Adaptive Optics For Vision Science

Adaptive Optics for Vision Science Principles, Practices, Design, and Applications Edited by JASON PORTER, HOPE M. QUEENER, JULIANNA E. LIN, KAREN THORN, AND ABDUL AWWAL

A JOHN WILEY & SONS, INC., PUBLICATION

Front cover art: In an adaptive optics system, a lenslet array (left circle) is used to measure an aberrated wavefront (top circle) that is then corrected by a deformable mirror (right circle) to produce a flattened wavefront (bottom circle). Lenslet array and deformable mirror images are courtesy of Adaptive Optics Associates, Inc. and Boston Micromachines Corporation, respectively. Copyright © 2006 by John Wiley & Sons, Inc., Hoboken, NJ. All rights reserved. Published simultaneously in Canada No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400, fax (978) 750-4470, or on the web at www.copyright.com. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, (201) 748-6011, fax (201) 748-6008, or online at http://www.wiley.com/go/permission. Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their best efforts in preparing this book, they make no representations or warranties with respect to the accuracy or completeness of the contents of this book and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose. No warranty may be created or extended by sales representatives or written sales materials. The advice and strategies contained herein may not be suitable for your situation. You should consult with a professional where appropriate. Neither the publisher nor author shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages. For general information on our other products and services or for technical support, please contact our Customer Care Department within the United States at (800) 762-2974, outside the United States at (317) 572-3993 or fax (317) 572-4002. Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic formats. For more information about Wiley products, visit our web site at www.wiley.com. Library of Congress Cataloging-in-Publication Data: Adaptive optics for vision science : principles, practices, design and applications / edited by Jason Porter . . . [et al.]. p. cm. “A Wiley-Interscience publication.” Includes bibliographical references and index. ISBN-10: *978-0-471-67941-7 ISBN-10: 0-471-67941-0 1. Optics, Adaptive. I. Porter, Jason. TA1520.A34 2006 621.36′9–dc22 2005056953 Printed in the United States of America 10 9 8 7 6 5 4 3 2 1

Contents FOREWORD

xvii

ACKNOWLEDGMENTS

xxi

CONTRIBUTORS

PART ONE 1

INTRODUCTION

Development of Adaptive Optics in Vision Science and Ophthalmology David R. Williams and Jason Porter 1.1

1.2

Brief History of Aberration Correction in the Human Eye 1.1.1 Vision Correction 1.1.2 Retinal Imaging Applications of Ocular Adaptive Optics 1.2.1 Vision Correction 1.2.2 Retinal Imaging

PART TWO WAVEFRONT MEASUREMENT AND CORRECTION 2

xxiii

1

3 3 3 5 9 9 11

31

Aberration Structure of the Human Eye Pablo Artal, Juan M. Bueno, Antonio Guirao, and Pedro M. Prieto

33

2.1 2.2

33 34

Introduction Location of Monochromatic Aberrations Within the Eye

v

vi

CONTENTS

2.3

2.4

2.5

2.6 2.7

3

Temporal Properties of Aberrations: Accommodation and Aging 2.3.1 Effect of Accommodation on Aberrations and Their Correction 2.3.2 Aging and Aberrations Chromatic Aberrations 2.4.1 Longitudinal Chromatic Aberration 2.4.2 Transverse Chromatic Aberration 2.4.3 Interaction Between Monochromatic and Chromatic Aberrations Off-Axis Aberrations 2.5.1 Peripheral Refraction 2.5.2 Monochromatic and Chromatic Off-Axis Aberrations 2.5.3 Monochromatic Image Quality and Correction of Off-Axis Aberrations Statistics of Aberrations in Normal Populations Effects of Polarization and Scatter 2.7.1 Impact of Polarization on the Ocular Aberrations 2.7.2 Intraocular Scatter

40 40 42 43 44 45 45 46 47 48 51 52 53 53 55

Wavefront Sensing and Diagnostic Uses Geunyoung Yoon

63

3.1

63 65 65 66 68

3.2

3.3

3.4

Wavefront Sensors for the Eye 3.1.1 Spatially Resolved Refractometer 3.1.2 Laser Ray Tracing 3.1.3 Shack–Hartmann Wavefront Sensor Optimizing a Shack–Hartmann Wavefront Sensor 3.2.1 Number of Lenslets Versus Number of Zernike Coefficients 3.2.2 Trade-off Between Dynamic Range and Measurement Sensitivity 3.2.3 Focal Length of the Lenslet Array 3.2.4 Increasing the Dynamic Range of a Wavefront Sensor Without Losing Measurement Sensitivity Calibration of a Wavefront Sensor 3.3.1 Reconstruction Algorithm 3.3.2 System Aberrations Summary

68 71 73 74 75 76 77 79

CONTENTS

4

Wavefront Correctors for Vision Science Nathan Doble and Donald T. Miller

83

4.1 4.2 4.3 4.4

83 84 86 88

4.5

4.6 5

Introduction Principal Components of an AO System Wavefront Correctors Wavefront Correctors Used in Vision Science 4.4.1 Macroscopic Discrete Actuator Deformable Mirrors 4.4.2 Liquid Crystal Spatial Light Modulators 4.4.3 Bimorph Mirrors 4.4.4 Microelectromechanical Systems Performance Predictions for Various Types of Wavefront Correctors 4.5.1 Description of Two Large Populations 4.5.2 Required Corrector Stroke 4.5.3 Discrete Actuator Deformable Mirrors 4.5.4 Piston-Only Segmented Mirrors 4.5.5 Piston/Tip/Tilt Segmented Mirrors 4.5.6 Membrane and Bimorph Mirrors Summary and Conclusion

89 90 91 92 95 98 99 101 106 107 109 111

Control Algorithms Li Chen

119

5.1 5.2 5.3 5.4

119 119 122 124 124 127 127 128 128 129 130

5.5

6

vii

Introduction Configuration of Lenslets and Actuators Influence Function Measurement Spatial Control Command of the Wavefront Corrector 5.4.1 Control Matrix for the Direct Slope Algorithm 5.4.2 Modal Wavefront Correction 5.4.3 Wave Aberration Generator Temporal Control Command of the Wavefront Corrector 5.5.1 Open-Loop Control 5.5.2 Closed-Loop Control 5.5.3 Transfer Function of an Adaptive Optics System

Adaptive Optics Software for Vision Research Ben Singer

139

6.1 6.2

139 140 140 140 141

Introduction Image Acquisition 6.2.1 Frame Rate 6.2.2 Synchronization 6.2.3 Pupil Imaging

viii

CONTENTS

6.3

6.4

6.5

6.6

6.7

7

Measuring Wavefront Slope 6.3.1 Setting Regions of Interest 6.3.2 Issues Related to Image Coordinates 6.3.3 Adjusting for Image Quality 6.3.4 Measurement Pupils 6.3.5 Preparing the Image 6.3.6 Centroiding Aberration Recovery 6.4.1 Principles 6.4.2 Implementation 6.4.3 Recording Aberration 6.4.4 Displaying a Running History of RMS 6.4.5 Displaying an Image of the Reconstructed Wavefront Correcting Aberrations 6.5.1 Recording Influence Functions 6.5.2 Applying Actuator Voltages Application-Dependent Considerations 6.6.1 One-Shot Retinal Imaging 6.6.2 Synchronizing to Display Stimuli 6.6.3 Selective Correction Conclusion 6.7.1 Making Programmers Happy 6.7.2 Making Operators Happy 6.7.3 Making Researchers Happy 6.7.4 Making Subjects Happy 6.7.5 Flexibility in the Middle

142 142 143 143 143 143 144 144 144 145 147 147 148 149 149 150 150 150 150 151 151 151 151 152 152 153

Adaptive Optics System Assembly and Integration Brian J. Bauman and Stephen K. Eisenbies

155

7.1 7.2 7.3

155 156 157 158 159 163 170 174 174

7.4

Introduction First-Order Optics of the AO System Optical Alignment 7.3.1 Understanding Penalties for Misalignments 7.3.2 Optomechanics 7.3.3 Common Alignment Practices 7.3.4 Sample Procedure for Offline Alignment AO System Integration 7.4.1 Overview 7.4.2 Measure the Wavefront Error of Optical Components 7.4.3 Qualify the DM

175 175

CONTENTS

7.4.4 7.4.5 7.4.6 7.4.7 7.4.8 7.4.9 7.4.10 7.4.11 8

184 184 185 189

8.1 8.2 8.3 8.4 8.5

189 189 191 192 194

Introduction Strehl Ratio Calibration Error Fitting Error Measurement and Bandwidth Error 8.5.1 Modeling the Dynamic Behavior of the AO System 8.5.2 Computing Temporal Power Spectra from the Diagnostics 8.5.3 Measurement Noise Errors 8.5.4 Bandwidth Error 8.5.5 Discussion Addition of Wavefront Error Terms

PART THREE

RETINAL IMAGING APPLICATIONS

194 196 198 199 200 200 203

Fundamental Properties of the Retina Ann E. Elsner

205

9.1 9.2 9.3 9.4 9.5 9.6 9.7

206 209 210 218 220 225

9.8 10

177 180 181 182 183

System Performance Characterization Marcos A. van Dam

8.6

9

Qualify the Wavefront Sensor Check Wavefront Reconstruction Assemble the AO System Boresight FOVs Perform DM-to-WS Registration Measure the Slope Influence Matrix and Generate Control Matrices Close the Loop and Check the System Gain Calibrate the Reference Centroids

ix

Shape of the Retina Two Blood Supplies Layers of the Fundus Spectra Light Scattering Polarization Contrast from Directly Backscattered or Multiply Scattered Light Summary

228 230

Strategies for High-Resolution Retinal Imaging Austin Roorda, Donald T. Miller, and Julian Christou

235

10.1 Introduction

235

x

CONTENTS

10.2 Conventional Imaging 10.2.1 Resolution Limits of Conventional Imaging Systems 10.2.2 Basic System Design 10.2.3 Optical Components 10.2.4 Wavefront Sensing 10.2.5 Imaging Light Source 10.2.6 Field Size 10.2.7 Science Camera 10.2.8 System Operation

236 237 237 239 240 242 244 246 246

10.3 Scanning Laser Imaging 10.3.1 Resolution Limits of Confocal Scanning Laser Imaging Systems 10.3.2 Basic Layout of an AOSLO 10.3.3 Light Path 10.3.4 Light Delivery 10.3.5 Wavefront Sensing and Compensation 10.3.6 Raster Scanning 10.3.7 Light Detection 10.3.8 Frame Grabbing 10.3.9 SLO System Operation

249 249 249 251 252 253 254 255 255

10.4 OCT Ophthalmoscope 10.4.1 OCT Principle of Operation 10.4.2 Resolution Limits of OCT 10.4.3 Light Detection 10.4.4 Basic Layout of AO-OCT Ophthalmoscopes 10.4.5 Optical Components 10.4.6 Wavefront Sensing 10.4.7 Imaging Light Source 10.4.8 Field Size 10.4.9 Impact of Speckle and Chromatic Aberrations

256 257 259 262 264 266 266 267 267 268

10.5 Common Issues for all AO Imaging Systems 10.5.1 Light Budget 10.5.2 Human Factors 10.5.3 Refraction 10.5.4 Imaging Time

271 271 272 272 276

10.6 Image 10.6.1 10.6.2 10.6.3 10.6.4 10.6.5 10.6.6

276 276 276 278 279 283 283

Postprocessing Introduction Convolution Linear Deconvolution Nonlinear Deconvolution Uses of Deconvolution Summary

247

CONTENTS

PART FOUR 11

VISION CORRECTION APPLICATIONS

13

289

Customized Vision Correction Devices Ian Cox

291

11.1

291

Contact Lenses 11.1.1 Rigid or Soft Contact Lenses for Customized Correction? 11.1.2 Design Considerations—More Than Just Optics 11.1.3 Measurement—The Eye, the Lens, or the System? 11.1.4 Customized Contact Lenses in a Disposable World 11.1.5 Manufacturing Issues—Can the Correct Surfaces Be Made? 11.1.6 Who Will Benefit? 11.1.7 Summary 11.2 Intraocular Lenses 11.2.1 Which Aberrations—The Cornea, the Lens, or the Eye? 11.2.2 Correcting Higher Order Aberrations— Individual Versus Population Average 11.2.3 Summary 12

xi

293 295 297 298 300 301 304 304 305 306 308

Customized Corneal Ablation Scott M. MacRae

311

12.1 Introduction 12.2 Basics of Laser Refractive Surgery 12.3 Forms of Customization 12.3.1 Functional Customization 12.3.2 Anatomical Customization 12.3.3 Optical Customization 12.4 The Excimer Laser Treatment 12.5 Biomechanics and Variable Ablation Rate 12.6 Effect of the LASIK Flap 12.7 Wavefront Technology and Higher Order Aberration Correction 12.8 Clinical Results of Excimer Laser Ablation 12.9 Summary

311 312 317 317 319 320 321 322 324

From Wavefronts To Refractions Larry N. Thibos

331

13.1 Basic Terminology 13.1.1 Refractive Error and Refractive Correction 13.1.2 Lens Prescriptions

331 331 332

325 325 326

xii

CONTENTS

13.2 Goal of Refraction 13.2.1 Defi nition of the Far Point 13.2.2 Refraction by Successive Elimination 13.2.3 Using Depth of Focus to Expand the Range of Clear Vision 13.3 Methods for Estimating the Monochromatic Refraction from an Aberration Map 13.3.1 Refraction Based on Equivalent Quadratic 13.3.2 Virtual Refraction Based on Maximizing Optical Quality 13.3.3 Numerical Example 13.4 Ocular Chromatic Aberration and the Polychromatic Refraction 13.4.1 Polychromatic Wavefront Metrics 13.4.2 Polychromatic Point Image Metrics 13.4.3 Polychromatic Grating Image Metrics 13.5 Experimental Evaluation of Proposed Refraction Methods 13.5.1 Monochromatic Predictions 13.5.2 Polychromatic Predictions 13.5.3 Conclusions

14 Visual Psychophysics With Adaptive Optics Joseph L. Hardy, Peter B. Delahunt, and John S. Werner 14.1

Psychophysical Functions 14.1.1 Contrast Sensitivity Functions 14.1.2 Spectral Efficiency Functions 14.2 Psychophysical Methods 14.2.1 Threshold 14.2.2 Signal Detection Theory 14.2.3 Detection, Discrimination, and Identification Thresholds 14.2.4 Procedures for Estimating a Threshold 14.2.5 Psychometric Functions 14.2.6 Selecting Stimulus Values 14.3 Generating the Visual Stimulus 14.3.1 General Issues Concerning Computer-Controlled Displays 14.3.2 Types of Computer-Controlled Displays 14.3.3 Accurate Stimulus Generation 14.3.4 Display Characterization

334 334 335 336 337 339 339 353 354 356 357 357 358 358 359 360

363 364 364 368 370 370 371 374 375 377 378 380 381 384 386 388

CONTENTS

14.4

14.3.5 Maxwellian-View Optical Systems 14.3.6 Other Display Options Conclusions

PART FIVE 15

16

DESIGN EXAMPLES

xiii

390 390 391

395

Rochester Adaptive Optics Ophthalmoscope Heidi Hofer, Jason Porter, Geunyoung Yoon, Li Chen, Ben Singer, and David R. Williams

397

15.1 Introduction 15.2 Optical Layout 15.2.1 Wavefront Measurement and Correction 15.2.2 Retinal Imaging: Light Delivery and Image Acquisition 15.2.3 Visual Psychophysics Stimulus Display 15.3 Control Algorithm 15.4 Wavefront Correction Performance 15.4.1 Residual RMS Errors, Wavefronts, and Point Spread Functions 15.4.2 Temporal Performance: RMS Wavefront Error 15.5 Improvement in Retinal Image Quality 15.6 Improvement in Visual Performance 15.7 Current System Limitations 15.8 Conclusion

397 398 398 403 404 405 406 406 407 409 410 412 414

Design of an Adaptive Optics Scanning Laser Ophthalmoscope Krishnakumar Venkateswaran, Fernando Romero-Borja, and Austin Roorda

417

16.1 16.2 16.3 16.4

417 419 419 420 420

Introduction Light Delivery Raster Scanning Adaptive Optics in the SLO 16.4.1 Wavefront Sensing 16.4.2 Wavefront Compensation Using the Deformable Mirror 16.4.3 Mirror Control Algorithm 16.4.4 Nonnulling Operation for Axial Sectioning in a Closed-Loop AO System 16.5 Optical Layout for the AOSLO 16.6 Image Acquisition

421 421 423 425 426

xiv

CONTENTS

16.7 Software Interface for the AOSLO 16.8 Calibration and Testing 16.8.1 Defocus Calibration 16.8.2 Linearity of the Detection Path 16.8.3 Field Size Calibration 16.9 AO Performance Results 16.9.1 AO Compensation 16.9.2 Axial Resolution of the Theoretically Modeled AOSLO and Experimental Results 16.10 Imaging Results 16.10.1 Hard Exudates and Microaneurysms in a Diabetic’s Retina 16.10.2 Blood Flow Measurements 16.10.3 Solar Retinopathy 16.11 Discussions on Improving Performance of the AOSLO 16.11.1 Size of the Confocal Pinhole 16.11.2 Pupil and Retinal Stabilization 16.11.3 Improvements to Contrast 17 Indiana University AO-OCT System Yan Zhang, Jungtae Rha, Ravi S. Jonnal, and Donald T. Miller 17.1 17.2 17.3

17.4

17.5 17.6

17.7

Introduction Description of the System Experimental Procedures 17.3.1 Preparation of Subjects 17.3.2 Collection of Retinal Images AO Performance 17.4.1 Image Sharpening 17.4.2 Temporal Power Spectra 17.4.3 Power Rejection Curve of the Closed-Loop AO System 17.4.4 Time Stamping of SHWS Measurements 17.4.5 Extensive Logging Capabilities 17.4.6 Improving Corrector Stability Example Results with AO Conventional FloodIlluminated Imaging Example Results With AO Parallel SD-OCT Imaging 17.6.1 Parallel SD-OCT Sensitivity and Axial Resolution 17.6.2 AO Parallel SD-OCT Imaging Conclusion

429 431 431 432 432 432 432 434 438 438 439 440 441 441 443 443 447 447 448 453 453 454 455 457 458 459 460 461 461 461 463 463 466 474

CONTENTS

18

Design and Testing of A Liquid Crystal Adaptive Optics Phoropter Abdul Awwal and Scot Olivier 18.1 Introduction 18.2 Wavefront Sensor Selection 18.2.1 Wavefront Sensor: Shack–Hartmann Sensor 18.2.2 Shack–Hartmann Noise 18.3 Beacon Selection: Size and Power, SLD versus Laser Diode 18.4 Wavefront Corrector Selection 18.5 Wavefront Reconstruction and Control 18.5.1 Closed-Loop Algorithm 18.5.2 Centroid Calculation 18.6 Software Interface 18.7 AO Assembly, Integration, and Troubleshooting 18.8 System Performance, Testing Procedures, and Calibration 18.8.1 Nonlinear Characterization of the Spatial Light Modulator (SLM) Response 18.8.2 Phase Wrapping 18.8.3 Biased Operation of SLM 18.8.4 Wavefront Sensor Verification 18.8.5 Registration 18.8.6 Closed-Loop Operation 18.9 Results from Human Subjects 18.10 Discussion 18.11 Summary

xv

477 477 478 478 483 484 485 486 487 488 489 491 492 493 493 495 495 496 499 502 506 508

APPENDIX A: OPTICAL SOCIETY OF AMERICA’S STANDARDS FOR REPORTING OPTICAL ABERRATIONS

511

GLOSSARY

529

SYMBOL TABLE

553

INDEX

565

Foreword

The rationale for this handbook is to make adaptive optics technology for vision science and ophthalmology as broadly accessible as possible. While the scientific literature chronicles the dramatic recent achievements enabled by adaptive optics in vision correction and retinal imaging, it does less well at conveying the practical information required to apply wavefront technology to the eye. This handbook is intended to equip engineers, scientists, and clinicians with the basic concepts, engineering tools, and tricks of the trade required to master adaptive optics-related applications in vision science and ophthalmology. During the past decade, there has been a remarkable expansion of the application of wavefront-related technologies to the human eye, as illustrated by the rapidly growing number of publications in this area (shown in Fig. F.1). The catalysts for this expansion have been the development of new wavefront sensors that can rapidly provide accurate and complete descriptions of the eye’s aberrations, and the demonstration that adaptive optics can provide better correction of the eye’s aberrations than has previously been possible. These new tools have generated an intensive effort to revise methods to correct vision, with the wavefront sensor providing a much needed yardstick for measuring the optical performance of spectacles, contact lenses, intraocular lenses, and refractive surgical procedures. Wavefront sensors offer the promise of a new generation of vision correction methods that can correct higher order aberrations beyond defocus and astigmatism in cases where these aberrations significantly blur the retinal image. The ability of adaptive optics to correct the monochromatic aberrations of the eye has also created exciting new opportunities to image the normal and diseased retina at unprecedented spatial resolution. Adaptive optics has strong roots in astronomy, where it is used to overcome the blurring effects of atmospheric turbulence, the fundamental limitation on the resolution of xvii

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FOREWORD

FIGURE F.1 Number of publications listed in PubMed (National Library of Medicine) that describe work where wavefront sensors were used to measure the full wave aberration of the human eye. Types of wavefront sensors included in this graph: Shack–Hartmann, spatially resolved refractometer, crossed-cylinder aberroscope, laser ray tracing, scanning slit refractometer, video keratography, corneal topography, phase retrieval, curvature sensing, and grating-based techniques.

ground-based telescopes. More recently, adaptive optics has found application in other areas, most notably vision science, where it is used to correct the eye’s wave aberration. Despite the obvious difference in the scientific objectives of the astronomy and vision science communities, we share a technology that is remarkably similar across the two applications. Recognizing this, together with Jerry Nelson and other colleagues, we created a center focused on developing adaptive optics technology for both astronomy and vision science. The Center for Adaptive Optics, with headquarters at the University of California, Santa Cruz, was founded in 1999 as a National Science Foundation Science and Technology Center. Initially under the leadership of Jerry Nelson and more recently of Claire Max, the Center for Adaptive Optics is a consortium involving more than 30 affiliated universities, government laboratories, and corporations. The Center has fostered extensive new collaborations between vision scientists and astronomers (who very soon discovered they were interested in each others’ science as well as their technology!). This handbook is a direct result of the Center’s collaborative energy, with chapters contributed by astronomers and vision scientists alike.

FOREWORD

xix

We wish to thank all of the contributors for generously sharing their expertise, and even their secrets, within the pages of this book. Especially, we congratulate Jason Porter, lead editor, and Hope Queener, Julianna Lin, Karen Thorn, and Abdul Awwal, coeditors, for their tireless dedication to this significant project. DAVID R. WILLIAMS University of Rochester, Rochester, New York Center for Adaptive Optics

CLAIRE MAX University of California, Santa Cruz Center for Adaptive Optics

Acknowledgments

I have been extremely privileged to have worked on this book and would like to thank everyone who contributed to its development, technical and scientific content, character, and completion. I am indebted to all of the authors and reviewers from the vision science, astronomical, and engineering communities who took the time and energy to write outstanding chapters in the midst of their busy research and personal lives. Thank you to George Telecki and Rachel Witmer at John Wiley & Sons, Inc. for sticking with us over the past two years, for believing in the importance of publishing a book on this topic, and for their patience and willingness to answer any and all questions that came their way. In addition, I could not have completed the project without the energy and efforts of my fellow co-editors, Hope Queener, Julianna Lin, Karen Thorn, and Abdul Awwal. I would particularly like to thank Hope and Julianna for their tremendous dedication to compiling a book with such a high level of scientific and technical competence and integrity (and for all of the many hours and late nights required to do so!). I am also grateful for the support, ideas, and encouragement I received from David Williams and the members of his lab (including Joe, Jess, Dan, Li, Sapna, and Alexis), and the Center for Visual Science and StrongVision administrative staff (including Michele, Debbie, Teresa, Sara, and Amy). A very special thanks goes to my family (Jen, Kevin, Debbie, Sarah, and Kyle) and friends (Mike, Lana, Frank, and others who are too numerous to mention) for their support, love, belief, encouragement, and prayers, and for helping to keep me refreshed and alive. Scientifically, I will always be grateful to Claire Max, who fi rst opened my eyes to the exciting field of adaptive optics during an internship at Lawrence Livermore National Lab that subsequently led me to fi nd a path to David Williams’ lab. In addition, I will always be indebted to my mentor, David Williams, for his guidance, instruction, support, encouragement, and confidence in me on so many levels in and outside of the office—it has been a xxi

xxii

ACKNOWLEDGMENTS

pleasure to work for one of the pioneers in the fields of vision science and adaptive optics. Finally, I would like to thank the National Science Foundation and the Center for Adaptive Optics for not only supporting this project but for also supporting and continuing the long tradition of vision scientists and astronomers working together to better science, health, and technology. JASON PORTER This editorial work was made possible by the support of the National Science Foundation’s Center for Adaptive Optics and the associated scientific community. The University of Houston College of Optometry provided time and computing resources. The University of Rochester’s Center for Visual Science provided time, space, and computing resources. I wish to particularly acknowledge the tremendous efforts of co-editors Jason Porter and Julianna Lin. As the project neared completion, the helpful responses from Larry Thibos, Marcos Van Dam, Jack Werner, and Joe Hardy were greatly appreciated. HOPE M. QUEENER I would like to extend a very heart-felt thank you to all of the authors, reviewers, collaborators, and supporters who dedicated so much of their time to making this book a reality. In particular, I would like to thank Jason Porter and Hope Queener for their staunch determination and perseverance, particularly toward the end of this project. I would also like to thank my husband, Gregory Brady, and my family (Y. S. Lin, G. Y. C. Lin, I. Lin, K. Su, S. Su, and little Stephen) for their love and support, even in the midst of the editing cycle. Financial and logistical support for this project was provided by the Center for Adaptive Optics. Additional support was provided by David Williams, the University of Rochester, and the Center for Visual Science. JULIANNA E. LIN

Contributors

AUTHORS Pablo Artal, Laboratorio de Optica (Departamento de Fisica), Universidad de Murcia, Murcia, Spain Abdul Awwal, Lawrence Livermore National Laboratory, Livermore, California Brian J. Bauman, Lawrence Livermore National Laboratory, Livermore, California Juan M. Bueno, Laboratorio de Optica, Universidad de Murcia, Murcia, Spain Li Chen, Center for Visual Science, University of Rochester, Rochester, New York Julian Christou, Center for Adaptive Optics, University of California, Santa Cruz, Santa Cruz, California Ian Cox, Bausch & Lomb, Rochester, New York Peter B. Delahunt, Posit Science Corporation, San Francisco, California Nathan Doble, Iris AO, Inc., Berkeley, California Stephen K. Eisenbies, Sandia National Laboratories, Livermore, California Ann E. Elsner, School of Optometry, Indiana University, Bloomington, Indiana Antonio Guirao, Laboratorio de Optica, Universidad de Murcia, Murcia, Spain Joseph L. Hardy, Posit Science Corporation, San Francisco, California Heidi Hofer, College of Optometry, University of Houston, Houston, Texas Ravi S. Jonnal, School of Optometry, Indiana University, Bloomington, Indiana xxiii

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CONTRIBUTORS

Scott M. MacRae, Department of Ophthalmology, University of Rochester, Rochester, New York Donald T. Miller, School of Optometry, Indiana University, Bloomington, Indiana Scot Olivier, Lawrence Livermore National Laboratory, Livermore, California Jason Porter, Center for Visual Science, University of Rochester, Rochester, New York Pedro M. Prieto, Laboratorio de Optica, Universidad de Murcia, Murcia, Spain Jungtae Rha, School of Optometry, Indiana University, Bloomington, Indiana Fernando Romero-Borja, Houston Community College Central, Houston, Texas Austin Roorda, School of Optometry, University of California, Berkeley, Berkeley, California Ben Singer, Center for the Study of Brain, Mind and Behavior, Princeton University, Princeton, New Jersey Larry N. Thibos, School of Optometry, Indiana University, Bloomington, Indiana Marcos A. van Dam, W. M. Keck Observatory, Kamuela, Hawaii Krishna Venkateswaran, Alcon Research Ltd, Orlando, Florida John S. Werner, Department of Ophthalmology, Section of Neurobiology, Physiology and Behavior, University of California, Davis Medical Center, Sacramento, California David R. Williams, Center for Visual Science, University of Rochester, Rochester, New York Geunyoung Yoon, Department of Ophthalmology, University of Rochester, Rochester, New York Yan Zhang, School of Optometry, Indiana University, Bloomington, Indiana EDITOR-IN-CHIEF Jason Porter, Center for Visual Science, University of Rochester, Rochester, New York CO-EDITORS Abdul Awwal, Lawrence Livermore National Laboratory, Livermore, California

CONTRIBUTORS

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Julianna E. Lin, Center for Visual Science, University of Rochester, Rochester, New York Hope M. Queener, College of Optometry, University of Houston, Houston, Texas Karen Thorn, 20 Todman Street, Brookyn, Wellington, New Zealand

FIGURE 1.7 Images of the cone mosaics of 10 subjects with normal color vision, obtained with the combined methods of adaptive optics imaging and retinal densitometry. The images are false colored so that blue, green, and red are used to represent the S, M, and L cones, respectively. (The true colors of these cones are yellow, purple, and bluish-purple). The mosaics illustrate the enormous variability in L/M cone ratio. The L/M cone ratios are (A) 0.37, (B) 1.11, (C) 1.14, (D) 1.24, (E) 1.77, (F) 1.88, (G) 2.32, (H) 2.36, (I) 2.46, (J) 3.67, (K) 3.90, and (L) 16.54. The proportion of S cones is relatively constant across eyes, ranging from 3.9 to 6.6% of the total population. Images were taken either 1° or 1.25° from the foveal center. For two of the 10 subjects, two different retinal locations are shown. Panels (D) and (E) show images from nasal and temporal retinas, respectively, for one subject; (J) and (K) show images from nasal and temporal retinas for another subject. Images (C), (J), and (K) are from Roorda and Williams [52]. All other images were made by Heidi Hofer. (See page 16 for text discussion.) (From Williams and Hofer [57]. Reprinted with permission from The MIT Press.)

830 nm

633 nm

Drusen

Some Nerve Fiber Layer

543 nm Retinal Vessels

488 nm Macular Pigment, Nerve Fiber Layer

514 nm Some Macular Pigment

633, 543, and 488 nm Images Combined

FIGURE 9.9 Images centered on the human macula, acquired with laser illumination over a range of wavelengths. The bottom right panel is the combination of three colors: red (633 nm), green (543 nm), and blue (488 nm). (See page 219 for text discussion.)

FIGURE 9.10 A color fundus photograph of the patient in Figures 9.4 and 9.5, showing that the larger retinal vessels are seen, but that the choroidal ones (other than the largest ones that feed and drain the neovascular membrane) are obscured. (See page 219 for text discussion.)

τ = −0.35 min

Kλ = +0.10 D

λ = 575 nm

τ = −0.74 min

Kλ = +0.22 D

λ = 600 nm

1-mm pupil offset

λfocus = 555 nm

Composite

Luminance

FIGURE 13.9 Image formation for a polychromatic source in the presence of chromatic aberration. Top row is for an eye with longitudinal chromatic aberration only. Bottom row is for an eye with longitudinal and transverse chromatic aberration produced by 1 mm of horizontal pupil offset from the visual axis (or, equivalently, 15° of eccentricity). The point source emits three wavelengths of light (500, 575, and 600 nm) and the eye is assumed to be focused for 550 nm. Chromatic errors of focus and position indicated for each image are derived from an analysis of the Indiana Eye model of chromatic aberration. (See page 354 for text discussion.)

τ = +0.62 min

τ=0

Kλ = −0.18 D

λ = 525 nm

PART ONE

INTRODUCTION

CHAPTER ONE

Development of Adaptive Optics in Vision Science and Ophthalmology DAVID R. WILLIAMS and JASON PORTER University of Rochester, Rochester, New York

This chapter briefly reviews the history of ocular aberration measurement and correction that paved the way to the development of adaptive wavefront correction of the eye. While the focus of this book is on the engineering of adaptive optics systems for the eye, this chapter describes recent applications of adaptive optics and the scientific discoveries that adaptive optics has made possible, encouraging the future development of this technology.

1.1 BRIEF HISTORY OF ABERRATION CORRECTION IN THE HUMAN EYE 1.1.1 Vision Correction The fi rst use of a transparent stone as a crude magnifying glass is not known, though it has been suggested that this could have been as early as 5000 bc [1]. It is also unclear who fi rst fi xed simple lenses to the head. Though corrective spectacles rank among the most important medical inventions in history, their origins are obscure [2]. Most sources attribute spectacles to an unknown Italian near the end of the thirteenth century. In any case, the invention of spectacles seems to have been based on empirical observation of the effects

Adaptive Optics for Vision Science, Edited by Porter, Queener, Lin, Thorn, and Awwal Copyright © 2006 John Wiley & Sons, Inc.

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DEVELOPMENT OF ADAPTIVE OPTICS IN VISION SCIENCE

of glass held before the eye rather than theoretical insight, as their invention preceded the fi rst clear description of how the image was formed on the retina [3] by at least 300 years. Kepler, an astronomer perhaps best known for his laws of planetary motion, pointed out that the retinal image is inverted and also clearly described the benefits of concave lenses for myopic correction and convex lenses for hyperopic correction. This was a time of rapid advances in the field of optics. The telescope was invented around this same time, though again there is controversy about the inventor [4]. Most scholars attribute the invention to Hans Lipperhey, a Dutch spectacle maker who produced a telescope in 1608. Galileo would soon use one of the first telescopes to observe the moons of Jupiter and sunspots. The link between astronomy and the eye apparent in Kepler’s scientific contributions and Lipperhey’s telescope is a recurring theme in the history of vision science, culminating in the recent translation of adaptive optics from astronomy to vision science. Galileo had a competitor, Christoph Scheiner, who was also an astronomer with interests in physiological optics. Scheiner demonstrated empirically that the retinal image was inverted by cutting a hole in the back of an excised animal eye and viewing the retinal image directly [5]. Scheiner also constructed what was arguably the fi rst wavefront sensor for the eye. Scheiner’s wavefront sensor evaluated the fate of light passing through only two locations in the eye’s entrance pupil. Modern ophthalmic wavefront sensors extend this concept by measuring the direction that light takes as it passes through hundreds of different locations in the eye’s pupil. Scheiner made two holes in an opaque disk. When held close to the eye, the perceived image was doubled if the eye was defocused and single only if the eye was in focus, providing subjective information about the eye’s most important aberration. It would be nearly 200 years before a clear understanding developed of astigmatism, the eye’s second most important monochromatic aberration. Thomas Young recognized the existence of astigmatism in his own eye and determined that his astigmatism was predominantly lenticular in origin by noting that it persisted even when he immersed his eye in water, largely neutralizing the cornea [6]. In 1827, Sir George Biddell Airy, yet another astronomer, fabricated the fi rst spherocylindrical lenses to correct astigmatism. This ultimately lead to the current ophthalmic practice of prescribing aberration corrections with 3 degrees of freedom corresponding to a defocus correction, the cylindrical power, and the cylinder axis. Helmholtz argued that the normal eye contained more monochromatic aberrations than just defocus and astigmatism, based in part on his own subjective observations of a bright point source viewed in the dark [7]. These monochromatic, higher order aberrations were often referred to as “irregular astigmatism” to distinguish them from the regular astigmatism that could be corrected with a cylindrical lens. Roughly one and a half centuries after Helmholtz’s description of the higher order aberrations in human eyes, we are now equipped with the adaptive optics (AO) technology that can systematically measure and correct them.

BRIEF HISTORY OF ABERRATION CORRECTION IN THE HUMAN EYE

5

1.1.2 Retinal Imaging The main hurdle to obtaining the fi rst view of the inside of the living eye was that such a small fraction of the light entering the pupil returns back out of it. The reflectance of the back of the eye is only 0.1 to 10% depending on wavelength (400 to 700 nm) [8, 9], and the pupil restricts the amount of light that can exit the eye by another factor of about 100. These two factors together reduce the light returning through the pupil by 10 −3 to 10 −5 depending on wavelength. Purkinje appreciated that under some conditions the pupil of the eye could be made to appear luminous instead of black [10, 11]. Brücke demonstrated the glow in the pupil that could be seen through a tube held in a candle flame and pointed at an eye [12]. In 1851, Helmholtz revolutionized the field of ophthalmology with the invention of the ophthalmoscope [13]. He called his invention the Augenspiegel or “eye mirror.” Jackman and Webster obtained the fi rst photographs of the human retina in vivo by attaching the camera to the patient’s head to reduce image motion during the 2.5-minute exposures that were required [14]. Subsequently, the fundus camera was improved by blocking the unwanted reflection from the corneal surface, increasing fi lm sensitivity, and the electronic flash lamp, which allowed exposures brief enough to avoid eye movement blur. The scanning laser ophthalmoscope (SLO), invented by Robert Webb [15], allowed the use of detectors such as the avalanche photodiode or the photomultiplier tube, which increased the sensitivity of retinal imaging systems well beyond that which could be achieved with photographic fi lm. The use of raster scanning in this instrument instead of flood illumination provided the advantage of real-time video imagery. Moreover, the instrument can be equipped with a confocal pinhole to reject light that originates from retinal planes other than the plane of interest, providing an optical sectioning capability. The application of optical coherence tomography (OCT) to the eye enabled even greater improvements in axial resolution [16, 17], with axial resolutions as high as 1 to 3 mm in vivo [18]. 1.1.2.1 Microscopic Retinal Imaging In Vivo While conventional fundus cameras, SLOs, and OCT systems provide a macroscopic view of the living retina, they do not have the transverse resolution needed to reveal retinal features on the spatial scale of single cells. Photoreceptor cells had been observed in the living eyes of animals with good optics and large receptors, such as the cane toad and the snake [19, 20]. At about this time, my laboratory was characterizing the human photoreceptor mosaic in vivo, relying on the subjective observations of aliasing effects caused by imaging interference fringes on the retina [21]. Artal and Navarro wondered whether another form of interferometry, akin to stellar speckle interferometry [22], could be used to obtain objective information about the cone mosaic [23]. They collected images of the speckle patterns generated by illuminating small patches of living human retina with coherent light. The average power spectrum of

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multiple images revealed a local maximum corresponding to the fundamental frequency of the cone mosaic, providing the fi rst evidence that information about the granularity of the cone mosaic can be obtained from images of the living human eye. Marcos and Navarro improved this technique and were able to relate objective measures of cone spacing to visual acuity [24]. Unfortunately, this method does not allow for the direct observation of the photoreceptor mosaic because the photoreceptors are obscured by the interference of the coherent light reflected from multiple layers of the retina. To avoid this problem, Miller et al. constructed a high-magnification fundus camera capable of illuminating the retina with incoherent light [25]. By dilating the pupil to reduce the influence of diffraction and with careful correction of defocus and astigmatism, they obtained the fi rst direct images of the human cone mosaic in vivo. This could be achieved in only a subset of young eyes with very good optical quality. Better images awaited methods to measure and correct not only defocus and astigmatism but also the higher order monochromatic aberrations of the eye as well. 1.1.2.2 Adaptive Optics Modern measurements of the wave aberration began with Smirnov, who employed a subjective vernier task to measure the retinal misalignment of rays entering through different parts of the pupil, providing a description of the third- and fourth-order aberrations [26]. Smirnov recognized that his method could, in principle, allow for the fabrication of contact lenses that corrected higher order aberrations in the eye but thought that the lengthy calculations required to compute the wave aberration made this approach impractical. He could not forsee the rapid development of computer technology that would eventually make it possible to compute the eye’s wave aberration in a matter of milliseconds. Following Smirnov’s pioneering work, a number of investigators devised a variety of different methods to characterize the wave aberration [27]. Walsh, Charman, and Howland demonstrated an objective method that greatly increased our understanding of the properties of the eye’s wave aberration [28]. However, this technology was sophisticated for its time, and while its value was appreciated by some scientists, it was not ready for clinical adoption. This situation changed abruptly when Junzhong Liang, working as a graduate student in Josef Bille’s laboratory at the University of Heidelberg (see Fig. 1.1), demonstrated that it was possible to adapt the Shack–Hartmann wavefront sensor, typically used in optical metrology, to measure the eye’s wave aberration [29]. This proved to be the key development that paved the way to closed-loop adaptive optics systems for the eye. The simplicity of the Shack–Hartmann method and the fact that it is the wavefront sensor used in most astronomical adaptive optics systems made it easier to translate adaptive optics to the eye. The method was also amenable to automation, as our group at Rochester eventually demonstrated in collaboration with Pablo Artal’s group at the University of Murcia [30]. We were able to measure the wave aberration in real time, showing that the most significant temporal fluctuations in the eye’s wave aberration were

BRIEF HISTORY OF ABERRATION CORRECTION IN THE HUMAN EYE

7

FIGURE 1.1 Josef Bille and Junzhong Liang, two of the inventors of the fi rst Shack– Hartmann wavefront sensor for the eye, on the day Liang defended his Ph.D. thesis in 1992.

caused by focus changes associated with the microfluctuations of accommodation. Adaptive correction of the eye’s wave aberration has its origins in astronomy, specifically in Horace Babcock’s proposed solution to the problem of imaging stars through the turbulent atmosphere [31]. Babcock introduced the idea of an adaptive optical element that could correct the time-varying aberrations caused by atmospheric turbulence. Due to the technical complexity of measuring atmospheric aberrations and fabricating and controlling a deformable mirror to correct them, the first successful demonstration of adaptive optics in astronomy was not made until 1977 by Hardy and his colleagues [32]. Many of the major ground-based telescopes around the world are now equipped with adaptive optics, which can sometimes achieve images with higher resolution than those obtained with the Hubble space telescope [33, 34]. In 1989, Andreas Dreher and colleagues, also working in Joseph Bille’s laboratory in Heidelberg, described the fi rst attempt to use a deformable mirror to improve retinal images in a scanning laser ophthalmoscope [35]. They were able to use a deformable mirror to correct the astigmatism in one subject’s eye based on a conventional spectacle prescription. In 1993, Junzhong Liang, who had previously demonstrated the fi rst Shack–Hartmann wavefront sensor in Heidelberg, joined my laboratory as a postdoctoral fellow. We developed a high-resolution wavefront sensor that provided a more complete description of the eye’s wave aberration, measuring up to 10 radial Zernike orders [36]. These measurements showed that higher order aberrations can be significant sources of retinal image blur in some eyes, especially

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when the pupil is large. Liang, Don Miller, and I then constructed the fi rst closed-loop adaptive optics system that could correct higher order aberrations in the eye (see Fig. 1.2) [37]. We may have never built this instrument were it not for the availability of the fi rst deformable mirror made by Xinxtics, Inc., a small startup that Mark Ealey had just launched. Liang and I had also received encouragement and advice from Bob Fugate, head of the Starfi re Optical Range (a satellite-tracking telescope equipped with adaptive optics). This fi rst system required about 15 min for each loop of measuring and correcting the wave aberration, with 4 or 5 loops required to complete the correction. Wavefront sensing was not yet automated for the eye and each frame of Shack–Hartmann spots required tedious adjustment of the centroid estimates to correct errors made by the imperfect centroiding algorithm available at the time. Tedious though our fi rst experiments with adaptive optics were, we were able to improve contrast sensitivity beyond what was possible with a conventional spectacle correction, and we obtained higher contrast images of the cone mosaic than Miller had previously obtained without adaptive optics. Real-time correction of the wave aberration [38, 39] was not possible until the development of automated wavefront sensing, which allowed the fi rst realtime measurement of the eye’s wave aberration [30]. Fortunately for vision science, an adaptive optics system operating with a closed-loop bandwidth of only a few hertz is adequate to capture the most important temporal changes in the fi xating eye with diminishing returns for higher bandwidths [40].

FIGURE 1.2 The University of Rochester’s adaptive optics ophthalmoscope, the fi rst system capable of measuring and correcting higher order aberrations in the eye. The instrument used a 37-actuator deformable mirror made by Xinxtics, Inc. From left to right, Don Miller, Junzhong Liang, and David Williams in 1996.

APPLICATIONS OF OCULAR ADAPTIVE OPTICS

1.2 1.2.1

9

APPLICATIONS OF OCULAR ADAPTIVE OPTICS Vision Correction

By compensating for the eye’s higher order aberrations with adaptive optics, Liang, Williams, and Miller showed that the contrast sensitivity of the eye in monochromatic light could be improved over that obtained with the best spectacle correction [37]. This result spurred the rapid development of wavefront-guided refractive surgery, in which measurements of the eye’s wave aberration control an excimer laser to correct the eye’s static higher order aberrations as well as defocus and astigmatism (see also Chapter 12). The most enthusiastic proponents had hopes that this surgical procedure could improve essentially everyone’s vision beyond 20/20. However, there are practical limitations on the visual benefit of correcting higher order aberrations. First of all, the benefit is limited to conditions under which the pupil is larger than about 4 mm, that is, dim illumination. Yoon and Williams subsequently confi rmed the increases in both contrast sensitivity and acuity with higher order aberrations corrected with adaptive optics, but due to chromatic aberration in the eye, these increases were reduced in white light compared with the increases that can be obtained in monochromatic light (see Fig. 1.3) [41]. Changes in the wave aberration with accommodation reduce the visual benefit to some extent [42]. In addition, there is the possibility that the reduction in the depth of field caused by correcting higher order aberrations could actually decrease visual performance in some circumstances. Finally, the visual benefit is only as good as the technology available to correct the wave aberration, and factors such as errors in the alignment of the excimer laser with the eye and biomechanical changes in the eye following surgery reduce the accuracy of higher order correction. Although wavefront correction will never provide everyone with supervision, it remains clear that there are eyes that can benefit from the correction of higher order aberrations, Indeed, in some eyes, the correction of higher order aberrations can produce truly dramatic improvements in visual performance by restoring poor vision to near normal levels. Examples include eyes with kerataconus and penetrating keratoplasty as well as normal eyes that happen to have large amounts of higher order aberrations. Chapter 11 addresses the prognosis for contact lenses that correct higher order aberrations, and Chapter 12 discusses current progress in customized vision correction with refractive surgery. 1.2.1.1 Automated Refraction with Adaptive Optics Adaptive optics may eventually play a role in the clinical refraction of the eye. A consortium lead by Lawrence Livermore National Laboratories, including the University of Rochester, Bausch & Lomb, Wavefront Sciences, Boston Micromachines, and Sandia National Laboratories, has developed a phoropter equipped with adaptive optics for automatically refracting the eye. This device allows a

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DEVELOPMENT OF ADAPTIVE OPTICS IN VISION SCIENCE

FIGURE 1.3 The measured contrast sensitivity (lower panels) and visual benefit (upper panels) for two subjects when correcting various aberrations across a 6-mm pupil: both monochromatic and chromatic aberrations (fi lled circles, dashed black line), monochromatic aberrations only (open circles, solid gray line), and defocus and astigmatism only (× symbols, solid black line). The visual benefit is the ratio of the contrast sensitivity with higher order aberrations corrected to the contrast sensitivity with only defocus and astigmatism corrected. Note that chromatic aberration, which is present in normal viewing, tempers the visual benefit of correcting higher order aberrations. These two observers show about a twofold increase in contrast sensitivity in white light, which is the modal benefit computed from population data of the wave aberration. (From Yoon and Williams [41]. Reprinted with permission of the Optical Society of America.)

patient to view a visual acuity chart through adaptive optics. The AO system automatically measures and corrects the aberrations of the patient’s eye and provides a prescription for glasses, contact lenses, or refractive surgery. The phoropter measures the eye’s wave aberration with a Shack–Hartmann wavefront sensor. Wavefront correction in the phoropter is achieved with a microelectromechanical systems (MEMS) deformable mirror.

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11

1.2.1.2 Image Quality Metrics Assessed with Adaptive Optics The effective use of wavefront sensing to refract the eye depends on the ability to transform the wave aberration, which is usually defi ned by several dozen Zernike coefficients, into the values of sphere, cylinder, and axis that generate the best subjective image quality. This transformation is not as trivial as it might fi rst appear because the amounts of defocus and astigmatism required to optimize image quality depend on higher order aberrations as well as neural factors [43]. Adaptive optics has played a useful role in this effort because it can be used not only to correct aberrations but also to generate them (see also Section 5.4.3). This has allowed Li Chen at Rochester to measure the effect of aberrations on vision with psychophysical methods and helped to generate plausible metrics for image quality [44]. 1.2.1.3 Blur Adaptation The use of adaptive optics to generate aberrations has also been used in clever experiments by Pablo Artal to reveal the eye’s adaptation to its own point spread function (PSF) [45]. Artal measured the subjective blur when subjects viewed a scene through their normal wave aberration, as well as through rotated versions of their normal wave aberration. Li Chen and Ben Singer developed the method to rotate the wave aberration using the deformable mirror in the adaptive optics system. Despite the fact that the amount of aberration in all conditions was constant, the subjective blur varied significantly, with the least blur occurring when the subject was seeing the world through his or her own wave aberration. These experiments reveal the neural mechanisms that influence subjective image quality and show that the nervous system has learned to at least partially discount the blur produced by the particular pattern of aberrations through which it must view the world. 1.2.2 Retinal Imaging The use of adaptive optics to increase the resolution of retinal imaging promises to greatly extend the information that can be obtained from the living retina. Adaptive optics now allows the routine examination of single cells in the eye, such as photoreceptors and leukocytes, providing a microscopic view of the retina that could previously only be obtained in excised tissue. The ability to see these structures in vivo provides the opportunity to noninvasively monitor normal retinal function, the progression of retinal disease, and the efficacy of therapies for disease at a microscopic spatial scale. 1.2.2.1 Photoreceptor Optics Revealed with Adaptive Optics Retinal Imaging The benefit of adaptive optics for photoreceptor imaging can be seen in Figure 15.7. Adaptive optics has also proved useful in studying the optical properties of single cones in vivo, properties that are difficult if not impossible to study in excised retina. Cone photoreceptors appear bright in high-resolution images because they act as waveguides radiating the light

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incident on them back toward the pupil in a relatively narrow beam with a roughly Gaussian profi le. Images of the cone mosaic have high contrast over a wide range of wavelengths [46] as shown in Figure 1.4. The angular dependence of the light radiated from the cones is closely related to the Stiles–Crawford effect measured psychophysically. The Stiles– Crawford effect describes the loss in sensitivity of the eye to light incident on the mosaic with increasing obliquity from the optical axes of the receptors, which point roughly toward the pupil center [47]. This tuning function is measured with a relatively large number of cones and is therefore the combination of the waveguide properties of single photoreceptors and the disarray in individual cone pointing direction. Though psychophysical methods have suggested that the disarray is likely to be small [48], it has not been possible to disentangle these factors with direct measurements in the human eye. With adaptive optics we have succeeded in measuring the angular tuning properties of individual human cones for the fi rst time, and the disarray in individual cone axes that contributes to the angular tuning properties of the retina as a whole [49]. Figure 1.5 shows images of the same patch of cones when they are illuminated with light entering different locations of the pupil. The image at

FIGURE 1.4 Images of the cone mosaic obtained at 1° eccentricity in the temporal retina with wavelengths of 550, 650, and 750 nm. The top row shows the registered raw images and the bottom row shows the same images deconvolved to remove any differential effects of the eye’s PSF with wavelength. Note that the image contrast is relatively independent of wavelength. Scale bar is 10 mm. (From Choi et al. [46]. Reprinted with permission of the Optical Society of America.)

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FIGURE 1.5 A series of 9 images obtained at the same retinal location (1° eccentricity) on the same retina, but with the entry point of light in the pupil shifted in 1-mm increments in the horizontal and vertical pupil meridians. The central image corresponds to a location near the pupil center, which returns the most light. Due to the directionality of the cone reflectance, the light returning when the retina is illuminated obliquely is reduced. Each image is the registered sum of 8 raw images. The pupil used to illuminate the retina had a diameter of 1.5 mm, while the pupil used to image the retina had a diameter of 6 mm. Scale bar is 5 arcmin. (From Pallikaris et al. [50]. Reprinted with permission of the Association for Research in Vision and Ophthalmology.)

the center corresponds to illumination at the pupil center, and the surrounding images correspond to oblique illumination in 1- and 2-mm increments from the pupil center. By comparing the intensities of each cone under the different illumination conditions, we can determine the directional sensitivity of each cone.

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Figure 1.6 shows the pointing direction of each cone relative to the center of the pupil for two subjects. For each subject, all the cones are tuned to approximately the same direction. The disarray in cone pointing direction is only about 0.11 times the width of the tuning function for a single cone, implying that the Stiles–Crawford effect is a good estimate of the angular tuning of single cones. Additional experiments using adaptive optics have revealed new optical properties of the cone photoreceptors. Pallikaris et al. observed large differences in the reflectance of different cones and that the reflectance of the same cone changed sometimes several-fold over time [50]. These changes were found in all three cone classes and were not caused by changes in the directionality of individual cones. While the changes Pallikaris et al. observed occurred over time scales of minutes to days, Don Miller’s group has recently demonstrated that there are also short-term fluctuations in cone reflectance [51]. They have also shown that these changes can be induced by photopigment bleaching. The cause or causes of these temporal variations remains a matter of investigation, but they may ultimately provide a valuable optical diagnostic of functional activity with each cell. 1.2.2.2 Imaging the Trichromatic Cone Mosaic One of the fi rst demonstrations of the scientific value of retinal imaging with adaptive optics was its use

FIGURE 1.6 Pupils of two subjects with the origin corresponding to the geometric center of the pupil. Each dot represents the location where the optical axis of a single cone intersects the pupil plane. These locations are tightly clustered, with standard deviations of 180 and 160 mm, respectively, indicative of the small amount of disarray in the alignment of cones within the retina. (From Roorda and Williams [49]. Reprinted with permission of the Association for Research in Vision and Ophthalmology.)

