplitude and phase of the beacon, then an outgoing wave propagating in a direction exactly opposite to the beacon would be perfectly corrected. This result can be derived from the reciprocity relations for wave propagation. Most adaptive-optics systems apply only the conjugate of phase, and the direction of the outgoing beam may be offset from that of the beacon. For this case the residual log-amplitude and phase variances after correction by an adaptive-optics system are given by Eq. (21)


94

Sasiela and Shelton

with F(γ κ) ⫽ 1 as

冤σ 冥 σ 2φc 2 χc

⫻

⫽ 0.2073 k 20

冮

L

dz C 2n (z) ∫ dκជ κ ⫺11/3

0

冷冤冥冤冥冤冥冷冷 sin 冤κ2kL冥 cos 冤κ2kz冥 ⫺ exp[i κជ ⋅ dជ ] sin 冤κ (L2k⫺ z)冥冷

冤

κ2L κ2z κ 2 (L ⫺ z) cos cos ⫺ exp[i κជ ⋅ dជ ] cos 2k 0 2k 0 2k 0 2

2

0

2

2

0

0

2

冥

(117)

For the case in which the offset between the outgoing and beacon beams, dជ, is zero, the variance is

冤σ 冥 σ

2 φc

2 χc

⫽ 0.2073 k 20

冮

L

dz C 2n (z) ∫ d κជ κ ⫺11/3

0

冤κ2kz冥 sin 冤κ2kL冥 κ z κ L sin 冤 cos 冤 2k 冥 2k 冥

冤

2

2

sin 2

2

0

0

2

2

2

2

0

0

冥

(118)

A simple relation is obtained from adding the phase and log-amplitude variances: σ 2φc ⫹ σ 2χc ⫽ 0.2073 k 20

冮

L

dz C 2n (z) ∫ d κជ κ ⫺11/3 sin 2

0

冤冥 κ2z 2k 0

⫽ 0.5631 k 7/6 0 µ 5/6

(119)

This is the same value one obtains for scintillation of a point source propagating towards the ground, which was analyzed in Sec. VI.E. Therefore, total variance is equal to the variance of the scintillation on the beacon. This expression includes the effect of diffraction and is valid for distributed turbulence. The corrected beam in a phase-only adaptive-optics system can be distorted by several causes. Consider the distortion in a system that has an anisoplanatic error produced by offsetting the outgoing beam by θ from the beacon. The filter function for the total beam distortion can be found from Eq. (117) as Ft (κជ) ⫽ sin 2

冤κ2kz冥 ⫹ 2 cos 冤κ2kz冥(1 ⫺ cos[κជ ⋅ dជ ]) 2

2

2

0

(120)

0

The first term is the same one that was encountered above with correctly aligned beams, and the variance it causes is given in Eq. (107). It will be assumed that diffraction is negligible, enabling the first cosine to be replaced by unity.



95

The integral can be evaluated by using the Mellin transform in Eq. (T6) and the definition of isoplanatic angle in Eq. (46) to obtain

冢冣

θ σ ⫽σ ⫹ θ0 2 c0

2 χ

5/3

(121)

The angle-dependent variance is equal to the variance of angular anisoplanatism.

H.

Power Spectral Density of Tilt

In this section, the power spectral density for tilt of a collimated wave is derived. Plane wave results have been discussed briefly in Tatarski [1971] and presented in more detail in Greenwood and Fried [1976]. Greenwood and Fried made a simplifying assumption in order to obtain simple, analytic results. Fields [1983] coined the term ‘‘parallel approximation’’ for this simplification. One consequence of the parallel approximation is that the rate at which spectra decay at high frequencies is underestimated. Tyler [1986] subsequently analyzed plane wave tilt spectra without making this approximation. However, his results remain in integral form, containing an integral over a dummy variable related to spatial frequency and an integral over altitude. In a subsequent report, Vaughn [1986] provided numerical techniques to solve the integrals presented by Tyler, but that solution relies heavily on numerical integration. A similar approach has been used in tilt spectra associated with a point source (spherical wave analysis). This leads to integral expressions that also must be evaluated numerically [Hogge and Butts, 1976, Butts, 1980]. Power spectral density of tilt is found here using the general expression for power spectral density given in Eq. (26), with the filter function for Zernike tilt given in Eq. (33). It is assumed that one is in the near field so that the cosine term can be replaced by unity. This assumption breaks down at sufficiently high frequencies, where the exact equation must be used. The effect of including the cosine term is to lower the high-frequency spectrum. The spectral density is S T (ω) ⫽ 2.606 k ω 2 0

冮

0

冢冣冤

16 ⫻ k0 D

2

L

dz

C 2n (z) v 2 (z)

冮

∞

dc

cU(c ⫺ 1) √c2 ⫺ 1

0

J 2 (ωcD/2v(z)) ωcD/2v(z)


冥

2

⫺11/3

冢冣 ωc v(z)

(122)

96

Sasiela and Shelton

The result of evaluating this expression is a series solution for the power spectral density:

冢冣

2 S T (ω) ⫽ 52.53 ω

∞

2/3

冱 n⫽0

(⫺1) n n!

冦冤n ⫹ 3, n ⫹ 5, ⫺n ⫹ / 冥冢冣 v n ⫹ 5/2 , ⫺n ⫹ 1/3

⫻ Γ

5

⫹Γ

冤n ⫹

ωD 2

6

n ⫹ 17/6 , ⫺n ⫺ 1/3 /3 , n ⫹ 16/3 , ⫺n ⫹ 1/2

10

2n

(123)

⫺2n⫺1/3

冥冢冣 ωD 2

2n⫹2/3

冧

v⫺2n⫺1

For the slew-dominated case in which the velocity is proportional to axial position, the velocity moments in the above expression are infinite. The low-frequency asymptote comes from the first term of the first series, and the dependence is ω⫺2/3. For large frequencies, an asymptotic series can be found; thus, the power spectral density at high frequencies is S T (ω) ⬃ 52.53 D ⫻

冦冮

冢冣

1/2

2v(z) √ π ωD

0

n⫽0

冢冣

1

dz C 2n (z)

冱

17/3

2 ωD

L

n0

⫹

2/3

v 14/3 (z) cos

ωD 冤冢v(z) 冣 ⫺ π4冥

冤

n ⫹ 5/2 , n ⫹ 17/6 (⫺1) n Γ n ⫹ 10/3 , ⫺n ⫹ 1/2 , ⫺n ⫹ 5/2 n!

冥冢冣 2 ωD

(124)

2n

v2n⫹14/3

冧

The high-frequency asymptote comes from the first series term, and it varies as ω⫺17/3. If velocity is constant along the path, spectral density for small values of ωD/2v is S T (ω) ⫽ 3.453 ⫻

µ 0 D 2/3 v

冦F冤 2

3

冢冣 2v ωD

2/3

冢冣冥

ωD /2 , /6 ; 3, 5, ⫺ /3 ; ⫺ 2v

5

1

2

冢冣冤

ωD ⫺ 0.5394 2v

2/3

2

F3


2

(125)

冢冣冥冧

ωD /6 , /2 ; /3 , /3 , /3 ; ⫺ 2v

17

1

10

16

4

2


97

The spectral density for large values of ωD/2v is an asymptotic series given by S T (ω) ⫽ 52.53 n0

⫹

冱 n⫽0

fD v

µ 0 D 2/3 v

冢冣冦冢冣冤冢冣冥 2v ωD

17/3

1

2v ωD √π

冤

1/2

cos

ωD π ⫺ v 4

n ⫹ 5/2 , n ⫹ 17/6 (⫺1) n Γ n ⫹ 10/3 , ⫺n ⫹ 1/2 , ⫺n ⫹ 5/2 n!

⬎1

冢

f⫽

ω 2π

冥冢冣冧 2v ωD

2n

(126)

冣

Typically, the number of terms of the asymptotic series, n 0 , necessary to get a good approximation is less than 6. Power spectral density is plotted in Fig. 6. The spectrum is normalized to give a value of unity when the fD/v is unity. In the figure both Taylor series (first 40 terms) and asymptotic solution (first 5 terms) are plotted. For fD/v ⬍ 1, the asymptotic series was set equal to unity since the series is very inaccurate

Figure 6 Log-linear plot of the power spectral density of tilt. The velocity and turbulence are constant along the path. Notice the agreement of asymptotic and Taylor series over a considerable range. This curve is independent of the turbulence distribution along the propagation path.


98

Sasiela and Shelton

in that region, and above 5 the Taylor series was discarded because it becomes inaccurate with the 40-term approximation. The two series give virtually the same answer in the parameter range between 1 and 5. Notice the strong influence of the cosine term on the spectrum. If velocity is allowed to vary along the path, various cosine terms for different velocities tend to average to zero, and the spectrum does not exhibit nearly as strong an oscillatory behavior. I. Scintillation from a Finite Size Receive Aperture and a Finite Size Source Scintillation in a receiver is considerably reduced from that of a point source receiver if the aperture is a few inches in diameter. Also, the scintillation from a finite size object, such as a planet, is considerably lower than that from a point source, such as a star. Both of these averaging effects can be calculated using the filter function approach. In these problems one is interested in the average scintillation over an aperture or at a point receiver from a finite source. One has to average the intensity, but the statistical expressions are given in terms of the log-intensity. The Rytov approximation, upon which this work is based, is valid for low levels of scintillation, and in that approximation the normalized intensity is well approximated by the log-intensity. Therefore, the average of the log-intensity approximates the average of the intensity. The scintillation from stars is low, and the approximation holds for that practical problem. The exact solution was found by Fried [1967], using a numerical integration. The scintillation averaged over an aperture, using the filter function in Eq. (39) for a source at infinity in the formula for variance given in Eq. (15), is σ 2χA ⬇ 0.2073 k 20 ⫻

冤

冮

L

dz C 2n (z) ∫ d κជ f(κ)sin 2 [P(γ, κ, z)]

0

2 J 1 (γκD/2) γκD/2

冥

2

(127)

For propagation toward the receiver, the diffraction parameter is P(γ, κ, z) ⫽ γκ 2 z/2k 0 . For most situations in which diameter is larger than the source, the parameter in the previous equation is very large, and the answer can be approximated by the first term of the asymptotic series, which is σ 2χA ⬇

4.34 D 7/3

冮

L

0

dz C 2n (z)

z2 γ 1/3

(128)

For propagation from space to ground γ ⫽ (L ⫺ z)/L. If the source is well above the atmosphere, then this is close to unity. In this case the scintillation is



99

equal to σ 2χA ⬇

4.34 µ 2

(129)

D 7/3

The aperture-averaged scintillation does not depend on the wavelength. That is the reason that no scintillation-induced color is evident in a star image as seen in a telescope. The aperture-averaged scintillation depends on the second moment of turbulence, which is close to the five-thirds moment that is needed to calculate the isoplanatic angle. By modifying the aperture function using masks in their isoplanometer, Walters et al. [1979] were able to have the aperture-averaged scintillation closely approximated by the five-thirds moment of turbulence. In this manner the instrument is used to estimate isoplanatic angle. If turbulence is constant along the path, for plane wave propagation the variance is σ 2χA ⬇

1.45 C 2n L 3

(130)

D 7/3

Scintillation with no averaging is given in Eq. (107) as σ 2χ ⫽ 0.5631 k 7/6 0 µ 5/6

(131)

The scintillation reduction ratio is σ 2χA σ 2χ

⬇ 7.7 ⫽

µ2 µ 5/6 D 7/3 k 7/6 0

冢冣 Dc

⫽ 0.902

µ2 µ 5/6

冢冣 λ D2

7/3

D

D ⬎⬎ D c

,

7/6

(132)

where D c , the characteristic diameter for scintillation averaging, is given by D c ⫽ 0.957

冢冣 µ2

3/7

µ 5/6

√λ

(133)

For the HV-21 model, the values of the turbulence moments are µ 2 ⫽ 1.91 ⫻ 10⫺5 and µ 5/6 ⫽ 5.45 ⫻ 10⫺10. At a wavelength of 0.5 µm the characteristic diameter is 6 cm. If turbulence is constant along the path, Eq. (132) becomes σ 2χA σ 2χ

⬇ 0.109

冢冣 √ λL D/2

7/3

,

D ⬎⬎ √ λL


(134)

100

Sasiela and Shelton

This formula agrees with the power-law dependence given in Tatarski [1971]. For this case the characteristic diameter is D c ⫽ 0.774 √ λL

(135)

Next, the case of a source of finite angular extent viewed by a point receiver is considered. In this case the filter function for a finite size source of diameter D s given in Eq. (39) is rewritten in terms of the angle subtended by the source, θ s , where θs ⫽

Ds

(136)

L

The filter function is thus F(γ κជ) ⫽

冤

2 J 1 (γκθ s z/2) γκθ s z/2

冥

2

(137)

The analysis proceeds in the same manner as above. Once again the first term of the asymptotic series is the most significant for larger angles, and it is σ 2χA ⬇

4.34 θ s7/3

冮

L

dz C 2n (z)(γz)⫺1/3

(138)

0

The negative moment of turbulence is finite. The leading term for scintillation from a finite source, just as for aperture averaging, does not depend on the wavelength. For propagation from space to the ground γ ⫽ (L ⫺ z)/L. If the source is well above the atmosphere, then this is close to unity. For large source sizes the scintillation reduction ratio is σ 2χA σ 2χ

⬇ 7.7

µ ⫺1/3 7/6 µ 5/6 θ 7/3 s k0

⫽ 0.902

冢冣冢冣

µ ⫺1/3 λ µ 5/6 θ 2s

7/6

⫽

θc

θs

7/3

, θ s ⬎⬎ θ c

(139)

where θ c , the characteristic angle of the source for scintillation reduction, is θ c ⫽ 0.957

冢冣 µ ⫺1/3 µ 5/6

3/7

√λ

(140)

For the HV-21 model µ ⫺1/3 ⫽ 5.50 ⫻ 10⫺13, and the characteristic angle is equal to 35.2 µrad at a wavelength of 0.5 µm. If turbulence is constant along the path,



101

the reduction factor is σ 2χA σ

2 χ

⬇ 2.48

冢冣 λ Lθ 2s

7/6

,

θ s ⬎⬎

√

λ L

(141)

The characteristic angle in this case is θ c ⫽ 1.48

√

λ L

(142)

J. Focal Anisoplanatism for Point Sources Analytic expressions can be obtained for the phase variance with point, distributed, and offset beacons. The analysis approach is to use a general formula for variance, which applies to a wide variety of problems, and insert filter functions appropriate for each situation. The resulting integrals can be evaluated analytically using Mellin transform techniques. The general formula for a function of the phase variance between two waves that can have different propagation characteristics was given in Sec. II. Different problems are solved by using the same general formula and inserting different filter functions of the transverse turbulence spectrum. Filter functions appropriate to the synthetic beacon problem are derived. This formula is applied to find analytical expressions for the phase variance with a single point source beacon in this section, for a distributed beacon in Sec. VI.K and for an offset beacon in Sec. VI.L. In Sec. VII, systems using more than one synthetic beacon are analyzed. For large apertures, a large phase residual remains in using a single synthetic beacon in the atmosphere. This error can be reduced by using multiple beacons. Because wavefront sensors measure phase gradient rather than tilt, the measurement of the wavefront from each beacon has a tilt error. The measurements from the various beacons must be ‘‘stitched’’ together. The analysis is complicated because tilt errors are correlated. Tilt errors and their correlation are calculated and inserted into a least squares reconstructor to obtain estimates of the errors associated with multibeacons systems. The net results of this analysis are simple formulas to obtain the figure variance for systems with one or more beacons. Phase variances are obtained for a 60-cm system that was fielded in Maui [Primmerman et al., 1991; Murphy et al., 1991] and for a larger 4-m system. Focal anisoplanatism refers to errors made in using a phase disturbance measured by a focused beam to correct for turbulence effects on a collimated beam or one focused at a different altitude. Some authors refer to this as focus anisoplanatism. If a point source is very far away, the error is very small. For closer sources, such as those using a synthetic beacon in the ‘‘guidestar’’ concept,


102

Sasiela and Shelton

the effect can be significant and may result in a low Strehl ratio. For these systems, the maximum beacon altitude proposed is at the height of the sodium layer at 90 km, which can provide significant scattering of yellow light [Humphreys et al., 1991]. Fugate et al. [1991] used another type of beacon. The phase variance from focal anisoplanatism is calculated in this section. First, an expression for variance from turbulence below the beacon is obtained that has both piston and tilt components present. Next, expressions for piston and tilt variances are obtained. Subtracting the last variances from the total, gives piston- and tilt-removed phase variances due to focal anisoplanatism from turbulence below the beacon. The phase variances from turbulence above the beacon are added to these variances to get the total variances. The technique to find the variance is the same one that has been used in previous problems. The filter function for a distributed, circular, uniform beacon of diameter D s and offset dជ that is at a range L and altitude H is inserted into the formula for phase variance of a single wave given in Eq. (15). Assume one is operating in the near field, allowing the cosine term to be replaced by 1. Doing the angle integration, one obtains for the aperture-averaged phase variance σ 2⫺d ⫽ 1.303 k 20

冦

⫻ 1⫺

冮

L

dz C 2n (z)

0

冮

∞

dκ κ ⫺8/3

0

2 J1 (κ D s z/2L) 2 J1 (κDz/2L) κ D s z/2L

冤

⫹ 2

κ Dz/2L

J1 (κD s z/2L) κ D s z/2L

J0

冢冣 κbz L

(143)

冥冧 2

The subscript d on the variance refers to a distributed source. The negative subscript means that it is the component from turbulence below the beacon. The component above the beacon will be added to this to obtain the total variance. If the d subscript is missing, it means a point source is being considered. Piston and tilt are present in σ 2⫺d . This equation is the starting point for the evaluation of the total phase variance for all cases considered in this and the next two sections. The problem is solved in steps since the results for a point source, a distributed source, and a displaced source are of interest for different situations. The phase variance of a point source on the aperture center with piston and tilt present can be found from Eq. (143) by setting the source diameter and displacement equal to zero to obtain σ 2⫺ ⫽ 2.606 k 20

冮

L

0

dz C 2n (z)

冮

∞

0

冤

dκ κ ⫺8/3 1 ⫺


2 J1 (κ Dz/2L) κDz/2L

冥

(144)


103

The integrals can be evaluated, and the phase variance is equal to σ 2⫺ ⫽ 0.5 k 20 µ ⫺5/3

冢冣冢 D L

5/3

⫽

0.348D Lθ 0⫺

冣

5/3

(145)

The partial isoplanatic angle is defined as (θ ⫺0 )⫺5/3 ⫽ 2.91 k 20 µ ⫺5/3

(146)

The phase variance results from an angular offset of the collimated and focused rays. The phase variance varies with radius; however, one can consider the average phase variance to be due to an angular offset equal to that of the ray that emanates from the point that is about 0.7 of the radius from the center. This angle is 0.348D/L. This phase variance with piston included is finite, unlike the result for unfiltered turbulence, which is infinite. The infinite result comes from the zero spatial wavelength term. This infinity cancels out in the subtraction of the collimated beam phase from that of a focused beam. Even though the above result is finite, it might be possible that the major component of this variance is due to the piston, which is no practical interest. It can be shown that at satellite altitudes the piston contribution is less than 10% of the variance. At 10 km it can be as large as 20%. Therefore, the above simple expression is a reasonable zero-order approximation to phase variance. A 60-cm aperture looking at a point source at 300 km would have the same phase variance as two beams with angular offset of 0.7 µrad. For typical isoplanatic angles, this will produce a very small variance. For a 4-m system with a beacon at 90 km the corresponding offset is 15 µrad, which is typically larger than the isoplanatic angle. Often one is interested in phase variance with piston or piston and tilt removed. The phase variance due to piston and tilt can be calculated separately and subtracted from the total phase variance to obtain these variances. Piston and tilt phase variances are found by using Eq. (19) with filter functions given in Eq. (31). σ 2⫺Z ⫽ 0.2073 k 20 ⫻

冤

冮

L

dz C 2n (z) 4ν 2 ∫ d κជ κ ⫺11/3

0

J ν (κD/2) κD/2

冤

冥

z Jν (κ D(1 ⫺ z/L)/2) 2 J1 (κD s z/2L) ⫺ exp i κជ ⋅ bជ L κ D(1 ⫺ z/L)/2 κ D s z/2L

冥

(147) 2

where D s is the source diameter, and bជ is displacement of the source from boresight. The value of ν is 1 for piston and 2 for tilt. First consider the case of zero source size and no offset. The piston phase variance due to turbulence below the beacon is


104

Sasiela and Shelton

冮

σ 2⫺P ⫽ 0.5 k 20 D 5/3

L

dz C 2n (z)

0

冦 F [⫺ / , ⫺ / ; 2; (1 ⫺ / ) ] ⫺ 1 ⫹ √(1π ⫺2 z/L) Γ 冤 / 冥冧 5/3

⫻

2

1

5

11

6

z

6

L

2

⫺8/3

(148)

7

/3

23

6

This can be integrated numerically; however, to express this as turbulence moments, the expression in braces is expanded in a Taylor series about the point. Some care must be taken in expanding this series to obtain a short series that is a good approximation to the exact calculation. It has been found that a good approximation to the exact result for typical turbulence models and aperture diameters is σ 2⫺P ⬇ 0.0833 k 20 D 5/3

µ ⫺2 (H )

(149)

L2

The integrations can be performed for tilt to obtain σ 2⫺T ⫽ 0.8345 k 20 D 5/3 ⫻

冦

冮

L

dzC 2n (z)

0

1 ⫹ (1 ⫺ z/L) 5/3 √ π 2⫺11/3

冤 /冥 7

Γ

/3

29

(150)

6

冧

⫺ (1 ⫺ z/L ) 2 F1 [1/6 , ⫺11/6 ; 3; (1 ⫺ z/L ) 2 ]

After expanding this in a Taylor series in the same way as piston, the tilt variance is closely approximated by σ 2⫺T ⬇ 0.368 k 20 D 5/3

µ ⫺2 (H ) L2

(151)

The piston-removed variance is σ

2 ⫺PR

冢

0.348D ⫽ Hθ 0⫺

冣

5/3

⫺ σ 2⫺P

(152)

The piston- and tilt-removed variance is σ 2⫺PTR ⫽

冢

0.348D Hθ ⫺0

冣

5/3

⫺ σ 2⫺P ⫺ σ 2⫺T

(153)

With the above results, a good approximation to the piston-removed phase is



σ 2⫺PR ⫽

冢

0.348D Hθ ⫺0

冣

5/3

⫺ σ 2⫺P ⬇

105

冤

k 20 D 5/3 µ ⫺5/3 (H) 2

L 5/3

⫺ 0.167

µ ⫺2 (H ) L2

冥

(154)

Similarly, a good approximation to the piston- and tilt-removed phase is σ 2⫺PTR ⫽ ⬇

冢

0.348D Hθ ⫺0

冣

冤

5/3

⫺ σ 2⫺P ⫺ σ 2⫺T

k 20 D 5/3 µ ⫺5/3 (H ) 2

L 5/3

⫺ 0.903

µ ⫺2 (H ) L2

冥

(155)

For changes in zenith angle, when the beacon is kept at the same altitude, it is clear from the variation of each term in brackets that the zenith dependence is simply sec(ξ). The effect of turbulence above the beacon is obtained by modifying the standard results for piston or piston- and tilt-removed phase variance. The phase variance with piston removed is σ

2 ⫹PR

冢冣

D ⫽ 1.033 ⫹ r0

5/3

(156)

where the coherence diameter looking up from the beacon altitude is (r ⫹0 )⫺5/3 ⫽ 0.423 k 20 µ 0⫹

(157)

The expression for phase variance from turbulence above the beacon with piston and tilt removed is σ 2⫹PTR ⫽ 0.134

冢冣 D r ⫹0

5/3

(158)

The contribution due to the unsensed turbulence increases with increasing aperture size, and one will find that the beacon altitude must be increased for larger diameters in order to keep this distortion manageable. The total phase variance with piston removed is σ 2PR ⫽ σ 2⫺PR ⫹ σ 2⫹PR

(159)

The total phase variance with piston and tilt removed is σ 2PTR ⫽ σ 2⫺PTR ⫹ σ 2⫹PTR

(160)

In Fig. 7 the total, piston-removed, and piston- and tilt-removed phase variances due to turbulence below the beacon are plotted for a single beacon operating with a 60-cm-diameter aperture with the HV-21 turbulence model. In Fig. 8 the


106

Sasiela and Shelton

piston-removed, and piston- and tilt-removed phase variances of the entire atmosphere are plotted for the same conditions.

K. Focal Anisoplanatism for Distributed Sources The artificial beacon source is adequately approximated by a point source for very distant beacons. This, however, is a poor approximation for some closer beacons. In trying to use resonant sodium backscatter from the sodium layer at about 90 km, one wants to project as large a beacon as possible that does not significantly increase the phase variance over that of a point source. A large beacon is desirable because the return from a given area saturates at a certain power. To increase the return signal not only must the beacon power be increased but the focused spot size must be increased. The expression for the phase variance for a distributed source centered on the aperture with piston and tilt present can be found from Eq. (143) by setting bជ equal to zero to obtain, for D s /D ⬍ 1, σ 2⫺d ⫽ 1.303 k 20

冮

L

dz C 2n (z)

0

冦

⫻ 1⫺8

冮

∞

dκ κ ⫺8/3

0

J1 (κ D s z/2L) J1 (κDz/2L) κ D s z/2L

κDz/2L

冤

⫹ 2

J1 (κ D s z/2L) κ D s z/2L

冥冧 2

(161)

Use Mellin transform techniques to evaluate the integral and obtain σ

2 ⫺d

冢

0.348D ⫽ Lθ 0⫺ ⫻

冣

5/3

(162)

冤 F 冤⫺ / , ⫺ / ; 2; 冢 D 冣冥 ⫺ 0.872 冢 D 冣冥, 2

1

5

11

6

Ds

2

Ds

5/3

Ds

6

D

⬍1

The first few terms of this solution are σ 2⫺d ⬇

冢

0.348D Lθ ⫺0

冣冤 5/3

冢冣冢冣 Ds

1 ⫺ 0.872

⫺ 0.0177

D

Ds D

4

5/3

⫹ 0.764

冢冣冢冣冥

⫺ 0.000287

Ds

2

D

Ds

6

(163)

D

The first three terms in brackets give a good approximation to the phase variance when D s /D ⬍ 1.



107

Figure 7 Phase variance of turbulence below the beacon. The total, piston-removed, and tilt- and piston-removed phase variances are plotted.

Figure 8 Piston-removed phase variance and piston- and tilt-removed phase variance caused by turbulence of the entire atmosphere.


108

Sasiela and Shelton

When D s /D ⬎ 1, the variance is σ

2 ⫺d

⫽

冢

0.348D s Hθ

⫺ 0

冣冦 5/3

冤

冢冣冥冧

D ⫺0.128 ⫹ 2 F1 ⫺ /6 , ⫺ /6 ; 2; Ds 5

11

2

,

Ds D

⬎1

(164)

The first few terms of this solution are σ

2 ⫺d

⬇

冢

0.348D s Lθ 0⫺

冣冤 5/3

冢冣冢冣

D 0.872 ⫹ 0.764 Ds

D ⫺ 0.0177 Ds

4

2

冢冣冥

D ⫺ 0.000287 Ds

6

(165)

The normalized variance is plotted in Fig. 9. Initially, the phase variance caused by the turbulence below the beacon decreases as source size increases, then it increases. L. Focal Anisoplanatism for Offset Sources The beacon may not be placed in the exact location that the correction is wanted, resulting in a decreased Strehl ratio. This misplacement can be caused by errors in positioning the beacon and in knowing which direction to point the beacon. If a laser beam is to be projected in the point-ahead direction, which is typically

Figure 9 Phase variance below the beacon. Effect of beacon diameter on the phase variance due to focal anisoplanatism normalized to the phase variance due to focal anisoplanatism from a point source with piston and tilt present.



109

about 50 µrad, there is an error in pointing the mirrors due to tilt anisoplanatism. This error can be as much as a microradian. The phase variance with a point source offset from the origin is obtained by setting the source diameter equal to zero in Eq. (161). After performing the angular integration one obtains σ 2⫺0 ⫽ 1.303 k 20

冮

冤

L

dz C 2n (z)

0

⫻2 1⫺

冮

∞

dκ κ ⫺11/3

0

2 J1 (κDz/2L) κ Dz/2L

J0

(166)

冢冣冥 κbz L

For this case, phase variance from turbulence below the beacon is σ

2 ⫺0

⫽ 0.5 k µ 2 0

⫺ 5/3

冢冣冤 D L

5/3

冢冣冥

2b 2 F1 ⫺ /6 , ⫺ /6 ; 1; D 5

11

2

2b ⬍1 D

,

(167)

The variance normalized to that of an on-axis source is σ 2⫺0 σ 2⫺0 (b ⫽ 0)

冤

冢冣冥冢冣冢冣 2b D

⫽ 2 F1 ⫺5/6 , ⫺11/6 ; 1; 2

b ⫽ 1 ⫹ 6.12 D

2

b ⫺ 0.848 D

(168)

4

2b ⬍1 D

⫹ ⋅ ⋅ ⋅,

When the displacement is zero, the hypergeometric function is equal to unity, and one obtains the previous result: σ (b ⫽ 0) ⫽ 0.5 k µ 2 ⫺0

2 0

⫺ 5/3

冢冣 D L

5/3

(169)

When the displacement is larger than the radius, the variance normalized to that with no offset is σ 2⫺0 σ 2⫺0 (b ⫽ 0)

⫽ 1.836

冢冣冤 2b D

5/3

5 5 2 F1 ⫺ /6 , ⫺ /6 ; 2;

冢冣冥 D 2b

2

,

2b ⬎1 D

(170)

This is plotted in Fig. 10. For larger source displacements than those plotted, one can use the approximation σ 2⫺0

冢冣

2b ⬇ 1.836 2 σ ⫺0 (b ⫽ 0) D

5/3

,

2b ⬎3 D


(171)

110

Sasiela and Shelton

Figure 10 Total phase variance below the beacon for focal anisoplanatism with a displaced point source. The variance is normalized to that of a source on axis. Piston and tilt are present. It is plotted versus twice the displacement divided by the aperture diameter.

The phase variances in this and the last sections can be used to determine the importance of beacon size and placement; however, the piston and tilt terms are included. The evaluation of the integrals to determine the phase variance with these components removed is complicated and lengthy and is not given here.

VII. ANALYSIS OF A MULTIBEACON SYSTEM If an astronomical source is very bright, one can use part of the light coming from the object in a wavefront sensor to measure the phase gradient due to turbulence. The wavefront can be reconstructed, and the distortions can be corrected with a deformable mirror. Unfortunately, the sky is not populated with a sufficient number of bright objects to be able to correct much of the visible sky. To overcome this limitation, one can use synthetic beacons produced by projected laser beams. They can be located at any angular position in the sky and can, therefore, provide a means to get a corrected image of dim objects. The errors made in correcting the turbulence with one or more synthetic beacons are analyzed. The approach differs from previously published results [Thompson and Gardner, 1987, 1989; Foy and Tallon, 1989; Gardner et al., 1989,



111

1990; Welsh and Gardner, 1989a, b; Welsh et al., 1989; Welsh and Thompson, 1991] in that analytic expressions are obtained for the phase variances. The results are expressed as a function of aperture diameter, turbulence moments, zenith angle, operating wavelength, and beacon height. Results for a different system configuration are obtained by simply changing the parameters in analytic expressions. Much of the analysis presented here is taken from Sasiela [1994b]. Parenti and Sasiela [1994], Gavel et al. [1994], Esposito et al. [1996], and Stroud [1996] contain more recent analysis using some of the ideas presented here. The single beacon results can be extended to a multibeacon system. With several beacons, each beacon may be used to correct the phase over a portion of the aperture, thus reducing the isoplanatic error. However, since the wavefront sensor measures phase gradient, not actual phase, the phase surface reconstructed from the phase gradients under each beacon must be ‘‘stitched’’ with the other phases. Errors are introduced into this process because of the incomplete knowledge of the overall tilt in each section of the aperture. This error comes from two sources; the turbulence-induced tilt is incorrectly measured in each section due to focal anisoplanatism, and the measurement of the relative position of the beacons has an error due to anisoplanatic effects. The magnitude of each error is calculated and a least-squares estimate of the error in putting together the phases of the individual sections is calculated. The multibeacon analysis is performed for point source beacons. The model used to calculate the phase variance assumes that it is composed of two major parts as shown in Fig. 11. The first part is the focal anisoplanatic error which, of itself, has the phase variance due to the unsensed turbulence above the beacons, plus the error made in sensing the turbulence below the beacons. If there were only one beacon, the sum of these two errors with piston and tilt removed would give the total phase variance with piston and tilt removed. For the case of more than one beacon, there is a tilt error made in measuring the turbulence below the beacons that can be different for each section. These tilt errors combine in the stitching operation of putting together the phases of each

Figure 11 Contributions to the phase variance of a multibeacon system include focal anisoplanatism below the beacon, unmeasured turbulence above the beacon, and errors in determining the relative positions of the beacons.


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section to increase the error. The tilt error is composed of two parts: the first is due to the tilt difference between the collimated beam and the focused beacon beam. The second is due to the misplacement of the beacons above the center of each section. It will be assumed that the relative beacon positions are measured with the full aperture of the receiver. Because the ray paths through the atmosphere are different for the various beacons, there will be a measurement error associated with the evaluation of the relative beacon positions. Both tilt jitters are stitched together to obtain the phase profile across the entire aperture. Each component mentioned above is evaluated separately using Mellin transform techniques combined with the use of the appropriate filter functions. Analytic expressions were found for the components of the focal anisoplanatism with piston and tilt removed above and below the beacon altitude in Sec. VI. The tilt components below the beacon, also evaluated in Sec. VI, are used in the stitching analysis. It is assumed in this analysis that the sections are circular in order to get analytic expressions for the results. In actuality, the butted sections are square on some sides. The error resulting from this assumption should be small. Next, the error made in measuring the beacon position will be shown to be equivalent to finding the tilt anisoplanatism of focused beams. An analytic expression for this tilt error will be found, and it is shown that it is considerably less than tilt error due to focal anisoplanatism. To stitch the individual sections together, a least-squares estimation procedure is used. This approach is similar to that of stitching together tilt errors measured in a wavefront sensor in an adaptive-optics system to obtain the phase profile [Herrmann, 1980]. The difference in this case is that the tilts can be correlated; this modifies the formula for the error. Expressions for the correlation coefficients of the focal anisoplanatic tilts are derived. From these correlation coefficients, the error propagator for stitching of the tilts is found. The error propagator is the multiplier of tilt phase variance whose product gives the phase variance of the stitching process. Finally, the phase variances due to focal anisoplanatism and stitching are added to get the total phase variance. This computation is performed for various altitudes. The results are in good agreement with the results obtained by ray tracing. A. Beacon Position Measurement Error If more than one beacon is used, the tilt of individual sections must be measured. Errors in the measurement are important because the tilt component of turbulence is 87% of the turbulence-induced phase variance with piston excluded. There are



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two errors in measuring tilt. The first arises from the difference in ray paths between a collimated and focused beam. This was calculated in Sec. VI. The other is due to the beacons not being directly over the center of the sections. Here, it will be assumed that the beacons are projected up into the sky by some system, and their relative positions are measured by the full aperture of the system. Because of the difference in paths between the measurement rays through the turbulence, there will be a measurement error in the apparent relative position of the beacons. The variance of tilt difference between focused displaced rays whose focus points are separated in space by bជ is found by using Eq. (20) with diffraction neglected and with the filter function for tilt of a ray focused at z ⫽ L given by

冮

T 2Z ⫽ 0.2073 k 20 ⫻

冢

L

冢冣冣冤冦冧冥 16 k0 D

dz C 2n (z) ∫ dκជ κ ⫺11/3

0

J2 (κ D[1 ⫺ z/L]/2) κD[1 ⫺ z/L]/2

2

2

(172)

bជz 2 1 ⫺ cos κជ ⋅ L

The tilt variance is evaluated using Mellin transform techniques to give T 2Z ⬇

10.68 D 1/3 ⫻

冮

zt

dz C 2n (z)

0

冤冢冣冢冣 bz DL

2

⫺ 1.472 ⫹

12.16 D 1/3

冮

L

zt

⫺1/3

⫺7/3

冢冣冢冣冢冣冢冣冢冣冢冣冥冤冢冣冢冣冢冣冢冣冢冣冢冣冢冣冢冣冢冣冥

1⫺

z L

bz DL

⫺ 2.067

14/3

dz C 2n (z)

1⫺

z L

1⫺

z L

bz DL

1⫺

⫺3

⫹ 0.339

5/3

⫺ 0.6657

DL ⫺ 0.00308 bz

⫺ 3.06 ⫻ 10

⫺6

⫺ 6.15 ⫻ 10⫺6


4

7/3

DL bz

DL bz

z L

6

bz DL

1⫺

DL bz

z 1⫺ L 13/3

z L

1/3

1⫺

4

z 1⫺ L

19/3

1⫺

z L

6

8

⫺13/3

z L

2

(173)

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Figure 12 Jitter variance caused by turbulence below the beacon resulting from focal anisoplanatism for a 60-cm aperture with 30-cm beacon spacing at 0.5-µm wavelengths for HV-21 turbulence.

The transition altitude between the low- and high-altitude solutions is zt ⫽

L b/D ⫹ 1

(174)

Tilt variances are converted to phase variances by multiplying by (k 0 D/4) 2. The tilt variance due to this effect is less than one-third that due to the tilt component of focal anisoplanatism. The smaller effect results from the low weighting given to turbulence-induced tilts close to the beacon altitude for this problem compared to the almost unity weighting of the difference in tilt between focused and collimated beams. Therefore, it will be neglected, thereby simplifying the analysis. It was assumed in the analysis that the beacon positions were measured with the full aperture. If a smaller aperture is used or other sources of jitter are introduced, the error in estimating the position of the beacons can be considerably larger. In Figs. 12 and 13, the phase variances of the tilt component of focal anisoplanatism are plotted for 30-cm and 2-m sections of the aperture for Hufnagel-Valley 21 turbulence with the system operating at 0.5-µm wavelength. B. Correlation Function of the Tilt Component of Focal Anisoplanatism In calculating the error in stitching together the tilt errors of the individual sections, the correlation of the tilt is very important in determining the resulting tiltremoved phase variance. For instance, Fig. 14 shows the results from stitching together perfectly correlated one-dimensional tilts and tilts that are partially correlated. The figure error is the phase with piston and tilt removed. In the perfectly correlated case, the resultant phase is composed solely of



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Figure 13 Jitter variance caused by turbulence below the beacon resulting from focal anisoplanatism for a 4-m aperture with 2-m beacon spacing at 0.5-µm wavelength for HV-21 turbulence.

tilt, and the tilt-removed variance is zero. The correlation function of parallel components of tilt is a function of the displacement angle with respect to the tilt. The correlation function versus angle is calculated in Sasiela [1994b] and plotted in Fig. 16 for parallel, perpendicular, and 45° displacements. The perpendicular components of tilt are also correlated in certain directions. There is no correlation if the displacement is parallel to either tilt component. The correlation function for a displacement 45° to the tilt is plotted in Fig. 17. C.

Stitching Model and Results

Using the correlation functions that were found, one can find the error propagator for the tilt stitching process. A model of an aperture with four beacons is given

Figure 14 Stitching difference between correlated and uncorrelated tilts.


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Figure 15

Aperture model for stitching.

in Fig. 15. The circled numbers are phase values in the center of the nine subapertures. The eight tilts with arrows are measured in each of the four subapertures. The measurement model has the gradient, m, equal to a matrix multiplying the N phase points plus a measurement noise, n g , in the tilts given by m ⫽ Aφ ⫹ n g

(175)

It is assumed that the signal-to-noise ratio is infinite, and the measurement noise is due to the phase-variance error incurred in propagating through turbulence. Each gradient is the average of the difference of the four phase points around that gradient, and is given by

Figure 16 Correlation function for parallel components of tilt in which the displacement is parallel, perpendicular, and at 45° to the tilt.



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Figure 17 Correlation function for perpendicular components of tilt in which the displacement is 45° to the tilt.

A ⫽ 0.5

⫺1

1

0

⫺1

1

0

0

0

0

0

⫺1

1

0

⫺1

1

0

0

0

0

0

0

⫺1

1

0

⫺1

1

0

0

0

0

0

⫺1

1

0

⫺1

⫺1

0

1

1

0

0

0

0

0

1

0

0

0

0

0

1

1

0

0

1

1

⫺1 ⫺1

0 0

0

0

0

0

0

⫺1 ⫺1 0

⫺1 ⫺1

⫺1 1

(176)

The noise correlation matrix of the gradients is C n ⫽ 〈n gn Tg 〉

(177)

and is explicitly equal to

Cn ⫽

1

i

p q

i

1 q

0 0 0

p 0 0

r

r 0

p q

1

i

0

r

0 0

q p

i

1

r

0

0 0

0 0

0

r

1

i

p q

0 0

r

0

i

1

q p

0

r

0

0 p q

1

i

r

0 0

0 q p

i

1


(178)

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Figure 18 Figure variance of a 60-cm aperture with four beacons at various altitudes for the HV-21 turbulence model at 0.5-µm wavelength.

where the correlation values i ⫽ 0.558, p ⫽ 0.778, q ⫽ 0.594, and r ⫽ 0.109 are taken from Fig. 16 and 17. The least-squares solution for the phase is ⫹ T ⫺1 φ˜ ⫽ (A T C ⫺1 n A) A C n m ⫽ Lm

(179)

where the superscript ⫹ denotes the generalized inverse. A generalized inverse is required because the matrix has some singular values. It is easy to show that the least-squared phase has zero piston. The tilt- and piston-removed phase is ⫹ T ⫺1 φ˜ TPR ⫽ Λ(A T C ⫺1 n A) A C n m ⫽ ΛLm

(180)

L is the tilt removal matrix that is a combination of x- and y-tilt removal given by Λ ⫽ 1 ⫺ t Tx t x ⫺ t Ty t y

(181)

where t x ⫽ 0.5[⫺1 0 1

⫺1 0 1

⫺1 0 1]

(182)

1 1 1]

(183)

and t y ⫽ 0.5[⫺1 ⫺1 ⫺1

0 0 0

The error in the estimate can be shown to be Error ⫽ LC n LT

(184)

and the error propagator, which is the fraction of the tilt error appearing in the stitched phase, is



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Figure 19 Figure variance of a 4-m aperture with four beacons at various altitudes for the HV-21 turbulence model at 0.5-µm wavelength.

E⫽

1 ⫹ T Trace[Λ(AT C ⫺1 n A) ⌳ ] N

(185)

The error propagator with four beacons is 0.426, which means that less than half the phase variance due to tilt jitter is effective in causing a figure variance of the stitched beam. The error propagator is 0.465 and 0.957 for 9- and 16-beacon systems, respectively. Setting the correlation function of perpendicular components of tilt to zero changes the error propagator by only a few percent. The performance of the system can be calculated from the above result and Eqs. (155), (151) and (158). The total piston- and tilt-removed phase variance is σ 2 ⫽ σ 2PTR ⫹ Eσ 2⫺T

(186)

The diameter of a section rather than the aperture diameter has to be inserted into the expression for σ 2⫺PTR . The performances of a four-beacon 0.6-m and 4-m system operating in the visible at 0.5-µm wavelength at zenith and at 45° off zenith is found and plotted for various altitudes in Fig. 18 and in Fig. 19, respectively.

REFERENCES Butts, R. R., Spectra of Turbulence Induced Wavefront Aberrations, AFWL-TR-80-107 (Air Force Weapons Laboratory, 1980). Esposito, S., Riccardi, A., Ragazzoni, R., Focus anisoplanatism effects on tip-tilt compensation for adaptive-optics with use of a sodium laser beacon as a tracking reference, J. Opt. Soc. Am. A, 13, (1996), 1916–1923.


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Fields, D. A., High frequency behavior of the tilt spectrum of atmospheric turbulence, Appl. Opt., 22, (1983), 645–647. Foy R., Tallon M., ATLAS Experiment to Test the Laser Probe Technique for Wavefront Measurements, SPIE, vol. 1114, Active Telescope Systems (1989). Frehlich, R., Laser Scintillation Measurements of the Temperature Spectrum in the Atmospheric Surface Layer, Journal of the Atmospheric Sciences, 49, (1992), 1494– 1509. Fried, D. L., Aperture Averaging of Scintillation, J. Opt. Soc. Am., 57, (1967), 169–175. Fugate, R. Q., Fried, D. L., Ameer, G. A., Boeke, B. R., Browne, S. L., Roberts, P. H., Ruane, R. E., Tyler, G. A., Wopat, L. M., Measurement of atmospheric distortion using scattered light from a laser guide star, Nature, 353, (1991), 144–146. Gardner C. S., Welsh B. M., Thompson L. A., ‘‘Sodium Laser Guide Star Technique for Adaptive Imaging in Astronomy,’’ SPIE, vol. 1114, Active Telescope Systems (1989). Gardner, C. S., Welsh, B. M., Thompson, L. A., Design and Performance Analysis of Adaptive Optical Telescopes Using Laser Guide Stars, Proc. IEEE, 78, (1990), 1721–1743. Gavel, D. T., Morris, J. R., Vernon, R. G., Systematic design and analysis of laser-guidestar adaptive-optics systems for large telescopes, J. Opt. Soc. Am. A, 11, (1994), 914–924. Gradshteyn, I. S., Ryzhik, I. M., Table of Integrals, Series, and Products, Academic Press, New York, 1980. Greenwood, D. P., Tarazano, D. O., A Proposed Form for the Atmospheric Microtemperature Spatial Spectrum in the Input Range, RADC-TR-74-19 (ADA 776294/1GI) (Rome Air Development Center, 1974). Greenwood, D. P., Fried, D. F., Power spectra requirements for wave-front-compensative systems, J. Opt. Soc. Am., 66, (1976), 193–206. Greenwood, D. P., Bandwidth specifications for adaptive optics systems, J. Opt. Soc. Am., 67, (1977), 390–393. Gurvich, A. S., Time, N. S., Turovtseva, V. F., Turchin, V. F., Reconstruction of the temperature fluctuation spectrum of the atmosphere from optical measurements, Izvestiya, Atmospheric and Oceanic Physics, (1974). Herrmann, J., Least-Squares Wave Front Errors of Minimum Norm, J. Opt. Soc. Am. 70, (1980), 28–35. Hill, R. J., Clifford, S. F., Modified spectrum of atmospheric temperature fluctuations and its application to optical propagation, J. Opt. Soc. Am., 68, (1978), 892–899. Hogge, C. B., Butts, R. R., Frequency Spectra for the Geometric Representation of Wavefront Distortions Due to Atmospheric Turbulence, IEEE Trans. Antennas Propaga, AP-24, (1976), 144–154. Humphreys R. A., Primmerman C. A., Bradley L. C., Herrmann J., Atmospheric-Turbulence Measurements using a Synthetic Beacon in the Mesospheric Sodium Layer, Optics Letters, 16, 1991, 1367–1369. Ishimaru, A., Fluctuations of a beam wave propagating through a locally homogeneous medium, Radio Sci., 4, (1969), 293–305. Marichev, O. I., Integral Transforms of Higher Transcendental Functions, Ellis Horwood Limited, Chichester, England, 1983.