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in identifying the photopigment in single human cones in vivo [52, 53]. We had known for nearly 200 years that human color vision depends on three fundamental channels in the retina [54], now referred to as the short wavelength (S), middle wavelength (M), and long wavelength (L) sensitive cones. The packing arrangement and relative numbers of two of the three cone classes (the L and M cones) remained unclear. With adaptive optics, we succeeded in classifying large numbers of living human foveal cones by comparing images of the photoreceptor mosaic when the photopigment was fully bleached to those when the photopigment was selectively bleached with different wavelengths of light. We used 650- and 470-nm light to selectively bleach L cones and M cones, respectively. Heidi Hofer subsequently improved this method and increased the number of eyes that were characterized [55]. Julian Christou improved it still further by showing that deconvolution of the retinal images could decrease the error rate in classifying the pigment in each cone [56]. Figure 1.7 shows the combined results of the Hofer and Roorda studies, in which the S, M, and L cones have been pseudo-colored blue, green, and red, respectively. These results illustrate the random, or nearly random, packing arrangement of the cones as well as the large variation from eye to eye in the ratio of L to M cones in the normal retina. The retinas span a 45fold range of L to M cone ratio. The ability to directly observe the ratio of L to M cones in the living eye allowed us to settle a controversy about the role of cone numerosity on color appearance. We established that color appearance does not vary with L to M ratio [58, 59]. Pokorny and Smith had previously suggested that experience with the chromatic environment rather than cone numerosity establishes the subjective boundaries between hues [60]. Neitz et al. have reported some experimental support for this view, showing that the color boundary between red and green, termed unique yellow, can undergo modification over a period of many days of exposure to an altered chromatic environment [59]. Adaptive optics has also provided an opportunity to explore the color appearance produced by stimulating individual cones or small groups of cones with tiny, brief flashes of monochromatic light [61]. It has been known since Holmgren that the color appearance of such stimuli fluctuates from flash to flash, presumably depending on the specific photoreceptors that are excited by each flash [62]. Adaptive optics allows us to make much more compact light distributions on the retina, enhancing these color fluctuations and making them easier to study in the laboratory. Hofer et al. could explain the variation in color appearance with a model in which different cones containing the same photopigment produce different chromatic sensations when stimulated [61]. These experiments showed that the color sensation produced by stimulating a cone depends on the circuitry each cone feeds rather than simply on the photopigment the cone contains. 1.2.2.3 Tracking Eye Position with Adaptive Optics Eye movements, even when the eye is fi xating, constantly translate the retina relative to the fi xation

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DEVELOPMENT OF ADAPTIVE OPTICS IN VISION SCIENCE

FIGURE 1.7 Images of the cone mosaics of 10 subjects with normal color vision, obtained with the combined methods of adaptive optics imaging and retinal densitometry. The images are false colored so that blue, green, and red are used to represent the S, M, and L cones, respectively. (The true colors of these cones are yellow, purple, and bluish-purple). The mosaics illustrate the enormous variability in L/M cone ratio. The L/M cone ratios are (A) 0.37, (B) 1.11, (C) 1.14, (D) 1.24, (E) 1.77, (F) 1.88, (G) 2.32, (H) 2.36, (I) 2.46, (J) 3.67, (K) 3.90, and (L) 16.54. The proportion of S cones is relatively constant across eyes, ranging from 3.9 to 6.6% of the total population. Images were taken either 1° or 1.25° from the foveal center. For two of the 10 subjects, two different retinal locations are shown. Panels (D) and (E) show images from nasal and temporal retinas, respectively, for one subject; (J) and (K) show images from nasal and temporal retinas for another subject. Images (C), (J), and (K) are from Roorda and Williams [52]. All other images were made by Heidi Hofer. (See insert for a color representation of this figure.) (From Williams and Hofer [57]. Reprinted with permission from The MIT Press.)

target. The high transverse resolution of retinal imaging systems equipped with adaptive optics provides a unique opportunity to record these eye movements with very high accuracy. Putnam et al. showed that it is possible to record the retinal location of a fi xation target on discrete trials with an error at least 5 times smaller than the diameter of the smallest foveal cones [63]. We used this capability to measure the standard deviation of fi xation positions

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17

across discrete fi xation trials, obtaining values that ranged from 2.1 to 6.3 arcmin, with an average of 3.4 arcmin, in agreement with previous studies [63, 64]. Interestingly, the mean fi xation location on the retina was displaced from the location of highest foveal cone density by an average of about 10 arcmin (as shown in Fig. 1.8), indicating that cone density alone does not drive the location on the retina selected for fi xation. This method may have interesting future applications in studies that require the submicron registration of stimuli with respect to the retina or delivering light to retinal features as small as single cells. Whereas the method developed by our group can only record eye position on discrete trials, Scott Stevenson and Austin Roorda have shown that it is possible to extract continuous eye movement records from video-rate images obtained with an adaptive optics scanning laser ophthalmoscope (AOSLO) [66]. Eye movements cause local warping of the image within single video frames as well as translation between frames. The warping and translation information in the images can be used to recover a record of the eye movements that is probably as accurate as any method yet devised. This is illustrated in Figure 1.9, which compares the eye movement record from the AOSLO with that from a Dual Purkinje Eye Tracker. The noise in the AOSLO trace is on the order of a few arc seconds compared to about a minute of arc for the Dual Purkinje Eye Tracker. Note also the greatly reduced overshoot following a saccade in the AOSLO trace. These overshoots are thought to be partly artifacts caused by lens wobble following the saccade and do not reflect the true position of the retinal image. The AOSLO is not susceptible to this artifact because it tracks the retinal position directly rather than relying on reflections from the anterior optics. Adaptive optics will no doubt prove to

FIGURE 1.8 Area of highest cone density is not always used for fi xation. Shown are retinal montages of the foveal cone mosaic for three subjects. The black square represents the foveal center of each subject. The dashed black line is the isodensity contour line representing a 5% increase in cone spacing, and the solid black line is the isodensity contour line representing a 15% increase in cone spacing. Dots indicate individual fi xation locations. Scale bar is 50 mm. (From Putnam et al. [63]. Reprinted with permission of the Association for Research in Vision and Ophthalmology.)

DEVELOPMENT OF ADAPTIVE OPTICS IN VISION SCIENCE

Eye Position (arcmin)

18

20 18 16 14 12 10 8 6 4 2 0

Dual Purkinje Eye Tracker – Vertical AOSLO Retinal Image Vertical

Lens Wobble Artifact in dPi Signal

Small Retinal Image Shift in AOSLO Video Due to Lens Wobble

2.5

2.7

2.9

3.1

3.3

3.5

Time (s) FIGURE 1.9 Comparison of an eye movement trace obtained from an AOSLO (black line) and that obtained from a Dual Purkinje Eye Tracker (gray line), operating simultaneously. Note the reduced noise in the AOSLO trace and the dampened overshoot compared with the Dual Purkinje trace. (Courtesy of Scott Stevenson and Austin Roorda.)

be a useful tool for new studies of eye movements that require high accuracy. Roorda’s group is already using adaptive optics to conduct psychophysical experiments in which the effect of small eye movements on a vernier acuity task can be measured to determine whether the brain compensates for these small movements. Moreover, the process of recovering the eye movement signal makes it possible to register successive video frames and remove the warping artifacts within single frames. Roorda’s group, in collaboration with David Arathorn and Curt Vogel at Montana State University, have demonstrated dewarped and stabilized video images of the retina obtained with adaptive optics. It may eventually be possible to perform these computations in real time, allowing real-time stabilization of the retinal image. An alternative approach, under development by Dan Ferguson and Dan Hammer at Physical Sciences Corporation, is to couple a separate eye tracker to an adaptive optics scanning laser ophthalmoscope for the purposes of image stabilization [67]. These approaches may herald a new generation of psychophysical experiments in which the location of a stimulus on the retinal cone mosaic can be controlled in real time with an error less than the diameter of a single cone. 1.2.2.4 Imaging Retinal Disease One of the most exciting applications of adaptive optics is for the diagnosis and treatment of retinal disease. The advantage offered by AO is that the microscopic structure of the diseased

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retina can be imaged in vivo and tracked in single eyes, monitoring the progression of the disease, or the efficacy of therapy over time. Investigators are only just beginning to explore AO imaging of retinal disease, but there are already several studies that reveal its value. Photoreceptor Degeneration The fi rst discovery about the abnormal eye made with adaptive optics arose from imaging the cone mosaics of color blind eyes [68]. Dichromatic color vision results from the functional loss of one cone class. However, a central question has been whether red–green color blind individuals have lost one population of cones (rendering a patchy cone mosaic) or whether they have normal numbers of cones fi lled with either of two instead of three pigments. Evidence has accumulated favoring the latter view in which the photopigment in one class of cone is replaced, but the issue has not been resolved directly. The Rochester group obtained images from two dichromats, one of which showed a remarkable loss of cones in his retina (while the other dichromat had a normal appearing mosaic). Images from these two subjects are shown in Figure 1.10. The images are from the same retinal eccentricity, about 1° in the temporal direction. The image on the right is from a dichromat who has a novel mutation in one of his cone visual pigment genes, resulting in the loss of the corresponding cone class. The areas where normal cones do not appear have been shown to be functionally blind, using a method developed by Walt Makous. In this method, the subject’s psychometric functions for detecting microflashes delivered to the retina through adaptive optics are shallower than normal and fall well short of 100% detection at moderately high light levels. Both these features are quantitatively consistent with the fraction of the retinal area that is lacking normal cones [69]. The subject on the left is missing the gene for one of his cone visual pigments, and his mosaic is normal in appearance. This fi nding suggested that previous models of dichromacy do not hold for all subjects. One

FIGURE 1.10 Images of the retinas of two dichromats. One dichromat is missing the gene for his L cones and his mosaic is complete, as shown on the left. The other has all three photopigment genes, but his M pigment gene has a mutation that results in the loss of normal cones of that class, as shown on the right. Scale bar is 20 mm.

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DEVELOPMENT OF ADAPTIVE OPTICS IN VISION SCIENCE

of the intriguing aspects of the eye of the patient missing functional cones is that his visual acuity is 20/16, normal, despite the loss of 30% of his cones. This result highlights the insensitivity of conventional clinical tests and points to the role that adaptive optics may eventually play in the early detection of diseases that produce the dropout of photoreceptors or perhaps other neurons, such as ganglion cells. Mosaic in Cone–Rod Dystrophy Adaptive optics has also proven valuable in imaging retinal degeneration. Jessica Wolfi ng and her colleagues at Rochester and UC, Berkeley have imaged a patient with cone–rod dystrophy, which is characterized by a bull’s-eye lesion in the macula [70]. Adaptive optics images were taken in the bull’s-eye lesion as well as the relatively spared central macula using Rochester’s adaptive optics ophthalmoscope and Austin Roorda’s AOSLO. Adaptive optics retinal images of the cone–rod dystrophy patient were compared to those of an age-matched normal subject (Fig. 1.11). In the center of the bull’s-eye lesion, which appeared relatively spared using conventional ophthalmoscopy, adaptive optics revealed a nearly continuous cone photoreceptor mosaic. In the central 1.25°, the cones were larger than normal and cone density was decreased (Fig. 1.12). At the fovea, the patient’s cone density was 30,100 cones/mm 2 instead of the normal average of 199,200 cones/mm 2 [71]. From 2.5° to 4°, the area corresponding to the atrophic bull’s-eye lesion, patches devoid of waveguiding cones were interspersed with highly reflective areas.

FIGURE 1.11 (Top) Adaptive optics retinal images of the right eye of the cone–rod dystrophy patient taken at (a) the fovea, (b) 1°, (c) 2.5°, and (d) 4° nasal to fi xation, respectively (from left to right). (Bottom) Images of an age-matched normal subject at the same respective eccentricities temporal to fi xation. Scale bar is 25 mm.

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FIGURE 1.12 Cone density and cone diameter versus retinal eccentricity in normal subjects compared with the cone–rod dystrophy patient. Measurements of cone density were made only in areas with a complete photoreceptor mosaic (central 1.25° for the cone–rod dystrophy patient). Light gray shaded region: normal range of cone densities measured using microscopy [71]. Dark gray circles: mean and standard deviation of cone density of normal subjects measured from adaptive optics images using a direct counting procedure. Black diamonds: cone density for the cone–rod dystrophy patient. Dark gray bars (bottom panel): mean normal cone diameter. Black bars (bottom panel): mean cone–rod dystrophy cone diameter.

Longitudinal studies of disease progression or investigations of early stage and presymptomatic cone–rod dystrophy patients would help determine the mechanism underlying the increased cone diameter and decreased cone density observed in this study, providing insight into developmental and degenerative processes in normal and diseased retina. This study demonstrates the viability of applying adaptive optics retinal imaging to provide quantitative cell counts in retinal disease, an important diagnostic for tracking the progression of the disease as well as for the efficacy of therapy. 1.2.2.5 High-Resolution Imaging of Vascular Structure and Blood Flow Accurate measurements of blood vessel structure and function are critical for numerous retinal diseases. Current technologies for directly measuring blood flow in small vessels are limited by the axial and transverse resolutions of retinal imaging. Doppler OCT methods improve axial resolution but still suffer from poor lateral resolution. Also, direct methods to measure flow rely on injecting contrast agents such as fluorescein, which carries a risk of a reaction that can be fatal. In addition, the observations can be made only during

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the brief dwell time of the dye in the retina. Joy Martin in Austin Roorda’s group has shown that adaptive optics allows for the noninvasive measurement of blood flow without the need for contrast agents [72]. This allows for imaging over much longer observational periods in individual patients. In vivo studies of blood flow in these smallest of vessels may provide valuable information about early vascular changes in diseases such as diabetic retinopathy. Figure 1.13 shows measurements of blood velocity in different capillaries surrounding the fovea. 1.2.2.6 AO-Assisted Vitreo-Retinal Surgery Some aspects of retinal microsurgery may benefit from the high resolution provided by adaptive optics, an application of adaptive optics that has yet to be explored. For example, an AO-equipped surgical microscope may improve the ability to target microaneuryms in diabetic retinopathy. Other procedures that might benefit include the removal of micron-thick structures from the surface of the retina (such as epiretinal membranes), the removal of tissue (such as choroidal neovascularization [CNV] lesions) from the subretinal space, the treatment of retinal vein occlusions by cannulation or by incising the thin connective tissue sheath separating retinal veins and arteries. The capability to stabilize the image, described earlier, would be valuable in the delivery of therapeutic laser beams through an AO-equipped surgical microscope, since, for example, it would be possible to target the desired location and minimize damage to adjacent retinal locations.

FIGURE 1.13 Measurements of blood velocity in individual capillaries surrounding the foveal avascular zone in a normal patient. (From Carroll et al. [73]. Reprinted with permission of the Optical Society of America.)

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1.2.2.7 Future Directions for Improving Resolution and Contrast To date, high-resolution imaging with adaptive optics has been successful at imaging photoreceptors because they are waveguides, which glow brightly by preferentially sending incoming light back toward the pupil. Blood flow can be monitored in single capillaries because of the differences in absorptance between red and white blood cells. However, many of the retinal structures of greatest interest have low contrast as well as low reflectivity. For example, the ganglion cells, which play the critical role of conveying signals from the retina to the brain and are the cells devastated by glaucoma, reflect more than 60 times less light back than the photoreceptors [74]. They and the Müller cell matrix within which they are entangled are necessarily transparent to avoid compromising the quantum efficiency of the photoreceptors beneath them. There are additional hurdles to increasing retinal image contrast and resolution. The maximum permissible light exposure dictated by safety considerations sets an upper bound on the signal, and a lower bound on the contrast that can be detected in the presence of noise. Another constraint is speckle arising from the interference of light backscattered from different retinal structures. Speckle can be mitigated by decreasing the temporal coherence of the illumination, but the concomitant increase in spectral bandwidth requires the correction of the eye’s chromatic aberration to reap the full resolution potential of adaptive optics. Imaging inner retinal cells poses challenges that probably cannot be met by adaptive optics alone. The best ophthalmic adaptive optics systems today already approach diffraction-limited imaging using the fully dilated pupil. Modest improvements will no doubt accrue through improved wavefront sensing, better deformable mirror technology, more careful calibration, and optimized control algorithms. Small resolution gains could be achieved by decreasing wavelength, but this approach is problematic because of the eye’s susceptibility to damage at short wavelengths [75]. The most significant gains are likely to arise by combining adaptive optics with new imaging technologies. Roorda et al. have demonstrated the value of marrying confocal scanning with adaptive optics (an AOSLO), which can optically section the retina with an axial resolution of about 100 mm, rejecting unwanted photons from deeper or shallower layers of the retina than the one of interest (see also Chapter 16) [76]. Adaptive optics not only increases resolution, it also increases the signal by coupling more light into the confocal pinhole. The combination of adaptive optics with optical coherence tomography (AO-OCT) may eventually provide even better axial resolution, as small as ~2 mm. A number of groups are actively pursuing this direction (see also Chapter 17) [77–79]. AOOCT could produce a point spread function of less than 2 mm in all three spatial dimensions, which would be smaller than the cell bodies of the smallest retinal cells. Though the diffraction limit on resolution set by the relatively low numerical aperture of the eye is a formidable barrier, it may eventually be possible to surpass it using techniques such as structured illumination [80, 81], which has successfully exceeded the diffraction limit in microscopy.

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DEVELOPMENT OF ADAPTIVE OPTICS IN VISION SCIENCE

FIGURE 1.14 Image on the left is a reflectance image of monkey cones obtained with 808-nm illumination in a high-magnification confocal scanning laser ophthalmoscope. Image on the right shows the same area but with fluorescence imaging of rhodamine dextran injected into the lateral geniculate nucleus and retrogradely transported to the ganglion cell bodies.

Additional technologies, such as polarization imaging [82, 83] and multiphoton imaging [84–86], may also prove useful. A promising approach is to use selective fluorescent markers to distinguish cells from the surrounding tissue. An exciting example of this approach is the recent demonstration of in vivo fluorescence imaging of apoptosis of single primate ganglion cells stained with annexin-5 [87]. Dan Gray and Bill Merigan have demonstrated a high-magnification SLO that can image individual ganglion cells in the living primate retina using rhodamine dextran retrogradely transported from the lateral geniculate nucleus into ganglion cell bodies (see Fig. 1.14) [88]. When fluorescence methods are combined with adaptive optics, it may be possible to quantify the extent of damage to particular classes of ganglion cells due to retinal disease, and perhaps even the rescue of these cells pursuant to pharmacological intervention. High-resolution imaging with adaptive optics will benefit greatly from the development of new and more selective molecular markers, especially those that signal specific biochemical events within individual cells. There is also a great need to identify intrinsic signals within the retina and new ways to transport extrinsic markers to their targets noninvasively. Adaptive optics retinal imaging is poised to capitalize on new tools for noninvasive, optical interrogation of the functional activity of single cells and the communication between them.

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PART TWO

WAVEFRONT MEASUREMENT AND CORRECTION

CHAPTER TWO

Aberration Structure of the Human Eye PABLO ARTAL, JUAN M. BUENO, ANTONIO GUIRAO, and PEDRO M. PRIETO Universidad de Murcia, Murcia, Spain

2.1

INTRODUCTION

The image-forming properties of any optical system, in particular the eye, can be described completely by the wave aberration. It is defi ned as the difference between the perfect (spherical) and the actual wavefronts for every point over the eye’s pupil. A perfect eye (without aberrations) forms a perfect retinal image of a point source (Airy disk). In reality, however, imperfections in the refracting ocular surfaces generate aberrations that produce a larger and, in general, asymmetric retinal image. The monochromatic aberrations of the complete eye, considered as one single imaging system, can be measured using a large variety of wavefront sensing techniques (see also Chapter 3). Every ocular surface contributes differently to the overall quality of the retinal image. The relative contribution to the eye’s aberrations of the main ocular components (the crystalline lens and the cornea) can be obtained by the combined use of ocular and corneal aberration data. The monochromatic aberrations of the eye depend on a variety of factors that will be reviewed in this chapter: in particular, accommodation, aging, and retinal eccentricity. Beyond monochromatic aberrations, in normal whitelight illumination, chromatic aberrations also play an important role that will

Adaptive Optics for Vision Science, Edited by Porter, Queener, Lin, Thorn, and Awwal Copyright © 2006 John Wiley & Sons, Inc.

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be discussed. The effect of polarization and scatter in the retinal image is also considered in the latter sections of this chapter. In adaptive optics (AO) applications, it may be useful to have a statistical description of the aberrations of the system to be corrected. In the case of the atmosphere in astronomical applications, this approach is widely used and is very successful. Although, in the case of the eye, the situation is rather different, we considered a similar statistical approach for the aberrations in a population of normal eyes. If the aberrations of the eye are known, it is possible to correct them using a wavefront correcting device that compensates for the eye’s aberrations in real time. This is a direct application of AO to the eye. In the ideal case, the system of corrector + eye becomes permanently aberration free, producing perfect retinal images. In different laboratories, AO in the eye has been demonstrated using deformable mirrors or liquid crystal spatial light modulators as corrector devices [1–5]. These systems are still laboratory prototypes that include a wavefront sensor and a corrector, allowing vision science investigators to perform visual psychophysics (see also Chapter 14) or to record high-resolution retinal images (see also Chapters 1 and 10) through near to aberration-free ocular optics. However, for practical applications, aberration correction for the eye probably needs to be performed using simpler approaches than those already demonstrated in research laboratories. The most promising options are customized ablations in refractive surgery and customized contact lenses or intraocular lenses. These cases are examples of static corrections that are permanent and fi xed, such as customized corneal ablations (see also Chapter 12) or intraocular lenses (see also Chapter 11), or are reversible and mobile, such as customized contact lenses (see also Chapter 11). These technologies will be reviewed in other chapters of this book, but their correct implementation in the eye depends heavily on the understanding of the nature of ocular aberrations.

2.2 LOCATION OF MONOCHROMATIC ABERRATIONS WITHIN THE EYE The optical aberrations in the normal eye depend on many factors and conditions. They vary from individual to individual [6], with pupil size [7, 8], the age of the subject [9–12], accommodation [13, 14], retinal eccentricity [15, 16], refractive state, and so forth. In normal young subjects at the fovea, the average root-mean-square, or RMS, wavefront error of higher order aberrations for a 5-mm pupil diameter is approximately 0.25 mm (or around l/2). To have an intuitive rough idea of the relative importance of higher order aberrations in normal eyes, in a system only affected by defocus, 0.25 mm of aberration would be approximately equivalent to 0.25 diopters (D) for a 5-mm pupil. This value is very large in the context of precision optics but rather modest in ophthalmic optics. Beyond

LOCATION OF MONOCHROMATIC ABERRATIONS WITHIN THE EYE

35

defocus and astigmatism, spherical aberration, coma, and trefoil are the most significant aberrations in normal eyes. Why is the eye affected by these aberrations? Where are the sources of the aberrations in the eye? These questions can be answered by simultaneously measuring the aberrations induced by the anterior surface of the cornea and the total ocular aberrations in the same eye. Then, the aberrations of the internal ocular optics, that is, those produced by the posterior corneal surface and the lens, can be determined. This allows one to determine the relative contributions of the different optical elements of the eye to the total wavefront. Advanced applications of AO will need or will use such a detailed tomographic structure of the aberrations. The aberrations associated with the anterior surface of the cornea can be computed from its shape as measured with corneal topography instruments. The simplest approach to calculate the anterior corneal aberrations is to obtain a “remainder lens” by subtracting the best conic surface fit to the measured cornea, and calculating the aberrations by multiplying the residual surface profi le by the refractive index difference between air and the cornea. Another option is to trace rays through the corneal surface to compute the associated aberrations [17]. Figure 2.1 shows a schematic representation of the complete procedure. The corneal elevations, representing the distance (zi) from each point of the corneal surface to a reference plane tangential to the vertex of the cornea, are represented with a Zernike polynomial expansion (see Thibos et al. for a description of these polynomials and standards for reporting [18]): N

z ( r, θ ) = ∑

n

∑a

m n

Znm ( r, θ )

n= 0 m=− n

(2.1)

using a Gram–Schmidt orthogonalization method [19]. The wave aberration associated with the corneal surface (W) is obtained as the difference in optical path length between the principal ray that passes through the center of the pupil and a marginal ray: W = nz + n ′l ′ − n ′l

(2.2)

where n, n′ are the refractive indices of air and the cornea, respectively, and z, l′, and l are the distances represented in Figure 2.1. By using the Zernike representation for the corneal surface [Eq. (2.1)], the corneal wave aberration is also obtained as another Zernike expansion: N

W ( r, θ ) = ∑

n

∑c

m n

Znm ( r, θ )

(2.3)

n= 0 m=− n

where the coefficients cmn are linear combinations of the coefficients amn [17]. On the other hand, the aberrations of the complete eye can be measured using a variety of different subjective and objective techniques. Although they are described more extensively in Chapter 3, some of them are the method

36

ABERRATION STRUCTURE OF THE HUMAN EYE

N

z(r, q) =

n

Σ Σ anm Znm (r, q)

Corneal Elevations (z) Z

n = 0 m = −n

…fit using a Zernike polynomial expansion

z r

Y

Corneal Wave Aberration

z(r, q) Y n

X

q

W = nz + n′I ′− n ′I

n′

Z r θ X

I′ I Z

W(r, q) =

N

n

Σ Σ cnm Znm (r, q)

n = 0 m = −n

FIGURE 2.1 Schematic representation of the procedure to calculate the aberrations of the anterior surface of the cornea. Corneal elevations (z) provided by a videokeratoscope are fit to an expansion of Zernike polynomials. A ray tracing procedure is used to calculate the corneal wave aberration (W) as the differences in optical path between the marginal and principal rays, also expressed as a Zernike polynomial expansion (see the text for details).

of “vernier” alignment [20], the crossed-cylinder aberroscope [6], the Foucault knife-edge technique [21], calculations from double-pass retinal images [22, 23], the pyramid sensor [24], and, probably the most widely used method today, the Shack–Hartmann wavefront sensor [25–27]. Since we can now measure the wave aberrations of the complete eye and of the cornea, the relative contributions of the different ocular surfaces to retinal image quality can be evaluated. In particular, the wave aberration of the internal ocular optics, that is, the posterior surface of the cornea plus the crystalline lens, is estimated simply by directly subtracting the corneal from the ocular aberrations. Figure 2.2 shows a schematic representation of this procedure. In a simple model, the aberrations of the internal optics (c′mn) can be obtained by direct subtraction if the Zernike coefficients for both the cornea (cmn) and the eye (c ″mn) are known. It is assumed that the changes in the wave aberration are small for different axial planes, that is, from the corneal vertex to the pupil plane.

LOCATION OF MONOCHROMATIC ABERRATIONS WITHIN THE EYE m m c′m n = c″n − cn

c″nm

cm n

Internal Wave Aberration

Ocular Wave Aberration

Corneal Wave Aberration

37

FIGURE 2.2 Schematic representation of the combination of corneal and ocular wave aberrations to estimate the wave aberration of the internal optics.

A few experiments have been performed to determine the precision of the data obtained from the combination of ocular and corneal aberrations. In addition to measuring ocular and corneal aberrations, Artal et al. also directly measured the wave aberration for the internal optics using a Shack– Hartmann wavefront sensor when the aberrations of the corneal surface were canceled by immersing the eye in saline water using swimming goggles [28]. This idea was similar to that of Young in 1801 [29] and more recently Millodot and Sivak [30], who instead used current wavefront sensing technology. The comparison of the aberrations obtained from independent measurements is an indication of the validity of the combination approach. In particular, the aberrations of the cornea, measured both directly from its shape and by subtraction of the aberrations of the internal optics from the whole eye, were found to be similar within the experimental variability. This result provided a strong proof of consistency for these types of procedures to calculate corneal and internal aberrations despite the experimental and methodological difficulties involved. The relative contribution of the aberrations of the cornea and the internal optics in different eyes has been evaluated in several recent studies. Figure 2.3 shows an example of the wave aberrations and the associated point spread functions (PSFs) for the cornea, the internal optics, and the complete eye in a normal young eye. The magnitude of aberrations is larger both in the cornea and the internal optics than in the complete eye. This indicates an active role of the lens in partially reducing the aberrations produced by the cornea. Figure 2.4 shows the Zernike terms for the aberrations of the cornea (solid symbols) and the internal optics (white symbols) for a number of young normal subjects [28]. It is remarkable that the magnitude of several aberration terms is similar for the two components, but they tend to have opposite signs. This indicates that the internal optics may play a

38

ABERRATION STRUCTURE OF THE HUMAN EYE

Anterior Cornea

Internal Optics

Complete Eye

2 1.5 1 0.5 0 −0.5 −1 −1.5 −2

FIGURE 2.3 Example of wave aberrations for the cornea, the internal optics, and the complete eye in one normal young subject. The associated point spread functions (PSFs) were calculated at the best image plane from the wave aberrations and subtend 20 min of arc of visual field. The aberrations of the internal optics partially compensate for the corneal aberrations.

FIGURE 2.4 Zernike terms for the cornea (solid symbols) and the internal optics (open symbols) for a number of normal young subjects.

significant role in compensating for the corneal aberrations in normal young eyes. This behavior may not be present in every young eye, depending on the amount of aberrations or the refractive error [31]. Determining the location of the aberrations in the eye has important implications for aberration correction in adaptive optics and also for current clinical procedures, such as wavefront-guided refractive surgery (see also Chapter 12). In normal young subjects, customized ablation should be performed based on the aberrations of the complete eye. If the ablation is based on only

LOCATION OF MONOCHROMATIC ABERRATIONS WITHIN THE EYE

1.4

39

Ocular Wave Aberration

1.2

RMS (µm)

1.0

After Correction Cornea Based

0.8 0.6 0.4 0.2 0.0

Before Correction After Correction Ocular Based

FIGURE 2.5 Aberration maps for the eye before ideal surgery (middle), after an ideal surgery based on the aberration data of the complete eye (bottom), and after an ideal surgery based on corneal aberration data alone (top). The aberration maps are placed approximately at their corresponding value of total aberration, expressed using the RMS.

the corneal aberrations, the fi nal aberrations of the eye could be larger than before the ablation. Figure 2.5 shows a schematic example: the aberration maps for the eye before an ideal surgery (middle map, “Before correction”), after an ideal surgery based on the aberration data of the complete eye (bottom map, “After correction, ocular based”), and after an ideal surgery based on corneal aberration data alone (top map, “After correction, cornea based”). The aberration maps are placed approximately at their corresponding value of total aberration (expressed using the RMS). If a perfect (ideal) ablation is performed, the eye becomes limited only by diffraction (i.e., without aberrations, represented by a flat aberration map). However, if the same perfect (ideal) ablation is performed using only corneal aberration data (i.e., correcting only the corneal aberrations), the remaining eye has the aberrations corresponding to the internal surfaces that, in many cases, can be more severe in the eye than before the treatment. Another similar example is cataract surgery after the implantation of an intraocular lens (IOL). These lenses usually have good image quality when measured on an optical bench, but the fi nal optical performance in the implanted eye was typically lower than expected [32]. The reason is that the ideal substitute of the natural lens is not a lens with the best optical performance when isolated, but one that is designed to compensate for the aberrations of the cornea [33]. This is shown schematically in Figure 2.6. Intraocular and contact lenses should ideally be

40

ABERRATION STRUCTURE OF THE HUMAN EYE

Corneal Wave Aberration IOL Wave Aberration

Ocular Wave Aberration

FIGURE 2.6 Schematic illustration of the coupling between aberrations of the cornea and the intraocular lens (see the text for details).

designed with an aberration profi le matching that of the cornea or the lens to maximize the quality of the retinal image.

2.3 TEMPORAL PROPERTIES OF ABERRATIONS: ACCOMMODATION AND AGING Ocular aberrations change with time due to a variety of factors. Perhaps the most important ocular changes are those due to accommodation [14]. To focus on objects at different distances, the crystalline lens automatically changes both its shape and power, and consequently, the ocular aberrations. Eye movements and changes in the tear fi lm and humors also produce rapid, although in general, small changes of the aberrations. Hofer et al. performed a very detailed study on the dynamics of the aberrations under normal conditions and their impact on adaptive optics correction [34]. In addition, within a different and much longer time scale, it has also been demonstrated that aberrations increase with normal aging. In this section, both the effect of accommodation and aging on aberrations are considered in some detail. 2.3.1

Effect of Accommodation on Aberrations and Their Correction

The dynamic changes of ocular aberrations during accommodation can be measured using a real-time wavefront sensor [3, 34]. Due to the continuous

TEMPORAL PROPERTIES OF ABERRATIONS: ACCOMMODATION AND AGING

41

changes of the aberrations over time, an ideal, perfectly static correction will not provide stable, aberration-free optics. For example, when an eye that is perfectly corrected for distance vision accommodates to near objects, the aberrations will change and this eye will no longer be aberration free. As an example, Figure 2.7 shows how selected aberration terms (spherical aberration and horizontal and vertical coma) change dynamically during accommodation in one subject for a pupil diameter of 5.5 mm. The actual wavefronts for 0 and 2 D are also included in the figure. The changes in spherical aberration are well correlated with accommodation in most subjects. Other aberration terms may remain relatively more stable, despite the overall changes in the wave aberration. For AO, one direct implication of the change of aberrations with accommodation is the need for dynamic corrections. Figure 2.8 shows an example: The wave aberrations are depicted for far (0 D), 1 D, and 2 D of accommodation for one normal subject (upper row), along with the residual aberrations after a perfect static correction for far vision at the same three vergences (bottom row). While the eye would become aberration free for far vision, as soon as the subject accommodates to near objects, the eye will become aberrated again. For a moderate vergence, as is the case for 2 D, the eye after correction has an amount of aberration similar to, and for some subjects maybe even larger than, those present in the eye prior to any correction. In addition, it must be pointed out that in normal viewing, the precision of accommodation is not perfect, and there is a defocus term beyond the residual aberrations. This will also prevent the eye from producing diffraction-limited retinal images. These results clearly indicate that, due to the dynamic nature of the ocular optics, a static, perfect correction (as attempted in customized refractive surgery) would not remain perfect for every condition occurring during

0.2

Vert. Coma

Microns

0.15 0.1

Near

Horiz. Coma

0.05 0

−0.05 0 −0.1

−0.15 −0.2

0.5

1

1.5

2

2.5

Far Spherical Aberration

−0.25

Accommodation (Diopters) FIGURE 2.7 Values of some aberration terms (spherical aberration and coma) as a function of accommodation for subject PA (5.5-mm pupil diameter).

42

ABERRATION STRUCTURE OF THE HUMAN EYE

Far

1D

2D

Without Correction

“Perfect” Correction for Far Vision FIGURE 2.8 Wave aberration maps in subject SM with (bottom) and without (top) perfect correction for different object vergences.

normal accommodation. Of course, real-time adaptive optics systems may cope with this problem. 2.3.2

Aging and Aberrations

Normal aging affects different aspects of the ocular optics. Elderly eyes typically experience increased light absorption by the ocular media, smaller pupil diameters (senile miosis), and nearly a complete reduction of accommodative capability. In addition, Artal et al. fi rst showed that the mean ocular modulation transfer function (MTF) in a group of older subjects was lower than the average MTF for a group of younger subjects [9]. This result, although obtained in a rather small population, suggested that the ocular aberrations increase with age. More recent measurements in a larger population show a nearly linear decline of retinal image quality with age [10]. This result suggested a significant increase in the optical aberrations of the eye with age, in agreement with other studies in which aberrations were measured directly [35, 36]. In addition, intraocular scatter also increases noticeably in older eyes [37]. Different factors could contribute to the age-related increment in aberrations, such as changes in the aberrations of the cornea [38] and the lens or their relative contributions. The increment in the corneal aberrations is too small to account for the complete reduction of retinal image quality observed with age. This suggests that mechanisms other than changes in the cornea should be primarily responsible for the increase in the ocular aberrations with age. An obvious candidate could be an increase in the aberrations of the crystalline lens caused by the continuous changes in the lens with age. As the lens grows, its dimensions, surface curvatures, and refractive index change,

CHROMATIC ABERRATIONS

Cornea

Internal

43

Whole Eye

FIGURE 2.9 Example of wave aberrations (represented module-p) for the cornea, the internal optics, and the complete eye in one normal older subject. The associated PSFs were calculated at the best image plane from the wave aberrations and subtend 20 min of arc of visual field.

altering the lens aberrations. Glasser and Campbell found a large change in the spherical aberration of excised older lenses measured in vitro [39]. Another important factor to be considered is the nature of aberration coupling within the eye. The amount of aberrations for both the cornea and the internal optics was found to be larger than for the complete eye in young subjects, indicating a significant role of the internal ocular optics in compensating for the corneal aberrations to yield an improved retinal image [28, 40]. During normal aging, the relatively small corneal changes cannot account for the degradation in the retinal image quality. However, the lens dramatically changes both its shape and effective refractive index with age, leading to changes in its aberrations. In this context, it has been shown more recently that as the aberrations of the lens change with age, this compensation is partially or even completely lost [12]. This explains the overall increase in aberration and the reduction of retinal image quality throughout the life span. As an example, Figure 2.9 shows wave aberrations (and their associated PSFs) for the cornea, internal surfaces, and the complete eye for a typical older eye. This should be compared with the same type of results shown in Figure 2.3 for a young eye. In the young eye, the corneal and internal optics aberrations had similar magnitude and shape but were opposite in sign, producing an eye with lower overall aberrations. However, in the older eye, this fi nely tuned compensation is not present.

2.4

CHROMATIC ABERRATIONS

Beyond the monochromatic aberrations, chromatic aberrations in optical systems arise from chromatic dispersion or the dependence of refractive index on wavelength. Chromatic aberrations are also present in the eye. Composed of mostly water, the chromatic behavior of the eye is frequently modeled by

44

ABERRATION STRUCTURE OF THE HUMAN EYE

considering its dispersion curve, although other more elaborate models include dispersion curves based on actual measurements in real subjects. Chromatic aberrations are traditionally divided into longitudinal chromatic aberration (LCA) and transverse chromatic aberration (TCA). The former is the variation of axial power with wavelength while the latter is the shift of the image across the image plane with wavelength. Both LCA and TCA have been widely studied in the eye. These two types of chromatic aberrations can be understood as the wavelength dependence of the lower order terms of the wave aberration: LCA is the change in focus and TCA is the change in tip/tilt or prism. Only recently have the fluctuations of higher order aberrations with wavelength been studied [41]. Chromatic aberrations limit the actual retinal image quality of the eye since the real world is usually polychromatic and, therefore, its image becomes distorted in the retina in a color-dependent fashion. Furthermore, since AO systems do not present, in general, the capability for chromatic compensation, chromatic aberrations can reduce the expected benefit of this technology for improved vision. This problem should not be present in AO systems for eye examination because the light used for imaging is usually monochromatic. Another potential impact of the chromatic aberrations on AO systems comes from the current tendency to shift the sensing light toward the infrared (IR) range. If the sensing and imaging beam wavelengths are not the same, the chromatic differences between the respective wavefronts can reduce the correction efficiency. These differences, however, are typically small except in defocus, which is very predictable and easy to calibrate. 2.4.1

Longitudinal Chromatic Aberration

Since the fi rst study published in 1947 by Wald and Griffi n [42], LCA has been extensively measured and modeled. A review of measurements prior to 1992 can be found in Thibos et al. [43]. Initially, the studies consisted of measuring the subjective best focus [42, 44]. Charman and Jennings measured LCA objectively with retinoscopic methods [45]. More recently, Marcos et al. used a subjective wavefront sensor [41]. Consistently across studies and across subjects, LCA has been found to be around 2 D across the visual spectrum (see also Fig. 10.2). Furthermore, this value seems to be stable with age [46]. It is feasible to construct a lens to compensate for the eye’s LCA and several designs have been proposed [44]. However, the achromatization process is extremely sensitive to positioning errors [47]. Small displacements of the lens or the eye produce artificial TCA that degrades image quality. According to these authors, a lateral misalignment of just 0.4 mm is enough to eliminate all of the potential benefit of the achromatization. For larger displacements, the achromatizing lens actually worsens image quality. Also, since correction is on-axis, the field of view is very limited, even with perfect alignment. Although centering can be very precise in clinical instruments and achromatization could be feasible for eye examinations, this is not the case

CHROMATIC ABERRATIONS

45

for vision correcting elements, such as spectacles or contact lenses. Therefore, achromatization for improved vision remains a difficult task. 2.4.2

Transverse Chromatic Aberration

The TCA measurement techniques typically involve vernier alignment tasks for two colors at the extremes of the visible spectrum (blue and red). Two types of TCA are usually defi ned for the eye [41]. The optical TCA is measured in Maxwellian view [48] or with a pinhole pupil [49] and therefore is related to the wavelength-dependent prism differences for the center of the pupil. The perceived TCA is measured in normal (Newtonian) view covering the whole pupil and represents the mean apparent prism differences across the pupil [50]. Differences are expected and have been experimentally found between these two types of TCA measurement, not only because they are defi ned for different-sized portions of the pupil but also because the perceived TCA is affected by the Stiles–Crawford effect. Unlike LCA, the TCA widely varies both in amount and direction among studies and subjects. Since a centered system should not present TCA on-axis, it is usually assumed that ocular TCA arises from the off-axis position of the fovea and from natural pupil misalignments [51]. However, although these two factors probably produce a part of the eye’s TCA, a recent study has demonstrated that these two effects cannot explain the TCA variability alone [51]. In terms of a Zernike expansion, both TCA and LCA can be understood as the wavelength dependence of the tip/tilt and defocus terms, respectively. Marcos et al. used a psychophysical method to measure the wave aberration up to seventh order for a series of wavelengths across the visible spectrum [41]. They found a very small increase in astigmatism, coma, spherical aberration, and higher order aberrations (expressed in microns) with wavelength. 2.4.3

Interaction Between Monochromatic and Chromatic Aberrations

In most applications of retinal imaging, monochromatic or narrow-bandwidth light sources are used or can be used. In these cases, wave aberration correction through AO or any other means should not be negatively affected by chromatic aberrations. On the contrary, when the aim is improved vision, chromatic aberrations would limit the actual improvement in retinal image quality for a polychromatic scene [41]. In fact, it has been recently argued that higher order aberrations limit the impact of chromatic aberrations in white light [11]. Therefore, perfectly correcting the monochromatic wave aberration for a single wavelength would enhance the disparities between different color components of the scene, potentially allowing the effects of chromatic aberration to be observed by the subject. As a consequence, the potential impact of chromatic aberration should be considered when AO, or any other kind of wave aberration correction, is intended to improve vision.

46

2.5

ABERRATION STRUCTURE OF THE HUMAN EYE

OFF-AXIS ABERRATIONS

The previous sections are related to on-axis aberrations for foveal vision. However, when a point object is located more than 5° off-axis (outer extent of the foveal angular size), its image is formed on the peripheral retina. The plane containing the visual axis, the point object and its peripheral image, is called the tangential plane. The sagittal plane is normal to the tangential plane and contains the principal ray. From the point of view of the light coming from the off-axis point, the incident wavefront reaches a tilted optical system and rays in the sagittal and tangential planes refract dissimilarly. Thus, the oblique incidence of light on the eye produces off-axis (oblique or peripheral) aberrations. These aberrations increase with the angle of eccentricity, so the off-axis optical performance of the eye deteriorates as one moves away from the foveal center. A reduced eye model may help to give a preliminary view of the eye’s off-axis aberrations (although more sophisticated schematic models are necessary to describe the optics in real eyes). For a spherical refracting surface, the Seidel (or fourth order) off-axis aberrations are distortion (tilt), field curvature, astigmatism, and coma. Tilt is a prismatic effect that produces either an image shift for a point object or a shape distortion in the case of the image of an extended object. Astigmatism appears because the emerging refracted wavefront has two principal curvatures that determine two focal image points: the sagittal focus, where the sagittal (or horizontal) ray fan converges, and the tangential focus, where the tangential (or vertical) ray fan converges. As a point object moves away from the optical axis, these foci map out surfaces known as sagittal and tangential surfaces, which are parabolic in the Seidel approximation. The sagittal surface is behind (more hyperopic than) the tangential surface. The Sturm interval is defi ned as the difference between the sagittal and the tangential focal lengths and defi nes the amount of oblique astigmatism, which is the difference between the sagittal and tangential dioptric powers. For a spherical refracting surface with curvature, R, separating media of indices n and n′, the sagittal and tangential focal lengths are, respectively, FS =

n ′R n ′ cos βr − n cos βi

and FT = FS cos2 βr

(2.4)

where b i and b r are the off-axis incident and refracted angles, respectively. Thus, the Sturm interval for the reduced eye is

( ) n n′

2

FS sin 2 βr

(2.5)

OFF-AXIS ABERRATIONS

47

Field curvature is a defocus for off-axis objects and implies that the best image is not formed on the paraxial image plane but on a parabolic surface called the Petzval image surface. In real eyes, the retina (which may be approximated as being a sphere with a radius between 11 and 13 mm) constitutes a curved image plane that, in most individuals, compensates for field curvature. Oblique coma in the reduced eye depends on sin b i and increases linearly with off-axis position. With the exception of the attention paid to peripheral refractive errors, the off-axis optical performance of the eye has not been as well studied and characterized as has the optical quality for foveal vision. Astigmatism and defocus have traditionally been assumed to be the main off-axis aberrations in the eye. The role of oblique astigmatism was recognized by Young back in 1801 [29] and has been studied since the last quarter of the nineteenth century. 2.5.1

Peripheral Refraction

A wide literature exists on peripheral refraction, including experimental results on oblique astigmatism and field curvature [16, 52–58], accommodation [59, 60], and the development of various eye models [61–66]. Measurements have revealed a systematic (between linear and parabolic) increase in astigmatism with the field angle. Figure 2.10(a) shows an example of the Sturm interval in a subject for offaxis vision at an eccentricity of 45°. The three images show the sagittal and tangential foci, and the circle of least confusion. The dioptric value in each figure refers to the refractive position relative to the foveal refraction. The sagittal focus lies behind the retina, the tangential focus lies in front of the retina, and the circle of least confusion is close to the retina, which, in this subject, compensates quite well for field curvature. Figure 2.10(b) shows the sagittal focus in a subject for 15°, 30°, and 45° eccentricities from the fovea. The dioptric value indicates the oblique astigmatism at each angle. Figure 2.11(a) shows the oblique astigmatism of the eye across the visual field. Dashed lines indicate the range of experimental results obtained from a survey of the literature. The solid line represents the function PA = 0.01 × βv1.5

(2.6)

where PA is the value of oblique astigmatism (in diopters) and bv is the offaxis angle (in degrees), according to a fit proposed by Lotmar and Lotmar [55]. Oblique astigmatism increases systematically with eccentricity in all subjects in the human population and may reach values of up to approximately 2 D at 20°, and more than 6 D at 60°. However, the amount and type of astigmatism vary considerably between individuals. In typical schematic eyes, the sagittal surface lies behind the retina (equivalent to a hyperopic refractive error) and the tangential surface lies in front of the retina (equivalent to a myopic error). In real eyes, the relationship between the curvature

48

ABERRATION STRUCTURE OF THE HUMAN EYE (a)

Sagittal (+2 D)

(b)

15∞(2 D)

(-1 D)

30∞(3.5 D)

Tangential (-4 D)

45∞(6 D)

FIGURE 2.10 (a) Double-pass images in a −3-D myopic subject for a point object at a retinal eccentricity of 45°. Images were recorded for equal entrance and exit pupils of 1.5 mm and for three focus positions corresponding to the sagittal focus, the circle of least confusion, and the tangential focus. For a 1.5-mm pupil, higher order aberrations are small, so images show mainly the effect of oblique astigmatism. Each image subtends 80 arcmin. (b) Sagittal focus in a subject for 15°, 30°, and 45° retinal eccentricities, from double-pass images. Within parentheses is the amount of oblique astigmatism, in diopters, measured as the difference between the sagittal and the tangential powers.

of the retina, the axial length, and the refractive power of the ocular components produces large individual variations in the degree of peripheral defocus and astigmatism. The type of astigmatism may differ depending on the foveal refraction of the subject. Plots in Figure 2.11(b) are called skiagrams and represent the variation of the sagittal and tangential refractions (relative to the foveal refraction) over the horizontal field meridian for subjects classified according to their foveal refraction [58]. On average, there is a trend for the sagittal focus to move toward the opposite side of the foveal refraction. This indicates a peripheral emmetropization for the least myopic meridian. This emmetropization does not occur, however, for the circle of least confusion, which tends to be mainly myopic in the periphery. 2.5.2

Monochromatic and Chromatic Off-Axis Aberrations

Some investigators have studied retinal image quality as a function of eccentricity by measuring line spread functions [67] or double-pass retinal images [15, 16, 68]. Off-axis optical quality has been characterized by means of the modulation transfer function [69].

OFF-AXIS ABERRATIONS

49

8

(a) Astigmatisms (diopters)

7 6 5 4 3 2 1 0 −1 20

0

40

60

Off-Axis Angle (deg) (b) Sagittal Tangential

Refractive Error (diopters)

2 0 −2 −4 −6 −8 −10

Myopic 0

10

20

Emmetropic 30

40

50 0

10

20

30

Hyperopic 40

50 0

10

20

30

40

50

Off-Axis Angle (deg)

FIGURE 2.11 (a) Interval of distribution of oblique astigmatism across the visual field in the human population (dashed lines). The solid line represents the value of astigmatism obtained using the function 0.01 × b1.5 v , where b v is the off-axis angle (in degrees). (b) Refractive position of the sagittal and the tangential surfaces across the horizontal nasal meridian for three groups of myopic, emmetropic, and hyperopic subjects.

From the fi rst results, obtained with ophthalmoscopic methods used to estimate optical performance in the periphery, it was initially impossible to extract the magnitude of the monochromatic higher order aberrations, other than astigmatism and defocus, and to identify their role in the decay of optical quality. To estimate third-order aberrations, Guirao and Artal [16] developed a procedure based on a configuration of unequal entrance and exit pupils in a double-pass apparatus [70] and a geometrical method of coma image analysis. After determining and correcting the amount of peripheral astigmatism and defocus for each field angle, double-pass images with a tiny entrance pupil and a large exit pupil (or vice versa) showed the effect of higher order

50

ABERRATION STRUCTURE OF THE HUMAN EYE

aberrations, revealing a significant amount of coma. Coma estimated in four subjects showed a nearly linear increase across the visual field. Figure 2.12(a) shows double-pass images in a subject for three eccentricities with the best correction of defocus and astigmatism, indicating how the effect of coma increases across the visual field. Navarro et al., using a laser ray-tracing technique, measured aberrations up to fi fth order in four subjects across the horizontal meridian [71]. Individual variations were found to be due to a different coupling between the foveal and the peripheral aberrations in each subject. However, for large eccentricities, there was a linear increase in the higher order aberrations (mainly coma) that dominated the intersubject variability found in the aberration patterns. Atchison and Scott, using a Shack– Hartmann sensor to measure up to sixth-order aberrations in five subjects, found a slight increase in spherical aberration with angle and a large increase in coma, reaching values between two and three wavelengths at 40° from the fovea [72]. (a)

15∞

(b)

1

30∞

4-mm Pupil

45∞

1.0

3-mm Pupil Astigmatism and Defocus Corrected

0.1

20° 30° 50°

0.01 0

10 20 30 40 50 60 Spatial Frequency (c/deg)

MTF

MTF

0.8 Fovea

0.6

Fovea

0.4

20°

0.2

20°

0.0 0

Only Defocus Corrected

10 20 30 Spatial Frequency (c/deg)

FIGURE 2.12 (a) Double-pass images recorded with astigmatism and defocus corrected for a 1.5-mm entrance and 4-mm exit pupil in a subject for different retinal eccentricities. (b) MTF for different eccentricities calculated analytically by using exponential functions after a least-square fit of experimental data collected in typical young eyes. The left panel shows MTFs for a 4-mm pupil at the fovea and at 20°, 30°, and 40° eccentricities, for foveally fi xed accommodation when all aberrations are included. The right panel shows MTFs for a 3-mm pupil at the fovea (best-corrected) and at 20° for two conditions: When peripheral defocus is corrected, and when both defocus and astigmatism are corrected.

OFF-AXIS ABERRATIONS

51

In polychromatic natural viewing conditions, chromatic aberrations may also have a significant detrimental effect on peripheral image quality. Due to poor visual acuity in the human eye in the periphery, the subjective determination of longitudinal chromatic aberration is only possible within a few degrees of the fovea. For large field angles, an objective approach is necessary. Rynders et al., by focusing a monochromatic point source of four different wavelengths and measuring the objective refraction, obtained a small gradual increase in longitudinal chromatic aberration with eccentricity, from near 1.0 D (between 632.8 and 458 nm) at the fovea to approximately 1.6 D at 40° [73]. Transverse chromatic aberration also increases with eccentricity [49]. This increase is approximately linear and occurs at a rate of about 0.25 min of arc per unit of angle. 2.5.3 Monochromatic Image Quality and Correction of Off-Axis Aberrations The pattern of decay of monochromatic retinal image quality with eccentricity is summarized in Figure 2.12(b) using the modulation transfer function. These curves were calculated with the equation MTF ( f ) = ( 1 − C ) exp ( − Af ) + C exp ( − Bf )

(2.7)

where f is the spatial frequency and A, B, and C are the set of parameters for each angle. These parameters were obtained by fitting experimental data from typical young eyes for natural viewing conditions while subjects kept a foveal fi xed viewing distance (4-mm diameter pupil) and by applying a refractive correction for a 3-mm pupil. In the left panel of Figure 2.12(b), when computing the modulation transfer functions, all off-axis aberrations (including defocus and astigmatism) were left uncorrected. The right panel of Figure 2.12(b) shows the comparison between the modulation at 20° when the oblique astigmatism is corrected or not corrected. A fi nal question, relevant to vision science, is how much visual improvement can be achieved by correcting off-axis aberrations? There have been a few attempts to correct the off-axis aberrations in normal eyes, but little or no visual improvement was found for the most part. Peripheral grating resolution thresholds were not markedly improved after improving the optics [65]. An exception is the improvement in motion detection and orientation that seems to be limited by the refractive errors in the peripheral optics of the eye [74]. Thus, the general consensus is that the optics of the eye play a relatively small role in peripheral vision, with the limits set mainly by neural factors. Moreover, it has been shown that optical blur in the periphery can reduce aliasing due to receptoral and post-receptoral sampling, so an improvement of the off-axis optical quality of the eye could potentially produce peripheral aliasing and be detrimental to visual quality [15].

52

2.6

ABERRATION STRUCTURE OF THE HUMAN EYE

STATISTICS OF ABERRATIONS IN NORMAL POPULATIONS

The knowledge of the statistical properties of the eye’s aberrations would be a useful tool in the design of AO systems or any other wavefront corrector for visual optics applications. Toward this goal, several laboratories in the last few years have embarked on the systematic measurement of ocular aberrations in relatively large healthy populations [75–77]. There is reasonable agreement in results between these studies. All three of them showed good mirror symmetry between the aberrations inherent in left and right eyes of the same individual. Also, the average magnitude (i.e., absolute value) of the aberrations was found to decrease with increasing Zernike order in accordance with earlier or smaller scale studies [6, 26]. When the Zernike coefficients were averaged (preserving sign) across the population, most of the mean values were approximately zero. Defocus and astigmatism do not follow this rule, which is already well known from refraction studies. These aberrations are typically non-zero in the population. Among the higher order terms, the one exception to this general behavior was spherical aberration, found to have usually a significantly positive mean value. Both Porter et al. [75] and Thibos et al. [77] performed a principal component analysis to investigate the effectiveness with which Zernike polynomials efficiently represent the eye’s aberrations. Principal components are statistically uncorrelated functions. As a basis, they are the most efficient way of representing a wavefront from a statistical point of view. This kind of analysis is useful to study the appropriateness of the Zernike polynomials for representing the ocular aberrations. The former study shows the Zernike basis is close to optimal in this sense since little improvement is found in mode economy when principal components are compared with Zernikes (see Figs. 6 and 8 in Porter et al. [75]). This result is corroborated by the sparseness of the correlation matrix found in the latter work (see Fig. 11 in Thibos et al. [77]). A different approach was taken by Cagigal et al. [78] using the aberration data from the Murcia optics lab study [76]. Following a course of action typically taken in astronomy, they calculated the phase variance, the power spectrum, and the structure function of the eye as a phase screen and compared it with the classical Kolmogorov model. The Kolmogorov model is widely used in astronomy to describe the wavefront disturbances produced by the atmosphere. It is based on the assumption that the wavefront perturbations are produced by local changes in the refractive index of the media traversed by the light. Although the phase variance was found not to be constant radially across the pupil, as is the case according to the Kolmogorov statistics, this behavior was attributed to the lack of knowledge of the piston and tip/tilt terms, which are not assessable in the eye with current wavefront sensing technology. On the contrary, the behavior of the variance of the higher order terms, the power spectrum, and the structure function suggested that ocular aberrations follow a Kolmogorov distribution. According to this theory, the eye behaves as a statistically homogeneous medium with fi nite inner and outer

EFFECTS OF POLARIZATION AND SCATTER

53

scales for the inhomogeneities of the refractive index. Knowing the transversal size of these two scales would allow a correct simulation of a population of phase screens (i.e., eyes) with the correct statistical behavior. Using the experimental data available, the outer scale can be estimated to be around 5 mm. The spatial resolution of the wavefront is too low to estimate the inner scale, either because of an insufficient number of Zernike modes or a limited number of sampling points.

2.7 2.7.1

EFFECTS OF POLARIZATION AND SCATTER Impact of Polarization on the Ocular Aberrations

Ocular aberrations and retinal image quality may also be affected by an intrinsic property of light, such as its polarization. This may be especially important considering the birefringent nature of the ocular optics and the eventual components in AO systems. In particular, the eye changes the polarization state of the light that passes through the ocular media and is reflected by the retina in a complex way [79, 80]. The most important ocular polarization properties are located within the cornea and the retina, with the effect of the lens being smaller [81, 82]. The cornea is strongly birefringent and the introduced retardation increases as one moves away from the corneal center [83]. In addition, retinal dichroism could be responsible for differences in the intensity of the light reaching the retina as a function of the incident polarization state [84, 85]. Any technique that involves light passing through the ocular media (in single- or double-pass configurations) might be potentially affected by these changes, and ocular imaging could also be dependent on the spatially resolved polarization properties of the eye. Although information on ocular polarization was recognized early, its interaction with any apparatus used to estimate retinal image quality has always been thought to be negligible. Most techniques used to measure ocular aberrations and retinal image quality have just been limited to include a linear polarizer in the illumination pathway, without worrying about the effect of polarization on the measurements. Moreover, since polarized light is required when using spatial light modulators as wavefront correctors in AO systems [2], the study of the impact of polarization on the calculation of ocular aberrations is important. Figure 2.13(a) shows an example of two double-pass (DP) retinal images registered with parallel and crossed linear polarizers in a young eye [86]. Bueno and Artal [87] used a DP imaging polarimeter to study the effect of polarization on the estimates of retinal image quality [88]. They reported that the DP image was strongly affected by the combination of polarization states in the incoming and outgoing pathways. On the other hand, the ocular MTF was nearly independent on the polarization state of the incident light itself. Prieto et al. used a spatially resolved refractometer to measure the effect of

54

ABERRATION STRUCTURE OF THE HUMAN EYE (a)

Parallel Polarizers

Crossed Polarizers

Polarization Angle

(b) −π/4

0

π/6

π/3

FIGURE 2.13 (a) Examples of double-pass retinal images registered with parallel (left) and crossed (right) linear polarizers in the incoming and outgoing pathways. Each image subtends 59 min of arc of visual field and was taken in a young healthy eye. (b) Wave aberration maps (calculated for a 5-mm pupil in a normal young eye) for a fi xed incoming polarization state and four independent polarization states in the registration pathway. RMS values ranged from 0.26 to 0.30 mm.

changes in the polarization state of the incoming light on the eye’s wave aberration estimates obtained in a single pass [89]. Measurements carried out on four subjects revealed that the polarization state of the incident light had little influence on the measured wave aberration. Marcos et al. used two different wavefront sensors to study the effect of different polarization configurations in the aberration measurement of 71 eyes [90]. The distribution of light in the retinal spots depended on the in- and outgoing polarization configurations, but the measured aberrations did not. This was also confi rmed by Bueno et al. using an aberro-polariscope, which combined polarimetry and wavefront sensing [91]. As an example, Figure 2.13(b) shows the similarity in wave aberrations for a 5-mm pupil in a young normal eye for four independent polarization states placed in the registration pathway. In summary, different experiments have shown that there is little impact of polarization on ocular aberration measurements. Concerning the depolarization of light in the ocular media, there are a large variety of results in the previous literature. Whereas early studies found a complete depolarization of the light reflected at the retina [92], other authors show that polarization was substantially preserved [79]. Analysis of the degree of polarization (DOP) in DP images has shown that the parameter decreases toward the skirts of the image following the averaged intensity radial profi le [93]. This confi rms that skirts of the DP images contain depo-

EFFECTS OF POLARIZATION AND SCATTER

55

larized light that has been scattered and that depolarization increases with age. Moreover, the fact that the DOP of the light located at the center of the DP image is lower in older eyes indicates that guided light also suffers from scattering processes (retinal or ocular). At the pupil plane, the DOP also decreases toward the margins, but the maximum is not always located at the center of the pupil. The location of the maximum light intensity (Stiles– Crawford peak [94]) and the area with higher DOP are close to each other, confi rming that the DOP corresponding to the directional component returning from the photoreceptors is higher than the DOP associated with the diffuse component (i.e., the surrounding area that fi lls the entire pupil). 2.7.2 Intraocular Scatter Beyond the effect of aberrations, scattered light reduces the performance of any optical system in terms of imaging, including the eye. This is quite an important issue, as wavefront correctors will not remove the effects of scattered light in the eye. In addition, wavefront sensors will not capture the higher order details related to scattered light. In particular, intraocular scatter degrades retinal image quality and diminishes both visual acuity (VA) and the contrast sensitivity function (CSF). This is also generally related to glare, which forms a veil of luminance in the eye. Although, under normal conditions, the presence of intraocular scatter in a young healthy eye is low, this may become significant with aging [37] or after refractive surgery (corneal haze). The loss of transparency of the lens (ultimately resulting in cataract formation) is the main source of scatter in the older eye. Numerous techniques have been developed to measure ocular aberrations. Most methods used to measure light scatter in the human eye are subjective and require the collaboration of the subject. However, double-pass retinal images contain information on both aberrations and intraocular scatter. Whereas the central part of the DP image is associated with aberrations, the scattered light mainly comprises the tails of the image. However, it is quite difficult to actually separate out the scattered contribution. A new objective method to measure changes in intraocular scattering is based on measuring the degree of polarization of the light emerging from the eye using a DP imaging polarimeter [95]. The ocular media and the retina are the two sources of scatter in the eye. However, a technique to separate both contributions has not yet been reported. Scatter in the ocular media is mainly due to diffusion and the loss of transparency in the cornea, the lens, and the humors. Apart from the wavelength used, scatter within the retina will mainly depend on both the fraction of light passing along the photoreceptors and the layer of the retina where the light is reflected. In the light passing through the ocular media and reflecting in the retina, two components have been classically described: a directional (or guided, which maintains polarization) and a diffuse component (or scattered, which is depolarized) [96].

56

ABERRATION STRUCTURE OF THE HUMAN EYE

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CHAPTER THREE

Wavefront Sensing and Diagnostic Uses GEUNYOUNG YOON University of Rochester, Rochester, New York

3.1

WAVEFRONT SENSORS FOR THE EYE

Various wavefront sensing techniques have been developed for the human eye [1–12]. Wavefront sensing is a key technique required to better understand the optical quality of the eye and to develop advanced vision correction methods, such as adaptive optics, customized contact lenses, and customized laser refractive surgery. It is also a necessary technique for high-resolution imaging of the retina. The most commonly used wavefront sensors are the spatially resolved refractometer, the laser ray tracing technique, and the Shack–Hartmann wavefront sensor. Wavefront sensors measure the aberrations of the entire eye generated by both corneal surfaces and the crystalline lens, whereas corneal topography can only measure the aberrations induced by the anterior or both anterior and posterior corneal surfaces. Since wavefront sensing light needs to pass through the cornea and crystalline lens, the eye’s pupil is the absolute limiting aperture of wavefront sensing, which may require pupil dilation. Wavefront sensing may be very difficult if parts of the eye have opacities, such as cataracts and corneal scars. Wavefront sensors can be categorized by whether the measurement is based on a subjective or objective method and whether the wavefront sensor measures the light going into the eye or coming out of the eye, as shown in

Adaptive Optics for Vision Science, Edited by Porter, Queener, Lin, Thorn, and Awwal Copyright © 2006 John Wiley & Sons, Inc.

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WAVEFRONT SENSING AND DIAGNOSTIC USES

Figure 3.1. However, all wavefront sensors developed for vision science and ophthalmology are based on the same principle, which is an indirect measurement of local wavefront slopes and the reconstruction of the complete wavefront by integrating these slopes, as illustrated in Figure 3.2. The relationship between wavefront slope (the fi rst derivative of the wavefront) and the spot displacement, ∆xS and ∆yS with respect to the x and y directions, can be expressed as: ∂W ( x, y ) ∆xS = ∂x F ∂W ( x, y ) ∆yS = ∂y F

FIGURE 3.1 aberration.

(3.1) (3.2)

Representative wavefront sensors developed to measure the eye’s wave

FIGURE 3.2 Principle of measuring ocular aberrations with a wavefront sensor. A wavefront sensor measures local wavefront slopes and calculates the complete wavefront shape from the measured slopes.

WAVEFRONT SENSORS FOR THE EYE

65

where F is the focal length of the focusing optics. With the measured spot displacements in the x and y directions at each sampling point, the original wavefront can be calculated using different reconstruction algorithms. In this section, the principle of operation and advantages and disadvantages for each wavefront sensor will be summarized. 3.1.1

Spatially Resolved Refractometer

The spatially resolved refractometer [5, 6] has two light sources: a fi xed source that serves as a reference (the light from which passes through the center of the pupil) and a movable source (the light from which is moved to different locations in the pupil). For each location of the movable source, the subject’s task is to change the position of the movable light source on the retina until it is aligned with the reference spot formed by the fi xed light source. The same task is repeated at different locations of the movable light source in the pupil plane. The change in the angle of incidence of the movable light required to align the spots at different locations in the pupil is a measure of the local wavefront slope. The main advantage of this type of wavefront sensor is its large dynamic range. However, the subjective method of measurement has the disadvantage that the measurement performance depends on the subject’s ability to precisely complete the task. The measurement accuracy varies given the subject’s degree of training and attention. More importantly, the measurement process is very time consuming, which makes this method inappropriate for use in a clinical environment and for real-time control, such as in an adaptive optics system. 3.1.2

Laser Ray Tracing

An objective version of the spatially resolved refractometer is the laser ray tracing [7–10] device. This technique was developed by Navarro and Losada for the living human eye [7] and consists of a sequentially delivered light pencil (i.e., a beam with a small diameter) that comes from a point object and passes through different locations in the eye’s pupil. A charge-coupled device (CCD) camera acquires images of each spot pattern focused on the retina, and the displacement of each spot from the location of the reference (or chief) ray is computed. These displacement data provide information about the local wavefront slopes. This type of wavefront sensor can have a larger dynamic range compared to that of the conventional Shack–Hartmann wavefront sensor because each spot is acquired sequentially and processed independently. In the Tscherning wavefront sensor, which uses the same measurement principle as the laser ray tracing, the entire spot array pattern is captured at once. Therefore, this Tscherning approach has a smaller dynamic range of measurement than the laser ray tracing technique.

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A major disadvantage of the laser ray tracing system is the error caused by distortions in the spot intensity distribution when the beam falls on a patch of retina that is spatially nonuniform due to blood vessels, melanin distribution, or other factors that influence retinal reflectance. In the laser ray tracing system, the beam entering the pupil forms a spot on the retina, the location of which is determined by the slope of the wave aberration at the entry point in the pupil. Then, the light reflected off the retina passes back through the eye’s optics and forms an image of the retinal spot on a light detector. It is the position of this image outside of the eye that is carefully measured to infer the wavefront slope for each point in the pupil. However, the measured position of this image could be influenced by nonuniformities in the retina, especially since each pupil entry point casts the retinal spot on a different retinal location and each retinal location will have its own distinct spatial nonuniformities. This may produce an error in the aberration measurement, especially when measuring highly aberrated eyes that have much larger local refractive errors on the cornea. 3.1.3

Shack–Hartmann Wavefront Sensor

The Shack–Hartmann wavefront sensor [11, 12] contains a lenslet array that consists of a two-dimensional array of a few hundred lenslets, all with the same diameter and the same focal length. Typical lenslet diameters range from about 100 to 600 mm. Typical focal lengths range from a few millimeters to about 30 mm. The light reflected from a laser beacon projected on the retina is distorted by the wave aberration of the eye. The reflected light is then spatially sampled into many individual beams by the lenslet array and forms multiple spots in the focal plane of the lenslets, as shown in Figure 3.3. A CCD camera placed in the focal plane of the lenslet array records the spot array pattern for wavefront calculation. For a perfect eye (i.e., an aberrationfree or diffraction-limited eye), light reflected from the retina emerges from the pupil as a collimated beam, and the Shack–Hartmann spots are formed along the optical axis of each lenslet, resulting in a regularly spaced grid of spots in the focal plane of the lenslet array. In contrast, individual spots formed by an aberrated eye, which distorts the wavefront of the light passing through the eye’s optics, are displaced from the optical axis of each lenslet. The displacement of each spot is proportional to the wavefront slope at the location of that lenslet in the pupil and is used to reconstruct the wave aberration of the eye. This type of wavefront sensor has advantages common to those of laser ray tracing, such as the fact that both techniques are objective and can be operated in real time. These advantages allow for the automatic measurement of ocular aberrations, which is essential for routine clinical testing. A main difference between the Shack–Hartmann wavefront sensor and laser ray tracing is the method used to acquire the spot image. In contrast with laser ray tracing, where the incident beam is scanned sequentially over

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67

FIGURE 3.3 Schematic diagram of the measurement principle of a Shack–Hartmann wavefront sensor. Two Shack–Hartmann images for perfect (left) and real (right) eyes are also shown.

the entrance pupil to measure light going into the eye, the Shack–Hartmann wavefront sensor measures light coming out of the eye using a parallel process to acquire multiple spots over the exit pupil. By comparing the aberrations measured using laser ray tracing and other psychophysical methods with those from a Shack–Hartmann wavefront sensor in the same eyes, several investigators have demonstrated that there is no significant difference in the eye’s measured aberrations when using either ingoing or outgoing light [13, 14]. Since there is only one spot formed on the retina when using the Shack– Hartmann wavefront sensor, any intensity variation of the spot does not generate a measurement error because individual spots created by the lenslet array are equally affected. If this retinal location has nonuniform reflectance, the distortion in the light distribution on the detector will be essentially the same for all the pupil locations sampled, and the wave aberration measurement will not be corrupted by the nonuniformity. If the wavefront shape within one lenslet significantly varies, the spot pattern formed by that lenslet can be blurred and cause an error in estimating the centroid of the spot. This typically does not happen with lenslets that are a few hundred microns in diameter because such a small sized aperture is almost diffraction-limited after passing through the eye’s optics, provided the tear fi lm thickness is uniform. However, the major disadvantage of the Shack– Hartmann device, which will be addressed in this chapter, is its relatively small dynamic range that is limited by the lenslet spacing (or number of lenslets across the pupil) and the focal length of the lenslet array. Although this chapter will focus on optimization strategies for a Shack–Hartmann wave-

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front sensor, most of the design concepts can be applied to different types of wavefront sensors based on wavefront slope measurements.

3.2

OPTIMIZING A SHACK–HARTMANN WAVEFRONT SENSOR

There are a few design parameters that need to be properly determined. These parameters can be different depending on what application the wavefront sensor will be used. What makes optimizing the wavefront sensor difficult is that these parameters are not typically independent from each other. In other words, there are trade-offs between these parameters. The four most important parameters—the number of lenslets (or lenslet diameter), dynamic range, measurement sensitivity, and the focal length of the lenslet array—are discussed in this section. 3.2.1 Number of Lenslets Versus Number of Zernike Coefficients As will be discussed in Chapter 6, the conversion matrix used to reconstruct the wave aberration consists of the fi rst derivatives of individual Zernike polynomials with respect to the lateral directions, x and y, at each lenslet location. A method of singular value decomposition (SVD) is typically used to calculate its inverse matrix. However, this method becomes inaccurate if the reconstruction algorithm calculates too many Zernike polynomials compared to the number of lenslets and causes unpredictable errors in the reconstructed wave aberration. Therefore, it is important to understand the relationship between the number of lenslets and the maximum number of Zernike coefficients that can be calculated reliably with the reconstruction algorithm. Figure 3.4 shows the maximum number of Zernike coefficients that can be accurately calculated as a function of the number of lenslets sampling the pupil. Wave aberrations originally measured through tenth-order Zernike coefficients with a large number of lenslets (217) were theoretically sampled with different numbers of lenslets. Different numbers of Zernike coefficients were then calculated using the algorithm described in Chapter 6. The calculated coefficients were compared to the coefficients originally measured with the large number of lenlsets. The maximum number of Zernike coefficients that could be used to adequately represent the original wavefront was determined by minimizing the difference in root-mean-square (RMS) wavefront error between the resampled and recalculated wavefront and the originally measured wavefront. As Figure 3.4 illustrates, the maximum number of coefficients that the reconstruction algorithm can reliably calculate is approximately the same as the number of lenslets. For example, if Zernike coefficients up to the tenth order (corresponding to 63 total coefficients without piston, tip, and tilt, which cannot be measured accurately using Shack–Hartmann

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69

FIGURE 3.4 Maximum number of Zernike modes that can be calculated reliably for a given number of sampling points. The dashed line represents a slope of 1.

wavefront sensors) need to be calculated, at least 63 lenslets (sampling points) are required for a reliable reconstruction. When selecting the number of lenslets to use to sample the pupil, it is important to consider the total number of Zernike coefficients needed to effectively represent the true wave aberration. As shown in Figure 3.5, the true higher order aberration profi le can be represented more precisely when sampling the same area with a larger number of lenslets and consequently by calculating more Zernike coefficients. The required number of Zernike coefficients is related to the population of eyes to be measured with the wavefront sensor. For a population of normal eyes with no pathology (i.e., eyes without keratoconus or corneal transplantation), the majority of higher order aberrations are typically included in Zernike modes up to and including eighthorder Zernike coefficients, corresponding to 42 coefficients in total (excluding piston, tip, and tilt) [12]. The number of Zernike coefficients (excluding piston, tip, and tilt), J, through a given Zernike order, N, can be computed by the following equation: J=

( N + 1) ( N + 2 ) −3 2

(3.3)

This indicates that at least 42 lenslets are needed to reliably measure the higher order aberrations in these eyes. If a 6-mm pupil is considered, the

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FIGURE 3.5 Relationship between the reliability of wavefront representation and the number of lenslets (or lenslet diameter). More lenslets provide a more precise representation of the wavefront. (From Yoon et al. [21]. Reprinted with permission from SLACK Inc.)

TABLE 3.1 Total Number of Zernike Coefficients for Different Lenslet Diameters (6-mm Pupil) Lenslet Diameter (mm)

Maximum Number of Zernike Coefficients

200 300 400 500 600

657 (~ up to 34th order) 277 (~ up to 22th order) 145 (~ up to 15th order) 89 (~ up to 12th order) 61 (~ up to 9th order)

maximum size of each lenslet would be approximately 0.65 mm in diameter (providing a total of 49 lenslets in the pupil). On the other hand, highly aberrated eyes or eyes with abnormal and larger amounts of higher order aberrations generally need more Zernike coefficients to adequately represent the wavefront. In other applications of wavefront sensing, such as an adaptive optics system, an additional number of lenslets depending on the number of actuators and the control agorithm may be required for more effective deformable mirror actuator control. Table 3.1 summarizes the maximum number of Zernike coefficients that can be represented for different lenslet diameters over a 6-mm pupil. The highest number of approximate Zernike orders that can be calculated for a given number of lenslets are also included in the table.

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The maximum number of Zernike coefficients shown in Table 3.1 is based on the case where there is no other effect, such as various noise sources that may exist in an actual wavefront sensor, temporal variations in the tear fi lm surface profi le, and a partial occlusion of peripheral lenslets by the pupil boundary. Therefore, it is often desirable to oversample the wavefront with more lenslets than those specified in the table. 3.2.2

Trade-off Between Dynamic Range and Measurement Sensitivity

As mentioned in Section 3.1, a Shack–Hartmann wavefront sensor reconstructs the wave aberration by measuring the displacement of the focused Shack–Hartmann spots from a reference spot position, corresponding to the slope of the wavefront. In the Shack–Hartmann wavefront sensor, the relationship between the wavefront slope, q, and the spot displacement, ∆s, can be expressed as: ∆s = Fθ

(3.4)

where F is the focal length of the lenslet. If the focal length is constant, larger wavefront slopes will cause larger displacements of the spot. Measurement accuracy of the wavefront sensor is directly related to the precision of the centroid algorithm, that is, to the measurement precision of ∆s. A conventional centroid algorithm will fail to fi nd the correct centers of the spots if the spots partially overlap or fall outside of the virtual subaperture (located directly behind the lenslet) on the photodetector array (see Fig. 3.6) unless a special algorithm is implemented. These factors limit the dynamic range of

FIGURE 3.6 Limitations in a Shack–Hartmann wavefront sensor: multiple spots, overlapped spots, and spot crossover. (From Yoon et al. [21]. Reprinted with permission from SLACK Inc.)

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the wavefront sensor. As diagrammed in Figure 3.7, the dynamic range, q max, is the wavefront slope when the Shack–Hartmann spot is displaced by the maximum distance, ∆smax, within the subaperture, which is equal to one-half of the lenslet diameter for a given focal length lenslet array (when ignoring spot size). It is important to note that spot size needs to be considered in Eqs. (3.5) and (3.6) when the f-number of the lenslet (defi ned as the ratio of the focal length to the lenslet diameter) is relatively large. Equation (3.4) can be rewritten to defi ne the dynamic range of the sensor as: ∆smax F d = 2F

θ max =

(3.5) (3.6)

where d is the lenslet diameter. To increase the dynamic range of the sensor, a larger lenslet diameter and/or a shorter focal length lenslet needs to be used. Assuming that the lenslet diameter is determined by the required number of Zernike coefficients (as discussed in the previous section), the only way to increase the dynamic range is to shorten the focal length of the lenslet.

FIGURE 3.7 Trade-off between the dynamic range and measurement sensitivity of a Shack–Hartmann wavefront sensor: Increasing the dynamic range results in a decrease in measurement sensitivity and vice versa.

OPTIMIZING A SHACK-HARTMANN WAVEFRONT SENSOR

73

However, if the focal length is too short, this causes a decrease in measurement sensitivity. The term measurement sensitivity can be described as the minimum wavefront slope, q min, that the wavefront sensor can measure. Equation (3.4) can also be rewritten to defi ne the measurement sensitivity as:

θ min =

∆smin F

(3.7)

where ∆smin is the minimum detectable spot displacement. Typically ∆smin is determined by the pixel size of the photodetector, the accuracy of the centroid algorithm and the signal-to-noise ratio of the sensor. Therefore, a longer focal length lenslet is needed for better sensitivity if ∆smin is constant. By combining Eqs. (3.6) and (3.7), the relationship between dynamic range, q max, and measurement sensitivity, q min, can be described as:

θ min =

2∆sminθ max d

(3.8)

This equation defi nes the trade-off between the dynamic range and sensitivity of a Shack–Hartmann wavefront sensor. These quantities are inversely related, that is, increasing the dynamic range of the wavefront sensor results in a decrease in its sensitivity (i.e., an increase in q min) and vice versa for a constant lenslet spacing, d. 3.2.3

Focal Length of the Lenslet Array

As discussed in the previous section, understanding the trade-off between the dynamic range and measurement sensitivity in a Shack–Hartmann wavefront sensor is critical when selecting the focal length of the lenslet array. Using the longest focal length that meets both the dynamic range and measurement sensitivity requirements is ideal. To illustrate this, the fraction of subjects whose wave aberrations could be measured accurately using lenslet arrays of different focal lengths and different lenslet diameters was calculated based on a large population of 333 normal preoperative, laser refractive surgery eyes. The results for two different lenslet diameters (200 and 400 mm) are shown in Figure 3.8. The solid, dashed, and dotted lines represent the lenslet focal length required to measure a certain percentage of eyes with no aberration correction, a defocus correction, and a second-order (defocus and astigmatism) correction, respectively. A smaller lenslet diameter requires a shorter focal length to cover a certain fraction of subjects. For both lenslet sizes, precompensating for defocus (dashed line) allows for the use of longer focal length lenslets compared to that with no precompensation (solid line). In addition to precompensating for defocus, correcting astigmatism (dotted line) slightly improves the focal length requirement. However, the improvement is very small, especially in the case of the 200-mm lenslet diameter.

74

WAVEFRONT SENSING AND DIAGNOSTIC USES

Measurable Fraction of Subjects (%)

7-mm Pupil, n = 333 Normal Eyes 100

100 Lenslet Diameter = 200 mm

80

Second–Order Aberrations Corrected

60

Lenslet Diameter = 400 mm

80 60 40

40 20

Defocus Corrected

No Corrected

20

0 0

10

20

30

40

50

0 60 0

10

20

30

40

50

60

Focal Length of Lenslet (mm)

FIGURE 3.8 Measurable fraction of subjects that can be measured adequately when the wave aberration is sampled with different focal length lenslet arrays. Two different lenslet diameters, 200 mm (left) and 400 mm (right), were assumed. Solid, dashed, and dotted lines represent cases with no aberration correction, a defocus-only correction, and a defocus and astigmatism correction, respectively.

An appropriate focal length lenslet for measuring most normal eyes can be chosen from these plots when the lenslet diameter is given. 3.2.4 Increasing the Dynamic Range of a Wavefront Sensor Without Losing Measurement Sensitivity It is also important to choose the proper lenslet focal length to provide enough measurement sensitivity for the given application. It could be possible that the lenslet focal length is too short to measure small amounts of the wave aberration while adequate dynamic range can be obtained. For example, a much shorter focal length lenslet is required for eyes with abnormal corneal conditions (such as keratoconic and postcorneal transplant eyes) than for normal eyes, simply due to the presence of larger amounts of aberrations inherent in abnormal eyes. However, appropriate measurement sensitivity may not be obtained in this case. Therefore, it becomes more important to have the capability of increasing the wavefront sensor’s dynamic range without sacrificing its measurement sensitivity. A few methods have been proposed to increase the dynamic range of a Shack–Hartmann wavefront sensor without losing measurement sensitivity. One of the simple ways is to increase the magnification of the pupil size at the lenslet array. Since the wavefront is expanded across the magnified pupil without changing the magnitude of the wavefront, magnifying the pupil

CALIBRATION OF A WAVEFRONT SENSOR

75

reduces the averaged wavefront slope, which generates smaller spot displacements. However, a larger CCD camera is required to capture the spot array pattern over the magnified pupil, which increases the cost of the wavefront sensor. Ophthalmic trial lenses (spherical and cylindrical lenses) can be used to precompensate for the majority of the eye’s aberration, which are defocus and astigmatism. In this method, it is important to know the exact position and power of the trial lenses used for the precompensation if those lenses are not inserted in the pupil plane. This precompensation can also be performed with a software algorithm that allows the user to reposition individual centroid boxes irregularly according to an estimated amount of defocus and astigmatism. Computer algorithms, including the unwrapping [15] and iterative spline fitting methods [16, 17], have been suggested as ways of overcoming this limitation. These algorithms are capable of reassigning focused spots to their corresponding lenslets, although there is still a limitation on the maximum measurable wavefront curvature when crossover between adjacent spots occurs. Other methods use hardware to resolve the problem described above. One method, by Lindlein et al., uses a spatial light modulator to selectively block and unblock different parts of the wavefront in an effort to resolve ambiguities relating to the presence of multiple spots within a virtual subaperture [18]. This method fails when overlapping spots exist. An “adaptive” Shack–Hartmann sensor has also been proposed, where the static lenslet array is replaced with a liquid crystal display (LCD) device [19]. The LCD can be programmed to generate an array of Fresnel microlenses with different focal lengths. Due to the pixelated structure of the Fresnel lenses, light diffracted into the higher diffraction orders may reduce the reliability of determining centroid locations, affecting higher order aberration measurements. Another method that has recently been proposed is to increase the size of the virtual subaperture by using a translatable plate with discrete clear apertures that are the same size as the lenslets [20]. For example, if the translatable plate selectively blocks every other lenslet in both the horizontal and vertical directions, this increases the virtual subaperture by a factor of 2, resulting in a twofold increase in dynamic range. The translatable plate will then need to be translated to capture spots that are blocked in the previous frame. In a two-dimensional space, two translations in both the horizontal and vertical directions are required to acquire a complete set of Shack–Hartmann spots.

3.3

CALIBRATION OF A WAVEFRONT SENSOR

In calibrating a wavefront sensor, there are two main parts: software and hardware calibration. The software calibration should be done to confi rm that the wavefront reconstruction algorithm correctly calculates the expected Zernike coefficients from the spot displacement data. This process also includes determining the accuracy of the centroid algorithm to precisely detect the center of each spot. Once the software is calibrated, the hardware

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needs to be calibrated by measuring a known wave aberration. This process calibrates systematic errors in the wavefront sensor (i.e., system aberrations generated by various optical components, such as lenses and beam splitters, misalignments, and any manufacturing errors in the lenslet array’s focal length and spacing). Detailed calibration procedures are described below. 3.3.1

Reconstruction Algorithm

Figure 3.9 shows a flowchart illustrating how to calibrate the reconstruction algorithm. Although this calibration process is designed for calibrating a Shack–Hartmann type of wavefront sensor, it can easily be used for other types of wavefront sensors, with some minor modifications. The basic idea of this process is to run the reconstruction algorithm on simulated data and check if the Zernike coefficients output from the reconstruction algorithm are the same as the input coefficients that were used to generate the simulated data. Any combination of Zernike coefficients for lower and higher order aberrations that reliably represent the wave aberration in an optical system can be assumed as an input. Using these coefficients, a wave aberration, W(x, y), can be calculated as:

FIGURE 3.9 Flowchart illustrating the calibration process for the reconstruction algorithm using a simulated spot array pattern.

CALIBRATION OF A WAVEFRONT SENSOR

77

J

W ( x, y ) = ∑ c j Z j ( x, y )

(3.9)

j =3

where cj and Zj (x, y) are the Zernike coefficients and their corresponding Zernike polynomials, respectively, and J is the total number of Zernike coefficients to be included in calculating the wave aberration. This input wave aberration, which fi lls the entire pupil, can then be divided into subapertures that have the same diameter as each lenslet. The local wave aberration, Wk (x, y), for each subaperture is used to build the subpupil function, pk (x, y), defi ned as: 2π pk ( x, y ) = exp  −i Wk ( x, y )    λ

(3.10)

where l is the wavelength of the light source used by the wavefront sensor. Then, a Fourier transformation is applied to each subpupil function, pk (x,y), to simulate the spot pattern produced for each individual lenslet. This yields the ideal Shack–Hartmann spot array pattern simulated from the known set of Zernike coefficients, in the absence of noise. Noise sources that can be measured or statistically modeled can be added to the simulated spot array pattern to test the reliability of the centroiding algorithm to the noise. Using this simulated image, the centroid and reconstruction algorithms are applied to detect the center of each spot and to calculate the output Zernike coefficients. Note that the total number of calculated Zernike coefficients cannot be larger than the maximum number of Zernike coefficients that can be calculated reliably with the given number of lenslets (as discussed in Section 3.2.1). The output coefficients are compared with the ones used to generate the simulated spot pattern. If there is no error in the reconstruction processes, both the signs and magnitudes of the output coefficients should be the same as the input ones. This calibration process is also useful for confi rming that the coordinate system and sign convention of the reconstruction algorithm are correct.

3.3.2

System Aberrations

Although the reconstruction algorithm may be well calibrated, there are other factors in an actual wavefront sensor that may yield inaccuracies in the measured wave aberration. Potential error sources include (1) a misalignment of optical components, such as relay lenses used in an optical system, (2) residual aberrations of the optics themselves, and (3) manufacturing errors in the lenslet array. These factors can significantly affect measurement performance if they are not correctly calibrated. Calibrating lenslet array parameters, such as its focal length and spacing, is especially important for achieving precise measurements of the aberrations.

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A good technique for calibrating the aberrations inherent in a wavefront sensor is to measure an aberration profi le with both an interferometer, one of the most reliable methods available to measure aberrations, and with the developed wavefront sensor. Figure 3.10 shows a schematic diagram of an optical layout for measuring a wave aberration simultaneously with these two instruments. This setup consists of a set of relay optics that produces another plane conjugate with the pupil of the optics to be measured. The collimated beam from the illumination path of an interferometer passes through the relay optics and is reflected by either an aberration generator or a flat mirror. The reflected beam then returns to the two sensors, via a beamsplitter, after passing through the same relay optics. In practice, aberrations are present in all of the optical components, including the lenslet array, of the optical system. Each wavefront sensor also measures slightly different aberrations because light is reflected off of the beamsplitter for the interferometer, as opposed to being transmitted through the beamsplitter for the wavefront sensor. Therefore, the aberrations of the entire system fi rst need to be measured using a flat mirror. Following this measurement, the flat mirror is then replaced with an aberration generator that has an irregular surface profi le that aberrates the wavefront of the incident collimated beam. This aberration generator can be a custom optic or a deformable mirror. The distorted wavefront is then measured with the two sensors. However, the measured aberration includes both the aberrations from the aberration generator and the aberrations from the system previously measured with the flat mirror. Therefore, the true aberration of the aberration generator is the difference between the measured aberrations from the aber-

FIGURE 3.10 Schematic diagram of an optical setup to calibrate the measurement performance of a Shack–Hartmann wavefront sensor by comparing it to an interferometric method. The system aberrations measured with a flat mirror in each sensor should be subtracted from the measured aberration of the aberration generator.

REFERENCES

79

ration generator (measured with each wavefront sensor) and the system aberrations (measured with the flat mirror). The true aberration from the interferometer measurements is used as the reference to evaluate the measurement results from the Shack–Hartmann wavefront sensor. When the CCD camera used to capture the spot array pattern is precisely place at the focus of the lenslet array, there is typically a discrepancy between the two sensors in the magnitude of the individual Zernike coefficients. The signs of each coefficient should be the same if the coordinate systems used for both instruments are the same and the calibration of the reconstruction algorithm (described in Section 3.3.1) was successful. If the magnitude of each Shack–Hartmann Zernike coefficient tends to be larger than the interferometric measurement, the focal length of the lenslet array used in the reconstruction algorithm needs to become longer by the ratio of the coefficient magnitude from the Shack–Hartmann sensor to that from the interferometer.

3.4

SUMMARY

In this chapter, we reviewed various wavefront sensing techniques and their advantages and disadvantages. When developing a wavefront sensor, it is very important to carefully consider certain parameters, such as the number of sampling points and the dynamic range that are required, in order to develop the wavefront sensor that works best for each particular application. There are also practical issues to improve the reliability of the measured aberration. Those include calibrating the eye’s chromatic aberration, the effect of accommodation on the higher order aberrations, the temporal variability of the aberrations and the potential shift of the pupil center between dilated and natural pupil conditions. This effort results in the most effective and reliable outcome when using wavefront sensing techniques. There has been a significant increase in the interest and application of wavefront sensing techniques in the vision science and ophthalmological communities. It is becoming a more critical tool that allows us to correct higher order aberrations to improve visual performance using laser refractive surgery and customized optics (described in Chapters 11 and 12) and to obtain highresolution images of the retina (described in Chapters 10, 15, 16, and 17).

REFERENCES 1. Howland HC, Howland B. A Subjective Method for the Measurement of Monochromatic Aberrations of the Eye. J. Opt. Soc. Am. 1977; 67: 1508–1518. 2. Iglesias I, Lopez-Gil N, Artal P. Reconstruction of the Point-Spread Function of the Human Eye from Two Double-Pass Retinal Images by Phase-Retrieval Algorithms. J. Opt. Soc. Am. A. 1998; 15: 326–339.

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3. Smirnov MS. Measurement of the Wave Aberration of the Human Eye. Biophysics. 1961; 6: 687–703. 4. Walsh G, Charman WN, Howland HC. Objective Technique for the Determination of Monochromatic Aberrations of the Human Eye. J. Opt. Soc. Am. A. 1984; 1: 987–992. 5. Webb RH, Penney CM, Thompson KP. Measurement of Ocular Local Wave-front Distortion with a Spatially Resolved Refractometer. Appl. Opt. 1992; 31: 3678–3686. 6. He JC, Marcos S, Webb RH, Burns SA. Measurement of the Wave-front Aberration of the Eye by a Fast Psychophysical Procedure. J. Opt. Soc. Am. A. 1998; 15: 2449–2456. 7. Navarro R, Losada MA. Aberrations and Relative Efficiency of Light Pencils in the Living Human Eye. Optom. Vis. Sci. 1997; 74: 540–547. 8. Navarro R, Moreno E, Dorronsoro C. Monochromatic Aberrations and PointSpread Functions of the Human Eye across the Visual Field. J. Opt. Soc. Am. A. 1998; 15: 2522–2529. 9. Pallikaris IG, Panagopoulou SI, Molebny VV. Evaluation of TRACEY Technology for Total Eye Refraction Mapping. Reproducibility Tests. Invest. Ophthalmol. Vis. Sci. 2000; 41: S301. 10. Pallikaris IG, Panagopoulou SI, Siganos CS, Molebny VV. Objective Measurement of Wavefront Aberrations with and without Accommodation. J. Refract. Surg. 2001; 17: S602–S607. 11. Liang J, Grimm B, Goelz S, Bille J. Objective Measurement of Wave Aberrations of the Human Eye with the Use of a Hartmann–Shack Wave-front Sensor. J. Opt. Soc. Am. A. 1994; 11: 1949–1957. 12. Liang J, Williams DR. Aberrations and Retinal Image Quality of the Normal Human Eye. J. Opt. Soc. Am. A. 1997; 14: 2873–2883. 13. Moreno-Barriuso E, Merayo-Lloves JM, Marcos S, Navarro R. Ocular Aberrations after Refractive Surgery Measured with a Laser Ray Tracing Technique. Invest. Ophthalmol. Vis. Sci. 2000; 41: S303. 14. Salmon TO, Thibos LN, Bradley A. Comparison of the Eye’s Wave-front Aberration Measured Psychophysically and with the Shack–Hartmann Wave-front Sensor. J. Opt. Soc. Am. A. 1998; 15: 2457–2465. 15. Pfund J, Lindlein N, Schwider J. Dynamic Range Expansion of a Shack– Hartmann Sensor by Use of a Modified Unwrapping Algorithm. Opt. Lett. 1998; 23: 995–997. 16. Groening S, Sick B, Donner K, et al. Wave-front Reconstruction with a Shack– Hartmann Sensor with an Iterative Spline Fitting Method. Appl. Opt. 2000; 39: 561–567. 17. Unsbo P, Franzn LK, Gustafsson J. Increased Dynamic Range of a Hartmann– Shack Sensor by B-spline Extrapolation: Measurement of Large Aberrations in the Human Eye. Invest. Ophthalmol. Vis. Sci. 2002; 43: U465. 18. Lindlein N, Pfund J, Schwider J. Algorithm for Expanding the Dynamic Range of a Shack–Hartmann Sensor by Using a Spatial Light Modulator Array. Opt. Eng. 2001; 40: 837–840. 19. Seifert L, Liesener J, Tiziani H. The Adaptive Shack–Hartmann Sensor. Opt. Comm. 2003; 216: 313–319.

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20. Pantanelli SM, Yoon G, Jeong T, MacRae S. Large Dynamic Range Shack– Hartmann Wavefront Sensor for Highly Aberrated Eyes. Invest. Ophthalmol. Vis. Sci. 2003; 44: U1. 21. Yoon G, Pantanelli S, MacRae SM. Optimizing the Shack–Hartmann wavefront sensor. In: Krueger RR, Applegate RA, MacRae SM, eds. Wavefront Customized Visual Correction: The Quest for Super Vision II. Thorofare, NJ: SLACK, 2004, pp. 131–136.

CHAPTER FOUR

Wavefront Correctors for Vision Science NATHAN DOBLE Iris AO Inc., Berkeley, California DONALD T. MILLER Indiana University, Bloomington, Indiana

4.1

INTRODUCTION

Aberrations of the ocular media and diffraction generated by the fi nite size of the eye’s pupil fundamentally limit our ability to resolve fi ne retinal structure when looking into the eye. Conversely, with the light path reversed, diffraction and aberrations limit visual acuity to well below the spatial bandwidth imposed by the neural visual system, such as that dictated by the sampling of the photoreceptor mosaic. Conventional corrective methods, such as spectacles, contact lenses, and refractive surgery, provide a static amelioration of prism, sphere, and cylinder, which correspond to the lower order Zernike aberrations of tilt, defocus, and astigmatism. Image quality in the eye, however, can be significantly increased by dilating the pupil to minimize diffraction and subsequently correcting the ocular aberrations across the large pupil using, for example, an adaptive optics (AO) system. In recent years, AO has been successfully applied to correct both the lower and higher order ocular aberrations in a variety of retinal camera architectures. These include conventional fundus cameras [1–3], confocal scanning laser ophthalmoscopes (cSLOs) [4], and optical coherence tomography

Adaptive Optics for Vision Science, Edited by Porter, Queener, Lin, Thorn, and Awwal Copyright © 2006 John Wiley & Sons, Inc.

83

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(OCT) [5–7]. The increase in contrast and resolution permits observation of retinal structure at the single-cell level, which could not otherwise be seen in the living eye. AO has also been used to improve vision by controlling the type and amount of aberrations to which the retina is exposed. Specifically, AO provides a means to directly assess the visual impact of individual types of aberration [8, 9] and allows patients to experience beforehand the predicted visual benefit of invasive surgical procedures, such as refractive surgery [10, 11]. In general, the ability of AO to improve resolution in the eye makes it a key enabling technology for probing the microscopic living retina and enhancing vision. The extent to which AO can effectively improve resolution, however, fundamentally depends on its ability to accurately measure, track, and correct the ocular aberrations. This chapter focuses on the last step, correction. While all of the steps are critical, the performance limiter of current AO systems for vision science appears to be the wavefront corrector. This limitation, coupled with the expense of wavefront correctors ($7000 to >$100,000), motivates the need for discussion of this device. This chapter attempts to bring together wavefront corrector information that is important for the design of AO systems for vision science. Much of this is not readily found in the adaptive optics literature, which is heavily centered on atmospheric applications. Section 4.2 introduces the principal components of an AO system. Section 4.3 presents the primary types of wavefront correctors, and Section 4.4 surveys versions that have been applied or are in the process of being applied to vision science AO systems. Section 4.5 contains theoretical performance predictions for the most common types of wavefront correctors. The section summarizes results already in the literature with new predictions for corrector types not yet evaluated, all within the mathematical framework of Fourier optics. Predictions are based on modeling the correctors’ principal operation in conjunction with measured wave aberrations collected on two large populations. This theoretical analysis extends that already reported for segmented piston-only devices [12].

4.2

PRINCIPAL COMPONENTS OF AN AO SYSTEM

The late 1980s saw the fi rst AO wavefront corrector, a membrane mirror, applied to the human eye for the correction of astigmatism, which was obtained by a conventional refraction [13]. This was followed in the early 1990s by the fi rst Shack–Hartmann wavefront sensor (SHWS) [14] applied to the eye [15]. The fi rst complete AO system, which successfully corrected the eye’s most significant higher order aberrations, was built in the mid-1990s [1]. A conceptual schematic of AO as employed for retinal imaging and vision testing is shown in Figure 4.1. As illustrated, AO systems consist of three main components:

PRINCIPAL COMPONENTS OF AN AO SYSTEM

85

FIGURE 4.1 Basic schematic of an AO system for the eye. Light from a laser beacon (not shown) is focused onto the retina, some of which reflects and fi lls the dilated pupil of the eye. Light exiting the eye reflects off the wavefront corrector (deformable mirror) and is directed by a beamsplitter into the wavefront sensor. Centroid positions are obtained from the raw sensor data and are used in the wavefront reconstructor to determine the appropriate drive voltages for the wavefront corrector. The wavefront sensing and the corrector control process repeat in a continuous fashion. The system is considered a real-time, closed-loop AO system if it operates sufficiently fast to track temporal changes in the ocular aberrations. Once a good wavefront correction is obtained, images are collected of the retina using the science camera, or vision experiments are conducted with the visual stimulus. The illumination source for the science camera is not shown. (Figure courtesy of M. Helmbrecht, Iris AO, Inc.)

• The wavefront sensor measures the optical aberrations in the pupil plane of the eye. While there are numerous types of wavefront sensors, the Shack–Hartmann wavefront sensor is almost universally used for the eye [15, 16]. More recently, other alternatives have been investigated, including pyramid sensing [17] and interferometry [18]. Chapter 3 provides a detailed discussion of the wavefront sensing techniques. • The control computer converts the raw output of the wavefront sensor into voltage commands that are sent to the wavefront corrector. For the eye, the computational requirements to operate in a real-time, closedloop fashion are modest and can be achieved with a desktop computer. Details of this stage can be found in Chapter 5. • The wavefront corrector compensates for the measured aberrations by generating a surface shape that is ideally conjugate to the aberration profi le. (If a reflective wavefront corrector is used, the mirror surface

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shape should be equal to, but with half of the amplitude of, the incoming wavefront.) The wavefront corrector is placed in a plane conjugate to both the pupil of the eye and the wavefront sensor. The most common wavefront corrector consists of a continuous reflective surface and an array of adjoining computer-controlled actuators or electrodes that physically or electrically push and pull on the surface, transforming it into a desired shape. Wavefront correctors have been commercially available for many years, though their construction is still a field of active research. Many different types of correctors exist, including those based on conventional piezoelectric, bimorph, liquid crystal, and microelectromechanical system (MEMS) technologies [19, 20].

4.3

WAVEFRONT CORRECTORS

Wavefront correctors alter the phase profi le of the incident wavefront by changing the physical length over which the wavefront propagates or the refractive index of the medium through which the wavefront passes. Most wavefront correctors are based on mirror technology and impart phase changes by adjusting their surface shape (i.e., change their physical length while keeping the refractive index constant). Other devices, such as those based on liquid crystal technologies, rely on localized changes in refractive index with the physical length being held constant. Using a nomenclature similar to that by Tyson [19] and Hardy [20], there are four broad categories of wavefront correctors that one is most likely to consider for an AO system designed for the eye. These categories are depicted in Figure 4.2 and are described below. Discrete Actuator Deformable Mirrors Deformable mirrors of this type have a continuous mirror surface whose profi le is controlled by an underlying array of actuators [Fig. 4.2(a)]. Pushing one actuator produces a localized (also termed zonal) Gaussian-like deflection of the mirror surface, termed the influence function. The deflection extends to adjacent actuators, where it typically changes the surface height by 10 to 15% of the peak deflection. The 10 to 15% deflection is commonly referred to as the coupling coefficient, as it describes the degree of cross-coupling between actuators. The influence functions are not identical nor are they independent, the extent of which depends on many factors, such as the thickness and material properties of the top facesheet, and the characteristics of the underlying actuator, such as the modulus of elasticity and the type of surface/actuator junction. A detailed treatment of the impact of these additional parameters is given by Tyson [19] and Hardy [20]. Segmented Correctors Mirrors of this type consist of an array of adjacent, planar mirror segments that are independently controlled [Fig. 4.2(b)].

WAVEFRONT CORRECTORS

(a)

V1

V2

V3

(b)

V4 (c)

87

V5

V6

V1

V2

V3

V4

V5

V6

(d)

FIGURE 4.2 Four main classes of wavefront correctors. (a) Discrete actuator deformable mirrors consist of a continuous, reflective surface and an array of actuators, each capable of producing a local deformation in the surface. (b) Piston-only segmented correctors consist of an array of small planar mirrors whose axial motion (piston) is independently controlled. Liquid crystal spatial light modulators modulate the wavefront in a similar fashion but rely on changes in refractive index rather than the physical displacement of a mirror surface. Piston/tip/tilt-segmented correctors add independent tip and tilt motion to the piston-only correctors. (c) Membrane mirrors consist of a grounded, flexible, reflective membrane sandwiched between a transparent top electrode and an underlying array of patterned electrodes, each of which is capable of producing a global deformation in the surface. (d) Bimorph mirrors consist of a layer of piezoelectric material sandwiched between a continuous top electrode and a bottom, patterned electrode array. A top mirrored layer is added to the top continuous electrode. An applied voltage causes a deformation of the top mirrored surface.

Piston-only segmented mirrors have one degree of freedom that corresponds to a pure, vertical piston mode. Piston/tip/tilt segmented mirrors have two additional degrees of freedom (tip and tilt) for slope control. This results in much better wavefront fitting and significantly reduces the number of segments needed to achieve the same correction as that for piston-only mirrors. Segmented mirrors are considered zonal correctors in that each segment induces a localized wavefront correction. Mirror segments are not coupled, producing a coupling coefficient of zero. The influence function of piston-only mirrors is described by a “top hat” function. Piston/tip/tilt mirrors are more complicated and have three influence functions per segment. Unlike discrete actuator deformable mirrors, the segmented mirrors have gaps between the segments that can trap and scatter incident light. These reduce the efficiency and quality of the correction. Gaps also necessitate co-phasing of the segments, that is, matching the reflected wavefront phase at the segment boundaries to ensure a continuous wavefront profi le. These gaps are typically characterized by their fi ll factor, the ratio of the actual mirrored surface to

88

WAVEFRONT CORRECTORS FOR VISION SCIENCE

the total corrector surface area. Fill factors vary considerably among devices with some approaching 100% and others well below 50%. Liquid crystal spatial light modulators (LC-SLMs) are another type of piston-only, segmented corrector. Instead of mirrored segments that physically move, LC-SLMs change the refractive index over a range that permits optical path changes of at least one wavelength. The resulting effect on the wavefront is essentially the same as for the segmented mirrors. Control of the refractive index is achieved electronically (via electrodes) [21] or optically (by imaging an intensity pattern directly onto the liquid crystal housing) [22]. Membrane Mirrors Mirrors of this type consist of an edge-clamped, flexible, reflective membrane (analogous to a drumskin) sandwiched between a transparent top electrode and an underlying array of patterned electrodes [Fig. 4.2(c)]. When no voltage is applied, the membrane is in principle flat. Application of a voltage to one electrode induces an electrostatic attraction that deflects the entire membrane. Hence membrane mirrors are often viewed as modal correctors, as opposed to discrete actuator deformable mirrors and segmented mirrors that are quasi- or fully zonal. Because the edges of the membrane are clamped and cannot move, only the central two-thirds of the membrane is useful for correction. Most membrane mirrors rely on electrostatics, though other activation mechanisms are possible, such as magnetic [23, 24], thermal [25], or voicecoil [26] methods. Bimorph Mirrors Bimorph mirrors are another form of modal corrector [19, 20] and at their most basic level consist of two dissimilar layers with one or both being piezoelectric. These are sandwiched between a continuous top electrode and a bottom, patterned electrode array [Fig. 4.2(d)]. A mirrored layer of high optical quality is added to the top continuous electrode. Application of a voltage across the top and bottom electrodes changes the underlying surface area of the two dissimilar layers and results in a bending of the entire mirror. The magnitude of the deformation is dependent on the electric field and the dielectric properties of the material. Bimorphs are particularly adept at providing large dynamic ranges at low spatial frequencies.

4.4

WAVEFRONT CORRECTORS USED IN VISION SCIENCE

Several types of wavefront correctors have been used in vision AO systems for the correction of ocular aberrations. These include macroscopic discrete actuator deformable mirrors, LC-SLMs, bimorph mirrors, and a variety of MEMS-based mirrors. The LC-SLMs represent a form of segmented corrector; the MEMS-based mirrors typically represent a form of membrane mirror or discrete actuator deformable mirror. The use of each of these is surveyed below.

WAVEFRONT CORRECTORS USED IN VISION SCIENCE

4.4.1

89

Macroscopic Discrete Actuator Deformable Mirrors

A discrete actuator deformable mirror was employed in the first vision science AO system that successfully corrected the most significant higher order aberrations of the eye and with which the University of Rochester authors performed high-resolution retinal imaging and vision testing [1]. This initial Rochester AO system consisted of a Shack–Hartmann wavefront sensor and a 37-channel discrete actuator deformable mirror that was manufactured by Xinxtics, Inc. [27]. The mirror has a dynamic range of 4 mm (8 mm in reflection) for an applied voltage of ±100 V. A photograph of a larger version of the mirror is shown in Figure 4.3. The initial Rochester system took tens of seconds to converge and so more accurately depicted an active, rather than adaptive, optics system. An improved 37-channel Xinxtics mirror with lead–magnesium–niobate (PMN) actuators was used in the next generation of this AO system and with which a closedloop bandwidth of approximately 1 Hz was achieved [2]. The same mirror was also used in the fi rst AO confocal laser scanning ophthalmoscope (see also Chapter 16) [4], and the fi rst AO time-domain and spectral-domain OCT cameras (see also Chapter 17) [5, 6].

FIGURE 4.3 The discrete actuator deformable mirror manufactured by Xinxtics, Inc. The device shown has 97 actuators that are arranged in a rectilinear pattern behind the 8-cm mirror. Maximum stroke of the actuators is ±2 mm, produced by an applied voltage of ±100 V. (Reprinted with permission of Xinxtics, Inc.)

90

WAVEFRONT CORRECTORS FOR VISION SCIENCE

Discrete actuator deformable mirrors with substantially more actuators have yielded improved aberration correction in the eye. A 97-channel, electrostrictive PMN Xinxtics mirror was successfully integrated into the Rochester AO system and routinely compensates the wave aberration across a 6.8-mm pupil diameter to a root-mean-square (RMS) wavefront error of 5 cm in diameter), and limited actuator stroke (i.e., displacement of the mirror surface) are notable limitations. The large size leads to long optical path lengths that are necessary to magnify the small pupil of the eye (10 mm), most of which is available for lower order aberration correction. This is particular attractive for correcting the large amounts of second-order aberrations that are typically present in the human eye. The available stroke, however, decreases rapidly as the square of the radial Zernike order, reducing the effectiveness of the mirror to compensate higher spatial frequency aberrations. A detailed description of the optical characteristics of bimorph mirrors can be found in Horsley et al. [40] and Dalimier and Dainty [57]. Their performance is further discussed in Section 4.5.6. 4.4.4

Microelectromechanical Systems

Microelectromechanical systems (MEMS) mirrors are a group of correctors that offer enormous potential for aberration correction in the eye, especially from a commercialization standpoint. MEMS technology leverages the considerable investments made in the integrated circuit industry and has the potential to provide low-cost, compact devices. Fabricated predominantly from silicon, MEMS mirrors promise the ability of batch fabrication and a design space that allows for extremely high dynamic ranges, high temporal frequency operation, and large degrees of freedom (i.e., number of actuators or electrodes). Depending on their method of fabrication, MEMS mirrors can be subdivided into two main classes—bulk and surface micromachined devices [41]. Bulk micromachined mirrors are a form of membrane mirror [42] that consist of a grounded, flexible membrane lying between a continuous transparent electrode and an underlying array of patterned electrodes [Fig. 4.2(c)]. Applying a voltage between one of the patterned electrodes and the top electrode causes the entire membrane to deform, much like striking a drum skin. These modal correctors have a broad influence function that is similar to that of bimorph mirrors. Consequently, their available stroke decreases rapidly with spatial frequency. As an example, if 6 to 7 mm of mirror deflection is available for low-order aberration correction (second order), then less than 0.25 mm will be available for the correction of fi fth-order terms in the Zernike expansion [43]. Figure 4.5 shows a 37-actuator device from OKO Technologies [42]. Agile Optics (formerly Intellite) also manufactures a range of MEMS-based membrane mirrors [44]. Interestingly, the fi rst wavefront corrector applied to the eye was a sixelectrode membrane mirror that was integrated into a custom scanning laser ophthalmoscope [13]. Since objective wavefront sensors, such as the SHWS, had not yet been developed for the eye, the authors were limited to correcting the astigmatism in one subject’s eye based on a subjective refraction. Bartsch et al. [45] and Fernandez et al. [3, 43] used a more powerful membrane mirror from OKO Technologies (37 electrodes) in conjunction with a SHWS to provide real-time correction of the aberrations in the human eye. The other class of MEMS mirrors are the surface micromachined devices, which are typically a form of discrete actuator deformable mirror. These mirrors are fabricated by depositing successive layers of material onto a sub-

WAVEFRONT CORRECTORS USED IN VISION SCIENCE

93

FIGURE 4.5 (Left) Enlarged view of a 37-electrode, 15-mm aperture OKO Technologies MEMS mirror [43]. (Reprinted with permission of Flexible Optical B.V.) A maximum deflection of 8 mm is possible for an applied voltage of 210 V. (Right) The mDM140 MEMS deformable mirror manufactured by Boston Micromachines Corporation [47, 48]. The device has a clear aperture of 3.3 to 4.4 mm, 140 actuators on a square grid, and greater than 3.5 mm of stroke. (Reprinted with permission of Boston Micromachines Corporation.)

strate and then patterning each one using masks and sacrificial layers. The structure is then built up from a stack of such layers. With this fabrication process, it is possible to make several different types of deformable mirrors all of which can be scaled to the desired number of actuators while keeping the overall physical size less than or equal to that of a dilated pupil (~8 mm). Doble et al. [46] evaluated a surface micromachined MEMS deformable mirror from Boston Micromachines Corporation (Fig. 4.5) [47, 48] and provided the fi rst images of the human photoreceptor mosaic using a wavefront corrector other than a discrete actuator Xinxtics mirror. Actuation in the Boston device is realized with electrostatics, as opposed to piezoelectrics in the Xinxtics mirror, and provided 2 mm of stroke for an applied voltage of 220 V. Like the Xinxtics mirror, actuator deformation is local and generates an influence function with a coupling coefficient of 15%. As with all electrostatic devices, the lack of hysteresis (which is present in piezoelectric devices) is a considerable advantage. Other industrial groups are also fabricating MEMS-based mirrors. Gehner et al. describe the fabrication and use of a 200 × 200 piston-only segmented MEMS device built directly onto complementary metal–oxide– semiconductor (CMOS) circuitry [49]. The device has one optical wavelength of stroke as it is intended to be used in conjunction with phase wrapping. Figure 4.6 shows a high stroke, modal MEMS membrane mirror described by Kurczynski et al. [50]. These devices are made from low-stress silicon and

94

WAVEFRONT CORRECTORS FOR VISION SCIENCE

are operated by electrostatic attraction. The mirrors have a 10-mm diameter active area, actuated by 1024 electrodes, and have demonstrated ±20 mm of wavefront deformation for lower-order modes while operating at less than 20 V in a closed-loop system. Figure 4.7 shows a scanning electron micrograph of a 37-segment piston/ tip/tilt MEMS-based mirror described by Doble et al. [51]. High-quality mirror surfaces are bonded to the actuator platforms shown on the left of Figure 4.7. Fill factor values greater than 98% have been achieved with over 7 mm of stroke available for an applied voltage of 60 V. The array shown in the figure is approximately 2.5 mm in diameter. Each 700-mm diameter segment requires three independent voltages that control the axial position and the slopes of the segment. Table 4.1 lists many of the vision AO systems currently in use. The list includes 20 AO systems with 9 employed in flood illumination retinal cameras, 6 in scanning laser ophthalmoscopes, 4 in optical coherence tomography systems, and 1 in a phoropter. Various types of mirrors are being used with actuator counts ranging from 13 to 144. All employ Shack–Hartmann wavefront sensors to measure the wave aberration of the eye. Typical (approximate) performance parameters as reported by the authors are given for each system.