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Murphy, D. V., Primmerman, C. A., Zollars, B. G., Barclay, H. T., Experimental Demonstration of Atmospheric Compensation Using Multiple Synthetic Beacons, Opt Lett, 15, (1991), 1797–1799. Noll, R. J., Zernike polynomials and atmospheric turbulence, J. Opt. Soc. Am., 3, (1976), 207–211. Parenti R. R., Sasiela R. J., Laser Guide-Star Systems for Astronomical Applications, J. Opt. Soc. Am. A, Special Issue on Atmospheric Compensation Technology, Vol. 11, No. 1, Jan 1994, 288–309. Primmerman, C. A., Murphy, D. V., Page, D. A., Zollars, B. G., Barclay, H. T., Compensation of Atmospheric Optical Distortion Using a Synthetic Beacon, Nature, (1991), 141–143. Range, R. M., Holomorphic Functions and Integral Representations in Several Complex Variables, Springer-Verlag, Berlin, Germany, 1986. Sasiela R. J., Shelton J. D., Mellin Transform Techniques Applied to Integral Evaluation: Taylor Series and Asymptotic Approximations, J. Math. Phys., 2572–2617, (1993a). Sasiela R. J., Shelton J. D., Transverse Spectral Filtering and Mellin Transform Techniques Applied to the Effect of Outer Scale on Tilt and Tilt Anisoplanatism, J. Opt. Soc. Am. A, 10, 646–660, (1993b). Sasiela, R. J., Electromagnetic Wave Propagation In Turbulence: A Mellin Transform Approach & other applications of Mellin transforms, Springer-Verlag, 1994a. Sasiela R. J., Wavefront correction using one or more synthetic beacons, J. Opt. Soc. Am. A, Special Issue on Atmospheric Compensation Technology, Vol.11, No. 1, Jan 1994b, 379–393. Slater, L. J., Generalized hypergeometric functions, Cambridge University Press, LondonNew York, 1966. Strohbehn, J. W., Laser Beam Propagation in the Atmosphere, Springer-Verlag, Berlin, 1978. Stroud, Philip A., Anisoplanatism in adaptive optics compensation of a focused beam with use of distributed beacons, J. Opt. Soc. Am. A, 14, (1996), 868–874. Tatarski, V. I., Wave Propagation In a Turbulent Medium, Dover Publications, Inc., New York, 1961. Tatarski, V. I., The Effects Of The Turbulent Atmosphere On Wave Propagation, U. S. Department Of Commerce, 1971. Thompson L. A. and C. S. Gardner, Experiments on Laser Guide Stars at Mauna Kea Observatory for Adaptive Imaging in Astronomy, Nature, vol. 328 (16 July 1987). Thompson L. A., Gardner C. S., Excimer Laser Guide Star Techniques for Adaptive Imaging in Astronomy, SPIE, vol. 1114, Active Telescope Systems (1989). Tyler, G. A., Turbulence-induced adaptive-optics performance degradation: evaluation in the time domain, J. Opt. Soc. Am. A, 1, (1984), 251–262. Tyler, G. A., The Power Spectrum for G-tilt and Z-tilt, tOSC Report No. TR-700 (the Optical Sciences Company, Placentia, California, 1986). Valley, G. C., Long- and short-term Strehl ratios for turbulence with finite inner and outer scales, Appl. Opt., 18, (1979), 984–987. Vaughn, J. L., Calculation of the Power Spectra of Z-tilt and G-tilt, tOSC Report No. TR710 (the Optical Sciences Company, Placentia, California, 1986).


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Walters, D. L., Favier, D. L., Hines, J. R., Vertical Path Atmospheric MTF Measurements, J. Opt. Soc. Am., 69, (1979), 828–837. Welsh B. M., Gardner C. S., Thompson L. A., Effects of Nonlinear Resonant Absorption on Sodium Laser Guide Stars, SPIE, vol. 1114, Active Telescope Systems (1989). Welsh B. M., Gardner C. S., Nonlinear Resonant Absorption Effects on the Design of Resonance Fluorescence Lidars and Laser Guide Stars, Applied Optics, Vol. 28, No. 19, (1 October 1989a). Welsh, B. M., Gardner, C. S., Performance Analysis of Adaptive-Optics Systems using Laser Guide Stars and Slope Sensors, J. Opt. Soc. Am., 6, (1989b), 1913–1923. Welsh, B. M., Thompson, L. A., Effects of Turbulence-Induced Anisoplanatism on the Imaging Performance of Adaptive-Astronomical Telescopes using Laser Guide Stars, J. Opt. Soc. Am. A, 8, (1991), 69–80. Winker, D. M., Effect of a finite outer scale on the Zernike decomposition of atmospheric optical turbulence, J. Opt. Soc. Am. A, 8, (1991), 1568–1573. Yura, H. T., Tavis, M. T., Centroid anisoplanatism, J. Opt. Soc. Am. A, 2, (1985), 765– 773.


4 Wavefront Sensors Joseph M. Geary University of Alabama in Huntsville, Huntsville, Alabama

I. INTRODUCTION The imagery obtained from ground or spaceborne optical telescopes systems depends upon the shape of the wavefront emerging from their exit pupils. There are both extrinsic and intrinsic factors influencing wavefront shape (such as medium and/or optical system induced aberration). An instrument which measures wavefront shape is called a wavefront sensor. Wavefront sensor measurements provide a single pass end-to-end diagnosis of an optical imaging system. The aberration content of the beam in the exit pupil is determined and expressed mathematically in terms of a polynomial. Such information provides a convenient means for calculating far-field behavior such as Strehl ratio, encircled energy, and modulation transfer function. A comparison can then be made between theoretical and empirical values of these parameters. This chapter provides information on various techniques employed by wavefront sensors, and how such sensors can be used in optical metrology.

II. DESCRIPTION OF WAVEFRONT Before discussing wavefront sensors it is appropriate to understand and define the subject of the measurement, i.e., the wavefront. As is known from shadow casting phenomena and the pinhole camera, light travels in straight lines. The pinhole camera also helps define what is meant by a ‘‘ray of light.’’ Consider a point source of light emitting rays in all directions. Next, consider the ensemble of rays having a certain optical path length (OPL):


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Figure 1 Example of basic monochromatic wave structure propagating along a ray.

OPL ⫽ (length) ⫻ (refractive index)

(1)

The OPL is a radius in this case and the ray tips lie on the surface of a sphere (centered on the point source). This surface is a basic example of what is meant by a wavefront. (Note that rays and their associated wavefronts are always orthogonal to one another.) The wavefront is also called a phasefront. Light is an electromagnetic wave phenomenon. A ray can be thought of as the path along which the electric field strength, u, propagates in a sinusoidal manner with velocity c as described by Eq. (2) and illustrated in Fig. 1: u ⫽ u 0 sin φ ⫽ u 0 sin

冤2π(xλ⫺ ct)冥

(2)

The field amplitude u is cyclic. Phase φ refers to some point in this cycle. For example, when φ ⫽ 90°, u ⫽ u 0. In an optical system, the temporal variation of the electric field is generally ignored. Ray paths connect the object and image, and the phase variations along those paths can be considered ‘‘frozen-in.’’ What is of interest is the phase differences (or optical path difference, OPD) between different parts of the ray path or between different rays. Such phase differences are constant in time. Consider the two rays from the point source in Fig. 2. (Note: It is the nature of a point source that all rays leave the source having the same phase.) Along the ray path, the cyclic nature of light is indicated by the number of wavelengths that fit the OPL. At the surface of the wavefront, both rays have the same phase, φ0. If the point source is moved to infinity, then the wavefront observed will be flat. The rays (normal to this surface) will be parallel. (The parallel rays are said to form a collimated beam.) The spherical wavefront and flat wavefront are ideal constructions against which other wavefronts will be compared. In that context they will be called reference wavefronts.


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Figure 2 Phase relationship between two rays.

III. INTERACTION OF WAVEFRONTS WITH OPTICAL SYSTEMS Consider the ideal system in Fig. 3. At the entrance pupil of the system is a diverging spherical wavefront centered on the object point. It is a surface of constant OPL. At the exit pupil of the system there is a collapsing or converging spherical wavefront centered on the image point. It too is a surface of constant OPL, not only with respect to the image point but also with respect to the object point as well. A perfect optical imaging system is such that the OPL from the object point through the system to the image point is the same for any ray path! Define an OPL on the axial ray between object and exit pupil, and let all other rays have this same OPL. Only a wavefront with a spherical surface can converge to a point. For an imperfect optical system these other rays do not terminate on the surface of a sphere (centered on the image point), hence no point image can be formed. This is illustrated in Fig. 4. There is still a wavefront. All the rays terminating on its surface have the same phase. But this wavefront surface is nonspherical. Such a wavefront is said to be aberrated. The image formed by the aberrated wavefront will be spread out in a tiny volume about the ideal image point and cause a loss in resolution.

Figure 3 Wavefront manipulation by a perfect optical imaging system.


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Figure 4 Rays from a converging nonspherical wavefront cannot form a point image.

IV. WAVEFRONT DESCRIPTION An aberrated wavefront can be described by comparing it to the ideal spherical wavefront, which we will call the reference wavefront (1, 2). The reference wavefront is set up with its vertex tangent to the exit pupil and with its center of curvature coincident with the ideal image point. For each point in the exit pupil, we measure the optical path difference, W, between the spherical reference surface (SRS) and the aberrated wavefront (AWF) along the radius of the spherical reference surface. This is shown in Fig. 5. A function W(x,y) is obtained over the pupil, which is now used as a description of the aberrated wavefront. The OPD function W(x,y) can be cast in a mathematical form by a polynomial. This

Figure 5 Exit pupil aberration W(x,y).


Wavefront Sensors

127

is useful because each term in the polynomial describes a specific aberration and how much of it is present. Two sets of polynomials have traditionally been used to describe aberrations in the exit pupil. In optical design the Seidel polynomial (1) series is typically used. In optical testing the aberration content of a measured wavefront must be deciphered. The procedure commonly used is to fit the data with a Zernike polynomial (2).

V.

ACCESSIBLE MEASUREMENT PARAMETERS

Wavefronts cannot be perceived because it is light intensity rather than phase that interacts directly with matter. Detectors such as our eye respond to brightness levels, not to differences in optical path length. Detectors in wavefront sensors also respond to light level. However, this response is related to some kind of length measurement. Figure 6 shows an aberrated wavefront and associated reference sphere in the exit pupil of a perfect imaging system. Also indicated in the figure are the three physical parameters that are accessible for measurement: (a) the optical path difference, W(x,y); (b) the differential phase, dW(x,y), between adjacent sampling points in the pupil; (c) the transverse ray aberration, T. The equation relating local wavefront tilt in the pupil to T is (3).

冢冣

dW nr ⫽⫺ T dρ R

Figure 6 Physical parameters measurable by wavefront sensors.


(3)

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Figure 7 (a) Local tilt as a function of sampling location in pupil; (b) reconstructed wavefront estimate.

where dW/dρ is local wavefront tilt, R is the radius of curvature of the reference sphere, n is the refractive index, and r is the radius of the pupil. Wavefront sensors estimate the overall shape of the phasefront from a finite number of discrete measurements. These measurements are usually made at uniform spatial intervals. If the wavefront sensor measures T, Eq. (3) can be used to find the local wavefront tilt. All this yields is the local wavefront tilt as a function of the transverse ray aberration defined at specific pupil locations. The situation is illustrated in Fig. 7a along a radius. Since the wavefront is really continuous, the local tilts must be stitched together so that a contiguous wavefront profile is generated as shown in Fig. 7b. This process is called wavefront reconstruction and generates an estimate of W(y). If the wavefront sensor measures the differential wavefront dW, we obtain the local incremental OPD as a function of pupil position, as illustrated in Fig. 8a. To get the wavefront W(y), these dWs are sequentially stacked as shown in Fig. 8b. VI. WAVEFRONT SENSOR A wavefront sensor (WFS) is an instrument used to obtain the OPD function W(x,y). A generic system consists of an optical head, detectors, electronics, com-


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Figure 8 (a) Differential phase, dW, as a function of pupil location; (b) reconstructed wavefront estimate.

puter controlled data acquisition and storage, and a sophisticated analysis software program. The latter is used to fit the OPD data, make various calculations (e.g., far-field performance), and display data in graphical format. This section will examine four WFSs utilized by NASA on the Hubble space telescope project, each of which measures one of the three basic parameters discussed in Sec. V. The Hubble OTA generates an f/24 beam. Hence, the WFS will be viewing a nominal point source. For a comparison of the relative performance of the WFS discussed here (except for Sec. VI. A.2), the reader is referred to Ref. 4.

A.

Direct Wavefront Measurements

This section will examine WFSs which measure the pupil OPD W(x,y) directly. The basic data will be in the form of an interferogram generated by the interaction of two wavefronts: a reference wavefront and an object wavefront. Whenever two coherent wavefronts overlap, interference takes place. The generic expression for interference is given by (5) I ⫽ I1 ⫹ I2 ⫹ 2√I 1 I 2 cos Φ(x,y)

(4)

The shape of the fringe is defined by Φ(x,y). Each fringe in the interference pattern is a contour of constant OPD or W(x,y) ⫽ mλ, where m is a fringe order


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Figure 9 (a) Structure of PDI plate; (b) principle of operation.

number and λ is the wavelength. Please note that Φ(x,y) ⫽ kW(x,y), where k ⫽ 2π/λ. 1. Point Diffraction Interferometer (PDI) Wavefront Sensor The heart of the PDI WFS is the PDI plate, which is a self-referencing interferometer (6). The PDI structure is illustrated in Fig. 9a. It is a monolithic device consisting of two concentric circles (thin-film coated onto a thin transparent substrate). The fat annular region is semitransparent and acts like a neutral density filter. The small inner circle (pinhole) is a clear diffraction aperture and plays a spatial filtering role. PDI operation is illustrated in Fig. 9b. The aberrated wavefront is focused onto the PDI disk. Most of the beam passes through unhindered except for a reduction in intensity. The tiny part of the beam interacting with the pinhole is diffracted into a clean spherical wavefront; i.e., it has become the reference wavefront. Immediately on the far side of the PDI, interference takes place between the aberrated main beam and the reference beam. As with any interferometer good fringe contrast occurs when the intensities of the two wavefronts are about equal. For the PDI this depends on the size of the diffraction core in the far-field pattern relative to the size of the pinhole. The diffraction core size depends on the F-number and the amount of aberration on the incident wavefront. This intensity balancing also comes into play when the PDI plate is translated axially or laterally (for focus and tilt fringes respectively). Fringe contrast will decrease as these translation distances increase. Figure 10 is an optical schematic of a PDI WFS used to measure the ESA Hubble simulator (OSL). The configuration is quite simple. It consists of a lens, the PDI plate, and a CCD. The lens in front of the PDI plate serves two functions.


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Figure 10 PDI wavefront sensor configuration for convergent slow beam.

First, it decreases the F-number (f/#) of the incident beam (or increases speed) into the operational f/# mid-range of the PDI. Second, it forms an image of the pupil (of the optical system responsible for the converging beam) onto the recording medium. Because of its location ahead of the PDI, aberrations picked up within the ‘‘speed lens’’ will be added to those of the incident beam. This difficulty can be eliminated by designing a well-corrected speed lens so that inherent aberrations are negligible. A lower cost approach would involve use of an off-the-shelf component whose aberration content for the given F-number is measured. This speed lens aberration would be saved as an error file for later subtraction from the actual beam data. The fringe pattern recorded on the CCD is transferred via a frame-grabber to a PC where the fringe analysis software is in residence. A standard commercially available code was used for the system depicted in Fig. 10. The OPD data from the interferogram was fitted with a 36-term Zernike polynomial from which aberration content of the beam could be extracted. 2. Radial Shear Interferometer Wavefront Sensor The radial shear interferometer (4) (RSI) wavefront sensor is a modified MachZehnder interferometer (5) with afocal telescopes in each arm. One telescope expands the beam while the other compresses it, as shown in Fig. 11. (Obviously both telescopes must themselves be of high quality in order for this method to work properly.) Interference occurs in the region of overlap, as also indicated in


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Radial shear interferometer. Insert shows resulting interferogram.

Fig. 11. If the variations in phase of the expanded beam over the region of the compressed beam footprint are insignificant (a tenth wave or less), then this subregion of the expanded beam can be considered a reference wavefront. Wiggles in the fringe pattern can then be attributed solely to phase variations in the compressed beam which are due to the incident wavefront, W(x, y). At the recording plane, the fringe pattern is coincident with a pupil image. The resulting interferograms can be analyzed to yield the structure of the wavefront W(x, y) over the pupil. An RSI WFS resides aboard the Hubble space telescope as a separate instrument inside each of the fine guidance sensors. This WFS is designated as the optical control subsystem (OCS). The fine guidance sensor proper is unaffected by the eight waves of spherical aberration from the primary mirror. Unfortunately, this is well outside the operational range of the OCS. COSTAR correction applies to the axial scientific instruments, not to the radial instruments such as the FGS. B. Indirect Wavefront Measurements This section examines wavefront sensor schemes that measure either the local slope, dW/dy (or dW/dx), or the differential wavefront, dW, as a function of pupil coordinates. Recall that the former provides information like that contained in Fig. 7a; the latter, like that contained in Fig. 8a. 1. Shack-Hartmann Wavefront Sensor The Shack-Hartmann WFS is a modification of the classical Hartmann test (typically used to test astronomical primaries) (7). The Hartmann test measures the


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Figure 12 Principle of operation of Hartmann test: (a) flat wavefront incident on Hartmann plate; (b) aberrated wavefront incident on plate.

parameter T (transverse ray aberration), as discussed in Sec. V. A simple Hartmann test is illustrated in Fig. 12. In Fig. 12a a perfect (flat) wavefront is incident on a perfect lens. A small hole in a movable plate defines a position in the pupil where a ‘‘ray’’ of light is admitted. This ray proceeds to the paraxial focal plane and arrives dead center. Now suppose the wavefront is imperfect. The ‘‘ray’’ admitted by the hole plate strikes the paraxial plane off-center, as shown in Fig. 12b. The separation between this ray pierce and the optical axis is T. This is directly related to the local slope of the wavefront at the hole location in the pupil through Eq. (3). The ensemble of such measurements over the entire pupil will yield information from which the shape of the wavefront in the pupil can be reconstructed. A simplified schematic of the Shack-Hartmann technique is presented in Fig. 13. Instead of a single hole, the Shack-Hartmann plate contains an array of holes each with its own ‘‘perfect’’ little lens (8). Each lenslet is identical and serves the same function as in the single hole Hartmann configuration in Fig. 12. Each lenslet samples the local wavefront tilt at its particular location in the pupil. Associated with each lenslet is a position sensitive detector (PSD). The ‘‘ray’’ from each lenslet strikes its PSD with an offset, T, directly related to the local slope of the wavefront at the lenslet location. A Shack-Hartmann wavefront sensor is shown in Fig. 14. The focus of a collimating lens is made coincident with the focus of the test system having a certain f/#. The beam is collimated and transferred to a lenslet array. The collimating lens also images the system exit pupil onto the lenslet array. A relay lens reimages the array of focal spots formed


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Principle of operation of Shack-Hartmann wavefront sensor.

by the lenslets onto a CCD. The CCD is segmented so that a subarray of pixels is associated with each lenslet. This subarray behaves as a PSD and measures the local T for a particular lenslet. Figure 15 shows a photograph of a ShackHartmann WFS (9). Goddard employed a Shack-Hartmann WFS from the Adaptive Optics Association (AOA) for use on the Hubble recovery program. It was used to validate COSTAR performance (10). 2. Lateral Shear Interferometer Wavefront Sensor Lateral Shear interferometer WFSs are based on lateral shear interferometry (7), the simplest example of which is provided by a parallel plate. This is illustrated

Figure 14

Shack-Hartmann wavefront sensor.


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Figure 15 Photograph of an AOAShack-Hartman WFS.

in Fig. 16. A nominally collimated but aberrated beam is incident on a parallel plate. The two reflected beams are parallel but laterally shifted. In the region of overlap there is a lateral shear interference pattern. This pattern provides information on OPD differences (as per Fig. 8a) at a pupil location but only in the direction of shear. To reconstruct a wavefront, two patterns sheared in orthogonal directions are needed. Diffraction gratings (5) can also generate lateral shear interferograms. For example, when a focused beam falls on a grating, the outgoing power is divided


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Figure 16 Lateral shear interferometry: (a) by reflection from parallel plate; (b) interferogram showing a defocused wavefront; (c) sheared profile pair.

among several diffracted orders, as illustrated in Fig. 17a. If the grating is at the focus of a following lens as per Fig. 17b, the various diverging cones are collimated. If we place an observation screen downstream of this lens, we see a number of overlapping circles of light. This is illustrated in Fig. 17c. In the overlap areas lateral shear interference fringes can be observed. An example of a lateral shear interferogram for an f/3.1 wavefront with 45 waves of spherical aberration using a Ronchi grating (55 lines per millimeter) is shown in Fig. 18 for three different axial positions near paraxial focus. A wavefront sensor based on lateral shear interferometry using a grating

Figure 17 (a) Diffracted orders from focused beam incident on a grating: (b) collimation of the diffracted orders; (c) overlapping orders in plane of observation.


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Figure 18 Grating lateral shear interferograms for three axial positions for a spherically aberrated wavefront (11).

is illustrated in Fig. 19. The optical schematic in Fig. 19a shows a focused beam from a Hubble simulator being captured by an auxiliary optic which collimates the beam. This beam is directed to a grating which creates several collimated diffraction orders. These orders are collected by a focusing lens and imaged in the far field (as an array of focal spots). In the focal plane a spatial filter blocks all orders except the zero and ⫾first orders. These passed orders form a lateral shear interferogram (similar to those shown in Fig. 18) at a pupil image where a CCD records the fringe information. As with the parallel plate, sheared interferograms are needed in orthogonal directions in order to have enough information to reconstruct the wavefront. This can be obtained simultaneously by placing a ‘‘crossed’’ grating at the grating location in Fig. 19a and using a square spatial filter. A WFS breadboard based on this crossed grating idea was built by ITEK for potential use by the Goddard IVT on the Hubble project. It was called the RTSI (real time shearing interferometer). A photograph of the back end of the RTSI (from the crossed grating to the CCD) is shown in Fig. 19b. Figure 20 shows crossed grating (55 l/mm) lateral shear interferograms for the same input beam as in Fig. 18 at three axial positions.

VII. INTENSITY-BASED WAVEFRONT SENSING All of the wavefront sensors discussed thus far have involved measurements of W, dW, or T. The metric was a dimensional value expressed in waves, microns, or millimeters. The amount of light present in the system (expressed in terms of irradiance or intensity) was of little concern as long as there was enough signal. Wavefront sensing schemes will now be investigated where irradiance (or intensity) is the key ingredient of the measurement.


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(a)

(b) Figure 19 Grating based lateral shear interferometer wavefront sensor: (a) optical schematic; (b) photograph of ITEK RTSI.


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Figure 20 Crossed grating lateral shear interferograms.

A.

Axial Intensity

People with backgrounds in optics are familiar with the appearance of the star image lateral irradiance distribution in and around paraxial focus for a system with and without aberration. An example is the Airy pattern in Fig. 21. Its lateral profile is shown in Fig. 22a. Less familiar perhaps is the axial intensity distribution for this very same pattern shown in Fig. 22b (11).

Figure 21 Star image or Airy pattern for aberration less imaging system.


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Figure 22

(a) Radial and (b) axial intensity profiles for a system with no aberration.

Figure 23 shows corresponding point spread functions at equal axial focus offsets on either side of the focal plane. Note the positions where the axial intensity of the point spread functions are zero. In the axial intensity plot in Fig. 22b, zero intensity occurs for a focal shift of one wave as measured in the exit pupil. Most readers are probably not very familiar with the behavior of axial intensity profiles in the presence of spherical aberration. Experimental work on axial intensity stems largely from the 1980s (12–14). Much of the work concerned the influence of spherical aberration. Although the axial intensity pattern in the presence of spherical aberration changes shape and shifts away from paraxial focus, it nonetheless remains symmetric. This fact was not lost on those who saw this as a potential metrology tool (14). The separation between the plane of symmetry and the paraxial focal plane is directly related to the amount of spherical aberration present through the equation δ ⫽ ⫺8(f/#) 2W 040

(5)

where δ is the axial offset, f/# is the F-number, and W040 is the spherical aberration coefficient as determined in the exit pupil. Figure 24a shows an experimental axial intensity plot for a system with significant spherical aberration. The F-number of the system is f/10.3. The operating wavelength is 0.6328 micron. The separation between the plane of symmetry and the paraxial focal plane is 3.07 mm. (Note: Paraxial focus is established by a separate axial intensity scan with a small aperture in the entrance pupil.) Using Eq. (5), we find that the amount of spherical aberration is 5.7 waves! Figure 24b shows a theoretical plot with the same amount of spherical aberration.


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Figure 23 PSF images in different focal planes for system without aberration. (From M. Cagnet, M. Franzon, and J. C. Thierr, Atlas of Optical Phenomena, Springer-Verlag, New York, 1962.)

Recent work on axial intensity (15) shows promise of utilizing this technique in a wavefront sensing scheme. Reference 15 discusses measurement of coma and astigmatism as well as spherical aberration in systems with and without central obscurations using uniform and Gaussian pupil irradiance distributions. A possible wavefront sensing configuration is illustrated in Fig. 25. A nominally collimated but aberrated wavefront is incident on a high quality focusing lens. The star image pattern is examined with a microscope objective (40⫻– 60⫻). In the magnified image, the central irradiance is sampled by an optical fiber. The output end of the fiber is fed into a PMT/radiometer. The signal from the radiometer is fed into the Y-axis of an XY-recorder. Strictly speaking, one should move the optical fiber entrance face longitudinally to generate the axial intensity scan. Its axial motion would be monitored by a linear transducer whose


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Figure 24 Axial intensity where W 040 ⫽ 5.7 waves: (a) experimental; (b) theoretical. (Courtesy Dr. Qian Gong).

signal would drive the X-axis of the XY-recorder. However, it is also possible to move the microscope objective axially over a much shorter range if the magnification changes at the fiber optic input face remain very small. Although axial intensity can pull out the magnitude of pure Seidel aberrations, it also shows promise of doing so for mixed aberrations. However, the angular orientation of the asymmetric aberrations in the pupil cannot be determined from the axial scan alone. Additional information of some sort is needed. Techniques for doing so are currently being explored.

Figure 25 Wavefront sensor configuration for measuring aberration content via axial intensity scans.


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Figure 26 Measuring irradiance at equidistances on either side of paraxial focus.

B.

Curvature Sensor (16,17)

Consider the situation in Fig. 26. We have a flat wavefront incident upon a perfect lens. At equal distances on either side of the focal plane, irradiance measurements will be made. The distance from the focal plane is large enough so that diffraction effects can be neglected; i.e., geometric optics dominates. In these measurements planes there is a one-to-one correspondence with positions in the pupil; i.e., they are geometrically scaled versions of the pupil. As a consequence, the flat wavefront can be replaced with a rectilinear array of parallel rays which pierce the pupil on a uniform grid, as shown in Fig. 27. At the two measurement planes, the ray pierces will also be uniform, as shown in Fig. 28, and the density of ray pierces will be identical. If irradiance measurements were being made in these two planes, the values obtained would be the same. Irradiance profiles across a diameter for both planes is shown in Fig. 29. If the two profiles were subtracted the result would be zero. Now suppose that that some defocus (say some slight convergence) is introduced on the incident wavefront. The image point moves away from the focal plane and toward the lens, as shown in Fig. 30. Now the distribution of ray pierces in the pupil is still uniform (as per a Hartmann test). This is still true for both the measurement planes. However, the ray pierce density is no longer the same for both. The density is higher in the front measurement plane. This should be obvious because the same number of rays are distributed over a smaller area (but there is still a one-to-one correspondence with points in the pupil). Since the ray density is higher, so is the measured irradiance. The irradiance in the rear


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Figure 27

Uniform grid of rays at entrance pupil.

measurement plane will be lower. The two irradiance profiles are illustrated in Fig. 31. When subtracted, the result is no longer zero. Defocus as described in the pupil is the parabolic function W ⫽ W020y 2

(6)

The first derivative yields a linear function: dW ⫽ 2W020y dy

Figure 28

Ray pierces at the equidistant image planes on either side of focus.


(7)

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Figure 29 wavefront.

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Irradiance profiles across beam footprints at the two image planes for flat

This is local wavefront tilt and is the quantity that a Hartmann sensor measures indirectly through T. Taking the second derivative yields a constant: d 2W ⫽ 2W 020 dy 2

(8)

As seen in Fig. 31, the irradiance difference is a constant. Hence, there is a relationship between the second derivative of the wavefront and the irradiance difference between two equally spaced measurement planes. This is the basis of a ‘‘curvature’’ wavefront sensor developed at Kitt Peak by Francois and Claude Roddier (16–18). In a simple breadboard experiment using the test arrangement of Fig. 26, the Roddiers have demonstrated the ability to measure spherical aberration (whose second derivative is parabolic). The dotted line in Fig. 32 is the theoretical

Figure 30 Effect of converging wavefront.


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Figure 31 Irradiance profiles across beam footprints at the two image planes for convergent wavefront.

OPD profile in the pupil for 1.3 waves. The experimental curve extracted from the irradiance measurements in the two defocus planes is given by the solid line. The Roddiers are currently using this technique to obtain atmospheric wavefront behavior for ground-based astronomical telescopes. C. Phase Retrieval In 1977, Southwell (19) suggested that it should be possible to determine W(x,y) in the exit pupil by measuring irradiance in both the exit pupil and far field. The

Figure 32 Comparison of experimental and theoretical OPD plots for spherical aberration using curvature sensing.


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idea is to make a guess as to the nature of the phase in the pupil. This guess is quantified by arbitrarily selecting values for the Zernike coefficients for a polynomial of N terms. This initial phase estimate coupled with the measured pupil irradiance is then taken to the far field via a wavefront propagation code. The computer calculated far field is then compared to the measured far field. On the first iteration, the two far fields will not look very much alike. So a change is made in the Zernike coefficients according to some protocol, and the whole process is repeated again and again. The point of this is to walk the calculated far field ever closer to the measured far field. When the match is reasonably close (by comparing things like the volume under the PSF out to the first dark diffraction ring) the process is terminated. This final set of selected Zernike coefficients is then taken to represent the real phase existing in the pupil. As you can appreciate, the measurement side of this technique is very simple. Two irradiance maps are required. Some optics and two CCD cameras provide this as illustrated in Fig. 33, but this is just the tip of the iceberg. Massive computer power and very clever and sophisticated software are needed. Phase retrieval is largely a mathematician’s game. The major concern with this wavefront sensing scheme is uniqueness (20– 22). Given a random phase perturbation, it has yet to be proven with mathematical rigor that a specific final set of Zernikes alone yields the correct far field. There might be other Zernike combinations that could provide a reasonably close approximation. Empirically, however, phase retrieval appears to yield good agreement with known calibrated input wavefronts within the limits of measurement error. This is best illustrated by example. An experiment (23) was set up to mimic the Hubble telescope. The same F-number was used along with the same central obscuration ratio (including spiders). The entrance pupil was illuminated with a truncated Gaussian (HeNe). Pupil aberrations were directly measured by two methods: axial intensity (Sec.

Figure 33 Generic optical scheme for capturing pupil and image irradiance data for phase retrieval.


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Table 1 Image # 1 2 3 4

Focus offset (mm)

W 040

⫺0.62 0.64 4.45 5.72

5.59 5.52 5.42 5.54

VII.A) and point diffraction interferometry (Sec. VI.A.1). These yielded values for Seidel spherical aberration of 5.45 and 5.50 waves, respectively. Next, irradiance measurements were made both in the pupil and at a defocused image plane. This data was fed into the phase retrieval software. The retrieved Seidel spherical aberration at four defocus planes is presented in Table 1. As can be seen, these

Figure 34 Comparison between measured (left) and computed (right) far-field images at four defocus planes.


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estimates of aberration using the phase retrieval method are in good agreement with the direct measurements. The phased retrieved information just obtained was then used (along with the pupil irradiance measurements) to predict imagery in the far field. This is shown in the right-hand side of Fig. 34. The left-hand side of Fig. 34 shows photographs of the far-field patterns obtained at the four separate image planes about paraxial focus. A comparison of the measured and computed defocused images shows good agreement between the two. Fledgling phase retrieval had been used on the Airborne Laser Lab project by the Air Force (24) and on studies of the battlefield environment by the Army (25). However, its biggest boost came with the Hubble’s troubles. Soon after launch, Hubble operators discovered that something had gone terribly wrong. Imagery was severely degraded. Having isolated the problem to the telescope’s primary mirror, Hubble scientists and engineers employed (and refined) phase retrieval techniques to determine what aberrations were causing the problem, how big they were, and whether they were positive or negative (26). The result was that the Hubble was afflicted with about eight waves of overcorrected spherical aberration. This information proved useful in several ways. First, it was employed in computer image processing schemes to improve Hubble imagery. Second, it was used in determining the ‘‘on-orbit’’ or ‘‘as-built’’ conic constant of the primary mirror. Third, it was used in designing the optical fix called COSTAR. Fourth, phase retrieval was employed for ground verification tests of the optical systems deployed for the first servicing mission.

REFERENCES Much of what is found in this chapter is based on the SPIE Tutorial Text Number 18 entitled Introduction to Wavefront Sensors. The reader is referred to this book for additional information. 1. W. Welford, Aberrations of Optical Systems, 2nd ed. Academic Press (1989). (Seidels are discussed on p. 89.) 2. R. Shannon and J. Wyant, Applied Optics and Optical Engineering, Vol. 11, Chap. 1 (Wyant and Creath), Academic Press (1992). (See p. 15 for Seidels; p. 28 for Zernikes.) 3. J. Geary, Introduction to Wavefront Sensors, SPIE Press, TT-18 (1995). 4. J. Geary, M. Yoo, P. Davila, A. Wirth, A. Jankevics, M. Ruda, and R. Zielinski, ‘‘Comparison of wavefront sensor techniques,’’ Proc. SPIE, 1776, 58–72 (1992). 5. J. Hecht and E. Zajac, Optics, Addison-Wesley (1975). 6. R. Smartt and W. Steel, ‘‘Theory and application of point diffraction interferometers,’’ Japan J. Appl. Phys., 14, 351–356 (1975), Suppl. 14-1. 7. D. Malacara, ed., Optical Shop Testing, 2nd ed., Chap. 5, Wiley (1991). 8. B. Platt and R. Shack, ‘‘Production and use of a lenticular-Hartmann screen,’’ Optical Sciences Newsletter 5, 1, 15 (1971).


150 9. 10.

11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26.

Geary L. Schmutz,‘‘Hartmann sensing at AOA,’’ SPIE Proc., 779, 13–17 (1987). P. Davila, W. Eichhorn, and M. Wilson, ‘‘Hartmann wavefront sensing of the corrective optics for the Hubble space telescope,’’ SPIE Proc., 2198, 1261–1272 (1994). M. Born and E. Wolf, Principles of Optics, 6th ed., Pergamon Press (1980). V. Mahajan, ‘‘Strehl ratio for primary aberrations: some analytical results for circular and annular pupils,’’ JOSA, 729, 1258–1266 (1982). J. Stamnes, Waves in Focal Regions, Adam Hilger (1986). J. Geary and P. Peterson, ‘‘Spherical aberration: a possible new measurement technique,’’ Opt. Eng., 25, 2, 286–291 (1986). Q. Gong and S. Hsu, ‘‘Aberration measurement using axial intensity,’’ Opt. Eng., 33, 4, 1176–1186 (1994). F. Roddier,‘‘Curvature sensing and compensation: a new concept in adaptive optics,’’ Appl. Opt., 27, 7 (1988). F. Roddier, ‘‘Curvature sensing: a diffraction theory,’’NOAO R&D Note, 87, 3 (1987). C. Roddier and F. Roddier, ‘‘Wavefront reconstruction from defocused images and the testing of ground-based optical telescopes,’’ JOSA-A, 10, 11, 2277 (1993). W. Southwell, ‘‘Wavefront analyzer using a maximum likelihood algorithm,’’ JOSA, 67, 3 (1977). S. Robinson, ‘‘On the problem of phase from intensity measurements,’’ JOSA, 68, 1, 87–92 (1978). A. Devaney and R. Chidlaw, ‘‘On the uniqueness question in the problem of phase retrieval from intensity measurements,’’ JOSA, 68, 10, 1352–1354 (1978). J. Foley and R. Butts, ‘‘Uniqueness of phase retrieval from intensity measurements,’’ JOSA, 71, 8, 1008–1014 (1981). J. Geary, Introduction to Wavefront Sensors, SPIE Press, TT-18. See Sec. 10.4.4 (1995). D. Nahrstedt and W. Southwell, ‘‘Maximum likelihood phase-retrieval algorithm: applications,’’ Appl. Opt., 23, 23 (1984). T. Liepmann, ‘‘Laser atmospheric phase and amplitude measurement,’’ SPIE Proc., 1221 (1989). J. Feinup, J. Marron, T. Schulz, and J. Seldin, ‘‘Hubble space telescope characterized by using phase-retrieval algorithms,’’ Appl. Opt., 32, 10, 1747–1767 (1993).


5 Deformable Mirror Wavefront Correctors Ralph E. Aldrich Technology Consultant, Acton, Massachusetts

I. INTRODUCTION The wavefront corrector is the brute force end of the complex parallel servo system which performs optical wavefront measurement and correction. The function of the wavefront corrector is simple: accept position commands received from the wavefront sensor/reconstructor, which are already mapped to the actuator array, and shape the optical figure of the corrector as commanded with the minimum error possible. Wavefront correctors are generally reflective, the single exception being the liquid crystal corrector. The critical issue here is wavelength insensitivity. In general, transmissive systems have too much dispersion for use over broad wavelength ranges. Further, solid state materials exhibit induced changes in refractive index much too small for useful applications. Hence, most correctors are deformable mirrors or movable, segmented mirrors. The wavefront corrector has two parts: (1) the deformable mirror itself, a sophisticated electromechanical device, and (2) a comparatively large, brainless, and, unfortunately, expensive electronic driver amplifier array, one driver for each actuator in the corrector if the system corrects anything other than thermal effects. Early studies on atmospheric correction led to the derivation of the Greenwood frequency, r 0, defined as the distance over which the phase of the atmosphere changes by 1 rad. An analysis of correctability levels achievable shows that, for essentially all applications, this distance defines as well the optimum number of actuators: The number of actuators across the diameter of the beam


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is simply the diameter of the telescope divided by r 0. If the actuator count is lower, the residual wavefront quality drops precipitously. At the same time a statistically significant improvement in wavefront quality can only be achieved by doubling the actuator count across the diameter, i.e., quadrupling the number of channels in the adaptive optic system. This is not justified by either system economics or reliability. As in all systems, the rule of ‘‘keep-it-simple’’ dominates; hence, the wavefront corrector system should have the fewest possible number of actuators consistent with the level of correction required. This choice of actuator count, will generally result in a level of wavefront correction to ⬍λ/20 rms or better, adequate for all but the most demanding requirements. A. The Dream Unrealized ‘‘It shouldn’t take a lot of power to move a photon around.’’ This comment has been attributed to Alan MacGovern, the current head of the DoD optical tracking facility on Mt. Haleakala, Maui, HI. It could not be more true, but the actual realization of wavefront correction could not be further from that truth. The reason is that wavefront correctors are complex electromechanical devices which operate at kilohertz rates. Hence, they must be stiff to avoid resonances, and the stiffness requires power to deform the surface of the corrector. Further, stiffness implies weight, hence heavy duty supports, powerful drivers, and all the other trappings of a large system. Perhaps the extreme example of this is the first complete adaptive optics system ever fielded, the Compensated Imaging System built for the Air Force in the late 1970s and installed on the AMOS telescope on Maui. The wavefront corrector weighed less than 3 pounds, but its package weighed more than 30 pounds and the electronic drivers more than 300! B. Make or Buy? There are two vital issues facing the scientist or engineer who wishes to apply adaptive optics: (1) Is it worthwhile developing a custom system? (2) If not, where does one obtain the system? The temptation is always to try and make your own, but as many groups have discovered, this is not as simple as it may seem (Kibblewhite et al., 1992; Graves et al., 1992). The Department of Defense (DoD) spent many millions of dollars in the 1970s and 1980s to develop adaptive optics components and systems both for the viewing of space objects through the atmosphere and for high energy laser propagation for the Star Wars program. The majority of that technology is now in the public domain and is being practiced by various commercial suppliers. The world market for adaptive optics components is extremely limited; thus, the number of possible suppliers for wavefront correctors is small, keeping prices comparatively high. Also, development funding is limited. Mark Ealey,


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Xinetics CEO, recently commented that in the last four years, the company had sold only 18 deformable mirrors (Defense News, May 1997). The total world market for deformable mirrors is currently less than $50,000,000, even at current prices. Is it any wonder that costs remain high and competition is meager? Currently, the only regular suppliers of wavefront correctors as standard components are Xinetics, Inc., Littleton, MA (Ealey and Wellman, 1994), LASERDOT, Marcoussis, France (Gaffard et al., 1994), and ThermoTrex, Inc., San Diego, CA (Cuellar et al., 1992). In addition, TURN Ltd., Moscow, Russia (Safranov, 1996), offers custom fabrication, but their components have yet to be widely accepted and have not been evaluated against the other established suppliers. Several laboratories have designed and/or built their own wavefront correctors with varying degrees of success, including Lawrence Livermore National Laboratory (Swift et al., 1990) and the University of Chicago (Kibblewhite et al., 1993). Two other companies which frequently appear in the literature are Litton/Itek Optical Systems and United Technology Optical Systems (UTOS). These two pioneers in the field of wavefront correctors no longer exist. UTOS has closed its doors due to the massive cutbacks in defense spending and Itek has been sold to Raytheon Optical Systems, Inc. (ROSI), which is not supplying components commercially at this time. Fortunately, a good deal of the heritage of both companies resides in Xinetics. While each installation tends to be relatively customized, and it may make sense to perform the system integration in house, the components are sufficiently complex that it is generally worthwhile to purchase them from available suppliers. Further, it is inadvisable to specify custom components if it is at all possible to purchase a standard component with reasonable performance. For example, it will be much more expensive to have a system designed and built using 60 channels of correction if the components for a 69 channel system are already available as standard parts. This is particularly true of the wavefront corrector, whose performance is hardware-specific rather than software-specific, as is the case of a wavefront reconstructor. Similarly, one should attempt to accept standard specifications for performance parameters such as peak wavefront distortion and operating frequency. Even though it might seem trivial to make the actuator slightly longer to get more stroke or to squeeze a few more milliamps out of the drivers, it really is not. It may seem strange to begin a discussion of wavefront correctors with a dissertation on cost, but that is the key issue in the purchase of an adaptive optics system. The technology is sufficiently mature that any system which has reasonable engineering parameters can be implemented. The issue is, who will pay for it, and is the gain worth the pain? There are highly experimental concepts which may pay off with greatly reduced costs in the future, but at present, rather straightforward options exist for the size, number of channels of correction, peak wavefront correction, and operating speed. The available devices use piezoelectric or


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electrostrictive actuators to create a stiff structure having a high resonant frequency. Unfortunately, these materials have limited voltage response and high dielectric constants; hence, they require comparatively high power electronics to drive them. While it may seem surprising, these electronics comprise a major, if not the major, portion of the cost of a wavefront corrector system, since they do not simply supply power to the actuators, but serve multiple other functions, such as protecting the corrector from damage due to certain system failure modes and reporting system status to the main computer. Wavefront corrector electronics will be discussed in more detail later. Most adaptive optics components have been developed on DoD contracts and later adapted to other applications. Their nonrecurring costs are already amortized. Don’t fall into the trap. Buy, don’t build, and buy standard. C. Types of Deformable Mirrors The patent literature contains literally dozens of patents for different types of wavefront correctors made using different approaches. However, they fall into a limited number of categories. First, there is the piezoelectrically actuated mirror which uses a spring to provide a compressive force (pretensioning) to return the mirror to its ‘‘down’’ position (Fig. 1). Such devices were developed to a high degree of sophistication by United Technologies Optical Systems (UTOS) primarily for use with high energy lasers which required actively cooled faceplates. An advantage of this type of mirror is that the actuators are replaceable with special tooling. The disadvantage is that the constant high compressive forces on the actuators causes piezoelectric creep, so the surface figure of the faceplate is somewhat unstable. With the demise of UTOS in the early 1990s, these devices became unavailable commercially. Second, there is the piezoelectrically actuated segmented mirror developed by ThermoTrex (Fig. 2). A somewhat similar device is described by a British group (Sharples et al., 1992). The major advantages of these mirrors are their modularity, which allows comparatively easy replacement of segments, and the fact that there is no external stress on the actuators at any time. The major disadvantage is that a white light wavefront sensing interferometer is required to set the baseline position of the actuators since there is no inherent means for resolving 2π ambiguities. Much has been made of the gaps between segments, but the total gap area is approximately 2% of the corrector area and is not a significant detriment to system performance. The third approach, originated by Itek (Ealey and Washeba, 1990) and commercialized by Xinetics (Ealey and Wellman, 1994) and LASERDOT (Gaffard et al., 1994), uses an array of multilayer piezoelectric or electrostrictive actuators mounted on a rigid base to deform a continuous facesheet by both push-



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Figure 1 Pretensioned mirrors developed by United Technologies Optical Systems use springs of various types to pull on the faceplate, eliminating tensile stress on the actuator. (From Marlow, 1993)

Figure 2 Segmented mirrors have been developed by ThermoTrex Corp. as an alternative means for eliminating compressive stress on the actuator while allowing easy actuator replacement and size scalability. (From Cuellar et al., 1992)


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Figure 3 The Xinetics deformable mirror features a continuous facesheet attached only at the actuators which both push and pull. (From Ealey and Wellman, 1994)

ing and pulling (Fig. 3). This concept has a long history (Everson et al., 1981) but was brought to fruition only by the development of the diffusion bonded, multilayered, stacked actuator. Other types of actuators include the tubular actuators used by the University of Chicago. The major disadvantage of this mirror type is that it is not generally repairable, although Itek developed a proprietary means for removing the facesheet without damaging it or the actuators (S. Daigneault, private communication). The major advantages of this structure are the comparatively small number of components, the self-referencing facesheet, and an inherently high resonant frequency. The final approach is the bimorph (curvature) corrector, obtainable from LASERDOT (Gaffard et al., 1994). This device is extremely simple and easy to build, but is limited in the number of degrees of freedom available for correction. Its major advantages arise in applications requiring only low order and comparatively slow corrections. A more detailed description of these devices can be found in Sec. III. D. Repairable versus Reliable—The Great Debate The great debate in wavefront corrector circles has always been the issue of repairability. The issue initially arose due to the natural desire to be able to fix an expensive component which might fail in the field, rather than have to spare and replace complete subsystems. The wavefront corrector approach taken by



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UTOS was a direct response to that desire. The Itek approach was different: make the actuators reliable and there is no need for repairability. This approach worked well for early, simple devices such as the Monolithic Piezoelectric Mirror (MPM) (Feinleib et al., 1974). Some of these devices have operated for more than a decade with no change in their properties. However, the development of the diffusion bonded actuator added fuel to the fire. That actuator was developed simultaneously with the advanced deformable mirror structure for which it was intended, and due to pressure from the Star Wars program was rushed into production before the processes were well understood and controlled. Thus, there were a series of problems, primarily related to infant mortality (microfracture propagation) in the actuators of early correctors. These process problems have largely been eliminated and tests have been developed to qualify the actuators before installation in a wavefront corrector. As a result, reliability is greatly improved, and with the recent reductions in cost and fabrication time, it has become standard either to provide no spare or, in time-critical applications, to spare the entire corrector component. If a spare is available, replacement can usually be effected in an hour or less for any of the deformable mirror types currently available. Nonetheless, actuator reliability remains a problem for all manufacturers, and a large (⬎200 channel) deformable mirror is likely to experience at least one actuator failure. II. SPECIFYING YOUR DEFORMABLE MIRROR When specifying a wavefront corrector it is vitally important not to fall into the scientist’s trap and overspecify. Rather, one should become very familiar with possible offerings from the various suppliers both by perusing the literature and by direct contact. Then specify only those things which you must have, keeping to currently available designs if at all possible. Currently available components are described below. However, you should not go into detail and begin to specify parameters such as the actuator material. Table 1 provides a checklist for a reasonable set of specifications. Note that most of the internal features of the device are not specified. This is intentional since it allows the supplier to utilize his best practices and standard designs wherever possible, even if your requirement leads to a custom product. It is most likely that if you require a device having, say, 150 actuators, you would be better off buying the next larger standard size of corrector (243 actuators) and populating only the central 150 channels of electronics. A.

Actuator Arrays

Many different actuator arrays have been proposed, including various square, triangular, hexagonal, and modal formats. A detailed discussion of the implica-


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Table 1 Deformable Mirror Specification Number of actuators Array geometry (only if critical) Actuator spacing (use external optics to adjust beam size if possible) Minimum allowable stroke Uniformity of response Linearity/hysteresis requirements (if any) Response time Surface figure, nonoperating Corrected surface figure (specify wavefront to be corrected) Surface roughness or scratch/dig Mirror coating Mirror package size and weight constraints (if any) Mirror mounting (if required) Electronics package size, weight and format (as required) Cable lengths Environment (storage and operational) Temperature max. and min. Maximum relative humidity Minimum atmospheric pressure Thermal control requirements (if any)

tions of various actuator geometries can be found in Wild et al. (1994). Certain specific wavefront errors can be corrected using extremely simple devices. For example, both focus and spherical aberration can be removed using a single actuator and a properly formed facesheet. However, for random errors such as are produced by atmospheric turbulence, the only significant issue is the number of actuators, i.e., the number of degrees of freedom for correction. For any but very small numbers, this is independent of the array geometry. During the 1970s Itek developed a code which analyzed wavefronts and selected the actuator array which provided the best correction with the minimum number of actuators. Extensive testing showed that if the required number of degrees of freedom exceeded 40, the effect of actuator geometry was insignificant. Frequently triangular (hexagonal close packed) arrays are proposed rather than square arrays since they allow packing more actuators in a given area, but the additional complexity of fabrication does not generally warrant this approach, although triangular arrays are available from the suppliers. Also, modal arrays can have serious disadvantages. Perhaps the most famous failure of the modal array is the Hubble Space Telescope. The deformable primary had modal correction designed so that no spherical aberration could be



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introduced, and the manufacturing error in the system was pure spherical aberration! Current wavefront correctors are all built using square geometries which map easily to the square detector arrays of the wavefront sensors. This is particularly important if an analog reconstructor is to be used. With a digital reconstructor, mapping can be arbitrary since it is determined in software. However, no other pattern offers a significant advantage. B.