FIGURE 4.6 Closeup of the transparent electrode membrane device described by Kurczynski et al. [50]. Wire bonds to the ceramic package are visible on four sides of the device with the single wire on the right being the connection to the top transparent electrode. The membrane is circular and is positioned beneath the transparent electrode. Holes in the transparent electrode are positioned outside the active area and mitigate squeeze fi lm damping. The mirrors have a 10-mm diameter active area, actuated by 1024 electrodes, and have demonstrated ±20-mm wavefront deformation for lower-order modes, while operating at less than 20 V in a closed-loop system. The scale bar is indicated in centimeters. (Figure courtesy of P. Kurcyznski.)

PERFORMANCE PREDICTIONS FOR VARIOUS TYPES

95

FIGURE 4.7 (Left) Scanning electron micrograph of a piston/tip/tilt 37-segment MEMS mirror described by Doble et al. [51]. The actuators are shown without the high-quality mirror segments that are normally bonded to their top surface. (Right) Interferogram of a fully assembled subarray with random tip and tilts applied. Each mirror segment is larger than its underlying actuator. This design provides a fi ll factor greater than 98%. The segments are on the order of 700 mm in diameter. Strokes of 7 mm have been achieved for an applied voltage of 60 V. (Photographs courtesy of Iris AO, Inc.)

4.5 PERFORMANCE PREDICTIONS FOR VARIOUS TYPES OF WAVEFRONT CORRECTORS As surveyed in the previous section, numerous types of wavefront correctors have been applied to the eye and have successfully reduced the degrading impact of the ocular aberrations, providing the eye with optical quality exceeding that with which it is endowed. However, none have been reported to provide sufficient correction to yield diffraction-limited imaging for large pupils (≥6 mm), where the aberrations are most severe and the benefit of AO is the largest. Typically reported residual RMS values are ~100 nm for a 6.8mm pupil, corresponding to l/6 in the visible (although some subjects correct better than this) [2]. Even at smaller pupil sizes, many of the devices have not reached diffraction-limited imaging. This raises a fundamental concern as to what characteristics of the correcting device, such as actuator number and stroke, are required to achieve diffraction-limited imaging, and to optimally match corrector performance and cost to that required of a particular imaging task in the eye. Wavefront correctors, including many of those described in the previous section, have been largely designed for imaging through atmospheric turbulence. Specifically, their actuator number, stroke, influence functions, and speed are weighted toward the spatial and temporal properties of the atmosphere [19, 20]. Ocular aberrations, on the other hand, have different properties that should lead to different optimal corrector designs. As an example,

96

Flood

AO OCT Nonimaging

Murcia

Murcia/Vienna Imperial Col/ City, UK Galway, Ireland San Diego UC Davis/ LLNL

b

a

AO cSLO

Flood Flood Flood

AO phoropter AO cSLO

MEMS (144)

Bimorph (35) Membrane (19) Discrete actuator (109) MEMS (144) and bimorph (35) MEMS (144) MEMS (144) and bimorph (35) Bimorph (19 or 37) Bimorph (13) Bimorph (18)

Discrete actuator (97) MEMS (144) MEMS (144) Discrete actuator (37) MEMS (144) Discrete actuator (37) Discrete actuator (37) MEMS (144) and bimorph (35) Membrane (37), LC-SLM Membrane (37) Membrane (37)

Correctorb

All systems use a SHWS for measuring the aberrations of the eye. Number of wavefront corrector actuators specified in parentheses.

LLNL (will be operated at Doheny) Chengdu, China Paris Moscow/ Kestrel Corp. Schepens

AO cSLO AO cSLO AO cSLO Flood AO OCT

Berkeley/ Houston Indiana

Flood Flood Flood AO OCT

Flood

TYPE

BMC

CILAS

AOPTIX OKO Tech ITEK BMC AOPTIX BMC BMC AOPTIX

Xinxtics BMC BMC Xinxtics BMC Xinxtics Xinxtics BMC AOPTIX OKO Tech Hamamatsu OKO Tech OKO Tech

Company [2, 28] [46]

neural threshold where rMTF ( f ) =

(13.26)

2π

∫ abs ( OTF ( f , φ )) dφ

and OTF(f, f) is the optical

0

transfer function for spatial frequency coordinates f (frequency) and f (orientation). • SFcOTF = cutoff spatial frequency of the radially averaged optical transfer function (rOTF). Cutoff frequency is defi ned here as the intersection of the radially averaged OTF (rOTF) and the neural contrast threshold function. If the curves intersect more than once, the intersection with the lowest spatial frequency is chosen (excludes spurious resolution). The radially averaged OTF is determined by integrating the full twodimensional OTF over all orientations. Since the OTF is a complexvalued function, integration is performed separately for the real and imaginary components. Conjugate symmetry of the OTF ensures that the imaginary component vanishes, leaving a real-valued result. Since phase shifts in the OTF are taken into account when computing the rOTF, this metric is sensitive to spatial phase errors in the image: SFcOTF = lowest spatial frequency for which rOTF < neural threshold

(13.27)

2π

where rOTF( f ) =

∫ OTF ( f , φ ) dφ

and OTF(f, f) is the optical transfer

0

function for spatial frequency coordinates f (frequency) and f (orientation). • AreaMTF = area of visibility for the rMTF (normalized to the diffraction-limited case). The area of visibility is the region lying below the radially averaged MTF and above the neural contrast threshold function [25, 26]: cutoff

AreaMTF =

∫

cutoff

rMTF ( f ) df −

0 cutoff

∫ 0

∫

TN ( f ) df

0 cutoff

rMTFDL ( f ) df −

∫ 0

(13.28) TN ( f ) df

METHODS FOR ESTIMATING THE MONOCHROMATIC REFRACTION

351

where TN is the neural contrast threshold function, which equals the inverse of the neural contrast sensitivity function. When computing the area under the rMTF, phase-reversed segments of the curve count as a positive area to be consistent with our defi nition of the SFcMTF as the highest frequency for which the rMTF exceeds the neural theshold. This allows spurious resolution to be counted as beneficial when predicting visual performance for certain tasks (e.g., contrast detection). Metrics based on the volume under the MTF have been used in studies of chromatic aberration [27] and visual instrumentation [26]. • AreaOTF = area of visibility for the rOTF (normalized to the diffraction-limited case): cutoff

AreaOTF =

∫

cutoff

∫

rOTF ( f ) df −

0 cutoff

∫

TN ( f ) df

0 cutoff

∫

rOTFDL ( f ) df −

0

(13.29) TN ( f ) df

0

where TN is the neural contrast threshold function defi ned above. Since the domain of integration extends only to the cutoff spatial frequency of the SFcOTF, phase-reversed segments of the curve do not contribute to the area under the rOTF. This is consistent with our defi nition of the SFcOTF as the lowest frequency for which the rOTF is below the neural theshold. This metric would be appropriate for tasks in which phasereversed modulations (i.e., spurious resolution) actively interfere with performance. • SRMTF = Strehl ratio computed in the frequency domain (MTF method): ∞

∞

∫ ∫ MTF ( f , f x

SRMTF =

−∞ −∞ ∞ ∞

∫ ∫ MTF

DL

y

) dfx dfy (13.30)

( fx , fy ) dfx dfy

−∞ −∞

The Strehl ratio computed by the MTF method is equivalent to the Strehl ratio computed in the spatial domain for a hypothetical PSF with even symmetry (i.e., PTF = 0). • SROTF = Strehl ratio computed in the frequency domain (OTF method): ∞

∞

∫ ∫ OTF ( f , f x

SROTF =

−∞ −∞ ∞ ∞

∫ ∫ OTF

DL

y

) dfx dfy (13.31)

( fx , fy ) dfx dfy

−∞ −∞

The Strehl ratio computed by the OTF method quantifies the relative intensity of the PSF at the coordinate origin, rather than at the peak (as in the SRX).

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• VSMTF = visual Strehl ratio computed in the frequency domain (MTF method). This metric is similar to the SRMTF, except that the optical MTF is weighted by the neural contrast sensitivity function (CSF N): ∞

∞

∫ ∫ CSF

N

−∞ −∞ ∞ ∞

VSMTF =

∫ ∫ CSF

N

( fx , fy ) ⋅ MTF ( fx , fy ) dfx dfy (13.32)

( fx , fy ) ⋅ MTFDL ( fx , fy ) dfx dfy

−∞ −∞

This metric differs from the VSX by quantifying image quality at the coordinate origin, rather than at the peak of the PSF. VSMTF is equivalent to the VSX for a hypothetical PSF that is well centered with even symmetry computed as the inverse Fourier transform of the MTF (which implicitly assumes PTF = 0). • VSOTF = visual Strehl ratio computed in the frequency domain (OTF method). This metric is similar to the SROTF, except that the optical OTF is weighted by the neural contrast sensitivity function (CSF N): ∞

∞

∫ ∫ CSF

N

( fx , fy ) ⋅ OTF ( fx , fy ) dfx dfy

−∞ −∞ ∞ ∞

VSOTF =

∫

(13.33)

∫ CSFN ( fx , fy ) ⋅ OTFDL ( fx , fy ) dfx dfy

−∞ −∞

This metric differs from the VSX by emphasizing image quality at the coordinate origin, rather than at the peak of the PSF. • VOTF = volume under the OTF normalized by the volume under the MTF: ∞

∞

∫ ∫ OTF ( f , f x

VOTF =

y

−∞ −∞ ∞ ∞

∫ ∫ MTF ( f , f x

) dfx dfy (13.34)

y

) dfx dfy

−∞ −∞

This metric is intended to capture phase shifts in the PTF. Since MTF ≥ real part of OTF, this ratio is always ≤1. • VNOTF = volume under the neurally weighted OTF, normalized by the volume under the neurally weighted MTF. This metric is similar to the VOTF, except that the optical OTF and MTF are weighted by the neural contrast sensitivity function (CSF N): ∞

∞

∫ ∫ CSF ( f , f x

VNOTF =

y

−∞ −∞ ∞ ∞

∫ ∫ CSF ( f , f x

y

) ⋅ OTF ( fx , fy ) dfx dfy (13.35)

) ⋅ MTF ( fx , fy ) dfx dfy

−∞ −∞

This metric is intended to capture the visually significant phase shifts in the PTF.

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METHODS FOR ESTIMATING THE MONOCHROMATIC REFRACTION

Any metric of optical quality based on the OTF can be converted into a metric of visual quality by replacing the optical OTF with its neural counterpart, computed as the Fourier transform of the neural PSF described in Section 13.3.2.2. 13.3.3

Numerical Example

As a numerical example, virtual refraction was performed for a hypothetical eye that is free of all aberrations except Zernike spherical aberration, for which c 40 = 0.25 mm for a pupil diameter of 6 mm. The results are displayed in Figure 13.8 in the form of through-focus curves with lens power on the abscissa and metric value on the ordinate. The optimum lens power determined from these curves is shown for each wavefront metric, PSF metric, and OTF metric in the three stem plots in the lower right corner. As this example shows, the optimum value of the correcting lens can vary over a substantial

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FIGURE 13.8 Through-focus curves of virtual refraction. Each graph shows how a particular monochromatic metric of image quality varies with the power of a lens added to an eye’s wave aberration function. The three graphs on the bottom row, right side, summarize the lens power that optimizes each metric.

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range (−0.4 D to +0.75 D in this case) depending on the metric used to assess optical quality.

13.4 OCULAR CHROMATIC ABERRATION AND THE POLYCHROMATIC REFRACTION The human eye suffers from significant amounts of chromatic aberration caused by chromatic dispersion, which is the variation of the refractive index of the eye’s refractive media with wavelength. Chromatic dispersion causes the focus, size, and position of retinal images to vary with wavelength. Variation in the focusing power of the eye with wavelength is called longitudinal (or axial) chromatic aberration (LCA) and is measured in diopters. In effect, the eye’s far point varies with wavelength, which means that only one wavelength of light emitted by a point source can be well focused on the retina at any moment in time. Retinal image size for extended objects also varies with wavelength, called chromatic difference of magnification (CDM), and is specified as a fractional change. For any given point on an extended object, the image is spread across the retina like a tiny rainbow or colored fringe. This phenomenon is called transverse (or lateral) chromatic aberration (TCA) and is specified as a visual angle (i.e., an angle subtended at the eye’s nodal point by the colored fringe). The effects of LCA and TCA on the polychromatic PSF are illustrated in Figure 13.9. In general, LCA tends to smooth the λ = 525 nm Kλ = −0.18 D

λ = 575 nm

λ = 600 nm

Composite

Kλ = +0.10 D

Kλ = +0.22 D

λfocus = 555 nm

τ = −0.35 min

τ = −0.74 min

1−mm pupil offset

Luminance

τ=0

τ = +0.62 min

FIGURE 13.9 Image formation for a polychromatic source in the presence of chromatic aberration. Top row is for an eye with longitudinal chromatic aberration only. Bottom row is for an eye with longitudinal and transverse chromatic aberration produced by 1 mm of horizontal pupil offset from the visual axis (or, equivalently, 15° of eccentricity). The point source emits three wavelengths of light (500, 575, and 600 nm) and the eye is assumed to be focused for 550 nm. Chromatic errors of focus and position indicated for each image are derived from an analysis of the Indiana Eye model of chromatic aberration. (Figure also appears in the color figure insert.)

OCULAR CHROMATIC ABERRATION AND THE POLYCHROMATIC REFRACTION

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PSF since dark rings in the PSF for one wavelength are fi lled in by the bright rings of another wavelength. However, this smoothing effect is not as effective when TCA is present because the various PSFs at different wavelengths are not concentric. Clinical refractions are invariably performed with white light, which means patients are required to make subjective judgments about the quality of their vision based on some spectral aggregate of their sensations. It is generally presumed that these judgments are based primarily on stimulus luminance, rather than hue or saturation qualities of the colored image. Under this assumption, monochromatic methods for an objective refraction can be extended into the polychromatic domain with the aid of an optical model of the eye’s ocular chromatic aberration. One such model is the Indiana Eye, a reduced eye model (i.e., a single refracting surface) that accounts for a large experimental literature on ocular chromatic aberration [28, 29]. The variation of refractive error with wavelength of this model is shown in Figure 13.10. One need for such a model is to determine the focus shift associated with referencing measurements taken at some convenient wavelength (e.g., infrared) to a visible wavelength in focus. A chromatic aberration model is also needed when conducting a virtual refraction in polychromatic light, as described next. Virtual refractions simulate the placement of lenses of different powers in front of the eye for the purpose of determining the lens that maximizes retinal image quality. As illustrated in Figure 13.10, when an eye views through a spherical lens, the LCA curve shifts vertically. Positive lenses change the eye–lens system in the myopic direction (which corresponds to a negative refractive error clinically); hence the curve shifts downward. Conversely, negative lenses shift the LCA curve upward. This shifting of the curve changes the balance between the state of focus and the relative luminance of each wavelength component of polychromatic light. For example, shifting the curve upward reduces the amount of defocus in the shorter wavelength but increases the amount of defocus in the longer wavelengths. Whether this yields a net gain of image quality depends strongly on the luminance spectrum of the source. For this example, a blue source would benefit from a negative lens but a red source would not. In theory, the best lens is that which optimizes the polychromatic PSF according to some metric of optical quality. In the example of Figure 13.10, the eye is optimally focused for 555-nm monochromatic light according to some monochromatic metric. The LCA curve of the Indiana Eye model is passed through this point to quantify the amount of hyperopia expected at longer wavelengths and the amount of myopia expected at shorter wavelengths. In white light, the optimum PSF (according to some polychromatic metric of image quality) occurs when this particular eye views through an additional spherical lens of power −0.25 D. By comparison, when either a +0.25- or a −0.75-D lens is added, the PSF deteriorates markedly. Notice that when the negative lens is introduced to optimize the polychromatic PSF, the

356

FROM WAVEFRONTS TO REFRACTIONS

FIGURE 13.10 Polychromatic refraction shifts the longitudinal chromatic aberration function vertically. If the eye is emmetropic (i.e., refractive error = 0) at some reference wavelength (555 nm in this example), then the same eye will appear to be myopic (i.e., refractive error < 0) when viewing through a positive lens. At the same time, the wavelength in focus (i.e., zero crossing) will shift to a longer wavelength. Conversely, the eye will appear to be hyperopic when viewing through a negative lens and the wavelength in focus will shift to a relatively short wavelength. Thus the lens value (−0.25 D in this example) that optimizes retinal image quality for polychromatic light (according to some polychromatic metric) corresponds to a unique wavelength in focus (515 nm in this example) when the eye is well focused for polychromatic light emitted by a distant object.

eye is no longer well focused for 555 nm. Instead, 515 nm becomes the wavelength in focus (according to the chosen metric) when white light is optimally focused for this eye. 13.4.1

Polychromatic Wavefront Metrics

The wave aberration function is a monochromatic concept. If a source emits polychromatic light, then aberration maps for each wavelength are treated separately because lights of different wavelengths are mutually incoherent and do not interfere. For this reason, metrics of wavefront quality do not

OCULAR CHROMATIC ABERRATION AND THE POLYCHROMATIC REFRACTION

357

generalize easily to the case of polychromatic light. This lack of generality is a major limitation of virtual refraction based on wavefront quality. One possible approach, which would require justification, is to compute the weighted average of monochromatic metric values computed for a series of wavelengths, Metric poly = ∫ S ( λ ) Metric ( λ ) dλ

(13.36)

where the weighting function S(l) is the luminance spectrum of the source. 13.4.2

Polychromatic Point Image Metrics

The luminance component of a polychromatic point spread function, PSFpoly, is a weighted sum of the monochromatic point spread functions, psf(x, y, l), PSFpoly = ∫ S ( λ ) psf ( x, y, λ ) dλ

(13.37)

where the weighting function S(l) is the luminance spectrum of the source. Given this defi nition, PSFpoly may be substituted for PSF in any of the equations given in Section 13.3.2.2 to produce new, polychromatic metrics of image quality. In addition to these luminance metrics of image quality, other metrics can be devised to capture the changes in color appearance of the image caused by ocular aberrations. For example, the chromaticity coordinates of a point source may be compared to the chromaticity coordinates of each point in the retinal PSF and metrics devised to summarize the differences between image and object. Evaluation of Eq. (13.37) for a discrete series of wavelengths requires recalculating the PSF for each wavelength. This can become prohibitively time consuming, especially in a virtual refraction paradigm in which the calculations have to be repeated for a variety of added lens powers. A useful simplification in this case is to neglect the scaling of the diffraction-limited PSF with wavelength. Under this assumption, one may precompute a sequence of defocused PSFs for a given wave aberration map and reuse each one for every combination of through-focus lens power and longitudinal chromatic aberration that produces the same net defocus. Another useful simplification is to assume that all of the Zernike coefficients except defocus (c 02) are independent of wavelength. 13.4.3

Polychromatic Grating Image Metrics

Given the polychromatic PSF defi ned above in Eq. (13.37), a polychromatic optical transfer function OTFpoly may be computed as the Fourier transform of the PSFpoly. Substituting this new function for the OTF and its magnitude for the MTF in any of the equations given in Section 13.3.2.3 will produce new metrics of polychromatic image quality defi ned in the frequency domain.

358

FROM WAVEFRONTS TO REFRACTIONS

13.5 EXPERIMENTAL EVALUATION OF PROPOSED REFRACTION METHODS The metrics of optical quality defi ned above have many potential uses, only one of which is to determine the refractive error of the eye. Which method works best for any particular task can only be determined empirically. To judge the success of an objective method of refraction requires a “gold standard” for comparison. The most clinically relevant choice is a so-called subjective refraction in which the clinician adjusts the spherical and astigmatic power of correcting lenses to maximize the patient’s visual acuity. Acuity is quantified by the dimensions of the smallest letters a patient is able to read correctly on a letter chart illuminated by white light. Using this gold standard of subjective clinical refraction, several recent experimental evaluations of the refraction methods described above are summarized below. 13.5.1

Monochromatic Predictions

In the Indiana Aberration Study [30] subjective refractions were performed to the nearest 0.25 D on 200 normal, healthy eyes from 100 subjects using the conventional, hyperfocal refraction procedure outlined in Sections 13.2.2 and 13.2.3. Accommodation was paralyzed with one drop of 0.5% cyclopentalate during the refraction. The refractive correction was taken to be that spherocylindrical lens combination that optimally corrected astigmatism and located the hyperfocal point of the corrected eye at optical infi nity. This prescribed refraction was then implemented with trial lenses and worn by the subject for subsequent measurements of the eye’s wave aberrations (l = 633 nm). This experimental design emphasized the effects of higher order aberrations by minimizing the presence of uncorrected second-order aberrations. Since all eyes were optimally corrected during aberrometry (according to the psychophysical criterion of maximum visual acuity), the predicted refraction computed from the aberration map was PSE = PJ0 = PJ45 = 0. The level of success achieved by the various methods described above was judged on the basis of precision and accuracy at matching these predictions. Accuracy in this context is defi ned as the dioptric difference between the population mean refraction and the prediction from virtual refraction based on monochromatic metrics of optical quality. Precision is a measure of the variability in results and is defi ned for PSE by the standard deviation of the population values. For the astigmatic components of refraction, precision was defi ned as the geometric mean of the major and minor axes of the 95% confidence ellipse computed for the bivariate distribution of PJ0 and PJ45. The two methods for fitting the aberration map with an equivalent quadratic surface gave strikingly different results. The least-squares method and Eq. (13.2) predicted a mean spherical refractive error of PSE = −3/8 D. In other words, this method predicted the eyes were, on average, myopic when in fact

EXPERIMENTAL EVALUATION OF PROPOSED REFRACTION METHODS

359

they were well corrected. To the contrary, the method based on paraxial curvature matching and Eq. (13.3) predicted an average refractive error close to zero for our population. The mean error for predicting astigmatic errors was less than 0.1 D and precision was less than 0.4 D by both methods. The accuracy of the 31 methods for predicting the spherical component of refraction based on metrics of PSF and OTF quality varied widely from −0.50 to +0.25 D. A rank ordering of the accuracy of all 33 methods (2 based on wavefront fitting and 31 based on optical quality) indicated that paraxial curvature matching was the most accurate method, closely followed by maximizing the wavefront quality metrics PFWc and PFCt. However, these results should not be taken as defi nitive for a variety of reasons [13], the most important of which is that conventional hyperfocal refractions are biased in the sense described in Section 13.2.3. For this reason, we anticipate that it will be easier to accurately predict the result of subjective refractions designed to optimize image quality for distant objects by focusing the retina at infi nity rather than the hyperfocal point. A different way to judge the success of various methods described above for converting wave aberrations into refractive errors is to compare the visual performance of the patient when viewing through the lenses prescribed by the different methods. This is the approach taken by Cheng et al. [31] and by Marsack et al. [32] in systematic studies of the change in visual acuity produced when selected, higher order aberrations are introduced into an eye. The experimental design of the Cheng study was somewhat simpler in that monochromatic aberrations were used to predict monochromatic visual performance, whereas Marsack used monochromatic aberrations to predict polychromatic performance. Nevertheless, both studies concluded that changes in visual acuity are most accurately predicted by the wavefront quality metric PFSt and by the image quality metric VSOTF. Furthermore, both studies concluded that three of the least accurate predictors were RMSw, HWHH, and VOTF. In addition, the Cheng study demonstrated that, as expected, those metrics that accurately predicted changes in visual acuity also predicted the lens power that maximized acuity in a through-focus experiment. This was an important result because it established experimentally the anticipated link between variations in monochromatic acuity and monochromatic refractive error. 13.5.2

Polychromatic Predictions

Ultimately, the goal is to use a wave aberration map to predict the lens prescription that will optimize retinal image quality and visual performance for everyday objects emitting polychromatic light. Unfortunately, polychromatic light introduces several new factors that must be taken into account in the virtual refraction procedure. First, there is the need for an accurate optical model of the eye’s chromatic aberration in order to compute polychromatic

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FROM WAVEFRONTS TO REFRACTIONS

metrics of image quality. Although many studies have demonstrated a remarkable consistency between eyes in the longitudinal (focusing) aspect of chromatic aberration [28], significant amounts of individual variation in the transverse aspect of chromatic aberration are known to exist [33]. Thus the development of polychromatic wavefront aberrometers might be required to take account of individual variation in ocular chromatic aberration [27]. Such technology may help determine the wavelength that is in focus when the eye is optimally focused for polychromatic light. Wavelength in focus is critical for modeling polychromatic images because it determines how much defocus is present for all other wavelengths present in the source. The luminance spectrum of the source is another important variable that can have a significant impact on polychromatic virtual refractions since it acts as a weighting function for computing metrics of optical quality in Eqs. (13.36) and (13.37). Although defi nitive results have not yet been published, this is an active area of research that should yield useful results in the near future.

13.5.3

Conclusions

The various methods for objective refraction described in this chapter are able to predict the outcome of subjective refraction with various degrees of accuracy and precision. The majority of the variability between methods may be attributed to the spherical component of refraction, PSE, rather than the astigmatic component. This suggests that uncertainty regarding the wavelength in focus when the eye is viewing a polychromatic target is a major limiting factor in evaluating the various methods. Recent experiments using monochromatic light suggest that the wavelength in focus for a typical whitelight source is approximately 570 nm for most subjects [34]. The full implementation of the polychromatic metrics of image quality described above should provide a sound basis for interpreting these experimental results. Predicting the results of conventional, hyperfocal refraction is particularly challenging because it involves not only the optimum correcting lens but also the eye’s functional depth of focus. Thus, computational methods are also required that can identify the depth of focus of an eye through wavefront analysis. The experimental literature on depth of focus suggests that individual variability and task dependence will be major factors to be addressed by these computational methods. Variability in subjective refraction, the gold standard used to judge the accuracy of predictions, is another likely source of disagreement between objective and subjective methods of refraction. If such variability makes the current gold standard a moving target, then it is conceivable that wavefront-based methods of objective refraction will become the preferred gold standard of the future. Acknowledgment Support for the writing of this chapter and the experiments reported therein was provided by NIH/NEI grant R01 EY05109.

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REFERENCES 1. Krueger RR, Applegate RA, MacRae SM, eds. Wavefront Customized Visual Correction: The Quest for Super Vision II. Thorofare, NJ: SLACK, 2004. 2. MacRae SM, Krueger RR, Applegate RA, eds. Customized Corneal Ablation: The Quest for Super Vision. Thorofare, NJ: SLACK, 2001. 3. Thibos LN, Wheeler W, Horner DG. Power Vectors: An Application of Fourier Analysis to the Description and Statistical Analysis of Refractive Error. Optom. Vis. Sci. 1997; 74: 367–375. 4. Raasch TW. Spherocylindrical Refractive Errors and Visual Acuity. Optom. Vis. Sci. 1995; 72: 272–275. 5. Schwendeman FJ, Ogden BB, Horner DG, Thibos LN. Effect of Sphero-cylinder Blur on Visual Acuity. Optom. Vis. Sci. 1997; 74/12S: 180. 6. Thibos LN, Applegate RA, Schwiegerling JT, et al. Standards for Reporting the Optical Aberrations of Eyes. In: Lakshminarayanan V, ed. OSA Trends in Optics and Photonics, Vision Science and Its Applications, Vol. 35. Washington, D.C.: Optical Society of America, 2000, pp. 232–244. 7. ANSI. American National Standard for Ophthalmics—Methods for Reporting Optical Aberrations of Eyes. ANSI Z80.28–2004. Merrifield, VA: Optical Laboratories Association, 2004. 8. Williams DR, Applegate RA, Thibos LN. Metrics to Predict the Subjective Impact of the Eye’s Wave Aberration. In: Krueger RR, Applegate RA, MacRae SM, eds. Wavefront Customized Visual Correction: The Quest for Super Vision II. Thorofare, NJ: SLACK, 2004, pp. 78–84. 9. Ciuffreda KJ. Accommodation, the Pupil and Presbyopia. In: Benjamin WJ, ed. Borish’s Clinical Refraction. Philadelphia: W.B. Saunders, 1998, pp. 77–120. 10. Atchison DA, Smith G. Optics of the Human Eye. Oxford: ButterworthHeinemann, 2000. 11. Applegate RA, Sarver EJ, Khemsara V. Are All Aberrations Equal? J. Refract. Surg. 2002; 18: S556–S562. 12. Guirao A, Williams DR. A Method to Predict Refractive Errors from Wave Aberration Data. Optom. Vis. Sci. 2003; 80: 36–42. 13. Thibos LN, Hong X, Bradley A, Applegate RA. Accuracy and Precision of Methods to Predict the Results of Subjective Refraction from Monochromatic Wavefront Aberration Maps. J. Vis. 2004; 4: 329–351. 14. Mahajan VN. Aberration Theory Made Simple. In: O’Shea DC, ed. Tutorial Texts in Optical Engineering, Vol. TT6. Bellingham, WA: SPIE Optical Engineering Press, 1991. 15. Howland HC, Howland B. A Subjective Method for the Measurement of Monochromatic Aberrations of the Eye. J. Opt. Soc. Am. 1977; 67: 1508–1518. 16. Corbin JA, Klein SA, van de Pol C. Measuring Effects of Refractive Surgery on Corneas Using Taylor Series Polynomials. In: Rol PO, Joos KM, Manns F, Stuck BE, Belkin M, eds. Ophthalmic Technologies IX. Proceedings of the SPIE. 1999; 3591: 46–52. 17. Bracewell RN. The Fourier Transform and Its Applications, 2nd ed. New York: McGraw-Hill, 1978.

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18. Röhler R, Howland HC. Merits of the Gaussian Moment in Judging Optical Line Spread Width—Coment on a Paper by W. N. Charman and J. A. M. Jennings. Vision Res. 1979; 19: 847–849. 19. Charman WN, Jennings JA. Objective Measurements of the Longitudinal Chromatic Aberration the Human Eye. Vision Res. 1976; 16: 999–1005. 20. Charman WN, Jennings JA. Merits of the Gaussian Moment in Judging Optical Line Spread Widths. Vision Res. 1979; 19: 851–852. 21. Marcos S, Moreno E, Navarro R. The Depth-of-Field of the Human Eye from Objective and Subjective Measurements. Vision Res. 1999; 39: 2039–2049. 22. Campbell FW, Green DG. Optical and Retinal Factors Affecting Visual Resolution. J. Physiol. 1965; 181: 576–593. 23. Thibos LN, Bradley A. Modeling Off-Axis Vision—II: The Effect of Spatial Filtering and Sampling by Retinal Neurons. In: Peli E, ed. Vision Models for Target Detection and Recognition. Singapore: World Scientific, 1995, pp. 338–379. 24. Thibos LN. Calculation of the Influence of Lateral Chromatic Aberration on Image Quality Across the Visual Field. J. Opt. Soc. Am. A. 1987; 4: 1673–1680. 25. Charman N, Olin A. Image Quality Criteria for Aerial Camera Systems. Photogr. Sci. Eng. 1965; 9: 385–397. 26. Mouroulis P. Aberration and Image Quality Representation for Visual Optical Systems. In: Mouroulis P, ed. Visual Instrumentation: Optical Design and Engineering Principle. New York: McGraw-Hill, 1999, pp. 27–68. 27. Marcos S, Burns SA, Moreno-Barriuso E, Navarro R. A New Approach to the Study of Ocular Chromatic Aberrations. Vision Res. 1999; 39: 4309–4323. 28. Thibos LN, Ye M, Zhang X, Bradley A. The Chromatic Eye: A New ReducedEye Model of Ocular Chromatic Aberration in Humans. Appl. Opt. 1992; 31: 3594–3600. 29. Thibos LN, Bradley A. Modeling the Refractive and Neuro-sensor Systems of the Eye. In: Mouroulis P, ed. Visual Instrumentation: Optical Design and Engineering Principle. New York: McGraw-Hill, 1999, pp. 101–159. 30. Thibos LN, Hong X, Bradley A, Cheng X. Statistical Variation of Aberration Structure and Image Quality in a Normal Population of Healthy Eyes. J. Opt. Soc. Am. A. 2002; 19: 2329–2348. 31. Cheng X, Bradley A, Thibos LN. Predicting Subjective Judgment of Best Focus with Objective Image Quality Metrics. J. Vis. 2004; 4: 310–321. 32. Marsack JD, Thibos LN, Applegate RA. Metrics of Optical Quality Derived from Wave Aberrations Predict Visual Performance. J. Vis. 2004; 4: 322–328. 33. Rynders MC, Lidkea BA, Chisholm WJ, Thibos LN. Statistical Distribution of Foveal Transverse Chromatic Aberration, Pupil Centration, and Angle psi in a Population of Young Adult Eyes. J. Opt. Soc. Am. A. 1995; 12: 2348–2357. 34. Coe CD, Thibos LN, Bradley A. Psychophysical Determination of the Wavelength of Light That Is Focused by a Polychromatic Subjective Refraction. Invest. Ophthalmol. Vis. Sci. 2005; 46: e-abstract 1188.

CHAPTER FOURTEEN

Visual Psychophysics with Adaptive Optics JOSEPH L. HARDY and PETER B. DELAHUNT Posit Science Corporation, San Franciso, California JOHN S. WERNER University of California Davis Medical Center, Sacramento, California

Psychophysics is the study of the relations between human performance and physical stimulus variables. For nearly 200 years, visual psychophysicists have worked to quantify these relations for visual tasks and light stimuli. Vision scientists combine psychophysical techniques with measures of anatomy and physiology to gain an understanding of how the visual system processes information. To the extent that adaptive optics (AO) approaches diffraction-limited correction of the optics of the eye, it offers an exciting new tool for addressing fundamental questions about human vision. Bypassing the aberrations of the eye with AO, psychophysicists will be able to present precisely controlled stimuli directly to the visual nervous system. This will help address questions about the relative contributions of optical and neural factors in defi ning the limits of visual performance, as well as fundamental questions about neural information processing. Additionally, combining physiological and anatomical data from AO-based imaging techniques with measures of visual performance, vision scientists will be able to advance further the understanding of the relation between form and function in the visual nervous system. This chapter is a short introduction to psychophysics and psychophysical methods. A comprehensive treatment of these topics would be impossible in Adaptive Optics for Vision Science, Edited by Porter, Queener, Lin, Thorn, and Awwal Copyright © 2006 John Wiley & Sons, Inc.

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a single chapter. Instead, key terms, concepts, and techniques for designing and implementing psychophysical experiments are introduced, with an emphasis on information relevant in the context of AO. The fi rst part of the chapter describes the notion of a psychophysical function and introduces a few examples. The second part of the chapter addresses psychophysical methods— experimental techniques for measuring psychophysical functions. Finally, the third part describes some of the equipment and procedures for producing visual stimuli for psychophysical experiments.

14.1 14.1.1

PSYCHOPHYSICAL FUNCTIONS Contrast Sensitivity Functions

Psychophysics seeks to measure the relations between human performance and physical variables. Such a relation is quantified by a psychophysical function. Figure 14.1 presents an example of a commonly measured psychophysical function from the spatial domain in vision that is important for vision scientists interested in AO, the contrast sensitivity function (CSF). Here, the

Contrast Sensitivity

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1 10 Spatial Frequency (cpd)

100

FIGURE 14.1 Contrast sensitivity is plotted as a function of spatial frequency. Data points were fitted with a double-exponential function. Inset shows a luminancevarying stimulus defi ned by a spatial Gabor function (i.e., a sinusoid windowed by a Gaussian). The use of a Gabor function is important for limiting the bandwidth in the spatial frequency domain.

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sensitivity to the contrast of a Gabor pattern varying in luminance (radiance1 of a light source fi ltered by the human spectral efficiency function) is plotted as a function of spatial frequency. Gabor patterns are one-dimensional sinusoidal gratings weighted by a two-dimensional Gaussian function. Sensitivity refers to a characteristic of the individual, but it is defi ned by a physical variable, the inverse of the contrast required to produce a criterion level of performance, for example, 75% correct on a detection task. There are several ways to defi ne contrast, V, but here it is defi ned as: VM =

Lmax − Lmin Lmax + Lmin

(14.1)

where L max is the maximum luminance in the pattern and L min is the minimum luminance in the pattern. This is known as the Michelson contrast, and it can vary between 0.0 and 1.0. The Michelson contrast is most appropriate when discussing the contrast of patterns that contain both luminance increments and decrements in equal proportion and vary from a background of a spaceaveraged mean luminance value, as is the case with sinusoidally varying and Gabor patterns. An alternate defi nition of contrast following Weber relates the luminance of a stimulus (L stim) to the luminance of the background (Lback): VW =

Lstim − Lback Lback

(14.2)

This defi nition of contrast can take any value. This metric is more appropriate for stimuli that are luminance increments or decrements on a background where the maximum and minimum luminance values are not uniformly distributed about a space-averaged mean luminance and where large positive or negative values are often meaningful. The other physical variable in the CSF is spatial frequency. The spatial frequency of a pattern is usually represented by the number of cycles per degree of visual angle (c/deg). Visual angle (qvis) is defi ned as:

θ vis = 2 tan −1 ( a 2l )

(14.3)

where a is the length of the stimulus along an axis orthogonal to the direction of viewing and l is the distance from the stimulus to the nodal point of the eye. The distance between the front surface of the cornea and the nodal point of the eye is approximately 7 mm for an average adult. Retinal image size can be calculated from visual angle if the focal length of the eye is known. A typical focal length for the human eye is 1/60 m. 1

Power per unit projected area per unit solid angle reflected from a surface toward the eye, expressed in watts per meters squared per steradian; W/m 2 /sr.

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VISUAL PSYCHOPHYSICS WITH ADAPTIVE OPTICS

As is often the case in contrast sensitivity measurements, the patterns used to generate the CSF in Figure 14.1 were Gabor patches. The spatial luminance profi le of a Gabor pattern is    x 2 y2    L ( x, y ) = L0 1 + V M exp  −  2 + 2   cos ( 2π fx + φ )    σ σ    x y 

(14.4)

where L 0 = mean luminance V M = grating contrast sx = horizontal standard deviation of the Gaussian window sy = vertical standard deviation of the Gaussian window f = spatial frequency ø = grating phase These types of patterns are used regularly in studies of spatial vision because the “soft edges” produced by the Gaussian window eliminate the high spatial frequencies present in sinusoidal gratings that are cut off abruptly. If a small, sinusoidally varying grating with hard edges (rectangular-wave window) is presented in a contrast sensitivity experiment, the observer may be more sensitive to the frequency components produced by the edges than to the frequency of the sinusoid being tested. This could lead to an overestimation of the sensitivity of the observer to certain spatial frequencies. Spatial and temporal vision are often characterized using a CSF such as shown in Figure 14.1 because it can be useful in predicting the sensitivity of the visual system to more complex patterns when such patterns are represented by their Fourier decomposition [1, 2]. This approach has proven useful, notwithstanding the rather limited range over which the assumptions of linear systems analysis are valid for the human visual system. Various spatial and temporal variables need to be considered when discussing the CSF, and contrast sensitivity can be thought of as a family of functions rather than a single characteristic of the visual system. For example, contrast sensitivity varies as a function of the space-average luminance (we are generally more sensitive to contrast at higher light levels). Contrast sensitivity also depends on the temporal frequency of stimulus motion or fl icker, as well as on the chromatic properties of the stimulus. In addition, many individuals are more sensitive to vertical and horizontal gratings than to oblique (45° or 135° from horizontal) gratings of the same frequency. This phenomenon is referred to as the oblique effect [3]. The region of retina tested is also a critical variable for contrast sensitivity. Sensitivity is highest in the fovea, especially for high spatial frequencies [4]. In addition to these stimulus variables, observer variables need to be taken into account. There is a great deal of individual variability in sensitivity to contrast patterns, even among healthy people of approximately the same age. Both optical [5] and neural [6] factors are known to contribute to this variability. Additional variability can be expected if the observer pool includes

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participants with diseases known to reduce contrast sensitivity [7]. Finally, scotopic (rod-mediated vision) and photopic (cone-mediated vision) CSFs vary with observer age [8]. The CSF is particularly important in AO applications due in part to its close relation to the optical modulation transfer function (MTF) of the eye. The optical MTF of the eye refers to the proportion of the contrast present in a stimulus that is preserved in the retinal image, as a function of spatial frequency. The CSF of the individual observer can be thought of as a product of the optical MTF and an MTF due to neural filtering of spatial information by the visual system. Under a wide variety of photopic conditions, the human CSF shows peak sensitivity at intermediate spatial frequencies, usually around 2 to 6 c/deg of visual angle, with sensitivity falling off rapidly at higher and lower frequencies [1]. Low-spatial-frequency attenuation is caused by neural factors. Specifically, lateral inhibition in the visual pathways is thought to be responsible for this reduction in sensitivity to low-spatial-frequency patterns [2]. Under normal viewing conditions, reduced sensitivity to high spatial frequencies in the human CSF is due to both optical and neural factors [9]. Image blur due to higher order monochromatic aberrations will reduce sensitivity at high spatial frequencies. As the spatial frequency of the image increases, neural processing efficiency decreases somewhat, further reducing contrast sensitivity. The neural sampling properties of the visual system seem to be fairly well matched to the optical quality of the normal eye [10]. In other words, the Nyquist sampling limit2 of the cone mosaic in the fovea is fairly closely matched to the highest spatial frequencies passed by the eye’s optics. Thus, while contrast sensitivity at high spatial frequencies may be improved through adaptive optics [11], this does not necessarily imply that this additional information can be used effectively by the visual system to improve visual function under natural viewing conditions. It is possible, for example, that higher-spatial-frequency information is detected by mechanisms that are optimally tuned to lower spatial frequencies. If the visual system interprets this signal as being produced by image elements at these lower frequencies, the result is aliasing. Under these circumstances, increasing the contrast of the highest spatial frequencies available in the retinal image would not be a benefit to vision but would act as noise. However, there is some evidence that the visual nervous system is equipped to process spatial frequencies higher than those normally passed by the eye’s optics [12]. AO can be used to address such fundamental issues in vision science. Of particular interest in defi ning the limits of spatial vision is the resolution limit. This can be estimated by the spatial frequency at a sensitivity of 0.0 using a linear extrapolation from the high-frequency limb of the CSF. More commonly in clinical applications, the resolution limit is described by visual 2

The Nyquist sampling theorem states that the number of uniformly spaced samples needed to specify a waveform of a particular frequency is two per cycle.

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acuity measured with an eye chart [13]. This measure of acuity is often referred to as Snellen acuity [14]. These charts are used to measure the smallest gap in a letter that can be detected at a specified distance. Resolution, or visual acuity, VA, is given by: VA = l ′ l

(14.5)

where l′ is the standard viewing distance (20 ft in the United States or 6 m in Europe) and l is the distance at which the smallest identifiable test stimulus subtends a visual angle of 1′ (1/60 of a degree). For example, if the smallest line on the eye chart a patient can read has letters with gaps that subtend 2′ of visual angle at 20 ft (equivalently, 1′ at 40 ft), the patient’s visual acuity is 20/40. More recently, clinicians have begun using a different, but closely related, system for defi ning visual acuity, called the logarithm of the minimal angle of resolution, or logMAR. As the name suggests, the notation in this system refers to the logarithm of the visual angle (in minutes of arc) of the smallest identifiable features. A logMAR value of 0.0 is equivalent to 20/20 Snellen acuity. 14.1.2

Spectral Efficiency Functions

Visual performance depends on the number of quanta received from a stimulus, but because we are not equally sensitive to all wavelengths of light, and this sensitivity varies between photopic and scotopic levels of illumination, purely physical metrics do little to describe the effectiveness of a stimulus for vision. Instead, the radiance of a stimulus fi ltered by the spectral efficiency of the visual system is used. This quantity is known as luminance. The International Commission on Illumination (Commission Internationale de l’Eclairage, CIE) has developed a system for specifying luminance according to the spectral sensitivity of the human observer. The spectral sensitivity function used by the CIE is called the standard observer’s visibility function or Vl when specifying lights under photopic conditions and V′l when specifying lights viewed under scotopic conditions. Luminance is thus defi ned as: Lv = K ∫ Lλ Vλ dλ

(14.6)

where Ll is the radiance contained in the wavelength interval d l and Vl is the relative photopic spectral sensitivity function for the standard observer of the CIE. For scotopic conditions the same formula applies except that V′l is used instead of Vl . These spectral efficiency functions are shown by smooth curves in Figure 14.2. The functions Vl and V′l are tabulated by Wyszecki and Stiles [15] and can be downloaded from the website of the Colour and Vision Research Laboratory (CVRL) at University College London (http://cvrl.ioo. ucl.ac.uk/). The K in Eq. (14.6) is related to the units in which luminance is specified, the most common in current usage being the candela per square

Log Relative Sensitivity (Quanta at the Cornea)

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0

-1.0

-2.0

-3.0

-4.0

-5.0 400

500

600

700

Wavelength (nm) FIGURE 14.2 The solid and dashed curves show the log relative sensitivity of the CIE standard observer under scotopic (CIE V′l ) and photopic (CIE Vl ) conditions, respectively. The CIE curves are often plotted on an energy basis but shown here on a quantal basis. These curves are tabled as normalized values having a peak of 1.0, but when normalized to measured data as shown here, one can see the absolute sensitivity differences for scotopic and photopic vision more clearly. The fi lled symbols represent detection data obtained following 30 min dark adaptation for stimuli modulated at 2 Hz. The open symbols represent data from the same observer using heterochromatic fl icker photometry (minimizing fl icker by adjusting the radiance of a monochromatic light presented in 14 Hz counterphase to a 3.3 log troland, broadband standard) with a 1° diameter foveal stimulus. (After Werner [16].)

meter (cd/m 2), sometimes referred to as the nit. For these units, the value of K is 683 for photopic luminance or 1700 for scotopic luminance. In the literature one may fi nd luminance specified in different units by different investigators; conversion factors are provided by Wyszecki and Stiles [15]. There are a few points to note about luminance specifications. First, there is no subjectivity inherent in the measurement of luminance. One simply measures the radiance at each wavelength and multiplies this value by the relative sensitivity of the standard observer at that wavelength. These products are summed across wavelengths for broadband stimuli. Alternatively, one may

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directly measure luminance with a photometer—a meter that has been calibrated to have the spectral sensitivity of the CIE standard observer. Second, while there is no subjectivity in the measurement of luminance, it was the original intent of the CIE to develop a metric that would be closely related to the brightness of a visual stimulus. Brightness, however, depends on many variables such as the preceding or surrounding illumination, and these variables are not taken into account in specifying luminance. Thus, the luminance of a stimulus is often of limited value in specifying brightness. The term luminance should be reserved for the specification of radiances, and the term brightness should be reserved for a description of the appearance of a stimulus. Also, any individual may have a spectral sensitivity function that differs from that of the CIE standard observer. For this reason, precise studies often require stimuli in which the luminosity function is measured individually. This can be particularly important for studies with older observers for whom spectral efficiency is much lower than the CIE observer at short wavelengths [17]. Finally, while visual stimuli are often specified in terms of luminance, the illuminance of a stimulus on the retina is the critical value for visual performance. Retinal illuminance depends on the eye’s pupil size, 3 which varies with light level for a given observer and across observers for a given light level [18]. 14.2

PSYCHOPHYSICAL METHODS

When measuring psychophysical functions, such as the CSF, many different techniques can be employed, depending on the goals and context of the research program. Some of these techniques, such as magnitude estimation and hue scaling, are designed to quantify the perceptual experience created by a visual stimulus, such as its brightness or hue. These issues may become relevant in the context of AO in the future when assessing the more subjective consequences of improved retinal image quality. However, in the short term, the more critical questions for the AO researcher will probably concern the extent to which the limits of visual performance can be extended through higher order wavefront correction. So, rather than asking how a stimulus appears, AO researchers using psychophysical methods will often be asking whether the stimulus can be seen, discriminated from other stimuli, or identified. This section of the chapter will address some of the theoretical and procedural issues associated with answering these questions. 14.2.1

Threshold

The limiting value of a physical variable (e.g., number of quanta, contrast) for a criterion level of performance on a psychophysical task is referred to as 3 When apparent pupil area (natural or due to an artificial pupil) is known, stimuli can be specified in terms of retinal illuminance. The most common measure is the troland (luminance in cd/m 2 × area of the eye’s pupil in mm).

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the threshold. The inverse of threshold is sensitivity. Historically, thresholds were thought to be discrete barriers to perception. The threshold was considered either the stimulus strength above which the stimulus could always be detected (absolute threshold) or the difference between two stimuli beyond which they were always distinguishable from one another (difference threshold). However, this notion of threshold does not take into account the inherent variability in physical stimuli and the physiological mechanisms transducing these stimuli. Due to external and internal variability, a stimulus of a particular intensity may be detectable on a given experimental trial and undetectable on another seemingly identical trial. To complicate matters further, when observers are forced to make a choice about the presence or absence of lowintensity stimuli that they claim to be invisible, they will often display abovechance performance. Thus, rather than discussing thresholds as discrete barriers above which a stimulus is just detectable or for which two stimuli are just noticeably different, we consider thresholds in a statistical sense.

14.2.2

Signal Detection Theory

Our modern understanding of a sensory threshold has been greatly influenced by signal detection theory (SDT). This theory attempts to explain the variability in measures of detection in a systematic and quantitative fashion. In SDT, thresholds are assumed to depend on two independent processes: a sensory process and a detection process. According to SDT, the sensory process is noisy. As a consequence, the output of this process in response to a given stimulus input is variable. Critically, the sensory process also outputs a variable spontaneous response due to noise alone when no stimulus is presented. The task of the decision process is to distinguish between sensory activity produced in response to a stimulus (signal plus noise) and sensory activity produced in response to no stimulus (noise alone). However, in the SDT framework, the decision process only has access to a single output value from the sensory process, and thus a priori cannot distinguish between signal-related and noise-related activity. SDT presents a theoretical framework for understanding how sensory systems might be able to solve this problem. To understand SDT, it is helpful to consider a detection experiment. Imagine that in this experiment, on a given trial, a target stimulus (e.g., a low-contrast grating) is either presented or not. The observer’s task is simply to state “yes” if it is believed that the stimulus has been presented or “no” if it is believed that the stimulus has not been presented (a yes/no procedure). There are four possible results on any given trial. One possible result is that the observer says yes and the stimulus was presented on that trial. This is called a hit. Another possibility is that the observer responds yes and stimulus was not presented. This is called a false alarm. A correct rejection occurs when the stimulus was not presented and the observer indicates no. Finally,

372

VISUAL PSYCHOPHYSICS WITH ADAPTIVE OPTICS

when the stimulus was presented and the observer says no, this is called a miss (Fig. 14.3). Consider the trials on which no stimulus is presented, and the observer is simply viewing a blank field. During these trials, according to SDT, the sensory process will be generating a response that is entirely due to noise. Since this response is due to random activity in the sensory system, the output produced by noise alone will vary in strength from trial to trial. If we assume that this random noise response is generated by many independent sources, then it may be represented by a Gaussian probability density function, as shown in Figure 14.4. This distribution of response strengths is called the noise distribution. Now consider the probability of a given response strength on trials when a stimulus is presented. If the effects of the signal on the system are independent of the noise effects, and the response strengths from these sources are additive, then the resulting signal-plus-noise probability density function will also be a Gaussian function with similar variance. The signal-plus-noise distribution will be shifted to the right along the sensory strength axis relative to the noise-alone distribution (Fig. 14.4). How far the signal-plus-noise distribution is shifted will depend on the effectiveness of the stimulus for producing a sensory response. In SDT, the same output from the sensory system may be expected during some trials in which no stimulus is presented—that is, a noise-alone trial— and some trials in which a stimulus is presented—that is, a signal-plus-noise trial. If the same output can be produced in these two kinds of trials, how does the observer decide to say yes or no on a given trial? Notice that the likelihood that a given sensory response strength was generated by signal plus noise, rather than noise alone, increases as the signal strength increases. A sensible strategy would be to say yes when the sensory strength is above a certain value and to say no when it is below that value. This value is termed the criterion. The dashed vertical line in Figure 14.4 represents a criterion

Yes

No

Stimulus Present

Hit

Miss

Stimulus Absent

False Alarm

Correct Rejection

FIGURE 14.3 The possible outcomes from a trial in a detection experiment utilizing a yes/no procedure.

PSYCHOPHYSICAL METHODS

373

′

FIGURE 14.4 Theoretical sensory response strength probability density functions for noise-alone and signal-plus-noise trials. According to SDT an observer will report that a stimulus is detected only when the sensory strength is above criterion (represented by the vertical dashed line). The portion of the noise distribution fi lled with lines oriented to the right represents false alarms. The portion of the signal-plus-noise distribution fi lled with lines oriented to the left represents hits. d′ is defi ned in Eq. (14.7).

that a hypothetical observer in our experiment might select. During some trials in which the stimulus was absent (noise-alone), the sensory output will be higher than the criterion. Thus, an observer using this criterion will sometimes say that the target stimulus was present when it was not—a false alarm. In other trials, when a stimulus was presented (signal-plus-noise), the sensory output will be lower than the criterion, and the observer will fail to detect the stimulus—a miss. Suppose that the observer wished to make fewer false alarms and therefore more correct rejections. To do this, the observer could adopt a higher criterion (which could be represented by moving the dashed vertical line in Fig. 14.4 to the right) so that the noise-alone outputs would exceed it less frequently. If the observer did this, there would be an increase in the number of trials in which the observer would fail to detect the stimulus when it was presented— hit rates would decrease. In other words, there is a trade-off between hits and false alarms. The relation between hits and false alarms provides a way of measuring the criterion that an observer adopts. When the effect of the signal remains constant, a relatively low frequency of hits and false alarms indicates a high or cautious criterion, whereas a high frequency of hits and false alarms indicates a relatively low or lax criterion.