Stroke Requirements

Stroke requirements for a wavefront corrector should always be minimized. This means that system errors should be removed by passive correctors and that wavefront tilt should be corrected using a separate tip/tilt corrector. Such devices are fully developed and are available from a variety of sources, depending on the amount of tilt which must be removed and the required operating rate. An entirely functional tip/tilt corrector can be built using only a pair of miniature shaker actuators and a pair of capacitance sensors, mounted orthogonally on a rigid mirror having a central pivot. C.

Uniformity of Response

Allow a margin for gain variation. Variations in sensitivity of actuators can be significant. Even in the case of the Itek SELECT actuator (Thorburn and Kaplan, 1991) variations of ⫾10% can be anticipated. This can be thought of as a variation in system gain which can be compensated by having a higher than required frequency response, provided all actuators meet the minimum stroke requirement. D.

Response Time

In selecting a maximum operating frequency, two competing factors need to be addressed: on the one hand, a high system operating rate is often required and, on the other, mirror resonances and actuator heating degrade system performance. First, specify the lowest possible frequency limit. Then ensure that the lowest resonant frequency is much higher than that limit to provide adequate phase margin in the system control loop. The problem is not with the actuators; they can respond at rates limited only by the speed of sound in the material, typically 3 ⫻ 10 3 m/s and their RC time constant where the resistance is the resistance of the electrodes. Typically actuators can reach their commanded position in 0.1 ms or less. The resonance limitation is the result of the mechanical structure, a combination of the facesheet, base and support. Further, continuous driving of an actuator at high frequencies can cause significant heating due to hysteresis in


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the active material and I 2 R losses in the electrodes. For this reason actuators should not be driven for significant periods of time at frequencies above 10 kHz. In conventional atmospheric correction, heating is negligible.

E. Packaging and Mounting The major mechanical issues lie with the structure of the deformable mirror, the mounting and the electronics. Bimorph deformable mirrors tend to have very low frequency resonances since the fundamental frequency is related to the overall diameter of the thin plates. By comparison, the fundamental resonance of a corrector using a discrete array is determined by the disc diameter defined by twice the actuator spacing, and is very high, generally ⬎20 kHz. Mountings are also critical; very lightweight mounts will have low resonant frequencies. Also, if it is necessary to mount the device so that the edge is essentially free, the flexible potting will allow the corrector to move in its mount. In the discrete actuator mirror these problems are avoided by the use of heavy corrector bases and stiff mounts with thin, athermalized potting for support and containment. The system mounting should continue this approach or use other kinematic techniques which place no loads on the deformable structure.

F. Electronics The electronics generally limit the response of the wavefront corrector. For practical purposes the actuator is a pure capacitor; thus, the peak current is given by I p ⫽ 2πfCV

(1)

where f is the frequency, C the capacitance, and V the voltage needed to obtain the desired stroke. A typical actuator with a stroke of 4 µm will require approximately 100 mA at 100 V for operation at 1 kHz. Hence the peak power is 10 W. However, in average operation the driver needs to supply only 100–200 mW. This has led to some attempts to sequentially address the actuators (Kibblewhite et al., 1993). However, this approach has not won general acceptance, and most devices today which correct for atmospheric turbulence employ individual drivers. Note that Table 1 does not include a line for the input signal. It is far less costly to tailor the output of the wavefront computer to provide a signal matching the standard used by the corrector supplier (usually ⫾10 V analog) than it is to modify existing driver designs. Some drivers offer the option of a digital input.



G.

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Thermal Control Requirements

Wavefront correctors are designed to operate in the environment found from sea level to the highest astronomical sites. However, if there are specific environmental requirements to be met, these should be discussed in detail with the supplier. In general, there is no requirement to control the temperature of the wavefront corrector beyond that experienced in the operating environment. Power dissipation in the mirror package is quite low and environmental cooling of the package by radiation and convection is adequate. However, power dissipation in the drivers averages 0.1–1.0 W/actuator. Hence, the drivers should be placed some distance away from the optical path to avoid creating spurious turbulence. Most driver packages are forced convection cooled and may be placed more than 30 ft from the mirror package. Since the actuators have rather high capacitance and operate at low frequencies, they neither produce nor pick up significant electromagnetic interference (EMI). Also, if unshielded cable, such as ribbon cable is used, cable capacitance will not cause a significant voltage loss. The mirror coating should be highly reflective, particularly if the incident energy is high, as with a laser or when sunlight may illuminate the aperture. Typically the deformable mirror can withstand an absorbed heat load of 10 mW/ cm 2 with no cooling and no negative impact on the surface figure. If a cooling airflow through the actuator array of 10 m/s is provided, absorbed heat loads of 50 mW/cm 2 can be tolerated.

III. TYPES OF CORRECTORS—A HISTORICAL PERSPECTIVE The following is a description of the major types of deformable mirrors which have been developed in the past 25 years. Two device types, the monolithic piezoelectric mirror (Itek) and the pretensioned device (UTOS), are no longer available, although a number of these devices continue to be used in various laboratories. A.

Monolithic Devices

Monolithic devices have the advantage that the corrector element itself is very simple and, hence, inexpensive to fabricate. However, all monolithic devices developed to date have idiosyncrasies which limit their application to specific areas. 1. The Monolithic Piezoelectric Mirror (MPM) The first wavefront corrector to be fielded in an operational wavefront correction system was the Monolithic Piezoelectric Mirror (MPM) developed by Itek Corpo-


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Figure 4 The Monolithic Piezoelectric Mirror (From Ealey and Washeba, 1990).

ration in the early 1970s (Feinleib and Lipson, 1975). This was certainly not the first device used for wavefront correction, but all others to that time had been used in controlled laboratory experiments. The device had approximately 150 channels and was powered by high voltage drivers mounted directly on the telescope. A typical MPM structure is shown in Fig. 4. The MPM is fabricated using a solid block of low hysteresis PZT, usually PZT-8. These blocks are made for Navy sonar systems by a number of manufacturers and are available off the shelf. First the top electrode is removed. Then an array of holes with the spacing of the electrode pattern is drilled through the block using a high speed diamond tool. A larger tool creates the wells which form the electrodes. The diameter and depth of these wells are important since the response of a monolithic device is dependent on both the d 33 and d 15 coefficients of the piezoelectric tensor (Fig. 5): the deeper the well and the closer the spacing of wells, the greater the contribution due to the d 15 coefficient due to E a and E b in the figure. The d 15 coefficient produces a tilt, while the d 33 coefficient produces a simple elongation. This is the only device which has taken advantage of the d 15 coefficient in its operation. However, material stiffness and limitations on the applied field to avoid depoling the material set a practical depth limit of 5 mm and a well diameter of 60% of the spacing. The d 15 coefficient may contribute up to 30% of the total response of the device. Electrical connections are formed by coating the entire inside of each well with a conductor and attaching an insulated wire which passes through the hole and out the back of the device. A thin (typically 0.6–1.0 mm) Pyrex facesheet is bonded to the front of the device, polished and coated. The active device is athermally potted in a mounting ring for attachment to the optical sys-



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Figure 5 Both the electric field through the thickness of the piezoelectric element and the electric field between adjacent actuators contribute to the response of an MPM. E z provides the d 33 contribution, and E a and E b provide the d 15 contribution.

tem. Because of the high voltages required for these devices (⬎⫾1000 V), shielded cables must be used to minimize EMI. Unfortunately the capacitance of the individual actuators is small, so the cables must be kept short to avoid losses due to capacitive voltage division. This requires that the drivers be mounted adjacent to the corrector, an extremely undesirable turn of events. The device is very compact, easy to build, and, except for the electronics, extremely inexpensive. However, higher performance devices have replaced the MPM in all current applications, and the device is no longer commercially available. Typical performance parameters of an MPM are shown in Table 2.

Table 2 Performance Parameters of a Monolithic Piezoelectric Mirror Parameter Diameter Thickness Actuator array Number of actuators Actuator spacing Response time Sensitivity (surface) Operating voltage Maximum interactuator voltage Surface figure Surface roughness

Value 75 mm 13 mm square, 14 per line 156 (21–349 reported) 3.8 mm (2.5–10 mm reported) ⬍0.1 ms 0.4 nm/V ⫾3000V 1500V ⬍60 nm rms ⬍1 nm rms

Source: From Ealey and Washeba (1990).


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Figure 6 The LASERDOT bimorph mirror (BIM) is an exceedingly simple structure. (Based on Gosselin et al., 1993, Fig. 2).

2. Bimorph Curvature Correctors A bimorph mirror (BIM) has been developed by LASERDOT for use with curvature sensing wavefront sensors (Gaffard et al., 1994; Gosselin et al., 1993). The BIM is a very simple device fabricated by bonding together two plates of PZT which are poled in the same direction (Fig. 6). A common segmented electrode array is formed between the plates before bonding, and the top surface of the device is optically finished. This device uses the d 31 coefficient of the PZT: when a voltage is applied to an electrode, that portion of one plate shrinks in diameter while the corresponding part of the other plate grows. The resulting moment causes a bending of the entire plate centered on the activated electrode. Because of its structure, the bimorph mirror is limited to correcting low order errors which do not change rapidly. The device can only be supported at the edge; hence, its resonant frequency is that of a large, thin plate. Also, the dimension of the electrodes must be large with respect to the thickness of the plates to have a useful effect. Typically the minimum dimension of an electrode is four times the plate thickness, and larger relative dimensions are desirable. Also, the bending is more efficient if the plates are thinner, but that exacerbates the resonant frequency problem. This type of device does have two significant advantages: its performance matches very well with a curvature sensor, and, unlike other wavefront correctors it can be configured to remove tilt as well as make higher order corrections. However, for large aperture systems where there may be many cycles of wavefront error across the diameter (many r 0 ) this type of device fails to provide adequate spatial resolution. Major features of the BIM are given in Table 3. B. Discrete Actuator Devices with Continuous Surfaces The vast majority of deformable mirrors currently available use continuous facesheets with actuators rigidly attached to the back of the faceplate in a regular, square array. The stiff attachment between the base and facesheet is the major factor in determining the shape of the correction which results. Most devices



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Table 3 Performance Parameters of a Bimorph Mirror Parameter Diameter Thickness Actuator array Number of actuators Operating voltage Stroke Maximum tilt Hysteresis Resonant frequency Surface figure (corrected) Surface roughness

Value 60 mm 2 mm 1 central and 2 outer rings 7 in active area and 6 outside ⫾400 V ⫾4 µm/actuator, ⫾10 µm focus ⫾0.2 mrad ⬍6% 3.8 kHz 0.03 µm rms ⬍2 nm rms

Source: From Gaffard et al. (1994).

have an influence function (the shape of the deformation when a single actuator is activated) that looks vaguely Gaussian. The smoothness of the curve is the result of the uniform properties of the facesheet which must be thick enough not to sag when two adjacent actuators are both activated by the same amount, but to deform consistently. Typically, at a nearest neighbor actuator the amount of deflection will be 5–10% of the peak deflection. Between the nearest neighbor actuator and the next in line, the deflection will be in the opposite direction since the facesheet pivots on the nearest neighbor attachment point, but returns to zero by the time it reaches the next in line. A typical influence function is shown in Fig. 7. This localization of the deformation simplifies the operation of the wavefront computer significantly since there is no crosstalk between adjacent channels of the system. The stiffness of the actuators ensures that each actuator acts as an independent entity, with the commanded position being consistent irrespective of positions of the surrounding actuators. However, this structural stiffness comes at a price: it requires more power to drive a stiff actuator. 1. Pretensioned Devices An excellent example of the state-of-the-art pretensioned correctors built by UTOS can be found in Fig. 8—the HIPACE 293 deformable mirrors. Note that, unlike the UTOS device described earlier, the actuators in this device are not replaceable, indicative of the advances made in actuator reliability in just a few years. These mirrors are no longer available. The key design parameters of the HIPACE 293 mirrors are given in Table 4.


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Figure 7 The influence function of a discrete actuator mirror is localized to the immediate vicinity of the driven actuator.

Figure 8 The most advanced UTOS pretensioned deformable mirror is the HIPACE 293. (From Lillard and Schell, 1994)



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Table 4 Performance of the UTOS HIPACE 293 Correctors Parameter Clear aperture Actuators in clear aperture Actuator spacing Edge control actuators Operational temperature range Operational lifetime Surface figure Surface roughness Stroke Interactuator stroke Small signal bandwidth Full power bandwidth Coupling Driver amplifier voltage Driver amplifier current Actuator material Actuator capacitance Maximum hysteresis

Value 110 mm 241 6.5 mm 52 0–25°C ⬎10 8 full power cycles ⬍33 nm rms 1.7 nm rms 11.5 µm 10 µm 6.7 kHz 412 Hz ⬍2% at adjacent actuators 150 V 180 mA PZT 0.4 µF 13.2%

Source: From Lillard and Schell (1994).

It is perhaps significant that these mirrors, the last reported on by UTOS, are very similar, except for the pretensioning arrangement and details of the electrical connections, to current generation devices developed by LASERDOT and Xinetics. This would indicate a convergence of ideas toward an optimal design concept for wavefront correctors of this type, and is indicative of the maturity of the device concept. 2. Free-Floating Faceplate Devices The design approach taken by Itek and continued by M. Ealey when he spun off from the parent company to found Xinetics is to use the actuator to both push and pull a continuous reflective facesheet. Similar approaches have been used by both LASERDOT and the University of Chicago. It is well known that the ceramics used to make actuators are much stronger (by a factor of at least 2) in compression than they are in tension. This led to an early view of these materials as weak in tension. However, this was proven to be untrue provided that the device was designed for loads compatible with the tensile strength of the materials and steps were taken to eliminate microfractures from the ceramic. The major difference between the actuators for this type of device and the UTOS systems


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is that the actuators are larger in cross section than those used by UTOS in the pretensioned approach. a. Xinetics. Wavefront correctors developed by both Xinetics and Itek share many of the same features: a rigid base, a diffusion bonded multilayer actuator, a machined facesheet, and printed circuit cards for making electrical connections to the interior of the corrector. There are, however, a number of differences in detail between the devices, reflecting the more advanced work at Xinetics. This includes actuators bonded to circuit cards and installed as modules (Fig. 9), a much less labor intensive operation than the Itek approach where the actuator array was first bonded to the base, then the cards were inserted and wired. The Xinetics approach should, in theory, allow closer spacing of actuators, although this has not been done to date. Diffusion bonded actuators employed by Itek were developed in concert with the Materials Research Laboratory at Pennsylvania State University using electrostrictive lead magnesium niobate (PMN) (Aldrich et al., 1984). Xinetics originally used PZT actuators purchased from outside vendors, but has since developed its own piezoelectric formulations of PMN which should be entering full production in the near future (Acton et al., 1996). Actuator technology is discussed in more detail in Sec. V. A significant feature of this type of deformable mirror is the use of a guard

Figure 9 The Xinetics deformable mirror is currently available in several sizes. (From Ealey and Wellman, 1994)



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Table 5 Characteristics of Xinetics Deformable Mirrors Parameter Number of channels Actuator spacing Actuator material Total stroke Actuator coupling Hysteresis Actuator response uniformity Actuator linearity Drive voltage Actuator capacitance Control band Surface figure (unpowered) Surface finish Scratch/dig Operating temperature

Value 37, 97, 349 7 mm PZT 4 µm ⬍15% 10% nominal 15% nominal 5% nominal 100 V 180 nF 300 Hz 160 nm 2 nm rms 60/40 ⫺25 to ⫹50°C

Source: From Ealey and Wellman (1994).

ring of actuators around the active area. This guard ring supports the edge of the facesheet and ensures that the influence function of each actuator is identical. In addition, if the mirror is to be polished after bonding to the actuator array (the usual Itek approach) the guard ring provides needed support for the edge of the facesheet. While correctors having more than 2000 actuators have been built by Itek using this technology as early as 1986 (Ealey, 1993), currently available devices typically offer less than 500 channels of correction. Performance of typical Xinetics offerings is given in Table 5. b. LASERDOT. The LASERDOT deformable mirror with multilayered actuators (Fig. 10) can be traced to the heritage of an approach originally described by Itek (Everson et al., 1981). LASERDOT has improved on the basic concept by bonding the PZT layers forming the actuator stacks and the base into a single block, then machining out individual square actuators. This results in a much higher actuator density than achieved by Itek. The LASERDOT device also has an alumina base and silicon facesheet with lower thermal expansion than the Macor (trademark Corning Glass) base and Pyrex (trademark Corning Glass) facesheet of the Itek device. This should provide significantly greater thermal stability to the corrector. Performance characteristics of the LASERDOT device are given in Table 6.


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Figure 10 LASERDOT has designed several sizes of deformable mirrors for astronomical applications. (From Gaffard et al., 1994)

Table 6 Performance of the LASERDOT Stacked Array Corrector Parameter Clear aperture Actuators in clear aperture Actuator spacing Surface figure Surface roughness Stroke Interactuator stroke First resonant frequency Operating voltage Actuator capacitance Actuator material Number of layers in stack Maximum hysteresis

Value 66.5 mm 52 (88 and 249 also available) 9.5 mm ⬍30 nm rms after correction ⬍2 nm rms ⫾5 µm ⫾2.5 µm ⬎10 kHz ⫾400 V 20 nF PZT 40 ⬍6%

Source: From Gaffard et al. (1994).



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Figure 11 The tubular actuator in this design is commercially available at low cost.

c. University of Chicago. The University of Chicago (Kibblewhite et al., 1994) employed a tubular PZT actuator rather than the multilayer stacks employed by several other manufacturers as part of an effort to reduce costs (Fig. 11). While it is true that tubular actuators are less expensive that multilayer devices, they suffer from a lack of sensitivity which precluded their use in earlier DoD sponsored activities. The tubular actuator has the voltage applied between the inner and outer walls, and utilizes the d 31 coefficient in the piezoelectric matrix, which has only about one half the value of the d 33 coefficient. Hence, to compensate, the actuator must be longer and the wall thin to enhance performance. The response of such an actuator is given by ∆⫽

d 31 Vl d

where V is the applied voltage, l is the length of the tube and d is the wall thickness. Such devices generally require approximately three times the operating voltage-length product of the multilayer devices. Other departures from previous work in this design are the attachment of glass balls to the tops of the actuators to act as localized attachments and the bonding of the facesheet to the actuators without further finishing, using approximately 20% of the actuator stroke to flatten the mirror. The glass balls were


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apparently employed because it was believed that a machined facesheet was expensive. This is not true for standard designs when compared to the touch labor involved in mounting the balls, unless graduate students are available to perform the work. A machinist using either a CNC milling machine with a diamond head or a step-and-repeat ultrasonic cutter can produce a machined facesheet in very little time once the basic setup is available. Many commercial shops have this capability. However, the concept of prefinishing the facesheet and mounting it on the actuator array has been generally adopted by the commercial manufacturers with improvements to achieve surface figures ⬍160 nm p-v. If better passive surface figure is required, the corrector face may be polished after assembly. While this device has been reported extensively in the literature it is not known to have been made available commercially. d. ThermoTrex Corp. (TTC). TTC has offered two types of deformable mirrors for the astronomical community. The first is a segmented mirror using tubular actuators (Fig. 12), based on earlier designs for segmented devices (Cuellar et al., 1992; Hulburd and Sandler, 1990). However, whereas the earlier devices utilized three separate actuators to control each segment, the current generation

Figure 12 The ThermoTrex segmented mirrors represent an alternative approach to continuous facesheet mirrors. (From Cuellar et al., 1992)



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Figure 13 A single tube with three electrodes equally spaced around the circumference provides an elegant means for achieving tip/tilt and piston control in a segmented mirror. (Adapted from Cuellar et al., 1992)

use a single tube whose electrode is segmented into three equal radial sections to provide tip, tilt and piston (Fig. 13). The segments of the mirror are mounted on the top of the tube, which is in turn attached to a mounting support. Individual segments are fabricated to the same length to high tolerances and are assembled individually in an adjustable mount. The individual segments are, of course, replaceable. This device has one significant feature which adds complexity to the system compared with continuous facesheet devices. Final flattening of the segmented mirror requires a white light interferometer. This interferometer provides the necessary reference to avoid 2π ambiguities in the piston positions of the segments. To accomplish this a portion of the stroke must be used to adjust the segments in piston. Despite this drawback, this is the only wavefront corrector currently offered having replaceable actuators, and it should be considered carefully for applications in which that is a significant issue. Table 7 summarizes the features of these devices. ThermoTrex is also developing an adaptive secondary mirror in conjunction with the Steward Observatory (Bruns et al. 1995, 1996). This mirror is unique in that it represents the first use of voice coil (electromagnetic) actuators (Fig. 14) to perform high speed correction on a high density of points. Voice coil activated force actuators have been in use since the early 1970s for adaptive correction of large mirrors. This device overcomes the high heat dissipation of the inductive actuator by using a moving-magnet system and liquid cooling the coils, in an approach reminiscent of some early high energy laser mirrors. However, in this case the coolant only flows through the support structure, greatly increasing the stability of the system. Position is monitored using both capaci-


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Table 7 Performance of the ThermoTrex Segmented Corrector Parameter Clear aperture Segments Segment size Stroke reserved for phasing Stroke available for correction Linearity First resonant frequency Operating voltage Actuator material Maximum hysteresis

Value 35 ⫻ 35 mm 25 (169 and 512 also available) 7 ⫻ 7 mm 2 µm 4 µm ⬍⫾1% (closed loop) 2.5 kHz ⫾225 V PZT ⬍21%

Source: From Cuellar et al. (1992).

tance sensors and a wavefront sensor. Since the capacitance sensors provide an accurate indication of position, shape can be maintained without input from the wavefront sensor and the wavefront sensor bandwidth can be significantly reduced. Since the electromagnetic actuator responds to current rather than to voltage, current drivers are required and the currents are quite high in the 1.1 mH coils. The major drawbacks of a voice coil activated system are, first, that the forces which can be applied are small compared with piezoelectric devices and, second, that the current must flow continuously in order to maintain a fixed posi-

Figure 14 Outline of a generic moving magnet voice coil actuator. The ThermoTrex actuator is built without a closure plate, hence is less efficient, but has a bias magnet installed behind the coil.



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Table 8 Features of the ThermoTrex Adaptive Secondary Mirror Parameter Mirror diameter Actuator type Number of actuators Actuator spacing Force efficiency Average power dissipation Facesheet thickness Maximum operating frequency Stroke

Value 640 mm electromagnetic, moving magnet 320 32 mm 0.36 N/W ⬇1 W/actuator 2 mm 1 kHz not specified

Source: From Bruns et al. (1996).

tion. Hence, heating is a problem addressed by liquid cooling in this case. The major features of this concept are given in Table 8. This device is still being developed. It is not known at this time whether such devices will be made available commercially. Since the secondary mirror of each telescope tends to be different, it is likely that each adaptive secondary would have to be a custom design, albeit with significant commonality of components.

IV. A LOOK UNDER THE HOOD—CORRECTOR SPECIFICS All deformable mirrors with the exception of the monolithic mirrors share certain key features. These include a stiff, heavy base structure, an array of actuators and a flexible or segmented facesheet which has an optical surface and a reflective coating. While the details of each part may differ significantly depending upon the design preferences of the individual mirror manufacturer, the reasoning which drives these designs is common to all. Unfortunately it is not possible to provide detailed trade-offs among the various design approaches here since much of the needed information is considered highly proprietary by the manufacturers. It is possible, however, to make some general comparisons. A.

The Facesheet

The common features of all facesheets are that they must be flexible without taking a set, be sufficiently hard to be optically polished by one means or another,


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and be strong enough to withstand significant local deformation. This last feature is not required for segmented mirrors. These requirements limit the choice of materials sharply: While early devices for high energy lasers were made using metal facesheets, all uncooled devices (and some cooled devices) have used glass or silicon. Pyrex (trademark Corning Glass) was the material of choice in the early years, but has now been largely replaced by low expansion glasses such as ULE, produced by Corning Glass, Corning, NY, and having essentially zero thermal expansion at room temperature (α ⬇ 3 ⫻ 10⫺8 /°C). Another choice is single crystal silicon which takes an extremely high quality optical polish and is somewhat stronger and stiffer than glass, but has a higher thermal expansion coefficient, ⬇2.2 ⫻ 10⫺6 /°C. The facesheet is thin, approximately 1 mm, for a device with an actuator spacing less than 1 cm. Since the resonant frequency of the facesheet in normal (random) operation is determined by the resonant frequency of that portion of the facesheet defined by the nearest neighbors to the driven actuator, the facesheet resonance will always be well beyond the upper limit of the system bandwidth. The thickness represents a compromise between the difficulty of fabrication and the desire for maximum flexibility with acceptable stress levels. Even so, facesheets experience much higher stresses than are normally considered allowable for bulk materials. For example, optical designs using bulk ULE generally use a maximum stress of 1100 psi as the design limit, while ULE facesheets routinely use stresses in excess of 2500 psi in their designs. This is possible since the small size of the facesheet allows selection of high quality material free of inclusions, voids and other defects, and the facesheet is processed to remove all scratches and microfractures. The needed quality is achieved by polishing all surfaces or by stress relieving using chemical processes, usually etching in an acid solution. Since both glass and silicon must be etched in solutions containing hydrogen fluoride (HF), special handling and safety equipment are required for this process. Despite the high stresses on the facesheet there is no record of a facesheet breaking under normal operating conditions. The structure of the facesheet is more complex than simply a flat plate since the facesheet cannot be bonded directly to the actuators. For one thing the actuators generally occupy a large portion of the interactuator spacing, and a bond over that area would make it impossible to achieve a smooth deformation, which ideally requires contact at a single point. As a result a shaped pad separates the facesheet from the actuator. These are round at the contact with the facesheet and frequently have a taper so that the contact area to the facesheet is small while the contact to the actuator is larger. Various approaches can be used for this: Facesheets may be machined, using CNC processes with diamond tools, or ultrasonic cut to provide pads which are integral with the facesheet. This is a more desirable process than bonding



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pads to a thin facesheet since the bond to the actuator is now separated from the back surface of the facesheet. These bonds are made using high strength epoxies, and, while thin, they shrink during cure and frequently shrink further as they age. The stresses caused by the shrinkage or by curing at elevated temperatures can produce a pronounced dimple if the contact area is more than a small fraction of the faceplate thickness, a dimple which may appear well after the mirror has been completed. The effect is magnified if epoxy fillets form around the perifery of the contact during bonding. These fillets are desirable for bond strength, but greatly magnify the residual stresses. An integral pad allows the stress to be dissipated in the pad and not transferred to the facesheet. Details of these structures are the result of careful mechanical analysis and the design rules are closely held by the manufacturers. Alternative approaches include bonding balls to the facesheet (Kibblewhite et al., 1993) and attaching machined pieces which may include pretensioning springs (Lillard and Schell, 1994). For most astronomical applications the optical coatings required are straightforward metal or enhanced metal reflectors, e.g., aluminum or silver with one or two protective layers. These can be deposited using conventional thermal or electron beam evaporation. However, if a specialized coating, particularly one involving a multilayered dielectric is required, care must be taken to ensure that coating stress is not a problem. The stresses generated in the coating by the deformation of the facesheet are never a problem; the difficulty arises since the high internal coating stresses resulting from conventional evaporative processes will deform the thin facesheet. To avoid this problem only low stress coatings should be used. One method for reducing coating stress is to use ion assisted deposition. This method yields extremely durable and high quality coatings that can be deposited at room temperature and with the internal stresses in the coating layers virtually eliminated. B.

The Base

The base of the deformable mirror is both stiff and heavy. This is needed to supply a rigid structure for the actuators to push against. Also the base must have a high resonant frequency so it is not excited by the operation of the actuators. This is not nearly the problem that might be visualized since the operation of a single actuator is insufficient to drive a base into a drumhead resonance. Resonance can occur only when a significant portion of the mirror is being driven in phase at large amplitude. While some deformable mirrors use a base which is made from a different material than the facesheet, it is generally found that the greatest thermal stability is achieved by having the base and facesheet made from the same material. Using a metal base simplifies the mounting since the kinematic mounting points may be


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machined directly into the base. However, small differences in thermal expansion between the base and facesheet can produce significant facesheet distortion (often an uncorrectable high spatial frequency ripple) if the deformable mirror operates at a temperature different from the temperature at which it was manufactured, or is subjected to temperature extremes during shipment or storage. The severity of the problem depends purely on the differential thermal expansion. Both approaches are currently in use: LASERDOT uses a silicon facesheet and an alumina base (Gaffard et al., 1994), while Xinetics uses glass for both (Ealey and Wellman, 1994). The thickness of the base is determined by stiffness issues and by the requirements of the mounting approach. The base is usually at least 10 times as thick as the facesheet to provide a reference surface which appears infinitely stiff. The thick base also simplifies the mount design. If a one piece metal base with integral mounts is used the base must be sufficiently rigid to withstand any mounting torques without distorting. If the base is separate from the mounting ring, this requirement may be eased by the stress absorbing characteristics of the bonding layer, usually a silicone rubber. The base may be a very simple structure, e.g., a featureless cylinder. In this case the actuators are bonded to the base with an epoxy, and electrical leads are carried out to the edge in the gaps between the actuators. Alternatively, wiring may be passed through holes in the base directly to the back. In another variation, Itek used slotted bases which carried printed circuit cards in the slots. Xinetics mounts the actuators to circuit cards and these cartridges are subsequently mounted to the base (Ealey and Wellman, 1994). A more complex structure is required by the segmented mirrors. Here the differential thermal expansion is not a problem since the facesheet is segmented and is free to move as the base expands or contracts. However, each segment requires an intermediate structure for mounting the actuators, and these segment structures are mounted to the true base. The intermediate structure also carries the electrical connections to the rear of the mirror. Because of this the part count of a segmented mirror is significantly higher than that of a continuous facesheet device. The increased part count is a feature of all replaceable actuator devices. An important aspect of the base is its thermal conductivity. Since the base has a large thermal mass compared to any other part of the deformable mirror it is more subject to thermal gradients and requires much more time to reach thermal equilibrium than other parts of the structure. Most devices are built with bases which have relatively low thermal conductivity. As a result steps should be taken to ensure that temperature changes occur slowly. Manufacturers can provide specific information on both allowable rates of change of temperature and which thermal gradients are permissable. To avoid global distortion due to thermal gradients, the deformable mirror should always be brought to thermal equilibrium before it is put into service.



C.

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Actuators

The size of the actuator is determined by the stroke and force requirements of the deformable mirror. In general, actuator stresses can be minimized and structural stiffness optimized by maximizing the cross section of the actuator. However, in the case of multilayered actuators this may lead to excessive actuator capacitance and drive current. Hence, the actuator must be designed with an eye on the electronic driver design. The actuator length should be the minimum needed to achieve the required stroke at the planned operating voltage. Allowance must be made for variations in sensitivity among actuators, so some extra length must be provided, but too much stroke capability can lead to excess strain in the facesheet and facesheet failure. Hence, the facesheet is designed to withstand the stresses of the most responsive actuator in the array. It is desirable that the actuators in the array be as nearly uniform in properties as possible. With tubular PZT actuators reasonable uniformity can be obtained from the suppliers. Multilayer actuators are frequently fabricated as subelements which are selected and bonded into a complete actuator to reduce the variation in sensitivity (Ealey, 1993). This increases the part count and touch labor in actuator manufacture, but dramatically increases the actuator yield since sensitivity variations, particularly of electrostrictive PMN actuators, can exceed ⫾20% in normal manufacture. The variation is larger in electrostrictors than in piezoelectrics since variations in layer thickness translate into variations in response which are proportional to the square of the electric field. In the case of piezoelectric actuators, all actuators undergo an identical poling process which helps provide a uniform performance baseline. D.

The Package

The mounting of the deformable mirror uses methods very like those used for mounting any moderately large optical element. Generally the base of the mirror is potted into a stiff metal ring which contains the mounting points for the optical bench and which supports the covers for the internal structure. Small devices can simply be potted using continuous silicone rubber. Larger devices or those which may experience temperature extremes require athermalized potting which should be segmented as well to avoid hoop stresses. The simple formula for the thickness of an athermal potting is α m r m ⫽ α b r b ⫹ α p (r m ⫺ r b ) where r m ⫽ radius of the mount internal diameter r b ⫽ radius of the base


(2)

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α m ⫽ thermal expansion coefficient of the mount α b ⫽ thermal expansion coefficient of the base α p ⫽ thermal expansion coefficient of the potting This equation is solvable only for reasonable potting thicknesses (r m ⫺ r b ) if α p is large compared with the other thermal expansion coefficients; hence, silicone rubber with an expansion coefficient of 50–100 ⫻ 10 ⫺6 /°C is a preferred potting material. Silicone rubber is soft, and so does not transfer stresses well, but if the potting layer is too thick the deformable mirror will not sit stably in its mount. Deformable mirrors have two covers: a front cover which protects the actuators and the edge of the facesheet from dust and mechanical damage, and a rear cover to which the electrical connectors are usually mounted. These covers can be removed with due care to allow inspection of the internal structure or to dry the mirror if has been subjected to a condensing atmosphere. In general, these covers do not form a hermetic seal. The covers should remain in place unless a problem arises which warrants their removal. Deformable mirrors are also shipped with a facesheet cover which protects the coated surface from damage during handling. This cover should be in place whenever the device itself or other components mounted nearby are being handled. Also, if the environment is dusty, it it desirable to install this cover whenever the deformable mirror is not in use. Devices using piezoelectric actuators are shipped with grounding plugs on the electrical connector. These are necessary since PZT has a high thermal expansion and temperature changes induce high voltages on the actuators. (A change in length induces a voltage, just as an applied voltage induces a change in length.) These voltages can damage the mirror or impart an unpleasant shock to an unsuspecting operator. Hence, grounding plugs should be installed whenever the deformable mirror is not connected to the electronics. Electrostrictive devices do not exhibit this reverse effect and may not be supplied with ground plugs, although they are desirable for ensuring that all actuators are discharged when undergoing testing which does not involve integration with the driver electronics. Some devices are provided with an extra cover with a hole located over each actuator. This cover assists in achieving the proper alignment of the deformable mirror with the remainder of the adaptive optics system. If this cover is offered as an option, it is well worth the minor cost of obtaining one.

E. Spares and Repairs Despite the ease of access to the interior of a deformable mirror, any attempt to make repairs should be approached with the utmost caution. Components in the driver electronics can generally be replaced with impunity using only such repair equipment as is available in any well-equipped electronics laboratory. However,



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the deformable mirror itself is another story. While it is reasonable to remove covers to attempt to ascertain the source of a problem, it is inadvisable to attempt repairs without specific instructions from the manufacturer. In general, repairs, even replacement of replaceable actuators, require special tooling and fixtures and follow specific procedures which are usually the province of specially trained staff. Therefore, it is strongly urged that a device requiring repair be returned to the manufacturer. At the very least, poking around the innards of a device will invalidate the warranty if the device is relatively new. Worse, without specific knowledge of how to proceed, you are more likely to do damage than to effect a repair. It should also be recognized that devices with nonreplaceable actuators usually require significant rebuilding for any but the simplest repairs, e.g., recoating the mirror or replacing the electrical connector. These repairs cannot be performed in the field. This logistics issue must be considered as a part of the procurement plan. If 100% availability is required, then the procurement of a spare mirror and spare driver boards should be given serious consideration. If downtime is permitted, then this may not be needed, or perhaps a few electronic components might be spared. As a general rule, electronics can be repaired in less than a day, but repairs to the deformable mirror itself can require upwards of 90 days, depending upon the work required. If new actuators must be built, the repair cycle can be as long as the original procurement cycle. For this reason, some purchasers in the past have spared a set of actuators, thus keeping them available for a future major rebuild.

V.

ACTUATOR TECHNOLOGIES

Actuators for modern deformable mirrors generally have one of two structures: tubular, as in the case of the devices described by Kibblewhite et al. (1993) and Cuellar et al. (1992), or multilayered, the common commercially available low voltage actuator. The one exception is the bimorph, mirror which is discussed separately. Most piezoelectric and electrostrictive actuators available to date are composed of ceramics—small, randomly oriented grains which are fired together at high temperature to form a solid block having nearly the density of the bulk material. However, a novel process for growing single crystal materials has recently been described (Park and Shrout, 1997). These materials have three to five times the sensitivity of conventional ceramics and can withstand much higher electric fields, giving them a total strain capability approaching 1.5%, 10 times that of conventional ceramics. These unique materials recently became available from TRS Ceramics, Inc., State College, PA.


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A. Material Selection The key issue in material selection is whether to specify piezoelectric or electrostrictive materials. Piezoelectric materials have significant hysteresis, which is virtually absent in an electrostrictor. However, the temperature dependence of the electrostrictive sensitivity is high, typically 1%/°C for lead magnesium niobate (PMN), the leading electrostrictor, an order of magnitude more than that of typical piezoelectrics. This also limits the operating temperature range of the electrostrictor. The usual room temperature electrostricitve PMN compositions can only be used between 0 and 40°C, while a high quality piezoelectric will operate from ⫺100 to ⫹80°C. The electrostrictors were developed for military applications which required a ‘‘go to’’ capability in the wavefront correctors. This open loop operation could only be achieved by having essentially zero hysteresis. Current applications with closed loop systems are generally better served by piezoelectrics such as lead zirconate titanate (PZT), lead zinc niobate (PZN), and the piezoelectric compositions of lead magnesium niobate (PMN) since they have a response which is linear with respect to the applied voltage, greatly simplifying the control problem. Table 9 shows a comparison of typical properties of piezoelectric and electrostrictive materials reported in the literature. Electrostrictors can have sensitivities much higher than piezoelectrics and theoretically have a response which is quadratic with respect to the applied electric field, but this is only strictly true at low fields. At higher fields dielectric saturation occurs and the response becomes linear or even sublinear. Also because of the square law dependence, an electrostrictive actuator moves in the same direction for either direction of applied field. Essentially an electrostrictive actuator can only push while a piezoelectric material can both push and pull. To overcome this problem, electrostrictors are normally biased to the midpoint of their stroke. This has the added advantage of limiting nonlinear operation to the extremes of the response curve, and if the actuator has adequate stroke margin, the nonlinear regions can be avoided altogether. For that reason the operating voltage

Table 9 Comparison of the Properties of Piezoelectric and Electrostrictive Materials Property Material Sensitivity Linearity Hysteresis Dielectric constant Operating temperature Range

Piezoelectric

Electrostrictor

PZT 0.2–0.4 nm/V/layer ⬍3% for fields less than 0.5 Hc 5–20% 1000–2000 ⫺100 to ⫹80°C

PMN 0.2–1.0 nm/V/layer ⬍5% except at extremes ⬍3% 10,000–35,000 0 to 40°C



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of a corrector using electrostrictive actuators might be specified as 20 to 120 V rather than ⫾100 V. Another major difference between piezoelectric and electrostrictive materials is their dielectric constants. Both have nominally high dielectric constants and thus apply large forces per unit area. This is important in enabling the actuator to bend a rather stiff facesheet. However, while the typical dielectric constant of PZT is 1100, that for PMN is 30,000! In effect, PMN pushes 30 times harder, and in fact stress induced changes in sensitivity are unknown in all practical PMN applications, but the material also requires 30 times the current to charge in a given period of time. This is not a significant problem for operation at low frequencies (⬍1 kHz), but can become a major factor at higher frequencies where the current requirement can tax electronic designs severely.

B.

Tubular Actuators

Tubular actuators (Fig. 15) are always made of piezoelectric materials since a thick (typically 0.5–1 mm) wall is required for structural support. The electrostrictors require thin layers, typically of the order of 100 µm, since their response is determined by the electric field rather than the applied voltage. Theoretically it is possible to build a multilayered tubular actuator using technology available from the capacitor industry, but these have never been demonstrated. Such actuators might have higher strength and reliability than multilayered stacks since there are no bonds between dissimilar materials placed under

Figure 15 Tubular actuators are generally longer than they are wide and have the minimum wall thickness consistent with required mechanical strength. The external electrode should always be grounded for electrical safety.


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stress during activation. However, such actuators would have to be built up one at a time, an expensive process. The typical tubular actuator is made from a piezoelectric material such as PZT. These actuators can be purchased from a number of vendors. Electrodes are plated on the inside and outside walls of the tube and the material is poled radially. Thus the activation is provided by the d 31 coefficient. A mechanical advantage is realized since the tube may be long compared to the wall thickness: d 31 Vl d ⫽ d 31 El

∆l ⫽

(3) (4)

A PZT-8 composition will have d 31 ⬇ ⫺110 ⫻ 10 ⫺12 m/V so an actuator with a 1 mm wall providing 4 µm at ⫾1000 V would have to be 1.8 cm long. This electric field, 104 V/cm, provides a practical limit for many PZT compositions if excessive hysteresis is to be avoided. Using that benchmark the actuator length for any desired stroke is simply l min ⫽

∆l 10 4 d 31

(5)

The straightforward way to design an actuator is to determine the cross section required for mechanical strength, and from that calculate the operating voltage using the limiting field for the tubular actuator: V max ⫽ E max t ⫽ 10 4 d

(6) (7)

Then determine the length using Eq. (5). Note that defining the upper limit on the allowed electric field reduces the analysis to a simple determination of stress in the material, i.e., the actuator cross section. Given that information and the stroke required, the design of the actuator is fully determined. Tubular actuators are comparatively easy to build and have been a standard product of the major PZT suppliers for many years. Greenware rods are cast or extruded, then fired to form the ceramic. Greenware is the term used by ceramists for the mixture of ceramic powder and binder which is formed into the basic shape of the part. Greenware is soft and can be shaped or machined easily. However, the part is shaped to its final dimensions after firing because the greenware shrinks during firing and densification. After firing, the rods are cut to length, centerless ground to the outside diameter and cored to the inner diameter. Subsequently electrodes are applied to the inner and outer walls and the cylinders are



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poled in an oil bath at elevated temperature. Poling may cause minor changes in dimension, but usually not enough to cause difficulty in fabrication of the mirror. C.

Multilayered Stacks

Stacked actuators are more sensitive per unit length than tubular actuators since they employ the d 33 coefficient, which is larger than d 31. A typical PZT will have d 33 ⬇ 230 ⫻ 10⫺12 m/V Thus a given stroke can be obtained with an actuator somewhat less than half the length of the minimum for a tube. Further, that length can be divided into an arbitrary number of layers, limited only by the exigencies of fabrication, to reduce the voltage. Consider once again an actuator having 4 µm of stroke. This will be achieved in a length of l min ⫽

∆l ⫽ 0.8 cm d 33 E max

(8)

using the same limitation on the electric field used above. This actuator design offers flexibility in choosing the operating voltage since the layer thickness and, hence, the number of layers, n, can be varied: V app ⫽

E max l min n

(9)

Two trade-offs limit the number of layers and hence provide a practical minimum to the operating voltage. First, the manufacturing process sets practical limits. A process based on tape casting will have an effective minimum layer thickness of approximately 0.1 mm due to required handling of the tapes. Other processes where the multilayer is built up by noncontact means, e.g., spray, dip or waterfall coating, can have layers as thin as 0.01 mm. These very thin layers, however, impose a severe burden on the drive electronics. The total energy required to obtain a given stroke with a particular actuator cross section and length is a constant: S ⫽ CV ⫽ 8.8 ⫻ 10⫺14

(10) εAV l

(11)

where ε is the dielectric constant of the material and A is the cross-sectional area of the actuator expressed in cm 2. Hence, when the actuator is subdivided into n layers, the applied voltage on each layer is reduced by a factor of n, but the actuator capacitance increases by a factor of n, and the current required to charge the actuator in a particular time increases by the same factor. For this reason,


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most deformable mirrors for atmospheric compensation operate in the vicinity of 100–400 V with peak currents of approximately 1–100 mA, which seems to be the point of maximum electronic cost efficiency. This maximum is not well defined, and is subject to considerable variation among both suppliers of components and the users. However, actuator designs developed by the component manufacturers generally account for these factors and provide designed-in performance margins as well. For example, a typical actuator will be 10% longer and have additional layers to provide a margin on the achievable stroke. Today, stacked actuators can be obtained from a limited number of vendors, but the actuators used in deformable mirrors are often manufactured by the corrector fabricator (Ealey and Davis, 1990). This is because the small size and specific performance requirements are generally not available in the open marketplace. Some manufacturers have had special relationships with material suppliers to obtain actuators meeting specific design requirements in the small quantities needed for wavefront correctors, but those designs are proprietary and are not available to the public. Multilayered actuators are usually round to eliminate stress concentrations at the corners. These arise since the electrodes do not generally cover the entire area of the actuator. Typical electrode patterns for square and round actuators are shown in Fig. 16. These patterns arise from the need for simple patterns during screen printing of the electrodes and subsequent alignment of the layers. In both cases a single screen produces the pattern for all layers, and substrate

Figure 16 Electrodes cover most of the surface of the actuator layer, but a small portion remains uncoated so that contacts can be formed directly on the outside surface of the actuator.



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Figure 17 The alternation of the electrode patterns provides the means by which the actuators are connected electrically in parallel but operate in series mechanically to provide high stroke at low voltage.

rotation provides the necessary variation. These patterns allow stripe contacts to be applied to the actuators which contact only alternate electrodes (Fig. 17). Alternatively, one could electrode the entire surface area and then print an insulator over every other electrode before applying the contact (Fig. 18). Such actuators have been built, but while they eliminate the problem of stress concentrations in the actuator, that benefit seems to be outweighed by the cost of the additional process steps and the reduced yield.

Figure 18 Actuator with external insulating stripes.


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D. Actuator Manufacture of Multilayered Stacks Actuator manufacture is a nontrivial process bordering on the arcane, and, in the case of multilayered stacks, is certainly the most critical part of the process of building the wavefront corrector. A great deal has been written about the preparation of raw materials, with modern processes largely developed at the Materials Research Laboratory at Pennsylvania State University by L. E. Cross, T. R. Shrout, and S. J. Jang, and their academic and industrial partners. The actual compositions, particle size distributions and firing processes used are proprietary and are closely held by the manufacturers, but one form of the general process can be outlined here. Initially, purified oxide powders of the various materials are mixed and cofired to form the powder of the active material. This may involve multiple firing and grinding/mixing steps and the formation of precursor materials. For example, the usual process for making lead magnesium niobate, a popular electrostrictive material, is to first form magnesium niobate (columbite), then mix the columbite with lead oxide and the dopants used to fix the actuator properties, then refire the mixture (Swartz and Shrout, 1982). The two step process is common since lead oxides sublime at relatively low temperatures and may be lost from the mix even if fired in a nominally closed container. The raw powder in final form is ground and sieved for subsequent use. For economic reasons and to ensure uniformity of properties, powders are generally made in rather large batches and stored for later use. Special handling processes are required to meet environmental requirements for the processing of lead containing materials. Fortunately, the finished actuators are extremely inert and may be handled without employing the precautions required during manufacture. To form the layers, the powder is mixed with a water soluble binder and dispersants to form a slurry which is cast as a long tape, usually several inches wide by several feet long. The simplest tape casting process uses a doctor blade, although spray and ‘‘waterfall’’ casting can be employed as well. The doctor blade is a bar held a fixed distance above the substrate upon which the slurry is deposited. The substrate may be either a glass plate or a Mylar film which passes under the blade at a constant rate, drawing with it the slurry which wets the substrate. The slurry is contained in a slurry pool behind the doctor blade. The thickness of the resulting tape is controlled by the gap in the doctor blade, and the solid content and the viscosity of the slurry. Tapes are dried under heat lamps or, for small quantities, they may be air dried. The dried tapes are cut to size and the electrode pattern is applied using standard screen printing processes. The composition of the ink is dependent upon the temperature of subsequent firing. Printed tape sections are stacked in a form and pressed with mild heat to fuse the binder, making a solid block from which many actuators can be cut. Round actuators are core drilled from this



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‘‘greenware’’ block and are fired in a two step process, first in air to remove the binder, then in a lead atmosphere to fuse the powder into a mechanically strong block having a density approaching theoretical limits. Porosity which remains in the actuators after this firing process can become a source of stress fractures and actuator failure during subsequent use; hence, this step is critical to achieving high yields. During this process the actuator will shrink 15% to 30% as it densifies. There are many other variations of the process, depending upon the manufacturer’s preferences. For example, in waterfall casting the layers can be built directly on one another, avoiding the need to handle and process individual layers. Subsequently contacts are applied to the actuator surfaces, often by screen printing, and if the actuator is piezoelectric it is placed in an oil bath and poled at elevated temperatures and under very high electric fields, near the dielectric breakdown of the material. Poling profiles (temperature, time, and field strength) vary widely for different material compositions. After completion the actuator segments are measured and critical parameters, e.g., capacitance, loss tangent, and sensitivity are recorded. Since the sensitivity of segments may vary rather widely, segments are often selected and bonded together to form a group of actuators with more nearly uniform stroke. This painstaking process of selecting and matching high and low sensitivity segments greatly increases the yield of the process and is justified on that basis. Historically, actuator failure within the finished wavefront corrector has been the single most important reliability issue. These failures appear to result from microcrack propagation which occurs whenever the material is under tension, and is the reason the UTOS wavefront correctors were built with mechanical restraints which provided the ‘‘pull.’’ This is difficult to achieve in modern high density mirrors where the active material may occupy half the volume between the faceplate and the base structure. Unfortunately, since the actuators are a rigid ceramic, they are not subject to the same kinds of failure and failure prevention as are metal parts. While most failures occur very early in the life of the actuator, a failure can occur at any time and at any level of stress, hence the process is not entirely controllable. Numerous attempts have been made to determine means for testing actuators to ensure their performance once they are installed in a wavefront corrector. These include acoustic testing, a common test for ceramic capacitors, operation for various times at both high and low stress levels, and various thermal and electromechanical tests, but none have been found to be completely trustworthy. A part of the problem is almost assuredly that the actuators must be handled to be installed in the deformable mirror, and a minor scratch during installation can result in an actuator failure at a later date. At present it appears that a small percentage of actuators can be expected to fail in any large multilayered actuator deformable mirror. Tubular actuators which lack the multiple boundaries between dissimilar


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materials appear to have greater reliability, as do the actuators for a curvature sensing mirror. Without further data available, one may conclude that in actuators, structural simplicity and reliability go hand in hand, but at the cost of reduced performance.

VI. ELECTRONICS The driver electronics for the deformable mirror system are the most complex, and by far the most expensive part of the deformable mirror system, typically accounting for two thirds of the total system cost. This apparent contradiction in an era of low cost computers is the simple result of low volume. No deformable mirror system or family of systems has ever been accepted as a standard and purchased in sufficient quantity to justify the nonrecurring costs of low cost production. Further, drivers which seem to be simple in concept become both complex and costly as the ‘‘accessories’’ are added. A driver for a single channel of a deformable mirror can consist of less than a dozen components, including a high voltage operational amplifier. But look at what else is typically required: There will be a feedback loop which limits the available current and shuts the driver down in case the actuator fails as a short circuit. This prevents damaging the mirror by power dissipation in the actuator. There will be a voltage divider, frequently with an A/D converter on the output to provide the main system computer with moment to moment information on the status of each corrector channel. Today analog inputs are generally insufficient since most wavefront computers are now digital, so each channel has its own D/A converter for the input. Frequently gain and offset calibrations are supplied, although in digital systems these are best provided as a look-up table and corrections made in software. All these additions serve the useful purposes of increasing system safety, providing desired information and making it more convenient to interface the corrector with the rest of the adaptive optics system, but each addition increases part count, real estate requirements, and cost while reducing system reliability. In an extreme example, the first 2000 channel mirror built had approximately 125 electronic components per control channel just for the driver, a total part count of more than 250,000! These drivers were incredibly safe, but were so complex as to be unreliable. By far the best approach is to perform all calibration functions digitally and to keep other requirements to an absolute minimum. Further, every attempt should be made to use existing hardware designs. Only on rare occasions does customizing justify the increased nonrecurring costs. Recently some driver designs have been hybridized. This significantly reduces the part count on the board, but does so by hiding many of the components inside an expensive hybrid which must be replaced should any of the internal



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components fail. However, there are cost savings associated with this approach, once the nonrecurring engineering (NRE) has been amortized. When one thinks of hybrid circuits, the first vision is of greatly reduced size. True, the circuit may be smaller, but in a wavefront corrector driver this may not lead to a smaller electronics package. The actuator is a low loss capacitor which must be charged and discharged at the operating rate, typically up to 1000 Hz. The required current is given by I peak ⫽ 2πfCV

(12)

where f is the frequency, C the capacitance of the actuator and V the voltage needed to drive the actuator to its intended position. The peak power consumption is thus P peak ⫽ √2V max I peak

(13)

For typical systems, I peak ranges from 1 to 100 mA for an actuator with a maximum operation voltage of 100–400 V. Thus each driver is a linear power amplifier with a peak rating of roughly 1–10 W per channel. Certainly every channel is not operating at its full rating all the time. In fact, a reasonable rule of thumb is that the average power dissipation in a driver package can be obtained on the assumption that each channel is operating at one third the maximum voltage and one fourth the frequency. This convenient rule applies whenever wavefront data is provided continuously to the corrector. Pulsed systems which collect a wavefront and then apply the data simultaneously to all actuators necessarily consume more power, since they operate at the maximum frequency at all times, and are generally required to reach their stable operating point in a small fraction of the time between pulses. Since the actuator is a capacitor, all this power must be dissipated within the driver amplifier. This represents a not insignificant design problem if space is constrained, since heat sinks and cooling systems, whether conductive or convective, require significant real estate. Typically the drivers for a 100 channel mirror will be installed in a 24–30-in-high rack, including power supplies and the control panel. This allows adequate space for forced-air cooling of all components. One issue which has received great, and perhaps misguided, emphasis in electronic driver design is actuator hysteresis (Kibblewhite et al., 1994). In fact, hysteresis is a problem only in those rare systems which operate in a fully open loop mode where the actuators are told to ‘‘go to’’ a specific position and that position is never monitored by the system. In general, hysteresis can be treated as a loss of system gain, and simply increasing the system gain a few percent will compensate, without having any negative effects on overall system stability. However, should it be desirable to have a highly linear system, a simple modification of the voltage control loop will provide the necessary input. Hysteresis in


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an actuator means simply that the length of the actuator is not a linear function of the voltage applied due to remnant polarization from its previous activation history. However, in all piezoelectric materials, the actuator length is linearly proportional to the charge on the actuator. Hence, rather than placing a voltage divider in parallel with the actuator to monitor the applied voltage, one can simply place a low loss capacitor which is much larger than the actuator capacitance in series with the actuator and monitor the voltage across that capacitor, since the charge on the two capacitors is necessarily equal. To minimize divider effects, the passive capacitor should have at least 10⫻ the capacitance of the actuator, and preferably 100⫻. If the entire array is driven to a reference point periodically to provide a calibration, the system will remain linearized to within 3–5%. This approach is generally necessary only if the hysteresis is very large. A second type of nonlinearity arises if electrostrictive actuators are used. At low voltages the response is quadratic, and at high voltages the dielectric constant saturates, producing a butterfly shaped curve (Fig. 19) which may or may not have significant hysteresis, depending upon the composition of the actuator material. Generally this nonlinearity can be dealt with by operating the actuator around a bias point which places the zero position of the corrector near the inflection point of the response curve, and providing a modest system gain margin to allow for the sublinear response at the extreme ends of the response range. Typically such an actuator might operate in a range of 10–120 V with a 60 V bias. Another often voiced false hope is the possibility of multiplexing actuators. The difficulty is that the power available to drive 10 multiplexed actuators is exactly the same as that required to drive the same number of actuators individually. Thus, if each actuator requires 100 mA, then the amplifier driving 10 multiplexed actuators must supply 1 A so that each actuator can be charged in one tenth of the time. There is no saving in power. However, multiplexing may be effective in very slow systems which cor-

Figure 19 Due to their quadratic response and dielectric saturation, electrostrictors exhibit a response in the shape of a butterfly’s wings.



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rect for errors such as thermally induced optical errors. There currents remain small and the saving in part count is significant. In a multiplexed system, each actuator must be supplied with a high impedance switch to hold the charge on the actuator between addressing times. Presuming that the switch, usually a FET, has a capacitance much smaller than the actuator, simple x-y addressing can be used. In this case the columns of actuators are accessed in sequence and the drivers (one for each row) provide a voltage signal for the actuators in that column. In this way the number of drivers required for an n ⫻ n array is reduced from n 2 to n. The limitation on the size of the array that can be addressed in this way depends upon the resistance of the switch, the capacitance of the actuator and the requirement for accuracy of surface figure maintenance. The voltage on the actuator will decay with an RC time constant characteristic of the actuator capacitance and the switch impedance. If a shape drift of x can be tolerated, then the refresh time for the array can be obtained from 1 ⫺ x ⫽ e ⫺t/RC

(14)

where t is the desired refresh time. If the drivers provide sufficient power to charge an actuator fully in t′ s, then the maximum number of rows which can be addressed by this approach is N⫽

t t′

(15)

For arrays requiring a high degree of accuracy, one can overdrive the actuator by half the voltage that it will decay during the refresh time, thereby reducing the average figure error due to intermittent actuation by a factor of approximately 2. The high cost of high power components generally obviates any part count advantage obtained by multiplexing a system designed for atmospheric compensation. If piezoelectric actuators having comparatively low actuator capacitance are used, e.g., short, tubular PZT actuators, then one might consider multiplexing (Kibblewhite et al., 1993).

VII. CONCLUSION The technology of wavefront sensors is evolving even though the support for new designs and especially new concepts is rather limited. Innovations will change many things in the next few years, hopefully leading to a more capable, more reliable, less expensive, easier to use product. Other applications will be found for the technology which may change our way of thinking about these devices. This ‘‘conventional’’ type of corrector may be replaced by MEMS or other emerging technologies. Nonetheless, the technologies described here will


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almost certainly retain a place in the sun for many years. The original MPM has been in use on Mount Haleakela for more than 15 years, despite the advent of much more advanced devices. There are sufficient discrete actuator devices in existence, most still in reasonable working condition, that their usage for the foreseeable future is ensured, and more are being purchased and installed in new adaptive optics systems every year. This is a mature technology, though certainly not a dead one. New innovations are still being announced regularly. For example, T. Shrout, at the Materials Research Laboratory at Penn State, recently described a new actuator material with 10 times the sensitivity of existing materials. But most technologies have a life cycle of 20–30 years, and current wavefront corrector technologies are roughly 15 years old. So don’t be surprised when that innovative upstart shoulders its way in and says, ‘‘Move over, Grandpa.’’ The future will arrive, but it is probably too far off to be worth waiting for.

VIII. A LAST WORD... So now you have your very own wavefront corrector. You have spent endless hours anguishing over the design of your system. You have read everything you can find about correctors, talked to all the suppliers, sifted their conflicting claims, and finally made a choice. You have filled out the ream of paperwork your organization requires before making any major purchase, argued your way through the maze of approval authorities, and waited for your turn in the budget queue. You have happily watched Procurement place the order, sweated out the delivery delays, and finally the great day dawns. No parent ever waited for a first child with more anxiety, and the temptation to just turn the thing on and try it out is overpowering. Don’t. Go at it slowly. You have as much to learn about your new toy as does a new parent about that first baby. First make a checklist for your setup and test procedure, and assemble and check out the test equipment. This will help pass the time while you are awaiting delivery and give you an excuse to call the supplier occasionally to find out what is going on while you ask about this or that operating detail. Look it over carefully, read the instruction manuals, which are, unfortunately, all too often not very instructive, and compare the part with the drawings. Will it fit where you plan to mount it? Is everything you need included with the delivery including cables and connectors? Do the bolt holes match your mount? If possible, have the supplier set up the equipment for you the first time and run an acceptance test. Watch carefully and use the operating manual as a checklist. Does the supplier do everything as described in the manual? If not, why not? Note changes and options for procedures for future reference. You won’t remember all this if you don’t write it down. Pick up the equipment and get used to the weight and handling characteristics. The last thing you want to



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do is drop an item that may take a year or more to replace. Make sure your test setup, preferably an interferometer with a real time readout (eyeballs are good for a start), is complete and safe...nothing to drop on the corrector, all power clearly identified, connectors keyed so they cannot be hooked up incorrectly, input and output sources tested and operating within the bounds specified, the mirror’s passive surface figure consistent with the supplier’s delivery data, all the electronic cards properly seated and no loose connections. Without connecting the corrector if possible, check out the electronics. Are the gains correct? Are the output voltages within limits specified as safe for the corrector? Do the monitor points provide good data? Hook up a dummy load. Do the drivers produce the expected voltages and currents at the expected frequencies for the inputs provided? Check the corrector with a capacitance meter or similar low voltage test hardware to be certain the internal wiring is in good condition and each actuator is connected electrically. Now carefully connect everything and turn it on step by step. No smoke or sparks? You just passed the first critical test. Next put a small signal on one actuator. Does it move? Yes? Was it the one you expected to see move? Does the influence function have the correct shape? Try another, and another until you have checked them all. Make sure they move both up and down. So far so good. Now repeat the supplier’s acceptance test using DC signals. Is everything consistent? Once everything checks out at this level, you will probably want to move to the real setup. Disconnect everything and replace the protective covers. You don’t want to trip over a dangling cable or drop a screwdriver at the wrong moment, a very expensive mistake. Besides hooking it back up again will give you confidence in your ability to handle the hardware. Note: Most handling should be a two person operation, one to hold the part and one to insert the mounting bolts. Install the corrector in the system mount. Does everything fit? If it doesn’t, fix it now! Reconnect everything and run individual actuator checks. Now comes the acid test. Using the wavefront computer and a laser, tell the mirror to flatten itself. Does it do it? How well? What do the residuals look like? Save this data (and all the previous data) as a reference in case you have problems later. Now give the system a passive wavefront error consistent with the type of error you expect to encounter. Does the mirror give a good correction? If not, is the problem with the mirror, the sensor/computer or (hope not) did you specify things wrong? Once you get through all this, you are finally ready to open the dome and see if you can correct a real wavefront. I hope it all works perfectly. Good luck!

REFERENCES D. S. Acton, P. Stomski, P. Wizinowich, J. Maute, T. Gregory, M. Ealey, and T. Price, ‘‘Keck Adaptive Optics: Test of a Deformable Mirror in a Freezing Environment,’’


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Adaptive Optics, Vol. 13, 1996 Technical Digest Series, Postconference Edition, July 8–12, 1996, Maui, Hawaii, sponsored by the Optical Society of America. R. E. Aldrich, S. M. Daigneault, C. E. Wheeler, and C. DeLuca, ‘‘Low Voltage Kiloactuator Deformable Mirror Program,’’ RADC-TR-84-168 (Aug. 1984). D. Bruns, T. Barrett, G. Grusa, R. Biasi, and D. Gallieni, ‘‘Adaptive Secondary Development,’’ Adaptive Optics, 1996 Technical Digest Series, Vol. 13, sponsored and managed by the Optical Society of America, p. 302. D. G. Bruns, D. G. Sandler, B. Martin, and G. Grusa, ‘‘Design and Prototype Tests of an Adaptive Secondary Mirror for the New 6.5 m Single Mirror MMT,’’ Proc. SPIE, 2534 (1995), p. 130. L. Cuellar, R. Arnold, and B. Cram, ‘‘Broadband Segmented Mirrors for Astronomical Adaptive Optics,’’ DoD Conference on Adaptive Optics, Kirtland AFB, NM (1992), not formally published, p. 809. M. A. Ealey, ‘‘Low Voltage SELECT Deformable Mirrors,’’ Proc. SPIE, 1920 (1993), p. 91. M. A. Ealey and P. A. Davis, ‘‘Standard SELECT Electrostrictive Lead Magnesium Niobate Actuators for Active and Adaptive Optical Systems,’’ Opt. Eng., 29, No. 11 (1990), p. 1373. M. A. Ealey and J. F. Washeba, ‘‘Continuous Facesheet Low Voltage Deformable Mirrors,’’ Opt. Eng., 29, No. 10 (1990), p. 1191. M. A. Ealey and J. A. Wellman, ‘‘Low Cost Deformable Mirrors with Actuator Replacement Cartridges,’’ Proc. SPIE, 2201 (1994), p. 680. J. H. Everson, R. E. Aldrich, and N. P. Albertinetti, ‘‘Discrete Actuator Deformable Mirror,’’ Opt. Eng., 20, No. 2 (1981), p. 316. J. Feinleib, S. G. Lipson, and P. E. Cone, ‘‘Monolithic Piezoelectric Mirror for Wavefront Correction,’’ Appl. Phys. Lett., 25, No. 6 (1974), p. 311. J. Feinleib and S. G. Lipson, ‘‘Monolithic Piezoelectric Wavefront Phase Corrector,’’ U. S. Patent No. 3,904,274 (Sept. 9, 1975). J.-P. Gaffard, P. Jagourel, and P. Gigan, ‘‘Adaptive Optics: Description of Available Components at LASERDOT, Proc. SPIE, 2201 (1994), p. 688. P. Gosselin, P. Jagourel, and J. Peysson, ‘‘Objective Comparisons between Stacked Array Mirrors and Bimorph Mirrors,’’ Proc. SPIE, 1920 (1993), p. 81. J. E. Graves, F. Roddier, D. McKenna, and M. Northcott, ‘‘Latest Results from the University of Hawaii Prototype Adaptive Optics System,’’ DoD Conference on Adaptive Optics, Kirtland AFB, NM (1992), not formally published, p. 511. W. Hulburd and D. Sandler, ‘‘Segmented Mirrors for Atmospheric Compensation,’’ Opt. Eng., 29, No. 2 (1990), p. 1186. E. Kibblewhite, M. F. Smutko, and M. Chun, ‘‘Deformable Mirrors for Astronomy,’’ Proc. SPIE, 1920 (1993), p. 115. E. Kibblewhite, M. F. Smutko, and F. Shi, ‘‘The Effect of Hysteresis on the Performance of Deformable Mirrors and Methods of Its Compensation,’’ Proc. SPIE, 2201 (1994), p. 754. E. Kibblewhite, W. Wild, B. Carter, M. Chun, F. Shi, and M. Smutko, ‘‘A Description of the Chicago Adaptive Optics System (ChAOS),’’ DoD Conference on Adaptive Optics, Kirtland AFB, NM (1992), not formally published, p. 522.



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R. L. Lillard and J. D. Schell, ‘‘High Performance Deformable Mirror for Wavefront Compensation,’’ Proc. SPIE, 2201 (1994), p. 740. W. C. Marlow, ‘‘On the Dynamics of Deformable Mirror Actuators,’’ Proc. SPIE, 1920 (1993), p. 63. S.-E. Park and T. R. Shrout, ‘‘Relaxor Based Ferroelectric Single Crystals for Electromechanical Actuators,’’ Materials Research Innovations, 1 (1997), p. 20. A. G. Safronov, ‘‘Active and Adaptive Bimorph Optics Mirrors, Technology and Design Principles,’’ Adaptive Optics, 1996 Technical Digest Series, Vol. 13, July 8–12, 1996, Maui, HI, sponsored by the Optical Society of America. R. M. Sharples, A. P. Doel, C. N. Dunlop, J. V. Major, and R. M. Myers, ‘‘MARTINI: Current Status and Future Developments,’’ DoD Conference on Adaptive Optics, Kirtland AFB, NM (1992), not formally published, p. 575. S. L. Swartz and T. R. Shrout, ‘‘Fabrication of Perovskite Lead Magnesium Niobate,’’ Mat. Res. Bull., 17 (1982), p. 1245. C. Swift, E. Bliss, D. Lenz, and R. Miller, ‘‘Deformable Mirror for Zigzag Solid-State Lasers,’’ Opt. Eng., 29, No. 10 (1990), p. 1199. W. G. Thorburn and L. Kaplan, ‘‘Low Voltage SELECT Deformable Mirror,’’ :Proc. SPIE, 1543 (1991), p. 1543. W. Wild, E. Kibblewhite, and V. Scor, ‘‘Quasi-hexagonal Deformable Mirror Geometries,’’ Proc. SPIE, 2201 (1994), p. 726.


6 Innovative Wavefront Estimators for Zonal Adaptive Optics Systems Walter J. Wild† University of Chicago, Chicago, Illinois

I. INTRODUCTION It may be appropriate to consider the deformable mirror in an adaptive optics system as its ‘‘heart’’ because without its continuous operation the system ceases to function. In further analogy with human anatomy, the ‘‘mind’’ of an adaptive system is the control algorithm responsible for generating the signals that drive the deformable mirror. The set of commands passed to each actuator of a deformable mirror for each sample interval is generated from an estimate of the instantaneous wavefront. The nature of these estimators is the basis of this paper. We shall confine our discussion to high-order zonal adaptive optics technology where local subaperture tilts are measured and the wavefront is reconstructed from these tilt data. Cunning designs for low-order modal systems exist (Roddier et al. 1991; Bruns and Meyer 1994) as well as very high order zonal systems (Angel 1994) suggested for ultrahigh Strehl correction required for ground-based extrasolar planet searches around nearby bright stars. In the low-order case a curvature wavefront sensor measures the second derivative of the wavefront; the principles are based on the irradiance transport equation governing the intensity field evolution in the focal region. In the second case the wavefront is measured directly using a self-referencing Mach-Zehnder interferometer utilizing a suitably bright and tilt-stabilized speckle in one leg of the interferometer (Colucci 1994).

†

Deceased.


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A significant number of systems presently being constructed utilize HartmannShack wavefront sensors and have ⬎100 actuators used in the deformable mirror. In the early analog days of adaptive optics, wavefront reconstruction was performed using resistive networks which solved the elliptic differential equation relating the phase differences to the phase. With maturing digital technology matrix-vector multipliers were built and linear matrix estimators of the form φ ⫽ Ms are used to drive the system (Sasiela and Mooney 1985), where φ is the estimated phase, M is the estimator installed in the digital reconstructor, and s is the ordered vector of subaperture slopes. By the 1990s most, if not all, zonal systems utilize a matrix multiplier to perform the reconstruction. Newer architectures being designed, primarily for military systems, will incorporate generalizations of the matrix multiplier approach to accommodate high scintillation conditions and branch points in the phase function; the nonsmooth nature of these wavefronts dictates that nonlinear or iterative methods be used to get the best estimate of the phase (Le Bigot et al. 1997). Our discussion centers on the choices for the matrix M.

II. LEAST-SQUARES ESTIMATION If a least-squares estimator is used, the matrices for any particular system depend on the pupil subaperture/actuator configuration, or geometry, which includes the obscuration, relative actuator and subaperture arrangement and ordering, and which, if any, modes such as tip, tilt, focus, etc., are to be projected out or retained in the estimator. If one or more buffer rings of actuators are to be used to establish some desired boundary or edge condition around the controlled or active actuators within the illuminated pupil, then appropriate slaving relationships between controlled and slaved actuators can be included in the matrix design. Other linear matrix estimators can include information about the adaptive optics control loop, or loops, and the physics of the atmosphere via the steady state covariance matrices. If Kolmogorov turbulence is assumed, these covariance matrices have analytical forms which depend on the separation between points in the pupil, r 0 , the wind velocity profile in the atmosphere (or of the dominant layer), and the system sample time. It is also possible to include the physics of the deformable mirror via the facesheet influence function as well as the beacon configuration. The least-squares estimator is the most commonly used, to date, in most adaptive optics systems. It is obtained by minimizing the Euclidean L 2 scalar norm (Luenberger 1969) Λ ⬅ 储 s ⫺ Aφ储 2 ⫽

冱 (s ⫺ [Aφ] ) i

i


i

2

⫽ Tr{(s ⫺ Aφ)(s ⫺ Aφ) T }

(1)

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where A is the geometry matrix (Wild et al. 1994a), discussed in more detail below, the subscripts on the right side are the indexed elements of the quantities, and φ ⫽ Ms is the reconstructed wavefront which is the basis of the signals sent to the deformable mirror. The Tr is the matrix trace operation, which is the sum of the diagonal elements of the enclosed outer product square matrix on the right of (1). The least-squares estimator is not inherently statistical. In the literature one usually finds this norm expressed as 储φ ⫺ ϕ储, where s ⫽ Aϕ is the fundamental governing equation that relates local slopes to local phases via the geometry matrix A, and where ϕ is the incoming atmospheric phase at some instant in time. The simple, i.e., non-SVD, approach for practical inversion used in the A⫹⫹ (Wild 1997) software, which generates various reconstructor matrices M, is M ⫽ lim (AT A ⫹ δI)⫺1 AT, δ→0

where δ ⬇ 10⫺6

Least-squares matrices based on simple sparse forms of A have been the most widely used in zonal adaptive optics systems. The least-squares matrices are the easiest to generate and do not require any a priori information about atmospheric statistics, beacon brightness or photon noise levels, winds, etc. In practice, whenever we are involved in developing any matrix for a new system, we will always begin with a least-squares matrix because they will work when all of the practical ordering and coordinate conventions have been worked out. The reason that we express the norm as in (1) is that the minimization of Λ leads to the analytic form of M in terms of A, and this minimization can be performed either via rigorous, but painful, matrix derivative methods or via simpler matrix perturbative techniques and manipulation. Utilizing 储φ ⫺ ϕ储 for the least-squares minimization directly requires somewhat esoteric concepts in linear vector space theory (Luenberger 1969), whereas our approach is purely algebraic. Minimization of (1) requires that ∂Λ/∂M ⫽ 0 with the system geometry specified by A; the basic least-squares estimator contains no a priori information about the control loop, the atmosphere, or the deformable mirror and beacon physics. The trick to finding the analytic expression for M is to work with the rightmost trace expression, setting M → M ⫹ ε Ω, for an arbitrary matrix Ω, and then setting the terms linear in ε to zero. The trick inherent in this, and all the other optimizations summarized below, is that square matrices within a trace operation commute, so quantities such as ssT and Ω can be located either on the left or right of a matrix equation. The vanishing of this matrix equation represents our desired goal. For the least-squares estimator, the matrix manipulations yield the elegant result M ⫽ A⫹ ⬅ (AT A)⫺1 AT

(2a)

The generalized or pseudoinverse of a matrix A is commonly designated as A⫹. Minimizing Eq. (1) is identical to maximizing the on-axis Strehl ratio of the system because the quantities on the right side of Λ correspond to the sum of


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the individual, or discrete, phase point variances over the pupil. It is important to realize that AT A is singular because of piston ambiguities. This singularity, from a practical programming point of view, is easily overcome by computing M via the Moore pseudoinverse limit M ⫽ lim (AT A ⫹ δI)⫺1 AT δ→ 0

(2b)

where δ ⬇ 10⫺6 is sufficient for double precision computation and such that errors from the finite size of δ do not appear in 16-bit integer normalization of M encountered in most reconstructors, and I is the identity matrix. Equation (2b) is very straightforward to compute and avoids the necessity of using, or even knowing how to use, singular value decomposition methods; in Matlab (2b) is found by using the pinv(A) command. Once A is established in terms of the pupil and ordering (2b) readily gives us the matrix M that can be used in hardware, though it is usually necessary to rescale the entries in M to be in n-bit integer format. Note that the least-squares estimator ignores the noise in the measurement. Including photon noise leads to the weighted, or damped, least-squares estimator. The only real value of this estimator is if the partially obscured subapertures around the pupil perimeters are used and weighted based on their illumination fraction (unity for a fully illuminated subaperture and zero for an unilluminated subaperture). Scintillation effects induce rapidly varying intensity fluctuations in the subapertures, but a real-time weighted least-squares matrix entails computing several matrix multiplies and an inverse in real time, which is not feasible; nor is it necessary, as one can choose from several alternative estimators to minimize scintillation effects, including matrix iterative estimators, optimal estimators, or branch-point reconstructors, all of which are addressed below.

III. OPTIMAL WAVEFRONT ESTIMATION We now go a step further than that embodied in (1) by incorporating ensemble statistical averages so that steady state covariance matrices now enter into the estimator solutions of the matrix equations from minimizing the generalization to Λ. This is called optimal wavefront estimation. Here the mirror control utilizes a priori ensemble wavefront phase statistics information, integrated over the path, and includes effects such as measurement noise, winds, scintillation, etc., and can yield improved AO system performance compared to standard least-squares techniques if conditions encountered by the system are matched by the assumptions. The temporal and spatial performance of an adaptive optics system, i.e., the essential physics, is embodied in a single mathematical descriptor of the form Γ ⫽ 〈[ϕ(t i⫹1 ) ⫺ φ dm (t i )][ϕ(t i⫹1 ) ⫺ φ dm (t i )]T 〉


(3)


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where Γ is a covariance matrix, or often called a structure function, of the difference between the atmosphere phase ϕ(t i⫹1 ) incident on the pupil at time t i⫹1 and the deformable mirror figure, φ dm (t i ), derived from measurements at time t i . Lowercase boldface quantities designate a vector, and uppercase boldface a matrix. The phase reflecting off the mirror is ϕ(t i⫹1 ) ⫺ φ(t i ), assuming presently that φ dm (t i ) ⬇ φ(t i ). The matrix M reconstructs the residual phase error ϕ(t i⫹1 ) ⫺ φ(t i ) using the instantaneous measured slope information, and k is the servo loop gain. Here s(t i⫹1 ) ⫽ Aϕ(t i⫹1 ) for subaperture slope vector s(t i⫹1 ) and geometry matrix A. Here we note that Tr(Γ) is not an obvious statistical generalization of Λ because (3) is the reconstruction error rather than the slope discrepancy in (1). The difference arises because of the use of ensemble averages, so the quantity ϕϕ T, which appears in the perturbative M → M ⫹ εΩ expansion of (1) and which cannot enter the solution, is replaced by 〈ϕϕ T 〉, which is a known quantity and does become part of the solution with the optimal estimators. Stated another way, the perturbative solution for the least-squares estimator requires that both Ω and ϕϕ T be isolated from the A and M matrices within the trace, on the left and right sides of the expression, so that an equation relating M to A can be isolated. This can be done using the form for Λ and algebraic matrix manipulations in (1), but not using 储 φ ⫺ ϕ储 as the minimizing norm. However, for the optimal estimators only the matrix Ω is unknown, so the norm 储 φ ⫺ ϕ储, more precisely written in (3) to reflect temporal effects, can be used. The discrete-time indices are used because we shall incorporate a model for the control loop in the derivation of optimal estimators M. From the physics point of view an adaptive optics system is best characterized by discrete quantities in virtue of the temporal integration intervals used in the wavefront sensor, as well as the discrete spatial subapertures. Furthermore, continuous (differential equation) control loop models are less amenable to finding reconstructors M without imposing some mathematical assumptions about the structure of M (Ellerbroek 1994). Two critical pieces of information and one constraint upon Γ are presently used to generate optimal estimators. The first is the servo loop where mirror updates governed via the difference equation (Wild et al. 1995a) φ(t i⫹1 ) ⫽ a 0 φ(t i ) ⫹ kM[s(t i⫹1 ) ⫺ Aφ(t i )]

(4a)

where a 0 is approximately unity (called a ‘‘lossy integrator’’) in an operational system. The second term is the reconstruction of the slope error arising from the wavefront reflecting off the deformable mirror with a shape from the previous time interval placed on its surface. Equation (4a) can be recursively solved for φ(t i ) in terms of the slope history, k, M and A: η

φ(t η ) ⫽ k

冱 (a I ⫺ kMA) 0

η⫺i

Ms(t i )

(4b)

i⫽0

where the integer η ⱖ 0 is the number of look-back steps retained in the analysis.


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The second piece of information is the a priori knowledge of the atmospheric phase statistics; this is a phase covariance matrix, which for times t i and t i ′ is X φii ′ ⫽ 〈ϕ(t i ) ϕ(t i ′ )T 〉. For a dominant layer with wind velocity v and sample time τ, under the Taylor frozen turbulence flow hypothesis and Kolmogorov phase statistics [X ϕ ii′ ] jj ′ ⫽ ⫺3.44

冷

x j ⫺ x j ′ ⫺ vτ(i ⫺ i′) r0

冷

5/3

(5)

for vector locations x j in the pupil. Equation (5) is for a translation flow field but can be generalized for almost any situation, including cross winds in different layers, etc. It may be evident that our matrix formulation is a discrete approximation to continuous, integral, approaches. However, as we hope to convey, the discrete—or matrix—development is better suited to the system and much more powerful. This is particularly true given that a matrix must be created for use in the digital reconstructor. Estimators M that maximize the Strehl ratio S are constructed by minimizing Tr(Γ). For S ⬎ 0.1, S ⬃ exp(⫺σ 2 ), where σ 2 ⫽ Tr(Γ)/Na , for Na controlled

Figure 1 AO servo loop configuration.



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actuators within the pupil. As in the least-squares derivation, a matrix equation will be obtained for M, and for optimal estimators it is parameterized as a function of A, aperture D, r 0 , phase-difference variance σ 2pd , τ, a 0 , k, v, and η. Here Γ can be decomposed as follows: Tr(Γ) ⫽ Tr 〈[φ dm (t i ) ⫺ φ(t i )][φ dm (t i ) ⫺ φ(t i )]T 〉 ⫹ Tr 〈[φ(t i⫹1 ) ⫺ φ(t i )][φ(t i⫹1 ) ⫺ φ(t i )]T 〉

(6)

⫹ Tr 〈[φ(t i ) ⫺ ϕ(t i )][φ(t i ) ⫺ ϕ(t i )]T → ⫹ cross terms The first term is the fitting error between the mirror surface and the reconstructed wavefront; the second is the time-delay error, and the third is the computation error arising from MA ≠ I. Finding an optimal M, as discussed above, entails minimizing terms in (6) for η ⱖ 0, using perturbative or Kronecker product matrix derivative methods (Graham 1981; Rogers 1980), and isolating the resulting matrix equation, using the various rules of matrix manipulation within the trace operator. That is, if the M → M ⫹ ε Ω perturbative algebraic optimization method is used, the key ingredient is to isolate the Ω matrix from all other factors by putting it on the right or left side in the various terms in the expansion of (6). (It is important to note that there are two symmetric sets of terms in the expansion of (6), one being the transpose of the other, and that it is only necessary to work with one set.) Direct differentiation of matrices with respect to another matrix requires using Kronecker-product-based formulae which are absolutely horrendous and extremely painful to work with, and so far we have avoided this level of mathematical rigor, which in the final analysis will still yield the same result. If only the first term in (6) is minimized with η ⫽ 0 the solution is ˜ ϕ AT {AX ˜ ϕ AT ⫹ X n }⫺1 M⫽X (7) ii

ii

ii

˜ φ designates that which is known as the Wiener-Kolmogorov estimator. Here X ii piston and, possibly, tip/tilt have been projected out of the phases in (5); X n ii is the noise covariance matrix; for uniform illumination levels X n ii ⬇ σ 2pd I, for phase-difference (slope) variance σ 2pd . For high noise levels, i.e., a dim beacon, σ 2pd /(D/r 0)5/3 is large, so most information resides in the lowest order wavefront modes. The limiting form of (7) as σ 2pd /(D/r 0)5/3 → 0 is the least-squares M in (2a). Scintillation statistics will affect the structure of X n ii . It is important to recognize that X n ii is not directly related to the subaperture photon intensity, but rather to the noise associated with the computed slopes from the noisy data. In other words, there is a transformation from Poisson photon measurement statistics to that associated with the slope estimates. Since we do not always know σ 2pd / (D/r 0)5/3 well prior to an observation, it suffices to recognize that σ 2pd /(D/r 0)5/3 is a single number, and so several optimal estimators for σ 2pd /(D/r 0)5/3 ⬃0.01 to 1.0 are usually generated and tested to determine which may be best during an observing session.


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Wild

The quantity σ 2pd is dependent on the wavefront sensor and beacon physics. Tyler and Fried (1982) give a formula for σ 2pd for a quadrant cell Hartmann wavefront sensor, as well as a general expression that includes the MTFs associated with propagation through the atmosphere, the wavefront sensor, and noise. For focused laser beacons these formulae include laser beam quality and the twoway propagation path through the turbulent atmosphere (Lutomirski et al. 1977). For η ⬎ 0 in (4b), minimizing Tr(Γ∞ ) generates more complex and algebraically unsolvable matrix equations in M because of the noncommutative nature of matrices. For η ⫽ 1 a predictive optimal estimator (Wild 1996), or POE, minimizing the second and third terms in (6), yields a nonlinear matrix equation, which can be cast into a form enabling Picard iteration to solve for M: M j⫹1 ⫽ [T(M j )⫹ U(M j )]T

(8)

where M 0 can be the η ⫽ 0 POE, or even the L 2-norm least-squares inverse of A. Here T is quadratic in M and U is cubic in M: T ⫽ ka 0 Aψ ii AT βT ⫺ k 2 AMAψ ii AT βT ⫹ k 2 Aψ ii AT ⫹ k 2 a 0 Aψ i⫺1, i AT ⫺ k 3 AMAψ i⫺1, i AT ⫹ kAψ i, i⫺1 AT βT ⫹ ka 0 X n ii βT ⫺ k 2 AMX nii βT ⫹ k 2 X nii ⫹ k 2 a 0 X n ii ⫺ k 3 AMX n ii ⫹ kX nii βT ˜ ϕ ⫹ k 3 Aψ i⫺1, i AT M T MA ⫹ ka 0 AX ˜ϕ U ⫽ k 2 Aψ ii AT βT M T MA ⫹ kAX ii i⫺1, i 2 2 2 T T 3 T ˜ ϕ k AX ˜ ϕ MA ⫹ k X n β M MA ⫹ k X n M MA ⫺ k AMAX i⫺1, i

i⫺1, i

ii

ii

˜ϕ ⫺ where β ⫽ ka 0 I ⫺ k 2 AM, and for any integers m and m′, ψ m, m′ ⫽ 3X m, m ′ ⫹ ˜ ˜ ˜ ˜ X ϕ m, m ′⫺1 ⫺ X ϕ m⫺1, m′ and X ϕn, n′ ⫽ X ϕ n⫺1, n′⫺1 . Here T(M j ) is the psuedoinverse of T, which is a function of the jth iteration of M. The complexity of this deterministic equation arises from the ensemble spatial and temporal statistical weighting over the pupil with winds and time evolution to obtain the estimator M. What neural network techniques attempt to do for wavefront prediction, deterministic matrix equations do rigorously, though all the various effects become manifested as the many terms in the resulting matrix equation. For example, the η ⫽ 1 POE reconstruction using the slope vector derived from a single poked phase point smears out some phase to the neighboring phase point in the direction of the wind velocity, for which over the time interval τ the fractional subaperture wavefront translation under the Taylor hypothesis is ∆ τ ⬅ v x τ/L, for subaperture dimension L and v oriented along the x-axis. Physically, the POE anticipates the reconstructed phase accounting for the winds and servo loop so as to minimize time delay errors. The η ⫽ 2 POE matrix equation, analogous to (8), is a monster expression which has T of fourth order and U of fifth order in M. For Strehl ratio as a function of ∆ τ for various estimators M, there are crossing points in the curves wherein the η ⫽ 1 POE maximizes Strehl ratio for the larger values of ∆ τ (Wild 1996).



207

POEs have been experimentally tested using the MIT Lincoln Laboratory Firepond adaptive optics facility over a 5.5-km horizontal path in winds parallel to the ground ( y-axis in subaperture space) corresponding to ∆ τ ⬃ 0.3. This first test was qualitative in nature. Two POEs were generated with ∆ τ ⫽ 0.15 and ∆ τ ⫽ ⫺0.15 (or y-component of wind velocity being 3 m/s and ⫺3 m/s). The experiment was to demonstrate that one POE outperforms the other over repeated trials, since the wind stays reasonably constant and oriented in one direction during the tests. About a factor of 2 difference in Strehl ratio (0.05 versus 0.1) was observed; the Rytov number was measured as ⬃0.15 during these tests using a NOAA instrument, but there was significant ‘‘microturbulence’’ visually observed consistent with a much higher Rytov number. Further quantitative tests are planned using the WCE system at Yerkes with a laser beacon 3 miles distant across Lake Geneva and other facilities at MIT Lincoln Laboratory. Minimizing Tr(Γ) to obtain M using the expansion in (6), rather than (3) directly, appears to be necessary to properly isolate time delay errors, which are the crux of the POE formulation. Forcing each error to be explicitly used serves to find a true optimal solution via compromise between terms. Finding an optimal M for finite range beacon AO systems, where focal anisoplanatism is known to correlate with tilt anisoplanatism, may still best be done using a generalization of (6) explicitly incorporating these two quantities. We speculate that minimizing the tilt anisoplanatism, either via an added term in (6) or using matrix-based MTF techniques, will generate an optimal PSF flattener estimator whereby over some angular subtense greater than θ 0 the corrected image (or focused laser spot) is space invariant; this behavior is obtained at the expense of a reduced on-axis Strehl. The AO system can be automatically adjusted in response to prevailing atmospheric and tracking conditions as well as deformable mirror influence functions. The first term in (6) incorporates deformable mirror physics, which includes the mirror being liquid cooled and other environmental circumstances. In our present studies the mirror influence function can be simply modeled by using an optimal estimator to mimic an influence function: the influence function for actuator i is given by the ith element of MA operating on a poked actuator point (a phase vector consisting of all zeros with element i set to unity). Here MA is the ‘‘PSF ’’ of the wavefront reconstruction process, and as σ 2pd /(D/r 0)5/3 increases the modeled influence function spreads out in actuator space. The Wiener-Kolmogorov optimal (minimum variance) estimator (Luenberger 1969), on the other hand, is arrived at by minimizing an ensemble average of the difference between the true atmospheric phase and the reconstructed phase, i.e., Γ ⫽ 〈储 ϕ ⫺ φ储 2 〉,

dΓ ˜ ϕ AT {AX ˜ ϕ AT ⫹ X n }⫺1 ⫽0→M⫽X ii ii ii dM


(9)

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Wild

As discussed in Wild (1997) a perturbative method suffices to implement the minimization in Eq. (9). Here M → M ⫹ εΩ is used and linear terms in ε are kept and commutation rules within the Tr(Γ) are used to isolate Ω. The resulting matrix equation is linear and readily solved for M. Optimal estimators have been used in various systems and have a reduced sensitivity to the accumulation of global deformable mirror waffling, and enable operation at reduced beacon brightness, though the Strehl ratio cannot be as high as when there is ample brightness. Herrmann (1992) has shown that these estimators maximize the Strehl ratio. Under the assumption that X n ii ⬇ σ 2pd I s, for phase difference error σ 2pd , the estimator in Eq. (9) is fully parameterized by the quantity σ 2pd /(D/r 0)5/3. Here in Matlab notation I s ⫽ diag(diag(ceil[(AAT )/5])). Predictive optimal estimators (POEs) arise by minimizing a temporal generalization of Γ used in Eq. (9) (Wild 1996): Γ ⫽ 〈储 ϕ(t i⫹1 ) ⫺ φ(t i ) 储 2 〉 where η

φ(t η) ⫽ k

冱 (a I ⫺ kMA) 0

η⫺i

Ms(t i )

(10a)

i⫽0

Here φ(t η ) is the solution of the servo loop difference equation φ(t i⫹1 ) ⫽ a 0 φ(t i ) ⫹ kM[s(t i⫹1 ) ⫺ Aφ(t i )]

(10b)

for η look-back terms in the series solution. Wild (1996) does a statistical decomposition of Γ to arrive at an iteratively soluble third-order matrix equation for M. The POEs reflect the discretized sampling of the system and can incorporate a priori information about winds and the servo loop parameters embodied in the two parameters a 0 and k. The servo loop enables anticipatory or predictive control under the assumption of the Taylor frozen flow hypothesis. The cross-covariance matrices that appear in the matrix equations that inevitably occur in minimizing Γ have the compact analytic form for Kolmogorov phase statistics whose matrix elements are ˜ ϕ ] jj ′ ⫽ ⫺3.44(Ip ⫺ P) [X ii′

冷

x j ⫺ x j ′ ⫺ vτ(i ⫺ i ′) r0

冷

5/3

(I p ⫺ P),

where P⫽

ppT (pTp)


(11)


209

and p is the piston vector over the active discrete phase points defined in A; in Matlab language p ⫽ diag(ceil[AT A)/5]) and I p ⫽ diag(p). Here v is the wind velocity and τ is the system sample time. Equation (4) can easily be generalized to projecting out any other modes in various basis set representations. The POEs overcome the MA ≠ I p assumption made by Ellerbroek (1994). The price that one pays for relaxing this assumption can be seen as follows. If η ⫽ 2 the resulting POE matrix equation, which must be solved to get a usable M, is 0 ⫽ ⫺k 3 AβMAψ ii AT M T βT βT ⫹ k 2 Aψ ii AT M T βT βT β2 ⫺ k 3 AMAψ ii AT M T βT βT β ⫹ k 2 Aψ ii AT M T βT β ⫺ k 2 AMAψ ii AT M T βT ˜ ϕ ⫹ k 2 Aψ i⫺1, i AT M T βT β2 ⫹ k 2 Aψ ii AT M T ⫺ kAX ii

⫺ k 3 AβMAψ i⫺1, i AT M T βT ⫺ k 3 AMAψ i⫺1, i AT M T βT β ˜ ϕ β2 ⫺ k 3 AβMAψ i⫺2, i AT M T ⫹ k 2 Aψ i⫺2, i AT M T β2 ⫺ kAX i⫺2, i

˜ ϕ ⫺ k 3 AMAψ i⫺2, i AT M T β2 ⫹ k 2 AMAX ϕ β ⫹ k 2 AβMAX i⫺2, i i⫺2, i ⫹ k 2 Aψ i, i⫺1 AT M T βT β ⫺ k 2 AMAψ i, i⫺1 AT M T βT βT ⫹ k 2 Aψ i⫺1, i AT M T β ˜ ϕ β ⫹ k 2 AMAX ˜ϕ ⫺ k 3 AMAψ i⫺1, i AT M T ⫺ k 2 AX i⫺1, i

i⫺1, i

⫹ k 2 Aψ i, i⫺2 AT M T βT βT ⫹ k 2 Aψ i, i⫺1 AT M T βT ⫺ k 3 AβMX n ii M T βT βT ⫹ k 2 X n ii M T βT βT β2 ⫺ k 3 AMX n ii M T βT βTβ ⫹ k 2 X n ii M T βT β ⫺ k 3 AMX n ii M T βT ⫹ k 2 X n ii M T ⫹ k 2 X n ii M T βT β2 ⫺ k 3 AβMX nii M T βT ⫺ k 3 AMX n ii M T βT β ⫹ k 2 X n ii M T β2 ⫺ k 3 AβMX nii M T ⫺ k 3 AMX n ii M T β ⫹ k 2 X n ii M T β ⫺ k 3 AMX n ii M T ⫹ k 2 X nii M T βT βT β ⫺ k 3 AMX n ii M T βT βT ⫹ k 2 X n ii M T βT βT ⫹ k 2 X n ii MT βT ˜ϕ ⫺ X ˜ϕ ˜ϕ ˜ϕ ⫽ where β ⬅ a 0 I p ⫺ kMA, ψ m, m′ ⬅ 3X ⫺ X , and X m, m ′ m, m′⫺1 m⫺1, m ′ n, n′ ˜ X ϕn⫺1, n′⫺1 . This equation is of fifth order in M, and we have not yet succeeded in finding stable iterative solutions, though the η ⫽ 1 POE matrix equation is stably soluble (Wild 1996). In the limit of η → ∞ the solution approaches that of Eq. (9), although k must be made extremely small. That is, it is unclear if there is any benefit in going beyond the η ⫽ 1 equation, which we have successfully solved and tested in numerous systems (Wild 1997). L2-norms that also include some desired property to be introduced into the operation of the system can be developed and minimized. It is important to recognize, intuitively, that adaptive optics systems with only one deformable mirror are limited in terms of how flexible they can be. One can do various forms of temporal, spatial, or angular averaging over directions in these dimensions; this is embodied in the numerous covariance matrices that occur and are exemplified by the instantaneous compromised shape on the deformable mirror surface. A


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Wild

particular example is the concept of establishing a compensated image which has space-invariant behavior over some a priori angular patch in the sky. At a loss of on-axis Strehl performance, such a space-invariant M arises from minimizing the norm Γ ⫽ 〈储 f{ϕ(t i⫹1 , θ)} ⫺ φ(t i , 0)储 2 〉

(12)

for some specified θ ⬎ θ 0 , or more generally, θ ⬎ θe, where we define θe as the effective isoplanatic angle, which is the angle at which there is a 1-rad2 meansquare error between the on-axis and the off-axis wavefront. The issue of isoplanatic angles in adaptive optics is confused by the nomenclature used and historical terminology, for the defined quantity θ 0 is strictly a mathematical entity that has physical sense only when D/r 0 ⫽ ∞, whereas θe, which depends on D/r 0 , is what governs the actual field of view performance of a compensating system (Chun 1998). Equation (12) can be used to develop estimators that also minimize focal anisoplanatic errors (a beacon optimal estimator or BOE) due to d 0. The latter in effect extrapolates lower order modes sensed due to a layer sampled within the focal cone to the volume beyond the sensed region. To illustrate how Eq. (12) is used, define the piston-removed covariance matrix, which includes winds as above: ˜ ϕ ⬅ (I p ⫺ P)〈ϕ(t i , ∆ j ) ϕ T (t i′ , ∆ j ′)〉(I p ⫺ P) X ii′, jj′ ⫽ ⫺3.44(I p ⫺ P)

冷

x ⫺ y ⫺ (i ⫺ i′) ντ ⫺ (∆ j ⫺ ∆ j ′ ) r0

冷

5/3

(I p ⫺ P)

for layer height H and off-axis angle of magnitude | ∆ | in the sky or celestial sphere. Here | ∆ | ⬎ θ e . The directional components relative to an arbitrary (x, y) axis in a plane parallel to the ground are ∆ j ⬅ (H | ∆ | cos(θ j ), H| ∆ | sin(θ j )) and if we designate Cartesian sky coordinates for index j in the covariance matrix, a generalized optimal estimator for zero wind and an open loop (η⫽0) servo has a closed-form expression for one on-axis and an arbitrary number of off-axis directional points: ˜ ) AT ⫹ X n }⫺1 ˜ ϕ ; (0, 0)(0, 0) AT {A(X ˜ ϕ ; (0, 0)(0, 0) ⫹ (1/J 2 ) Θ M⫽X ii ii ii where there are J points distributed within the region over which the compensation has a space-invariant PSF; this is the basis of the PSF ‘‘flattener’’ or PSFF ˜ has J 2 terms incorporating the many cross-covariance matrices estimators. Here Θ for the different angular combinations within the angular patch. The larger J is the better averaging over the pupil; we suspect that for adaptive optics geometries



211

with large numbers of actuators that large J is important. Interestingly, the region over which the J points are set up need not be circular, but can be adjusted according to some need. The PSFF is a way to control the adaptive optics to attain what has been called OTF synthesis in years past where the aberrating medium is the random atmosphere that exhibits known statistical behavior as embodied in the covariance matrices. If we designate Cartesian sky coordinates for index j in the covariance matrix, the PSFF optimal estimator for zero wind and an open loop (η ⫽ 0) servo has a closed-form expression for one on-axis and four off-axis directional points (corresponding to the coordinates (0, 0), (∆, 0), (0, ∆), (⫺∆, 0), and (0, ⫺∆)): ˜ ) AT ⫹ X n }⫺1 ˜ ϕ ; (0, 0)(0, 0) AT {A(X ˜ ϕ ; (0, 0)(0, 0) ⫹ (1/25)Θ M⫽X ii ii ii ˜ ⬅X ˜ ϕ ; (∆, 0)(∆, 0) ⫹ X ˜ ϕ ; (∆, 0)(⫺∆, 0) ⫹ X ˜ ϕ ; (∆, 0)(0, ∆) ⫹ X ˜ ϕ ; (∆, 0)(0, ⫺∆) ⫺ 4X ˜ϕ Θ ii

ii

ii

ii

ii ; (∆, 0)(0, 0)

˜ ϕ ; (⫺∆, 0)(∆, 0) ⫹ X ˜ ϕ ; (⫺∆, 0)(⫺∆, 0) ⫹ X ˜ ϕ ; (⫺∆, 0)(0, ∆) ⫹ X ˜ ϕ ; (⫺∆, 0)(0, ⫺∆) ⫹X ii ii ii ii ˜ ϕ ; (⫺∆, 0)(0, 0) ⫹ X ˜ ϕ ; (0, ∆)(∆, 0) ⫹ X ˜ ϕ ; (0, ∆)(⫺∆, 0) ⫹ X ˜ ϕ ; (0, ∆)(0, ∆) ⫺ 4X ii

ii

ii

ii

˜ ϕ ; (0, ∆)(0, ⫺∆) ⫺ 4X ˜ ϕ ; (0, ∆)(0, 0) ⫹ X ˜ ϕ ; (0, ⫺∆)(∆, 0) ⫹ X ˜ ϕ ; (0, ⫺∆)(⫺∆, 0) ⫹X ii ii ii ii ˜ ˜ ˜ ˜ ϕ ; (0, 0)(∆, 0) ⫹ X ϕ ; (0, ⫺∆)(0, ∆) ⫹ X ϕ ; (0, ⫺∆)(0, ⫺∆) ⫺ 4X ϕ ; (0, ⫺∆)(0, 0) ⫺ 4X ii

ii

ii

ii

˜ ϕ ; (0, 0)(⫺∆, 0) ⫺ 4X ˜ ϕ ; (0, 0)(0, ∆) ⫺ 4X ˜ ϕ ; (0, 0)(0, ⫺∆) ⫺ 16X ˜ ϕ ; (0, 0)(0, 0) ⫺ 4X ii ii ii ii For more points distributed within the patch there are correspondingly more terms in the expression to accomplish the angular averaging. It may be interesting to note that the PSFF is a generalization of the η ⫽ 0 POE. Experimental tests have still to be performed to establish the principle behind these (and the BOE) estimators. Of course, the next step is to consider L p-norm estimators (Gonin and Money 1989) for p ≠ 2, where

冢冱冣 η

储x储p ⬅

1/p

| xi | p

i⫽0

For even p, matrix Hadamard product identities can be used to derive an expression whose solution for M is not at all obvious. In general, prominent L p-norm solutions have the following properties: L0 L1 L2 Lp L∞

Elements of ϕ and φ agree in as many places as possible. Minimizes ∑ i | ϕi ⫺ φi |. Solutions are smooth and can make use of outer product formulation; assumes normally distributed errors. p ⬎ 2, penalize large deviations between elements of ϕ and φ. Minimax problem: minimizes max i |ϕ i ⫺ φ i |.