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One factor affecting the choice of criterion is how willing the observer is to accept risk. Some observers are simply more cautious than others and will thus choose a more stringent criterion. The perceived impact of hits, misses, false alarms, and correct rejections also plays a critical role in determining criterion. For example, if the reward for a hit is significantly greater than the cost of a false alarm, observers will tend to adopt less stringent criteria. An additional important factor affecting criterion is the observer’s expectancy about the frequency of trials in which a stimulus is presented. Observers will be more willing to say they saw a stimulus if signal-plus-noise trials are more frequent than noise-alone trials. The observer’s sensitivity to a stimulus in an experiment can be described in terms of the distance between the noise-alone distribution and the signalplus-noise distribution. This sensitivity parameter is called d-prime (d′). Assuming that the distributions are normal as depicted in Figure 14.4, d′ is the difference between the mean of the signal-plus-noise distribution (s + n) and the mean of the noise distribution (n¯) divided by the standard deviation of the noise distribution, (s n): d′ =

( s + n) − n σn

(14.7)

The quantity d′ is a measure of sensitivity, independent of the observer’s criterion and expectancies, and it can be calculated from the proportion of hits and false alarms. When d′ = 0, the observer is performing at chance levels. When stimulus strength is reduced, the value of d′ decreases. Correspondingly, with more intense stimuli, detection is easier and d′ increases. Threshold may be defi ned as the stimulus that produces a particular value of d′, often 1.0. In addition to estimating sensitivity, SDT offers a set of techniques for estimating response bias. This process is beyond the scope of this chapter, but theoretical presentations of SDT in the context of psychophysics are provided by Green and Swets [19] and by Van Trees [20]. Practical user’s guides to SDT are provided by Macmillan and Creelman [21] and Wickens [22]. 14.2.3

Detection, Discrimination, and Identification Thresholds

One type of threshold that is frequently measured is the detection threshold. The detection threshold is the stimulus strength necessary to elicit a specific level of performance on a task in which an observer is asked to state if, when, or where a stimulus was presented. The level of performance that corresponds to the threshold will depend on the methods employed. Another important type of threshold is the discrimination threshold. This is the difference along a particular stimulus dimension that is necessary for the observer to correctly differentiate two or more stimuli with a given probability. In some contexts, the discrimination threshold is referred to as a just noticeable difference (jnd). The increase or decrease in stimulus intensity

PSYCHOPHYSICAL METHODS

375

necessary to detect a difference reliably depends on the initial intensity of the stimulus [23]. Specifically, if I is the intensity of the stimulus and ∆I is the change in intensity necessary to detect the change (the jnd) and KW is a constant, then ∆ I I ≈ KW

(14.8)

This relation is referred to as Weber’s law and is a fundamental principle in psychophysics. In this equation, KW is known as the Weber fraction. Although Weber’s law has wide generality, it tends to break down for very low and high stimulus values. A common discrimination task is contrast discrimination. In such a task, the observer is presented with two patterns that differ only in their contrast. The observer may be asked to indicate which of the two patterns is higher or lower in contrast. The contrast discrimination threshold will be the contrast difference necessary to yield a criterion level of performance on this task. The contrast detection measure mentioned above can be thought of as a special case of contrast discrimination where the contrast of one of the patterns is zero. A third type of threshold is the identification threshold. In an identification task, an observer is asked to state which stimulus was presented on a particular trial. A familiar example of such a task is the acuity chart used in an optometrist’s office. In this task, the patient is asked to read lines of letters. The letters are made progressively smaller until the patient is no longer able to correctly identify a given proportion of the letters. The smallest line on the chart on which most of the letters can be identified accurately can be thought of as the threshold. This particular identification threshold is referred to as visual acuity. 14.2.4

Procedures for Estimating a Threshold

There are several procedures for estimating thresholds, and each method has advantages and disadvantages. In discussing methods for estimating thresholds, contrast detection threshold will be used again as a representative example. However, it should be noted that some variant of each of these methods can be used for discrimination and identification tasks, as well. The quickest way to estimate a threshold is usually the method of adjustment. In this technique, the observer controls the stimulus strength directly. In the case of a contrast detection task, the observer adjusts the contrast of the stimulus until it is just barely visible. When satisfied with the setting, the observer makes some indication to end the trial, often with the press of a button. This procedure is repeated several times, and the mean of all settings is taken as the estimate of threshold. This procedure is intuitive and is generally accomplished reliably and comfortably after some practice. However, the visual sensation that corresponds to “just barely visible” may differ between

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observers or even within the same observer under different conditions. In other words, the criterion that is used to decide when a stimulus is just detectable depends on the judgment of the observer and cannot be assessed objectively with this technique. When the observer in the threshold task is naïve to the purposes of the experiment and has no vested interest in the outcome of the study, differences in criteria will contribute to noise in the data and can make comparisons across observers or between conditions difficult to interpret. However, the problem becomes particularly insidious when the observer is also an experimenter on the project. No matter how hard one tries to maintain a constant criterion, the danger is always present that one’s knowledge of project aims can influence threshold adjustments. An alternative method for assessing thresholds is the yes/no procedure (see also Section 14.2.2). Accurate measures of threshold can be obtained using this technique; however, like the method of adjustment, the yes/no procedure is also susceptible to criterion effects. The accuracy and efficiency of this method will depend on the stability of the criterion used by the observer. While this is not an intractable problem, the net result is that a large amount of data is needed to pull apart the effects of criterion from the effects of the underlying sensory response. This concern is of particular importance for psychophysical experiments with AO systems, where the correction presumably has some temporal dependence, and where it would be desirable to test patients who are not experienced with lengthy psychophysical tasks. The N-alternative-forced-choice (NAFC) procedures are relatively efficient, criterion-free techniques for measuring sensory thresholds. In this class of procedure, the observer is given 2 or more response options on a given trial and is obliged to respond even if no stimulus is detected. For example, in a temporal 2-alternative-forced-choice (t2AFC) procedure, a stimulus is presented in one of two periods of time, which can be delineated with auditory (e.g., tones) or visual (e.g., the numbers 1 and 2) markers. The observer indicates, often with the press of a button, whether the stimulus appeared in interval 1 or 2. In such a procedure, the observer will tend to guess correctly 50% of the time when the stimulus is so far below threshold that it is never detected (assuming an equal number of presentations in each interval). The particular level of performance that is taken as threshold depends on underlying assumptions about the probability density function that will defi ne the relation between stimulus strength and the probability of the stimulus being detected (as described in the next section). Often, 75% correct is used to defi ne threshold-level performance. While 2AFC procedures are quite common, the larger the N used in an NAFC procedure, the more efficient it will be. This is simply because the correct guessing rate will approach 1/N with large numbers of trials, and correct answers yield more information the lower the guessing rate.

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Identification tasks can be analyzed as NAFC procedures. For example, reading letters on an eye chart could be considered a 26-alternative-forcedchoice procedure. In this case, the guessing rate would be 1/26 if all letters were equally identifiable and equally probable. 14.2.5

Psychometric Functions

The probability of correctly detecting a stimulus increases smoothly as a function of stimulus intensity, when operating in a stimulus regime that is close to threshold. This relation between task performance and stimulus strength is referred to as the psychometric function (not to be confused with psychophysical functions discussed above). The psychometric function can be understood in terms of the concepts of SDT. Take the example of a t2AFC detection task. Assuming that the observer is paying attention for both intervals and responding based on sensory information alone, the observer responds correctly when the sensory response produced during the stimulus (signal-plusnoise) interval is greater than the sensory response produced during the no-stimulus (noise-alone) interval. When the stimulus strength is low, the signal-plus-noise and noise-alone distributions are highly overlapping. There will be many trials in which the sensory response due to noise alone will be greater than that due to signal plus noise. However, as the signal strength increases, the probability will be much greater that the larger sensory response is due to the signal. Thus, as signal strength increases, the proportion of trials in which the observer makes a correct response increases. Notice that in the 2AFC procedure mentioned here, criterion plays no role. Several distributions model the psychometric function acceptably well. Two of the most commonly used distributions are the cumulative normal function and the Weibull function [24]. The cumulative normal form of the psychometric function is used in signal detection theory based on the assumption that there are multiple independent sources of variability with unknown distributions that are feeding into a unitary sensory process. The Weibull function is used in detection models that assume probability summation among multiple independent detection mechanisms. Figure 14.5 shows psychophysical data from a contrast detection experiment. The x axis is the log10 of the stimulus (Michelson) contrast, and the y axis is the proportion of correct responses. In this task, the observer was instructed to indicate whether a Gabor patch was presented on the left or right of a central fi xation mark (spatial 2AFC). The data are fitted with a cumulative normal distribution using a maximum-likelihood fitting procedure. This approach is called Probit analysis [25]. The model has two parameters: the mean and the standard deviation. The mean of the distribution is commonly taken as the threshold. For a 2AFC task and a cumulative normal distribution, the mean corresponds to 75% correct. The standard deviation corresponds to the slope of the psychometric function.

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Percent Correct

100%

75%

50%

-3.0

-2.5

-2.0

-1.5

-1.0

Log Contrast FIGURE 14.5 Psychometric function from a luminance contrast detection task. Squares represent the percent correct data from a 2-alternative-forced-choice procedure in which the observer indicated on which side of fi xation a Gabor patch was presented. The data are fitted with a cumulative normal distribution.

14.2.6

Selecting Stimulus Values

To accurately estimate the mean and/or standard deviation of a psychometric function, several data points based on many trials are needed. Obtaining these data points efficiently requires a judicious choice of stimulus values (e.g., contrasts) to be tested. The goal is to test a range and distribution of stimulus values that will yield performance values from near the guessing rate to approaching 100% correct, while including enough points in between to fit the appropriate distribution to the data. Testing several stimulus values that are too far below threshold, where the subject is always guessing, or too far above threshold, where the subject is always correct, results in wasted trials. There are two basic ways to approach the choice of stimulus values and the number of trials to be performed. One approach is to choose the values based on information acquired from experience with the task, perhaps from preliminary experiments using other methods (such as method of adjustment) or from published data. The other approach is to choose the stimulus values and the number of trials dynamically, based on the performance of the observer in a given session. The stimulus values and the number of trials at each value to be tested are chosen in advance in the method of constant stimuli. On a given trial, the stimulus value is chosen at random (without replacement) from the predefi ned choices. This is probably the best method to use when interested in knowing

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the exact shape of the psychometric function. With equal numbers of trials distributed evenly across the function, results obtained using this method can provide excellent fits to the psychometric function, given a large amount of data. When good prior information is available about the approximate mean and standard deviation of the psychometric function, the method of constant stimuli can be reasonably efficient and yields precise results. When such information is not available, the method of constant stimuli can be cumbersome, and a method that selects stimulus values dynamically based on performance may be preferred. Such procedures are referred to generally as staircase methods. When the observer is performing well, the task is made more difficult. When the observer makes errors, the task is made easier. As a result, a plot of stimulus strength (e.g., contrast) versus trial number resembles stairs that rise and fall intermittently. There are many variants of the staircase method. One simple choice is the M-up/N-down procedure [26]. One form of this procedure used often in conjunction with a 2AFC task is the 1-up/3-down version. Take the example of a contrast detection task again. In such an experiment, the staircase might begin at a fairly high contrast level and be reduced (“down”) each time the observer makes 3 consecutive correct responses. When the observer can no longer detect the stimulus and makes a single incorrect response, the contrast is increased (“up”). This point of inflection is called a reversal. As the experiment proceeds, the staircase converges toward a particular percent correct performance level (~79% for the 1-up/3-down rule). After a predetermined number of reversals, the experiment is terminated, and the last several reversals are averaged as an estimate of threshold. While the M-up/N-down staircase procedure is simple and intuitive, it is not the most efficient choice with respect to the number of trials necessary to achieve a given statistical confidence level on the estimate of threshold. More efficient adaptive psychometric methods have been developed [27–29], and the use of these efficient methods is now well established in the vision science literature. Two commonly used methods are called QUEST and ZEST. These adaptive psychometric methods rely on Bayesian statistics to combine prior information about the probability density function of the threshold with the ongoing results of an experiment to determine a maximum-likelihood estimate of the threshold. On each trial, the threshold is estimated and the stimulus value that corresponds to that estimate is presented to the observer. Based on whether the observer is correct or incorrect, the threshold estimate is adjusted. The net result is a systematic and sensible way of choosing stimulus values on each trial such that all trials are used efficiently. In addition, these methods offer a variety of justifiable termination rules. For example, a set number of trials for the experiment can be selected in advance, or the experiment can be terminated when a particular confidence interval is achieved. An example of the trial-by-trial estimates of threshold from an experiment performed using a QUEST procedure is shown in Figure 14.6. Two functions are plotted in this figure, each representing an estimate of threshold as a function

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Contrast

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Trial Number FIGURE 14.6 Trial-by-trial estimates of threshold for two independent measures from the same observer using the QUEST staircase procedure. Each point represents the value of contrast presented on a given trial in a temporal 2AFC contrast detection task.

of trial number for two statistically independent staircases. The two staircases were obtained in the same session, and in a given trial, the contrast presented was randomly selected from one of the two staircases. Randomly interleaving staircases in this way has the important advantage of helping to maintain the statistical independence of each trial [27], in addition to providing an additional estimate of the threshold. Each method has its relative merits. For example, QUEST is quite efficient and easily implemented in a short computer program. An implementation of this procedure is provided in the PsychToolbox software package for MATLAB [30, 31] (available at http://www.psychtoolbox.org). A version of the QUEST procedure written in the C programming language is provided by Farell and Pelli [32]. ZEST, a close relative of QUEST, is computationally more complex but somewhat more efficient. It should be noted that adaptive staircases can be used to estimate the slope of the psychometric function as well as the threshold [33].

14.3

GENERATING THE VISUAL STIMULUS

Measuring psychophysical functions requires careful control of the physical stimulus parameters. Some of the biggest practical concerns for vision scientists involve the production of visual stimuli. Many devices for producing

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highly controlled visual stimuli exist, and creative engineers and vision scientists are constantly coming up with new machines for this purpose. Historically, stimuli for psychophysical experiments were produced in elaborate optical systems, or with a bank of function generators, custom electronics, and oscilloscopes. In recent years, however, psychophysicists are relying more heavily on stimuli generated by computers, often displayed on monitors. Computer-controlled display systems are popular because they provide a high degree of control over the stimulus to be presented and are available at a reasonable cost. However, computer-controlled displays have some disadvantages. In particular, luminance and chromaticity limitations make them inappropriate for some applications. When these factors limit production of suitable stimuli, other systems need to be considered. 14.3.1

General Issues Concerning Computer-Controlled Displays

While almost any device that can generate light can be controlled by a computer, this section discusses commercially available display solutions that work with desktop or laptop computers, such as cathode ray tube (CRT) monitors, liquid crystal display (LCD) monitors, plasma screen monitors, and light projector systems. For the most part, these displays have primary commercial functions that are not optimal for vision science applications. However, with proper software control, accurate characterization, and minor hardware additions or modifications, these displays can act as flexible and robust tools for studying vision. Computer-controlled displays offer many advantages for the vision scientist. Once the display is characterized, the user can accurately specify color and luminance values on a pixel-by-pixel basis. The temporal and spatial resolution of most of these devices is adequate for many vision research applications. In addition, computers with high processing speeds and large amounts of memory can be purchased for relatively low cost. Finally, when using computer-controlled displays, the same system that generates the visual stimulus can also conveniently collect, store, and analyze experimental data. Several factors need to be weighed when considering which type of computer-controlled display will be best for a given research goal. One important consideration is the luminance output of the display. A typical commercially available CRT computer monitor, for example, produces a maximum luminance output of about 100 to 150 cd/m 2 when using all three color channels. While these values are sufficient for many purposes, the effective maximum luminance level will depend on several factors, including the desired chromaticities. The effective luminance of the display will be reduced when viewed through the components of an optical system. This issue can be of particular importance in AO systems where beamsplitters or polarizers are used, often greatly reducing the amount of light available from the display. Other types of display technology can provide higher luminance values but often at the cost of spatial or temporal resolution.

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There are important chromatic limitations to be aware of for most computer-controlled displays. Most emit relatively broad spectral bands of light (see Fig. 14.7 for an example) and this can be problematic for some research applications. Filters can be used with these displays to produce narrow-band light, but the luminance will be greatly reduced, making this solution impractical in most cases. It should also be noted that not all chromaticities in the visible spectrum can be produced using a computer-controlled display. The gamut, or range of displayable chromaticities, depends on the chromaticities of the light emitted from each of the color channels used in the display. Figure 14.8 shows the chromaticities available from a typical CRT display in CIE xy chromaticity coordinates. The gamut of the CRT is a triangle in CIE chromaticity space with vertices defi ned by the chromaticity coordinates of the phosphors. All chromaticities within this triangle can be obtained at a low luminance level, but the gamut is increasingly restricted at increasingly higher luminance levels. Additionally, the intensity resolution allowed by the computer should be considered. Most computers are supplied with 8-bit graphics cards that allow 256 (2 8) discrete levels of luminance for each color channel. This resolution level is insufficient for some applications (e.g., contrast threshold experiments) that require very low luminance contrast values or subtle chromatic variations. A variety of options are available to obtain greater luminance and color resolution, and these are discussed in a later section. Display devices vary in their temporal and spatial characteristics. Frame rates of 75 Hz or higher are desirable to reduce the perception of fl icker and

Power

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Wavelength (nm) FIGURE 14.7 The spectral power distributions of a typical set of CRT phosphors are shown. Note that the red phosphor has two major peaks.

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y Chromaticity

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x Chromaticity FIGURE 14-8 The chromaticities of typical CRT phosphors are shown by the open circles plotted in the CIE xy chromaticity space. The thick lines show the maximum chromaticity gamut available using these phosphors.

to allow for precise temporal modulation. Most commercially available computer-controlled displays are designed to operate at this or higher rates. However, some experiments require greater temporal resolution, and displays that operate at higher frame rates are available, although there may be a trade-off between temporal and spatial resolution. Additionally, frame rate is not the only consideration for the temporal resolution of a display. The persistence of the light source is also critical. For example, phosphors differ in how quickly they reach maximum light output when stimulated and return to minimal output when turned off. Also, spatial resolution should be at least high enough so that pixels cannot be resolved at the desired viewing distance. Generally, higher spatial resolutions are better. The highest spatial frequencies that can be produced will depend on the size of the pixels as measured in the retinal image. A display with relatively large pixels (low spatial resolution) can produce high-spatial-frequency patterns only if the observer is placed far enough away. However, this will limit the effective size of the display, making it difficult to display low-spatial-frequency information. Thus, only a limited range of spatial frequencies can be displayed at any given magnification. Display devices are useful for research purposes only if they can be characterized accurately. Characterization (or “calibration”) allows for accurate specification of chromatic and luminance values and is relatively straightforward to perform if the following assumptions hold:

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• Spectral Constancy The relative spectral power distribution of each color channel should be constant over the range of intensity values. • Color-Channel Independence The output of each color channel should be unaffected by the output of the other channels. • Spatial Homogeneity The luminous output of each pixel should be consistent across the display. • Spatial Independence The output of each pixel should be unaffected by neighboring pixels. It is recommended that these assumptions be checked within the operating range of the experimental application. Complicated software manipulations are possible to correct for some violations [34, 35]. A few other issues should be considered. Ideally, when the input to the red, green, and blue (RGB) channels is zero, no light should be emitted from the display. However, most display types produce some residual light at this setting. This can be minimized by adjusting the brightness and contrast settings but can be a persistent problem for some display types. Also, the intensity of the light from a pixel can vary with the viewing angle for some displays, as is the case with LCDs. This problem can be reduced if the head position of observers is stabilized. The space available for a display might be an issue for some applications. Flat screen displays take up less space than traditional CRTs while some projector-based displays require a lot of room. Finally, the cost of a display can range from a few hundred dollars for CRTs to many thousands of dollars for some projector-based displays and plasma screens. More detailed information on computer-controlled displays can be found elsewhere [36, 37]. 14.3.2

Types of Computer-Controlled Displays

Cathode ray tube (CRT) displays emit light when an electron beam excites a phosphor coating on a display screen. A monochrome monitor uses one electron beam to excite a single phosphor type that emits broadband light. A color monitor uses three electron beams to separately excite three different phosphors that are dominated by short-, middle-, or long-wavelength light. Higher luminance values can be obtained using some monochrome monitors (e.g., those designed for specialized applications such as medical imaging) compared to color monitors. For chromatic stimulation at higher luminance values, other display options might be more appropriate. The monitor is made up of thousands of picture elements known as pixels. In a color monitor, each pixel is made up of red, green, and blue components. By varying the intensity of the electron beam, the intensity of the light emitted can be manipulated. The electron beams scan the screen in a raster pattern from left to right, moving rapidly from the top to the bottom of the screen many times per second (e.g., 75 Hz). Some CRTs can operate at high frame rates (>150 Hz).

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However, specialized graphics cards and monitors are usually necessary to take advantage of these rates, and spatial resolution may be compromised when operating in these modes. The persistence of CRT phosphors is generally quite short with a relatively quick return to minimal light output; however, different phosphor types have different persistence times. Cathode ray tubes are an attractive option for vision research because their characteristics are well understood and they are relatively inexpensive. The assumptions of spectral constancy, spatial homogeneity, color-channel independence, and spatial independence hold to a close approximation for most CRTs over a useful range [36, 38]. Liquid crystal display (LCD) monitors are commonly used in laptop computers and are increasingly supplied as standard equipment with desktop computers. They use polarized light that is transmitted through aligned liquid crystals. The alignment of the crystals can be disturbed by applying an electric current. This results in decreased light transmittance. In an LCD display, a polarized light source is positioned behind an array of liquid crystal elements that are controlled separately to produce an image. Color images are created when three separate RGB light sources are used. The luminosity of LCDs can be much higher than CRTs because it is mainly a function of the choice of backlights. LCD displays are also much more compact than CRT displays. There are, however, some disadvantages from a vision science perspective. For one, the intensity of the display output can vary greatly depending on the viewing angle. Also, useful refresh rates tend to be lower than with CRTs since the persistence of the liquid crystal elements is longer than that of CRT phosphors. Another potential problem with LCD monitors is that the spectra of the RGB outputs can vary as the intensity changes [36], which may be problematic when attempting to characterize the display. Plasma displays produce light by exciting plasma gas pockets coupled to phosphors. Plasma gas contains about equal numbers of positive ions and electrons. When this balance is disturbed by an electric current, the gas becomes excited, causing photons to be released. The photons emitted by the gas are mostly in the ultraviolet (UV) part of the spectrum and thus invisible. However, this energy can be used to excite phosphors, which do emit visible light. In a plasma display, there are many separate gas cells that can be stimulated independently. Since the plasma cells can be stimulated independently with no scanning electron beam, spatial homogeneity and spatial independence can be excellent with these displays. This also allows for the possibility of very large displayable areas. The luminance output and chromaticity gamut will depend on the phosphors being used but will be similar to that of CRTs. Plasma technology is fairly new and currently relatively expensive. Projectors are an important option for vision scientists because they can produce chromatic images at high luminance levels. Many home entertainment systems employ self-contained projection and screen setups that look like large TVs. Projectors and screens can also be purchased separately. There are a number of issues to consider when projection systems are used for vision

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research. Keystone distortion can occur when the image is projected at an angle (e.g., a circle may appear as an oval). However, this distortion can normally be corrected by adjusting controls on the projector. Projectors often produce much higher luminance values at the center of the screen than at the edges. This can be countered to some degree by using specially designed screens (e.g., Fresnel/lenticular screens). Spatial resolution can be poor for some types of projectors. The resolution will also vary with the projection distance. Separate projector and screen systems can also require a lot of space. Three main types of projector technologies are currently available. CRT projectors use very bright CRT tubes combined with lenses to project images. Various configurations are used, but CRT projectors are becoming less popular now that improved image quality can be obtained using alternative technology. LCD projectors use the same technology as LCD displays but employ more powerful backlights combined with a lens to project the image. The spatial resolution is typically better than with CRT projectors. Digital light projectors (DLPs) use a large array of digital micromirror devices (DMDs) that can be tilted independently. The DMDs reflect light onto the screen when in the normal position or into a light trap when tilted. The mirrors can be tilted many times per second, and the intensity of the light reaching the screen depends on the proportion of time the mirror is in the normal position. DLPs come in two main formats. The fi rst format uses one DMD array and a color wheel with separate RGB segments. Color images are produced by synchronizing the wheel with the mirror activity. With this type of display, refresh rates are typically low and colors can appear to separate with fast eye movements. This effect is sometimes referred to as rainbowing. The second format uses three separate DMD arrays that reflect three different light sources (RGB) that are combined to form a color image. This type of DLP avoids the issue of rainbowing but is more expensive. Packer et al. [39] reported that DLPs have good contrast, high light levels, and offer the potential for larger color gamuts than CRTs. 14.3.3

Accurate Stimulus Generation

With any of the display options discussed above, several issues will need to be addressed in order to generate the desired stimuli accurately. Some of the most important concerns are the accurate control of chromaticity and luminance. Most computers are sold with a standard 8-bit graphics card. Higher intensity resolution capabilities are required for some applications, for example, the low-contrast stimuli that are used in contrast threshold experiments. Figure 14.9 illustrates the importance of adequate luminance resolution for correct rendering of low-contrast spatial sine-wave patterns. At a mean luminance level of 35 cd/m 2 , the graphics card with 8-bit resolution is unable to produce a smoothly varying sine-wave pattern at 1% contrast. This problem

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8 bits 10 bits 14 bits

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FIGURE 14.9 Quantization effects for a luminance-modulated sine-wave stimulus with a contrast of 1% and a mean luminance of 35 cd/m 2 . The 14-bit resolution (solid curve) provides a luminance profi le that is very close to a sine wave. The 10-bit resolution (thick dashed curve) results in contrast artifacts (the desired maximum and minimum luminance levels cannot be generated), and the luminance profi le is a relatively poor approximation of a sine wave. The 8-bit resolution (thin dashed curve) cannot produce a sine-wave profi le at this contrast level.

is exacerbated at lower luminance levels and lower contrasts. It should be noted that some graphics cards that provide a greater number of possible luminance levels allow only 256 color values to be written to the display at any one time. This contrasts with the 8-bit per color channel (24 bits) of simultaneous colors available in most commercial graphics cards. This is not a problem for many vision experiments but may be problematic when attempting to display more naturalistic images. When using these systems, color look-up tables (CLUTs) are used to select the 256 colors to be displayed on a given frame. Clever use of CLUTs can greatly reduce the inconvenience of having a limited number of simultaneous colors. CLUTs are discussed in greater detail by Robson [40]. A simple way to obtain higher color resolution is to replace the existing graphics card with a commercially available 10-bit card. If the experiment requires monochromatic stimuli only, then the Pelli attenuator [41] is an attractive option. This device combines the outputs of the three 8-bit color channels to produce 12-bit monochrome resolution. If a color monitor is used, the output typically uses the green phosphor only as this produces the highest luminance. Bit stealing is another technique that provides higher resolution for monochromatic stimuli using a color monitor [42]. The three elements of each pixel are dithered to provide fi ner luminance steps. This can, however,

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result in some chromatic artifacts. This problem is reduced if the observer is far enough from the screen. Cambridge Research Systems (CRS) provides several “visual stimulator” products for producing high luminance and color resolution stimuli. The Bits+ + (see http://www.crsltd.com) takes input from a standard Digital Video Interface (DVI) graphics card and converts each channel to 14 bits using digital-to-analog converters. The Bits++ has three operating modes. The basic mode converts three 8-bit color channels to three 14-bit channels using a color look-up table, limited to 256 simultaneous colors. The mono mode converts two 8-bit channels to one monochrome 14-bit channel. The third mode, color, overcomes the previous 256-color limitation of the basic mode at the expense of horizontal spatial resolution. Both mono and color modes have the potential for all values to be displayed concurrently (i.e., true 14 bit). The Vision Stimulus Generator (VSG) systems, manufactured by CRS, have proven popular with vision scientists. These add-in cards for the personal computer (PC) provide up to 15 bits of resolution per color channel, but only 256 sets of RGB values simultaneously. The latest generation of this product line, the ViSaGe, is an external device that connects to a dedicated second monitor in a dual-monitor PC system. Changing color look-up tables and synchronizing image frames usually requires some real-time processing, and an important advantage of these systems is that a dedicated microprocessor provides high-speed temporal modulation of the stimuli, separate from the host computer operating system. These devices have many desirable features for vision science applications but are more expensive than other computer graphics generators. There is a large variety of software packages available for a range of psychophysical experiments (a comprehensive list can be found at the Vision Science website: http://www.visionscience.com/vs-software). However, offthe-shelf software packages will not be sufficiently flexible for many users. Software can be written from scratch using programming languages such as Java or C++; however, it is much easier to incorporate existing software libraries into your code. Two popular libraries specifically designed for vision science are the VSG Software Library (http://www.crsltd.com/catalog/vsl/) and PsychToolbox [30, 31] (http://www.psychtoolbox.org). 14.3.4

Display Characterization

To allow accurate specification of luminance and chromaticity, the display device must be characterized (or calibrated). Two main steps are required to characterize a display device. First, the chromaticity of each phosphor must be measured with a colorimeter or spectroradiometer. Second, the relation between display luminance and the digital-to-analog-conversion (DAC) input values needs to be characterized for each color channel. For CRTs, display luminance can be modeled as a power function (often referred to as a “gamma”

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function in this context) of DAC values. A typical gamma function is shown in Figure 14.10. When the chromaticities and gamma functions of the three color channels are characterized, the DAC values necessary to create the desired stimulus chromaticity and luminance values can be calculated, provided that the assumptions of spectral constancy, color-channel independence, spatial homogeneity and spatial independence hold. The necessary computations are described by Wandell [43] and Nakano [44]. While it is often useful to specify display outputs in terms of luminance and chromaticity values, it is increasingly desirable to make these specifications in terms of more physiologically meaningful units. Reliable measurements of the absorption spectra of human cone photoreceptors are available [45–48], though there are important individual differences in the peak sensitivity, optical density, and ocular media through which the light is fi ltered before reaching the receptors. Using this information together with the above display measurements, conversions can be made between RGB values and cone coordinates (or other related color spaces). A detailed discussion of display characterization and color conversions can be found in Brainard, Pelli, and Robson [36]. The appropriate measurement device must be chosen for proper display characterization. For monochrome display applications, a photometer may be appropriate. A photometer is a device that measures the radiance of a light source weighted by a fi lter that emulates a human spectral efficiency function. The output of this device is in terms of luminance. For color applications, a colorimeter or spectroradiometer will be required. A colorimeter is a light

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Input (DAC) Value FIGURE 14.10 Luminance of a CRT display is plotted as function of DAC value. The best fitting gamma function is shown. Luminance values have been normalized to unity.

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measuring device that uses three fi lters based on a set of color-matching functions derived from psychophysical experiments. Colorimeters perform best when measuring broadband light. They do not perform as well when measuring light that is narrow band or has sharp spectral peaks. Spectroradiometers measure light over a large number of steps across the visible spectrum. Because each measurement combines multiple estimates of light (e.g., 101 measurements at 4-nm steps from 380 to 780 nm), with some noise introduced at each estimate, the signal-to-noise ratio can be low, especially for lower intensities. Colorimeters and spectroradiometers are discussed in greater detail by Mollon [49]. On most displays, a number of adjustments (e.g., contrast and brightness) and display modes are available. Any adjustments should be made prior to making a display characterization, and no further adjustments should be made until the next characterization. The measuring device should be placed at the same distance and viewing angle as the position of the observer in the experiment. Regular characterization is recommended to take account of changes in the display device over time. Characterization is advised each time the display is moved since the surrounding electromagnetic fields can affect performance. 14.3.5

Maxwellian-View Optical Systems

Maxwellian-view optical systems were fi rst introduced by James Clerk Maxwell almost 150 years ago. The basic Maxwellian-view system uses a bright light source (such as a tungsten or xenon lamp) that is imaged by a lens onto the plane of the observer’s pupil. The observer sees the lens uniformly fi lled with light. The principle advantage is that high retinal illuminances can be obtained with spatially uniform narrow-band light. More detailed information on these systems can be found in Wyszecki and Stiles [15] and Westheimer [50]. A hands-on account of how to build a Maxwellian-view optical system is provided by Boynton [51]. Various additional devices can be used with Maxwellian-view systems to produce a variety of stimuli. Interference fi lters or monochrometers can be employed to produce narrow-band light. Beam choppers and polarizers can be used to create temporally modulated stimuli. Multiple channel systems can be used to produce complex stimuli. Maxwellian-view systems can be used to project discrete spots of light onto the retina by using a series of pinholes. These systems have been successfully used with AO systems for vision research purposes. For example, Hofer, Singer, and Williams [52] used a Maxwellianview system together with an AO system to stimulate single cones. 14.3.6

Other Display Options

Interferometry can be used to produce gratings on the retina that are independent of the optical aberrations in the eye. Two small points of coherent

REFERENCES

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light are passed through the pupil of the eye and interfere to produce gratings on the retina. This technique has been used to test the limits of visual resolution in the absence of optical aberrations [6, 10, 12, 53]. Other light sources might also be considered for visual stimulation [e.g., lasers and light-emitting diodes (LEDs)]. Finally, the vision scientist can use real objects to test visual performance, if the visual environment is carefully controlled (e.g., Kraft and Brainard [54]).

14.4

CONCLUSIONS

Psychophysics seeks to examine the relations between physical stimuli and visual performance and to understand the anatomical and physiological mechanisms that process those stimuli to create the world that we perceive. By increasing our understanding of the ocular optics forming the retinal image, the wavefront measurement technologies that are an integral part of AO are providing the psychophysicist with new tools for understanding the inputs to the visual system. Additionally, by correcting the higher order aberrations of the eye, AO allows very high resolution images to be placed on the retinas of human observers, providing the opportunity to present complex stimuli to the visual system more precisely than previously possible. In turn, what is learned through psychophysical means could be important to an engineer whose primary interest is in developing AO technologies. By understanding the fundamental neural limits to visual performance, we can assess the potential commercial viability of optometric and ophthalmic devices that rely on modern wavefront measurement and AO correction. For example, if it could be established that the visual system would benefit greatly from correction of higher order aberrations, this would drive the demand for new technologies that take advantage of AO to improve visual performance. Acknowledgments We thank Vicki J. Volbrecht, Lewis O. Harvey, Jr., John A. Wilson, David H. Brainard, Cynthia Angel, and William P. Hardy for their helpful comments on the manuscript. Supported by the National Institute on Aging (grant AG04058), the National Eye Institute (grant EY014743), and the University of California Campus Laboratory Exchange Program.

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23. Weber EH. Der Tastsinn und das Germeinful [The Sense of Touch and General Sensation]. In: Wagner R, ed. Handworterbuch der Physiologie, Vol. 3. Braunshweig: Vieweg, 1846; pp. 481–588. 24. Harvey LO. Efficient Estimation of Sensory Thresholds. Behav. Res. Meth. Instru. Comp. 1986; 18: 623–632. 25. Finney DJ. Probit Analysis. Cambridge, UK: Cambridge University Press, 1971. 26. Levitt H. Transformed Up-Down Methods in Psychoacoustics. J. Acoust. Soc. Am. 1971; 49: 466–477. 27. Watson AB, Pelli DG. QUEST: A Bayesian Adaptive Psychometric Method. Percept. Psychophys. 1983; 33: 113–120. 28. Harvey LO Jr. Efficient Estimation of Sensory Thresholds with ML-PEST. Spat. Vis. 1997; 11: 121–128. 29. King-Smith PE, Grigsby SS, Vingrys AJ, et al. Efficient and Unbiased Modifications of the QUEST Threshold Method: Theory, Simulations, Experimental Evaluation and Practical Implementation. Vision Res. 1994; 34: 885– 912. 30. Brainard DH. The Psychophysics Toolbox. Spat. Vis. 1997; 10: 433–436. 31. Pelli DG. The VideoToolbox Software for Visual Psychophysics: Transforming Numbers into Movies. Spat. Vis. 1997; 10: 437–442. 32. Farell B, Pelli DG. Psychophysical Methods, or How to Measure a Threshold, and Why. In: Carpenter RHS, Robson JG, eds. Vision Research: A Practical Guide to Laboratory Methods. Oxford: Oxford University Press, 1999, pp. 129– 136. 33. Kontsevich LL, Tyler CW. Bayesian Adaptive Estimation of Psychometric Slope and Threshold. Vision Res. 1999; 39: 2729–2737. 34. Brainard DH, Brunt WA, Speigle JM. Color Constancy in the Nearly Natural Image. I. Asymmetric Matches. J. Opt. Soc. Am. A. 1997; 14: 2091–2110. 35. Post DL, Calhoun CS. An Evaluation of Methods for Producing Desired Colors on CRT Monitors. Color Res. Appl. 1989; 14: 172–186. 36. Brainard DH, Pelli DG, Robson T. Display Characterization. In: Hornak J, ed. Encyclopedia of Imaging Science and Technology. New York: Wiley, 2002, pp. 172–188. 37. Cowan WB. Displays for Vision Research. In: Bass M, ed. Handbook of Optics: Vol. 1. Fundamentals, Techniques, and Design. New York: McGraw-Hill, 1995, pp. 27.1–27.44. 38. Brainard DH. Calibration of a Computer Controlled Color Monitor. Color Res. Appl. 1989; 14: 23–34. 39. Packer O, Diller LC, Verweij J, et al. Characterization and Use of a Digital Light Projector for Vision Research. Vision Res. 2001; 41: 427–439. 40. Robson T. Topics in Computerized Visual-Stimulus Generation. In: Carpenter RHS, Robson JG, eds. Vision Research: A Practical Guide to Laboratory Methods. Oxford: Oxford University Press, 1999, pp. 81–105. 41. Pelli DG, Zhang L. Accurate Control of Contrast on Microcomputer Displays. Vision Res. 1991; 31: 1337–1350.

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42. Tyler CW. Colour Bit-Stealing to Enhance the Luminance Resolution of Digital Displays on a Single Pixel Basis. Spat. Vis. 1997; 10: 369–377. 43. Wandell BA. Foundations of Vision. Sunderland, MA: Sinauer, 1995. 44. Nakano Y. Appendix Part III: Color Vision Mathematics: A Tutorial. In: Kaiser PK, Boynton RM, eds. Human Color Vision, 2nd ed. Washington, DC: Optical Society of America, 1996, pp. 544–562. 45. Smith VC, Pokorny J. Spectral Sensitivity of the Foveal Cone Photopigments between 400 and 500 nm. Vision Res. 1975; 15: 161–171. 46. Vos JJ, Estevez O, Walraven PL. Improved Color Fundamentals Offer a New View on Photometric Additivity. Vision Res. 1990; 30: 937–943. 47. Stockman A, Sharpe LT, Fach C. The Spectral Sensitivity of the Human ShortWavelength Sensitive Cones Derived from Thresholds and Color Matches. Vision Res. 1999; 39: 2901–2927. 48. Stockman A, Sharpe LT. The Spectral Sensitivities of the Middle- and LongWavelength-Sensitive Cones Derived from Measurements in Observers of Known Genotype. Vision Res. 2000; 40: 1711–1737. 49. Mollon JD. Specifying, Generating, and Measuring Colours. In: Carpenter RHS, Robson JG, eds. Vision Research: A Practical Guide to Laboratory Methods. Oxford: Oxford University Press, 1999, pp. 106–128. 50. Westheimer G. The Maxwellian View. Vision Res. 1966; 6: 669–682. 51. Boynton RM. Vision. In: Sidowski JB, ed. Experimental Methods and Instrumentation in Psychology. New York: McGraw-Hill, 1966, pp. 273–330. 52. Hofer HJ, Singer B, Williams DR. Different Sensations from Cones with the Same Photopigment. J. Vis. 2005; 5: 444–454. 53. Sekiguchi N, Williams DR, Brainard DH. Aberration-Free Measurements of the Visibility of Isoluminant Gratings. J. Opt. Soc. Am. A. 1993; 10: 2105–2117. 54. Kraft JM, Brainard DH. Mechanisms of Color Constancy under Nearly Natural Viewing. Proc. Natl. Acad. Sci. 1999; 96: 307–312.

PART FIVE

DESIGN EXAMPLES

CHAPTER FIFTEEN

Rochester Adaptive Optics Ophthalmoscope HEIDI HOFER, JASON PORTER, GEUNYOUNG YOON, LI CHEN, BEN SINGER, and DAVID R. WILLIAMS University of Rochester, Rochester, New York

15.1

INTRODUCTION

The Rochester Adaptive Optics Ophthalmoscope uses a Shack–Hartmann wavefront sensor with 221 lenslets and a continuous faceplate deformable mirror (Xinxtics, Inc.) with 97 lead–magnesium–niobate (PMN) actuators to measure and correct the ocular wave aberration over a 6.8-mm pupil. This mirror has high enough spatial resolution to correct aberrations up to eighthorder radial Zernike modes and enough stroke, ±2 mm per actuator, to correct a maximum peak-to-valley wavefront error of 8 mm. The system operates at rates up to 30 Hz, resulting in a 0.7-Hz closed-loop bandwidth, which is high enough to track most of the temporal fluctuations in the eye’s wave aberration [1]. Temporal performance is in good agreement with predictions based on theory. Residual root-mean-square (RMS) wavefront error is typically brought below 0.1 mm for a 6.8-mm pupil. The system incorporates both a floodilluminated retinal camera and a visual stimulus display for psychophysical experiments.

Adaptive Optics for Vision Science, Edited by Porter, Queener, Lin, Thorn, and Awwal Copyright © 2006 John Wiley & Sons, Inc.

397

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ROCHESTER ADAPTIVE OPTICS OPHTHALMOSCOPE

15.2

OPTICAL LAYOUT

15.2.1

Wavefront Measurement and Correction

Figures 15.1 and 15.2 show a schematic diagram and a picture (respectively) of the optical layout of the Rochester Adaptive Optics Ophthalmoscope. This system is a modification of the original system demonstrated by Liang, Williams, and Miller [2] that is largely identical to the system described by Hofer et al. [3]. The major difference between the previous systems and the current system is that the correcting element was upgraded from a 37-channel deformable mirror (DM) to a 97-channel mirror. The Shack–Hartmann wavefront sensor beacon is a collimated superluminescent diode (SLD) with a central wavelength of ~825 nm and a beam diameter of ~1 mm. The power of the SLD at the eye’s pupil is always kept at 5 mW or less during all experiments, more than 120 times smaller than the maximum permissible exposure for continuous viewing at this wavelength [4]. SLDs have very short coherence lengths (approximately 30 mm for our SLD), resulting in much less speckle in the

R

Eye 825-nm Superluminescent Diode

P

Pellicle Beamsplitter

SL

R

D R

Dichroic Mirror R

P

Off-axis Parabolic Mirrors

P

Pellicle Beamsplitter

Fixation Target

Deformable Mirror

R

Krypton Flash Lamp

Computer

R

Bleaching Lamp

P

Dichroic Mirror

CCD R

Lenslet Array

P R

CCD Wavefront Sensor

Retinal Imaging Camera

Focusing Lens R

DMD Visual Stimulus Display

FIGURE 15.1 Schematic diagram of the Rochester Adaptive Optics Ophthalmoscope. Planes marked R and P are conjugate with the eye’s retina and pupil, respectively.

OPTICAL LAYOUT

399

Deformable Mirror

Wavefront Sensor

Retinal Camera FIGURE 15.2 Photograph of the Rochester Adaptive Optics Ophthalmoscope in its retinal imaging mode. The major system components are labeled and the path that light follows once exiting the subject’s pupil is indicated with a black line. The stimulus display arm of the system is not shown in this image.

Shack–Hartmann spots than when using a coherent laser source. Because it is a near-infrared source, it also provides more comfortable and safer viewing conditions for the subject due to the eye’s decreased sensitivity at longer wavelengths. Before entering the eye, the SLD shares a common path with the illumination path from the retinal imaging flash lamp. To avoid losing more light from the flash lamp than necessary, the SLD is coupled into this path using a customized dichroic optic that reflects light over a narrow band centered at the SLD wavelength and transmits light of shorter (visible) or longer (infrared) wavelengths. A fi xation target conjugate with the retina also shares the flash lamp path and is coupled with an uncoated pellicle beamsplitter. To avoid unnecessary reflections in retinal images or the Shack– Hartmann spot images, the SLD–flash lamp path is coupled with the rest of the system using a 50/50 pellicle beamsplitter just before the eye’s entrance pupil. To eliminate the corneal reflection in the Shack–Hartmann images, an off-axis scheme for the SLD illumination is employed [5]. A 97-channel PMN deformable mirror (Xinxtics, Inc.) is used for the correcting element. (See also Chapter 4 for a description of this type of mirror.) It is very large, about 8 cm in diameter, and requires a very long path to magnify the eye’s pupil to nearly fi ll the entire mirror diameter. The use of

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ROCHESTER ADAPTIVE OPTICS OPHTHALMOSCOPE

curved mirrors instead of very large focal length lenses reduces the chromatic aberration of the system and allows the optical path to be folded to fit on the optical table without requiring extra optical components. To minimize aberrations, the two mirrors are 5° off-axis parabolic sections, with the foci at the two retinal planes located on either side of the deformable mirror. The mirror has ±2 mm stroke for each actuator, allowing a wavefront shift of ±4 mm in the reflected beam. The wavefront sensor uses a square lenslet array with a 24-mm focal length and 0.4-mm lenslet spacing. There is a small magnification difference between the entrance pupil of the system and the plane of the lenslet array, resulting in a sampling distance of 0.384 mm in the eye’s pupil. The Shack–Hartmann spots are recorded with a 12-bit, cooled, frame transfer CCD camera (PentaMAX-512EFT) with 15-mm square pixels in a 512 × 512 array. This camera has a maximum frame rate of 15 Hz that can be increased to 30 Hz if 2 × 2 binning is used, which results in an effective pixel size of 30 mm. For wavefront correction and reconstruction, only the Shack–Hartmann spots formed by the central 221 lenslets are considered. Figure 15.3 illustrates the configuration of the lenslets and mirror actuators relative to the eye’s pupil. Although the wavefront is corrected using a direct-slope method that allows mirror actuator voltages to be determined directly from Shack–Hartmann spot displacements (without the need for reconstructing the wavefront) [6], this sampling density allows Zernike modes up to and including the 10th radial order to be reconstructed for a 6.8-mm pupil. The wavefront is usually

FIGURE 15.3 The configuration of lenslets and mirror actuators relative to the eye’s pupil in the Rochester Adaptive Optics Ophthalmoscope.

OPTICAL LAYOUT

401

reconstructed in order to estimate the RMS wavefront error and Strehl ratio, which are monitored in real time to assess the quality of the wavefront correction. Before beginning adaptive correction, the subject roughly aligns his or her pupil by looking at the SLD superimposed on the fi xation target. The pupil is held steady by means of a dental bite plate. The subject’s spherical refractive error, if any, is subjectively removed by moving the bite plate and the fi rst lens of the system in tandem on an axial slide until the target appears to be in best focus. If the refractive error is very large or if a significant amount of astigmatism is present, trial lenses are also placed immediately in front of the eye. Lower order aberrations (defocus and astigmatism) are removed in this manner prior to adaptive correction so that the limited range of the deformable mirror is used only to reduce higher order aberrations and is not wasted on aberrations that can be removed by other means. The subject’s pupil position and focus offset are then refi ned by looking at the Shack–Hartmann spot image and the defocus coefficient reported by the wavefront sensor (see Fig. 15.4 for an example of the diagnostic display).

FIGURE 15.4 The diagnostic display and user interface for controlling and monitoring the adaptive optics system performance.

402

ROCHESTER ADAPTIVE OPTICS OPHTHALMOSCOPE

Once the initial alignment is complete, subjects are able to start each instance of adaptive correction themselves, when ready, with a keystroke on a small computer keypad. Aberrations are usually corrected over a 6.8-mm pupil diameter but can be corrected over only the central 6 mm if the subject’s pupil is not large enough. During the experiment, a shutter prevents the SLD light from entering the eye at any time except when the wavefront is being actively measured. Once the subject begins correction, the light from the SLD enters the eye, the wavefront is measured, and the deformable mirror actuator voltages are updated at a rate between 15 and 30 Hz, depending on the needs of the particular experiment. Within 5 to 10 iterations, which require only a fraction of a second to complete, the residual RMS wavefront error usually reaches its minimal value. Correction is automatically terminated after the residual RMS wavefront error reaches a prespecified value or after a maximum number of iterations, whichever comes fi rst. Immediately following correction, the adaptive optics computer can send a signal to either the flash lamp or a computer producing a visual stimulus, allowing a retinal image to be acquired or a stimulus to be presented. Figure 15.4 shows a picture of the user interface for the program used to monitor and control the adaptive optics system. The window named Spots (top left, Fig. 15.4) displays an image of the Shack–Hartmann spot array pattern. Superimposed on this image are the centroid locations and a grid of search boxes that defi nes the areas used for the centroid calculations (see also Section 6.3). Once the aberrations have been minimized and the spot locations have become stable, the size of the search boxes can be minimized to reduce the computational time required for calculating the centroid locations, thereby decreasing the delay of the system. Next to this window is the Wave Aberration window (top center, Fig. 15.4), which displays a contour plot of the current wave aberration reconstructed from the spot centroids. (In Fig. 15.4, this window shows a wavefront that is nearly perfectly flattened following adaptive compensation with the deformable mirror.) The Mirror window (middle center, Fig. 15.4) shows gray-scale values for each actuator that reflect the current actuator voltages sent to the deformable mirror and also displays whether the mirror is updating continuously (“in loop”) or is in a static state (“not in loop”). Clicking on individual actuators allows the operator to access the numerical values of the actuator voltages as well as to change the voltage sent to a particular actuator. The Console window (top right, Fig. 15.4) shows diagnostic information, such as the correction rate, the timing of individual frame measurements, when a visual stimulus or flash lamp was triggered, the current spherocylindrical correction required to minimize the second-order aberrations, and any response keys pressed by the subject. The Traces window (bottom, Fig. 15.4) shows a running trace of the values of the RMS wavefront error, Strehl ratio, and a specific Zernike coefficient. (The Zernike coefficient typically displayed is defocus. However, the user can choose to display any desired coefficient.) This information is primarily used to assess the correction

OPTICAL LAYOUT

403

performance of the system but may also be used to determine the best initial subject refraction. Other system parameters, such as the exposure time, binning, gain, maximum number of frames for correction, and/or minimum RMS wavefront error for stopping the correction, are controlled by dropdown menus at the top of the screen. (See Chapter 6 for a more detailed discussion of the adaptive optics system computer interface.) 15.2.2

Retinal Imaging: Light Delivery and Image Acquisition

The retina is imaged with flood illumination from a krypton flash tube. The imaging wavelength is controlled by interference fi lters. Typically used for retinal imaging is 550- and 650-nm light; however, both shorter (such as 500 nm) and longer wavelengths (such as 900 nm) are available with a different choice of interference fi lter. The flash illuminates a 1° circular field and subjects are asked to look at specific locations on the fi xation target to image different retinal eccentricities. The flash duration is set to 4 ms to avoid motion blur in retinal images due to eye movements. The intensity of the flash is controlled by changing the voltage across the tube or by changing the size of an aperture stop in the pupil plane of the flash lamp path. Changing the extent of the pupil through which light from the flash may enter the eye has a significant effect on the contrast of cone photoreceptors in the retinal images due to their waveguide properties. This phenomenon, where light entering through the pupil margins is less likely to be coupled into the cones than that entering through the pupil center, is known as the Stiles–Crawford effect [7]. Usually images of the best contrast are obtained when the diameter of the pupil aperture controlling the flash illumination is not larger than 2 to 3 mm. With these aperture sizes, the energy of a 550-nm flash entering the eye is typically 0.3 to 0.6 mJ per flash. The light from the flash exiting the eye follows the same path through the system as the wavefront sensor beacon until it reaches a dichroic mirror just prior to the lenslet array. This mirror directs all light of wavelengths above or below the SLD wavelength toward the retinal imaging charge-coupled device (CCD) camera. The imaging path consists of an aperture stop in a pupil conjugate plane, a 60-cm focal-length achromat mounted on an electronically driven movable stage (for focusing the retinal image), and a camera to collect the retinal image. The pupil aperture stop is set so that light from only the central 6 mm of the pupil is collected in the retinal image, avoiding edge effects from the deformable mirror correction. The lens responsible for bringing the retinal image into focus on the camera is mounted on a movable stage because there is generally a difference in focus between the wavelength used for wavefront sensing and that used for imaging due to the longitudinal chromatic aberration of the human eye. In this system configuration, the focus difference is compensated by translating the lens toward the camera. The focus offset generally required to bring the photoreceptor layer into best focus when imaging at 550 nm is approximately 0.8 diopter (D). The lens may also

404

ROCHESTER ADAPTIVE OPTICS OPHTHALMOSCOPE

be moved to focus on more anterior retinal structures, such as the blood vessels and capillaries. One of the drawbacks of using this particular scheme to compensate for the effects of the eye’s longitudinal chromatic aberration is that the magnification of the path changes as the lens position is changed. In an attempt to fi x the magnification of the retinal image, we plan to add an additional two lenses after the pupil plane in the imaging arm to gain access to an additional retinal plane. The telescope, consisting of the second of these new lenses and the current focusing lens, will then be mounted with the CCD camera on the same translatable plate so that the entire imaging arm of the system will move together. This will allow the focus to change while the magnification remains fi xed. The camera used for acquiring the retinal image is a cooled, back-illuminated CCD camera from Princeton Instruments. It contains a 512 × 512 chip with 24-mm square pixels. Magnifications resulting from typical positions of the focusing lens make one pixel correspond to 0.10 to 0.13 min of arc. For comparison, cone photoreceptors are approximately half an arcminute in diameter in the central fovea and are approximately 1 arcmin at 1° retinal eccentricity. 15.2.3

Visual Psychophysics Stimulus Display

Psychophysical experiments are conducted using a digital light projector (DLP) in the adaptive optics system. (See also Chapter 14 for more information on DLPs.) The DLP is coupled into the optical path via a mirror that is inserted between the focusing lens and the retinal imaging camera in the system’s imaging path. Visual stimuli displayed by the projector are reflected by the dichroic mirror and follow a reversed path back through the adaptive optics system. After passing through the off-axis paraboloids and the deformable mirror, these stimuli are projected onto the subject’s retina. The projector used to display visual stimuli (Compaq MP1600) contains a digital micromirror device (DMD) chip. When looking through the adaptive optics system, subjects directly view the visual stimuli displayed on the DMD chip. The color wheel and all projection optics in the path between the DMD chip and the mirror coupling the projected light into the adaptive optics system were removed from the projector. The color wheel could not be completely separated from the projector because the projector would function only if it detected a working color wheel. To remedy this problem, we removed the color wheel and attached it to a specially designed circuit, connected to the projector, which activates the wheel whenever the projector is powered. Therefore, the projector detects an operational color wheel despite its absence from the optical path. The DMD chip contains 1024 × 768 pixels, with a pixel size of 17 mm on an edge and a center-to-center spacing of 18 mm. Each pixel is a reflective micromirror that can rapidly tilt to alter the gray-scale value of its particular location in the image. Due to magnifications inherent in the optical system

CONTROL ALGORITHM

405

and focusing lens, one pixel on the DMD typically corresponds to 0.075 to 0.1 min of arc on the retina. This relationship places a minimum of nearly five pixels across the smallest of foveal cones. Each pixel has 8 bits of intensity resolution and has a response time of approximately 20 ms. Currently, we do not use any technique to enlarge the bit depth of the DMD chip, although this could be done by customizing the temporal control of the DMD chip. There are several advantages for using a DLP over a conventional cathode ray tube (CRT) monitor. (For comparisons between devices used to display visual stimuli, refer to Chapter 14.) DLPs can be controlled just like regular CRT monitors but can be made extremely bright and have good contrast levels [8]. The Compaq MP1600 has a brightness of 600 lm, and a contrast ratio of 150 : 1. (This contrast ratio is defi ned as the ratio of the maximum to minimum light outputs measured with an ANSI checkerboard pattern.) Visual stimuli are generally displayed on the DMD with custom software written using MATLAB® (The MathWorks, Inc.). For experiments involving monochromatic stimuli, the wavelength of the stimulus can be controlled by placing a narrow-band fi lter (10 or 25 nm bandwidths) immediately in front of the DMD chip. Adjusting the focusing lens can offset the difference in chromatic aberration between the 825-nm wavefront sensing wavelength and the wavelength of the fi lter in front of the DMD chip. A more elegant solution is to place an appropriate trial lens in the pupil conjugate plane of the imaging arm, as shifts in the position of the focusing lens will induce changes in the magnification of the DMD on the retina. Once subjects are aligned, they are able to initiate the adaptive correction and subsequent psychophysical procedure by pressing the appropriate key on a small computer pad or by pressing the appropriate button on an altered gaming joystick. Depending on the task, aberrations are typically corrected in real time throughout the duration of the procedure. For some psychophysical experiments, the adaptive optics system may also act as an aberration generator to simultaneously remove the subject’s native aberrations while superimposing a new subset or pattern of aberrations [9].

15.3

CONTROL ALGORITHM

The Rochester Adaptive Optics Ophthalmoscope uses a mirror control scheme in which the applied actuator voltages are determined by the directslope method [6]. In this method, an influence function is computed for each mirror actuator (see also Section 5.3 for a more detailed description). Each influence function specifies the Shack–Hartmann spot displacements caused by the movement of a single mirror actuator as a function of its applied voltage. These influence functions are then combined to construct a single matrix relating the Shack–Hartmann spot displacements directly to the actuator voltages. This method has the advantage that it is extremely quick, requiring only a single matrix multiplication, and does not require the reconstruction

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ROCHESTER ADAPTIVE OPTICS OPHTHALMOSCOPE

of the wavefront shape in terms of Zernike polynomials or other basis functions. Therefore, it is less sensitive to fitting errors and edge artifacts. The direct-slope control method is also somewhat self-calibrating and impervious to small misalignments between the deformable mirror and the lenslet array. A simple proportional control scheme is used to correct the wavefront and maintain correction. In each iteration, the actuator voltages required to null the wavefront are calculated from the Shack–Hartmann spot displacements and then a fraction of these voltages, the gain, is applied. The gain of the system is an adjustable parameter, with 30% gain usually providing the best results for our system. In a noisy system or a system with an inadequate sampling rate, a gain too high will result in an unstable correction. However, a gain too low will also result in a poor correction, as the system requires a long time to complete the correction and lacks the agility to deal with rapid changes in the wavefront. The integration time of the wavefront sensor CCD camera is an adjustable system parameter that also affects both the rate and delay time of the system. If the light level of the Shack–Hartmann spots is high, then four pixels can be binned into one without affecting the accuracy of the Shack–Hartmann spot centroiding process, and 33 ms camera integration time is adequate. In this case, the system corrects at a rate of 30 Hz. Under low light levels or if the quality of the Shack–Hartmann spots is poor, a longer integration time may be necessary to reduce noise. The wavefront sensor camera runs in a double-buffer mode, which means that one frame transfers to the computer while the other frame is integrating. Thus there is a delay time due to image transfer equal to the camera integration time. In addition, there is a delay due to the time it takes to calculate the Shack–Hartmann spot centroids from the image data. When binning the camera pixels and using 33 ms camera integration time, the total delay between the end of the frame integration and the application of the new actuator voltages is 67 ms. This control method can also be used to induce a particular pattern of aberrations as well as to remove the wavefront error. In this case, the Shack– Hartmann spot displacements that would result from the desired aberration profi le are computed and the mirror actuator voltages necessary to null the difference between these and the actual locations of the Shack–Hartmann spots are applied.

15.4 WAVEFRONT CORRECTION PERFORMANCE 15.4.1

Residual RMS Errors, Wavefronts, and Point Spread Functions

Within a fraction of a second from the start of adaptive correction, the eye’s residual RMS wavefront error can usually be reduced to 0.06 to 0.10 mm over a 6.8-mm pupil, depending on the individual subject. Figure 15.5 illustrates

WAVEFRONT CORRECTION PERFORMANCE

407

GYY, 6.8-mm pupil

Without Aberration Compensation

With Aberration Compensation

Wave Aberration

Point Spread Function

FIGURE 15.5 Improvement in the eye’s wave aberration and PSF obtained for one subject by correcting the eye’s aberrations with the Rochester Adaptive Optics Ophthalmoscope. Panels on the left show the measured wave aberration for one subject over a 6.8-mm pupil before and after aberrations were compensated for. Contour lines occur at single wavelength intervals (l = 550 nm). Panels on the right show the associated PSFs calculated from the wave aberration at a wavelength of 550 nm. Correcting the eye’s aberrations greatly improves the compactness of the PSF.

the improvement afforded by the adaptive optics system on the wave aberration and its associated point spread function (PSF) for one subject. The PSF was calculated from the measured wave aberration for a wavelength of 550 nm. Without adaptive correction, the RMS wavefront error over a 6.8-mm pupil was 1.3 mm and the PSF was very irregular and distended. After aberrations were corrected, the residual RMS wavefront error was reduced to 0.09 mm and the PSF was distinctly sharpened. 15.4.2

Temporal Performance: RMS Wavefront Error

The expected performance given the usual system parameters can be calculated from a simple temporal model if a few assumptions are made. The first assumption is that the deformable mirror has adequate spatial resolution to reconstruct the eye’s aberration profi le, and the second is that the sampling rate of the system is high enough to accurately measure the dynamics of the eye’s aberrations. The fi rst assumption should be approximately true since the 97-channel mirror has enough spatial resolution to reconstruct aberrations up to eighth-order radial modes and the typical eye does not contain substantial aberrations in modes of higher orders than this to significantly impact image

408

ROCHESTER ADAPTIVE OPTICS OPHTHALMOSCOPE

quality [10]. The second assumption is believed to be true because the eye’s dynamics show only negligible activity above approximately 6 Hz [1]. Thus a rate of only 12 Hz should be needed to adequately capture the eye’s dynamics, which is well below the sampling rates used. The effects of noise are subsequently ignored. The model consists of the following components: a wavefront sensor (integrator), frame readout and slope calculation (simple delay), and mirror update (discrete proportional control). Since the speed of the mirror is 4 kHz and is much faster than any other element in our system, we assume that the mirror responds instantaneously to changes in voltage signals. These components have the following Laplace transforms (see also Section 8.5.1 for more details on modeling temporal performance): • Wavefront sensor: 1 − e − sT sT

(15.1)

(integration over the exposure time T = 33 ms) • Delay: e − sτ c

(15.2)

(t c = 67 ms and includes the time for CCD frame transfer and slope calculation) • Simple digital proportional mirror control: z transform of deformable mirror: K ⋅z ( K = mirror loop gain ) z−1 Continuous Laplace transform equivalent:

(15.3) K 1 − e − sT

(15.4)

(new mirror voltages are applied once every T = 33 ms) • Zero-order hold: 1 − e − sT sT

(15.5)

(mirror voltages are also held constant over each sampling interval T = 33 ms) These terms can be combined to calculate the total closed-loop and openloop system transfer functions, which allows the system correction bandwidth to be predicted. The correction bandwidth of the system is defi ned as the temporal frequency where these functions are equal. Fluctuations in the eye’s aberrations of frequencies lower than this are reduced by the system, while

IMPROVEMENT IN RETINAL IMAGE QUALITY

409

1

Averaged Power

Open-Loop Correction Closed-Loop Correction

0.1

0.01 Closed-Loop Bandwidth ~0.7 Hz

0.001 0.1

1

10

Temporal Frequency (Hz)

FIGURE 15.6 Log-log plot of the temporal power spectra of the eye’s residual RMS wavefront error during open-loop and closed-loop aberration correction. The correction bandwidth of the system is defi ned as the temporal frequency where these curves cross, here just slightly higher than 0.7 Hz. Fluctuations in the eye’s aberrations of frequencies lower than this are reduced by the system, while fluctuations with higher frequency components are somewhat exacerbated. These data were taken with a system gain of 30% and a rate of 21 Hz. Results are averaged across three subjects, with 10 runs per subject.

fluctuations with higher frequency components are somewhat exacerbated. The gain, K, used in the model (K = 28%) was the optimal gain as determined by Bode analysis given the various system parameters. Figure 15.6 shows the open-loop and closed-loop temporal power spectra of the measured residual RMS wavefront error. Empirically, we determined that a 30% mirror loop gain provided optimal performance, consistent with the model’s predicted optimal gain. The model predicts a correction bandwidth of 0.9 Hz, nearly what we observe empirically (~0.7 Hz).

15.5

IMPROVEMENT IN RETINAL IMAGE QUALITY

Wavefront measurements provide a theoretical estimate of the benefit of adaptive correction on the eye’s optical performance. However, the actual benefit may be worse than predicted by wavefront measurements, due to unaccounted non-common-path errors, errors in wavefront reconstruction, or other reasons. The actual benefit of adaptive correction is reflected in the improvement in the quality of retinal images acquired before and after adaptive correction. Figure 15.7 shows two single representative images of

410

ROCHESTER ADAPTIVE OPTICS OPHTHALMOSCOPE

Without Adaptive Optics (Single Image)

With Adaptive Optics (Single Image)

With Adaptive Optics (Sum of 31 Images)

AP, 1 Deg Retinal Eccentricity

FIGURE 15.7 Improvement in the quality of retinal images afforded by the Rochester Adaptive Optics Ophthalmoscope. The left panel shows a single image of the photoreceptor mosaic acquired without adaptive compensation; this is the best image quality that can be achieved with conventional means. Even though this subject possesses superior optical quality, only hints of the structure of the mosaic can be seen without aberration correction. The middle panel shows a single image of the same section of the mosaic acquired after the aberrations have been compensated. After the aberrations have been corrected, individual receptors can be clearly resolved. The right panel shows the further improvement in image quality that can be achieved by averaging (31) multiple images. All images were acquired at 1° temporal retinal eccentricity at a wavelength of 550 nm.

precisely the same retinal location, ~1° temporal retina, taken for one subject with and without aberrations corrected, as well as an image showing the further improvements in image quality that can be achieved by adding together many individual images acquired after aberrations have been corrected. This particular subject has superior optical quality compared with the majority of subjects, yet there is still a dramatic improvement in the quality of the retinal images taken with adaptive optics. Cones that are barely detectable in the leftmost image are clearly visible in the images acquired after aberrations were corrected. The improvement in image quality is so dramatic that it is possible to routinely image cones in nearly all subjects and, for some subjects, even obtain clear images of cones in the central fovea, as seen in Figure 15.8.

15.6

IMPROVEMENT IN VISUAL PERFORMANCE

After correcting the eye’s higher order aberrations with the Rochester Adaptive Optics Ophthalmoscope using the initial 37-channel deformable mirror (instead of the 97-channel mirror currently used), subjects’ contrast sensitivity

IMPROVEMENT IN VISUAL PERFORMANCE

411

FIGURE 15.8 Montage illustrating a foveal patch subtending approximately 2° in a living human retina. Several 1° retinal images, taken with adaptive optics at a wavelength of 550 nm, were combined to form this montage. The approximate location of the subject’s foveal center is marked with a +. Photoreceptor size becomes visibly larger with increasing distance from the foveal center. Scale bar represents 100 mm. (From Roorda and Williams [12]. Reprinted with permission from SLACK Inc.)

for a large (6-mm) pupil was increased by about a factor of 2 in broadband illumination [11]. In monochromatic light, which avoids the eye’s chromatic aberration, improvement in contrast sensitivity was higher, a factor of approximately 3 to 5 times above that when only correcting defocus and astigmatism. These improvements were well matched by those expected given the improvement calculated in the eye’s modulation transfer function from the measured wave aberration before and after correction. These measurements were also performed under open-loop correction only. While no similar contrast sensitivity data have been acquired with the Rochester Adaptive Optics Ophthalmoscope since incorporating the 97-channel deformable mirror, it is reasonable to assume that even greater improvements would have been seen if the same experiment were performed now with the higher resolution mirror and closedloop correction. Figure 15.9 shows the improvement in visual acuity when correcting the eye’s higher order aberrations with the Rochester Adaptive Optics Ophthalmoscope. The figure shows visual acuity for three subjects in monochromatic (550-nm) and white light for a 6-mm pupil with and without higher order aberrations corrected. These data were acquired before incorporating the 97-channel deformable mirror; however, the measurements were taken under open-loop conditions (only a static aberration compensation was employed).

412

ROCHESTER ADAPTIVE OPTICS OPHTHALMOSCOPE

FIGURE 15.9 Improvement in visual acuity afforded by the Rochester Adaptive Optics Ophthalmoscope. This plot shows visual acuity for three subjects in monochromatic (550-nm) and white light for a 6-mm pupil with higher order aberrations corrected and with only defocus and astigmatism corrected (without correction). Acuity was measured using a four-alternative-forced-choice orientation discrimination procedure with an illiterate E. These measurements were made under open-loop conditions (i.e., only a static aberration compensation was employed). Even so, all subjects showed a marked improvement in acuity after aberrations were compensated for with the adaptive optics system, achieving an ultimate acuity of approximately 20/10. Acuity was slightly worse after aberrations were corrected in white light (white bars) than in monochromatic light (black bars) due to the impact of the eye’s chromatic aberration. Monochromatic acuity before aberration correction is not shown because it is not significantly different from acuity in white light (due to the impact of the eye’s higher order aberrations).

Even so, all subjects showed a marked improvement in acuity after higher order aberrations were corrected, achieving an acuity of approximately 20/10. Acuity was slightly worse after aberrations were corrected in white light than in monochromatic light due to the impact of the eye’s chromatic aberration.

15.7

CURRENT SYSTEM LIMITATIONS

There are several factors that limit the correction that can be provided by the Rochester Adaptive Optics Ophthalmoscope or impact the ability to image the retina or perform psychophysical studies. Both the limited range of the Shack–Hartmann sensor and the deformable mirror affect the correction ability. For subjects with very high wavefront errors, there can be potentially an overlap of the spots in the Shack–Hartmann spot image. This makes it

CURRENT SYSTEM LIMITATIONS

413

impossible to tell which spot corresponds to which lenslet and, therefore, to calculate the appropriate mirror voltages or reconstruct the wavefront. However, in practice, this is not a major limitation since subjects for whom this is problematic tend to have wavefront errors so large that the deformable mirror does not have enough stroke to correct them. Based on aberration data from a population of 70 pre-LASIK patients, it is estimated that a deformable mirror stroke of 26 mm (~53 mm in the wavefront) is needed (7.5-mm pupil) to correct the eye’s total wavefront error without the need for correcting defocus and astigmatism with trial lenses (or other means) beforehand (see also Section 4.5). The Rochester Adaptive Optics Ophthalmoscope has a range of 8 mm in the wavefront, allowing for few subjects to be corrected in this manner. If the investigator does not mind performing a refraction beforehand, then a range of only 5.5 mm (11 mm in the wavefront) is needed. In this case, the Rochester Adaptive Optics Ophthalmoscope will be able to correct the higher order aberrations in approximately 80% of subjects. The Rochester Adaptive Optics Ophthalmoscope, while providing impressive correction of the eye’s wavefront error, also does not use the best control methods available. Currently, only proportional control is used. Using a more sophisticated algorithm, such as a method incorporating integral as well as derivative control, should result in an even quicker and better correction (see Chapter 5 for more on different control algorithms). The changing magnification of the visual stimulus/imaging path as different amounts of chromatic aberration are compensated for in the Rochester Adaptive Optics Ophthalmoscope is a factor that potentially limits correction ability as well as causing inconvenience during psychophysical and imaging experiments. Correction ability is limited because the changing position of the focusing lens makes it difficult to maintain alignment along the length of the lens travel. In addition, since the optical configuration changes as the lens moves, noncommon path aberrations also change. This makes it difficult to characterize the non-common-path aberrations and correct them. This configuration causes inconveniences during imaging or psychophysical experiments because the angle subtended by a pixel of either the retinal imaging CCD camera or the visual stimulus display depends upon the focus offset used. Even for the same wavelength, individual subjects may require slightly different focus offsets to bring a retinal image or visual stimulus in focus. This means that retinal image magnification or stimulus magnification must be recalibrated in real time as a function of the focus adjustment needed. For retinal imaging applications, the flood-illuminated scheme utilized in the Rochester Adaptive Optics Opthalmoscope is excellent for obtaining en face images of blood vessels or the photoreceptor mosaic. However, this scheme is not well suited for other applications that require the optical sectioning of retinal tissue or the imaging of other retinal structures since there is no means of rejecting light originating from other depth planes. Because of the particular CCD camera and light source used for retinal imaging, the Rochester Adaptive Optics Ophthalmoscope is also unable to

414

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acquire real-time images of the retina. The Rochester Adaptive Optics Ophthalmoscope does have the ability to deliver visual stimuli in real time. However, if this is to be done while correcting aberrations, the eye’s wavefront error must also be measured in order to maintain correction while the visual stimuli are being displayed. This means the SLD wavefront sensor beacon is visible to the subject during the psychophysical task and is superimposed on the visual stimulus field. While the SLD is infrared and rather dim, it is still visible to the subject and could potentially interfere with the perception of visual stimuli during some experiments. To minimize its influence, the beacon is often displaced slightly from the center of the stimulus so that aberrations are measured and corrected just slightly off-axis relative to the visual axis [5]. This does not significantly impact the correction afforded by the adaptive optics system since the eye’s aberrations do not vary significantly over small field angles (see also Chapter 10).

15.8

CONCLUSION

Dynamically correcting the eye’s aberrations using the Rochester Adaptive Optics Ophthalmoscope provides excellent optical system performance and significant improvements in retinal image quality and visual performance. The system measures the eye’s aberrations using a SLD (l = 825 nm) and a Shack–Hartmann wavefront sensor with 221 lenslets (F = 24 mm, d = 400 mm) and corrects for them using a 97-channel Xinxtics continuous faceplate deformable mirror with ±2 mm of mirror stroke. Aberration measurement and correction take place at a rate of up to 30 Hz over a 6.8 mm pupil diameter, providing a closed-loop bandwidth of approximately 0.7 Hz. Residual RMS wavefront errors after correction are typically better than 0.1 mm and can be obtained in 0.25 to 0.50 s. After achieving an adequate correction, visual psychophysics or retinal imaging are conducted over the central 6 mm of the 6.8-mm pupil to avoid edge artifacts in the adaptive optics (AO) correction. A DLP is used as the stimulus for the visual psychophysics experiments. Flood-illuminated retinal imaging is done using a krypton flash lamp (with 4-ms flashes) combined with appropriate interference fi lters. The Rochester Adaptive Optics Ophthalmoscope continues to serve as an excellent instrument for conducting clinical and basic scientific research.

REFERENCES 1. Hofer H, Artal P, Singer B, et al. Dynamics of the Eye’s Aberrations. J. Opt. Soc. Am. A. 2001; 18: 497–506. 2. Liang J, Williams DR, Miller DT. Supernormal Vision and High-Resolution Retinal Imaging through Adaptive Optics. J. Opt. Soc. Am. A. 1997; 14: 2884– 2892.

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3. Hofer H, Chen L, Yoon GY, et al. Improvement in Retinal Image Quality with Dynamic Correction of the Eye’s Aberrations. Opt. Express. 2001; 8: 631–643. 4. ANSI. American National Standard for the Safe Use of Lasers. ANSI Z136.1. Orlando, FL: Laser Institute of America, 2000. 5. Williams DR, Yoon G. Wavefront Sensor with Off-Axis Illmination. US Patent 6,264,328 B1. July 24, 2001. 6. Jiang W, Li H. Hartmann-Shack Wavefront Sensing and Control Algorithm. In: Schulte-in-den-Baeumen JJ, Tyson RK, eds. Adaptive Optics and Optical Structures. Proceedings of the SPIE. 1990; 1271: 82–93. 7. Stiles WS, Crawford BH. The Luminous Efficiency of Rays Entering the Eye Pupil at Different Points. Proc. R. Soc. Lond. B. 1933; 112: 428–450. 8. Packer O, Diller LC, Verweij J, et al. Characterization and Use of a Digital Light Projector for Vision Research. Vision Res. 2001; 41: 427–439. 9. Chen L, Singer B, Guirao A, et al. Image Metrics for Predicting Subjective Image Quality. Optom. Vis. Sci. 2005; 82: 358–369. 10. Liang J, Williams DR. Aberrations and Retinal Image Quality of the Normal Human Eye. J. Opt. Soc. Am. A. 1997; 14: 2873–2883. 11. Yoon GY, Williams DR. Visual Performance after Correcting the Monochromatic and Chromatic Aberrations of the Eye. J. Opt. Soc. Am. A. 2002; 19: 266–275. 12. Roorda A, Williams DR. Retinal imaging using adaptive optics. In: Krueger RR, Applegate RA, MacRae SM, eds. Wavefront Customized Visual Correction: The Quest for Super Vision II. Thorofare, NJ: SLACK, 2004, pp. 43–51.

CHAPTER SIXTEEN

Design of an Adaptive Optics Scanning Laser Ophthalmoscope KRISHNAKUMAR VENKATESWARAN Alcon Research Ltd, Orlando, Florida FERNANDO ROMERO-BORJA Houston Community College, Houston, Texas AUSTIN ROORDA University of California, Berkeley, Berkeley, California

16.1 INTRODUCTION Images taken of human retinas with a scanning laser ophthalmoscope (SLO) can rarely resolve features as small as cone photoreceptors, nor can the axial sections reveal the independent retinal layers. This is because image quality in SLOs is limited by the aberrations of the eye, which leave lateral and axial resolution to be about 5 and 300 mm, respectively [1, 2]. For comparison, cone photoreceptors in the center of the fovea are as small as 2 mm [3] and the retinal thickness is about 300 mm. Combining adaptive optics (AO) and introducing the smallest possible confocal pinhole at the imaging plane, the imaging resolution of the SLO is increased dramatically [4]. For example, in a typical human eye, using a light source of wavelength 660 nm, a 5.81-mm pupil, an 80-mm confocal pinhole, and adaptive optics, we can achieve a Adaptive Optics for Vision Science, Edited by Porter, Queener, Lin, Thorn, and Awwal Copyright © 2006 John Wiley & Sons, Inc.

417

418

DESIGN OF AN ADAPTIVE OPTICS SCANNING LASER OPHTHALMOSCOPE

lateral resolution of about 2.5 mm and axial resolution as low as 100 mm. This approaches an order of magnitude decrease in the volume resolution element. This chapter provides a detailed description of a current AO scanning laser ophthalmoscope. Henceforth in our discussions, we will term the AO system combined with the SLO as AOSLO (see also Chapter 10). Figure 16.1 shows the actual layout of the AOSLO [4]. The AOSLO occupies a 1.5 m × 1.4 m area on an optical table. The different components in the AOSLO can be considered as different modules, which derive signals from each other and operate in a closed loop. Based on the order in which the light passes through the system, the main components are (1) light delivery optics, (2) wavefront compensation optics, (3) raster scanning mirrors, (4) wavefront sensor, (5) light detector, and (6) image recording. Another major aspect of the design is the relay optics, which optically connect these different components. The details of the relay optics are explained in the design process as we discuss the optical layout starting from light delivery to light detection. Detailed references are provided at the end of this chapter. We also present some of the results from the present AOSLO and discuss a few ways of improving the AOSLO system, which will be useful for developing next-generation AOSLO imaging systems. In all our discussions on Zernike polynomials, we adhere to the VSIA (Vision Science and Its Applications) standards on reporting aberrations [5].

1. Light Delivery 5. Light Detection

FIBER OPTIC r

L L 6. Frame Grabbing

CP p

p

L

p

M1

2. Wavefront Sensing

PMT r

L

r

r

CCD

LA

r

3. Wavefront Compensation DM DM

COMPUTER

M2

p

M3 r

M4

p r

M5 VS

HS

4. Raster Scanning M6 M7

p r

M8

p

Eye r

FIGURE 16.1 Optical layout of the adaptive optics scanning laser ophthalmoscope. The retinal and pupil planes along with the components are labeled. L, lenses; M, mirrors; VS, vertical scanner; HS, horizontal scanner; BS, beamsplitter; DM, deformable mirror; Acousto-Optic Modulator AOM; CP, confocal pinhole; AP, artificial pupil; LA, lenslet array; FO, fiber optic; PMT, photomultiplier tube; CCD, chargecoupled device camera; r and p are the retinal and pupil planes, respectively.

RASTER SCANNING

16.2

419

LIGHT DELIVERY

The light from a diode laser is coupled to a single-mode optical fiber. The wavelength of the light delivered is 660 nm. The tip of this optical fiber provides a point source, which is then collimated using a 30-mm focal length achromatic doublet. The lens is followed by a variable neutral density fi lter, which allows us to control the laser light levels at the imaging plane. The beam is then focused with a 150-mm focal length achromat to an acousto-optic modulator (AOM). The optical path after the AOM is aligned so that the fi rst diffracted order of the AOM is passed through the system. This allows us to control the light into the system and is used primarily to limit exposure to the retina to only those times when data is being recorded. The signal, which drives the AOM, is the same signal that is used to gate each recorded line in the video image. The fi rst-order diffracted beam output from the AOM is recollimated using a second 150-mm focal length achromatic lens. The collimated beam is passed through an iris diaphragm, which blocks the zerothorder diffracted beam from the AOM and serves as the entrance pupil for the system. After the entrance pupil, the beam is introduced into the primary SLO path with a glass wedge beamsplitter, preventing ghost reflections. The beamsplitter is about 5% reflective, which allows for 95% of the returning light to be collected for wavefront sensing and imaging. Safety limits on the exposure of the retina to radiation prevent us from increasing the source power arbitrarily. The light levels to which the retina will be exposed are kept 10 or more times lower than the maximum permissible exposure specified by the American National Standards Institute for the safe use of lasers [6]. The AOSLO presently operates with about 30 mW of laser power at the corneal plane at a wavelength of 660 nm and a duty cycle of 40%. The most significant light loss occurs in the eye itself. The reflectivity, scattering, and absorption of light in the eye varies between individuals [7], and the signal-to-noise ratio in the fi nal image depends highly on the optical properties of the subject’s eye. The amount of light reflected off the human retina constrains the total light available for wavefront sensing and imaging. Although SLOs capture images in a different manner than traditional imaging systems, there is still a possibility that the coherence of the light source will generate a type of speckle, affecting the photometry and reducing the signal-to-noise ratio of the images. We are currently investigating low coherence laser sources to remove these artifacts.

16.3

RASTER SCANNING

The beam is scanned on the retina with a resonant and galvanometric scanner combination. The horizontal scanning mirror, which is a resonant scanner that operates at a 16 kHz line frequency, is the master timer for the system.

420

DESIGN OF AN ADAPTIVE OPTICS SCANNING LASER OPHTHALMOSCOPE

The vertical scanning mirror synchronizes with the horizontal scanning mirror in order to provide 525 lines per frame. The vertical scanner operates in a sawtooth pattern where about 480 lines make up the image frame, and the remaining 45 lines occur in the time it takes the vertical scanner to return to the top of the frame. With a 16 kHz line frequency and 1/525 frame frequency, the SLO runs at a rate of about 30 frames per second. Both the horizontal and vertical scanners are placed conjugate to each other and also to the entrance pupil plane of the eye, which is the pivot point of the raster scanning beam. The amplitude of the scan in the present setup can be adjusted to a field size from about 3° × 3° to 1° × 1°. The signals acquired by the frame grabber board are recorded as image frames. The horizontal synchronization pulse (hsync) and the vertical synchronization pulse (vsync) that defi ne the frame are provided for the frame grabber by converting the analog outputs from the scanner units into transistor transistor logic (TTL) pulses. The mirror size of the horizontal and vertical scan mirrors is 3 × 3 mm and 12 × 5 mm, respectively. The resonant scanner used is a product of Electro-Optics Products Corp. This will be discussed in more detail in the section on image recording.

16.4

ADAPTIVE OPTICS IN THE SLO

16.4.1 Wavefront Sensing To compensate for the aberrations introduced by the eye’s optics, we use adaptive optics. The fi rst step in AO is to measure the aberrations, which is typically done with a Shack–Hartmann wavefront sensor. The lenslet array is made up of 24-mm focal length lenslets, each with an actual diameter of 400 mm. There is magnification of 1.21× between the pupil of the eye and the wavefront sensor so the lenslets project to a size of 331 mm in the eye. The sampled wavefront is decomposed mathematically into Zernike modes describing the wavefront. In the present AOSLO geometry, the wavefront sensor has square subapertures, 17 across the diameter and a total of 241 lenslets inside a 7-mm pupil (see Fig. 16.2). The centroid of the focused spot from each lenslet is estimated by calculating the fi rst moment of intensity at every spot location. The accuracy of the centroid location algorithm is dependent on the signal-to-noise ratio of the focused spots at the focal plane of the different lenslets. With all the light loss due to the absorption and scattering of light in the optics of the AOSLO remaining the same, the retinal absorption of the human retina dictates the required exposure time for Shack–Hartmann slope measurements, which in turn dictates the frequency of the closed-loop system in the AOSLO. As mentioned earlier, the retinal absorption is different for different people. For more detailed discussions on wavefront sensing, the reader is referred to Chapter 3.

ADAPTIVE OPTICS IN THE SLO

421

FIGURE 16.2 Shack–Hartmann wavefront sensor’s output image. A total of 241 lenslets sample the wavefront. Each spot is the focused spot of the wavefront sampled by the lenslet.

16.4.2

Wavefront Compensation Using the Deformable Mirror

A 37-channel Xinxtics deformable mirror (DM) is placed in the optical path conjugate to the entrance pupil of the eye. By placing the DM before the raster scanners, the size of the mirrors required for relaying the light through the system is minimized. Minimization of the mirror sizes allows for smaller reflection angles, which reduces the inherent aberrations in the system. The diameter of the DM is 47 mm, and therefore the pupil has to be magnified to fi ll the mirror aperture. The size of the DM is the primary reason for the large size of the instrument. Aberrations are compensated on both the in-going and the out-going light paths. Correcting the wave aberrations on the way into the eye helps in presenting a compact spot on the retina and results in increased resolution of features in the retina. Correcting the aberrations on the way out helps to focus the light to a compact spot in the confocal plane, resulting in higher axial resolution with increased light throughput from the scattering layer being imaged. 16.4.3

Mirror Control Algorithm

The wavefront sensor computes the wave aberration, which is in turn mapped onto the DM actuator array to compensate for the aberrations.

422

DESIGN OF AN ADAPTIVE OPTICS SCANNING LASER OPHTHALMOSCOPE

Let 2K be the number of slope measurements (or twice the number of lenslets) and M the number of actuators in the DM. In the current AOSLO configuration, 2K >> M and therefore the system is said to be overdetermined. The wavefront modeled using 2K slope measurements from the Shack– Hartmann sensor is projected on to a deformable mirror with M actuators using an influence matrix (A). Figure 16.3 shows the wavefront sensor geometry superimposed on the DM actuator position geometry. The average slope of the wavefront over every lenslet can be written as a superposition of the product of the actuator element voltage v and the elements a of the influence matrix A such that s1 = a11v1 + a12 v2 + a13v3 + . . . a1 M vM s2 = a21v1 + a22 v2 + a23v3 + . . . a2 M vM ... s( 2 K ) = a( 2 K )1v1 + a( 2 K ) 2 v2 + a( 2 K ) 3v3 + . . . a( 2 K ) M vM

(16.1)

where s represents the slope measurements, and v represents the 37 actuator voltages on the Xinxtics mirror. In our case, the scattered light from the object is imaged using a 5.81-mm pupil, which is then projected onto the lenslet array of diameter 7.00 mm. Zernike polynomials up to eighth order (a total of 45 modes with piston, tip, and tilt set to zero) are fit to the estimated wavefront slopes and projected onto the deformable mirror. The voltages generated to compensate for the aberrations are sent to the mirror driver in the sequence as prescribed by the DM manufacturer.

5.89-mm Pupil Size 47-mm DM Aperture

}

400-µm Lenslet Spacing

}

7-mm DM Actuator Spacing

FIGURE 16.3 Centers of the lenslets of the Shack–Hartmann wavefront sensor projected onto the DM actuator array. The physical aperture of the DM projects to a pupil size of 5.89 mm and serves as the limiting aperture of the system. In the AOSLO, there are 241 lenslets inside the pupil and 37 actuators on the DM.

ADAPTIVE OPTICS IN THE SLO

423

In our current system, the reconstruction is done with several intermediate stages, which permits us to closely monitor the system performance. First, the wave aberration is fit with a Zernike expansion. From the Zernike coefficients, several metrics such as the root-mean-square (RMS) wavefront error, Strehl ratio, point spread function (PSF), and values of specific modes can be displayed to monitor the performance. Once the fit has been made, the desired voltages for the actuators on the deformable mirror are computed as one half of the value of the wavefront computed at each actuator position. Before sending the voltage values, they are multiplied by a gain factor (90 dB sensitivity up to a depth of 363 mm. The noticeable drop between the 25th and 50th A-scans is attributable to nonuniformities in the roughly Gaussian intensity profi le of the 843-nm SLD and the spherocylindrical lens. The 75th A-scan is more than 20 dB below the central A-scan. While this drop is impractically large for retinal imaging due to the high loss of light in the eye, the results do illustrate that the instrument can acquire a single B-scan composed of 150 A-scans in 1 ms and achieve shortburst rates of 75,000 A-scans/s. These rates are 5 and 2.5 times higher than current SD-OCT retinal cameras and could be more effectively realized in the eye if a more uniform illumination source with a retinal irradiance comparable to the maximum of the current system were used. The FWHH of the axial PSF was also measured and converted to that in retinal tissue assuming a retinal refractive index (nret) of 1.38. The theoretical FWHH axial resolution in retinal tissue for the 843-nm SLD (∆l = 49.4 nm) is 2

( 843 nm ) 2 ln 2 λ 2 2 ln 2 FWHH = = = 4.6 µ m π n ret ∆λ π ( 1.38 ) ( 4 9.4 nm )

(17.2)

The measured resolution obtained with the planar mirror in the sample channel is 7.6 mm (air), which corresponds to 5.5 mm in the retina. The differ-

466

INDIANA UNIVERSITY AO-OCT SYSTEM

ence between the theoretical and experimental values is likely due to a residual dispersion mismatch and the dependence of the SLD spectrum on output power, which was set below its maximum as a precautionary measure. As a third measurement, the specular inner limiting membrane (ILM) reflection at the base of the foveal pit was captured and used to determine an in vivo resolution of 5.7 mm. 17.6.2

AO Parallel SD-OCT Imaging

Images acquired with the parallel SD-OCT camera have been largely used to validate the instrument performance and to compare it to current OCT systems (such as the Stratus OCT3 and a research-grade scanning SD-OCT system) and to AO conventional flood-illuminated systems. To this end, we targeted two distinct patches of retina at eccentricities of 1° and 2.4°, of which images were collected using an AO conventional flood-illuminated system (Fig. 17.10), a conventional OCT system (Fig. 17.13), and an AO parallel SDOCT system (Fig. 17.14).

Stratus OCT3

Scanning SD OCT

2.4 deg 1 deg

NFL GCL & IPL INL OPL ONL IS/OS RPE

FIGURE 17.13 (Top) Stratus OCT3 and (bottom) scanning SD-OCT B-scans collected in the same subject. Images are centered on the fovea and bisect the superior and inferior retinal fields (Stratus OCT3) and nasal and temporal fields (scanning SD-OCT). B-scans are 4.9 mm (16.3°) wide and 0.75 mm in depth. White rectangles depict 100-mm-wide by 560-mm-deep subsections that are centered at eccentricities of 1° and 2.4° and were imaged using an AO conventional flood-illuminated system and an AO parallel SD-OCT system. A magnified view of these subsections is shown in Figure 17.14 with the corresponding AO parallel SD-OCT B-scans. The retinal layers include the nerve fiber layer (NFL), ganglion cell layer (GCL), inner plexiform layer (IPL), inner nuclear layer (INL), outer plexiform layer (OPL), outer nuclear layer (ONL), the junction between the inner and outer photoreceptor segments (IS/OS), and the retinal pigment epithelium (RPE). (Adapted from Zhang et al. [3]. Reprinted with permission of the Optical Society of America.)

EXAMPLE RESULTS WITH AO PARALLEL SD-OCT IMAGING

Without AO With AO (focus at cones) (focus at cones)

Stratus OCT3

467

Scanning SD OCT

1∞ 61 µm (dof)

NFL

2.4∞

GCL & IPL INL OPL

61 µm (dof)

ONL IS/OS RPE

Linear Images of IS/OS Interface and RPE

No AO

AO

FIGURE 17.14 (Left two columns) B-scan images acquired with the AO parallel SD-OCT instrument (shown in Fig. 17.1) with and without AO at eccentricities of 1° and 2.4° (superior). (Right two columns) Stratus OCT3 and scanning SD-OCT Bscans are shown at the same retinal eccentricities (from white rectangular boxes in Fig. 17.13). All images were acquired on the same subject and are 100 mm wide and 560 mm deep. (bottom) The interface between the inner and outer segments and RPE are enlarged and displayed as an amplitude on a linear scale (as opposed to a logarithmic scale). Images without AO are normalized to the corresponding AO images, including the enlarged images. Depth of focus (dof) is 61 mm and is defi ned in the text. The retinal layers include the nerve fiber layer (NFL), ganglion cell layer (GCL), inner plexiform layer (IPL), inner nuclear layer (INL), outer plexiform layer (OPL), outer nuclear layer (ONL), the junction between the inner and outer photoreceptor segments (IS/OS), and the retinal pigment epithelium (RPE). (Adapted from Zhang et al. [3]. Reprinted with permission from the Optical Society of America.)

Figure 17.13 shows the Stratus OCT3 and scanning SD-OCT B-scans that radially bisect the foveal pit. The axial resolution of the Stratus and scanning SD-OCT instruments is 10 and 6 mm, respectively. The white rectangular boxes depict the targeted retinal patches that are centered at 1° and 2.4° eccentricities, and are 100 mm × 560 mm (width × height) in size. Labels for the intraretinal layers are also shown and depict current anatomical interpretations of high-resolution OCT images [15]. Most of these layers are also

468

INDIANA UNIVERSITY AO-OCT SYSTEM

suggested in the Stratus OCT3 image but are not as well defi ned due to increased speckle and lower resolution [16]. Figure 17.14 shows the corresponding AO SD-OCT images (extracted from a sequence of short bursts) that were acquired at the two eccentricities. The corresponding subsections of the Stratus OCT3 and scanning SD-OCT images (white rectangles in Fig. 17.13) are also shown for comparison. Thickness measurements of the retina (i.e., the distance between the ILM and the posterior edge of the cone outer segments) are in reasonable agreement. Differences in the AO SD-OCT retinal thickness measurements of 20.6 and 8.8% relative to the Stratus at the two eccentricities are likely within the error imposed by image interpretation, sampling errors, and differences in retinal location. A comparison of retinal thickness between the AO SD-OCT image and the scanning SD-OCT image was not performed as the latter was acquired from a different meridian, albeit at roughly the same eccentricity. The images from the three cameras contain grossly similar bright and dark bands that occur at similar depths in the retina. Interestingly, the stratification of the intraretinal layers appears most defi ned in the AO parallel SD-OCT

Collage of AO-OCT Images at 2.4° ecc.

Collage of AO-OCT Images at 1° ecc.

NFL GCL & IPL INL OPL ONL IS/OS RPE

FIGURE 17.15 Two collages created by digitally pasting together an alternating sequence of AO parallel SD-OCT images acquired at eccentricities of 1° and 2.4°. The collages are roughly 3.25° to 3.5° wide. Focus is at the cone photoreceptor layer. Images in each alternating sequence were taken from the same short-burst series. The collage at 2.4° eccentricity was generated from two images (each with 70 A-scans from the central region of a B-scan; dashed white rectangle represents the combined two images and is 140 A-scans) that were repeated about 7 times. The collage at 1° eccentricity contains three images (each with ~70 A-scans; dashed white rectangle represents 210 A-scans) that were repeated about 5 times. The collages were axially registered to each other by aligning the reflection from the IS/OS junction. The retinal layers include the nerve fiber layer (NFL), ganglion cell layer (GCL), inner plexiform layer (IPL), inner nuclear layer (INL), outer plexiform layer (OPL), outer nuclear layer (ONL), the junction between the inner and outer photoreceptor segments (IS/OS), and the retinal pigment epithelium (RPE). (Adapted from Zhang et al. [3]. Reprinted with permission of the Optical Society of America.)

EXAMPLE RESULTS WITH AO PARALLEL SD-OCT IMAGING

469

images (left two columns of Fig. 17.14) even though the AO parallel SD-OCT images are somewhat darker, which suggests a reduced signal-to-noise ratio (SNR). Some of the differences between AO parallel SD-OCT images and the standard OCT images are caused by image resizing necessitated by differences in A-scan density. While the layers are clearly less distinct than those in Figure 17.13, they can be better visualized in a collage. The collage was created by compressing a few AO parallel SD-OCT images along their lateral dimension to roughly the same dimensions as the sections shown in Figure 17.13, and then digitally pasting together duplications of those images to form a collage, an example of which is shown in Figure 17.15. By simulating a larger field of view, the two collages (1° and 2.4°) more clearly show the stratification of the retinal layers that is typically observed with high-resolution OCT. Bright reflections are visible at the nerve fiber layer, inner segment/outer segment junction, and RPE; typically, in OCT images, bright reflections occur at the interfaces between media with different refractive indices. Note the distinct physical separation between the inner nuclear layer and adjacent plexiform layers as well as the anterior (inner segment/outer segment junction) and posterior sides of the outer segments. The outer segments of the photoreceptors are slightly longer at the smaller eccentricity as expected. Collectively, the results from Figures 17.13 to 17.15 indicate that the AO parallel SD-OCT instrument is sufficiently sensitive to detect reflections from essentially all major layers of the retina (nerve fiber layer to the retinal pigment epithelium). Speckle is an unfortunate by-product of the coherent nature of OCT detection and is indeed readily visible in the images acquired from all three OCT instruments (Fig. 17.13 to 17.15). For the AO parallel SD-OCT images, it clearly hinders our ability to correlate retinal reflections to microscopic retinal structures, especially those that approach the size of an individual speckle. Speckle in the AO-OCT images is about the average size predicted theoretically (2.9 and 5.7 mm in the lateral and axial directions, respectively) for the imaging configuration used here. As evident in Figure 17.14, the commercial and research-grade OCT instruments generate speckle of noticeably larger size and of different shape. These differences originate from differences in the pupil diameter and coherence lengths of the SLD light sources. In Figure 17.14, smaller speckle appears clearly less disruptive of retinal features and illustrates an advantage of larger pupils and shorter coherence length sources. However, even with the large 6-mm pupil and 5.7-mm coherence length of the AO parallel SD-OCT instrument, our results show that fully developed speckle is still present and substantially limits the microscopic retinal structures we can observe. This is despite the fact that the instrument has achieved the necessary 3D resolution, sensitivity, and speed required to observe these structures. While speckle is clearly disruptive, some microscopic structures are apparent (particularly within the instrument’s depth of focus) when the camera is focused at the location where cones are clearest in the flood-illuminated image. [The depth of focus for the Indiana AO parallel SD-OCT instrument is about 61 mm (see Fig. 17.14) and is defi ned as two times the Rayleigh range

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INDIANA UNIVERSITY AO-OCT SYSTEM

for a Gaussian beam and a 6-mm pupil.] Specifically the bright reflection from the interface between the inner and outer segments of the photoreceptors appears spatially segmented, having a quasi-regular pattern with a periodicity of several microns. This unique pattern is not observed in any other part of the OCT images, for example, in any of the other retinal layers. Due to the random position of the slit on the retina, the portion of the mosaic sampled by the camera varied from acquisition to acquisition. The quasi-regular pattern largely disappears into a thin line when there is no AO correction (Fig. 17.14). The presence of some pattern information without AO should be expected, as there are hints of structure in the cone mosaic in the floodilluminated images without AO (Fig. 17.10) for subjects with normal optical quality when defocus and astigmatism are meticulously corrected. While the images showing an enlarged view of the RPE and the interface between the inner and outer segments (bottom of Fig. 17.14) appear to contain structural information specific to the cones, only the AO image contains structures whose regular spacing (~7 mm) matches that measured using flood-illuminated imaging (Fig. 17.10). The structural spacing without AO is noticeably smaller, suggesting it is corrupted by speckle, whose average size is similar to the spacing. Note the increased brightness of the IS/OS interface and the RPE (to a lesser extent) when the aberrations are compensated. This is particularly evident in the enlarged amplitude images of Figure 17.14 and would be even more dramatic if displayed as an intensity on a linear scale. As a potential means to reduce the contrast of the speckle noise, we investigated the impact of averaging images within a short-burst sequence. Micromovements of the retina that involuntarily occur between images (with a 2-ms spacing) might be sufficiently large to spatially alter the speckle pattern, while still sufficiently small to preserve much of the retinal signal at the cellular level. Figure 17.16 shows examples of averaging across three short-burst sequences. For the left two columns, averaging reduces speckle contrast with some increase in clarity of the IS/OS junction when AO compensation is employed. However, averaging seems to reduce the contrast in many of the other retinal layers, which is expected given that these layers are out of focus and therefore should carry little cellular (or high frequency) information. Based on these preliminary observations, speckle noise is the likely source of the high contrast in these layers. As an example, the contrast of the GCL and IPL layers decreased by 1.8 and 1.6 dB for the left two columns in Figure 17.16. The rightmost column is an example with almost no contrast change after averaging. Analysis of the short-burst images reveals that the retina was effectively stationary during the 15-ms short-burst sequence and produced no change in the speckle pattern. This latter example is rather atypical in that small amounts of retinal motion are usually present. While many more images than that used in Figure 17.16 are necessary to achieve appreciable speckle contrast reduction, these preliminary results illustrate a possible approach, provided that lateral motion of the retina during a single exposure is smaller than the size of the structures of interest.

EXAMPLE RESULTS WITH AO PARALLEL SD-OCT IMAGING

Without AO

With AO

(focus at cones)

(focus at cones)

471

Single Image

NFL GCL & IPL INL Averaged Image

OPL ONL IS/OS RPE

FIGURE 17.16 (Top row) Single B-scans acquired with AO parallel SD-OCT with and without AO compensation at 2.4° eccentricity (superior). (Bottom row) Average B-scans computed from seven (left and right) and three (middle) images from the same short-burst sequence, at the same eccentricity as the top row. Images are displayed on a logarithmic scale. Images are 100 mm wide and 560 mm deep. Major layers of the retina are labeled on the right. The retinal layers include the nerve fiber layer (NFL), ganglion cell layer (GCL), inner plexiform layer (IPL), inner nuclear layer (INL), outer plexiform layer (OPL), outer nuclear layer (ONL), the junction between the inner and outer photoreceptor segments (IS/OS), and the retinal pigment epithelium (RPE). The figure shows that micromovements of the retina allow averaging to reduce speckle contrast, but they may have been insufficient for significantly reducing speckle in this data.

As additional substantive evidence that the quasi-regular pattern present in the B-scans corresponds to reflected light at the IS/OS junction, we analyzed both flood-illuminated and SD-OCT data in the Fourier domain [17]. To account for the pseudo-hexagonal distribution of the cones and the random position of the narrow slit at the cones due to eye motion, a virtual 2.8 mm × 100 mm slit was projected onto the 1° and 2.4° flood-illuminated cone images. One-dimensional power spectra were computed along the slit length using only the narrow slice of the mosaic sampled by the slit. Because only the

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INDIANA UNIVERSITY AO-OCT SYSTEM

(a) 2.4°

8

10

Flood OCT, cones OCT, OPL OCT, ONL OCT, IPL

7

8

10

Flood OCT, AO OCT, no AO

7

10

Power

(b) 2.4°

6

10

6

10

5

8

10

Flood OCT, AO OCT, no AO

7

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10

(c ) 1.0°

6

10

5

10

10

4

5

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4

10

10

3

4

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10 20 40 60 80 Frequency (cyc/deg)

20 40 60 80 Frequency (cyc/deg)

20 40 60 80 Frequency (cyc/deg)

FIGURE 17.17 Average power spectra obtained by 1D Fourier transformation of the conventional flood-illuminated AO images (thin solid line) in Figure 17.10 after being sampled by the 2.8 mm × 100 mm slit and (a) of cross-sectional slices through the interface between the inner and outer segments (thick solid line), outer plexiform layer (dash-dot-dashed line), outer nuclear layer (dash-dash-dotted line), and inner plexiform layer (dotted line) at 2.4° eccentricity. OCT curves are an average of six shortburst sequences, each containing 7 B-scans. (b,c) Power spectra are shown with (thick solid line) and without (dotted line) AO at eccentricities of 2.4° and 1°, respectively. All parallel SD-OCT curves are normalized to have the same power at 0 cyc/deg.

general location is known at which the slit sampled the retinal patch in the actual AO parallel SD-OCT experiment, a rolling average of the power spectra was computed. This was realized by laterally shifting the virtual slit across the portion of each cone mosaic image (Fig. 17.10) where the slit was known to fall. The resultant power spectra (thin solid lines) are shown in the three plots of Figure 17.17 for the two eccentricities. Note the cusps in the spectra occurring at frequencies corresponding to 5 and 7.1 mm for the 1° and 2.4° eccentricities, respectively. Average power spectra [Fig 17.17(a)] generated from six short-burst sequences of OCT images are also shown for crosssectional slices through the interface between the inner and outer segments, outer plexiform layer, outer nuclear layer, and inner plexiform layer. As the figure reveals, only the OCT cone curve contains noticeable energy localized near that of the spatial frequency of the cones from the conventional floodilluminated image. The power spectra in Figures 17.17(b) and 17.17(c) show

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EXAMPLE RESULTS WITH AO PARALLEL SD-OCT IMAGING

the impact of AO at both 1° and 2.4° eccentricities when focused on the cones. Spectra were averaged across six short-burst sequences, each containing seven B-scans at each of the two eccentricities. Cusps in the AO-OCT power spectra are again observed. The cusp in the AO-OCT power spectrum at 2.4° agrees very well with the spatial frequency of cone photoreceptors observed in the corresponding flood-illuminated power spectrum; the cusp in the AOOCT power spectrum at 1° is of significantly smaller amplitude but occurs at the spatial frequency of the cone photoreceptors observed in the 1° floodilluminated power spectra. The gain in spatial resolution afforded by AO should also be accompanied by a gain in SNR as correcting aberrations produces a more concentrated focus of the retinal reflection at the CCD detector. The change in the SNR of the photoreceptor and nerve fiber layer (NFL) reflectances were studied for three imaging scenarios: (1) the image was focused at the cones without AO correction, (2) the image was focused at the cones with AO dynamic correction, and (3) the image was focused near, but not at, the NFL with AO dynamic correction. The same 2.4° retinal patch was imaged in all three scenarios. To reduce the speckle contamination that would lead to a wrong comparison, the SNR of the cone photoreceptor layer and NFL (SNRcone and SNR NFL , respectively) were calculated from the average of 20 contiguous Ascans obtained during each short-burst image. Figure 17.18 shows typical profi les (average of 20) for the three scenarios, in which the reflectivity is the inverse Fourier transformation of the interference fringes collected by the CCD. Note that comparing the SNR visually from Figure 17.18 does not precisely reflect the true trend because of differences in the noise floor in these profi les, though they differ by less than 1.3 dB.

Without AO (focused on cones)

With AO (focused on cones)

With AO (focused near NFL)

cones Reflectivity (dB)

40

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Depth in Retina (µm)

FIGURE 17.18 Average profi le of 20 contiguous A-scans centered about the brightest region in the B-scan. Images were obtained at a retinal eccentricity of 2.4°. The figure shows that the SNR of the photoreceptor layer increases significantly when the camera is focused on it and the ocular aberrations are compensated by AO. (Adapted from Zhang et al. [3]. Reprinted with permission from the Optical Society of America.)

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INDIANA UNIVERSITY AO-OCT SYSTEM

TABLE 17.1 Average SNR for the Cone Photoreceptor Layer (SNRcone) and NFL (SNRNFL) for Three Imaging Scenarios* Without AO (Focused on Cones)

With AO (Focused on Cones)

With AO (Focused near NFL)

44.4 40.5

41.3 51.9

46.4 38.8

SNR NFL (dB) SNRcone (dB)

* Adapted from Zhang et al. [3]. Reprinted with permission of the Optical Society of America.

Table 17.1 lists SNRcone and SNR NFL for the three imaging scenarios. The table shows, as expected, that the detected retinal reflection is highly dependent on focus and the presence or absence of ocular aberrations. For example, an 11.4-dB increase in SNRcone is observed when aberrations are corrected with dynamic AO for the case when the image is focused on the cone layer. This increase reflects the influence of the correction of 0.50 mm RMS wavefront error (astigmatism, third order and higher) by AO. It is expected that there will be an additional increase in SNRcone if AO acts on both the light entering and exiting the eye instead of only the latter, as in this AO parallel SD-OCT camera. It is also worth pointing out that there is a 13.1-dB drop in SNRcone when the focus is shifted from the layer at which the clearest floodilluminated images of cones were acquired to a layer 200 mm anterior (near the NFL) in the presence of dynamic AO correction. This decrease in SNR was due to a 0.55-D (200-mm) change in defocus since AO was dynamically correcting the ocular aberration during the experiment. From a theoretical standpoint, light efficiency through the OCT slit is predicted to decrease by 12.3 dB when 0.55 D of defocus are added and the system is assumed to be diffraction limited for a 6-mm pupil. The NFL and outer nuclear layer also show consistent differences in reflectivity associated with focus and AO correction, although to a lesser extent.

17.7

CONCLUSION

This chapter provides a technical overview of the Indiana AO-OCT retinal camera. The camera was designed for sequentially collecting 3D highresolution SD-OCT and 2D high-resolution conventional flood-illuminated images of the microscopic retina in the living human eye. The overview includes a detailed description of the camera, general performance procedures for preparing subjects and collecting retinal images, a performance assessment of the AO system, and fi nally, imaging results. For AO conventional flood-illuminated imaging, the high acquisition rates (60 Hz; 500 Hz) coupled with the high lateral resolution due to AO provides the ability to quickly navigate through the retina, recognize individual cells

REFERENCES

475

of relative high contrast without image warp and motion blur, and monitor retinal dynamics occurring at the cellular level (e.g., capillary blood flow). The 3D resolution of AO SD-OCT substantially surpasses that of either methodology alone. The camera was found to have sufficient 3D resolution (3.0 mm × 3.0 mm × 5.7 mm), sensitivity (up to 94 dB), and speed (100,000 Ascans/s for single shot of 100 A-scans) for imaging the retina at the single-cell level. This system provided the fi rst observations of the interface between the inner and outer segments of individual cones, resolved simultaneously in both the lateral and axial dimensions. The camera sensitivity was sufficient for observing reflections from essentially all neural layers of the retina. The signal-to-noise ratio of the detected reflection from the photoreceptor layer was highly sensitive to the level of ocular aberrations and defocus. A critical limitation of the current AO SD-OCT instrument is that highcontrast speckle noise hinders our ability to correlate retinal reflections to specific cell-sized retinal structures. While speckle is a serious problem, a meaningful solution will permit OCT to reap the full benefit of AO that conventional flood-illuminated systems and scanning laser ophthalmoscopy now enjoy, but with the additional benefits of considerably higher axial resolution and sensitivity. The AO parallel SD-OCT results presented here already reveal subcellular structure in the cone photoreceptor layer that have not been reported with either flood-illuminated or SLO systems. More recently, a scanning AO SD-OCT method was developed that allowed the observation of cones in volume images [18]. These fi rst results will surely improve as speckle reduction techniques are applied and real-time 3D imaging is implemented. Acknowledgments The authors thank previous group members Karen Thorn, Junle Qu, and Huawei Zhao, as well as Thomas Milner, Robert Zawadzki, and Weihua Gao for advice on the project. Assistance from Marcos van Dam with the AO diagnostics is much appreciated. The authors also thank William Monette and Daniel Jackson’s group for electronics and machining support. Financial support was provided by the National Eye Institute grant 5R01 EY014743. This work was also supported in part by the National Science Foundation Science and Technology Center for Adaptive Optics, managed by the University of California at Santa Cruz under cooperative agreement No. AST-9876783. REFERENCES 1. Miller DT, Qu J, Jonnal RS, Thorn K. Coherence Gating and Adaptive Optics in the Eye. In: Tuchin VV, Izatt JA, Fujimoto JG, eds. Coherence Domain Optical Methods and Optical Coherence Tomography in Biomedicine VII. Proceedings of the SPIE. 2003; 4956: 65–72. 2. Liang J, Williams DR, Miller DT. Supernormal Vision and High-Resolution Retinal Imaging through Adaptive Optics. J. Opt. Soc. Am. A. 1997; 14: 2884– 2892.

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3. Zhang Y, Rha J, Jonnal RS, Miller DT. Adaptive Optics Parallel Spectral Domain Optical Coherence Tomography for Imaging the Living Retina. Opt. Express. 2005; 13: 4792–4811. 4. Cense B, Nassif NA, Chen TC, et al. Ultrahigh-Resolution High-Speed Retinal Imaging Using Spectral-Domain Optical Coherence Tomography. Opt. Express. 2004; 12: 2435–2447. 5. ANSI. American National Standard for the Safe Use of Lasers. ANSI Z136.1. Orlando, FL: Laser Institute of America, 2000. 6. Thorn KE, Qu J, Jonnal RJ, Miller DT. Adaptive Optics Flood-Illuminated Camera for High Speed Retinal Imaging. Invest. Ophthalmol. Vis. Sci. 2003; 44: 1002. 7. Hecht J. Understanding Fiber Optics. Upper Saddle River, NJ: Prentice Hall, 1998. 8. Hofer H, Chen L, Yoon GY, et al. Improvement in Retinal Image Quality with Dynamic Correction of the Eye’s Aberrations. Opt. Express. 2001; 8: 631–643. 9. Liang J, Grimm B, Goelz S, Bille J. Objective Measurement of Wave Aberrations of the Human Eye with the Use of a Hartmann-Shack Wave-front Sensor. J. Opt. Soc. Am. A. 1994; 11: 1949–1957. 10. Jiang W, Li H. Hartmann-Shack Wavefront Sensing and Control Algorithm. In: Schulte-in-den-Baeumen JJ, Tyson RK, eds. Adaptive Optics and Optical Structures. Proceedings of the SPIE. 1990; 1271: 82–93. 11. Malacara D. Optical Shop Testing, 2nd ed. New York: Wiley, 1992. 12. Thibos LN, Hong X, Bradley A, Applegate RA. Accuracy and Precision of Methods to Predict the Results of Subjective Refraction from Monochromatic Wavefront Aberration Maps. J. Vis. 2004; 4: 329–351. 13. Curcio CA, Sloan KR, Kalina RE, Hendrickson AE. Human Photoreceptor Topography. J. Comp. Neurol. 1990; 292: 497–523. 14. Leitgeb R, Hitzenberger CK, Fercher AF. Performance of Fourier Domain versus Time Domain Optical Coherence Tomography. Opt. Express. 2003; 11: 889– 894. 15. Nassif N, Cense B, Park BH, et al. In Vivo Human Retinal Imaging by UltrahighSpeed Spectral Domain Optical Coherence Tomography. Opt. Lett. 2004: 29; 480–482. 16. Sander B, Larsen M, Thrane L, et al. Enhanced Optical Coherence Tomography Imaging by Multiple Scan Averaging. Br. J. Ophthalmol. 2004; 89: 207–212. 17. Miller DT, Williams DR, Morris GM, Liang J. Images of Cone Photoreceptors in the Living Human Eye. Vision Res. 1996; 36: 1067–1079. 18. Zhang Y, Rha J, Cense B, et al. Motion-free volumetric retinal imaging with adaptive optics spectral-domain optical coherence tomography. In: Manns F, Sderberg PG, Ho A, eds. Ophthalmic Technologies XVI. Proceedings of the SPIE. 2006; 6138 (submitted).