212

Wild

L p-norm applications to the phase unwrapping problem (Ghiglia and Romero 1996), which can be applied to adaptive optics, have been studied. Specialized reconstructor hardware may be necessary for p ≠ 2, though it may be worthy of further study. It would be interesting to examine if for a given turbulence profile there is an optimal choice for p. A superposition of various estimator approaches may be useful for various DoD applications. For example, the point ahead compensation task using a Rayleigh beacon may benefit from using an η ⫽ 1 POE generalized to also include PSFF and BOE statistical averaging. This may be particularly viable for very high order zonal systems where the complexities of all of this statistical averaging make reasonable sense. Solving the η ⫽ 1 POE under these conditions will be extremely taxing because of the many covariance matrices and the large size of the matrices, though the cost of such computations will be negligible compared to the complex hardware needed to develop more conventional solutions for the task.

IV. ITERATIVE WAVEFRONT ESTIMATORS Inspection of Eq. (4a) indicates a fundamental property of an adaptive optics system: It functions as an iterative digital-analog algebraic processor. That is, (4a) can be recognized as the basic iterative solution scheme for systems of linear equations where the kernel is M and k is the acceleration parameter. In the algebraic paradigm the slope vector s(t i⫹1 ) remains constant throughout the iterations, whereas for the evolving atmosphere s(t i⫹1 ) undergoes constant change; these changes are correlated if the Taylor hypothesis holds. From this analogy, Wild et al. (1995a, b) realized that an entirely new class of iterative estimators exist analogous to the kernels used in linear numerical algebra. These include the sparse Jacobi kernel and the less sparse successive overrelaxation and the preconditioned kernels. The iterative estimators exhibit increased time delay though diminished sensitivity to low subaperture illumination levels; i.e., the error propagators are smaller. Wild et al. (1995a) suggest an optimal k exists when these estimators are used. Here ‘‘iterative’’ does not mean a hardware system that necessitates iterative feedback to generate the phase reconstruction, but is adopted in analogy with their origin in the theory of numerical solutions to large linear systems of equations. The adaptive optics system itself is the iterative processor, and the wavefront estimator itself is an unchanging matrix that, once computed, resides within the digital reconstructor digital memory without any alteration during the operating session. The iterative estimators are, in analogy with the kernels of numerical linear algebra for geometry matrix A (Axelsson 1994; Wild et al. 1995a),



213

Richardson M ⫽ AT and Jacobi M ⫽ D⫺1 AT for D ⫽ diag(AT A); these are sparse. Iterated Jacobi (p ⬎ 1 iterations): p

M⫽

冱 (a I ⫹ kA A) A T

0

i

T

i⫽0

Successive overrelaxation (SOR): M ⫽ [(1/ω) D ⴚ L]⫺1 AT. Symmetric SOR (SSOR):

冤

2⫺ω M⫽ ω

冥冦冤

冥冤

1 1 D ⫺ L D⫺1 D ⫺ LT ω ω

冥冧

⫺1

AT

(where AT A ⫽ D ⫹ L ⫹ L T and L the lower triangular part of AT A). The iterated Jacobi estimator converges to the least-squares estimator as p → ∞ when a 0 ⫽ k ⫽ 1. For the SOR and SSOR estimators the parameter 0 ⱕ ω ⱕ 2 specifies the degree of convergence per iteration. There is no closed-form matrix expression for the Gauss-Seidel kernel.

V.

MODE REMOVAL

Tip and tilt are projected out of the wavefront estimators to diminish overall actuator stroke requirements on the deformable mirror. Often a quadrant or centroiding sensor (gradient or G-tilt) is used to measure these modes and a separate servo drives the fast steering mirror. If signals derived from the wavefront sensor are used this gives the Zernike (best fit or Z-) tilt signals for driving the fast steering mirror. A G-Z tilt variance, or centroid anisoplanatism, arises because Gtilt sensors do not sense the wavefront tilt component present in the higher-order coma terms. Let a be a column vector of the Zernike-Noll coefficients and the columns of the matrix Z represent modes evaluated at the discrete phase points. The reconstructed phase is φ ⫽ Za. Operating on φ with A gives Aφ ⫽ AZa ⫽ s, or aˆ ⫽ (AZ)⫹ s, where (AZ)⫹ is the least-squares inverse of AZ. Consequently, φ ⫽ Z(AZ)⫹ s ⫽ A⫹ (AZ)(AZ)⫹ s, where A⫹ A ⫽ I, and it is apparent that (AZ)(AZ)⫹ retains only those Zernike modes in the data s which are desired to be reconstructed via A⫹. To reject particular Zernike modes, as determined by the number of columns in Z, use the formula φ ⫽ A⫹ [I ⫺ (AZ)(AZ)⫹] s. Furthermore; aˆ ⫽ Z⫹ φ ⫽ Z⫹ A⫹ s, whereby φ ⫽ Zaˆ ⫽ ZZ⫹ A⫹ s. The reconstruction is obtained by A⫹ and the desired Zernike modes are retained by the right inverse ZZ⫹ ≠ I; they are projected out via φ ⫽ [I ⫺ ZZ⫹] A⫹ s. The right inverse retains information about the nonorthogonality of the modes over the discrete grid (Herrmann 1980).


214

Wild

These formulae can be generalized for any M in place of A⫹. In the first case modes are either retained ( projected in) or excluded ( projected out) in slope space (G-space), while in the second case these operations are done in the reconstructed phase space (Z-space). These operations are summarized in the following table:

Space

Modes projected in

Modes projected out

G Z

φ ⫽ M(AZ)(AZ) s φ ⫽ ZZ⫹ Ms

φ ⫽ M[I s ⫺ (AZ)(AZ)⫹]s φ ⫽ [I p ⫺ ZZ⫹]Ms

⫹

Here I p and I s are identity matrices but with diagonal entries that are zero when the corresponding diagonal elements in AT A and AAT are zero, respectively. That is, for inactive actuators or subapertures outside the pupil these identity matrices must be accordingly modified in order that the mode projection operations are valid over the active pupil region. Some AO systems utilize additional rows in their fast digital wavefront reconstructors to include the elements of (AZ)⫹ for selected modes (tip, tilt, focus, etc.), so operating on s, i.e., (AZ)⫹ s, gives modal coefficients that can be sent to a D/A converter to drive a fast steering mirror, focus element, etc. The coefficients of the low-order modes can be displayed as a evolving histogram, generated from real-time data, which can be used to keep the AO system optically aligned.

VI. COVARIANCE MATRICES The Zernike modes will not be orthogonal over a discrete actuator grid even if the pupil is circularly symmetric. Starting with 〈φφ T 〉 ⬅ X φ ⫽ Z〈aa T 〉 Z T ⫽ ZC z Z T, after some manipulations, C z ⫽ (AZ)⫹ AX φ A([AZ] T )⫹, where C is the Zernike coefficient covariance matrix and X φ is the phase covariance matrix in (4a). The diagonal elements of C z are the variances of the various Zernike modes up to some specified order; these variances decrease approximately monotonically with increasing mode order, and if too many modes are included aliasing (undersampling) effects will be present and propagate to lower-order modes (Herrmann 1981). The Zernike modes are not orthogonal or statistically independent over the discrete pupil grid and correlations exist for the analytic Noll terms. Open-loop slope data can be used to generate C z experimentally via measurement of the slope covariance matrix X s ⫽ AX φ AT, and the correspondence between the theory and data is an indicator of how well matched the turbulence spectrum is, for example, to the Kolmogorov model.



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VII. MODAL CONTROL The Karhunen-Loeve (KL) modes are defined to be orthogonal and statistically independent over the specified actuator grid and pupil embodied in A, although there are theorems in matrix theory which imply that functional modes cannot be found that also have orthogonal derivatives (slopes); the latter entails simultaneous diagonalization in phase and slope space. To obtain the KL modes it is necessary to diagonalize C z via a unitary matrix U, i.e., U T U ⫽ I, whereby D K ⫽ UC z U T for diagonal D K , and the KL coefficient set is contained in the vector b ⫽ Ua and the KL eigenvectors (modes) are the columns of the matrix K ⫽ ZU T. The various KL modes can be projected in or out of the slopes or phases as discussed above by replacing the matrix Z by K. Optimal modal control of a zonal adaptive optics system therefore entails developing a set of reconstruction matrices that are a linear combination of the KL modal wavefront estimators; each KL modal matrix M Ki may be weighted by a servo gain parameter k i . The choice of the set {k i } is derived via the minimization of Tr(Γ). Here the KL modal matrices can be a priori generated and servo control established from calibration measurements leading to X φ , to C z , and then D K and b and then obtaining the {k i } from rapid solution of the set of algebraic equations that arise from the minimization of Tr(Γ); the algebraic set arises because of the statistical independence of the KL coefficients whereby 〈b i b Ti ′ 〉 ⫽ 0 for i ≠ i ′. For predictive modal KL control, fast processing of the resulting series of matrix equations is required. Ellerbroek et al. (1994) were the first to consider using multiple optimized control bandwidths, though they restricted their attention to MA ⫽ I estimators. Automatic control of the {k i } for KL-based estimators can be similarly performed in the context of optimal estimation: as σ 2pd /(D/r 0 ) 5/3 and v evolves different matrices can be installed within a cycle time of the system; here v is computable and σ 2pd /(D/r 0 ) 5/3 can be inferred from slope data. Whereas modal control entails adjusting the ensemble {k i } for the linear combination of pregenerated control matrices, with optimal estimators only one matrix is used at any instant and is chosen from a set of precomputed matrices.

VIII. STABILITY ISSUES Closed-loop AO stability can be formulated in terms of the error propagator, g n , or discrete Lyapunov matrix equation approach. The former entails evaluating a series matrix expression (Wild et al. 1995a) for arbitrary noise covariance matrix:


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g n σ 2pd ⫽ lim

η→ ∞

冢冣冦冱冱 k2 Na

η

(13)

η

⫻ Tr

(a 0 I ⫺ kAT M T ) η⫺i ′ (a 0 I ⫺ kMA) η⫺i″ MX ni ′ i ′ M T

i ′⫽0 i ″⫽0

冧

based on (4a), and the Lyapunov approach requires solving a linear matrix equation. Instability regions exist when the solved matrix is not positive-definite. A closed-form matrix expression for (13) does not appear to exist in general; a compact formula does exist if X n ii′ represents correlated subaperature noise between measurements in time, but, interestingly, is more difficult to derive when 〈n(t i ) n T (t j )〉 ⫽ X n ij ⬇ σ 2pd δ ij I, i.e., uncorrelated noise in time and space. For multiconjugate adaptive optics systems we anticipate having a system of coupled Lyapunov equations to govern system stability (Gajic and Qureshi 1995). Finding optimal M κ estimators, for κ adaptive modules, may entail developing optimization techniques similar to that discussed above, with constraints that embody stability and other aspects, such as G-Z tilt variance minimization, via Lagrange multiplier techniques. It is well known that there are close relationships between optimal estimation, prediction, and stability (Bibby and Toutenburg 1977; Willsky 1978).

IX. MATRIX GENERATION Our discussion so far has been entirely theoretical. The practical generation of an array of numbers that is used in a system relies on applying the physics of the control loop, beacon, noise sources, and atmosphere to the generation of the covariance matrices, and the pupil and geometry to the generation of the A matrix. For the least-squares and iterative estimators only the A matrix is needed. In the early days the authors’ experience in making M matrices was developed using cryptic Fortran code, and specialized subroutines were used to install different slaving rules for actuators around the perimeters of the pupil but outside the active illuminated region. Since 1992 the University of Chicago adaptive optics group, led by Professor Edward Kibblewhite, has initiated a sustained effort to developing user friendly graphics user interface Macintosh-based software to generate the many different types of estimators for virtually any possible zonal adaptive optics configuration. Our dedicated software programmer, Ms. Vijuna Scor, has enabled us to adopt our vision into a working reality. The first generation package was given the name A⫹ , and the goal in developing A⫹ was to have it do about as much as could be done in a single piece of software.



217

Figure 2 Six geometries supported by A⫹ and A⫹⫹ , clockwise from the upper left: Fried, WCE, Southwell, Hudgin, and two quasi-hexagonal arrangements.

In Fig. 2 are shown the six geometries supported by A⫹ . It is very easy to generate pupil configurations, to project in or out any combination of Zernike terms (up to the 400th), install slaving rules (direct, extrapolated, or optimal), and perform many other matrix generation and testing processes. However, as this code was developed we made substantial inroads in the theory of wavefront reconstruction, and so increasing numbers of features were added to A⫹ . The effect, however, was to keep upgrading the code as expeditiously as possible, and the result, while remarkably bug-free, has reached a state where further expansion is extremely difficult. As a consequence of our experience in developing A⫹ , a second generation of software—appropriately called A⫹⫹ —was initiated in 1995. The philosophy in developing A⫹ was based on first knowing what was expected based on the years of using and stressing A⫹ to run in all sorts of different modes. For example, every adaptive optics system has a unique design in regard to how data are measured and ordered. The WCE interlaces x- and y-slopes, while most other systems pipeline x-slopes followed by y-slopes. The effect is to reorder columns in the M matrix. The MIT Firepond (SWAT) system uses a Hudgin geometry


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which rasters x-slopes in a different direction than the y-slopes, and this affects the placement of columns in the A matrix. In A⫹⫹ we have worked on a feature which will enable any ordering scheme to be set up to accommodate the different hardware realizations. These features in fact are the core of any software package that attempts to be flexible and which will be useful to the widest possible audience. By being a priori aware of the unique needs of the many users via experience with A⫹ , we have been able to lay out the design for A⫹⫹ and so, from a programming viewpoint, know where to go and how to work on developing really good code which will then be expandable and much more easily maintainable and much less prone to the almost unavoidable infestation of bugs. Our second philosophical attitude in the development of A⫹⫹ is to organize tasks in a queue and to leave the details of all programming up to Ms. Scor to remove the usual pressures that make software development prone to errors. The A⫹⫹ software is still under development but is fully operational. It continues to improve and have more features. Given the large number of zonal systems in existence and under development, and the many issues that are being brought to light, A⫹⫹ will be a valuable tool for the community. For example, one topic that we are addressing is the inclusion of five-parameter actuatorsubaperture misregistration and appropriate DM influence functions to generate the A matrix. While A⫹ currently enables up to a 17 ⫻ 17 actuator array to be handled, A⫹⫹ goes up to 100 ⫻ 100 limited only by memory and speed considerations. A⫹⫹ also has superior graphics and handling capabilities.

X. OTHER WAVEFRONT RECONSTRUCTORS Primmerman et al. (1995) discuss the special challenges facing adaptive optics systems for imaging and laser power propagation over long horizontal paths. Such situations are of interest to various military agencies, and in particular to the USAF Airborne Laser (ABL) effort, and potentially for low elevation angle astronomy such as imaging Mercury in deep twilight. The ABL demonstrator system is challenged to project high energy laser radiation from a high altitude modified 747 aircraft to a moving target ⬃200–500 km distant. Estimators such as POEs may be beneficial to alleviating the induced winds over the path induced by relative aircraft and target motions. However, it is speculated—based on extensive propagation code simulations—that there are branch points leading to discontinuities in the phase function due to extreme scintillation conditions. These branch points manifest themselves in the slope data but are not reconstructed using the linear matrix estimators discussed above because they impose a smoothness criterion onto the wavefront (this may not be strictly true with



219

the iterative estimators). Consequently, alternative reconstruction algorithms are needed. Le Bigot et al. (1997) have developed a branch point reconstructor which in the limit of no noise reconstructs the wavefront perfectly, and which is weighted to accommodate noise effects in addition to branch points. The technique developed by Le Bigot et al. (1997) is based on a modification of the ‘‘power method’’ for finding an eigenvector and an eigenvalue of a hermitian operator, though in this technique only a positive eigenvalue will be selected. The branch point reconstructor finds a phasor u such that Πu ⫽ λu, where λ is real and positive and Π is the propagation matrix based on summing over paths of increasing length over the weighted slopes. This reconstructor is linear in phasor space but is nonlinear in phases. We are exploring other somewhat speculative and esoteric ideas in wavefront estimation and adaptive optics system control. The area of multiconjugate system control remains open to study, and we speculate that there may be concepts in tensor theory that might be applicable to the simultaneous and interlocking control of many deformable mirrors; this may lead to what we call a tensor reconstructor. Our first study of the formal connections between tensors and matrices led us to find a concise formulation of the Riemann curvature tensor. Modal control of a zonal system has a certain appeal because of the intuitive aspect associated with the modal basis set. The Zernike basis set over the unit circle of course first comes to mind, but there are advantages to working with orthogonal statistically optimal (uncorrelated modes) basis sets such as the Karhunen-Loeve (KL) functions (Lane and Tallon 1992). As recognized long ago, the KL functions cannot be analytically computed for Kolmogorov turbulence, though the lower orders have a strong resemblance to the Zernike polynomials. In the matrix formulation computing and handling these quantities is almost trivial because the KL modal matrix is obtained by diagonalizing the Zernike covariance matrix (Roddier 1990). The beauty of this approach is that all the pupil information and localized geometry relationships are handled via the matrix A. If we wish to drive an adaptive optics system such that performance can be adjusted as external atmospheric conditions evolve, two techniques immediately come to mind. The first is a serial method wherein a body of optimal estimators sampled throughout the hyperspace of parameters, i.e., σ 2pd /(D/r 0)5/3, wind velocity v, and possibly including a 0 and k, are developed a priori. As the atmosphere (or slew for satellite tracking) changes, the estimator is simply changed to a new one in the memory address of the hardware: an executive controller keeps track of which estimator is in use and to be chosen. For optimal modal control a parallel scheme can be developed wherein a set of modal matrices are a priori generated, and appropriately weighted linear combinations comprise the


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estimator M. Ellerbroek et al. (1994) were the first to consider two embedded control laws and the optimal gains for each. Each modal matrix represents one KL mode projected into the least-squares or optimal estimator (Wild 1997). The gain for each matrix may be found via the recipe: 1. 2.

3. 4.

5.

6.

7.

Generate a reconstruction matrix M Ki for each KL mode i; the system reconstruction matrix is M ⫽ ∑ Ji k i K i for mode gain k i . Compute the matrix Γ ⫽ 〈[ϕ(t i⫹1 ) ⫺ φ dm (t i )][ϕ(t i⫹1 ) ⫺ φ dm (t i )]T 〉. This approach can incorporate the influence function of the deformable mirror. Here Eqs. (10a, b) are used for computing Γ. The measured slope covariance matrix (over perhaps 1 s of data) is ˜ φ AT. transformed to phase covariances: 〈ss T 〉 ⫽ X s ⫽ AX ⫹ ˜ φ AT ([AZ] T )⫹, The modal covariances are computed: C z ⫽ (AZ) AX T D K ⫽ UC z U , where Z is the Zernike mode matrix, K and KL mode matrix, and U is a unitary matrix where K ⫽ ZU T obtained by diagonalizing the Zernike covariance matrix C z to get D K . Here D K ⫽ 〈bb T 〉 for the KL modal coefficient vector b; 〈b i b Ti ′ 〉 ⫽ 0 for i ≠ i′. The matrix Γ has N 2a elements for Na discrete phase points and there ˜ φ ⫽ ZU T D K UZ T to are J unknowns k i for J modes selected. Use X transform from Kolmogorov to KL covariance matrices. Also note that we can generate KL covariance matrices from the symmetric matrix ˜ϕ ˜ϕ , . X ⫹X m, m ′⫺1 m⫺1 m′ Compute d Tr(Γ)/d k ⫽ 0 to maximize Strehl, for gain vector k; the result is an overdetermined system of nonlinear algebraic equations for k. Find the solution of k via a fast nonlinear algebraic equation solver. As the external circumstances {σ 2pd /(D/r 0 )5/3, v} change, modify change k, or more generally the larger parameter set {A, k, τ, a 0 , η}, to retain optimal (e.g., maximum Strehl) system performance of the system.

Another possible method of automatically evolving the matrix M is to work with an η ⫽ 1 POE and as {σ 2pd /(D/r 0 )5/3, v} changes to apply a few iterations to the POE matrix equation using the existing POE as the initial starting point. For an AO system with 17 ⫻ 17 subapertures an iteration may be done with a fast computer in native C code in perhaps 1 s, so this is a viable technique. Here the M is iterated in the background and then shifted into memory once it is updated. Other algorithms using the Sherman-Morrison algorithm exist, although it appears that this approach is better suited for optimal estimators rather than the more general POEs. For situations such as encountered by the Keck telescope where there is an irregular boundary to the pupil which rotates relative to the wavefront sensor,



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matrices need to be generated for about every 5° and automatically swapped as observations are made. Near the zenith the frequency of switching matrices will be the greatest. For the Gemini Altair adaptive optics system the offset science and illuminated wavefront sensor subapertures, due to the conjugation at 6.5-km altitude for the natural guidestar system, means that a large number of estimators need to be a priori generated spanning both angle and beacon-science target separation.

XII. SLAVING AND STABILITY The application of slave actuators is a subject that has yet to be addressed quantitatively. Slaving around the outer perimeter and within the central obscuration is desired to attain smoothly varying boundary conditions as the deformable mirror surface evolves in a closed-loop setting. Experience at the SOR using the first generation (Gen. I; Fugate et al. 1994) adaptive optics system in 1989–1990 indicated that complex ad hoc rules will cause the system to rapidly go unstable as the gains are increased. Slaving may cause instabilities to form and propagate throughout the actively controlled region. Partially illuminated subapertures, with lower SNR, will generate noisy slope estimates that in turn will propagate errors locally and which will be spread either proportionately or linearly increased into the slaved region, depending on if direct (Fig. 3a) or linear extrapolation (Fig. 3b) slaving rules are implemented. These errors will also induce unnatural ridging into the mirror because such slaving rules, while they appear logical, are fundamentally unnatural and bear no relationship to the statistical properties of the atmosphere. Initial error propagator analyses (Wild 1998a) confirm that as a function of servo gain extrapolated linear slaving rules will have an error propagator that increases much faster than that associated with simple direct slaving or optimal extrapolated slaving (Wild et al. 1995b) based on atmospheric statistics over the controlled pupil. For the closed-loop servo system Eq. (3b), for F ⬅ a 0 I p ⫺ kMA, the error propagator is (Wild et al. 1995a) gn ⫽

Ω

Ω

冦冱冱F

k2 Tr lim Ω→ ∞ Na

Ω⫺j′

MM T (F T ) Ω⫺j″

j ′⫽0 j ″⫽0

冧

(14)

Figure 4 shows this quantity computed as a function of k for the four cases (shown as curves from the bottom to top): (1) no slaving at all; (2) optimal extrapolated slaving (OES) using the same slaved actuators shown in Fig. 3; (3) direct slaving as shown in Fig. 3a; (4) linear extrapolated slaving as shown in Fig. 3b. The linear extrapolated slaving approach has an error propagator about an order of magnitude greater than the OES approach. For these curves a 0 ⫽ 0.999.


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Wild

Figure 3 (a) Direct slaving pattern; (b) Linear extrapolated slaving pattern. In the former case slaved actuators have the same phase estimate as the controlled actuator, while in the latter case the slaves are linearly extrapolated from the two controlled actuator within each looped area.

For the Gemini system a fairly large number of actuators must be controlled when the star beacon is up to 1 arc-min from the science object (an offset of three subapertures) which are not adjacent to illuminated subapertures because the deformable mirror is conjugate to a layer at 6.5-km altitude. Figure 5a shows the case when the beacon is on axis with respect to the science target, while Fig. 5b shows the case for a 1-arc-min separation, where the highlighted subapertures



223

Figure 4 Error propagator g n for various slaving techniques.

are illuminated by the beacon source and the circle defines the on-axis science pupil. Two rings of slaved actuators surround the science pupil. Four reconstruction matrices based on least squares with and without OES were generated. It is noted from Fig. 6 that while g n is larger for the slaved cases there is no evidence of any instabilities based on purely mathematical properties of the estimator, or because of slaving defects as encountered for linear extrapolated slaving above. The OES technique has been successfully employed in recent tests with the Starfire Optical Range 3.5-m telescope. Other techniques exist based on imposed

Figure 5 Deformable mirror conjugate to a finite altitude atmospheric layer; (a) Full pupil; (b) beacon 1′ off axis. The half filled actuators are slaved via OES to the controlled actuators within the illuminated pupil.


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Figure 6 Error propagator g n with a 0 ⫽ 0.999 for the filled and gibbous pupils in Figure 5a and 5b. From the bottom to the top at left: (1) least squares for gibbous pupil, (2) least squares for filled pupil, (3) least squares with OES for gibbous pupil, (4) least squares with OES for filled pupil. In all cases tilt is projected out from phases (Wild 1997) and for OES σ 2pd /(D/r0)5/3 ⫽ 0.03.

various forms of local smoothing or curvature constraints (Ellerbroek, personal communication); we intend to publish a detailed collaborative paper discussing the various forms of slaving and quantitative behavior using data obtained with the SOR 3.5-m adaptive optics system. The use of formula (6) ties in with Lyapunov stability theory developed for adaptive optics systems that satisfy Eq. (10b) (Wild 1998a). The Lyapunov linear matrix equation for discrete systems specifies that a system is asymptotically stable if the unknown matrix in that equation remains positive-definite. We have found that instability as predicted when solutions to this equation are no longer positive-definite and they occur at the same locations when the error propagator begins to diverge. This is shown in Fig. 7 for several different estimators (Wild 1998a).

XIII. VERY HIGH ORDER COMPENSATION Sivokon and Vorontsov (1998) and Wild (1998b) have suggested following a conventional Shack-Hartmann-based adaptive optics system by an intensitybased phase retrieval system to attain high Strehl performance. By removing most of the severe aberrations with the zonal system the residual wavefront aberrations



225

Figure 7 Error propagator curves and corresponding gains for Lyapunov stability for the pupil geometry shown at the lower left. g n curves at lower left from bottom to the top are for estimators (1) v ⫽ 0, η ⫽ 1 POE, (2) optimal estimator, Eq. (2), (3) SSOR iterative estimator, and (4) least squares. The flat curves show the interval of positive-definiteness for the matrix solution the corresponding Lyapunov matrix equation which indicates domains of stability. Each line is actually at unity but offset for clarity. When they terminate corresponds to when g n for the same estimator begins to diverge.

are much reduced and perhaps well estimable using phase retrieval techniques. Wild (1998b) has found an exact solution of the phase retrieval problem under the assumption of small phase errors and three (or more) intensity measurements with known small phase offsets. One potential application is in extrasolar planet searches using successively higher order adaptive optics systems in tandem. Presently we shall summarize our derivation of the phase retrieval inversion in the small phase limit. The intensity-phase integral equation in discrete matrix form is i ⫽ (Fp) 䡩 (F* p), for p ⫽ w 䡩 e i ϕ ⬃ w 䡩 (1 ⫹ iϕ) in the small phase limit, for amplitude vector w, phase vector p and its complex conjugate p, and 2D Fourier matrix F and its complex transpose F*. Here 䡩 is the matrix Hadamard product, which in our context is simply a way of writing functional multiplication to enable us to


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Wild

keep the nomenclature simple. Upon decomposing F into the real and imaginary components the intensity i becomes i ⫽ [FR w ⫺ FI (w 䡩 ϕ)] 䡩 [FR w ⫺ FI (w 䡩 ϕ)]

(15)

⫹ [FI w ⫹ FR (w 䡩 ϕ)] 䡩 [FI w ⫹ FR (w 䡩 ϕ)]

If a small but known phase offset ϕ → ϕ ⫹ ϕk, for image i k , is applied Eq. (15) becomes 1 (i k ⫺ i ⫺ c k ) ⫽ [FR (w 䡩 ϕ k )] 䡩 [FR (w 䡩 ϕ)] 2 ⫹ [FI (w 䡩 ϕ k )] 䡩 [FI (w 䡩 ϕ)] where c k ⫽ [FR (w 䡩 ϕ k)] 䡩 [FR (w 䡩 ϕ k )] ⫹ [FI (w 䡩 ϕ k )] 䡩 [FI (w 䡩 ϕ k )] ⫹ 2[FR (w 䡩 ϕ k )] 䡩 [FI w] ⫺ 2[FI (w 䡩 ϕ k )] 䡩 [FR w] the latter being a known quantity. For three or more phase offsets Eq. (15) can be inverted algebraically to give w 䡩 ϕ ⫽ (FI ⫹ FR )⫺1

冦冢2 D1 冣䡩 [(i ⫺ i ⫺ c ) 䡩 (F ⫺ F )(w 䡩 ϕ ) ⫺ (i ⫺ i ⫺ c ) 䡩 (F ⫺ F )(w 䡩 ϕ )]冧 k

k

I

R

k′

kk′

k′

k′

I

R

k

where D kk′ ⬅ [FI (w 䡩 ϕ k ′)] 䡩 [FR (w 䡩 ϕ k )] ⫺ [FI (w 䡩 ϕ k )] 䡩 [FR (w 䡩 ϕ k′)]. The matrix (FI ⫹ FR )⫺1 has a condition number of unity and is invertible. Generalizations for higher terms in the expansion of the phase exist and are discussed in Wild (1998b). It is significant that the linearized phase retrieval problem has an exact solution, though it remains to be determined if the properties and noise characteristics associated with these formulae will enable a better understanding of the properties of the full phase retrieval problem (Kuznetsova 1988); further research is required. Note that pupil intensity fluctuations (scintillation) via w have been explicitly included in the mathematics. A. Branch Point Detection 1. Curl Annihilation Matrices The existence of branch points in the phase function (Fried and Vaughn 1992) is a concern over the highly scintillated paths encountered at low elevation angles, and in DoD applications such as the Airborne Laser. The presence of a branch point is accompanied by a nonzero curl in the slopes around the point so that



227

the reconstructed phase must be in the form of a Riemann sheet with a cut appropriately placed in the reconstructed phase. Iterative algorithms exist (Szeto 1997; Le Bigot et al. 1998) to perform the phase estimation from subaperture slope data making use of the fact that while the phase function is discontinuous the exponential function eiϕ is not. There are issues concerning finding the location of branch points unambiguously and how many can be treated with existing adaptive optics systems. We are investigating a new class of estimators called curl Annihilation matrices (CAMs), which localize the global curl so that it can be reconstructed. In essence the geometry matrix is designed for a pupil with zero weighting given to subapertures containing the branch point to the edge of the pupil; the cut subapertures are arbitrary. For branch point pairs with opposing slope curls, the cut subapertures need only be those joining the branch points. It can be shown that for a global curl circulation in slopes with the cut introduced into the A matrix the Riemann sheet is well reconstructed. A least-squares estimator will demonstrate significant waffle effects, while an optimal estimator will yield a smooth reconstruction in the noiseless case. Though it appears that for a fast adaptive optics system that the computational burden associated with generating the matrices as the branch point(s) move is insurmountable, there are two solutions. The first is to have a large matrix farm in memory and to select matrices with the cuts associated with branch points in specific subapertures. The other is to just have a standard estimator assuming no zero weighted subaperture cuts, but to ‘‘smash’’ the slopes along a cut to zero (equivalent to setting columns of the M matrix to zero). While the latter does not do a perfect reconstruction, it approximates the openloop subaperture cut case. In the closed-loop servo environment wherein several frames elapse before the branch point propagates to a neighboring subaperture the adaptive optics system itself acts as an iterative processor analogous to the iterative estimators (Wild et al. 1995a), and so the smashed slope CAMs may work well. Further mathematical research into the behavior of branch point reconstructors and CAMs for closed-loop servo systems is needed to appraise their performance and to seek optimal estimation techniques in the presence of noise and scintillation. We are in the process of doing an experiment with the WCE at Yerkes (Wild et al. 1995b) where we alternate between two matrices, one being a CAM with a zero weighting subaperture cut going to the center of the pupil, and the other a matrix with no cuts. Presumably, as branch points propagate across the pupil, using our horizontal path telescope configuration, when they enter the cut region the instantaneous Strehl should improve because the hidden phase will be reconstructed. A fast frame transfer CCD camera will be used to acquire compensated imagery, and we might expect a statistically larger number of high Strehl points in the temporal (short exposure) data with the CAM than without it. This experiment is being designed, and results will be reported in future publications.


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REFERENCES Angel, J. R. P., Nature 368, 203 (1994). Axelsson, O. Iterative Solution Methods (Cambridge University Press, 1994). Bibby, J., and H. Toutenburg, Prediction and Improved Estimation in Linear Models (Wiley, Chicester, 1977). Bruns, D. G., and E. Meyer, Proc. SPIE 2201, 962 (1994). Chun, M., Publ. Astron. Soc. Pac. 110, 317 (1998). Colucci, D., Ph.d. thesis, University of Arizona (1994). Ellerbroek, B. L., J. Opt. Soc. Am. A11, 783 (1994). Ellerbroek, B. L., C. VanLoan, N. P. Pitsianis, and R. J. Plemmons, J. Opt. Soc. Am. A11, 2871 (1994). Fried, D. L., and J. L. Vaughn, Appl. Opt. 31, 2865 (1992). Fugate, R. Q., B. L. Ellerbroek, C. H. Higgins, M. P. Jelonek, W. J. Lange, A. C. Slavin, W. J. Wild, D. M. Winker, J. M. Wynia, J. M. Spinhirne, B. R. Boeke, R. E. Ruane, J. F. Moroney, M. D. Oliker, D. W. Swindle, and R. A. Cleis, J. Opt. Soc. Am. A11, 310 (1994). Gajic, Z., and M. Qureshi, Lyapunov Matrix Equation in System Stability and Control (Academic Press, San Diego, 1995). Ghiglia, D. C. and L. A. Romero, J. Opt. Soc. Am. A13, 1999 (1996). Gonin, R. and A. H. Money, Nonlinear Lp-norm Estimation (Marcel Dekker, New York, 1989). Graham, A., Kronecker Products and Matrix Calculus: With Applications (Wiley, New York, 1981). Herrmann, J., J. Opt. Soc. Am. 70, 28 (1980). Herrmann, J., J. Opt. Soc. Am. 71, 989 (1981). Herrmann, J., J. Opt. Soc. Am. A9, 2257 (1992). Kuznetsova, T. I., Sov. Phys. Usp. 31, 364 (1988). Lane, R. G. and M. Tallon, Appl. Opt. 31, 6902 (1992). Le Bigot, E. O., W. J. Wild, and E. J. Kibblewhite, in press (1997). Le Bigot, E.-O., W. J. Wild, and E. J. Kibblewhite, Opt. Lett. 23, 7 (1998). Luenberger, D. G., Optimization by Vector Space Methods (Wiley, New York, 1969). Lutomirski, R. F., W. L. Woodie, and R. G. Buser, Appl. Opt. 16, 665 (1977). Primmerman, C. A., et al., Appl. Opt. 34, 2081 (1995). Roddier, N., Opt. Eng. 29, 1174 (1990). Roddier, F., M. J. Northcott, and J. E. Graves, Publ. Astron. Soc. Pac. 103, 131 (1991). Rogers, G. S., Matrix Derivatives (Marcel Dekker, New York, 1980). Sasiela, R. J., and J. G. Mooney, Proc SPIE 551, 170 (1985). Sivokon V. P. and M. A. Vorontsov, J. Opt. Soc. Am. A15, 234 (1998). Smutko, M., Publ. Astron. Soc. Pac. 109, 807 (1997). Szeto, R. K.-H., J. Opt. Soc. Am. A14, 1412 (1997). Tyler, G. A., and D. L. Fried, JOSA 72, 804 (1982). Wild, W. J., Opt. Lett. 21, 1433 (1996). Wild, W. J. Proc. SPIE 3126, 278 (1997). Wild, W. J. Opt. Lett. 23, 570 (1998a).



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Wild, W. J. Opt. Lett. 23, 573 (1998b). Wild, W., E. Kibblewhite, V. Scor, Proc. SPIE 2201, 726 (1994a). Wild, W., E. Kibblewhite, F. Shi, B. Carter, G. Kelderhouse, R. Vuilleumier, H. Manning, Proc. SPIE 2201, 1121 (1994b). Wild, W. J., E. Kibblewhite, and R. Vuilleumier, Opt. Lett. 20, 955 (1995a). Wild, W., E. J. Kibblewhite, R. Vuilleumier, V. Scor, F. Shi, and N. Farmiga, Proc. SPIE 2534, 194 (1995b). Wild, W., et al., Proc. SPIE 2534, 194 (1995b). Willsky, A. S., Proc. IEEE 66, 996 (1978).


7 Micromachined Membrane Deformable Mirrors Gleb Vdovin Delft University of Technology, Delft, The Netherlands

I. INTRODUCTION The high price of modern adaptive optics is explained by the lack of a uniform fabrication technology for their optical, mechanical and electrical constituents. Expensive adaptive optics are acceptable for applications where specialized custom-made systems are needed. The technology to date has been developed in the direction of system quality improvement, with little attention being given to system cost. Extension of adaptive optics into the field of mass-produced optoelectronics will only be possible with a reduction of the fabrication costs by a few orders of magnitude. The functionality and quality of devices and systems must be preserved or even improved during this transition. The market sector for low-end optoelectronics, such as bar-code scanners, CD player optical pickups, amateur telescopes, optical communication systems and displays, is comparable to the aforementioned high-end military and research markets. To cope with large volume demand, serial or even mass-production technologies should be developed for the fabrication of adaptive optics. A high system quality for low cost can be achieved by using semiconductor fabrication technology, which has recently been extended into the fields of micromechanics and microoptics. IC-compatible microfabrication of microoptoelectromechanical systems (MOEMS) (1) (see Fig. 1) has emerged to serve the demand for lowcost mass-produced optics compatible with standard silicon microelectronics. This new branch is based on well-developed silicon technology widely used for sensor production (2) together with recent developments in micromachining of optical components (3,4). This technology is receiving more and more attention,


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Figure 1 The field of microoptoelectromechanical systems is formed by the merging of three existing technologies.

and very promising results have recently been obtained in the fabrication of mechanical, electrical and optical sensors and actuators. Silicon-based microfabrication in its present state can form the technological foundation for inexpensive implementation of complete adaptive optical systems, as it allows for cheap fabrication of wavefront sensors, control computers and wavefront correctors in the framework of a uniform technology.

II. MATERIALS FOR MICROMACHINING Silicon and related materials, such as silicon nitride and silicon oxide, form the material’s base of silicon microfabrication. These materials have excellent mechanical properties and can be formed with standard IC processing on the surface of a silicon wafer in the form of thin patterned layers. Mechanical properties of the most frequently used construction materials, including those used for microfabrication, are shown in Table 1.

III. TILT CORRECTORS AND SCANNERS Wavefront tilts are statistically important in aberrated optical fields of different origins. Actually, wavefront tilts represent the main aberration component in ground-based astronomical systems with input pupil size of the order of tens of


Diamond TiC SiC Nitride (Si 3 N 4 ) Al 2 O 3 Iron Si SiO 2 fiber Steel W Stainless steel Mo Al

Yield strength (10 10 d/cm 2)

Knoop hardness (kg/mm 2)

Young’s modulus (10 12 d/cm 2 )

Density (g/cm 3)

Thermal conductivity (W/cm°C)

Thermal expansion (10⫺6 /°C)

53 20 21 14 15.4 12.6 7.0 8.4 4.2 4.0 2.1 2.1 0.17

7000 2470 2480 3486 2100 400 850 820 1500 485 660 275 130

10.35 4.97 7.0 3.85 5.3 1.96 1.9 0.73 2.1 4.1 2.0 3.43 0.70

3.5 4.9 3.2 3.1 4.0 7.8 2.3 2.5 7.9 19.3 7.9 10.3 2.7

20 3.3 3.5 0.19 0.5 0.803 1.57 0.014 0.97 1.78 0.329 1.38 2.36

1.0 6.4 3.3 0.8 5.4 12 2.33 0.55 12 4.5 17.3 5.0 25

Micromachined Membrane Deformable Mirrors

Table 1 Properties of Materials for Microfabrication

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centimeters. As flexible adaptive mirrors have very poor tilt correction performance, specialized devices have been developed for dynamic correction of tilts in traditional adaptive optics and microoptics. Bulk micromachining was applied to the fabrication of a resonant micromachined scanner with a light aperture of a few millimeters by K. E. Petersen of IBM Research Laboratories in 1980 (5). The scheme of the micromachined tilt corrector is shown in Fig. 2. The corrector mirror is etched together with the torsion bars in the bulk of a silicon wafer, using anisotropic etching. The etched mirror is fixed over the electrode structure, supported by a ridge in the center. The support structure with electrodes is fabricated on a glass wafer. The small static angular deflection of the mirror, ϕ, in response to the voltage, Vc , applied to one of the electrodes is given by ϕ⫽

ε 0 (Vc ⫹ Vb ) 2lb 3 (1 ⫹ ν) 16K Ed 2t 4

(1)

where ε 0 is the permittivity of the vacuum, l is the length of the silicon torsion bar, b is the dimension of the square mirror, ν ⫽ 0.09 (Poisson’s ratio of silicon), K is a constant dependent on the cross section of the torsion bar (K ⫽ 0.24 for the pyramidal bar cross section, typically obtained with anisotropic etching), d is the distance between the mirror and the electrode, and t is the thickness of the wafer. Equation (1) is valid for small angles of deflection. When the deflection bϕ/2 is comparable to the thickness of the air gap d, the mirror behavior becomes unstable and the mirror lands on the electrode. The static angular sensitivity of the device for Vc ⬍⬍ Vb is proportional to the bias voltage: dϕ ε V lb 3 (1 ⫹ ν) ⬇ 0 b dVc 8K Ed 2t 4

a

(2)

b

Figure 2 Bulk micromachined mirror of an electromechanical scanner (left) and cross section of the device.



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In the case of Vb ⫽ 0, the angle of deflection is proportional to the applied voltage and the tilt angle does not depend on the sign of the control voltage Vc . Introduction of the bias voltage Vb linearizes the response and allows for bidirectional control of the mirror deflection by changing the control voltage Vc on a single electrode. The resonant frequency of the scanner is given by fr ⫽

1 2π

√

12K Et 3 ρlb 4 (1 ⫹ ν)

(3)

where ρ is the density of silicon. The quality factor Q cannot be analytically estimated because it depends on viscous air damping (humidity, pressure, temperature) and proximity effects. Experimentally measured values of Q lie in the range of 10–20, while the intrinsic Q of the silicon mechanical structure is at least one order of magnitude higher. The stress in the torsion bar at the maximum mirror deflection—when the mirror is contact to the electrode—is given by σmax ⫽

冢冣

38K E td 2(1 ⫹ ν) lb

(4)

The static deformation of the scanner mirror due to electrostatic biasing is given by δmax ⫽

3ε 0V 2b 4 32Ed 2t 3

(5)

and the dynamic deformation of the mirror due to the bar torque, while operated at the resonant frequency with angular amplitude φ max, is given by δd ⫽ 0.226

冢冣

ρb 5 2φ max f 2(1 ⫺ ν 2) Et 2

(6)

Micromachined scanners can be scaled to light apertures of up to a few square centimeters, with angular sensitivity in the order of 10⫺3 rad/V in a frequency range of at least a few hundred hertz.

IV. GENERAL PRINCIPLES OF MMDM Bulk silicon micromachining is well suited for the fabrication of micromachined membrane deformable mirrors (MMDM) [6,9,10]. An MMDM is formed by a


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thin flexible reflective membrane that is stretched over an array of electrostatic actuators—see Fig. 3. Electrically grounded conductive membrane can be locally deflected by applying voltages to these actuators so that individual responses superimpose to form the necessary optical figure. The mirror consists of two parts: • The die with the flexible mirror membrane which is mounted over the actuator structure. • The actuator structure, obtained by patterning the Al interconnection layer on the surface of a Si wafer, insulated by a comparatiavely thick– up to 1µm–layer of silicon dioxide [9]. Complex electrode structures can be integrated in silicon together with control and switching electronics. Electrode structures can be patterned on a glass wafer [6] or directly in the metalization layer of a printed circuit board (PCB). The latter option allows for inexpensive hybrid integration of adaptive optical devices together with analog control electronics based on discrete components. The active part of a MMDM is formed by a low-stress nitride membrane. To make the membrane reflective and conductive, the etched side is coated with a thin layer of evaporated metal, usually aluminum or gold. Aluminum adheres well to many of the materials used in the semiconductor industry and micromachining. Gold coatings demand special treatment of the surface, usually an intermediate layer of Cr is deposited to improve the adhesion. Reflective membranes, fabricated with this technology have a good optical quality.