CHAPTER EIGHTEEN

Design and Testing of a Liquid Crystal Adaptive Optics Phoropter ABDUL AWWAL and SCOT OLIVIER Lawrence Livermore National Laboratory, Livermore, California

18.1

INTRODUCTION

Conventional phoropters are used by ophthalmologists and optometrists to estimate and correct for the lower order aberrations of the eye, defocus and astigmatism, in order to derive a prescription for their patients. An adaptive optics phoropter measures and corrects the aberrations in the human eye using adaptive optics techniques, which are capable of dealing with both the standard lower order aberrations and higher order aberrations, including coma and spherical aberration. This chapter describes the design and testing of an adaptive optics (AO) phoropter based on a Shack–Hartmann wavefront sensor to measure the aberrations of the eye, and a liquid crystal spatial light modulator to compensate for them. The goal is to produce near diffractionlimited image quality at the retina, which will enable the investigation of the psychophysical limits of human vision. We will later show some preliminary results from testing human subjects. Corrective lenses can generally improve Snellen visual acuity to better than 20/20 in normal eyes by correcting the lower order aberrations, defocus, and

Adaptive Optics for Vision Science, Edited by Porter, Queener, Lin, Thorn, and Awwal Copyright © 2006 John Wiley & Sons, Inc.

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DESIGN AND TESTING OF A LIQUID CRYSTAL ADAPTIVE OPTICS PHOROPTER

astigmatism, also known as sphere and cylinder [1]. Higher order aberrations remain untreated, however, and continue to affect visual performance. One of the goals of designing an AO phoropter is that it can correct higher order aberrations to improve acuity beyond what can be achieved with conventional spectacles or contact lenses. This improvement has been coined “supernormal vision” [2]. The same design can be extended to produce in vivo images of the human retina that are sharper, with higher resolution than conventional fundus photography [2, 3]. Conventional deformable mirror (DM) devices, such as continuous faceplate mirrors, are used in vision science and astronomical applications. In addition to being expensive, these DMs typically have much larger apertures than the eye. This leads to a large optical system in order to magnify the eye’s dilated pupil to the larger size of the deformable mirror. This combination of cost and size limits the suitability of an AO system using a conventional DM for clinical trials and eventual commercialization. Recently, new DM technologies have been developed based on both liquid crystal (LC) devices and microelectromechanical system (MEMS) mirrors, which are both compact and less expensive than the conventional DM devices. The AO group at the Lawrence Livermore National Laboratory (LLNL) has previously demonstrated very high order wavefront correction using LC and other DM technology [4, 5]. This chapter demonstrates the use of new LC technologies in the area of vision correction. We start with a discussion of important design parameters related to the wavefront sensor, light source, and spatial light modulator (SLM). Then, we describe the testing of each subsystem followed by the testing of the combined system. Results from human subjects testing are discussed at the end, along with suggestions for future design improvement.

18.2 18.2.1

WAVEFRONT SENSOR SELECTION Wavefront Sensor: Shack–Hartmann Sensor

The Shack–Hartmann wavefront sensor (SHWS) serves as the wave aberration measuring device [6, 7]. It consists of an 8-bit digital camera coupled with a lenslet array. The principle of operation of an SHWS can be described as follows: When a plane wave is incident upon the sensor, it produces a regular array of spots, each of which are located on the optical axes of the corresponding lenslets. The position of the initial array of spots is called the reference position. When a wavefront with aberrations is incident on the sensor, the focal spot of each subaperture (i.e., lenslet) shifts relative to the reference position by a factor proportional to the local slope of the wavefront. The position of the focal spot is determined by a centroid operation. The difference of the spot position from its reference position yields an estimate of the phase in each subaperture location.

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For an SHWS, there is a minimum and maximum phase that can be measured. The minimum phase is determined by the sensitivity of the SHWS, and the maximum phase is determined by the dynamic range of the SHWS. The sensitivity and dynamic range of the SHWS are discussed next. (For a more detailed discussion of sensitivity and dynamic range, see Chapter 3). Sensitivity Sensitivity is a measure of the smallest wavefront slope that can be accurately measured with a given lenslet array and charge-coupled device (CCD) camera. This relationship between the local slope of the incident wavefront and the Shack–Hartmann spot shift can be estimated from the dimensions of the SHWS. For our system, the lenslet diameters are 203 mm, with a focal length of 5.8 mm. This focal length was evaluated at a wavelength of 632.8 nm, which differs from the design wavelength (785 nm) chosen for the wavefront sensor beacon. However, the change in focal length for the longer wavelength was determined to be minimal. The SHWS has 20 × 20 subapertures. The camera pixels are 16 mm 2 . The sensitivity (or scale) of the SHWS in tilt angle per pixel is approximated by dividing the pixel size, dpix, by the focal length, F, of the lenslets: Sensitivity ≈

dpix F

=

16 µ m pixel = 2.76 mrad pixel 5.8 mm

(18.1)

Thus a centroid will shift by one pixel when the local tilt of the wavefront at a subaperture shifts by 2.76 mrad, as shown in Figure 18.1. As a result of this tilt, two parallel rays reaching two neighboring subapertures will be delayed by a phase difference, ∆f. This difference is (approximately) equivalent to the distance of the line segment formed along a ray between the perpendicular to the tilted wavefront and the detector (see Fig. 18.1). From similar triangles, the angle subtended by ∆f is equal to the angle subtended by a pixel at the detector plane. Therefore, the phase difference between two neighboring subapertures caused by a single pixel of tilt is

FIGURE 18.1

Sensitivity of a Shack–Hartmann wavefront sensor.

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DESIGN AND TESTING OF A LIQUID CRYSTAL ADAPTIVE OPTICS PHOROPTER

∆φ =

dpix F

d

(18.2)

where d is the diameter of the subaperture. Substituting the appropriate values yields ∆φ = 2.76 mrad × 203 µ m = 0.56 µ m

(18.3)

For a wavelength of 785 nm, this phase difference would correspond to (2p radians) × (0.56 mm/0.785 mm) = 4.48 radians. Dynamic Range The dynamic range of a Shack–Hartmann wavefront sensor is an important parameter of an AO system. The first step is to have an estimate of the range of aberrations that will need to be measured and compensated for. A statistical study of the aberrations of the human eye indicates that a dynamic range of 4 diopters (D) will be required to handle more than 90% of the population with pupil sizes under 6 mm, after correcting the lower order aberrations with trial lenses [8]. The second step is to determine the dynamic range of an SHWS. Dynamic range is defi ned as the maximum phase difference that an SHWS can measure, without having the focused lenslet spot leave its search box area (see Fig. 18.2) and fall behind a neighboring subaperture. The maximum wavefront slope will occur when the lenslet spot is at the edge of its search box. The spot size, S, produced by a square-sized subaperture at a wavelength of 785 nm is S≈

2 Fλ = 44.86 µ m = 2.8 pixels d

(18.4)

Now, drawing a perpendicular from the oblique ray to the center of the subaperture (see Fig. 18.2), we again form two similar triangles. Here, the phase difference of interest is the optical path difference between the oblique ray

FIGURE 18.2

Dynamic range of a Shack–Hartmann wavefront sensor.

WAVEFRONT SENSOR SELECTION

481

and the ray hitting the center of the aperture, and is denoted by ∆f. Since this phase difference subtends the same angle, q, as the distance (d/2 − S/2) of the larger triangle, we obtain tan θ = [ ( d 2 − S 2 ) F ]

(18.5)

Thus, the maximum phase difference that can be measured by each subaperture is Dynamic range = ∆φ = ( tan θ ) ( d 2 − S 2 ) = [( d 2 − S 2 ) F ] ( d 2 − S 2 ) 2 = (d 2 − S 2 ) F

(18.6)

Given the system parameters listed in Figure 18.2, the dynamic range of our wavefront sensor is 1.08 mm per subaperture. The dynamic range could be improved if a spot tracker is used, where the new spot position is tracked independent of the search box positions. In order to relate the above number to the dioptric power of a lens, consider that the peak-to-valley phase delay produced by a spherical lens can be described by the equation Phase delay ≈ 0.5 Psph r 2

(18.7)

where Psph is the dioptric value of the lens and r is the pupil radius. Here, a 5.8-mm pupil (measured at the pupil plane of the eye) was used to limit the magnitude of the higher order aberrations. Since our optical system contained a 2 : 1 scaling factor between the pupil plane of the eye and the CCD plane of the SHWS, the equivalent subaperture size at the pupil plane is 2 × 203 mm = 406 mm, and approximately 14 subapertures can fit within a 5.8-mm pupil. Thus, from the center of the SHWS, a maximum of 7 subapertures will be able to detect a total of 1.08 × 7 = 7.56 mm of phase difference. Using r = 2.9 mm in Eq. (18.7), Phase delay, φ = 0.5 ( Psph ) ( 2.9 ) µ m = 7.56 µ m 2

(18.8)

Solving for Psph yields Psph = 1.8 D Light Sensitivity Next, we estimate the camera response in terms of digital numbers (DN) given a certain amount of incident power. A digital number is an integer increment within the available bit range of a digital device. For instance, a camera with 8-bit pixels can represent an integer range from 0 to 255 in each pixel. Based on the camera manufacturer’s spectral responsivity

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DESIGN AND TESTING OF A LIQUID CRYSTAL ADAPTIVE OPTICS PHOROPTER

curve, the camera (Dalsa CA-D1) produces 10 DN for an incident illumination of 1 nJ/cm 2 at 785 nm. This is equivalent to 1011 photons/pixels per DN (see calculation below). Of the 5 mW of power incident on the retina from the wavefront sensor beacon, approximately 0.02% is reflected. This 1 nW of power, when incident on the SHWS for 100 ms, produces 190 DN (calculation to follow) and is sufficient for detecting the wavefront. The following detailed calculations show how we derived the digital number (DN) corresponding to 1 nW of light exposure: 1. To convert from nJ/cm 2 to photons/pixel, we fi rst calculate the number of photons in 1 nJ of energy at 785 nm: Energy in 1 photon at 785 nm = hv = hc λ = ( 6.626 × 10 −34 Js ) ( 3 × 10 8 m s ) ( 7.85 × 10 −9 m ) = 2.53 × 10 −19 J Number of photons in 1 nJ = ( 1 × 10 −9 J ) ( 2.53 × 10 −19 J ) = 3.95 × 10 9 photons

(18.9) (18.10)

Next, we calculate the number of pixels per cm 2 : 2

Number of pixels per cm 2 = ( 1.0 cm ) ( 16 µ m ) = 390, 625 pixels

2

Thus, the number of photons producing 10 DN per pixel is (3.95 × 109)/ (390,625) = 10,109, and approximately 1011 photons will produce 1 DN per pixel. 2. The total power coming to the camera from the eye is 1 nW. This is integrated for 100 ms. There are 20 × 20 subapertures. We assume that all of the energy contained within a single lenslet (i.e., subaperture) is focused to approximately 5 pixels on the CCD camera. Thus, the energy/ pixel of the camera is Energy pixel = ( 1 nW × 0.1 s ) ( 20 × 20 × 5 pixels ) = ( 0.1 nJ ) ( 400 × 5 pixels ) Photons pixel = ( 0.1 nJ × 3.95 × 10 9 photons nJ ) ( 400 × 5 pixels ) = 197,449 photons pixel Consequently, the digital number for 1 nW of light exposure is DN for 1 nW = ( 197,449 photons pixel ) ( 1010.9 photons DN pixel ) = 190 DN

(18.11)

WAVEFRONT SENSOR SELECTION

483

FIGURE 18.3 Experimental curve of peak pixel intensity versus integration time. The bottom curve is obtained from the digital number (DN) reading from one pixel of the camera that is a relatively dim pixel peak. The top curve is taken from a relatively brighter pixel at a different pixel location.

This number was verified in our laboratory by varying the integration time of the camera while recording the peak pixel intensity of the focused spots from the various subapertures. For a 100-ms integration time, we obtained peak values in the range of 109 DN to 220 DN, which encompasses the value obtained in the actual calculation. The graph in Figure 18.3 shows data recorded from two pixels from the Shack–Hartmann spots, one with relatively low light levels and the other with relatively high light levels. When using a human eye, the transmission/reflection of the beamsplitter used in front of the eye will be an extremely important design parameter to minimize the loss of photons that have been reflected by the eye. 18.2.2

Shack–Hartmann Noise

The noise statistics of the camera were measured for various camera integration times. The statistics provide the mean noise level (or dark noise) as well as the variance of the noise level, which determines the minimum detectable noise power. For example, for an integration time of 110 ms, the mean noise floor is 20.7 DN, while the standard deviation of the noise at any pixel is 0.03 DN for a dark exposure. When exposed to uniform light, the root-meansquare (RMS) error of the centroid location for 100 frames calculated using a center-of-mass algorithm with dark subtraction is 0.04 to 0.05 pixels. Note

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DESIGN AND TESTING OF A LIQUID CRYSTAL ADAPTIVE OPTICS PHOROPTER

from Eq. (18.3) that one pixel movement of the centroid is equivalent to 0.56 mm of phase error. Thus, this centroid error corresponds to a phase error of approximately 0.0256 mm, or ~l/31 (for l = 785 nm). When using a Gaussian fit algorithm to determine the centroid location, the RMS error is 0.0047 pixels (or 0.0026 mm ≈ l/300). 18.3 BEACON SELECTION: SIZE AND POWER, SLD VERSUS LASER DIODE At the beginning of the project, a 785-nm laser diode (LD) was used as the wavefront sensing beacon. However, we later changed to a superluminescent diode (SLD) when testing human subjects due to its lower observed speckle noise in the Shack–Hartmann spots. Two typical Shack–Hartmann spot patterns are shown in Figure 18.4, one for an LD and one for an SLD. Visually, it is obvious that the Shack–Hartmann spots produced by the laser diode have more noise. To observe the effect quantitatively, we calculated the standard deviation of the position of the Shack–Hartmann spots for 100 consecutive frames of images for each subaperture location. The result is compared for both the LD and SLD, as shown in Figure 18.5. The calculations were performed only on the illuminated pixels. The subapertures at the edges were not included in the calculation. The mean RMS noise of the illuminated subapertures for the laser diode is 163 nm and is 95 nm for the SLD. A consequence of changing the light source is that we also had to change the SLM to one that had a higher reflectivity at 820 nm. However, most of the characterizations were performed with the 785-nm source.

FIGURE 18.4 Hartmann spots for the human eye with a laser diode (LD) and a superluminescent diode (SLD). The LD shows more speckle in each individual spot than the SLD spot array pattern.

WAVEFRONT CORRECTOR SELECTION

485

FIGURE 18.5 Standard deviation of the Shack–Hartmann spot positions for a human eye, measured using a LD (left) and an SLD (right). The LD exhibited 163 nm of noise versus only 95 nm for the SLD.

18.4

WAVEFRONT CORRECTOR SELECTION

The spatial light modulator that we used (Hamamatsu Model 7665) contains approximately 230,000 phase control points and serves as the wavefront corrector in the AO phoropter. The parallel aligned nematic liquid crystal spatial light modulator (PAL-SLM) is an optically addressable (intensity to phase) spatial light modulator, as shown in Figure 18.6. The PAL-SLM has an amorphous silicon layer, a dielectric mirror, and a liquid crystal (LC) layer sandwiched between two glass substrates with transparent electrodes. A write light beam impinges on the amorphous silicon side, and the read beam is presented on the LC side. The impedance of amorphous silicon becomes extremely high when no write light is present. When the write light is applied, the impedance of the amorphous silicon is lowered reducing the voltage drop across it. Consequently, the voltage across the liquid crystal layer is increased. The increase in voltage across the liquid crystal layer affects the molecular orientation of the crystals and changes its index of refraction, causing a phase modulation of the read beam. To control the optical intensity of the write beam, a laser diode is coupled with a liquid crystal display (LCD) used in the transmissive mode. This allows the projection of any intensity pattern on the write side of the PAL-SLM. The PAL-SLM module combines the laser diode and LCD with the PAL-SLM so that the entire system acts as an electronically addressable phase/intensity spatial light modulator. The SLM contains 480 × 480 individually addressable control points on a 20-mm × 20-mm surface, where each control point can provide up to 0.8 mm of phase modulation. The rise time is 140 ms and the fall time is 230 ms for the X7665, with a readout wavelength between 550 and 850 nm. This timing was measured by the manufacturer with a pulsed laser diode writing directly

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DESIGN AND TESTING OF A LIQUID CRYSTAL ADAPTIVE OPTICS PHOROPTER

LCD PAL SLM

Laser Diode VGA Signal

ITO Electrode Layer

5Vpp

LC

Dielectric Mirror

ITO Electrode Layer

Read Beam Write Beam

Readout Light

Optical Optical Alignment Alignment α-SiH Glass Layer Layer PhotoGlass conductor Substrate Substrate

FIGURE 18.6 The liquid crystal (LC) spatial light modulator (SLM). A picture of the device is shown in the left image, while a schematic of the SLM’s constituent layers is shown on the right.

on the write surface. However, in the actual SLM module, an LCD panel is used to modulate the write light and the LC panel has its own delay. Thus, the delay of the SLM practically may be much higher than the rise and fall times listed. Another important consideration of the SLM is its reflectivity at green wavelengths, the wavelength at which the image will be viewed, since the brightness of the target image is important for psychophysical experiments. Also note that the reflectivity of the SLM at 820 nm is important for the sensitivity of the SHWS. Choice of the proper SLM must take into consideration all of these variables.

18.5

WAVEFRONT RECONSTRUCTION AND CONTROL

As described in Section 18.2, the wavefront is calculated by comparing its deviation from a plane wave. To determine the plane wave position of the lenslet spots, the Shack–Hartmann wavefront sensor is illuminated with a planar wavefront. In each subaperture, the light will be focused on the optical axis of that subaperture. These positions are marked by calculating the centroid location of each spot and are called the reference centroid positions. When there is a phase gradient across the wavefront, the rays of light will be tilted and the focus point of each subaperture will be shifted by a proportional amount. Thus, when an aberrated wavefront is sampled by the Shack– Hartmann wavefront sensor, the locations of the new centroids are shifted according to the local phase gradient. These new centroid locations are calculated, and the difference between these locations and their reference

WAVEFRONT RECONSTRUCTION AND CONTROL

487

positions provides a measure of the local slope of the wavefront. This can be expressed as: ∂W ( x, y ) ∆xS = ∂x F

(18.12)

where ∂W/∂x is the slope of the wavefront in the x direction, ∆xS is the displacement of the centroids in the x direction, and F is the focal length of the lenslet. A similar equation can be derived for the wavefront slope in the y direction and a set of discrete linear equations can be written relating the slopes (in both x and y directions) to the derivatives and eventually to the displacement of the centroids (see Chapters 3 and 5). This leads to a discrete set of equations relating the phase to the slope. These equations can be solved by either a least-squares method or a Fourier transform method to yield the reconstructed wavefront. In our system, we utilize a Fourier transform technique to reconstruct the wavefront from the centroid differences [9]. Note also that the sampling interval of the wavefront slope is equal to the pitch of the subaperture array. Thus, using the Nyquist sampling theorem, we will only be able to detect phase variations corresponding to spatial frequencies that are half the sampling rate, as determined by the size of the subapertures. 18.5.1

Closed-Loop Algorithm

The algorithm for closed-loop error correction consists of the following steps: 1. Retrieve reference centroid locations from a planar wavefront. 2. Obtain test centroids from the aberrated wavefront. 3. Threshold both images for read noise and dark noise. (The dark noise level was estimated from a dark frame and read noise was estimated from the variance of constant illumination.) 4. Calculate centroid locations and the difference between the reference and the test centroid positions. 5. Input the difference data to the reconstructor and reconstruct the wavefront. 6. Estimate the correction to be applied, which is a function of the gain parameter of the control loop. This parameter determines how much of the error term is applied as a correction. (Lower gain typically implies that more iterations are necessary to achieve convergence.) 7. Convert the correction (in phase units) to SLM units using a lookup table or other simple formula. 8. Repeat steps 2 to 7 a prespecified number of iterations, or until the error is below a certain threshold value.

488

18.5.2

DESIGN AND TESTING OF A LIQUID CRYSTAL ADAPTIVE OPTICS PHOROPTER

Centroid Calculation

Accurate calculation of the centroid positions is one of the most important operations in wavefront reconstruction algorithms. We compared the variances in calculating centroid locations with different algorithms, such as the center of mass, Gaussian fitting, and the diminishing area of interest technique [10]. One hundred frames of Shack–Hartmann spots were collected and the variation of the centroid position of each subaperture was measured. Ideally, with no change of wavefront slope, the spots should remain stable, but because of noise the estimated position (in pixels) changes. These changes can be converted to phase differences using our 0.56 mm per pixel scaling factor derived in Eq. (18.3). The standard deviations of the centroid positions in the center-of-mass technique were on the order of 10 −2 pixels (0.0056 mm), whereas those obtained from Gaussian fitting were on the order of 10 −3 pixels or 0.00056 mm. The standard deviation of the diminishing area of interest (or pyramidal) centroid technique approaches a Gaussian fitting technique as the area of the smallest box approaches that of the spot size. For tightly focused Shack–Hartmann spots with high signal-to-noise ratios, the Gaussian method provided excellent repeatability in the presence of noise. The noise condition was varied by changing the signal level while keeping the noise level relatively constant, resulting in a variation in the signal-to-noise ratio. Although the Gaussian fitting technique yielded better results, it was not used ultimately because the method consumes a considerable amount of time (~5 s or more) and sometimes fails to converge. An RMS measurement reveals the variation in determining centroid locations over time; however, it may not reveal a systematic error. For real Shack– Hartmann spots from the eye, fi nding an appropriate threshold for each Shack–Hartmann spot is another important step. It should be noted that a typical subaperture after dark subtraction shows a significant amount of background noise around the central peak, as shown in Figure 18.7. One of the problems with fi nding an appropriate global threshold for excluding this background is the variance of intensities in the Shack–Hartmann spots among different subapertures when a human eye is used. Pyramidal thresholding [10] seems to do a better job in reducing the noise than simple thresholding over the whole plane. In the pyramidal technique, an initial estimate of the spot position (based on a center-of-mass calculation) is made using a bounding box (search box) that is equal in size to an array of 11 × 11 pixels per subaperture. Thereafter, the bounding box is reduced to a size of 9 × 9 pixels per subaperture, centered on the previous estimate and a new estimate is made. The process is repeated and the bounding box reduced until the box size is equal to the estimated spot size. A practical problem of applying the pyramidal technique is when the estimated spot size is very small, such as 2.4 pixels. However, when practical, this technique allows for more effective noise fi ltering by thresholding and creating a smaller bounding box around the spot.

SOFTWARE INTERFACE

489

FIGURE 18.7 Pixel intensity values of a Shack–Hartmann spot within a single, typical subaperture after dark subtraction. Note the presence of noise outside of the main lobe (with a radius of 1.7 pixels and a central value of 226 DN). The noise floor that has been subtracted is 20 DN.

To provide a visual estimate of the effect of various centroiding algorithms, wavefronts reconstructed when a Gaussian phase function was applied to the SLM are shown in Figures 18.8 and 18.9. The Gaussian pulse shape is recreated in all the pictures, however, the floor of the reconstructed wavefront demonstrates significant fluctuations from an estimated flat wavefront. The RMS wavefront error is calculated over the flat region. As the aperture size is reduced in four consecutive steps (as shown in Fig. 18.8), the pyramidal reconstruction starts to look very similar to the Gaussian curve fit results shown in Figure 18.9. A curve is plotted (Fig. 18.10) to show the reduction in noise as the width of the aperture decreases. Finally, other systematic error sources that may creep up are round-off errors in calculating centroid differences. This occurs when subtracting two numbers that may be very small, such as subtracting two centroid positions to fi nd the local slope.

18.6

SOFTWARE INTERFACE

There are two types of software needed to run the AO system: the control software and the diagnostic software. All software for this project was written using IDL (Research Systems Inc., Boulder, CO) and the interface was run from the IDL command line. Examples of some of the properties of the diagnostic software are to display the live position of the Shack–Hartmann spots within the subaperture boundaries, measure the phase response by writing phase functions and recording the spot intensities, measure the statistics of spot locations on any number of consecutive frames, or simply record consecutive Shack–Hartmann spots. The diagnostic software is useful in aligning the optics with the Shack–Hartmann spots and also in displaying the effect of the pupil function directly. The control software has the capability

DESIGN AND TESTING OF A LIQUID CRYSTAL ADAPTIVE OPTICS PHOROPTER

2

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Subaperture Size = 9 ¥ 9

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Subaperture Size = 7 ¥ 7

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Subaperture Size = 5 ¥ 5

Phase (rad)

Phase (rad)

FIGURE 18.8 Various stages of the pyramidal centroiding process. As the subaperture size (in pixels) decreases, the floor of the Gaussian phase function appears smoother.

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Center of Mass (11 ¥ 11)

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Gaussian Curve Fit

FIGURE 18.9 Comparison of the center-of-mass centroiding technique using 11 × 11 subapertures, versus centroid calculations using a Gaussian curve fit.

of displaying the reconstructed wavefront or point spread function, showing the RMS error or other statistics when the loop is closed, and so forth. Simulation software was also created that could run the experiments offl ine from the stored corrections sent to the SLM and the stored Shack–Hartmann spots.

AO ASSEMBLY, INTEGRATION AND TROUBLESHOOTING

491

RMS Wavefront Error (rad)

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Subaperture Size (Pixels) FIGURE 18.10 Effect of subaperture size on RMS wavefront error. Here, the error is calculated based on the RMS wavefront error on the flat region of the Gaussian phase function.

18.7

AO ASSEMBLY, INTEGRATION AND TROUBLESHOOTING

A schematic of the adaptive optics phoropter system is shown in Figure 18.11. The instrument uses a 5-mW superluminescent laser diode at 820 nm that is focused onto the retina of a human eye. (Initially, the system was equipped with a 785-nm laser diode used for characterizing the system. It was switched to an 820-nm SLD to reduce the speckle in the Shack–Hartmann spots from the human eye.) The laser beacon is reflected off the retina and out through the optics of the eye, thereby sampling its aberrations. The light reflected by the retina (~0.02% of the input intensity) is transmitted to a Shack–Hartmann wavefront sensor after being reflected by a wavefront corrector. This arrangement allows for a closed-loop correction of the optical aberrations. Here, the Shack–Hartmann wavefront sensor initially measures the wavefront when the wavefront corrector is flat or is in its nonaltered position. The deviation of the reconstructed wavefront from an ideal flat wavefront is estimated, and this produces the error term. This error term is used to calculate the correction required to compensate for the deviation. The wavefront is flattened after a few iterations by the wavefront corrector, or SLM. A control loop is used to update the correction applied to the wavefront corrector in a stable fashion. During the closed-loop operation of the AO system, a correction is applied until the error converges to a minimum value. When the correction is applied successfully in the closed-loop system, it should result in improved optical image quality and enhanced vision. After the system has converged to a stable, low aberration value, the subject views

492

DESIGN AND TESTING OF A LIQUID CRYSTAL ADAPTIVE OPTICS PHOROPTER

Mirror

Removable Reference

Lenslet Array

Lens

WS Mirror

Mirror

2 ft

4 ft

Dichroic Phoropter

Monitor for Acuity Test Lens

Eye

SLM Lens

Lens Polarizer

Fiber Source SLD

FIGURE 18.11 Prototype adaptive optics phoropter using a liquid crystal spatial light modulator. SLM, spatial light modulator; WS, wavefront sensor; SLD, superluminescent diode.

any of a variety of visual stimuli (e.g., sine-wave gratings) on a custom, highintensity cathode ray tube (CRT) computer display. This system is used to perform psychophysical tests examining the effects of a higher order correction on the limits of visual performance. Figure 18.11 illustrates two light paths. The darker gray beam shows light emerging from the eye and entering the wavefront sensor after being reflected by the wavefront corrector. The lighter gray beam represents rays from the visual stimulus traveling to the subject’s eye. 18.8 SYSTEM PERFORMANCE, TESTING PROCEDURES, AND CALIBRATION As a fi rst step, each separate subsystem is tested independently [11]. Then, they are combined in order of complexity, and the overall functionality of the

SYSTEM PERFORMANCE, TESTING PROCEDURES, AND CALIBRATION

493

system is tested. The testing of the SLM subsystem is described in Sections 18.8.1, 18.8.2, and 18.8.3, and the testing of the wavefront sensor is discussed in Section 18.8.4. Testing the combined system (i.e., registration and closedloop operation) is covered in Sections 18.8.5 and 18.8.6. 18.8.1 Nonlinear Characterization of the Spatial Light Modulator (SLM) Response A set of experiments were carried out to determine the phase modulation characteristics of the liquid crystal SLM. This characterization will determine the gray level needed to achieve a certain value of phase modulation. The SLM was characterized by applying a periodic rectangular wave of varying amplitude to the SLM and measuring the far-field pattern. The relative magnitude of the zeroth- and fi rst-order components provides an indication of the phase jump magnitude. For example, for a phase jump of p radians, the zeroth-order intensity becomes zero, while the fi rst-order intensity becomes maximal. The same phenomenon is observed across each of the subapertures of the Shack–Hartmann wavefront sensor, which instead of consisting of a single, big lens, uses a microlens array. Thus, a second spacevariant test was developed to determine the phase response of individual actuators, or SLM pixels. If a step function of varying amplitude is applied over a single aperture, each subaperture produces a far-field pattern corresponding to the Fourier transform of the phase jump. The history of all the wavefront sensor responses as a function of the amplitude of the step function was recorded. The phase response of individual actuators was calculated. A lookup table for the SLM was devised, combining the individual subaperture responses with the overall SLM response. The desired phase angle (i.e., desired compensation expressed in radians, where 2p represents one wave of modulation) is the input to the lookup table and the required gray level of the SLM to cause that phase change is the output. The phase-response behaviors obtained from the individual subapertures reveal that the SLM has space-variant phase modulation characteristics. A spatially varying lookup table was devised that compensated for the nonuniformity of the phase response across the SLM surface. The average phase response from the lookup table is expressed in the curve shown in Figure 18.12. The x axis represents the phase modulation, while the y axis represents the driving gray level for the SLM. For convenience, the phase modulation (as shown on the x axis) has been remapped to integer values in the 0 to 255 range; this was accomplished by multiplying the desired phase value by 40 and then rounding to the nearest integer. The SLM driving gray levels (y axis) are the values sent to the SLM. 18.8.2

Phase Wrapping

Note that the SLM can achieve only a fi xed range of phase modulations, corresponding to approximately one wave of compensation. Due to the limited

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DESIGN AND TESTING OF A LIQUID CRYSTAL ADAPTIVE OPTICS PHOROPTER

SLM Driving Gray Level

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Desired Phase Modulation (a.u.) FIGURE 18.12 SLM.

Graph illustrating the lookup table for the phase response of the

phase modulation of the SLM, the excess phase must be remapped to a phase between 0 to 2p. This is done by a simple modular operation, as demonstrated visually in Figure 18.13(b). This process is known as phase wrapping. In order to verify the phase wrapping technique, Gaussian inputs with peak phase differences varying from 1 to 3 waves were applied to the SLM. The corresponding reconstructed wavefronts were observed. The peak values of the input and output wavefronts were recorded. The plot of the output peak values versus the input peak values is shown in Figure 18.13. The phase wrapping technique was also verified in an earlier setup built at LLNL, known as an AO test bed [12]. In this system, a periodic pattern was written on the whole SLM, and the SLM was exposed to two different frequencies of light. When one of the frequencies of light matched the frequency used for the phase wrapping operation, the far-field image showed a single spot corresponding to a uniform phase delay, while the other frequency of light generated a diffraction pattern due to the periodic phase grating seen by that frequency. Thus, when operating the SLM at two different frequencies, the phase modulation characteristics should ideally be derived separately for both frequencies. Alternately, the appropriate phase correction could be approximated by multiplying the desired correction by the ratio of the two frequencies. The phase wrapping point, however, will change in this latter case. The simplest way to handle this complexity is to express the desired phase modulation in radians (rather than microns) for the second frequency. Then, one can perform the wrapping in the radian domain and then convert into radians for the fi rst frequency by scaling the wrapped phase using the frequency factor. This procedure allows us to use the same lookup table derived for the first

SYSTEM PERFORMANCE, TESTING PROCEDURES, AND CALIBRATION

495

Maximum Phase of Reconstructed Wavefront (Waves)

2.5

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(b) 0.0 1.0

1.5

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Maximum Phase of Aberration Applied to SLM (Waves) (a) FIGURE 18.13 (a) Linearity of the wrapped phase. (b) The wrapped input to the SLM results in a continuous response of the phase.

frequency of light to fi nd the necessary gray levels to send to the SLM to achieve the desired wrapped phase for the second frequency. 18.8.3

Biased Operation of SLM

A typical correction pattern reveals that both positive and negative corrections must be applied to compensate for an aberration. This is achieved by operating the SLM at a p-phase bias, allowing both positive and negative corrections to be applied. Here p refers to the phase of the SLM with a graylevel value corresponding to a phase delay of l/2, which for an 8-bit SLM with a linear response would be around a gray-level value of 128. Thus, negative phase shifts from −p to 0 would be achieved by sending the SLM graylevel values from 0 to 128. Similarly, positive phase shifts from 0 to +p would be achieved by sending the SLM gray-level values from 128 to 255. Phase shifts outside this range can be achieved by subtracting or adding integer multiples of 2p and sending the corresponding phase-wrapped value to the SLM. As a result of the bias point, the fi rst phase wrap occurs after a l/2 wave excursion and the second phase wrap will occur after 3l/2. 18.8.4

Wavefront Sensor Verification

To verify the correct operation of the wavefront sensor and the reconstruction algorithm, we measured the wavefront produced by a lens with a known focal

496

DESIGN AND TESTING OF A LIQUID CRYSTAL ADAPTIVE OPTICS PHOROPTER

length. For this experiment, a model eye consisting of a lens and a rotating disk at the position of the retinal plane was used. The distance between the lens and the rotating disk was fi xed at the focal length of the lens and both items were securely mounted on a rail, which could be easily inserted into the AO system. The rotation of the disk reduced the speckle in the Shack– Hartmann spots. A 0.25-D lens with an 8-mm diameter was selected on the phoropter, which was physically located in front of the model eye. Using the data from the Shack–Hartmann wavefront sensor, the wavefront was reconstructed using the Fourier reconstructor and the peak-to-valley wavefront error of the lens was measured to be exactly 2 mm for the 8-mm diameter lens (see Fig. 18.14). For comparison, the peak-to-valley phase difference was calculated using Eq. (18.7) and was also found to be 2 mm for an 8-mm pupil diameter. 18.8.5

Registration

The test for registration evaluated how well multiple subsystems (such as the SLM, the wavefront sensor and the laser beacon) worked together. Using an artificial eye, a systematic procedure was developed to register the SLM with the wavefront sensor. If the SLM is not properly registered with the wavefront sensor, then the correction calculated from the wavefront measurement will be misaligned with respect to the position of the aberrations on the incoming wavefront. Figure 18.15 illustrates an example of this problem. An initial aberration is applied to the SLM, and this elicits a compensating correction.

FIGURE 18.14 Reconstructed wavefront for a 0.25-D lens showing 2 mm of peak-tovalley phase difference.

SYSTEM PERFORMANCE, TESTING PROCEDURES, AND CALIBRATION

497

Actual Location of Aberration Peak

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SLM Plane FIGURE 18.15 Effect of misregistration between the wavefront sensor and corrector. The wavefront after one iteration of the closed-loop correction has become more aberrated because the wavefront sensor and corrector are not properly aligned. Instead, the correction has been applied to the wrong location. This position error contributes to the next iteration through the closed-loop system, resulting in a new incorrect position for the wavefront correction.

Due to misregistration, the wavefront sensor sees the aberration in one location but the AO system applies the correction at a different location. Figure 18.16 shows the results after 3, 5, and 11 iterations. A misregistration of the SLM in the x direction causes vertical lines to appear on the SLM, which increase in magnitude with each iteration. The bright spot in Figure 18.16 is the initial location of the aberration, also shown as the positive peak in Figure 18.15. In the leftmost panel of Figure 18.16, the negative correction is shown as a dark spot next to the initial positive peak. In the same way, the negative correction will then generate a positive correction at a neighboring location. This is shown as a bright spot located to the right of the dark spot [see Fig. 18.16 (center)]. Thus, with each iteration, the error propagates to the right resulting in a series of dark and bright lines [see Fig. 18.16 (right)]. To address this problem, the registration method starts by writing an asymmetric pattern of known size and shape on the SLM, as shown in the left panel of Figure 18.17. The asymmetry helps to identify any ambiguities in rotation and/or reflection. The actuation function is then compared to the wavefront reconstruction obtained from the wavefront sensor data, as shown in the right panel. Making this comparison provides information about the registration in terms of rotation, scale, and position of the SLM, with respect to the wavefront sensor. For improved accuracy, the comparison is performed at the pixel level of the SLM plane. Having compared the actuation function and the reconstructed wavefront, the size and position of the pattern is modified until the written and detected patterns corresponded.

498

DESIGN AND TESTING OF A LIQUID CRYSTAL ADAPTIVE OPTICS PHOROPTER

FIGURE 18.16 Effect of misregistration between the wavefront sensor and corrector. Note that the SLM shows the initial aberration as a bright spot. (Left) The negative correction applied shows up as a dark spot next to the initial aberration, shown as a bright spot. (Center) The dark spot generates a correction that appears as a neighboring bright spot. (Right) After several iterations, this misregistration leads to an alternating appearance of bright and dark spots on the SLM, and an improper correction.

FIGURE 18.17 Registration verification. (Left) An asymmetric pattern written on the SLM and (right) its reconstruction detected by the Shack–Hartmann wavefront sensor.

Figure 18.18 show a more detailed example of checking for correspondence between an image written by the SLM and an image detected and reconstructed by the Shack–Hartmann wavefront sensor. This process yields four parameters: the size (or scaling factor) of the pattern in the x and y dimensions; and its x and y positions relative to the whole SLM plane on which the pattern could be written. Neglecting rotation, these four parameters describe exactly the position of the pattern written on the SLM.

SYSTEM PERFORMANCE, TESTING PROCEDURES, AND CALIBRATION

499

FIGURE 18.18 Registration verification (enlargement of zoom area marked on Fig. 18.17). Finding the correspondence between (a) the pattern written on the SLM and (b) the pattern detected by the wavefront sensor. Four parameters are varied until these two patterns match.

18.8.6

Closed-Loop Operation

In order to verify the proper operation of all the subsystems, the AO system must be operated in a closed-loop fashion. In the closed-loop mode, the system operates to minimize the error in the wavefront. The system was tested with two specific aberrations: (i) an input on the SLM serving as a source of aberration and (ii) a trial lens from the phoropter at the pupil plane. In the fi rst test, an artificial aberration (in the form of a Gaussian function of fi xed width) was applied through the SLM. This was achieved by writing a graylevel Gaussian function to the SLM, as shown in Figure 18.19. Since the wavefront sensor will register the function as an aberrated wavefront, the control loop will change the surface of the SLM to reduce the wavefront error of the system. If the system is working properly, the wavefront error will be gradually reduced. The Shack–Hartmann wavefront sensor is used to measure the wave aberration for each iteration of the closed-loop operation. As shown in Figure 18.20, the aberration is gradually compensated by the correction applied through the control loop. The RMS wavefront error was reduced from 0.050 mm (or 0.40 rad of phase) to 0.025 mm (or 0.20 rad of phase), as shown in Figure 18.21. Next, an external aberration was applied to the system using a spherical trial lens. The reconstructed wavefront depicted in Figure 18.22 shows that the total aberration (peak-to-valley) produced by this 0.25-D lens was 2 mm or ~3 waves at 785 nm. Thus, to compensate for this aberration using the SLM, which has about 0.8 mm of phase modulation, phase wrapping was performed to extend the range of the compensation across multiple waves. When the loop was closed with the AO system operating, the wavefront was gradually flattened by the AO control loop, as shown in the four panels of Figure 18.22.

500

DESIGN AND TESTING OF A LIQUID CRYSTAL ADAPTIVE OPTICS PHOROPTER

Initial Input on SLM

Reconstructed Phase Phase (rad)

4 2 0 −2 −4 20 15

1

0 5 S SH 0 0 Nu ubap WS mb ert er, ure yA xis

10

5

15

20

SHWS re ertu Subap x Axis er, Numb

FIGURE 18.19 (a) An aberration input on the SLM and (b) its reconstructed phase for an 8-mm pupil.

Phase (rad)

2nd Iteration (0.34 s)

4th Iteration (1 s)

4

4

2

2

0

0

−2

−2

−4

−4

Phase (rad)

6th Iteration (1.7 s)

10th Iteration (3.1 s)

4

4

2

2

0

0 −2 −4 20 1

−2 −4

51

0

5 0 0 S SH Nu ubap WS mb ert er, ure yA xis

5

10

15

20

SHWS re ertu Subap x Axis er, Numb

FIGURE 18.20 Reconstructed wavefront (in radians of phase) after 2, 4, 6, and 10 iterations, with a pupil size of 8 mm.

The corresponding SLM phase modulations after the 2nd and 24th iterations of the closed-loop correction are shown in Figure 18.23. The right panel of Figure 18.23 demonstrates the phase wrapping boundaries at 0.5, 1.5, and 2.5 waves above the 0.5 wave bias level, which was necessary to correct 3

SYSTEM PERFORMANCE, TESTING PROCEDURES, AND CALIBRATION

501

FIGURE 18.21 Graph of the RMS wavefront error, showing convergence of the AO loop on a Gaussian input to a fi nal RMS error of 0.025 mm.

Phase (mm)

0th Iteration

3rd Iteration

2

2

1

1

0

0

−1

−1

20

20

15

10

5

0

0

5

10

15

20

Phase (mm)

5th Iteration 2

1

1

0

0

−1

−1

15

20 10

5

0

0

5

10

5

10

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20

5

10

15

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0 0

9th Iteration

2

20

15

10

15

20

15

10

5

S SH Nu ubap WS mb er er, tur yA e xis

0

0

SHWS re ertu Subap x Axis r, e b Num

FIGURE 18.22 (Top left) A 0.25-D lens in the adaptive optics system produces an initial aberration. (Top right) After three iterations, the aberration is partially compensated. The aberration is further reduced after the fi fth (bottom left) and ninth (bottom right) iterations. The lens was used with a pupil size of 8 mm.

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DESIGN AND TESTING OF A LIQUID CRYSTAL ADAPTIVE OPTICS PHOROPTER

FIGURE 18.23 SLM phase modulation applied by the control loop to correct defocus across an 8-mm pupil. Note that phase wrapping at 0.5, 1.5, and 2.5 waves is needed to achieve the full 3 waves of wavefront compensation.

FIGURE 18.24 Convergence of the closed-loop correction of the 0.25-D lens. The RMS wavefront error was reduced from 0.70 to 0.05 mm. Each iteration was 0.34 s in duration.

waves of aberration in the trial lens. The gray background shows the constant bias of 0.5 waves. Finally, as shown in Figure 18.24, the RMS wavefront error was gradually reduced from 0.7 mm (nearly one wave RMS) to 0.05 mm (nearly 1 – waves). A gain of 0.2 was used to improve the stability of the system. 16 18.9

RESULTS FROM HUMAN SUBJECTS

In order to optimally measure and correct for the aberrations of the human eye, subjects must be seated, have their eyes dilated, and they must bite down

RESULTS FROM HUMAN SUBJECTS

503

on a molded plastic bite bar to stabilize the motion of their head. The bite bar is also used to position the subject’s pupil in three dimensions (horizontally, vertically, and axially) because it is important that the subject’s pupil remain centered on the optical axis of the system during the entire experiment. A typical setup with human subjects is shown in Figure 18.25. Note that the light reflected from the eye is detected by the Shack–Hartmann wavefront sensor, shown next to the display with the letter C. The monitor displaying the letter is the source of the visual stimulus viewed by the subject after aberration correction. The fi rst step is to correct the subject’s refractive error using a conventional phoropter placed in front of the eye to eliminate the defocus and astigmatism error. This procedure allows us to use the limited phase modulation of the SLM for correcting higher order aberrations, instead of using the bulk of the SLM phase modulation to correct for defocus and astigmatism. Then, the subject’s pupil is aligned using the x-y-z positioners on the bite bar. By monitoring the pupil camera image, the subject’s pupil is placed in a plane conjugate with the SLM and the wavefront sensor. The subject is asked to fi xate on a target on the monitor. Head position may again have to be adjusted slightly to align the subject’s pupil to the Shack–Hartmann wavefront sensor, if it is not already aligned perfectly. After the subject is stabilized and all the Shack–Hartmann spot data are obtained, the subject is asked to maintain fi xation on the target while the loop is closed.

FIGURE 18.25 subject.

A complete view of the adaptive optics phoropter with a human

504

DESIGN AND TESTING OF A LIQUID CRYSTAL ADAPTIVE OPTICS PHOROPTER

The left panel of Figure 18.26 shows a typical Shack–Hartmann spot pattern obtained from a human eye, for a pupil size of 8 mm. This figure shows noise in the form of speckle due to retinal scatter generated by the laser diode source. We have since switched to a superluminescent diode (SLD) to reduce speckle. The SLD has a center wavelength of 820 nm and a bandwidth of 30 nm. Using this source, the wavefront across a 5.8-mm pupil was measured at different steps during the closed-loop correction (see Fig. 18.27). The reconstruction shown in this figure appears blotchy because the wavefront has been reconstructed from a smaller pupil support that spans only 13 × 13 subapertures, instead of 20 × 20 subapertures. Also, no smoothing was applied to the reconstructed wavefront.

FIGURE 18.26 (Left) A typical Shack–Hartmann wavefront sensor pattern and (right) the corresponding reconstructed wavefront for a human eye (8-mm pupil). The RMS wavefront error for this subject was 3.5 mm.

FIGURE 18.27 The wave aberration from a human subject using a zonal compensation after (left) 2, (center) 5, and (right) 10 iterations of closed-loop correction.

RESULTS FROM HUMAN SUBJECTS

505

As the loop converges, the RMS wavefront error was reduced from 0.56 to 0.15 mm, as depicted in Figure 18.28. This figure also shows that the RMS wavefront error fluctuates while it converges. The reason for this was traced to the speed of the response of the SLM in the software-integrated environment. Although a 400-ms delay was assumed, in reality the SLM was responding with a 2-s delay. Thus, with an update time of 0.4 s, the wavefront was being measured before the effect of the correction took place. As a result, the error measured by the Shack–Hartmann wavefront sensor was sometimes higher than the actual error to be corrected. This would cause more correction to be applied in the next step that could again be sometimes in the wrong direction. Thus the error was seen to fluctuate. When a 2-s delay was added to the loop, the system converged much more smoothly in two to three iterations, as shown in Figure 18.29. Once the subject’s vision has been corrected and the laser beam is turned off, any of a number of psychophysical tests can be performed (see also Chapter 14). For example, in one typical experiment, a subject is asked to determine in which of two temporal intervals a sinusoidal grating appears. The contrast of the grating is adjusted using an adaptive procedure in order to determine the contrast that will yield correct performance on 82% of the trials. This contrast is referred to as the subject’s contrast threshold for that stimulus. By determining contrast thresholds at a number of spatial frequencies, a contrast sensitivity function (CSF) can be obtained. The human CSF is a product of the optical modulation transfer function (MTF) of the eye and the neural CSF of the visual nervous system. By measuring the CSF before

FIGURE 18.28 Convergence of RMS wavefront error for a human eye over a 5.8-mm pupil. The initial RMS wavefront error was 0.56 mm and was corrected to an RMS error of 0.15 mm.

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DESIGN AND TESTING OF A LIQUID CRYSTAL ADAPTIVE OPTICS PHOROPTER

FIGURE 18.29 Smooth convergence of RMS wavefront error for a human eye over a 5.8-mm pupil, when the proper 2-s delays were inserted in to the control loop.

and after wavefront correction, the benefits for improving vision by improving the optical MTF can be estimated.

18.10

DISCUSSION

One of the problems of the SLM is its wavelength-dependent phase modulation. Normally, a correction is applied based on the wavelength of the beacon used for the Shack–Hartmann wavefront sensor. However, the subject typically views an object using a green wavelength of light that generates a different phase response from the SLM than the 820-nm beam of light. To compensate for this effect in our vision experiments, we calculated the fi nal correction to be applied to the SLM (in radians) by converting the desired value to green wavelengths and phase wrapping the result appropriately, using the new wavelength. Here, we assumed the change in wavelength caused a proportional change in the phase modulation generated by the SLM, which may not be true. This issue could be overcome by deriving the SLM phase modulation characteristics using a green laser on the SLM. Then, by employing a lookup table for the green wavelength, this could yield a more realistic correction for the viewing conditions. Since this was not done, it was difficult to carry out any psychophysical experiments. However, to verify the closed-loop performance of the system when imaging at a second wavelength, we replaced the human eye with a CCD camera placed at the position of the eye, so that it was looking at a test pattern displayed on the monitor. The CCD was adjusted to produce a sharply focused image of the test pattern, thus mimicking the

DISCUSSION

507

FIGURE 18.30 Images from a CCD camera viewing the pattern on a display when imaged through a cylindrical lens, placed horizontally across the image path. (Left) Image before adaptive optics correction. (Right) Image after correction with the LC SLM.

focusing ability of the human eye. Then, the pattern was aberrated using a cylindrical trial lens, as shown in Figure 18.30, and subsequently the loop was manually closed by gradually increasing the correction and checking on the imaged object. The image, which was severely aberrated, was subsequently improved by the AO correction. This proves that the AO system could, in fact, improve image quality. The problem of applying an automated closedloop correction stems from the difficulty of measuring the aberrations produced by the artificial eye, since the surface of the CCD camera does not provide a very good reflection of the beacon back to the Shack–Hartmann wavefront sensor. Additional problems were encountered due to a high noise level from the Shack–Hartmann spots at low light levels. Sometimes, low light levels would result in double spots. Changing the beamsplitter in front of the eye to allow for the transmission of more light would improve the noise situation, though double spots could also be an artifact of the low dynamic range of the Shack– Hartmann wavefront sensor. The discrepancy between the published speed of the SLM and the speed of the SLM in the AO loop also caused difficulty in smoothly converging to a stable correction. The effect was minimized by choosing a low gain and increasing the delay time between loop iterations. However, if the error is not properly minimized, then the psychophysical tests will be unpredictable. It is also possible that, due to the limited dynamic range of the wavefront sensor, aliasing can occur for the Shack–Hartmann spots, which would result in an incorrect measurement and hence an incorrect AO correction. This incorrectness can only be detected by a human observer, and not by the wavefront sensor measurements or by the graphs of the RMS error. The impact of this could be minimized, however, by reducing the pupil size.

508

DESIGN AND TESTING OF A LIQUID CRYSTAL ADAPTIVE OPTICS PHOROPTER

18.11

SUMMARY

This chapter describes a systematic approach to the design and characterization of an AO system using a spatial light modulator. A series of steps were taken to fully characterize the performance of each subsystem. These steps are necessary to estimate the accuracy and limitations of the system, devise necessary remedies, and ensure repeatability of the measurements. These steps include recording noise statistics, verifying the accuracy of the centroid calculation, performing accurate registration between the wavefront sensor and corrector, understanding the effect of misregistration between these two components, characterizing the nonlinear behavior of the SLM and determining its phase wrapping properties, translating the nonlinear response of the SLM using a lookup table, quantifying the performance of the wavefront sensor, and verifying the closed-loop system operation using internally and externally generated aberrations. After characterizing the AO system, we were able to measure and correct for the aberrations in human eyes, which is needed to perform psychophysical experiments. It is expected that growing interest on LC-based vision correction [13] may lead to better LC devices suitable for the vision community. Acknowledgments This work was performed under the auspices of the U.S. Department of Energy by the University of California, Lawrence Livermore National Laboratory under contract No. W-7405-Eng-48. Dr. Awwal acknowledges Scott Wilks, Brian Baumann, Don Gavel, Jack Werner, Joseph Hardy, Thomas Barnes, Steve Jones, and Dennis de Silva for their help on various stages of this project.

REFERENCES 1. Slataper FJ. Age Norms of Refraction and Vision. Arch. Ophthalmol. 1950; 43: 466–481. 2. Liang J, Williams DR, Miller DT. Supernormal Vision and High-Resolution Retinal Imaging through Adaptive Optics. J. Opt. Soc. Am. A. 1997; 14: 2884– 2892. 3. Roorda A, Williams DR. The Arrangement of the Three Cone Classes in the Living Human Eye. Nature 1999; 397: 520–522. 4. Max CE, Avicola K, Brase JM, et al. Design, Layout, and Early Results of a Feasibility Experiment for Sodium-Layer Laser-Guide-Star Adaptive Optics. J. Opt. Soc. Am. A. 1994; 11: 813–824. 5. Kartz MW, Olivier SS, Avicola K, et al. High Resolution Wavefront Control of High Power Laser Systems. Proc. of 2nd International Workshop on Adaptive Optics for Industry and Medicine. University of Durham, England: 1999, pp. 16–21.

REFERENCES

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6. Hardy JW. Adaptive Optics for Astronomical Selescopes. Oxford: Oxford University Press, 1998. 7. Tyson RK. Principles of Adaptive Optics. 2nd ed. Boston: Academic, 1998. 8. Cagigal MP, Canales VF, Castejón-Mochón JF, et al. Statistical Description of Wavefront Aberration in the Human Eye. Opt. Lett. 2002; 27: 37–39. 9. Poyneer LA, Gavel DT, Brase JM. Fast Wavefront Reconstruction in Large Adaptive Optics Systems with Use of the Fourier Transform. J. Opt. Soc. Am. A. 2002; 19: 2100–2111. 10. Hofer H, Artal P, Singer B, et al. Dynamics of the Eye’s Aberrations. J. Opt. Soc. Am. A. 2001; 18: 497–506. 11. Awwal AAS, Baumann BJ, Gavel DT, et al. Characterization and Operation of a Liquid Crystal Adaptive Optics Phoropter. In: Tyson RK, Lloyd-Hart M, eds. Astronomical Adaptive Optics Systems and Applications. Proceedings of the SPIE. 2003; 5169: 104–122. 12. Wilks SC, Thompson CA, Olivier SS, et al. High-Resolution Adaptive Optics Test Bed for Vision Science. In: Tyson RK, Bonaccini D, Roggemann MC, eds. Adaptive Optics Systems and Technology II. Proceedings of the SPIE. 2002; 4494: 349–355. 13. Prieto PM, Fernández EJ, Manzanera S, Artal P. Adaptive Optics with a Programmable Phase Modulator: Applications in the Human Eye. Opt. Express. 2004; 12: 4059–4071.

APPENDIX A

Optical Society of America’s Standards for Reporting Optical Aberrations* LARRY N. THIBOS,1 RAYMOND A. APPLEGATE,2 JAMES T. SCHWIEGERLING,3 ROBERT WEBB,4 and VSIA STANDARDS TASKFORCE MEMBERS 1 School of Optometry, Indiana University, Bloomington, Indiana 2 Department of Ophthalmology, University of Texas Health Science Center at San Antonio, San Antonio, Texas 3Department of Ophthalmology, University of Arizona, Tucson 4Schepens Research Institute, Boston, Massachusetts

Abstract In response to a perceived need in the vision community, an OSA taskforce was formed at the 1999 topical meeting on vision science and its applications (VSIA-99) and charged with developing consensus recommendations on defi nitions, conventions, and standards for reporting of optical aberrations of human eyes. Progress reports were presented at the 1999 OSA annual meeting and at VSIA-2000 by the chairs of three taskforce

* From LN Thibos, RA Applegate, JT Schwiegerling, et al. Standards for Reporting the Optical Aberrations of Eyes. In: V Lakshminarayanan, ed. OSA Trends in Optics and Photonics, Vision Science and Its Applications, Vol. 35. Washington, D.C.: Optical Society of America, 2000, pp. 232–244. Reprinted in its entirety with permission from the Optical Society of America. Adaptive Optics for Vision Science, Edited by Porter, Queener, Lin, Thorn, and Awwal Copyright © 2006 John Wiley & Sons, Inc.