Figure 3 Principle of MMDM.



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Silicon nitride is chosen as the membrane material for the following reasons: • Nitride is a mechanically strong material, compatible with both bipolar and CMOS microelectronic processes. When a wafer with membranes is carefully handled during etching and cleaning, fabrication of comparatively large–up to one inch in diameter–nitride membranes is possible with a relatively high yield. • Nitride deposition allows precise control of the stress in the nitride layer, so tensile stressed membranes can eaily be obtained. After the silicon substrate is etched away, the shape of stretched membrane depends only on the boundary conditions. Since it covers an opening in an optically—flat silicon wafer, the released membrane is at least as flat as the surface of the substrate. • When selective etchants are used, the surface of the nitride layer is not damaged by etching, thus having a very low roughness since it replicates the surface of the highly polished wafer. • Silicon is a good insulator. Thus the reflective layer (Al or Cr/Au) can be used as a capacitor plate for the electrostatic control of the membrane shape with a little chance of a short circuit, even if the membrane sticks to the electrode structure. Both isotropic (6) and anisotropic (7,8) etchants have been used to fabricate flexible membranes for micromachined deformable mirrors. Isotropic etching produces an aperture with a continuous smooth contour, but isotropic etchants normally have a lower selectivity, so the nitride membrane is also etched. Anisotropic etching—for example in a water/KOH solution—is a very selective process preserving the high quality of the nitride surface, but resulting in a square or a rectangular aperture. A special compensation mask should be used to obtain an approximation to a circular aperture (9). Assembly of the reflective membrane with the actuator structure should ensure a good uniformity of the air gap and introduce no additional stress or deformation into the mirror chip. Both bonding to a glass wafer (6), and hybrid assembly of two silicon chips (7) have been used. The fabrication sequences for glass-bonding technology and for IC-compatible hybrid flip-chip technology are illustrated in Fig. 4. It has been experimentally shown that a membrane model (stiffness neglected) can be used for static description of MMDM response. A thin plate model is not applicable to membranes with aspect ratios in the order of a few thousand. The deflection U(x,y) of the stretched membrane under an external load P(x,y) is given by the Poisson equation (10) ∆U(x,y) ⫽

⫺P(x,y) T


(7)

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a

b

c

Figure 4 Fabrication sequence for MMDM based on silicon-to-glass bonding (6) (left), IC-compatible hybrid assembly (7) (center), inexpensive PCB-based (right).

where, for the case of electrostatic actuation, P(x,y) is given by (11) P(x,y) ⫽

εε 0 (V(x,y)) 2 (d(x,y,P))2

(8)

T is the membrane tension given by T⫽

Ehδ 2 2(1 ⫺ ν)

(9)

where εε 0 is the dielectric constant of air, V(x,y) is the potential distribution on the actuator structure, d(x,y) is the distance between the membrane and the actuator structure (dependent in the general case on the membrane deflection), E is Young’s modulus of the membrane material, h is the thickness of the membrane, ν is the Poisson ratio of the membrane material, and δ is the in-plane membrane elongation due to stretching. Here the change of the membrane tension caused by membrane deflection is considered to be negligible. Equations (7)–(9) must be supplied with a set of boundary conditions, describing the shape of the membrane contour: S c ⫽ F(x,y). As the membrane is initially fabricated on the surface of a silicon wafer, which is not specially prepared as an optical component, the initial aberration may be unacceptably large. Fortunately, after etching the silicon substrate, the surface takes on the natural shape of a tensed membrane, which is only dependent



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on the wafer shape along the contour of support. The membrane tension eliminates all the radial components initially introduced in the wafer shape, leaving intact the azimuthal components which are dependent on the contour shape. This means that astigmatism is the most significant aberration of membranes micromachined in silicon. Membranes with a characteristic size of 10 mm are typically astigmatic initially with the amplitude of aberration in the range of 0.1– 2 µm. Released nitride membranes demonstrate an extremely low level of roughness, since the surface of the released membrane replicates the surface of a highly polished silicon wafer. This high quality is preserved during the anisotropic etching of the silicon in KOH since KOH leaves the nitride surface practically intact. Isotropic etchants, such as HF/HNO 3 , are in general more aggressive to the nitride but special ‘‘polishing’’ compositions do not reduce the surface quality. The static model of membrane deformation can be used to describe the mirror behavior at frequencies much lower than the frequency of the first resonance fr, given by fr ⫽ α

√

T ρhA

(10)

where α is a coefficient depending on the shape of the membrane. The values of α for different shapes of membrane are shown in Table 2.

Table 2 Values of the Coefficient α in Eq. (10) for the Frequency of the First Resonance of Tensed Membrane Circle Square Qadrant of a circle Sector of a circle 60° Rectangle 3 ⫻ 2 Equilateral triangle Semicircle Rectangle 2 ⫻ 1 Rectangle 3 ⫻ 1

α α α α α α α α α

⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽

2.404√π ⫽ 4.261 √2π ⫽ 4.443 2.56√π ⫽ 4.551 6.379√π/6 ⫽ 4.616 √13/6π ⫽ 4.624 2π√tan(30°) ⫽ 4.774 3.832√π/2 ⫽ 4.803 π√5/2 ⫽ 4.967 π√10/3 ⫽ 5.736

Source: From Ref. 12.


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Figure 5 Defocus corrector mounted on a PCB.

V. DEFOCUS CORRECTORS Inexpensive micromachining of approximately circular nitride membranes facilitates the fabrication of varifocal mirrors. In such a mirror the reflective membrane is deflected by a single electrode to provide the possibility of fast control of the optical power in the range of a few diopters. Optical quality, speed of response, aberrations and sensitivity of such a device are all complex functions of design parameters and technology. The most common case is represented by a circular nitride membrane, fabricated with anisotropic etching and suspended 50–100 µm over a conductive substrate, such as an Al-coated silicon chip or PCB metallization. The assembled circular membrane mirror, consisting of a 10-mm-diameter membrane mounted at a distance of 75 µm over a single control electrode is shown in Fig. 5. Typical interferograms of the initial and deformed mirror surfaces are shown in Fig. 6. 1.

2.

3.

Spherical aberration, caused by the nonlinearity of the force field between the actuator and the deformed membrane. The central part of the deflected membrane has a higher curvature because it is subjected to a higher electrostatic pressure. The pressure nonuniformity may reach 10–20% in the typical case of a mirror having a diameter of 1 cm, gap thickness of 70 µm and focal distance range ∞ . . . 1 m. Coma-like aberrations due to membrane misalignment with respect to the electrode. It was experimentally shown that the gap nonuniformity of 1.5% in a MMDM with 10 mm diameter reduces the Strehl ratio to a value of 0.1 at a focal distance of 1 m. Initial astigmatism of the membrane caused by the deformation of the silicon frame.



a

241

b

Figure 6 Interferogram of the initial mirror surface with an active diameter of 10 mm (left); the same surface deformed by a control voltage of 90 V, corresponding to a focal distance of 1 m (right).

Spherical aberration of a membrane defocus corrector can be reduced by increasing the gap between the electrode and the membrane. The control voltage required is proportional to the square of the gap thickness; therefore this method is not very useful in practice, since a ‘‘linearized’’ mirror will require very high control voltages. Spherical aberration can also be reduced by using the actuator structure, consisting of several concentric rings. As a rough approximation, the voltage applied to each of these rings must be inversely proportional to the square of the average distance between the chosen actuator and the deformed membrane to achieve a uniform distribution of the membrane curvature. Computer simulation of this situation and some practical recommendations can be found in Ref. 13. Aberrations due to the membrane misalignment can be reduced by fabrication of the spacer and the actuator structure in a single technological process. Mounting over PCB holders with a spacer patterned in the antisoldering layer provides a better gap uniformity than mounting over silicon structures with the help of an external spacer. Further adjustments can be achieved with simple 2 ⫻ 2 square actuator patterns. Individual actuator voltages can be preadjusted using a set of voltage dividers; the entire device can then be controlled by a single external voltage. The dynamics of the response of the defocus corrector depends on the membrane resonant frequency and the air damping. Usually the air damping defines cutoff frequency in the range 50–1000 Hz (7), which is much lower than the resonant frequency of the membrane. The damping depends on the ambient gas pressure and the distance to the electrode structure. Perforation of the actuator structure, facilitating air flow under the membrane, increases the cutoff frequency by reducing the air damping.


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Optical setups for the correction of image focus are shown in Fig. 7. In the first configuration, the adaptive mirror replaces the reflective coating on one of the pentaprism surfaces. The angle of axial incidence on the mirror surface equals ϕ ⫽ arcsin (n sin 22.5°) ⬃ 34°, where n is the refraction coefficient of the prism material, so the deformed mirror introduces a considerable amount of astigmatism into the wavefront transmitted by the system. The angular field is limited to approximately 24° by the prism system and the image is inverted. The described system was used in combination with a CCD camera, equipped with a 50 mm F1/5 lens focused to infinity (14). The system was quickly focused

a

b Figure 7 Configurations for focus adjustment (top); two consequent frames (time separation 40 ms) obtained with a CCD camera by focusing the MMDM to a distance of 75 cm and 50 cm with exposition of 30 ms and switching time 10 ms (bottom).



243

using the MMDM in the range from infinity (zero control voltage) to 30 cm (control voltage of 170 V). An example of focus switching between two objects placed at distances 75 cm and 50 cm, respectively, is shown in Fig. 7. The observed image quality at distances shorter than 1 m was slightly lower than that obtained by direct manual focusing of the camera lens, but still acceptable as all important details were resolved. Focusing in the range 1 m to ∞ produced images with a quality indistinguishable from direct lens focusing. The PCB-mounted micromachined varifocal mirrors are simple, inexpensive, fast and reliable. A typical single electrode corrector, having an aperture of 10 mm with a usable light aperture of 6–8 mm, demonstrates a reasonable optical quality (with peak-valley deviation from the reference parabola of less than one wavelength) in a focal distance range of ∞ . . . 75 cm under control voltage in the range of 0–150 V. The optical quality of the mirror can be further improved by using more complex actuator patterns to correct the aberrations caused by the membrane misalignment.

VI. MULTICHANNEL DEFORMABLE MIRRORS All components of a MMDM except the reflective membrane can be implemented using printed circuit board (PCB) technology. An example of a 55-channel MMDM fabricated using PCB technology is shown in Fig. 8. Hexagonal actuators are connected to conducting tracks on the back side of the PCB by means of vias (metalized holes). These holes reduce the air damping, extending the linear range of the frequency response of a micromachined mirror to at least 1

Figure 8 PCB-based MMDM with a clear aperture of 25 mm; the membrane on the left device is removed making visible the hexagonal arrangement of 55 electrostatic actuators.


a

b Figure 9 MMDM response to a meander input signal with the vias in the substrate open (top) and sealed (bottom).

Figure 10 Interferogram of the initial optical figure of a circular 37-channel MMDM (top left), mirror responses to different combinations of control voltages.



245

Figure 11 Biased operation of micromachined adaptive mirror. All possible mirror deflections are limited by the initial surface and the surface of maximum deflections.

kHz, which is much better than for similar devices mounted over planar silicon dies (7,13). Optically measured MMDM response to a meander input is shown in Fig. 9 for a device with open and closed vias in the substrate. The optical quality of a typical MMDM is demonstrated in Fig. 10. The reflective surface of a MMDM can be deflected only in only one direc-

a

b Figure 12 Experimentally registered far-field intensity distribution produced after reflection from MMDM biased with constant voltage applied to all actuators (top) and MMDM with bias voltage optimized to produce spherical figure (bottom).


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a

b Figure 13 Experimentally registered far-field intensity distribution produced by reflection of an aberrated beam from MMDM (top) and intensity distribution after the aberration was corrected with a 37-channel MMDM (bottom).

tion (toward the actuators). To achieve bidirectional movement of the membrane, the mirror should be electrically biased as shown in Fig. 11. The optical figure of a mirror biased by equal voltages applied to all actuators is distorted due to fabrication misalignments. The bias voltage could be optimized and the membrane set to a perfectly spherical shape by using simple optimization algorithms with a pinhole as a source of reference spherical wave. Figure 12 illustrates the improvement in the far-field quality achieved by the optimization of the bias voltage. Figure 13 illustrates the far-field correction achieved by the same mirror after aberration was introduced into the biased optical system. Multichannel deformable mirrors can be used to form precision aspherical shapes to correct aberrations of low-power laser beams, for image stabilization in astronomy and optical instrumentation and as low-resolution spatial light modula-



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tors (15). Introduction of etching compensation techniques and mounting mirrors on simple PCB carriers significantly reduces the complexity of fabrication, providing ‘‘inexpensive’’ adaptive optics. Of course this simple technology is not readily scalable to mirrors with hundreds of control channels. More complex devices will require the integration of switching and driver electronics together with the integrated electrode structures. The development in this direction can eventually lead to integration of whole functional blocks of adaptive optical systems.

REFERENCES 1. M. Mehregany, Microelectromechanical systems, IEEE Circuits and Devices, 14– 22 (July 1993). 2. S. Middelhoek, S. A. Audet, Silicon Sensors, Academic Press (1989). 3. K. H. Brenner, M. Kufner, S. Kufner, J. Moisel, A. Mu¨ller, S. Sinzinger, M. Testorf, J. Go¨ttert, J. Morf, Application of three-dimensional micro-optical components formed by lithography, electroforming, and plastic molding, Applied Optics 32, 6464–6469 (1993). 4. T. A. Kwa, R. F. Wolfenbuttel, Integrated grating/detector array fabricated in silicon using micromachining techniques, Sensors and Actuators A31, 259–266 (1992). 5. K. E. Petersen, Silicon torsional scanning mirror, IBM Journal of Research and Development 24, 631–637 (1980). 6. L. M. Miller, M. L. Argonin, R. K. Bartman, W. J. Kaiser, T. W. Kenny, R. L. Norton, E. C. Vote, Fabrication and characterization of micromachined deformable mirror for adaptive optics applications, Proc. SPIE 1954, 421–430 (1993). 7. G. V. Vdovin, P. M. Sarro, Flexible mirror micromachined in silicon, Applied Optics 34, 2968–2972 (1995). 8. G. V. Vdovin, P. M. Sarro, S. Middelhoek, Technology and applications of micromachined silicon adaptive mirrors, Optical Engineering 36, 1382–1390 (1997). 9. G. V. Vdovin, S. Middelhoek, P. M. Sarro, Thin-film free-space optical components micromachined in silicon, in Digest of IEEE/LEOS topical meeting on optical MEMS and their applications, pp. 5–6, Keystone, Colorado, 1996. 10. S. Timoshenko, S. Woinowsky-krieger, Theory of Plates and Shells, McGraw-Hill (1959). 11. S. Attwood, Electric and Magnetic Fields, Wiley (1949). 12. J. W. Rayleigh, The Theory of Sound, Dover (1945). 13. G. V. Vdovin, S. Middelhoek, M. Bartek, P. M. Sarro, D. Solomatine, Technology, characterization and applications of adaptive mirrors fabricated with IC-compatible micromachining, in Proc. SPIE 2534, 116–129 (1995). 14. G. Vdovin, Quick focusing of imaging optics using micromachined adaptive mirrors, Optics Communications 140, 187–190 (1997). 15. G. V. Vdovin, Spatial light modulator based on the control of the wavefront curvature, Optics Communications 115, 170–178 (1995).


8 Surface Micromachined Deformable Mirrors William D. Cowan Air Force Institute of Technology, Wright-Patterson AFB, Ohio

Victor M. Bright University of Colorado at Boulder, Boulder, Colorado

Microelectromechanical systems (MEMS) is a rapidly emerging technology that employs batch fabrication processes similar to those of the integrated circuit industry to fabricate miniature electromechanical parts. The use of MEMS techniques to fabricate deformable mirrors for adaptive optics represents a nearly ideal match of technology and application. Optical system apertures are readily scaled to MEMS dimensions. The small motion ranges of MEMS are well suited to optical phase modulation. The conventionally manufactured deformable mirrors currently in use are large, heavy, and power hungry. Perhaps most importantly, current deformable mirrors are very expensive. Using MEMS batch fabrication techniques the cost of deformable mirrors for adaptive optics can potentially be reduced by a factor of 1000. As deformable mirror cost decreases, the number of practical applications increases. For many applications, a low-cost deformable mirror may be the enabling technology. The small size, weight, and power dissipation of a microfabricated part are also critical for airborne or spaceborne adaptive optics applications.

I. SURFACE MICROMACHINING TECHNOLOGY Surface micromachined devices are constructed from thin films deposited on the surface of the wafer. The films are then selectively etched to form a MEMS


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device. This technology borrows heavily from integrated circuit fabrication processes, and thus surface micromachined MEMS have characteristic planar surfaces with topology greatly influenced by the underlying layers. Sacrificial layers of phosphosilicate glass (PSG) are often used to isolate adjacent polycrystalline silicon (polysilicon) film layers. Phosphosilicate glass is silicon oxide doped with phosphorous. The PSG layers are dissolved at the end of device fabrication, creating device components from the polysilicon layers. The thickness and patterning of the films can be controlled to finer tolerances than typically possible in bulk micromachining, making surface micromachining the preferred technique for the fabrication of the smallest MEMS devices. In general, surface micromachining fabrication processes are similar, differing only in the number of releasable layers, thickness of each layer, and the materials used for each layer. A popular commercial surface micromachining process for MEMS in the United States is the Multi-User MEMS Processes (MUMPs) (1). MUMPs is representative of many polysilicon surface micromachining processes. MUMPs offers three patternable layers of polysilicon and two sacrificial layers of PSG on a base layer of silicon nitride. A top layer of gold is provided as the reflective and/or conductive surface. Table 1 identifies the layer thickness for each of the films used in MUMPs. The order of the entries in Table 1 is consistent with the deposition order of the films on the silicon wafer substrate, with silicon nitride being the first layer. Gold is evaporated onto the device after all other layers have been deposited by low pressure chemical vapor deposition. The polysilicon layers and the 〈100〉-cut silicon substrate are highly doped with phosphorus (approximately 1020 atoms-cm⫺3) to decrease electrical resistance. After construction the micromachined device is ‘‘released’’ by removing the PSG layers in a bath of hydrofluoric acid. After the devices are released, differing residual material stresses in the layers may cause undesirable curvature of the mirror surfaces (2,3). Control of residual material stresses and device design to reduce stress induced

Table 1 Structural and Sacrificial Layers Used in MUMPs Layer name Nitride (silicon nitride) Poly-0 (bottom polysilicon layer) 1st oxide (sacrificial layer–phosphosilicate glass) Poly-1 (middle polysilicon layer) 2nd oxide (sacrificial layer–phosphosilicate glass) Poly-2 (top polysilicon layer) Metal (gold) Source: From Ref. 1.


Nominal thickness (µm) 0.6 0.5 2.0 2.0 0.75 1.5 0.5

Micromachined Deformable Mirrors

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micromirror curvature are the principal challenges in surface micromachining of deformable mirrors. In surface micromachining, the thin film layers conform closely to the topology of the previously deposited and patterned layers. Unless the designer ensures a layer is flat by controlling the patterning of the layers beneath it, the induced topology can have detrimental effects on optical performance. The addition of a chemical mechanical polishing (CMP) step prior to deposition of the final structural layer effectively eliminates print-through of the underlying layers, yielding flat mirror surfaces and greatly increasing design flexibility. CMP is employed in Sandia’s Ultra Planar Multi-level MEMS Technology (SUMMiT) surface micromachining process (4).

II. MICROMIRROR ACTUATION MECHANISMS The most common actuation mechanisms in surface micromachined devices are electrostatic attraction and thermal expansion. Although arrays of thermally actuated piston micromirrors have been demonstrated (5), at the time of this writing, electrostatic actuation prevails as the dominant drive mechanism for surface micromachined deformable mirrors. The basic electrostatic piston micromirror structure consists of two parallel-plate electrodes separated by a dielectric gap, as shown in Fig. 1. Usually the gap is filled with free space or air, but other

Figure 1 Schematic view of basic electrostatic piston micromirror.


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Cowan and Bright

dielectric media are possible. Thermal models suggest that filling the gap with thermally conductive gases, such as helium, will substantially increase the optical power handling capacity of micromirror systems (6,7). To allow piston travel the upper electrode or mirror plate is supported by spring flexures with a total linear spring constant, k. The fixed lower electrode is attached to the substrate. When a voltage (V ) is applied across the electrodes the attractive force (F) between the plates is found by integrating the charge difference across the overlapping electrode areas. For typical piston micromirror geometries, fringing fields and deformation of the mirror plates can be neglected (8). Integrating the charge yields the nonlinear electrostatic force, F⫽

Aε 0V 2 2g 2

(1)

where A is the overlapping electrode area, ε0 is the dielectric constant of air (8.854 ⫻ 10⫺12 F/m), V is the voltage across the electrodes, and g is the gap between the electrodes. Because the upper electrode moves it is represented by the asfabricated plate height minus the deflection of the plate, g ⫽ h ⫺ d. F⫽

Aε 0 V 2 2(h ⫺ d)2

(2)

For small deflections the counter force applied by the linear spring flexures is F ⫽ kd, from Hooke’s law. The force balance equation for the system is thus kd ⫽

Aε 0 V 2 2(h ⫺ d )2

(3)

Solving Eq. (3) for voltage yields V ⫽ (h ⫺ d)

√

2kd ε0 A

(4)

Equation (4) provides a ready means of calculating the control voltage required for a desired deflection once the electrode area and spring constant are known. The spring constant represents the largest source of error in modeling the piston micromirror system. The number of flexures, shape of the flexures, and the mechanical properties of the flexure material determine the total spring constant. Of the factors influencing the spring constant, the mechanical properties of the flexure material comprise the largest uncertainty. For pure piston travel, each of the flexures of length l supporting the movable upper electrode are constrained to have zero slope at both ends. Thus, an n flexure system is modeled as n/2 fixed-fixed beams of length L ⫽ 2l, loaded by a force (F) in the beam center. The applied force must then be divided among



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each pair of flexures so the force supported by each fixed-fixed beam is 2F/n. Substituting the deflection formula for a fixed-fixed beam yields (9) d⫽

FL 3 (2F/n)(2l )3 Fl 3 ⫽ ⫽ 16Ewt 3 16Ewt 3 nEwt 3

(5)

where E is the elastic modulus of the flexure material, w is the flexure width, and t is the flexure thickness. Again using Hooke’s law, d ⫽ F/k, the crosssectional spring constant from Eq. (5) is nEwt 3 l3

k cs ⫽

(6)

A first order approximation is used to account for the contribution of residual material stress to the total spring constant (9): ks ⫽

nσ(1 ⫺ ν)wt 2l

(7)

where σ is the residual material stress and ν is the Poisson ratio of the flexure material. The total spring constant k ⫽ k cs ⫹ k s can be written as k⫽n

冤

冥

Ewt 3 σ(1 ⫺ ν)wt ⫹ l3 2l

(8)

The residual stress, σ, is negative for compressive residual stress and positive for tensile residual stress, thus decreasing or increasing the spring constant accordingly. Typically, the stress term is small compared to the cross-sectional term and can be neglected for design estimates. Equations (8) and (4) provide a convenient means of estimating control voltage requirements. Once the spring constant is known, either from Eq. (8) or a fit of measured data to Eq. (4), the resonant frequency of the flexure beam system can be estimated. Using a simple suspended mass model the resonant frequency in Hertz is given by FRes. ⫽

1 2π

√

k M

(9)

where k is the total spring constant and M is the suspended mass. The mass is obtained by summing the volume-material density products of the layers comprising the suspended upper electrode. Equation (4) neglects a common phenomenon of electrostatic devices. As the deflection of the upper electrode approaches one-third the total gap distance, the nonlinear electrostatic force increases much more rapidly than the restoring linear force of the spring flexures. As a result the system becomes unstable, and


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the upper electrode snaps down to the fully deflected position (10). If the upper and lower electrodes come into contact with each other they can be permanently stuck together, destroying the device. This characteristic snap-through instability behavior limits controllable deflection to less than approximately one-third the sacrificial oxide layer thickness for surface micromachined devices. For segmented mirror designs the restricted deflection range has little impact because only a λ/2 deflection range is required for 2π modulation, and a modulo λ/2 control scheme can be employed (11). For continuous facesheet designs the restricted controllable deflection range may limit aberration correction ability. Instead of controlling the voltage applied across the electrodes, it is possible to segment the bottom electrode into N digitally scaled areas. By selectively energizing N electrode areas with a digital control word, 2N discrete deflections are obtained. The ‘‘digital deflection’’ approach eliminates the many digital-to-analog (D/A) converters required to individually control large arrays of micromirror devices, greatly reducing the overall cost of the adaptive optics system. The digital deflection scheme also eases fabrication requirements for fully integrated deformable mirror systems on a chip. The digital deflection micromirror concept is illustrated in Fig. 2, which depicts a 4-bit device. The only change from the basic micromirror structure shown in Fig. 1 is the segmentation of the underlying control electrode into four digitally scaled areas such that 8A 1 ⫽ 4A 2 ⫽ 2A 4 ⫽ 1A 8. Deflection of the device is controlled by switching the electrodes segments on (V volts) or off (0 volts). Assuming the same on-voltage (V ) is used for all electrode segments, solving Eq. (3) for deflection yields

Figure 2 Conceptual view of digital deflection micromirror.



d⫽

εV 2 (A 1 b 1 ⫹ A 2 b 2 ⫹ A 4 b 4 ⫹ A 8 b 8 ) 2k(t ⫺ d )2

255

(10)

where b 8 b 4 b 2 b 1 is a 4-bit digital control word comprised of 1s and 0s. The nonlinear gap term, (t ⫺ d )⫺2, in Eq. (10) precludes linear deflection versus control word, but discrete deflection is much closer to linear than analog voltage controlled deflection because V is held constant. As a result the digital control scheme yields better position resolution than obtained using a linear D/A with N bits for voltage control (6). The simulations shown in Fig. 3 demonstrate the efficacy of optical phase control using only a few bits. An 8 ⫻ 8 array of digital deflection micromirrors with 203-µm center-to-center spacing illuminated over a 1.6-mm circular aperture comprises the simulated system. Defocus aberrations (approximated by a spherical wavefront) are corrected by the mirror array with varying deflection constraints imposed on the piston elements. The smooth curve shows the optimal correction case (analog position control). The shape of the analog correction curve is defined by the residual phase error due to the finite micromirror element size. For comparison 3-, 4-, and 5-bit linear deflection are shown. Also shown are simulated correction results obtained using the measured deflections obtained with a 4-bit prototype digital deflection micromirror.

Figure 3 Simulated optical aberration correction performance for digital deflection micromirror arrays compared to the analog controlled case. The performance metric plotted is the peak intensity of the optical point spread function (relative to the unaberrated case) versus defocus (spherical) aberration radius of curvature.


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The digital deflection micromirror is analogous to digitally wired piezoelectric actuator stacks (12). Although coarse phase discretization may prove inadequate for some applications, the cost savings in control electronics, potential speed advantages, and reduced system complexity may enable new applications. The simulation results in Fig. 3 show that a 3-bit digital deflection mirror accomplishes ⬃90% of the defocus aberration correction possible. The greatest payoff of the digital deflection micromirror scheme will be realized when the mirror elements, electrode switching circuits, a register (of length N ⫻ #mirrors), and interface circuitry are integrated on a die. The footprint of N switching transistors and resistors is much smaller than any D/A circuit which can provide equivalent mirror deflection resolution. Processes capable of fabricating an integrated deformable mirror system on a chip using the digital electrostatic deflection scheme exist (4).

III. MICROMIRROR ARRAY DESIGN TRADES Yield, deflection uniformity, and optical flatness are critical if micromirror/actuator arrays are to fully enjoy the cost and performance benefits of microfabrication. Without a planarization step, surface micromachined segmented deformable mirror designs require a trade of micromirror/actuator size, flexure geometry, and wiring depth. For a given flexure size and wiring depth, the array fill factor is improved by making larger mirror elements. The increased electrode area of larger mirror elements also lowers the required control voltage for a given deflection, which is usually desirable. Because the optical input to a micromirror array is readily scaled, the only real limitation to making larger micromirror elements is the requirement to maintain optically flat mirror surfaces. As element size increases it becomes more difficult to control deformation of the mirror surface due to residual material stresses. Without planarization, wiring of the control electrodes for individually addressed mirror or actuator elements constrains array design. The fill factor for micromirror arrays fabricated in conformal processes can be improved by employing ‘‘self-planarization.’’ In this design approach, the underlying topography is controlled to minimize print-through deformation of the mirror surface. For the MUMPs process maximum gaps in any layer of 1.5 µm yield reasonably ‘‘good’’ self-planarization results (13). Despite its limitations, self-planarization is helpful because only the surface areas over underlying gaps are not planar. All planar regions at uniform height contribute to the optical fill factor of a micromirror array. Two approaches to optical aberration correction using segmented mirrors are discussed in the following sections: development of segmented mirrors with optimized optical characteristics, and use of a refractive lenslet array to focus



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light onto the center of mirror elements. Lenslet use greatly simplifies mirror design, because the lenslet array defines the fill factor of the hybrid correcting element and eliminates background interference with the mirror support structures and wiring. Mirror curvature problems are also mitigated because the mirror surfaces can be smaller. But analysis and experimental results for a lenslet system show that the lenslet/mirror hybrid correcting element is not a solution for all optical phase modulation applications. When the incident optical signal is highly aberrated, the lenslet/mirror geometry limits performance of the hybrid correcting element.

IV. BARE SEGMENTED MICROMIRROR ARRAY OPTICAL PERFORMANCE A surface micromachined micromirror array with hexagonal elements was used in the first reported experimental demonstration of optical aberration correction using a microfabricated mirror device (14). Measured correction results were limited by the optical characteristics of the hexagonal microelectromechanical deformable mirror (MEM-DM). The hexagonal MEM-DM exhibited low fill factor (⬃40%), high mirror curvature (⬃400 nm) due to residual metal stress, and high static background interference because the support structures were metallized. In addition, only 61 mirror elements were controlled. A square-shaped micromirror array was later implemented. One element of this array (hereafter referred to as M19) is shown in Fig. 4. The 12 ⫻ 12 array has 128 individually

Figure 4 Corner of 12 ⫻ 12 element segmented piston micromirror array without metallization.


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addressable elements with 203 µm spacing. The design is an optimized trade of optical fill factor, functional device yield, address wiring, and mirror plate flatness. The mirror plate is of trapped oxide construction for maximum stiffness. The surface of the mirror plate is degraded only by four (4 µm ⫻ 4 µm) etch access holes, and the vias attaching the two polysilicon layers. Four flexures (4 µm wide by 65 µm long) support the movable mirror plate. Address wiring runs lengthwise beneath the flexures to help minimize optically inactive area. The nominal fill factor of the array is 77%. Residual stresses in the polysilicon and oxide layers cause convex curvature of the mirror plate. Measured peak-to-valley curvature is ⬃165 nm across the middle of the unmetallized mirrors (3). The M19 MEM-DM was housed in a pin grid array package (PGA-144) for use in the adaptive optics test bed. The test bench as configured for M19 testing is depicted in Fig. 5. The 18mW helium-neon (HeNe) laser source passes through a variable attenuator before it is collimated and optionally passed through a lens La to generate a quadratic aberration. After reflection from the fold mirror (M1), the beam enters a beam splitter (BS1) which reflects the incident wave toward the MEM-DM. The power meter at BS1 guides normalization adjustment of the optical signal incident on the MEM-DM. An afocal telescope (lenses L 1 and L s) between BS1 and the MEM-DM compacts the beam to fill the controllable surface of the micromirror array. The MEM-DM is placed at the back focal plane of L s. An iris located one focal length away from L 1 between BS1 and M1 controls the beam diameter.

Figure 5 Adaptive optics test bed used for M19 MEM-DM testing (left). Image and PSF of unmetallized M19 MEM-DM (right).



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The beam reflected from (and diffracted by) the MEM-DM traverses the afocal telescope (L s, L l), BS1, and translating lenses L t1, and L t2. At BS2 light is picked off for a CCD camera which records the image of the MEM-DM surface. A second afocal telescope (L w1 and L w2) reduces the image size to fit onto the Image camera CCD. A recorded image of the 12 ⫻ 12 M19 array in the circular aperture is shown in Fig. 5. In the other leg of BS2, a Fourier transforming lens L F generates the far-field diffraction pattern of the light transmitted by BS2, which is equivalent to the point spread function (PSF) of an imaging system (15). Another lens L M magnifies the far-field pattern on the 256 ⫻ 256 pixel array of the PSF camera. A PSF camera image with incident plane wave and all micromirrors at the same level is also shown in Fig. 5. Lens focal lengths and nominal positioning dimensions for the bare array configuration are summarized in Tables 2 and 3. A series of PSF images was recorded as a varying bias voltage was applied to simultaneously deflect all array elements. The observed change in measured peak intensity as a function of bias voltage shows the amplitude modulation effect of interference between the moving mirror surfaces and static background structures (supports, substrate, wires, etc.). The observed 23% intensity variation is consistent with the amplitude modulation expected for a 77% fill-factor device (16). Defocus aberrations were created by inserting negative focal length lenses into the beam path at a distance of 1.6 m from the MEM-DM. The approximately spherical wavefronts produced by the aberrating lenses are referred to by the radius of curvature (ROC) at the MEM-DM. Aberrating lens (L a) focal lengths of f ⫽ ⫺3.5 m, and f ⫽ ⫺0.5 m produce ROCs of 0.8 m and 0.35 m, respectively. Setting the MEM-DM elements to twice the aberration radius of curvature yields best aberration correction results. The factor of 2 between aberration radius of

Table 2 Focal Lengths of Lenses in Adaptive Optical Test Bed for M19 Testing Lens labels

Focal length (mm)

La L1 Ls L t1, L t2 LF LM L w1 L w2

⫺500, ⫺3500 250 100 300 75 38 100 40


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Table 3 Optical Component Locations in Adaptive Optical Test Bed for M19 Testing Distance L a to Iris Iris to L 1 L 1 to L s L s to MEM-DM L 1 to L t1 L t1 to L t2 L t2 to L F L F to L M L t2 to L w1 L w1 to L w2 L w2 to image camera L M to PSF camera

Nominal distance (mm) 1700 250 350 100 550 600 375 115 400 140 40 770

curvature and MEM-DM radius of curvature is due to the ‘‘round-trip’’ optical path length difference obtained using reflective devices. Aberration correction results for the M19 MEM-DM experiment are summarized in the image comparison matrix shown in Fig. 6. For this set of data the incident optical power was normalized by setting the power at BS1 to 1.47 µW for all cases. Image intensity was adjusted by varying the PSF camera frame rate. The variation in PSF camera frame rate provides an indication of the wide dynamic range of the data. Note that the display images are cropped to a 129 ⫻ 129 pixel region around the main lobe, and the gray scale is reduced to 64 levels to better show details of the far-field pattern. Comparison of the images on the matrix diagonal qualitatively show the excellent aberration correction performance obtained. PSF data was normalized to the plane MEM-DM with incident plane wave case, and are noted in parentheses beside each image. Examination of the images and normalized peak intensities indicates consistent performance of the M19 MEM-DM. Introduction of aberrations by applying curvature to the MEM-DM with incident plane wave does not decrease the observed peak intensity as much as the insertion of an aberrating lens, because only the element to element phase shift is applied. The MEM-DM aberration correction obtained is in reasonable agreement with a straightforward FFT simulation. Simulated system performance and measured M19 MEM-DM aberration correction points showing the relative peak intensity as function of aberration radius of curvature are plotted in Fig. 7.



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Figure 6 PSF image matrix from M19 MEM-DM optical aberration correction experiment. Note that images are independently scaled by camera frame rate and gray scale has been reduced to 64 levels to better show detail of far-field diffraction pattern. The numbers in parenthesis denote the normalized peak intensity.

Figure 7 Modeled and measured relative peak intensity of the point spread function (PSF) as a function of defocus aberration radii of curvature for both the M19 MEM-DM and lenslet/MEM-DM cases (left). Amplitude and phase emerging from lenslet depicted in gray scale for 0.35-m defocus aberration radius of curvature (right).


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The simulated maximum peak intensity for the ROC ⫽ 0.70 m case and the corresponding measured peak intensity maximum are both 76%. For simulated correction of the strongest aberration (ROC ⫽ 0.35 m), residual phase errors, (or the uncorrected phase across a mirror element), reduce the maximum peak intensity obtainable to ⬃52% of the unaberrated case. The measured MEM-DM corrected value for this case of 27% suggests that complete correction was not obtained.

V. LENSLET/MEM-DM OPTICAL PERFORMANCE The lenslet/MEM-DM approach achieves high optical efficiency without a specialized microfabrication process because the lenslet array defines the fill factor of the hybrid correcting element. The effects of stress-induced mirror curvature are also mitigated because the mirror surfaces can be smaller (2). All of the chip area not dedicated to the reflective surfaces is available for the actuator structure and wiring without regard for the optical effects; thus, control of the underlying topography and/or a planarization step in the micromachining process are avoided. Because fielded electrostatic micromirror systems will likely require an optical window to protect the microscopic moving parts from dust, moisture, and handling, the addition of a lenslet array which also serves as a protective window is not especially burdensome. Scanning electron micrographs of the MEM-DM array and a single micromirror element designed specifically for use with a lenslet array are shown in Fig. 8. Spacing of the elements matches the 203-µm center-to-center spacing of a commercially available lenslet array. The array shown is comprised of 12 ⫻

Figure 8 Annotated scanning electron micrographs of MEM-DM designed for use with a matching lenslet array.



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12 elements on a 203-µm square grid. Four elements (2 ⫻ 2) in each corner of the array are wired together serving only as test devices (16 total). The remaining 128 elements are individually controllable. The micromirror arrays were fabricated in the MUMPs polysilicon surface micromachining process. Several design features are incorporated to improve the device deflection uniformity, yield, robustness, and optical characteristics of this particular MEMDM design. The stacked polysilicon (3.5 µm thick) mirror plate area is made as large as possible to decrease the maximum control voltage required. Stacking of the polysilicon layers greatly increases the stiffness of the plate. Under the 60µm-diameter gold reflective surface, a 0.75-µm-thick layer of oxide is also trapped between the two polysilicon layers. Mirror surface curvature due to the residual tensile stress of the metallization is minimized by keeping the reflective surface as small as possible and making the underlying structure as stiff as possible. Measured mirror curvature is less than 30 nm across the mirror surface. The four flexures of this MEM-DM design are 10 µm wide, 114 µm long, and 2 µm thick. The voltage required for 316-nm deflection is 13.8 V. Die were mounted in a 144 PGA package for insertion in the adaptive optics test bed. The packaged MEM-DM was mounted in the test bench depicted in Fig. 9. The setup is similar to bare MEM-DM test setup shown in Fig. 5 with a few exceptions (17). The lenslet surface (instead of the MEM-DM) is now imaged,

Figure 9 Optical test bed layout for lenslet/MEM-DM experiments. Note drawing is not to scale. Lens focal lengths and dimensions are the same as those listed in Tables 2 and 3, with the exceptions noted in the text. The image of lenslet surface near aperture edge shows the expected image inversion. Note that the PSF camera image is reduced to only 16 gray scale levels to make grating lobes visible.


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and the MEM-DM is installed one lenslet focal length (7.8 mm) behind the lenslets. Also, the magnifying lens for the PSF camera had a focal length of 11 mm, and no demagnifying lens was used for the image camera. In addition, the optical attenuator and power meter were not available for easy normalization of the incident optical power. Background interference effects were measured prior to experimentation. All elements in the array were simultaneously deflected by slowly increasing the array bias voltage while monitoring the PSF camera. The main lobe peak intensity was observed to vary by 30% due to interference effects with a single minimum occurring at about 3λ/4 deflection. In contrast, similar testing of this MEM-DM (⬃7% fill factor) without lenslets shows that severe background interference can completely null the main lobe of the PSF. Thus, lenslets improve the effective fill factor of the micromirror array and correspondingly reduce static background interference phenomenon. Limiting the deflection range to λ/2 (316 nm) avoids the background interference minimum. To model the lenslet/MEM-DM case, computation of the PSF is simplified by defining the initial disturbance as the field emerging from the lenslets after reflection from the MEM-DM. The lenslet/MEM-DM is treated as a single system that modulates both the phase and amplitude of the incident optical signal. For incident plane waves, the amplitude for each mirror is identical and the phase is determined by micromirror deflections. For aberrated incident waves the phase is defined as the residual phase, or phase remaining after correction, and the amplitude for each lenslet/micromirror element is defined by the results of the geometric analysis. To represent the gap between lenslets, corresponding portions of amplitude are set to zero. For the incident plane wave case the amplitude is uniform across all modulating elements, and computation of the PSF is straightforward. When a highly aberrated incident wave is applied, amplitude variation across the array due to geometric effects can significantly influence the peak intensity of the PSF. Comparison of the lenslet/MEM-DM and bare M-19 MEM-DM cases in Fig. 7, which have similar optical fill factor, shows the effect that lenslet and micromirror geometry can have on system performance. The amplitude and residual phase errors predicted by a computer model for an incident defocus aberration (0.35 m radius of curvature) with this lenslet/MEM-DM geometry are shown as gray scale images. For this case, the predicted peak intensity of the PSF for the lenslet/MEMDM case is 34%, compared to over 50% for the bare MEM-DM case which considers only the residual phase errors. Experiments performed with the lenslet/MEM-DM combination demonstrate the optical beam steering capability of segmented MEM-DMs. In one dimension an optical phased array of n elements, with uniform phase shift spacing (between elements) from 0 to 2π, steers a beam to an angle θ s ⫽ λ/nL, where L is the spacing of the elements (in this case the lenslet dimension), and λ is the operating wavelength (18). Rotated versions of six control voltage files approxi-



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mating tilted planes were used to steer the beam to 24 locations off boresight. Peak intensity locations in angular units are plotted in Fig. 10. The peak intensity of the steered beams follows the diffraction envelope of a single element, sinc 2 (πL sin θ/λ), where θ is the angle from boresight (19). For small angles where θ ⬇ sin θ this simplifies to sinc 2 (π/n), where again n is the number of uniformly spaced elements spanning 0 to 2π phase shift. For n ⱖ 4 the peak intensity of the steered beam remains above 81%. Measured mean peak intensity values show good agreement with 1-D theory. A two-dimensional full-width half-maximum (FWHM) was also computed as a measure of beam quality. From phased array theory the beam width increases as 1/cos θ for small steering angles (20). Over the small steering angles possible, negligible beam spreading is expected. All FWHM values were within 12% of the flat MEM-DM case, with no correlation to steering angle. Hence the variation is attributed to measurement noise. One potential application of the lenslet/MEM-DM system is projection of laser beacons for adaptive optics. With a sufficiently large segmented mirror system, the mirror can be partitioned to create multiple independently steered beams. Appropriate control of the MEM-DM permits the number, position, and relative intensity of laser beacons to be varied in real time. To illustrate this capability, the lenslet/MEM-DM is partitioned into two control surfaces, creating two beams which are independently steered. The steering angles were chosen so that θ x1 ⫽

Figure 10 Summary plot of lenslet/MEM-DM beam steering results (left). All positions within the 0.75-mrad circle employ at least four elements per 2π of phase shift with a corresponding efficiency of ⬎81%. Dual beam steering results (right) with annotated peak intensity.


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⫺θ y2 and θ y1 ⫽ ⫺θ x2, yielding beams with the same total angle from boresight. PSF camera images of the central region for the unsteered beam and one dual steered beam case are shown in Fig. 10. The observed peaks of the steered beams are at angles of θ x1 ⫽ 0.330 mrad, θ y1 ⫽ 0.792 mrad and θ x2 ⫽ ⫺0.792, θ y2 ⫽ ⫺0.396 mrad with respect to boresight. The total steering angles for Beam 1 and Beam 2 are 0.858 mrad and 0.885 mrad, respectively. The peak intensity of Beam 2 is 46% the peak intensity of the unsteered beam, while Beam 1 is 25% the peak intensity of the unsteered beam. The difference in measured peak intensities for the steered beams is attributed primarily to slight misalignment of the aperture on the lenslet/MEM-DM. There was no significant change in the FWHM of the steered beams (all FWHM within 20%), suggesting that beam quality is preserved. This experiment demonstrates that a single lenslet/MEM-DM can replace the optical power splitter and multiple steering elements required to generate multiple laser beacons. Aberration correction experiments were also conducted with the lenslet/ MEM-DM correcting element. Aberrated and corrected PSF images for three defocus aberrations are shown in Fig. 11. The 256 level gray scale of each image is independently adjusted to show detail. The measured FWHM in camera pixels and peak intensity relative to the unaberrated case are annotated beside each image for comparison. The relative peak intensities for several aberration correction cases are plotted in Fig. 7. While the measured relative peak intensities confirm the presence of a steep performance drop-off predicted by the model, the

Figure 11 PSF camera images showing plane wave incident on undeflected MEM-DM and quadratic aberration correction results. Note that each image is independently scaled to better show PSF structure. FWHM and peak intensity relative to the plane wave/flat MEM-DM case are annotated.



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measured data is somewhat scattered due to experimental errors. Normalization of the incident optical power after inserting the aberrations represents the largest source of experimental error.

VI. SURFACE MICROMACHINED CONTINUOUS FACESHEET DEFORMABLE MIRRORS The best reported atmospheric aberration correction results to date have been obtained with axially actuated continuous facesheet deformable mirrors. Microfabrication can potentially reduce the cost of continuous facesheet deformable mirrors by a factor of 1000 or more. The principal challenge is development of a microfabrication process which yields actuators with sufficient stroke and acceptable facesheet planarity. Because the facesheet is a large structure, residual material stresses can cause significant deformations of the facesheet surface when the structure is released. In addition to the planarity and residual stress problems, the thicknesses of the polysilicon structural layers are typically not well suited to good continuous facesheet deformable mirror fabrication. The brief examination of continuous facesheet deformable mirror design equations below reveals the process imposed constraints and suggests process changes for continuous facesheet DM fabrication. The influence function of a continuous facesheet deformable mirror element is a function of the force applied by the actuator (F), actuator spacing (b), and facesheet stiffness. Beam theory approximations show good agreement with measured influence functions and finite element models of macro-sized deformable mirrors. Using beam theory approximations the shape of the influence function is given by (21)

冤冢冣冢冣冥

F x y ⫽ CC 2 k FS b

3

x ⫺3 b

2

or

冤冢冣冢冣冥

F 1 x y ⫽ CF k FS 2 b

3

1 x ⫺ 3 b

2

(11)

depending on whether a clamped-clamped (CC) or clamped-free (CF) approximation is employed, where y is the deflection of the facesheet at a distance x from the actuator. The force, F, applied by an electrostatic parallel-plate actuator is given by Eq. (2). Depending on the approximation, the stiffness of the facesheet is either (21) k CC FS ⫽

16Et 3 b2

or

k CF FS ⫽

8Et 3 b2

(12)

where t is the thickness of the facesheet, and E is the elastic modulus of the


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facesheet material. Equations (11)–(12) and (2) permit computation of the influence function for a given microfabricated deformable mirror geometry. Which beam deflection approximation case is most applicable for calculation of the influence function is primarily determined by the relative stiffness of actuators and facesheet. If the actuators are very stiff compared to the facesheet, the clamped-clamped is more appropriate, and vice versa if the relative stiffness are reversed. These relative stiffnesses also define the interactuator coupling, which is computed as (21) Actuator coupling ⫽

1 ⫻ 100% 4(k act /k FS ) ⫹ 1

(13)

For a flexure beam actuator, k act is the spring constant computed in Eq. (8). Typical values for low control voltage flexure beam actuators and polysilicon facesheets yield high interactuator coupling. Stiffer actuator structures reduce interactuator coupling but require high control voltages. A more practical approach is to reduce the stiffness of the facesheet material by reducing its thickness and or elastic modulus and by increasing the distance between actuators. Reducing the stiffness of the facesheet also lowers the maximum operation frequency of the deformable mirror. The natural frequency of a continuous facesheet membrane supported by axial actuators is given by (22) fn ⫽

10.21t 2πb 2

√

E 12ρ(1 ⫺ υ 2 )

(14)

where ρ and υ are the density and Poisson ratio of the facesheet material respectively. For a polysilicon membrane (t ⫽ 1.5 µm, E ⫽ 170 GPa, υ ⫽ 0.22, ρ ⫽ 2330 kg/m 3 ) supported by actuators with 203-µm spacing, Eq. (14) yields a resonant frequency over 165 kHz, much higher than required for atmospheric adaptive optics systems. Facesheet stiffness can be substantially reduced to obtain microfabricated deformable mirrors with low control voltages. Despite the limitations of existing foundry processes, a number of prototype systems have been developed and functionally tested. The optical quality of these prototypes is degraded by incomplete self-planarization, etch access holes, and residual stress in the polysilicon facesheet layer. Controllable actuator stroke in MUMPs technology is limited to about 0.67 µm, limiting the magnitude of aberrations that can be corrected. Interferometric microscope images of a functional device with two different actuators pulled down are shown in Fig. 12. The measured influence functions for this device with 18 V and 21 V applied are shown in Fig. 13. Reference subtraction eliminates print-through from the measurements. The scan lines depicted start at the center of the controlled actuator and span two actuators (406 µm). Note that for the 21-V measurement the actuator is fully deflected. These measurements indicate that interactuator



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Figure 12 Interferometric microscope images of a MUMPs continuous facesheet deformable mirror with two different actuators fully pulled down by 21-V control signal. Measurements from scan line in (b) are shown in Fig. 13.

Figure 13 Measured influence functions for a MUMPs fabricated continuous facesheet deformable mirror with 18 V and 21 V applied to the actuator. The measurement scan corresponds to (b) in Fig. 12. Note that the scan lines have been scaled to the same vertical scale.


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coupling is about 40%, in good agreement with the coupling estimated by Eq. (14). The addition of a chemical mechanical polishing (CMP) planarization step prior to deposition of the facesheet layer yields a process almost tailor-made for continuous facesheet deformable mirror fabrication, but the same design limitations imposed by layer thicknesses apply. An initial prototype surface micromachined continuous facesheet mirror fabricated in the planarized SUMMiT process is shown in Fig. 14. Ignoring print-through effects which are not visible in the micrograph (3), the facesheet is very flat across the 5 ⫻ 7 element actuator array, due to an essentially stress-free facesheet layer. Interferometric microscope testing of the planarized continuous facesheet shows that the influence function is dominated by the relatively stiff facesheet, as expected. An influence function with 50 V applied to a single actuator is shown in Fig. 14b. These results suggest that with improvement of the planarization process to eliminate print-through, and thinning of the facesheet layer, a similar surface micromachining process could produce very good continuous facesheet deformable mirrors. Efforts to develop a custom micromachining process specifically for deformable mirror fabrication are ongoing and promise to resolve the limitations of foundry processes (23–25). In a custom process, employing thicker sacrificial oxides to both increase actuator stroke and improve self-planarization of the polysilicon facesheet, Boston University researchers demonstrated a peak-topeak surface roughness of 0.03λ at 633 nm (25). But post-release deformations of the polysilicon due to residual material stresses result in a final facesheet planarity of λ/6 to λ/2 depending on actuator size (25).

Figure 14 SUMMiT continuous facesheet deformable mirror. Scanning electron micrograph (a) shows facesheet surface. Measured influence function with 50 V applied to the underlying actuator (b).



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VII. THE FUTURE OF SURFACE MICROMACHINED DEFORMABLE MIRRORS The future of adaptive optics using surfaced micromachined deformable mirrors is a bright one. The low cost and design flexibility of surface micromachining may easily make it the dominant means of fabricating deformable mirrors. Practical segmented and continuous facesheet designs have been demonstrated, and fielded adaptive optics systems employing micromachined deformable mirrors are near term. Efforts to completely eliminate print-through and residual stress induced mirror curvature will likely be completed before this text is in print. Integration of moving mirror elements and control electronics on a single die will allow low cost fabrication of deformable mirrors with hundreds to thousands of individually controllable elements.

REFERENCES 1. D. A. Koester, R. Mahadevan, and K. W. Markus, ‘‘Multi-user MEMS processes (MUMPs): introduction and design rules, rev. 4,’’ Technical Report, MCNC MEMS Technical Applications Center, 3021 Cornwallis Road, Research Triangle Park, NC 27709, 1996. 2. W. D. Cowan, V. M. Bright, A. A. Elvin, and D. A. Koester, ‘‘Modeling of stressinduced curvature in surface-micromachined devices,’’ in Microlithography and Metrology in Micromachining III, Proc. SPIE, vol. 3225, pp. 56–67, September 1997. 3. W. D. Cowan, V. M. Bright, M. K. Lee, J. H. Comtois, and M. A. Michalicek, ‘‘Design and testing of polysilicon surface-micromachined piston micromirror arrays,’’ in Spatial Light Modulators, Proc. SPIE, vol. 3292, pp. 60–70, 1998. 4. R. D. Nasby, J. J. Sniegowski, J. H. Smith, S. Montague, C. C. Baron, W. P. Eaton, P. J. McWhorter, D. L. Heatherington, C. A. Apblett, and J. G. Fleming, ‘‘Application of chemical-mechanical polishing to planarization of surface micromachined devices,’’ in Technical Digest of the Solid-State Sensor and Actuator Workshop, pp. 48–53, Hilton Head, SC, June 1996. 5. W. D. Cowan and V. M. Bright, ‘‘Thermally actuated piston micromirror arrays,’’ in Optical Scanning Systems: Design and Applications, Proc. SPIE, vol. 3131, pp. 260–271, July 1998. 6. W. D. Cowan, M. K. Lee, V. M. Bright, and B. M. Welsh, ‘‘Evaluation of microfabricated deformable mirror systems,’’ in Adaptive Optical System Technologies, Proc. SPIE, vol. 3353, pp. 790–804, March 1998. 7. D. M. Burns and V. M. Bright, ‘‘Investigation of maximum optical power rating for micro-electro-mechanical device,’’ in Transducers ’97, International Conference on Solid-State Sensors and Actuators, Chicago, IL, pp. 335–338, 1997. 8. M. A. Michalicek, V. M. Bright, and J. H. Comtois, ‘‘Design, fabrication, and testing of a surface-micromachined micromirror device,’’ Proceeding of the ASME Dy-


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Cowan and Bright namic Systems and Control Division ASME 1995, DSC-vol. 57-2, 1995 IMECE, pp. 981–988, 1995. T. H-. Lin, ‘‘Implementation and characterization of a flexure-beam micromechanical spatial light modulator,’’ Opt. Eng., vol. 33, no. 11, pp. 3643–3648, November 1994. P. M. Osterberg, R. K. Gupta, J. R. Gilbert, and S. D. Senturia, ‘‘Qualitative models for measurement of residual stress, Poisson ratio, and Young’s modulus using electrostatic pull-in of beams and diaphragms,’’ in Technical Digest, Solid State Sensor and Actuator Workshop, Hilton Head, SC, pp. 184–188, 1994. B. Hulburd and D. Sandler, ‘‘Segmented mirrors for atmospheric compensation,’’ Opt. Eng., vol. 29, no. 10, pp. 1186–1190, October 1990. R. H. Freeman and J. E. Pearson, ‘‘Deformable mirrors for all seasons and reasons,’’ Applied Optics, vol. 21, no. 4, pp. 580–588, February 1982. R. K. Mali, T. G. Bifano, N. Vandelli, and M. N. Horenstein, ‘‘Development of microelectromechanical deformable mirrors for phase modulation of light,’’ Opt. Eng., vol. 36, no. 2, pp. 542–548, February 1997. M. C. Roggemann, V. M. Bright, B. M. Welsh, S. R. Hick, P. C. Roberts, W. D. Cowan, and J. H. Comtois, ‘‘Use of micro-electro-mechanical deformable mirrors to control aberrations in optical systems: theoretical and experimental results,’’ Opt. Eng., vol. 36, no. 5, pp. 1326–1338, May 1997. J. W. Goodman, Introduction to Fourier Optics, McGraw-Hill, New York, 1968. L. J. Hornbeck, ‘‘Deformable-mirror spatial light modulators,’’ SPIE Critical Review Series, vol. 1271, pp. 51–62, 1990. M. K. Lee, W. D. Cowan, B. M. Welsh, V. M. Bright, and M. C. Roggemann, ‘‘Aberration correction results using a segmented micro-electro-mechanical deformable mirror and refractive lenslet array,’’ Optics Letters, vol. 23, no. 8, pp. 645– 647, 15 April 1998. S. D. Gustafson, G. R. Little, V. M. Bright, J. H. Comtois, and E. S. Watson, ‘‘Micromirror arrays for coherent beam steering and phase control,’’ in Proc. SPIE, vol. 2881, pp. 65–74, 1996. E. Hecht and A. Zajac, Optics, Addison-Wesley, Reading, MA, 1975. M. I. Skolnik, Introduction to Radar Systems, McGraw-Hill, New York, 1980. M. A. Ealy and J. A. Wellman, ‘‘Deformable mirrors: design fundamentals, key performance specifications, and parametric trades,’’ SPIE Active and Adaptive Optical Components, vol. 1543, pp. 36–51, 1991. R. K. Tyson, Principles of Adaptive Optics, Academic Press, San Diego, 1991. R. K. Mali, T. G. Bifano, N. Vandelli, and M. Horenstein, ‘‘Development of microelectromechanical deformable mirrors for phase modulation of light,’’ Opt. Eng., vol. 36, no. 3, pp. 542–548, February 1997. T. G. Bifano, R. K. Mali, J. K. Dorton, J. Perreault, N. Vandelli, M. K. Horenstein, and D. Castanon, ‘‘Continuous-membrane surface-micromachined silicon deformable mirror,’’ Opt. Eng., vol. 36, no. 5, pp. 1354–1360, May 1997. T. Bifano, R. K. Mali, J. Perreault, M. Horenstein, D. Koester, ‘‘MEMS deformable mirrors for adaptive optics,’’ in Technical Digest of the Solid-State Sensor and Actuator Workshop, pp. 71–74, Hilton Head, SC, June 1998.


9 Liquid Crystal Adaptive Optics Gordon D. Love University of Durham, Durham, United Kingdom

I. INTRODUCTION In this section we will describe work on using liquid crystal devices as wavefront correctors and controllers. We will concentrate on describing the parameters of liquid crystals which are relevant to adaptive optics. We will also review some of the work on using electrically addressed liquid crystal devices to date. Wavefront control in adaptive optics is generally achieved by keeping the refractive index constant (propagation through air) and by tuning the actual path length, with a mirror. An optically equivalent alternative is to fix the actual path length and to tune the refractive index. This could be achieved using many different optical materials; a particularly convenient class of which is liquid crystals because they can be made into closely packed arrays of pixels which may be controlled with low voltages. The liquid crystal phase is a mesophase of matter in between the solid and liquid states. This phase is characterized by molecular bonding and ordering which is strong enough that the bulk material displays crystalline properties, such as birefringence, but weak enough that this order can easily be controlled by external effects, such as electric fields. In general there are a large range of liquid crystals which can be classified according to their molecular structure or physical properties. From the perspective of commercially available electro-optical devices there are two main categories: nematic liquid crystals (NLCs) and ferroelectric liquid crystals (FLCs). NLCs are used to make the most common type of liquid crystal device, the twisted nematic display. Optically these are polarization modulators whose twisted molecular structure selectively rotates the plane of polarization of light in between polarizers to produce intensity modulation. A simple modification to the cell structure allows the production of a homoge-


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neously aligned NLC cell, which is the type of interest for analog phase control. Optically these are linear waveplates whose retardance can be electrically controlled by low voltages. In effect one has a medium which can control the phase of linearly polarized light (that which is polarized parallel to the extraordinary index). Much of the work on using liquid crystals in AO has involved NLCs, and we will concentrate primarily on them in this section. The following subsection describes the important parameters of NLCs which are relevant to their use in adaptive optics. In particular we concentrate on electrically addressed NLCs. Next we review some of the work to date using both NLCs and FLCs. Finally we describe some of the areas within AO where LCs may be particularly suitable.

II. SPECIFICATIONS OF NEMATIC LIQUID CRYSTALS RELEVANT TO AO In this section we attempt to describe nematic liquid crystals in terms of their properties which are of interest to one wishing to construct an adaptive optics system. These are stroke or throw, number of actuators, actuator impulse function, fill factor, optical quality, transmittance, dispersion, temperature effects, hysteresis, polarization effects, optical power handling capacity and response time.

A. Stroke or Throw The stroke, or throw, required for atmospheric compensation is generally several visible waves. For example, if D/r 0 ⫽ 40 then the rms wavefront error after global tilt removal, assuming Kolmogorov turbulence, is about 0.5 µm. A wavefront corrector would typically require a total stroke of six times this value to allow for plus or minus 3 standard deviations, which is 3 µm. The throw of an NLC cell is equal to the (NLC birefringence) ⫻ (NLC thickness). A typical birefringence is 0.2, and hence the required thickness is 15 µm which can easily be produced. Indeed NLCs can produce much larger throws than this. There is no prescribed maximum, although good quality cells become difficult to make once the thickness is much more than 50 µm (i.e., a stroke of 10 µm). By placing a mirror behind the NLC cell these strokes can be doubled. Although the achievable stroke can be achieved easily, a related topic is the response time. This is a more complicated topic which we relegate to the end of this section. We also note that work is in progress on producing bulk-aligned NLC cells which would have very large throws (many tens of microns) which may have applications in active optics.


Liquid Crystal Adaptive Optics

B.

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Number of Actuators

One of the reasons why there is an interest in liquid crystals for AO is that the technology already exists to construct very large arrays of liquid crystals, e.g., there are around 300,000 pixels in a typical laptop PC display. Simple multiplexed LC arrays can suffer from flicker due to the pixels being actuated in a cyclic fashion (which is not a problem in displays if the flicker is faster than the response time of the eye). There is also some cross-talk between pixels. Active matrix addressing involves on-display circuitry to ensure a constant correct voltage at each pixel. However then the fill factor can be compromised. The most promising technique for the construction of large arrays involves VLSI silicon backplane devices, a review of which is given by Johnson et al. (1). Typical devices are made with 128 ⫻ 128 or 256 ⫻ 256 pixels, and recently devices have been made with high fill factors and with good optical quality (2). It is currently hard to imagine an astronomical AO system with so many actuators; however, if one has a very large number available, then one can ‘‘bin’’ pixels together to provide a very flexible system. Currently available LC-SLMs for AO use direct addressing, in which each pixel is hardwired, which has the advantage that the precise voltage across each pixel can be accurately controlled. It also limits the total number of pixels to a few hundred, although this is sufficient for many applications.

C.

Actuator Impulse Function

A conventional pixilated LC-SLM is analogous to a segmented mirror with an actuator impulse function that is very close to a perfect piston-only top-hat function. In contrast, a deformable mirror actuator tends to produce a linear (tip-tilt) or a gaussian function. Therefore an important point to note is that when comparing a LC-SLM with a deformable mirror one typically needs more NLC pixels than mirror actuators. According to Hudgin (3) the wavefront fitting error variance, σ 2fit, is given by

冢冣

r σ ⫽κ s r0 2 fit

5/3

where r s is the actuator spacing, and κ is a constant depending on the impulse function. κ is 1.26 for piston-only and 0.4 for an experimental case of a deformable mirror. For a given fitting error, there needs to be four times as many piston-only actuators as mirror actuators. An alternative description of fitting error is that the phase jump between adjacent actuators in a piston-only wavefront corrector must not be greater than around 1 radian; otherwise diffraction effects will become noticeable.


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At a NLC pixel boundary there is some smoothing of the impulse function which occurs typically on a scale of the order of the cell thickness (typically 10 µm) due to so-called fringing fields. We also note that it is possible to produce continuous phase profiles. For example tip-tilt NLC devices which are not segmented (4), and modally controlled NLC (5) devices have been demonstrated. Alternatively one could use an optically addressed NLC to produce continuous phase profiles. D. Fill Factor The fill factor is the ratio of the useful active area to the dead space in between pixels. Both directly addressed and VLSI arrays are available commercially with high fill factors (⬎90%). E. Optical Quality Much of the early work on using NLCs for adaptive optics involved modifying LC displays which were not designed to be used as high quality optical elements. However, now devices have been made which are optically flat, typically λ/5 PV in the visible, which utilize optical quality glass. A device has been constructed (6) with an optical flatness of λ/26 P-V, λ/142 rms (at 0.633 µm). Successful work is in progress on producing planarized VLSI cells (2). F. Transmittance An NLC device has a nonunity transmission essentially because it is constructed from a number of different layers of material, rather than because the NLC layer has a significant absorption. Thus constructing a highly transmitting device involves refractive index matching and antireflection coatings. Transmissions of greater than 90% are possible in the visible. NLC devices can also be made which transmit infrared light, by first of all constructing the cell walls (usually glass) from an IR transmitting material. NLC materials transmit light throughout the full IR spectrum except at specific wavelengths corresponding to molecular absorption bands (7,8). G. Dispersion A related topic to the transmittance is the dispersion. If one ignores dispersion then NLCs are optical path length adjusters. Errors will arise when compensating a wavefront because of dispersion in the birefringence (as opposed to dispersion of the actual refractive indices). For example (9), over a bandpass of 150 nm (500 to 650 nm) the maximum error due to dispersion would be of the order of



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5%. Dispersion in the infrared is typically less than this, except close to absorption bands where there is significant dispersion (7,8). H.

Temperature Effects

Materials exhibiting the liquid crystal phase only do so over a restricted temperature range, e.g., Merck BDH-E44 (a commonly used NLC) exhibits the nematic phase from (8) ⫺10°C to 100°C. There are therefore obvious disadvantages if one wishes to operate at a high-altitude observatory or if one wishes to use cooled optics. Research is required to determine whether one can heat an NLC without inducing further wavefront distortions. No detrimental results have been found by taking NLCs to extreme temperatures (10). The actual NLC birefringence is also a function of temperature, although the effect is fairly weak over typical conditions. For example (11), in BDH-E44 the birefringence changes from 0.25 to 0.24 when the temperature varies from 20°C to 30°C. I. Hysteresis For practical purposes, NLCs do not exhibit hysteresis, although it is possible to observe it under certain conditions (12). J. Polarization Effects As already mentioned, an NLC can only phase modulate linearly polarized light, which is a severe problem for low-light level applications where the use of a polarizer could not be tolerated. Two simple techniques exist to circumvent this problem. The first involves using the NLC in reflection mode in conjunction with a quarter-wave plate (13). The second simply involves cell construction with two orthogonal LC layers. Such a device has been successfully built and tested (14). K.

Optical Power Handling Capacity

The precise optical power handling capacity of a NLC is an area where more research is required. Several LC manufacturers quote a (surprisingly high) CW power of 500 W cm⫺2. L.

Response Time

One cannot specify the response time of an NLC without specifying the associated phase shift, since the two are inversely related. The rise time of an NLC is field dependent and is of the order of a few milliseconds for an applied voltage of 20 V. Using larger voltages gives submillisecond rise times for a stroke of


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several waves. If one only requires a small phase shift, and therefore a small voltage change, then the response time is much slower, as much as a second. Thus the fast turn-on times for all desired phase changes are achieved by applying a large voltage pulse (⬃20 V) for a short time and then applying the holding voltage. NLCs relax, or turn off, when the field is removed, and the relatively weak intermolecular forces mean that the decay time is much longer. Lowering the viscosity in order to promote fast molecular realignment tends to lower the birefringence, and hence achievable stroke. When the electric field is removed from a fully activated NLC then the rate of change of phase is initially rapid, before then slowing down. The use of this initial rapid decay is known as the transient nematic effect (15). It is widely stated that NLC turn-off times are proportional to the square of the cell thickness. This is true for the time for the molecules to fully relax, but is misleading from the point of view of adaptive optics. What is of interest is the rate of change of phase, not the speed of molecular rotation. Thus a thick cell’s molecules will rotate slowly, but they do not need to rotate through a large angle in order to achieve a given phase shift. Conversely, a thin cell’s molecules rotate quickly; however, a large rotation is required for the same phase shift. Hence the rate of change of phase shift is approximately independent of the cell thickness. Using the transient nematic effect, a typical rise time is 5 ms and a typical decay time is 40 ms for one wave, and therefore a maximum bandwidth of about 20 Hz. A slightly different cell configuration called the pi-cell (16) can give faster times using the transient nematic effect, up to 100 Hz for one wave. If one requires a total stroke of more than one wave, then the available bandwidths are much lower. The response times are also inversely proportional to temperature. The times quoted above are typically at room temperature. Near the NLC freezing point the times will be much slower, and conversely NLCs can operate much faster by heating. Wu (17) has presented results showing that an optimum temperature exists for minimizing NLC response times. Another technique for improving the temporal bandwidths of NLC is to use the so-called dual frequency effect (18). As previously described, NLC rise times are quite fast because of the relatively strong interaction with the driving field, however the relaxation time is slow because of the relatively weak intermolecular alignment forces. A technique to ‘‘drive’’ the NLC molecules to their offstate is to select a material whose dielectric anisotropy undergoes a sign change as a function of frequency. The molecules will align with their director parallel to the electric field for frequencies below the crossover frequency, and orthogonal to the field when the applied frequency is greater than the crossover frequency. Further research is needed to demonstrate fast NLC cells using the dual frequency effect which also are highly transmissive and of good optical quality. The results published to date are nevertheless promising; for example, ⬍1 ms rise and fall times for a stroke of λ/2 have been reported (19). The advantage of dual fre-



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quency addressing is that one should achieve fast response times over a much larger stroke range than using the transient nematic effect. Esposito et al. (20) showed that a dual frequency NLC is capable of tracking atmospheric turbulence in the infrared.

III. WAVEFRONT CORRECTION USING NEMATIC LIQUID CRYSTALS In this chapter we concentrate on the work undertaken using electrically addressed SLMs for wavefront correction. We do not include optically addressed SLMs because they generally have been used in nonconventional AO systems, e.g., all-optical correction schemes or nonlinear optical correction. They have been reviewed by Pepper (21). One could nevertheless in principle use an optically addressed SLM in a conventional AO system. The earliest references to wavefront correction using NLCs are from Russia. Work on using both electrically addressed (22) and optically addressed (23) NLCs was reported. In Italy, work on using dual frequency addressed NLCs was reported by Bonaccini (24) and Esposito (20). They also showed that measurements of an NLC cell’s capacitance could be used to monitor the NLC retardance. Dou and Giles (25) described a modified NLC TV and a Mach-Zehnder interferometer to produce a closed-loop system. Meadowlark Optics, Inc. has undertaken work on producing high-quality NLC arrays custom designed for wavefront correction. They currently produce a 69-segment device (6,26) and a 127-segment device which operates with unpolarized light (27). Figure 1 shows an example of wavefront control using the 69-segment SLM. Experimentally measured phase maps of the first 15 Zernike modes are shown, next to the theoretical plots for comparison. Quantitative analysis of the fitting error of the SLM has shown that its performance is close to an ideal wavefront corrector with the given actuator geometry. Gourlay et al. presented results with these devices in a closed-loop system (28). Work is in progress to construct silicon-backplane nematic LC-SLMs with good optical quality, a large number of pixels, and a large fill factor suitable for active and adaptive optics applications. Morris et al. published results of wavefront shaping with a 128 ⫻ 128 pixel device (29).

IV. WAVEFRONT CORRECTION USING FERROELECTRIC LIQUID CRYSTALS The second type of LC of interest for adaptive optics are ferroelectric LCs. Optically FLCs are linear waveplates whose retardance is fixed but whose optical


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Figure 1 Generation of Zernike wavefront aberrations using a 69-actuator nematic liquid crystal device. Plots 1a–15a show experimentally measured phase maps, and 1b–15b are the corresponding theoretical plots. The amplitude of each term was selected to be the rms value for Kolmogorov turbulence for D/r 0 ⫽ 10. The dark regions in between the pixels for the low Zernike modes are areas of no data due to the measurement process. (From Ref. 6.)



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axis can be electrically switched between two states. Phase-only modulation with a retarder whose axis is switchable is more complicated than with one whose retardance can be varied. The simplest method involves sandwiching a FLC whose retardance is half a wave in between two fixed quarter-wave plates (30). FLCs have the advantage that they can be switched at kHz frame rates, but the obvious disadvantage that they are bistable. The use of a binary algorithm in wavefront correction was suggested by Tam (31). Broomfield (32) and Love (33) described how the approach could use FLCs and be used in AO. The basic wavefront correction algorithm is: whenever the wavefront error is greater than half a wave (modulo 2π), then a correction of half a wave is applied. The advantage of this is extreme simplicity if one accepts partial wavefront correction. The theoretical limit of achievable Strehl ratio is 40%, which must be compared with the uncorrected Strehl ratio in the visible at a large telescope which is ⬍⬍1%. This can be extended to an arbitrary degree of correction by cascading devices (34). Neil et al. have described a holographic technique for producing wavefront control with a binary device (35). It has the same efficiency as the binary technique described above, but the advantage that the uncorrected light is spatially filtered out and does not contaminate the resultant images. Figure 2 shows an example of this technique used to generate atmospheric turbulence for testing an adaptive optics system. New FLC materials are being developed which will allow analog phase modulation. If they are incorporated into commercial SLMs, then they will be very attractive AO elements. Broadly speaking, their optical specifications will be similar to NLCs, except for two main differences. First, they will have much improved switching speeds. Second, their stroke will be limited to be one wave. This is because phase modulation using FLCs is achieved (30) by a modulation of the geometrical phase, rather than of optical path length, as with a mirror or a NLC phase modulator.

V.

APPLICATIONS OF LIQUID CRYSTALS IN AO

Why should one wish to use a liquid crystal device instead of a deformable mirror? Here we describe where LC-SLMs may be particularly useful. Figure 1 shows the precise shaping of wavefronts using an NLC-SLM. Having a device whereby one can ‘‘dial-in’’ Zernike wavefront aberrations is particularly useful in testing AO systems, and indeed optical systems in general. Areas where LCSLMs may be useful are as follows: 1. Full correction in the visible for large (8 m) apertures requiring many thousands of actuators.


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Figure 2 An interferogram (a) showing laboratory generated turbulence used to test adaptive optics systems. The wavefront was controlled using a 128 ⫻ 128 FLC using the method described by Neil (35). The interferogram is one frame from a time evolving sequence having D/r 0 ⫽ 13. (b). The corresponding experimentally measured PSF produced by focussing the wavefront in (a) onto a CCD camera.

2. 3. 4. 5.

Multiconjugate correction where optical implementation is vastly simplified by using transmissive optics. Intensity-only modulators for scintillation correction using FLCs (36). Correction of large space-based optical systems (37). Industrial and medical applications of AO requiring high-order, low cost devices. For example work on using liquid crystal devices in ophthalmic optical systems has been described by Artal (38) and Thibos (39).

In this section we have concentrated on describing the state-of-the-art specifications of liquid crystals. This is necessarily a snapshot of a rapidly evolving field. We have attempted to describe the advantages of using LCs and highlight



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their current limitations. There is certainly great potential for liquid crystal adaptive optics, and time will tell how the field will progress.

REFERENCES 1. K. M. Johnson, D. J. McKnight, and I. Underwood, ‘‘Smart spatial light modulators using liquid crystals on silicon,’’ IEEE J. Quant. Elec. 29(2): 699–714 (1993). 2. J. Gourlay, A. O’Hara, A. J. Stevens, and D. G. Vass, ‘‘A comparative investigation into planarized liquid crystal over silicon spatial light modulators,’’ J. Mod. Opt. 43(1): 181–198 (1996). 3. Richard Hudgin, ‘‘Wavefront compensation error due to finite corrector-element size,’’ J. Opt. Soc. Am. 67(3): 393–395 (1977). 4. G. D. Love, J. V. Major, and A. Purvis, ‘‘Liquid crystal prisms for tip-tilt adaptive optics,’’ Opt. Lett. 19(15): 1170–1173 (1994). 5. A. F. Naumov and G. Vdovin, ‘‘Multichannel liquid-crystal-based wave-front corrector with modal influence functions,’’ Opt. Lett. 23(19): 1550–1552 (1998). 6. G. D. Love, ‘‘Wavefront correction and production of Zernike modes with a liquid crystal SLM,’’ Appl. Opt. 36(7): 1517–1524 (1997). 7. S. T. Wu, U. Efron, and L. A. D. Hess, ‘‘Infra-red birefringence of liquid crystals,’’ Appl. Phys. Lett. 44(11): 1033–1035 (1984). 8. S. T. Wu, ‘‘Nematic liquid crystals for active optics,’’ in Optical Materials. Ed. S. Musikant, vol. 1, Marcel Dekker, New York (1990). 9. G. D. Love and J. V. Major, ‘‘A fast way to measure liquid crystal voltage characteristics,’’ Measurement Science and Technology 3: 615–18 (1992). 10. A. Graham, G. Kopp, C. Vargas-Aburto, and R. Uribe, ‘‘Preliminary space environment tests of nematic liquid crystals,’’ Proc. Soc. Photo-opt. Instrum. Eng. 2811 (1996). 11. S. T. Wu, A. M. Lackner, and U. Efron, ‘‘Optimal operation temperature of liquid crystal modulators,’’ Appl. Opt. 26(16): 3441–3445, (1987). 12. L. M. Blinov and V. G. Chigrinov, Electrooptic Effects in Liquid Crystals, Springer, 1996. 13. G. D. Love, ‘‘Liquid crystal phase modulator for unpolarized light,’’ Appl. Opt. 32(13): 2222–2223 (1993). 14. G. D. Love, S. R. Restaino, R. C. Carreras, G. C. Loos, R. V. Morrison, T. Baur, and G. Kopp, ‘‘Polarization insensitive 127-segment liquid crystal wavefront corrector,’’ OSA summer topical meeting on adaptive optics. (1996). 15. S. T. Wu and C. S. Wu, ‘‘Small angled relaxation of highly deformed nematic liquid crystals,’’ Appl. Phys. Lett. 53(19): 1794 (1988). 16. P. J. Bos and K. R. Koehler/Beran, ‘‘The pi-cell: a fast new liquid crystal switching device,’’ Mol. Cryst. Liq. Cryst. 113: 329–339 (1984). 17. S. T. Wu, ‘‘Phase retardation dependent optical response times of parallel aligned liquid crystals,’’ J. Appl. Phys. 60(5): 1836–1838 (1986). 18. M. Schadt, ‘‘Low-frequency dielectric relaxations in nematics and dual frequency addressing of field effects,’’ Mol. Cryst. Liq. Cryst. 89: 77–92 (1982).


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30. 31.

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33.

Love V. A. Dorezyuk, A. F. Naumov, and V. I. Shmal’gauzen, ‘‘Control of liquid crystal correctors in adaptive optical systems,’’ Sov. Phys. Tech. Phys. 34(12): 1389–1393 (1989). S. Esposito, G. Brusa, and D. Bonaccini, ‘‘Liquid crystal wavefront correctors: computer simulation results,’’ In ICO-16 Conference on Active and Adaptive Optics. 2– 5 August. European Southern Observatory (1993). D. M. Pepper, C. J. Gaeta, and P. V. Mitchell, ‘‘Real-time holography, innovative adaptive optics, and compensated optical processors using spatial light modulators,’’ in Spatial Light Modulator Technology, U. Efron, Ed. pp. 585–665, Marcel Dekker, New York (1995). A. A. Vasil’ev, A. F. Naumov, and V. I. Schmal’gauzen, ‘‘Wavefront correction by liquid crystal; devices,’’ Sov. J. Quantum. Elec. 16(4): 471–474 (1986). A. A. Vasil’ev, M. A. Vorontsov, A. V. Koryabin, A. F. Naumov, V. I. Schmal’gauzen, ‘‘Computer controlled wavefront corrector,’’ Sov. J. Quantum Elec. 19(3): 395–398 (1989). D. Bonaccini, G. Brusa, S. Esposito, P. Salinari, P. Stefanini, and V. Billotti, ‘‘Adaptive optics wavefront corrector using addressable liquid crystal retarders II,’’ Proc. Soc. Photo-Opt. Instrum. Eng. 1543: 133143 (1991). R. Dou and M. K. Giles, ‘‘Closed-loop adaptive optics system with a liquid crystal television as a phase retarder,’’ Opt. Lett. 20(14): 1583–1585 (1995). D. C. Dayton, S. L. Browne, S. P. Sandvern, J. D. Gonglewski, and A. V. Kudryashov, ‘‘Theory and laboratory demonstrations on the use of a nematic liquid crystal phase modulator for controlled turbulence generation and adaptive optics,’’ Appl. Opt. 37(24): 5579–5589 (1998). G. D. Love, S. R. Restaino, R. C. C., G. C. Loos, R. V. Morrison, T. Baur, and G. Kopp. ‘‘Polarization insensitive 127-segment liquid crystal wavefront corrector,’’ OSA summer topical meeting on adaptive optics, (1996). J. Gourlay, G. D. Love, P. M. Birch, R. M. Sharples, and A. Purvis, ‘‘A real time closed loop liquid crystal adaptive optics system: first results,’’ Optics. Comm. 137(1–3): 17–21 (1997). D. J. Cho, S. T. Thurman, J. T. Donner, and G. M. Morris, ‘‘Characteristics of a 128 ⫻ 128 liquid-crystal spatial light modulator for wave-front generation,’’ Optics. Comm. 23(12): 969–971 (1998). G. D. Love and R. Bhandari, ‘‘Optical properties of a QHQ ferroelectric liquid crystal phase modulator,’’ Optics Comm. 110: 475–478 (1994). E. C. Tam, S. Wu, A. Tanone, F. T. S. Yu, and D. A. Gregory, ‘‘Closed-loop binary phase correction of an LCTV using a point diffraction interferometer,’’ IEEE Phot. Tech. Lett. 2(2): 143–146 (1990). S. E. Broomfield, M. A. A. Neil, E. G. S. Paige, and I. D. Thomas, ‘‘Binary optical correction of wavefront aberration using spatial light modulators,’’ Proc. Soc. PhotoOpt. Instrum. Eng. 2534: 167–175 (1995). G. D. Love, N. Andrews, P. Birch, D. Buscher, P. Doel, C. Dunlop, J. Major, R. Myers, A. Purvis, R. Sharples, A. Vick, A. Zadrozny, S. R. Restaino, and A. Glindemann, ‘‘Binary adaptive optics: atmospheric wavefront correction with a half wave phase shifter,’’ Appl. Opt. 34(27): 6058–6066 (1995) & Addendum 35(3): 347–350 (1996).



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10 Wavefront Sensing and Compensation for the Human Eye David R. Williams, Junzhong Liang,* Donald T. Miller,† and Austin Roorda‡ University of Rochester, Rochester, New York

I. INTRODUCTION Methods to correct the optics of the human eye are at least 700 years old. Spectacles have been used to correct defocus at least as early as the 13th century (1,2) and to correct astigmatism since the 19th century (3). Since then there has been relatively little work on correcting additional aberrations in the eye. Recently, however, advances in measuring the aberrations of the human eye and in compensating for them with adaptive optics make it possible to provide the eye with unprecedented optical quality. An observer viewing the world through adaptive optics can have a sharper image of it than he has ever had before. Adaptive optics can correct the aberrations for light leaving the eye as well as for light entering it, which make it possible to obtain sharper images of the living retina as well. Based on subjective observations of a point source of light, Helmholtz argued that the eye suffered from a host of aberrations that are not found in conventional, man-made optical systems (4). There have been a number of methods developed to quantify these aberrations (5–11). Recently, Liang et al. (12) developed a technique based on the Shack-Hartmann wavefront sensor (13) that provides a rapid, automated, and objective measure of the wave aberration simulta-

* Current affiliation: Intel Corporation, Santa Clara, California † Current affiliation: Indiana University, Bloomington, Indiana ‡ Current affiliation: University of Houston, Houston, Texas


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neously at a large number of sample points across the eye’s pupil. Using this technique, Liang and Williams (14) provided what is arguably the most complete description to date of the wave aberration of the eye. They measured the eye’s aberrations up to 10 radial orders, quantifying the irregular aberrations predicted by Helmholtz and subsequent investigators (6,7,15). The effect of these aberrations on retinal image quality is illustrated in Fig. 1, which shows the point spread function (PSF) at various pupil sizes. The PSFs are calculated at 555 nm from wave aberration data obtained with a Shack-Hartmann wavefront sensor (14) for a single eye. Defocus and astigmatism, which can be corrected with spectacles, have been removed. At pupil diameters less than about 3 mm, diffraction is the single largest source of retinal image blur. A pupil of about 3 mm diameter, which is typical in bright viewing conditions, generally provides the best optical performance in the range of spatial frequencies that are important for normal vision, i.e., 0–⬃30 cycles/deg. While increasing the pupil size increases the high spatial frequency cutoff, aberrations dominate and optical quality is actually reduced. For example, the Strehl ratio of the eye with a 6-mm pupil is typically less than 0.1 even when defocus and astigmatism

Figure 1 Change in PSF with pupil size. The point spread function for a single eye is calculated for a range of pupil sizes using the aberrations measured with the Shack-Hartmann wavefront sensor.


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have been corrected. A substantial gain in retinal image quality could be achieved if all the eye’s aberrations could be corrected across the dilated pupil. For example, for a fully dilated 8-mm pupil, the width of the PSF could be reduced more than 2.7-fold over that obtainable with a 3-mm pupil. The advantage is illustrated with the MTFs in Fig. 2, which shows the higher cutoff frequency and substantially increased modulation at all spatial frequencies expected when the aberrations have been corrected across an 8-mm pupil. There have been various attempts to compensate for the additional aberrations besides the defocus and astigmatism corrected by spectacles. The aberrations of the eye can be avoided with interference fringes imaged on the retina (16–19). However, this technique is impractical for viewing stimuli other than sinusoidal gratings and is of no use for improving normal vision and the quality

Figure 2 Potential improvement in the eye’s MTF by correcting the eye’s higher order aberrations. Shown is the best MTF of the eye in normal viewing, obtained with various pupil diameters averaged across at least 12 eyes (14) with an optimal correction of defocus and astigmatism, and the ideal MTF of the eye for an 8-mm pupil blurred only by diffraction. The shaded region shows the range of contrasts and spatial frequencies that are inaccessible both for the case of projecting patterns on the retina and for the case of imaging the retina outside of the eye (22).


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of fundus images. Another approach is to use a contact lens to null the refraction at the first surface of the cornea. This approach has the advantages of simplicity and low cost. However, contact must be made with the eye. Also, the method can correct only the wave aberration at the first surface of the cornea, though the complete wave aberration of the eye depends on the aberrations of the lens as well as the cornea. A recent attempt to use a contact lens to increase the axial resolution of a confocal laser scanning ophthalmoscope (20) showed only modest improvement. Adaptive optics has the potential to correct the wave aberration of all the eye’s refracting media without making contact with the eye. Moreover, adaptive optics is well suited to cope with the large variation in the wave aberration from eye to eye. In normal eyes in which light scatter in the ocular media is not an important limitation of image quality, adaptive optics offers the possibility of obtaining diffraction-limited retinal image quality with a dilated pupil. Dreher et al. (21) used a deformable mirror to correct the astigmatism in one subject’s eye based on a conventional spectacle prescription. Liang, Williams, and Miller (22) have combined a Shack-Hartmann wavefront sensor with a deformable mirror to correct most of the significant aberrations in the eye. They have studied the visual performance of eyes looking through adaptive optics and have obtained images of the living retina through adaptive optics.

II. MEASURING THE WAVE ABERRATION OF THE EYE A. Shack-Hartmann Wavefront Sensing for the Eye Figure 3 shows the basic components of the wavefront sensor used by Liang and Williams (14). The position of the subject’s eye is fixed with respect to the optical system by having him clench his teeth on a dental impression. The optical axis of the wavefront sensor passes through a reference axis in the eye. Liang and Williams chose as a reference axis the line connecting the center of the dilated pupil to the foveal center. The subject’s pupil is typically dilated with a drug such as tropicamide or cyclopentolate hydrochloride, though this is not necessary if a measurement is desired across only a relatively small portion of the pupil. A polarizing beamsplitter directs a collimated, linearly polarized beam of a low power laser onto the pupil. The beam is focused to a small spot on the retina. This spot is equivalent to the guide star or laser beacon in astronomical applications of adaptive optics. The use of a small (ⱕ1.5 mm) diameter for the beam at the pupil ensures that the image on the retina is diffraction-limited and relatively unaffected by changes in the refractive state of the eye. Liang and Williams used light from a 5-mW, 632.8-nm He-Ne laser. In a typical measurement at 632.8 nm, the retina was illuminated for 1 s with a total energy


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Figure 3 Wavefront sensor for the eye. Collimated light from the laser produces a compact point source on the retina. If the eye has aberrations, the wavefront of the light returning from the retina is distorted upon exiting the pupil plane. This wavefront is recreated by relay optics at the lenslet array. The 2-D lenslet array samples this warped wavefront and forms an array of focused spots on the CCD chip. Each spot is displaced on the CCD array in proportion to the slope of the wavefront, from which the wave aberration itself can be calculated.

entering the eye of less than 3 µJ, which is about 200 times less than the ANSI maximum permissible exposure (23). We have tried wavelengths from 543 nm to 780 nm with success. Shorter wavelength light could presumably also be used, though the low reflectance of the retina at short wavelengths and laser safety considerations would demand increased exposure durations. Longer wavelengths than about 780 nm can be used but there is some reduction in spot definition, presumably due to increased penetration and scatter in the choroidal layer behind the retina. Superluminescent diodes in the near infrared are convenient because they are compact and the point source appears less bright for the subject. Such diodes also have short coherence lengths, which reduces artifacts in the wavefront sensor image caused by interference from multiple surfaces in the instrument. The polarizing beamsplitter transmits depolarized light from the retina and also rejects the unwanted, polarized reflection from the first surface of the cornea. An alternative way to avoid the corneal reflection is to use a conventional beamsplitter and to displace the beam in the eye’s entrance pupil slightly. The relay lenses image the subject’s pupil onto an array of lenslets. Each lenslet forms an aerial image of the retinal point source on a CCD camera. In Liang and Williams’ system, the local wavefront slope was measured at 217 locations simultaneously across the pupil. The wave aberration was computed from the array of local slopes with a least-squares technique (24) and was represented with the sum of 65 Zernike polynomials, corresponding to aberrations up to and including 10th order (25).


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B. Wave Aberration of the Normal Human Eye Figure 4a–c shows the CCD camera images from three eyes obtained with a dilated 7.3-mm pupil. Figure 4d–f shows the contour plots of the wave aberration computed from these data. Defocus and astigmatism have been removed from the wave aberration plots, yet substantial irregular aberrations remain. The peak to valley wavefront error across the 7.3-mm pupil is about 7 µm, 4 µm and 5 µm for JL, OP, and ML, respectively. Subject ML has an arc-shaped ridge on the upper third of the plot that corresponds to the location where his eyelid normally rests against the cornea. Despite the large differences in the wave aberration between eyes of different observers, the wave aberrations of left and right eyes of the same observer are sometimes rather similar when the left eye image is flipped about a vertical axis for comparison with the right eye (14). This indicates that the irregular aberrations are not random defects. Almost all of the variability in repeated measurements of an individual eye’s wave aberration arises in the eye itself and not the instrument. Liang and Williams found that the standard deviation in the wave aberration for repeated

Figure 4 Wavefront sensor images and wave aberration of eyes for a 7.3-mm pupil. (a–c) Wavefront sensor images collected with the CCD camera for three observers: JL, OP, and ML. The center-to-center spacing of lenslets in the pupil was 0.42 mm. (d–f) Corresponding wave aberration of the three eyes from measurements of the wavefront slopes. The contour interval in the wave aberration plots is 0.15 µm for OP and 0.3 for JL and ML µm. Defocus and astigmatism have been removed from the wave aberrations, showing the presence of substantial irregular aberrations. (From Ref. 14.)


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measurements was 0.046 µm or about λ/14 (λ ⫽ 0.633 µm). Possible sources of variability include fluctuations in defocus and other aberrations (despite the use of cyclopentolate hydrochloride), small movements of the eye and head, cardiovascular movement, and variations in the thickness of the tear film. Eye movements probably have a very small effect on the wavefront measurement because the shift of pupil position is small for the fixating eye. Small eye movements during each exposure actually are beneficial as they result in the illumination of different patches of retina that destroy most of the laser’s spatial coherence. Any coherence that remains does produce laser speckle in the images which changes from exposure to exposure, adding some variability to the measurements, particularly when short exposures (⬍ 1 s) are used. The eye’s aberrations often do not correspond to the classical aberrations of man-made optical systems (e.g., Seidel aberrations). The root mean square (RMS) error of individual Zernike orders is a measure of its role in degrading optical quality. Figure 5 shows the RMS wavefront error contributed by each

Figure 5 Zernike description of the eye’s aberrations. The upper curve (square symbols) shows the RMS wavefront error of each Zernike order for a 7.3-mm pupil averaged across 14 human eyes. Error bars indicate the standard deviation among eyes. For the second order Zernike modes, only astigmatism is shown. The average amount of astigmatism in these observers was 0.6 diopter, corresponding to a mean RMS value of 0.77 µm. The middle curve (triangular symbols) shows the data for a 3.4-mm pupil averaged across 12 eyes. The lower curve (round symbols) is for an artificial eye. The error bars shows the standard deviation of 10 repeated measurements (14).


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Zernike order for a 3.4-mm pupil (triangular symbols) and a 7.3-mm pupil (square symbols). The average RMS wavefront error decreases monotonically as the Zernike order increases for both pupil sizes in the human eye, though the pattern varies somewhat among individual observers. The total RMS error for the small pupil lies three to four times lower than that for the large pupil of real eyes. This illustrates the well-known fact that aberrations grow with increasing pupil size. An RMS error of λ/14 (0.045 µm at 0.633 µm) is a common tolerance for diffraction-limited performance in an optical system (25). For the 3.4-mm pupil, only Zernike orders up to third order exceed this tolerance. At the larger pupil size, however, the mean RMS value of each Zernike radial order from 2 to 8 is greater than λ/14. Figure 5 also shows the RMS wavefront error for an artificial eye (round symbols) measured with the same instrument. The artificial eye consisted of a achromatic doublet ( f ⫽ 16 mm) and a diffuser to mimic the retina. These measurements reveal some expected spherical aberration in the artificial eye, but nonetheless the mean RMS value of each order averages about one order of magnitude lower than in human eyes for the large 7.3-mm pupil. Evolution has not yet created optics in the human eye that can rival the high quality optics fabricated by man.

III. ADAPTIVE COMPENSATION A. Technique Figure 6 shows the experimental setup for measuring and correcting the eye’s wave aberration. It consists of an adaptive optics system combined with a system for vision testing and retinal imaging. A detailed description of the system can be found in Liang, Williams, and Miller (22). The adaptive optics portion of the system contains the Shack-Hartmann wavefront sensor described earlier and a deformable mirror with 37 actuators in a square array (Xinetics, Inc.). The stroke of the mirror beneath each actuator is ⫾2 µm, allowing a wavefront shift of 8 µm in the reflected beam. The mirror lay in a plane conjugate with both the eye’s pupil plane and the lenslet array of the wavefront sensor. Compensation was achieved with closed-loop feedback control. In each loop, six images, each of 300-ms duration and separated by 400 ms, were obtained with the wavefront sensor. The wave aberration in the system including the eye and mirror was computed from the sum of the six images and evaluated at the locations of the actuators. Wavefront correction was performed iteratively by setting the mirror to compensate one tenth of the calculated wavefront error in each loop. Iterative loops were repeated until the RMS wavefront error could be reduced no further, which usually required 10–20 loops. A value of one tenth


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Figure 6 Optical system for wavefront sensing and correction. The eye focused a collimated laser beam onto the retina. The light reflected from the retina formed an aberrated wavefront at the pupil. The distorted wavefront is measured by a Shack-Hartmann wavefront sensor. A deformable mirror, conjugate with the pupil, compensated for the eye’s wave aberration. After compensation was achieved, psychophysical or retinal imaging experiments were performed with a 6-mm pupil. For vision testing, a stimulus, such as a grating, was viewed through the deformable mirror. For imaging the retina, a krypton flash lamp delivered a 4-ms flash, illuminating a retinal disk 1° in diameter. A scientific grade CCD acquired images of the retina (22).

of the wave aberration was selected for each iteration to ensure a well-behaved correction without propagating noise through the system. B.

Quality of Correction

Figure 7 shows the wave aberration for two subjects (JL and DM) before and after adaptive compensation. Though still not completely planar, the wavefront after compensation is much flatter. In four subjects measured, adaptive compensation reduced the peak to valley wavefront error across a 6-mm pupil by a factor of 4, on average. To determine which aberrations were corrected by the deformable mirror, the wave aberration of the eye was decomposed into 65 Zernike modes up to


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Figure 7 The wave aberration for two eyes (JL and DM) without and with adaptive compensation for a 6-mm pupil. For the uncompensated case, trial lenses were used to correct the astigmatism. According to the wavefront sensor measurements, the astigmatism left uncorrected by the trial lenses is 0.28 diopter and 0.27 diopter for JL and DM, respectively. Defocus of the eye was corrected by adjusting to get the highest contrast images when the deformable mirror was flat (22).

10th radial order. A comparison of the measured RMS wavefront error of each Zernike order for the eyes without adaptive compensation with the error after adaptive compensation showed that lower Zernike orders up to and including fourth order were significantly reduced while aberrations beyond sixth order remain almost unchanged. These results show that with adaptive compensation we not only can correct the eye’s defocus and astigmatism but also coma, spherical aberration, and to a lesser extent additional, irregular aberrations in the eye. Figure 8 shows the point spread functions calculated from the wave aberrations shown in Fig. 7 at a wavelength of 633 nm. For the best correction for these two eyes shown here, adaptive compensation increased the Strehl ratio from 0.05 to 0.47 for subject JL and from 0.05 to 0.33 for DM. After compensation, the PSF for both JL and DM has a full-width at half-height (FWHH) of 2.0 µm, close to the value of 1.9 µm expected from diffraction alone. This is smaller than


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Figure 8 The eye’s point spread function for subjects JL and DM without and with adaptive compensation for a 6-mm pupil. The PSFs were computed from the corresponding wave aberrations shown in Fig. 7 (22).

the diameter of a foveal cone and is about two times narrower than the FWHH obtained with a diffraction-limited 3.0-mm pupil, the pupil size giving about the best image quality in normal viewing. C.

Vision through Adaptive Optics

1. Contrast Sensitivity Stimuli such as edges viewed through the compensating deformable mirror have a strikingly crisp appearance consistent with the supernormal quality of the retinal image. Liang, Williams and Miller showed that observers with normal vision had enhanced contrast sensitivity when looking through adaptive optics. For example, they could resolve fine, 55 c/deg gratings that were invisible under normal viewing conditions. For lower spatial frequency gratings that were visible without adaptive optics, the contrast sensitivity was significantly increased after adaptive compensation. For example, they measured about a sixfold increase in contrast


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sensitivity for 28 c/deg gratings. These measurements were made with monochromatic light. A smaller increase would be expected in polychromatic light due to the chromatic aberration of the eye. 2. Aliasing and the Neural Limits of Vision While the contrast sensitivity measurements suggest that visual performance is enhanced by adaptive optics, visual performance probably declines for some specialized visual tasks because of aliasing by arrays of neurons, such as photoreceptors. For example, when observers viewed a steady, 633-nm point source through the deformable mirror, it sometimes appeared green and sometimes red. The color fluctuation was much more robust with adaptive compensation than without. This effect, which is an example of chromatic aliasing (26), has been attributed to the selective excitation of different cone classes as eye movements shift the retinal location illuminated by the point source (27,28). With adaptive compensation, the FWHH of the eye’s PSF is often smaller than the diameter of a single foveal cone, increasing the fraction of time that only a single receptor is stimulated. Color discrimination is apparently impaired under these conditions. Performance may also decline for certain other fine foveal tasks, such as two-dot vernier acuity (29). D. Retinal Imaging through Adaptive Optics Recently, techniques have succeeded in recovering very high spatial frequency information from the living human retina. Estimates of the dimensions of microscopic structures in living eyes have been obtained using spatially coherent (30) and spatially incoherent light (31) under conditions in which only defocus and astigmatism in the eye were corrected. Using a technique proposed by Artal and Navarro (32), the spacing between cone photoreceptors in living retinas can be estimated based on the power spectra of retinal images in temporally and spatially coherent light, but laser speckle associated with coherent imaging and imaging blur due to the eye’s aberrations prevent the resolution of these cells in single exposures. Miller et al. (31) showed, using spatially incoherent light which had a temporal coherence length of 1.25 mm, that when defocus and astigmatism are carefully corrected the cone mosaic can sometimes be resolved in individual retinal images. Roorda et al. (33) was also able to resolve photoreceptor structure with a confocal scanning laser ophthalmoscope. Adaptive optics can substantially improve the quality of such images. In the apparatus of Liang, Williams, and Miller (22), a krypton flash lamp delivered a 4-ms flash, illuminating a retinal disc 1° in diameter. The 4-ms exposure helped to prevent motion blur due to the movement of retina during each exposure. The lamp output, which was broadband white light, was filtered with a 10-nm band-


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width interference filter, providing a temporal coherence length of 40 µm at a wavelength of 630 nm. Images are typically taken with peak wavelengths from 550 to 630 nm. To acquire images of the retina, a scientific grade CCD was positioned conjugate with the retina, in the plane that previously contained the grating stimuli used in contrast sensitivity measurements. Astigmatism and defocus were corrected before deforming the mirror to correct the residual aberrations. Figure 9 shows images of the same retinal location (1° from the center of the fovea) for subject JW. The image on the left is a single snapshot taken without compensation, but with the best correction of defocus and astigmatism. The middle image was taken with adaptive compensation. After adaptive compensation, the photoreceptor array is better resolved and has a higher contrast. While most photoreceptors can be seen in a single corrected image, there are some regions where the reflected light is weak and the signal to noise is too low to detect every cone. Eye movements and retinal light exposure hazards preclude increasing the signal by increasing the exposure time or flash intensity. However, by registering and adding a series of images, improved images of the photoreceptor array can be obtained. The rightmost image in Fig. 9 shows the average of 61 frames obtained at the same retinal location in which all frames were registered with cross-

Figure 9 Images before and after adaptive compensation for aberrations for JW’s right eye. All three images are of the same retina area at 1° from the central fovea. Images were taken with 550-nm light (25-nm bandwidth) through a 6-mm pupil. Each image has been normalized to its mean intensity and both are displayed within the same gray scale range. The dark vertical band across each image is an out-of-focus shadow of a blood vessel. The leftmost image shows a single snapshot taken when only defocus and astigmatism have been corrected. The middle image is a snapshot of the same retinal location when additional aberrations have been corrected with adaptive optics. The rightmost image shows the benefit in image quality obtained by registering and averaging multiple frames, 61 in this case.


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correlation. In the registered stack of images, virtually every cone is resolved across the field. By adjusting focus, retinal structures at different depths can be observed. Figure 10 shows two images of the same retinal location for a single subject JP. The capillaries lie in the inner layers of the retina (toward the vitreous) and appear as shadows, back-illuminated by the out-of-focus photoreceptor array. By focusing roughly 100 µm deeper into the retina, the photoreceptor array is clearly seen with the blood vessels appearing as dim shadows across the image. In these images, there is no evidence of a variation in image quality across the image. This indicates that the isoplanatic patch, which limits the compensated field of view to only a few arc-seconds for ground-based telescopes (34), is at least as large as the 1° field of this instrument. Adaptive optics provides a noninvasive technique to study the normal and pathological living retina at a microscopic spatial scale. A fundus camera equipped with adaptive optics provides unprecedented transverse resolution, so the living retina can be seen at a spatial scale previously accessible only in excised retina. By correcting the eye’s wave aberration for a 6-mm pupil, the PSF measurements suggest that the present system has created about an eightfold increase in the Strehl ratio and a twofold decrease in the PSF’s FWHH.

Figure 10 1° images of JP’s right eye of the same retinal location at two focal planes. In the left image, capillaries as small as 7.5 µm are seen. By focusing deeper into the retina, the underlying photoreceptor mosaic can be seen and the capillaries appear as faint shadows.


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IV. FUTURE DIRECTIONS A.

Real-Time Compensation

Under steady-state viewing conditions the eye exhibits temporal instabilities in focus. These so-called accommodative microfluctuations occur with RMS amplitudes between 0.2 and 2.0 diopters, depending on viewing conditions(35,36). The spectrum of the temporal variations in accommodation is a falling function of frequency with two dominant regions of activity, a low frequency component (⬍ 0.6 Hz) and a higher frequency component occurring somewhere between 1.0 and 2.5 Hz. There is negligible power to be found in frequencies beyond about 3 Hz (37). The optical system of Liang, Williams, and Miller (22) was incapable of tracking these fluctuations in focus, nor any other potential temporal dynamics in the eye’s aberrations. Their system may be more properly described as an active rather than an adaptive optical system, in that the closed-loop feedback they used involved tens of seconds per loop. Generally, cyclopentolate hydrochloride was used to paralyze accommodation in an attempt to minimize these changes. High retinal image quality could be obtained more consistently and more quickly with a higher bandwidth system that tracked the wave aberration in real time. There are no fundamental obstacles to achieving this. Hofer, Porter and Williams (38) have demonstrated a wavefront sensor for the eye that can operate at 5 Hz and are currently incorporating this system in the adaptive optics retinal camera at the University of Rochester. Their measurements suggest that the temporal bandwidth of the eye’s total wave aberration is about 3 Hz, similar to that which has been previously measured for defocus alone. It appears that a closedloop bandwidth of less than 3 Hz, corresponding to a frame rate of 10–20 times higher (30–60 Hz), is all that is required to track the eye’s temporal dynamics. Typical adaptive optics systems in astronomy must operate at rates an order of magnitude higher. B.

Alternative Devices for Wavefront Compensation

Less expensive and smaller devices for wavefront compensation would accelerate the development of ophthalmic instruments equipped with adaptive optics. Significant cost reduction could be obtained using bimorph mirrors (39), membrane mirrors (40), or microelectromechanical deformable mirrors (MEM-DM) (41– 43). Both segmented and continuous-membrane MEM-DMs are still in early stages of development. Liquid crystals are also currently being investigated as an inexpensive alternative in adaptive optics applications (44), including retinal imaging (45). Unlike monolithic deformable mirrors the minimum size of which is limited by


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an actuator spacing on the order of 7 mm, liquid crystal arrays can be made small enough to eliminate the need to magnify the image of the eye’s pupil, reducing the size of the instrument. Their low temporal bandwidth is not a problem for vision applications. The light throughput of a liquid crystal device is lower than with a deformable mirror, and no commercial product is presently available for adequately modulating both polarization components of the incident wavefront. Also, the phase wrapping required in the present commercial products, due to their thin liquid crystal layer, rejects the use of polychromatic illumination. The ability to compensate the eye’s wave aberration in polychromatic illumination is desirable both in applications to improve vision and retinal imaging. Chromatic aberration (46) caused by the dispersion of the ocular media produces a large, 2-diopter, change in the eye’s power across the visible spectrum from 400 to 700 nm. Despite this, human vision is surprisingly unaffected, and correcting axial chromatic aberration by itself does little to improve visual performance (47). This is partly because the M and L cone photopigments, which catch the light used for detailed spatial vision, reject the short wavelength light that is most blurred by chromatic aberration (48). It is also because uncorrected monochromatic aberrations dilute the benefit of correcting chromatic aberrations. Much larger benefits accrue when both monochromatic and chromatic aberrations are corrected simultaneously. Axial chromatic aberration could be corrected with an achromatizing lens (49), though the alignment of such lenses is critical and difficult to ensure (47,50). Alternatively, images in different spectral bands could be obtained in rapid succession with the deformable mirror compensating for the particular chromatic defocus in each spectral band. While adaptive optics has successfully been applied to the human eye, the scientific and medical benefits of this technology are only just beginning to be explored. Possible future applications, for retinal imaging and for improving vision are discussed below.

C. Future Applications in Retinal Imaging There are a number of applications of adaptive optics for basic research in visual science, since it is now possible to study microscopic structures in the living retina that could not be resolved before. Currently, studies of retina at the spatial scale of single cells require removal of the retina from the eye. Moreover, this causes irreversible changes in the tissue that confound many kinds of experiments. For example, it has been known for some time that photoreceptors behave like waveguides (51,52), but it has not been possible to measure the antennae properties of single primate photoreceptors in their undisturbed state within the


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eye. The ability to image living photoreceptors with adaptive optics offers the possibility to improve our understanding of the light-collecting and antenna properties of the retina. As a second example, though we have known for nearly 200 years that color vision is mediated by three cone classes, their spatial arrangement and relative numbers are not well-characterized in the human retina. Mollon and Bowmaker (53) and Packer, Williams, and Bensinger (54) have shown that it is possible to reveal the relative numbers and packing geometry in primate retina based on the spectra of the photopigment contained in the cones, but their techniques have yet to be applied successfully to human tissue. It is hard to obtain excised human retina in a fresh enough state to conduct such experiments. Roorda and Williams (55) have recently successfully identified all three cone classes in the living human eye by combining high resolution imaging through adaptive optics with retinal densitometry of the cone photopigments. The ability to reveal the trichromatic mosaic in living eyes offers the exciting possibility of measuring human color and spatial vision in retinal locations where the cone mosaic is known exactly. Adaptive optics may also provide earlier detection, better diagnosis, and more effective treatment of retinal disease. The ability to resolve photoreceptors in the living eye could increase our understanding of cone distrophies. Glaucoma, a disease in which blindness ensues from the gradual loss of optic nerve fibers, can only be detected with conventional imaging techniques after a significant amount of damage has already taken place. More accurate measurements of the nerve fiber layer thickness around the optic nerve head will allow earlier detection of the disease. Another example is age-related macular degeneration (AMD), which causes a progressive loss in the central vision of 10% of those over age 60. It has been suggested that there may be a subclinical loss of receptors associated with AMD that is not yet detectable in the fundus by current means (56). Adaptive optics could reveal the early stages of the development of this disease. In one form of AMD, deterioration of the retina begins with a formation of new blood vessels in the choroid and an eventual leakage of fluid beneath the retina around the fovea. This can eventually cause blindness in the most highly resolving location in the visual field. Treatment requires accurate delivery of a photocoagulating laser beam which could benefit from a higher resolution view of this critical retinal region. Moreover, in combination with improved eye-tracking technology, adaptive optics may make it possible to deliver with improved accuracy a more compact, and therefore less invasive, photocoagulating beam. Another disease that effects the retina, diabetes, can produce microaneurysms in the retinal vasculature, the treatment for which is also laser photocoagulation. Adaptive optics could provide an enhanced view of this retinal region without the use of invasive dyes such as those used in fluorescein and indocyanine green


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angiography, and allow earlier detection and laser treatment with greater precision. The ability to image microscopic structures in living retina may be particularly valuable for tracking retinal processes that vary over time at a microscopic spatial scale, such as the progression of retinal disease or the development of the normal retina. Current methods using excised retina require piecing together evidence from large numbers of retinas, whereas microscopic imaging of the living retina could reveal the whole temporal sequence in individual eyes. For example, in retinitis pigmentosa, mutated genes that are responsible for the production of rhodopsin lead to the death of rod photoreceptors and ultimately blindness. The evaluation of therapies for this disorder could be aided by higher resolution retinal imaging that would allow the integrity of individual receptors to be tracked over time. Digital imaging offers the possibility to post-process images to extract quantitative information about the retina (57). Early attempts to enhance fundus images employed simple spatial filtering techniques to increase contrast. More powerful techniques, both those that require a priori knowledge of the eye’s aberrations and those that do not, are just beginning to be explored. Image restoration techniques applied after blurring by aberrations has occurred tend to be sensitive to noise, whereas adaptive optics precludes blurring to begin with. For this reason, it is unlikely that image restoration alone could provide the high resolution images available with adaptive optics. Nonetheless, post-processing could be a useful supplement to adaptive optics, and some low resolution applications may not require adaptive optics at all. Miller et al. (58) have evaluated a method called bispectral imaging (BIM) that does not require a priori knowledge of the retina nor the aberrations of the eye. In simulations, the method proved very effective for images blurred by even aberrations (e.g., defocus, astigmatism, and spherical aberration), but was less effective when confronted with typical ocular aberrations, which usually included significant odd aberrations (e.g., coma). Phase diversity (59,60) offers a promising method to improve fundus images while simultaneously estimating the eye’s aberrations. Alternatively, the measurement of the eye’s wave aberration with, for example, a Shack-Hartmann wavefront sensor offers the possibility of using deconvolution (61) in conjunction with retinal imaging. An additional increase in the quality of retinal images could be obtained if a larger exit pupil were used. Liang, Williams, and Miller (22) used a 6-mm pupil; with an 8-mm pupil and 555-nm illumination, one could, in principle, produce a FWHH of the PSF of 1.18 microns, and a 2.7-fold increase in transverse resolution over that for a 3.0-mm pupil. Even greater benefits of adaptive optics could be achieved in axial resolution. The confocal scanning laser ophthalmoscope achieves its optical sectioning


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capability by focusing a scanning spot on the retina and collecting the returning light with a pinhole positioned so as to reject light from other locations than the one at the point of focus. The performance of this instrument is limited by the dimensions of this focused spot, which could be made much smaller with adaptive optics. Whereas transverse resolution increases linearly with pupil size, the axial resolution of the confocal scanning laser ophthalmoscope increases as the square of the pupil size. This offers a potential increase in axial resolution of a factor of 7 provided that diffraction-limited imaging is obtained across an 8-mm pupil. Dreher et al. (21) demonstrated an axial resolution improvement in a confocal SLO by correcting the astigmatism of the eye with a deformable mirror. The adaptive correction of additional aberrations will increase axial resolution closer to the theoretical limit. Optical sectioning of the retina could ultimately provide an axial FWHM as small as 19.4 µm, approaching the current capabilities of optical coherence tomography (62,63). Such capabilities would allow one to produce the highest resolution 3-D images yet of the living human fundus.

D.

Future Applications to Improve Vision

Measurements of the eye’s wave aberration could provide a quantitative description of retinal image quality before and after refractive surgery (64). It may be possible to use wavefront sensing to improve the surgical procedure, perhaps ultimately providing a superior visual outcome. In one respect, wave aberration measurements have an advantage over corneal topographic measurements because they characterize the performance of all the eye’s refracting media upon which good vision depends, not just the corneal surface. Wave aberration measurements might also aid in the design of contact lenses (65) and intraocular lenses. In principle, one could develop customized components to correct additional aberrations besides defocus and astigmatism, a technology that could be particularly helpful for eyes with large amounts of irregular aberrations. The evidence of Liang, Williams, and Miller (22) that the eye’s contrast sensitivity is improved with the correction of additional high order aberrations encourages further exploration of the potential benefits of adaptive correction of normal human eyes. If the optics of the eye were completely corrected, neural factors would set an upper bound for visual performance. Indeed, aliasing, which can be seen when looking through adaptive optics, is one kind of neural limit. It is often argued that evolution has given the human eye an optical quality that is optimized with respect to the grain of the retina (66). Adaptive optics is a new tool to examine this theory and to investigate under what circumstances observers can take advantage of supernormal retinal image quality. The largest benefits will accrue when the pupil is large, corresponding, for example, to indoor lighting


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conditions. In bright daylight conditions outdoors, the natural pupil of most normal eyes is sufficiently small (⬃3 mm) that diffraction dominates and monochromatic aberrations beyond defocus and astigmatism are relatively unimportant. Eyes with large amounts of irregular aberrations would benefit at all pupil sizes. Liang, Williams, and Miller (22) concentrated their efforts on using adaptive optics to correct higher order ocular aberrations because spectacles and contact lenses already correct defocus and astigmatism. Nonetheless, defocus and astigmatism remain the largest sources of retinal image blur and adaptive systems that correct only these aberrations could be very useful. One can imagine, though we cannot yet build, a pair of adaptive binoculars that automatically compensates for the defocus and astigmatism of each user. It is an even more difficult engineering challenge to incorporate adaptive optics into a pair of spectacles, though there have been attempts to do so (67,68). If it could be made sufficiently light, compact, and reliable, autofocus eyeware would be an attractive alternative to bifocals and progressive lenses for those of us over 40 years of age who suffer from presbyopia.

ACKNOWLEDGMENTS This research was supported by NIH Grants EY04367, EY01319, a RE&HPB Fellowship, and an ophthalmology development grant from Research to Prevent Blindness, Inc. The authors thank G. M. Morris, J. Porter, A. Russell, B. Singer, H. Tamaddon, W. Vaughn, and T. Williams of the University of Rochester, and R. Fugate of Starfire Optical Range, Phillips Laboratory/LTE for their technical assistance.

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11 Wide Field-of-View Wavefront Sensing Erez N. Ribak Technion–Israel Institute of Technology, Haifa, Israel

Standard adaptive optics systems measure and correct the wavefront error along the path to the reference star. Because the measured volume is very narrow, and because the correction is limited to the region of the measured path, the attained field of view is rather small—a few arc-seconds. The problem is even more severe when the reference star is not the measured star: either it is another natural star which could be far away or an artificial guide star at tens of kilometers above the telescope. In both cases the measured volume of atmosphere (Fig. 1) does not overlap with the observation path to the star and its vicinity, namely a truncated cone whose bases are the telescope and the star field. Even for bright objects, where the measurement is accurate, the corrected isoplanatic angle cannot extend beyond the few arc-seconds limit. One would like to have wider fields, of arc-minutes and more, over which turbulence can be corrected. A wider field of view can be achieved by multiconjugate adaptive optics. In this method, the atmospheric turbulence is measured at various elevations and corrected by using several optical elements, usually conjugate to the most offending layers. The idea was proposed very early on by Dicke [1975] and by McCall and Passner [1978] and became possible with the suggestion of laser guide stars by Hudgin in 1980 and Feinlieb in 1981 [Fried 1992], and by Foy and Labeyrie [1985] and Beckers [1988, 1989]. However, in the same manner that technical difficulties delayed the employment of adaptive optics in general, multiconjugate adaptive optics has been slow in application, and only two initial experiments have been performed in this direction [Murphy et al. 1991, Neyman and Thompson 1995]. Like the rest of adaptive optics, there are some issues that will have to be addressed before the method is put to use. Since no adaptive


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Figure 1 The measured volume of atmospheric turbulence is smaller than the volume to be corrected for a wide field. The overlap between measurement and required correction (crosshatched) is shown schematically for two atmospheric layers: using a natural guide star (ngs: top) and a laser guide star (lgs: bottom). If the deformable mirror is conjugated to this layer, it can only correct it partially. If the area outside the mirror is blocked, it leads to vignetting.

optics system has yet been designed and constructed with wide field correction, most of the discussion to follow will concentrate on principles, identified problems and suggested solutions.

I. SEPARATION OF THE ATMOSPHERE INTO THIN LAYERS Let us assume we have a correcting element, such as a flexible or segmented mirror working in reflection (Chap. 5), or a liquid crystal phase modulator [Love


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1993], in transmission mode (Chap. 9). To simplify the discussion we shall refer to all wavefront correctors as deformable mirrors. Now in order to have maximum effect these mirrors have to be conjugated to the most offending atmospheric layers, or be placed strategically, so as to maximize their impact on the performance of the system. Thus, the effective turbulence height profile C 2n (h) [Roddier 1981] has to be known. Because wavefront measurements tend to be noisy, the subaperture size must be as large as possible so as to collect more light (Chap. 4). However, it is limited by the size of r0 [Roddier 1981] for each layer. Thus, knowledge of the actual r0 of the separate layers is also useful, as well as the wind speed and direction in the various layers. Is it at all possible to locate optimal positions for the mirrors? Is atmospheric turbulence continuous, or is it layered? These questions have various answers, according to telescope site and specific constraints. Older measurements of the turbulence [Bufton et al. 1972] led to rather accurate models (e.g., Hufnagel-Valley [Hufnagel 1974]) by assuming continuous turbulence [Coulman 1985, Beland 1993]. However, Troxel et al. [1994] were able to show that the atmosphere can be modeled adequately by using only four distinct layers. They were able to achieve an atmospheric optical transfer function accurate to within 1% of that of a continuous atmosphere (Fig. 2). Careful inspection of the older measurements of C 2n (h) show [Bufton et al. 1972, Coulman 1985, Beland 1993] that the atmospheric turbulence profile (as a function of height) is indeed rather inter-

Figure 2 Atmospheric turbulence is layered, as various measurements imply. Even very smooth models which represent it properly are still equivalent to very few layers as shown by experiment [Tallon et al. 1992a] and simulation [Troxel et al. 1994].


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mittent. Other measurements [Vernin and Roddier 1973, Rocca et al. 1974, Roddier and Vernin 1977, Azouit and Vernin 1980, Sarazin 1987, Caccia et al. 1987, 1988, Tallon et al. 1992a, Roddier et al. 1993, Irbah et al. 1993, Sivaramakrishnan et al. 1995, Vernin and Munõz-Tunõn 1994, Acton et al. 1996] seem to indicate that at very good sites the number of layers may be as small as two or three, and it becomes larger for worse sites. However, even at a very good site, modeled with only one or two dominant layers, there is some residual turbulence, not included in these layers, which has to be dealt with [Racine and Ellerbroek 1995]. Thus, there are three approaches that the designer of a multiconjugate adaptive optics can take, once he had acquired a good knowledge of the turbulence profile C 2n (h), of r0 (h), and of vwind (h): a. Fit a continuous model to C 2n (h); optimize the conjugate planes for the deformable mirrors. This approach is good for the cases where the seeing is usually bad and the atmosphere cannot be well described by a set of few layers. It is also applicable when it is not practical to vary the adaptive optics system, and an average model of the atmosphere is preferable. b. Fit a few-layers model to C 2n (h); conjugate the deformable mirrors to these fitted layers. Here it is assumed that all the turbulence is concentrated in the corrected layers. c. Fit conjugation planes for the deformable mirrors to C 2n (h) (not necessarily at the roughest layers) [Racine and Ellerbroek 1995; Avila et al. 1998]. Parts of this task can be achieved by neural networks [Monterra 1996]. Cost functions for optimizing the conjugation planes of the deformable mirrors are some combination of the following: a. Spatial band of the observed object. For example, when one is interested only in coarse or fine details of the object, or a specific orientation, such as when looking for extrasolar planets [Angel 1994]. In this case, objects are not at the highest spatial frequencies, but they are very different in intensities. The adaptive optics point spread function should be optimized so as not to have side lobes which might scatter light from the main star to its faint companion. Another example is spectroscopy, where one needs to get as much light into the slit, but it might not be as important to minimize the image size in the lateral direction. b. Spectral band. Turbulence is much worse in the visible regime as compared to the infrared regime. Thus, optimization should be easier and requirements lighter when dealing with the infrared observations. c. Temporal band. Usually the turbulence sets the time scales in the servo loop. Lower turbulence tends to be faster than higher one, except when jet streams occur (usually at very good sites like Hawaii). However,



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g.

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when observing a fast changing object like the sun, difficulties might arise. This is because the finest (and most interesting) details evolve the fastest, and locking on to them might be difficult [Acton et al. 1996]). At very low light levels the long integration times might be a limiting factor. Isoplanatic angle. Usually an adaptive system achieves high resolution at the center of the field and low resolution at the edges; instead, one might require the same medium resolution over the whole field. In such cases it will usually be the lower turbulence which has to be corrected, since it is shared for all field objects (Fig. 3) [Acton et al. 1996]. Strehl ratio. This parameter was chosen by Racine and Ellerbroek [1995] and by Wilson and Jenkins [1996]. However, the Strehl ratio is usually defined for the center of the field, where it is the highest. Alternatively, one can use an average, lower Strehl ratio for the whole field. Ellerbroek [1994] discussed using the residual mean-square phase distortion and the associated optical transfer function. Location and availability of natural and laser guide stars. Natural stars (if not the observed star itself or a glint from a man-made object) occur at random places which cannot be chosen (apart from moving asteroids [Ribak and Rigaut 1994]). Laser guide stars are limited by available power and hence by number (Sec. II). Mechanical limitations. The optical train might not have enough room to accommodate all possible locations of the conjugate mirrors. This is

Figure 3 Different parts of the atmosphere are traversed by beams from various stars in the field. A common patch of the lower atmospheric layers is crossed by beams from all stars, and its correction will reduce the total phase error and the Strehl ratio for all these beams.


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especially true if the conjugate planes are too close to each other, and bulky deformable mirrors are used for wavefront correction (Sec. III). h. Optical design. Various factors might lead to limitations in the optical design of the system. Such factors might include variable magnification between conjugate planes, vignetting (Fig. 1), the choice of observing equipment, such as a spectrometer or a camera, etc. (Sect. III). i. Servo loop limitations, electronic and computational. Ellerbroek et al. [1994] examined carefully some different choices for servo loop realizations. Notice that the turbulence profile could change from night to night and the relative importance of the layers might shift during the observation [Racine and Ellerbroek 1995; Avila et al. 1998]. This might require a continuous adjustment of the conjugation planes, a complex mechanical task (Sec. III). An adaptive optics systems operating on the principles discussed above still has to be constructed. However, some simple designs involving multiple correction elements are already being implemented. The simplest of all are those systems that have a tip-tilt mirror separate from the deformable mirror, usually because the latter has a limited travel range. Currently, these two mirrors are conjugated to the telescope aperture. However, Thompson suggested that it might be better to conjugate them to different heights [Richardson 1992]. The best arrangement should be decided according to an optimization scheme as described above. In one option currently under development the secondary mirror of the telescope is also a deformable mirror [Lloyd-Hart et al. 1996]. It cannot be conjugated to any atmospheric layer, but a second corrector (perhaps only for tip and tilt) might be useful if conjugated to another layer. The other option is to use a Gregorian design for the telescope, in which case the secondary mirror can be conjugated to an atmospheric layer [Beckers 1993].

II. MEASUREMENTS OF THE VARIOUS LAYERS: NATURAL AND ARTIFICIAL GUIDE STARS The measurements of the turbulence elevation profile C 2n (h) and the corresponding r0 (h) and vwind (h) are usually statistical. However, to achieve a wide field of view one must measure the instantaneous values that wavefronts take as they propagate down the atmosphere, along and near the optical axis of the telescope (Fig. 1). In essence, this is a tomographic measurement of the refractive index of air inside the relevant region of space. In the astronomical case there are some factors which make tomography simpler than the general case: a. Turbulence layers tend to lie in horizontal planes, so the three-dimen-



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sional volume can be simplified to a limited set of parallel planes. b. Since the statistics of turbulence usually obey Kolmogorov’s law, each turbulence layer needs be measured only up to a resolution of r0 /2 of that layer. The same applies to the temporal resolution—one can integrate up to (r0 /vwind )/2 of each layer (more accurately, up to the Greenwood frequency [see Roddier 1981]. c. Again, because of Kolmogorov’s spectrum, unruly solutions can be easily voted against. Alternatively, constraining the solutions to Kolmogorov’s limitations may make the calculation faster. d. Because of Taylor’s hypothesis of frozen flow, earlier measurements of the turbulence layer provide us with some prior knowledge of the current shape of the wavefront [Dicke, 1975, Hudgin 1977, Lukin and Zuev 1985, Jorgensen and Aitken 1992, Schwartz et al. 1994]. Notice, however, that some evidence shows that the lower layers might not obey Kolmogorov’s 5/3 law [Bester et al. 1992], perhaps because of heat sources and sinks at the ground level, not accounted for in Kolmogorov’s theory. On the other hand, tomography is made more difficult because e. The amount of light available is usually not sufficient. Natural and artificial guide stars provide too little flux for wavefront measurements to be accomplished without excessive noise. f. The distribution of available light sources is random, too sparse, and arbitrary if one uses only natural stars, or not high enough, if laser guide stars are employed. If laser power is limited, then there might be fewer beacons than required. g. The sensors are all at the bottom of the telescope and can sample only that section of space visible from this location. The lasers also emanate from the vicinity of the telescope. The sensors and the lasers can hardly be distributed in other locations (for an exception, see Ragazzoni [1996]). This limits very much the variability required for tomography. h. The measurements must be solved to provide the wavefronts at a rate faster than the atmospheric changes (the Greenwood frequency) for the corrections to be effective. i. The global tilt of the wavefront accumulated over all layers (or for each layer separately) cannot be determined without employing some special means, such as two-color laser stars [Foy et al. 1995] or distant lasers [Ragazzoni 1996]. Ragazzoni and Rigaut [1998] suggest overlapping a natural star and laser star to find the global tilt. Wavefront sensing is very crucial for adaptive optics, and even more so for multiconjugate adaptive optics. A multitude of methods has been proposed, and new ones are still required. Once it is grasped that atmospheric tomography


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is the issue at hand, it also becomes apparent that a diversity of sources and a diversity of detectors is required to solve the problem. The first suggestion of multiconjugate adaptive optics [Dicke 1975] involved only a single guide star. Making use of a single phase contrast wavefront sensor and former measurements, alternate corrections are made at two mirrors conjugate to the aperture and a low layer (Fig. 4). This takes care of the cylinder of turbulence between the telescope and the star (see also Angel [1992]). Even this simple approach shows the attraction of multiconjugate adaptive optics: suppose that only the layer near the telescope (boundary layer) is corrected successfully. Then all other sources in the field will now be affected only by higher turbulence (Fig. 3). Correction of these high layers can be achieved by locking on other stars, further off in the field. This is because all the stars in the field of view share the bottom layer turbulence. Alternatively, correcting the higher turbulence alone may result in larger isoplanatic angles but worse Strehl ratio. The suggestion made by Hudgin in 1980 and by Feinlieb in 1981 to use laser guide stars for military purposes [Fried 1992], and a similar but independent proposal for astronomy [Foy and Labeyrie 1985] made the guide star problem simpler: there was no need any more to rely on the observed star itself as the reference beacon. It was grasped from the start [Foy and Labeyrie 1985, Tallon and Foy 1990, Gardner et al. 1990, Sandler 1992, Shamir and Crowe 1992] that many sources could be hung in the sky, and their geometrical distribution was in the hands of the user, limited only by technical constraints.

Figure 4 Correction of atmospheric layers by a single guide star and a single wavefront sensor operating as a phase contrast detector [Dicke 1975]. The telescope images the object at i1 and a lens there images the lower turbulence on the first deformable mirror (DM1). This mirror reimages the field at i2. A second deformable mirror (DM2, conjugate to the turbulence at 5 km) relays i2 to i3. Here it is split to the wavefront sensor and to the science sensor at i4.



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It seems that a web of guide stars, Rayleigh or sodium ones, is necessary to solve the tomographic problem successfully [Foy and Labeyrie 1985, Gardner et al. 1990, Tallon and Foy 1990, Jankevics and Wirth 1992, Sandler 1992, Shamir and Crowe 1992, Shamir et al. 1993, Tyler 1994, Ellerbroek 1994, Baharav et al. 1994, 1996, Ragazzoni et al. 1999]. However, it was found that a smart approach will have to be taken, and even then the system might not be efficient or effective if the atmosphere cannot be modeled by thin layers alone [Fried 1995]. If one wishes to have a wider isoplanatic patch, and is willing to give up on very high Strehl ratio, then a Rayleigh beacon might be sufficient. This is because such a beacon samples the lower atmosphere only (Figs. 1, 3), and that section of the atmosphere is responsible for most of the errors common to the wider field of view [Ellerbroek 1994, Tyler 1994]. This argument was proved by measurements [Christou et al. 1995]. Foy et al. [1989], Tallon and Foy [1990], and Tallon et al. [1992a, b] show that the number of sodium guide stars must be equal to or greater than the number of thin atmospheric layers. Three laser guide stars might be sufficient; four are definitely enough (Fig. 5). Gardner et al. [1990] and Welsh and Gardner [1991],

Figure 5 Placement of four laser guide stars. At the top of the atmosphere (about 20 km), the cross sections of the four cones overlap slightly to cover a contiguous area of the layer. Further down their redundancy (multiple overlap) increases.


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using arguments of isoplanicity with a continuous (nonlayered) atmosphere, arrive at similar numbers for sodium stars and much higher numbers (in the hundreds) for Rayleigh stars. Approaches using Rayleigh stars for solution of the lower layers and sodium stars for the higher layers [Shamir and Crowe 1992, Shamir et al. 1993, Ellerbroek 1994, Tyler 1994, Baharav and Shamir 1995] were shown to increase the field of view. Baharav and Shamir [1995] found out, however, that the residuals from the Rayleigh measurement are comparable to the sodium measurement alone and might make the Rayleigh beacons redundant. What is the optimal placement of the guide stars? If the number is low (three to four, according to Tallon and Foy [1990] and Gardner et al. [1990]), then the guide stars should be placed around the rim of the telescope field of view (Fig. 5). Square arrays with more beacons were considered by Tyler [1994] and Ellerbroek [1994]. In these studies, Rayleigh stars were placed under the corresponding sodium stars. Still, there is no study that compares the various arrangements. These placements depend strongly on the location of the wavefront sensors, what volume of space they measure, and whether this volume is shared with the other sensors. The location of the wavefront sensors is different for the different schemes. One approach is to make each sensor measure a wavefront from each laser guide star, through a section of the telescope aperture (Fig. 6). This samples the turbulence in nearby sections, which are then combined in a process dubbed butting or stitching [Herrmann et al. 1992]. This is not a very efficient process, because the separate patches have different tilts which cannot be measured [Sasiela 1994]. Tyler [1994] literally widens the basis of the earlier approaches. He shows the benefits of having each sensor measure the full cone from each star to the whole aperture; sharing the lower part of the atmosphere brings about tomography of the atmosphere (Fig. 7). Neyman and Thompson [1995] proposed and started testing the idea of using part of or the whole aperture for sending up the laser beams as well as for sensing them through parts or over the full aperture. How should one extract the phases of the atmospheric layers from the tomographic measurements? The approach proposed initially by Beckers [1988] was to shift and average the wavefront sensor measurements as they propagated down the turbulence through different paths. This is justified since the phase differences between the various measurements are rather small [Shamir et al. 1993]. Tallon and Foy [1990] and Tallon et al. 1992a,b] were more specific. They showed that a set of equations could be set up for each guide star and for each layer, with the appropriate shifts for the different layers. They assume that there are K turbulence layers at elevations h(k) {k ⫽ 1, 2, . . . , K}, and the guide star is at altitude H. The wavefront phase at layer k and coordinate r is ψ(k, r), and its propagation to the ground creates a wavefront φ(k, r) which is actually a convolution with a harmonic function η[h(k), r] ⫽ η(k, r) [Roddier 1981]:



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Figure 6 The telescope aperture is imaged on the Hartmann-Shack sensor. Each lenslet has only one guide star in its field of view. The images should fall near the corner of four pixels in detector array for better position sensing and for reduction of the number of pixels. The measured sections of the wavefronts are stitched together in software.

φ(k, r) ⫽ ψ(k, r) ∗ η(k, r)

(1)

Because of geometric effects, one measures a linear combination G(m, k) (essentially scaling and shifting) of the elements of ϕ(k, r) which depends on the location of the mth guide star:

冦

φ(m, k, r) ⫽ G(m, k)φ (k, r) 1 ⫺

冧

h(k) H

n

(2)

where the last factor arises because wavefront sensing measures the nth derivative of the phase (the phase itself, n ⫽ 1, its gradient, n ⫽ 2, or its curvature, n ⫽ 3). The measurements for all the guide stars are K

M(m, r) ⫽

冱 k⫽1

冦

G(m, k) φ(k, r) 1 ⫺

冧

h(k) H

n

(3)

These equations have to be inverted for φ(k, r) and deconvolved from the atmo-


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Figure 7 The telescope aperture is imaged on the Hartmann-Shack sensor. Four guide stars create four images near each lenslet focus on the detector array. Notice unused pixels when the foci of the lenslets are far apart, as compared to the fields of view of the lenslets.

sphere (Eq. 1) to yield ψ(k, r), the required phases. Johnston and Welsh [1992, 1994] suggested solving these equations by least squares fitting. Ragazzoni et al. [1999] solved directly for the Zernike modes. Sandler [1992] proposed to use an ordered array of beacon laser spots spaced so as to have matching fringes in the shearing interferometer wavefront sensor (see Chap. 5). The results of the experiment were inconclusive, but the idea brought about an opposite idea: using beacon laser fringes over tens of meters to match a set of Hartmann-Shack sensors. Baharav et al. [1994, 1996] describe the scheme as a means to separate lower and higher turbulence layers. The projected aperture of the telescope is broken into a number of Hartmann-Shack lenslets, each facing most of the fringe pattern in the sky (Fig. 8). As in the standard sensors, the whole fringe pattern will shift with slope errors, mostly contributed by the low-lying turbulence. At the same time, the images of the fringes will suffer from distortion from high turbulence. Thus the phase errors inside each subaperture can be traced by high-pass filtering and added with the neighboring subapertures to yield the high turbulence. The main disadvantage of the multiple-beacon schemes is their requirement for high power lasers. Creation of many point beacons merely multiplies the requirements from a single beacon and makes the projection system cumbersome. Either three to four lasers will have to be employed in parallel, with required power of approximately 60–100 W. Alternatively, the beam will have to be scanned across the sky (multiplexed stars) at a high rate. Using very conservative calculations, the fringe method requires 300–500 W of laser power, but it has a rather simple projection system, that of a simple interferometer (Fig. 8). Ribak



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Figure 8 The telescope aperture is imaged on the Hartmann-Shack sensor. Most of the fringe pattern is imaged at each lenslet focus on the detector array. The fringe pattern is shifted as a whole because of turbulence at the low atmosphere, since the conjugate lenslet faces only a small section of it. At the same time the fringes are distorted by the high atmosphere inside the large field of view of the lenslet. These effects (global shift and distortion) also exist for the few-guide-star case (Fig. 7).

[1998] suggested replacing the laser fringes with visible plasma fringes, created by interference of radio beams. Acton et al. [1996] applied phase-diversity methods [Gonsalves 1994] on solar features to separate low from high wavefront aberrations. This method, using simple images at focus and out of focus, is rather slow in processing. Although not fully successful, their results are very encouraging. They show that even low-contrast features, which evolve in time, are sufficient to tell apart the


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different layers. Love et al. [1996] constructed a whole adaptive system using phase diversity, still not at the high rate required by the atmosphere. Another result of Action’s research on the sun is that even low contrast features are enough for wavefront sensing. For nighttime astronomy, there might not be a special need for a spatial arrangement of laser stars or laser fringes. Papen et al. [1996] stress the inhomogeneity of the sodium layer and of light scattered off the sodium. Simple sodium lamps (such as those used for street lighting) might be sufficient to illuminate a very large section of the sky (see a similar approach by Wirth and Jankevics [1992]). Laser light is much more collimated than incoherent light, but it might be possible to use concentrators to get sufficient light in the relevant area. The natural clumpiness of the sodium will result in inhomogeneous backscattering, which could be sufficient for wavefront sensing. If the returned intensity is still too smooth, radio beams could be employed to modify it. For example, a radio interferometer could modulate spatially the illuminated sodium layer and create fringes in the returned light [Ribak 1998]. Other means were proposed to measure the different layers. Curvature sensing (Chap. 4) relies on intensity variations down the beam from phase variations. But that effect also means that intensity variations at the aperture of the telescope, better known as scintillations, are related to phase variations at the high atmosphere, kilometers above the telescope. This prompted Ribak [1994, 1996] and, independently, Glindemann and Berkefeld [1996] to propose to invert these intensities to find the original phases. Intensity variations are measured anyway by wavefront sensors and discarded as noise. Instead, they can be used for this purpose, especially for brighter sources and stronger turbulence. The placement of the wavefront sensor, at a plane conjugate to a specific layer, is significant. By moving the wavefront sensor up and down the optical train, it is possible to cancel scintillation effects in the layer conjugate to the measured one [Bregman et al. 1991, Fuchs et al. 1994]. But another advantage which emanates from these works is that two curvature sensors, placed at the conjugates of two significant layers, will each measure the other layer simultaneously. Notice how similar this now becomes to Dicke’s method [1975].

III. OPTICAL DESIGN How big a field of view can we expect with current technology? It turns out that, like in so many other cases, we might be limited more by the detectors than by other elements. Consider the following one-dimensional example: We have electronic cameras (charge-coupled devices, for example) comprising N ⫽ 8192 pixels (in one or a few tiled cameras). Suppose also that we have a D ⫽ 5 m telescope. At a wavelength of λ ⫽ 0.6 µm and at maximum resolution, each pixel will see one-half the maximum angular resolution or 0.61 ⫻ 6 ⋅ 10⫺7 /5 ⫽ 0.0732 microrad each. The whole field of view will be 8192 ⫻ 0.0732 ⫽ 586



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microradians or 2 arc-min. Larger fields can thus be achieved only by lowering the resolution per pixel, using larger cameras or smaller telescopes. Somewhat larger fields of view (with fewer pixels) might be required by the wavefront sensors which need to measure slightly beyond the edge of the projected telescope aperture. The angular size of the field means that laser guide stars at the sodium layer extend over approximately 60 m, and the measured—and corrected—patch at 15 km elevation is about 14 m. While planning an adaptive optics system, certain issues arise that might become more complex with a multiconjugate system. In some cases the multiconjugate system is a simple extension of the single conjugate system with similar requirements (for example, that the whole system could be pulled out to allow for wide field imaging at low resolution without the benefit of adaptive optics). In other cases special issues will have to be answered. Richardson [1992, 1994] has raised some important subjects, such as the quality of the images of the conjugate planes when relayed down the optical train. The tendency to use reflective (rather than refractive) optics, brought about by the need for good infrared imaging, is very demanding. For this regime, off-axis paraboloids and hyperboloids have to be considered, with their severe off-axis aberrations. Other issues stem from the requirement for conjugation to various—and variable—atmospheric turbulence planes. Variations of the elevation of the turbulence layers over minutes [Racine and Ellerbroek 1995, Avila et al. 1998] might require fast changes. Also, the varying size of r0 requires zooming capabilities on top of the reimaging capabilities (assuming that the deformable mirror has a fixed geometry with a limited number of elements). At the same time both the scientific detector and the wavefront sensor or sensors should stay at the same location or be moved in unison (to within a fraction of a pixel!). The movement of the optical elements has to be designed so as not to have them interfere with each other’s path [Richardson 1994]. Wilson and Jenkins [1996] have investigated theoretically the effect of conjugation to different layer heights. Most adaptive optics systems today tend to put the deformable mirror at a position conjugate to the aperture of the telescope. The study shows that this is useful only in spectroscopic measurements, when one wishes to maximize the amount of energy in the slit. For imaging applications, a much better solution results when the deformable mirror is conjugated to the worst layer, usually higher up. The main disadvantage of this method is that with a single guide star there is vignetting (Wells [1995], Wells et al. [1996]). The degree of overlap is further decreased when using a laser guide star. In this case not only is the area of the higher layer smaller than that of the aperture, but it is also scaled differently (Fig. 1). Finally, there is also the problem of mismatch (both in registration and in scale) between the wavefront sensor and deformable pixels when conjugated to different layers. It seems that a large number of these problems can be solved by oversampling the turbulence in four dimensions: outside the aperture, at different elevations, and at former time steps. Spatiotemporal


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prediction using the fractal nature of the wavefronts [Schwartz et al. 1994] should replace all but the highest missing frequencies and those upwind from the guide star(s). Another issue is the order of the correcting elements. When turbulence is weak, the geometric approximation (that beams only bend, but do not diffract or cross) is valid, and the phase errors from the different layers add linearly [Roddier 1981]. Thus they can be subtracted linearly without regard to which conjugate layer is corrected first. If turbulence is not so weak, it might be better to conjugate and correct first the lower atmospheric layers, and then reimage and correct the higher ones [Dicke 1975, McCall and Passner 1978]. In this manner the errors are undone in the opposite direction to the order of their occurrence [Johnston and Welsh 1992, 1994]. How should we deal with objects that are not at zenith? In such a case, the atmospheric layers do not lie normal to the direction of observation, and a corresponding tilt to the deformable mirror needs to be designed (Fig. 9). This

Figure 9 When the observed star is far from zenith, the atmospheric layers are tilted with respect to the optical axis of the telescope. As a result, the mirrors are no longer conjugate to their respective layers at all points.



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problem might not have a proper solution, and might limit wide field adaptive optics to the vicinity of the zenith.

IV. FUTURE DEVELOPMENT Multiconjugate adaptive optics is considered as the next step after standard adaptive optics has proven its utility, reliability, and usefulness for the astronomer. Only when prices drop and more experience and confidence is gained will there be room for expansion to more laser guide stars, more sensors and more mirrors. Until full multiconjugate adaptive optics is available, other methods will have to be employed. Between these, usage of speckle methods for improvement of resolution on the edges of images acquired by adaptive optics, and by deconvolution from a second wavefront sensor [Roggemann et al. 1995] or from phase diversity [Love et al. 1996]. However, the huge advantages of adaptive optics, the long integration time and wide spectral band will have to be compromised. The number of degrees of freedom for each layer is approximately twice the number of correlation cells at that layer (at the Nyquist frequency), or (2Di /ri )2, where Di is the projection of the aperture on the ith layer and ri the corresponding r0. For I layers we get

冱冢 I

F⫽

i⫽1

2Di ri

冣

2

(4)

Thus it seems that a better scheme should also include reduction in the number of pixels of the wavefront sensors to match this number of degrees of freedom. These pixels must be sampled at twice the corresponding frequency for that layer, 2vi /rt, where vi is the local wind speed. The total bandwidth will thus be I

B⫽

冱 i⫽1

8D 2i vi r 3i

(5)

A very small number of other parameters need to be measured (the height, wind speed and correlation length for each layer), and those at a lower rate. Another issue that needs to be looked into is the quality and sufficiency of laser guide stars. It is not easy to duplicate these lasers and other schemes require either too many laser spots or a fringe pattern at very high power, in contrast to the small number of degrees of freedom described above. Simpler schemes that would allow less powerful lasers should be sought. A severe problem with realization of the multiconjugate systems is their optical design. Wavefront correctors are mostly made of mirrors, which means folding the optical path as well as losing light. Refractive elements like liquid


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crystals are not available yet at the required quality. Thus, more effort should be invested to provide a better solution to the problem.

ACKNOWLEDGMENTS I wish to thank S. G. Lipson for a critical reading of this chapter.

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