511

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subcommittees on (1) reference axes, (2) describing functions, and (3) model eyes. The following summary of the committee’s recommendations is available also in portable document format (PDF) on OSA Optics Net at http://www.osa.org/. OCIS codes: (330.0330) Vision and color; (330.5370) Physiological optics

BACKGROUND The recent resurgence of activity in visual optics research and related clinical disciplines (e.g., refractive surgery, ophthalmic lens design, ametropia diagnosis) demands that the vision community establish common metrics, terminology, and other reporting standards for the specification of optical imperfections of eyes. Currently there exists a plethora of methods for analyzing and representing the aberration structure of the eye but no agreement exists within the vision community on a common, universal method for reporting results. In theory, the various methods currently in use by different groups of investigators all describe the same underlying phenomena and therefore it should be possible to reliably convert results from one representational scheme to another. However, the practical implementation of these conversion methods is computationally challenging, is subject to error, and reliable computer software is not widely available. All of these problems suggest the need for operational standards for reporting aberration data and to specify test procedures for evaluating the accuracy of data collection and data analysis methods. Following a call for participation [1], approximately 20 people met at VSIA-99 to discuss the proposal to form a taskforce that would recommend standards for reporting optical aberrations of eyes. The group agreed to form three working parties that would take responsibility for developing consensus recommendations on defi nitions, conventions and standards for the following three topics: (1) reference axes, (2) describing functions, and (3) model eyes. It was decided that the strategy for Phase I of this project would be to concentrate on articulating defi nitions, conventions, and standards for those issues which are not empirical in nature. For example, several schemes for enumerating the Zernike polynomials have been proposed in the literature. Selecting one to be the standard is a matter of choice, not empirical investigation, and therefore was included in the charge to the taskforce. On the other hand, issues such as the maximum number of Zernike orders needed to describe ocular aberrations adequately is an empirical question which was avoided for the present, although the taskforce may choose to formulate recommendations on such issues at a later time. Phase I concluded at the VSIA-2000 meeting.

REFERENCE AXIS SELECTION

513

REFERENCE AXIS SELECTION Summary It is the committee’s recommendation that the ophthalmic community use the line-of-sight as the reference axis for the purposes of calculating and measuring the optical aberrations of the eye. The rationale is that the line-of-sight in the normal eye is the path of the chief ray from the fi xation point to the retinal fovea. Therefore, aberrations measured with respect to this axis will have the pupil center as the origin of a Cartesian reference frame. Secondary lines-of-sight may be similarly constructed for object points in the peripheral visual field. Because the exit pupil is not readily accessible in the living eye whereas the entrance pupil is, the committee recommends that calculations for specifying the optical aberration of the eye be referenced to the plane of the entrance pupil. Background Optical aberration measurements of the eye from various laboratories or within the same laboratory are not comparable unless they are calculated with respect to the same reference axis and expressed in the same manner. This requirement is complicated by the fact that, unlike a camera, the eye is a decentered optical system with non-rotationally symmetric components (Fig. 1). The principle elements of the eye’s optical system are the cornea, pupil, and the crystalline lens. Each can be decentered and tilted with respect to other components, thus rendering an optical system that is typically dominated by coma at the foveola.

FIGURE 1 The cornea, pupil, and crystalline lens are decentered and tilted with respect to each other, rendering the eye a decentered optical system that is different between individuals and eyes within the same individual.

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APPENDIX A OPTICAL SOCIETY OF AMERICA’S STANDARDS

The optics discipline has a long tradition of specifying the aberration of optical systems with respect to the center of the exit pupil. In a centered optical system (e.g., a camera, or telescope) using the center of the exit pupil as a reference for measurement of on-axis aberration is the same as measuring the optical aberrations with respect to the chief ray from an axial object point. However, because the exit pupil is not readily accessible in the living eye, it is more practical to reference aberrations to the entrance pupil. This is the natural choice for objective aberrometers which analyze light reflected from the eye. Like a camera, the eye is an imaging device designed to form an in-focus inverted image on a screen. In the case of the eye, the imaging screen is the retina. However, unlike fi lm, the “grain” of the retina is not uniform over its extent. Instead, the grain is fi nest at the foveola and falls off quickly as the distance from the foveola increases. Consequently, when viewing fi ne detail, we rotate our eye such that the object of regard falls on the foveola (Fig. 2). Thus, aberrations at the foveola have the greatest impact on an individual’s ability to see fi ne details.

FIGURE 2 An anatomical view of the macular region as viewed from the front and in cross section (below). a: foveola, b: fovea, c: parafoveal area, d: perifoveal area. From Histology of the Human Eye by Hogan. Alvarado Weddell, W.B. Sauders Company publishers, 1971, page 491.

METHODS FOR ALIGNING THE EYE DURING MEASUREMENT

FIGURE 3 sight.

515

Left panel illustrates the visual axis and right panel illustrates the line of

Two traditional axes of the eye are centered on the foveola, the visual axis and the line-of-sight, but only the latter passes through the pupil center. In object space, the visual axis is typically defi ned as the line connecting the fi xation object point to the eye’s fi rst nodal point. In image space, the visual axis is the parallel line connecting the second nodal point to the center of the foveola (Fig. 3, left). In contrast, the line-of-sight is defi ned as the (broken) line passing through the center of the eye’s entrance and exit pupils connecting the object of regard to the foveola (Fig. 3, right). The line-of-sight is equivalent to the path of the foveal chief ray and therefore is the axis which conforms to optical standards. The visual axis and the line of sight are not the same and in some eyes the difference can have a large impact on retinal image quality [2]. For a review of the axes of the eye see [3]. (To avoid confusion, we note that Bennett and Rabbetts [4] redefi ne the visual axis to match the traditional defi nition of the line of sight. The Bennett and Rabbetts defi nition is counter to the majority of the literature and is not used here.) When measuring the optical properties of the eye for objects which fall on the peripheral retina outside the central fovea, a secondary line-of-sight may be constructed as the broken line from object point to center of the entrance pupil and from the center of the exit pupil to the retinal location of the image. This axis represents the path of the chief ray from the object of interest and therefore is the appropriate reference for describing aberrations of the peripheral visual field.

METHODS FOR ALIGNING THE EYE DURING MEASUREMENT Summary The committee recommends that instruments designed to measure the optical properties of the eye and its aberrations be aligned co-axially with the eye’s line-of-sight.

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APPENDIX A OPTICAL SOCIETY OF AMERICA’S STANDARDS

Background There are numerous ways to align the line of sight to the optical axis of the measuring instrument. Here we present simple examples of an objective method and a subjective method to achieve proper alignment. Objective Method In the objective alignment method schematically diagramed in Fig. 4, the experimenter aligns the subject’s eye (which is fi xating a small distant target on the optical axis of the measurement system) to the measurement system. Alignment is achieved by centering the subject’s pupil (by adjusting a bite bar) on an alignment ring (e.g., an adjustable diameter circle) which is coaxial with the optical axis of the measurement system. This strategy forces the optical axis of the measurement device to pass through the center of the entrance pupil. Since the fi xation target is on the optical axis of the measurement device, once the entrance pupil is centered with respect to the alignment ring, the line-of-sight is co-axial with the optical axis of the measurement system. Subjective Method

Alignment Ring

Measurement System

In the subjective alignment method schematically diagramed in Figure 5, the subject adjusts the position of their own pupil (using a bite bar) until two alignment fi xation points at different optical distances along and co-axial to the optical axis of the measurement device are superimposed (similar to

Alignment Camera BS BS FP FIGURE 4 Schematic of a generic objective alignment system designed to place the line of sight on the optical axis of the measurement system. BS: beam splitter, FP: on axis fi xation point.

517

Measurement System

DESCRIPTION OF ZERNIKE POLYNOMIALS

FP

FP

BS

BS

FIGURE 5 Schematic of a generic subjective alignment system designed to place the line of sight on the optical axis of the measurement system. BS: beam splitter, FP: fi xation point source.

aligning the sights on rifle to a target). Note that one or both of the alignment targets will be defocused on the retina. Thus the subject’s task is to align the centers of the blur circles. Assuming the chief ray defi nes the centers of the blur circles for each fi xation point, this strategy forces the line of sight to be co-axial with the optical axis of the measurement system. In a system with significant amounts of asymmetric aberration (e.g., coma), the chief ray may not defi ne the center of the blur circle. In practice, it can be useful to use the subjective strategy for preliminary alignment and the objective method for fi nal alignment. Conversion Between Reference Axes If optical aberration measurements are made with respect to some other reference axis, the data must be converted to the standard reference axis (see the tools developed by Susana Marcos at our temporary web site: //color.eri. harvard/standardization). However, since such conversions involve measurement and/or estimation errors for two reference axes (the alignment error of the measurement and the error in estimating the new reference axis), it is preferable to have the measurement axis be the same as the line-of-sight.

DESCRIPTION OF ZERNIKE POLYNOMIALS The Zernike polynomials are a set of functions that are orthogonal over the unit circle. They are useful for describing the shape of an aberrated wavefront

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in the pupil of an optical system. Several different normalization and numbering schemes for these polynomials are in common use. Below we describe the different schemes and make recommendations towards developing a standard for presenting Zernike data as it relates to aberration theory of the eye. Double Indexing Scheme The Zernike polynomials are usually defi ned in polar coordinates (r, q), where r is the radial coordinate ranging from 0 to 1 and q is the azimuthal component ranging from 0 to 2p. Each of the Zernike polynomials consists of three components: a normalization factor, a radial-dependent component and an azimuthal-dependent component. The radial component is a polynomial, whereas the azimuthal component is sinusoidal. A double indexing scheme is useful for unambiguously describing these functions, with the index n describing the highest power (order) of the radial polynomial and the index m describing the azimuthal frequency of the sinusoidal component. By this scheme the Zernike polynomials are defi ned as Znm ( ρ, θ ) =

{

N nm Rnm ( ρ ) cos mθ ; for m ≥ 0 − N mm Rnm ( ρ ) sin mθ ; for m < 0

}

(1)

where Nnm is the normalization factor described in more detail below and Rn|m| (r) is given by Rnm ( ρ ) =

( n− m ) 2

∑ s= 0

s

( −1 ) ( n − s ) ! ρ n− 2 s s ![ 0 . 5 ( n + m − s ) ]![ 0 . 5 ( n − m − s ) ]!

(2)

This defi nition uniquely describes the Zernike polynomials except for the normalization constant. The normalization is given by N nm =

2 ( n + 1) 1 + δ m0

(3)

where d m0 is the Kronecker delta function (i.e., d m0 = 1 for m = 0, and d m0 = 0 for m ≠ 0). Note that the value of n is a positive integer or zero. For a given n, m can only take on values −n, −n + 2, −n + 4, . . . n. When describing individual Zernike terms (Table 2), the two index scheme should always be used. Below are some examples. Good: “The values of Z3 −1(r, q) and Z24 (r, q) are 0.041 and −0.121, respectively.” “Comparing the astigmatism terms, Z2−2 (r, q) and Z22 (r, q) . . .” Bad “The values of Z7(r, q) and Z12 (r, q) are 0.041 and −0.121, respectively.” “Comparing the astigmatism terms, Z5 (r, q) and Z 6 (r, q) . . .”

DESCRIPTION OF ZERNIKE POLYNOMIALS

519

Single Indexing Scheme Occasionally, a single indexing scheme is useful for describing Zernike expansion coefficients. Since the polynomials actually depend on two parameters, n and m, ordering of a single indexing scheme is arbitrary. To avoid confusion, a standard single indexing scheme should be used, and this scheme should only be used for bar plots of expansion coefficients (Fig. 6). To obtain the single index, j, it is convenient to lay out the polynomials in a pyramid with row number n and column number m as shown in Table 1. The single index, j, starts at the top of the pyramid and steps down from left to right. To convert between j and the values of n and m, the following relationships can be used: j=

n(n + 2) + m 2

( mode number )

(4)

 −3 + 9 + 8 j  ( radial order ) n = roundup   2   ( angular frequency ) m = 2 j − n(n + 2)

(5) (6)

Coordinate System

0.02 0.01

36

32

28

24

20

16

8

12

−0.01

4

0 0

Expansion Coefficient Value

Typically, a right-handed coordinate system is used in scientific applications as shown in Fig. 7. For the eye, the coordinate origin is at the center of the eye’s entrance pupil, the +x axis is horizontal pointing to the right, the +y axis is vertical pointing up, and the +z Cartesian axis points out of the eye and coincides with the foveal line-of-sight in object space, as defi ned by a chief ray emitted by a fi xation spot. Also shown are conventional defi nitions of the polar coordinates r = x 2 + y2 and q = tan−1(y/x). This defi nition gives x = r cos q and y = r sin q. We note that Malacara [5] uses a polar coordinate system

−0.02

Single Index, j FIGURE 6 Example of a bar plot using the single index scheme for Zernike coefficients.

520

10

−4

16

6

−3

11

3

−2

17

7

1

−1

12

4

j=0

0

18

8

2

+1

13

5

+2

Row number is polynomial order n, column number is sinusoidal frequency m, table entry is the single-index j.

15

0 1 2 3 4 5

a

−5

Zernike Pyramida

n/m

TABLE 1

19

9

+3

14

+4

20

+5

521

DESCRIPTION OF ZERNIKE POLYNOMIALS Clinician’s View of Patient

r

y

x

y

r

y

r

q q z

q x

x Right Eye = OD

Left Eye = OS

FIGURE 7 Conventional right-handed coordinate system for the eye in Cartesian and polar forms.

in which x = r sin q and y = r cos q. In other words, q is measured clockwise from the +y axis (Figure 1b), instead of counterclockwise from the +x axis (Figure 1a). Malacara’s defi nition stems from early (pre-computer) aberration theory and is not recommended. In ophthalmic optics, angle q is called the “meridian” and the same coordinate system applies to both eyes. Because of the inaccessibility of the eye’s image space, the aberration function of eyes are usually defi ned and measured in object space. For example, objective measures of ocular aberrations use light reflected out of the eye from a point source on the retina. Light reflected out of an aberration-free eye will form a plane-wave propagating in the positive z-direction and therefore the (x, y) plane serves as a natural reference surface. In this case the wavefront aberration function W(x, y) equals the z-coordinate of the reflected wavefront and may be interpreted as the shape of the reflected wavefront. By these conventions, W > 0 means the wavefront is phase-advanced relative to the chief ray. An example would be the wavefront reflected from a myopic eye, converging to the eye’s far-point. A closely related quantity is the optical path-length difference (OPD) between a ray passing through the pupil at (x, y) and the chief ray point passing through the origin. In the case of a myopic eye, the path length is shorter for marginal rays than for the chief ray, so OPD < 0. Thus, by the recommended sign conventions, OPD(x, y) = −W(x, y). Bilateral symmetry in the aberration structure of eyes would make W(x, y) for the left eye the same as W(−x, y) for the right eye. If W is expressed as a Zernike series, then bilateral symmetry would cause the Zernike coefficients for the two eyes to be of opposite sign for all those modes with odd symmetry about the y-axis (e.g., mode Z2−2). Thus, to facilitate direct comparison of the two eyes, a vector R of Zernike coefficients for the right eye can be converted to a symmetric vector L for the left eye by the linear transformation L = M * R, where M is a diagonal matrix with elements +1 (no sign change) or −1 (with sign change). For example, matrix M for Zernike vectors representing the fi rst 4 orders (15 modes) would have the diagonal elements [+1, +1, −1, −1, +1, +1, +1, +1, −1, −1, −1, −1, +1, +1, +1].

522

APPENDIX A OPTICAL SOCIETY OF AMERICA’S STANDARDS

TABLE 2 Listing of Zernike Polynomials up to 7th order (36 terms) j = Index

n = Order

m = Frequency

0

0

0

1

1

−1

2r sin q

2

1

1

2r cos q

3

2

−2

4

2

0

3 ( 2 ρ 2 − 1)

5

2

2

6 ρ 2 cos 2θ

6

3

−3

8 ρ 3 sin 3θ

7

3

−1

8 ( 3ρ 3 − 2 ρ ) sin θ

8

3

1

8 ( 3ρ 3 − 2 ρ ) cos θ

9

3

3

8 ρ 3 cos 3θ

10

4

−4

10 ρ 4 sin 4θ

11

4

−2

10 ( 4 ρ 4 − 3ρ 2 ) sin 2θ

12

4

0

13

4

2

10 ( 4 ρ 4 − 3ρ 2 ) cos 2θ

14

4

4

10 ρ 4 cos 4θ

15

5

−5

12 ρ 5 sin 5θ

16

5

−3

12 ( 5 ρ 5 − 4 ρ 3 ) sin 3θ

17

5

−1

18

5

1

19

5

3

12 ( 5 ρ 5 − 4 ρ 3 ) cos 3θ

20

5

5

12 ρ 5 cos 5θ

21

6

−6

14 ρ 6 sin 6θ

22

6

−4

14 ( 6 ρ 6 − 5 ρ 4 ) sin 4θ

23

6

−2

24

6

0

25

6

2

26

6

4

14 ( 6 ρ 6 − 5 ρ 4 ) cos 4θ

27

6

6

14 ρ 6 cos 6θ

28

7

−7

4r7 sin 7q

29

7

−5

4 (7r7 − 6r5) sin 5q

30

7

−3

4 (21r7 − 30r5 + 10r3) sin 3q

31

7

−1

4 (35r7 − 60r5 + 30r3 − 4r) sin q

32

7

1

4 (35r7 − 60r5 + 30r3 − 4r) cos q

33

7

3

4 (21r7 − 30r5 + 10r3) cos 3q

34

7

5

4 (7r7 − 6r5) cos 5q

35

7

7

4r7 cos 7q

Znm (r,q) 1

6 ρ 2 sin 2θ

5 ( 6 ρ 4 − 6 ρ 2 + 1)

12 ( 10 ρ 5 − 12 ρ 3 + 3ρ ) sin θ

12 ( 10 ρ 5 − 12 ρ 3 + 3ρ ) cos θ

14 ( 15 ρ 6 − 20 ρ 4 + 6 ρ 2 ) sin 2θ 7 ( 20 ρ 6 − 30 ρ 4 + 12 ρ 2 − 1 )

14 ( 15 ρ 6 − 20 ρ 4 + 6 ρ 2 ) cos 2θ

STANDARD ABERRATOR FOR CALIBRATION

523

STANDARD ABERRATOR FOR CALIBRATION The original goal was to design a device that could be passed around or massproduced to calibrate aberrometers at various laboratories. We first thought of this as an aberrated model eye, but that later seemed too elaborate. One problem is that the subjective aberrometers needed a sensory retina in their model eye, while the objective ones needed a reflective retina of perhaps known reflectivity. We decided instead to design an aberrator that could be used with any current or future aberrometers, with whatever was the appropriate model eye. The fi rst effort was with a pair of lenses that nearly cancelled spherical power, but when displaced sideways would give a known aberration. That scheme worked, but was very sensitive to tilt, and required careful control of displacement. The second design was a trefoil phase plate (OPD = Z33 = kr 3 sin 3q) loaned by Ed Dowski of CDM Optics, Inc. This 3rd order aberration is similar to coma, but with three lobes instead of one, hence the common name “trefoil”. Simulation of the aberration function for this plate in ZEMAX® is shown in Figs. 8, 9. Figure 8 is a graph of the Zernike coefficients showing a small amount of defocus and 3rd order spherical aberration, but primarily C 33. Figure 9 shows the wavefront, only half a micron (one wave) peak to peak, but that value depends on k, above. We mounted the actual plate and found that it had even more useful qualities: As the phase plate is translated across the pupil, it adds some C 22 , horizontal astigmatism. When the plate is perfectly centered, that coefficient is zero. Further, the slope of C 22 (∆x) measures the actual pupil. Z 33 ( x − x0 ) = κ ( r − x0 ) sin 3θ = κ ( 3 xy2 − x 3 )

(7)

Zernike Coefficient Value (µm)

0.4 0.3 0.2 0.1 0 −0.1 −0.2 2

4

6

8

10 12

14 16 18

20

22 24

26

28 30

32 34 36

Zernike Coefficient Index (Zmax convention)

FIGURE 8 Zernike coefficients of trefoil phase plate from ZEMAX® model (note different numbering convention from that recommended above for eyes).

524

APPENDIX A OPTICAL SOCIETY OF AMERICA’S STANDARDS

FIGURE 9

Wavefront map for trefoil phase plate from the ZEMAX® model.

so

∂Z 33 ( x − x0 ) = 3κ ( y2 − x 2 ) = 3Z 22 ∂x

(8)

∂Z33 ( x − x0 ) = −6κ xy = −3Z22 ∂y

(9)

and similarly

m m This means that ∆Z33 = 3Z22∆x and then, since W = ∑ ∑ C n Zn , we get a new 2 term proportional to ∆x. Plotting the coefficient C 2 against ∆x, we need to normalize to the pupil size. That could be useful as a check on whether the aberrator is really at the pupil, or whether some smoothing has changed the real pupil size, as measured. Figures 10–13 confi rm this behavior and the expected variation with rotation (3q). Although the phase plate aberrator works independently of position in a collimated beam, some aberrometers may want to use a converging or diverging beam. Then it should be placed in a pupil conjugate plane. We have not yet built the mount for the phase plate, and would appreciate suggestions for that. Probably we need a simple barrel mount that fits into standard lens

STANDARD ABERRATOR FOR CALIBRATION

525

5 4 3

2

2

1

1 µm

Pupil Position (mm)

3

0

0

−1

−1 −2

−2

−3 −4

−3 −3

−2

−1

0

1

2

3

−5

Pupil Position (mm)

FIGURE 10

Wavefront map from the aberrator, using the SRR aberrometer.

5 4

2

3

1

2 1 µm

Pupil Position (mm)

3

0

0 −1 −2

−1 −2

−3 −4

−3 −3

−2

−1

−5 0

1

2

3

Pupil Position (mm)

FIGURE 11

The phase plate of Figure 10 has been moved horizontally 4 mm.

holders—say 30 mm outside diameter. We expect to use a standard pupil, but the phase plate(s) should have 10 mm clear aperture before restriction. The workshop seemed to feel that a standard pupil should be chosen. Should that be 7.5 mm? We have tested the Z33 aberrator, but it may be a good idea to have a few others. We borrowed this one, and it is somewhat fragile. Bill Plummer of Polaroid thinks he could generate this and other plates in plastic for “a few thousand dollars” for each design. Please send suggestions as to whether other designs are advisable ([email protected]), and as to whether we will want to stack them or use them independently. That has some implications for the mount design, but not severe ones. We suggest two Z33 plates like this one, and perhaps a Z 06, fi fth order spherical.

526

APPENDIX A OPTICAL SOCIETY OF AMERICA’S STANDARDS

Zernike Coefficient (µm)

0.6 0.4

c33

Trefoil

c02

Defocus

0.2 0 −0.2

Trefoil

c-3 3

c-2 2 Oblique Astigmatism

−0.4 −0.6

c22

Horizontal/Vertical Astigmatism 0

0.1

0.2

0.3

0.4

Arbitrary Horizontal Position (pupil radius units) FIGURE 12 for C 33.

Zerike coefficients are stable against horizontal displacement, except

Zernike Coefficient (µm)

0.4 Modulus

0.3

c33

0.2 0.1 0 −0.1 −0.2

c-3 3

−0.3 −0.4

0

10

20

30

40

50

60

Phase Plate Rotation (deg) FIGURE 13 Zernike coefficients C 33 and C 3−3 as a function of rotation of the phase plate about the optic axis.

At this time, then, our intent is to have one or more standard aberrators that can be inserted into any aberrometer. When centered, and with a standard pupil, all aberrometers should report the same Zernike coefficients. We do not intend to include positioners in the mount, assuming that will be different for each aberrometer.

PLANS FOR PHASE II

527

Another parameter of the design is the value of k. That comes from the actual physical thickness and the index of refraction. Suggestions are welcome here, but we assume we want coefficients that are robust compared to a diopter or so of defocus. The index will be whatever it will be. We will report it, but again any chromaticity will depend on how it’s used. We suggest that we report the expected coefficients at a few standard wavelengths and leave interpolation to users.

PLANS FOR PHASE II Reference Axes Subcommittee • develop a shareware library of software tools needed to convert data from one ocular reference axis to another (e.g., convert a wavefront aberration for the corneal surface measured by topography along the instrument’s optical axis into a wavefront aberration specified in the eye’s exit pupil plane along the eye’s fi xation axis.) • generate test datasets for evaluating software tools Describing Functions Subcommittee • develop a shareware library of software tools for generating, manipulating, evaluating, etc. the recommended describing functions for wavefront aberrations and pupil apodizing functions. • develop additional software tools for converting results between describing functions (e.g., converting Taylor polynomials to Zernike polynomials, or converting single-index Zernikes to double-index Zernikes, etc.). • generate test datasets for evaluating software tools. Model Eyes Subcommittee • build a physical model eye that can be used to calibrate experimental apparatus for measuring the aberrations of eyes. • circulate the physical model to all interested parties for evaluation, with results to be presented for discussion at a future VSIA meeting. Acknowledgements The authors wish to thank the numerous committee members who contributed to this project.

528

APPENDIX A OPTICAL SOCIETY OF AMERICA’S STANDARDS

REFERENCES 1. Thibos LN, Applegate RA, Howland HC, Williams DR, Artal P, Navarro R, Campbell MC, Greivenkamp JE, Schwiegerling JT, Burns SA, Atchison DA, Smith G, Sarver EJ. “A VSIA-sponsored effort to develop methods and standards for the comparison of the wavefront aberration structure of the eye between devices and laboratories,” in Vision Science and Its Applications (Optical Society of America, Washington, D.C., 1999), pp. 236–239. 2. Thibos LN, Bradley A, Still DL, Zhang X, Howarth PA. “Theory and measurement of ocular chromatic aberration,” Vision Research 30: 33–49 (1990). 3. Bradley A, Thibos LN. (Presentation 5) at http://www.opt.indiana.edu/lthibos/ ABLNTOSA95. 4. Bennett AG, Rabbetts RB. Clinical Visual Optics, 2nd ed. (Butterworth, 1989). 5. Malacara D. Optical Shop Testing, 2nd ed. (John Wiley & Sons, Inc., New York, 1992).

Glossary Term

Defi nition

Chapter

Aberration

The optical deviations of a wavefront from a reference plane or spherical wavefront that degrade image quality.

3

Ablation optical zone

The diameter of a laser refractive surgical ablation on the central cornea that is designed to correct the eye’s refractive error and/or higher order aberrations. This is also called the “optical zone.”

12

Acousto-optic modulator

A device that varies the amplitude, frequency, or phase of the light (such as a laser) propagating through it.

16

Adaptive optics

An optical system that adapts to compensate for optical artifacts (such as aberrations) introduced by the medium between the object and the image.

1

AFC

See N-alternative-forced-choice.

14

Ametropia

The degree of defocus measured in an eye (either myopia or hyperopia).

11

Adaptive Optics for Vision Science, Edited by Porter, Queener, Lin, Thorn, and Awwal Copyright © 2006 John Wiley & Sons, Inc.

529

530

GLOSSARY

Term

Defi nition

Chapter

AO

See adaptive optics.

1

AO loop

The repeating cycle of wavefront measurement and correction in an adaptive optics system.

6

AOM

See acousto-optic modulator.

16

Aperture stop

The physical constraint that limits the size of the light bundle from an on-axis field point that passes through an optical system.

7

Aphakic

The physiological eye that has had its natural lens removed; literally “without lens.”

11

A-scan

A single axial profi le (along the z-axis) of optical reflectivity as measured by optical coherence tomography (OCT).

17

Axial chromatic aberration

See longitudinal chromatic aberration.

13

Axial point spread function

Axial intensity distribution in the threedimensional image of a point object.

17

Bandwidth error

Wavefront error due to the temporal lag between the occurrence of wave aberrations and the time at which the adaptive optics system corrects them.

8

Bimorph mirror

A modal device that consists of a piezoelectric material sandwiched between a continuous top electrode and a bottom, patterned electrode array. A mirrored layer is added to the top continuous electrode. Application of a voltage across the top and bottom electrodes changes the underlying surface area of the two dissimilar layers and results in a bending of the entire mirror.

4

Boresighting

The process of co-aligning the fields of view (FOV) of various optical subsystems.

7

531

GLOSSARY

Term

Defi nition

Chapter

B-scan

A two-dimensional profi le (x–z plane) of optical reflectivity that is composed of a sequence of adjacent A-scans as measured by optical coherence tomography (OCT).

17

Calibration error

Wavefront error in the absence of any aberrations external to the adaptive optics system.

8

Cathode ray tube

A computer-controlled display that produces light when an electron beam excites a phosphor coating on the display screen. These displays come in monochrome and color varieties.

14

CDM

See chromatic difference of magnification.

13

Center-of-mass

If f(x, y) is the density function, then the centroid or center-of-mass (xc, yc), of a thin plate is given by

18

Mxc =

∫∫ xf ( x, y ) dx dy R

Myc =

∫∫ yf ( x, y ) dx dy R

where M is the mass, M =

∫∫ f ( x, y ) dx dy, R

and R is the region of interest. Centroid

The centroid of the wavefront sensor spot image is the center-of-mass, generally computed over a rectangular area of interest called a search box. See also center-of-mass.

6

Chief ray

The ray for a given field point that passes through the center of the aperture stop. There is a distinct chief ray for each field point.

7

Chromatic difference of magnification

The fractional change in retinal image size due to variation in wavelength.

13

532

GLOSSARY

Term

Defi nition

Chapter

Closed-loop control

In closed-loop (or feedback) control systems, the wavefront corrector precedes the wavefront sensor in the system’s optical path. The wavefront corrector compensates for the aberrated wavefront fi rst. The wavefront sensor then measures the residual wave aberration (thereby receiving feedback on the accuracy of the correction), the required corrections are computed, and the wavefront corrector is updated. This process is repeated iteratively until the desired correction or wavefront profi le is obtained.

5

Coherence length, lc

Average path length over which the phase of a light source remains constant; quantity that addresses the spectral purity of a source and is defi ned as:

15

lc ≅

λ2 ∆λ

where l is the center wavelength of the source and ∆l is its bandwidth. Colorimeter

A device that measures the radiance of a light source weighted by three chromatic fi lters based on a set of color-matching functions derived from psychophysical experiments, thereby permitting specification of the light source in terms of chromaticity and luminance.

14

Contrast sensitivity function

Sensitivity of an observer to the contrast of a Gabor pattern that is varying in luminance as a function of spatial frequency.

14

Conventional ablation

Refractive surgery where only sphere and cylinder (or defocus and astigmatism) are corrected.

12

Conventional or classic refractive surgery

See conventional ablation.

12

533

GLOSSARY

Term

Defi nition

Chapter

Corneal aberration

Wave aberration typically corresponding to the anterior surface of the cornea.

2

Coupling coefficient

The amount of dependence between the actuators of a deformable mirror. A coupling coefficient of 15% implies pushing one actuator with unit magnitude causes displacement of the mirror surface at the location of the adjacent actuators to be 0.15.

4

cpd

Cycles-per-degree (of visual angle), typically used to measure spatial frequency in a visual stimulus.

14

CRT

See cathode ray tube.

14

CRT projectors

A computer-controlled display that uses bright CRT tubes combined with lenses to project images onto a display screen.

14

CSF

See contrast sensitivity function.

14

Customized ablation

Refractive surgery where sphere, cylinder, and higher order aberrations are corrected using wavefront measurements that are unique to the eye being treated. Also called “personalized” or “wavefront-guided” refractive surgery.

12

Customized refractive surgery

See customized ablation.

12

Cycloplegic refraction

A measurement of sphere, cylinder, and axis that uses a cycloplegic agent that inhibits accommodation.

12

Cylinder

The astigmatic component of a spectacle prescription.

11

Deformable mirror

An adaptive optical element that creates a uniform wavefront by applying an optical distortion to compensate for an incident distorted wavefront.

1

534

GLOSSARY

Term

Defi nition

Chapter

Degree of polarization

Fraction of light that remains polarized after passing through an optical system. It is related to the amount of scatter.

2

Depth of focus

For the eye, the range of object distances for which image quality is not significantly degraded, which depends on a variety of optical and neural factors.

13

Detection threshold

The stimulus strength necessary to elicit a criterion level of performance on a task in which an observer is asked to state if, when, or where a stimulus was presented.

14

DFT

See discrete Fourier transform.

8

Digital light projector

A computer-controlled display that uses a large array of digital micromirror devices (DMDs) to control the intensity of light reaching the screen.

14

Digital micromirror device

A chip-based device that consists of thousands (or millions) of tiny mirrors, whose positions can be controlled by electrical signals.

14

Digital numbers

An integer increment within the available bit range of a digital device.

18

Direct slope control algorithm

The direct slope control algorithm, also called the direct gradient control algorithm, treats the wavefront distortion using a zonal approach in which the wavefront sensor and wavefront corrector used in an adaptive optics system work on a slope basis.

5

Discrete actuator deformable mirror

A continuous mirror surface whose profi le is controlled by an underlying array of actuators. Pushing one actuator produces a localized (also termed zonal) deflection of the mirror surface, termed the influence function.

4

Discrete Fourier transform

A mathematical transform for discretely computing a Fourier transform (which expresses a signal as an integral of sinusoidal basis functions).

8

535

GLOSSARY

Term

Defi nition

Chapter

Discrimination threshold

The minimum difference along a particular stimulus dimension that is necessary for the observer to correctly differentiate two or more stimuli with a given probability. Sometimes called a just noticeable difference (jnd).

14

DLP

See digital light projector.

14

DM

See deformable mirror.

1

DMD

See digital micromirror device.

14

DN

See digital numbers.

18

DOP

See degree of polarization.

2

Dynamic range

In the context of wavefront sensing, the maximum wavefront slope that can be measured reliably. For a Shack–Hartmann wavefront sensor, it is a function of spot size, subaperture size, and the focal length of the lenslets.

3, 18

Emmetropia

A condition of the eye where light passing through the optical surfaces of the eye comes to focus at the retinal plane.

11

Equivalent quadratic

Given a wave aberration map, the quadratic surface that best represents the map.

13

Farsightedness

See hyperopia

11

Fast Fourier transform

An efficient algorithm for computing the discrete Fourier transform.

8

FFT

See fast Fourier transform.

8

First-order optics

The optical theory related to ideal imaging, which applies to optical systems with very small fields and apertures. In Snell’s law, sin q is approximated as q. The discrepancy between the results of fi rst-order optics and real ray tracing represents optical aberrations. (Also called Gaussian optics or paraxial optics.)

7

536

GLOSSARY

Term

Defi nition

Chapter

Fitting error

Wavefront error due to the deformable mirror’s inability to correct spatial frequencies larger than the inverse of the interactuator spacing.

8

Fovea

The central, approximately 600-mm area of the human retina that contains the densest photoreceptor packing needed for optimal spatial resolution and color vision.

9

Full width at half maximum

Given a function that has a central peak in the y dimension, this dimension is the distance between the x intercept points on either side of the peak where the function’s y value is half that of the peak.

8

Fundus

The concave portion of an anatomical structure, which for the eye includes the retina and choroid.

9

FWHM

See full width at half maximum.

8

Gabor pattern

A one-dimensional sinusoidal luminance grating weighted by a two-dimensional Gaussian function.

14

Gain, K

The fraction of the measured aberrations that the wavefront corrector attempts to correct in a single iteration. A gain of 1 implies that the wavefront corrector attempts to correct all of the aberrations just measured by the wavefront sensor, while a gain of 0.3 indicates that the wavefront corrector attempts to correct 30% of the aberrations just measured by the wavefront sensor.

15

Gamut

The range of displayable chromaticities for a light source. The gamut depends on the chromaticities and maximum luminance outputs of the color channels used in the display.

14

537

GLOSSARY

Term

Defi nition

Chapter

Horizontal synchronization pulse

The change in voltage level of a video signal that triggers the end of one line and the start of a new line. On a video display or frame grabber, each line in an image starts at the end of the hsync pulse and ends with the start of the next hsync pulse.

16

Hsync

See horizontal synchronization pulse.

16

Hyperfocal point The far end of the eye’s depth-of-focus interval. 13 Hyperopia

A condition of the eye where light passing through the optical surfaces of the eye comes to focus behind the retinal plane. Also called “farsightedness.”

11

Identification threshold

The stimulus strength necessary to elicit a criterion level of performance on a task in which an observer is asked to state which of a set of possible stimuli was presented.

14

Influence function

The surface deformation produced by one actuator when a unit voltage is applied to this actuator on the deformable mirror.

4, 5

Inner segment

The portion of the photoreceptor that contains its cell body and eventually terminates on the next neural stage of bipolar cells or horizontal cells.

9

Internal aberration

Wave aberration typically corresponding to the posterior surface of the cornea and the crystalline lens.

2

Intraocular lens

A lens implanted in the eye.

2

Intraocular scatter

Scattered light produced by the ocular media that degrades the quality of retinal images or image formation beyond the effect of aberrations.

2

IOL

See intraocular lens.

2

538

GLOSSARY

Term

Defi nition

Chapter

Keratoconus

A disease of the eye characterized by a steepening and decentering of the central corneal surface relative to the line of sight, in association with a thinning of the central corneal tissue. The steepening and decentration of the cornea, the single most powerful refracting surface of the eye, produces significant increases in lower and higher order aberrations of the eye.

11

LASEK

See laser-assisted epithelial keratomileusis.

12

Laser-assisted epithelial keratomileusis

A variant of photorefractive keratectomy (PRK) in which a dilute alcohol solution is applied to the front cells of the cornea and the corneal epithelial cells are gently peeled back in a continuous layer to expose the cornea. The excimer laser is then applied to the cornea with the epithelial layer retracted back away from the treatment area. After the ablation, the epithelial cells are then gently placed back into their original position over the cornea and a bandage soft lens is applied. The theoretical advantages of LASEK over PRK are the preservation of the epithelium, as well as quicker, more comfortable recovery, although there has not been convincing evidence of this to date.

12

Laser in situ keratomileusis

A technique to reshape the cornea that creates a corneal flap with a microkeratome or femtosecond laser. The flap is lifted and the excimer laser is used to reshape the corneal surface. For myopia, the laser treatment removes more tissue centrally to flatten the central cornea. For hyperopia, the laser removes more tissue in the periphery to steepen the central cornea.

12

LASIK

See laser in situ keratomileusis.

12

LCA

See longitudinal chromatic aberration.

2

539

GLOSSARY

Term

Defi nition

Chapter

LCD

See liquid crystal display.

14

LCD projectors

A computer-controlled display that uses the same technology as liquid crystal displays (LCD) but employs more powerful backlights combined with a lens to project the image onto a display screen.

14

LC-SLM

See liquid crystal spatial light modulator.

4, 18

Lenslet

A single small lens in an array of small lenses used to sample the wavefront in a Shack–Hartmann wavefront sensor.

6

Limbal junction

The region of the eye’s structure where the opaque sclera meets the transparent cornea. The limbal region is translucent and defi nes the change in curvature from the flatter radius sclera to the steeper radius cornea.

11

Liquid crystal display

A computer-controlled display that produces a visual stimulus by modulating a polarized light source with individually addressable liquid crystal elements.

14

Liquid crystal spatial light modulator

A liquid crystal spatial light modulator (LC-SLM) uses the electro-optic effects of liquid crystals to achieve modulation. These devices rely on the rotation of a liquid crystal molecule to induce localized refractive index changes, which in turn causes phase changes in the incoming wavefront. Both reflective and transmissive devices are available.

4, 18

logMAR

The logarithm of the minimum angle of resolution, in minutes of arc.

14

logMAR visual acuity

A measure of visual acuity using a chart specifically designed such that each line of the chart decreases in size according to a logarithmic progression. LogMAR charts are considered to be ideal for standardized clinical testing of visual acuity.

11

540

GLOSSARY

Term

Defi nition

Chapter

Longitudinal chromatic aberration

The variation of axial power of an optical system with wavelength.

2, 13

Luminance

A photometric measure of the effectiveness of a light source for human vision, based upon the radiance fi ltered by a human spectral efficiency function. Luminance is often expressed in candelas per square meter (cd/m 2).

14

Manifest refraction

A measurement of sphere, cylinder, and axis that is done without using any pharmacologic dilating or cycloplegic drugs.

12

Manifest refractive spherical equivalent

The measurement of refractive error that combines both sphere and cylinder into a single value. It is derived by taking one half of the cylinder value and adding it to the sphere value.

12

Measurement error

Wavefront error due to the noise in the wavefront slope measurements.

8

Measurement sensitivity

In the context of wavefront sensing, the minimum wavefront slope that can be measured reliably.

3

Membrane mirror

A type of mirror that consists of an edgeclamped, flexible, reflective membrane (analogous to a drumskin) sandwiched between a transparent top electrode and an underlying array of patterned electrodes. Application of a voltage causes deformation of the entire membrane.

4

MEMS

See microelectromechanical systems.

4

Method of adjustment

A procedure for estimating thresholds in which an observer adjusts a stimulus until a criterion perceptual experience (e.g., just barely visible) is achieved.

14

541

GLOSSARY

Term

Defi nition

Chapter

Method of constant stimuli

A technique for measuring a threshold in which the stimulus values and the number of trials at each value are chosen in advance. On a given trial, the stimulus value is chosen at random from the predefi ned choices.

14

Microelectromechanical systems

Devices with both electrical and mechanical functionality developed from microfabrication methods. This technology leverages the batch fabrication process developed for the integrated circuit industry.

4

Modal wavefront control algorithm

This algorithm treats wavefront distortions as a sum of modal functions, such as Zernike polynomials for a circular aperture, or modes that come from the AO system itself.

5

Modulation transfer function

Transfer function characterizing the proportion of contrast present in the object that is preserved in the image formed by an optical system.

10, 13

Monostable multivibrator

A timer circuit with only one stable state.

16

MRSE

See manifest refractive spherical equivalent.

12

MTF

See modulation transfer function.

10, 13

Multiplexer

A circuit with many inputs and one output. By applying appropriate control signals, many inputs can be steered to the output.

16

Myopia

A condition of the eye where light passing through the optical surfaces of the eye comes to focus in front of the retinal plane. Also called “nearsightedness.”

11

N-AFC

See N-alternative-forced-choice.

14

542

GLOSSARY

Term

Defi nition

Chapter

N-alternativeforced-choice

A procedure for estimating thresholds in which the observer has N (2 or more) response options on a given trial and is obliged to respond even if no stimulus is detected. For example, in the temporal 2AFC procedure, the observer is asked to indicate in which of two intervals the test stimulus was presented.

14

National Television Subcommittee

A television standard adopted for image transmission (mainly in the United States).

16

Nearsightedness

See myopia

11

Noncommon path aberration

Aberration arising from an element(s) in a portion of the optical system not common to the wavefront sensor (such as in an imaging arm) and is, consequently, not detected by the sensor.

15

NTSC

See National Television Subcommittee.

16

OCT

See optical coherence tomography.

17

Ocular aberration

Wave aberration of the complete eye.

2

Open-loop control

In open-loop (or feed-forward) control systems, 5 the wavefront sensor precedes the wavefront corrector in the system’s optical path. The wavefront sensor measures the uncorrected wavefront fi rst. The required corrections are computed and then fed to the wavefront corrector, with no feedback from the wavefront sensor on the accuracy of the aberration correction.

Optic nerve head

The anatomical opening in the eye where the nerve bundles exit the retina and the retinal blood vessels communicate with their vascular supply. The optic nerve head is a relatively rigid, collagenous structure that has its own circulation, but contains no neural elements to initiate sight.

9

543

GLOSSARY

Term

Defi nition

Chapter

Optical axis

(1) The axis of symmetry in a rotationally symmetric optical system; (2) the actual or desired series of line segments defi ned by the chief ray of the on-axis field point.

7

Optical coherence tomography

An optical imaging modality, typically based on Michelson interferometery, that optically sections tissue with microns of axial resolution realized by interferometrically discriminating reflected or backscattered light by its time of fl ight.

17

Optical transfer function

Transfer function that characterizes the performance of an optical system. It is the Fourier transform of the point spread function or the autocorrelation of the generalized pupil function, and consists of the modulation transfer function (MTF) and the phase transfer function (PTF). When multiplied by the object spectrum, the image spectrum is obtained.

13

Optical zone

See ablation optical zone.

12

OTF

See optical transfer function.

13

Outer segment

The portion of the photoreceptor that contains a lipid membrane supporting photopigment in a configuration allowing light to interact and cause bleaching, leading to a visual signal.

9

PAL-SLM

See parallel aligned nematic liquid crystal spatial light modulator.

18

Parallel aligned nematic liquid crystal spatial light modulator

An optically addressable (intensity to phase) spatial light modulator with an amorphous silicon layer, a dielectric mirror, and a liquid crystal layer sandwiched between two glass substrates with transparent electrodes.

18

Phakic

The physiological eye that retains its natural lens; literally “with lens.”

11

544

GLOSSARY

Term

Defi nition

Chapter

Phase transfer function

Transfer function characterizing the variation of phase shift in the image as a function of spatial frequency.

13

Phase wrapping

The mechanism of folding the phase greater than 2p into an equivalent phase less than 2p using a modulo operation. When the amount of to-be-compensated phase is more than 2p, then the excess phase must be folded within the 2p range. This is essential if the device has a maximum modulation of a single wave (2p at the operating wavelength).

18

Phoropter

A refractive device used to determine a patient’s sphero-cylindrical refractive error. When coupled with adaptive optics, the system can also measure and correct for the patient’s higher order aberrations (e.g., for psychophysical testing).

18

Photomultiplier tube

A photon counting device.

16

Photometer

A device that measures the radiance of a light source weighted by a fi lter that emulates a human spectral efficiency function.

14

Photopigment

The photolabile pigment that breaks down when struck by light, leading to the intiation of a neural signal.

9

Photoreceptor

A specialized neural cell that captures light, converts light energy to chemical energy, and transmits a neural signal.

9

Photorefractive keratectomy

A technique that removes the cells on the front surface of the cornea (the corneal epithelium) and then applies the excimer laser treatment to the cornea. A bandage soft contact lens is then applied postoperatively to aid in healing.

12

Piston-only segmented mirror

An array of adjacent, planar mirror segments that are independently controlled and have one degree of freedom that corresponds to a pure, vertical piston mode.

4

545

GLOSSARY

Term

Defi nition

Chapter

Piston/tip/tilt segmented mirror

An array of adjacent, planar mirror segments that are independently controlled and have three degrees of freedom that correspond to a vertical piston mode and two additional degrees of freedom (tip and tilt) for slope control.

4

Plasma display

A computer-controlled display that produces light by exciting plasma gas pockets coupled to phosphors.

14

PMT

See photomultiplier tube.

16

Point spread function

The response of an optical system to a point source of light. It is calculated as the squared modulus of the Fourier transform of the generalized pupil function.

8, 13

Power spectrum

The power spectrum is the squared modulus of the Fourier transform of a signal. Unlike the Fourier transform, it ignores phase information only and has positive values. It quantifies how much each spatial frequency is represented in a given pattern, image, or stimulus. For example, the image of a picket fence would contain a lot of power at the frequency corresponding to the spacing of the pickets.

8

Power spectral density

A continuous function with dimensions of wavefront squared per hertz, obtained by dividing the power spectrum of the wavefront by the product of the number of diagnostic frames and the sampling period.

8

Presbyopic

A condition of the eye where the natural lens can no longer accommodate (or increase its positive power by steepening its surfaces) to allow near objects to focus at the retinal plane. Presbyopic eyes are in focus for distant objects, but hyperopic for near objects.

11

PRK

See photorefractive keratectomy.

12

546

GLOSSARY

Term

Defi nition

Chapter

PSD

See power spectral density.

8

Pseudophakic

The physiological eye that has had its natural lens replaced with an artificial lens, such as an intraocular lens (IOL).

11

PSF

See point spread function.

8, 13

Psychometric function

The probability of correct performance on a psychophysical task as a function of stimulus strength.

14

Psychophysical function

The value of a physical stimulus variable required to produce a criterion level of performance on a psychophysical task as a function of a second physical stimulus variable.

14

Psychophysical tests

These tests are done, in the context of AO systems, to evaluate the effect of adaptive optics in enhancing the visual perception and performance of an observer. This is as opposed to objective measurements obtained by the wavefront sensor. A laboratory test is designed to simulate a real-life task that the observer must perform. For example, a test could be conducted by displaying a series of sinusoidal patterns in front of the observer and evaluating his contrast threshold function with and without AO correction.

18

Psychophysics

The study of the relations between human performance and physical variables. Also, the methods and techniques used in this study.

14

PTF

See phase transfer function.

13

Pupil

In a given optical space, the image of the aperture stop.

7

Refractive error

In the context of optometry and ophthalmology, 13 the spherical and astigmatic focusing errors of the eye.

547

GLOSSARY

Term

Defi nition

Chapter

Refractive surgery

Surgery done to improve the eye’s optics and correct refractive error. This can be accomplished with corneal refractive surgery (such as an excimer laser reshaping of the cornea) or with an ocular implant placed inside the eye to correct for refractive errors.

12

Registration

The relative alignment between two planes. In the context of AO system design, registration describes a method to co-align the wavefront sensor with the wavefront corrector. In the absence of registration, the corrections will be applied at the wrong place, which in essence will increase the error instead of decreasing it.

7, 18

Relay

Any imaging optical system.

7

Retina

The thin, transparent layer of neural tissue that initiates a visual signal at the back of the eye and transmits it toward the brain.

9

Retinal pigment epithelium

The melanin-containing monolayer of cells that 9 provides metabolic support for and helps in renewal of the photoreceptors.

RMS

See root-mean-square.

2

Root-meansquare

In statistics, the magnitude of a varying quantity, calculated as the square root of the mean of the sum of the squared values of the quantity. For a wave aberration described using Zernike polynomials, the root-meansquare (RMS) wavefront error is defi ned as the square root of the sum of the squares of a given number of Zernike coefficients.

2

Sclera

The opaque (white) portion of the eye. It is composed of randomly oriented collagen fibrils.

11

SD-OCT

See spectral-domain optical coherence tomography.

17

548

GLOSSARY

Term

Defi nition

Chapter

Search box

A rectangular (typically square) region of interest used to compute the centroid (center of mass) of the image formed by a single lenset of a Shack–Hartmann wavefront sensor. See also center of mass.

6

Segmented corrector

A wavefront correction component that uses segmented optical reflection or transmision. See also segmented mirror and liquid crystal spatial light modulator.

4

Segmented mirror

An array of adjacent, planar mirror segments that are independently controlled. See also piston-only segmented mirror and piston/tip/ tilt segmented mirror.

4

Sensitivity

The inverse (reciprocal) of threshold.

14

Shack– Hartmann wavefront sensor

A wavefront sensor that uses a regular array of lenslets to image the incident wavefront on a CCD array. A uniform plane wave produces a regular array of spots. An aberrated wavefront produces focal spots that are displaced from these reference positions by a factor proportional to the local slope of the wavefront.

3, 18

SLM

See spatial light modulator.

18

Spatial light modulator

A device for imprinting an amplitude or phase pattern (or both) on an optical beam. By modulating the phase of an aberrated beam, the aberrations of the beam can be compensated. A spatial light modulator (SLM) can be used as an optical input device for processing information in an optical information processing system. A SLM can also be used as an optical matched fi lter for an optical correlator.

18

Speckle

A spatially random intensity distribution produced from the coherent interference of light that reflects from an optically rough surface or propagates through a turbulent medium.

17

549

GLOSSARY

Term

Defi nition

Chapter

Spectraldomain optical coherence tomography

A highly efficient form of OCT that records the optical spectrum of the interferometric signal using a spectrally dispersive element, e.g., diffraction grating, and a linear or aerial CCD in the detection channel.

17

Spectroradiometer

A device that measures the radiance of a light source over a large number of steps across the visible spectrum.

14

Sphere

The defocus component of a spectacle prescription designed to correct for myopia or hyperopia.

11

Spherical equivalent

See manifest refractive spherical equivalent.

13

Spherocylindrical

Refraction that includes the defocus (sphere) and astigmatism (cylindrical) values. For correction, a lens designed to correct both defocus and astigmatism where at least one of the surfaces is toric in shape.

11

Spot

In a Shack–Hartmann wavefront sensor, the term “spot” refers to the image formed by light focused from a single lenslet.

6

Staircase method

A psychophysical technique in which threshold is measured by selecting stimulus values dynamically based on observer performance throughout the course of the experiment.

14

Stiles– Crawford effect

Light entering the pupil at different locations does not elicit the same visual response due to the waveguide nature of the photoreceptors and their pointing direction.

15

Strehl ratio

The ratio of the peak intensity of the point spread function of an optical system relative to the peak intensity of a diffraction-limited point spread function.

8

Stroke

The dynamic range of the deformable mirror actuators, typically measured in microns.

4

550

GLOSSARY

Term

Defi nition

Chapter

TCA

See transverse chromatic aberration.

2

Threshold

The limiting value of a physical variable (e.g., number of quanta, contrast) for a criterion level of performance on a psychophysical task (see also detection, discrimination, and identification thresholds).

14

Transition zone

The additional width of a laser refractive surgical ablation that extends past the ablation optical zone and is used to blend the ablation optical zone with the untreated peripheral cornea.

12

Transverse chromatic aberration

Change in the apparent magnification of the optical system with wavelength.

2

Troland

A photometric unit used to quantify retinal illuminance, calculated by multiplying the luminance value (in cd/m 2) with the area of the pupil (in mm 2). Units of (cd/m 2) ∗ (mm 2).

14

VA

See visual acuity.

14

Vertical synchronization pulse

The change in voltage level of a video signal that triggers the end of one frame and the start of a new frame. On a video display or frame grabber, each frame in an image starts at the end of the vsync pulse and ends with the start of the next vsync pulse.

16

Virtual refraction

A computational method that captures the essence of a traditional refraction by successive elimination by mathematically simulating the effect of spherocylindrical lenses of various powers.

13

Visual acuity

One measure of the resolution limit of vision. Acuity is often expressed in terms of Snellen notation, equivalent to the ratio of a standard viewing distance (20 ft in the United States or 6 m in Europe) to the distance at which the smallest identifiable symbol subtends a visual

14

551

GLOSSARY

Term

Defi nition

Chapter

angle of 5′, with the lines and their interdigitated spaces having a thickness of 1′ (1/60 of a degree). Visual angle

The portion of the visual world included between two lines that converge at the pupil.

9

Visual benefit

The improvement in visual performance that an eye can gain by correcting lower or higher order aberrations. As a metric, it can be defi ned as the ratio of the modulation transfer function of the ideally corrected eye to the modulation transfer function of the partially corrected or uncorrected eye for any selected spatial frequency.

11

Vitreo-retinal interface

The intersection between the collagenous vitreous body and the innermost retinal component, the nerve fiber layer, which is often so reflective as to obscure the deeper layers.

9

Vitreous humor

The collagenous body, between the crystalline lens and the retina, that supports the globe from the inside, and is in contact with the retinal surface prior to aging changes.

9

Volume resolution element

Defi nes the size of the smallest possible volume 16 in a three-dimensional image calculated using the formula for the volume of a cylinder.

Vsync

See vertical synchronization pulse.

16

Wave aberration

Function defi ned as the difference between the aberration-free (spherical or reference) wavefront and the actual wavefront for every point over the pupil.

2

Wavefront

An imaginary surface that represents the direction of propagating light. The wavefront is always perpendicular to the direction of travel at all points in space.

3

552

GLOSSARY

Term

Defi nition

Chapter

Wavefront sensor

A wavefront sensor is used to measure the wave aberration of the light. See also Shack– Hartmann wavefront sensor.

3, 18

Wavefrontguided refractive surgery

See customized ablation.

12

WS

See wavefront sensor.

7, 8

Yes/no procedure

A procedure for estimating thresholds in which an observer indicates whether a stimulus was detected on a given trial.

14

Zernike polynomials

Orthogonal polynomials used to expand an aberration function defi ned within the unit circle. The Zernike coefficients represent standard deviations of individual polynomials.

3

Symbol Table Symbol

Represents

䉺

Autocorrelation

丢

Convolution operator

∇2

Laplacian operator

a

Axis of astigmatism (or cylinder)

bi

Off-axis incident angle

br

Off-axis refracted angle

bv

Off-axis viewing angle

c

Log-normal amplitude

∆

Step, difference, or change in a variable

∆f

Phase difference

∆l

Bandwidth of light source

∆l res

Spectral resolution of imaging spectrometer

∆fx

Acquisition (spatial frequency) bandwidth

Adaptive Optics for Vision Science, Edited by Porter, Queener, Lin, Thorn, and Awwal Copyright © 2006 John Wiley & Sons, Inc.

553

554

SYMBOL TABLE

Symbol

Represents

∆h

Deviation of off-axis chief ray when deformable mirror, wavefront sensor’s pupil, and subject’s pupil are not coincident

∆s

Spot displacement

∆smax

Maximum measurable spot displacement

∆smin

Minimum measurable spot displacement

∆xS

Centroid or spot displacement of a Shack–Hartmann wavefront sensor in the x (horizontal) direction

∆yS

Centroid or spot displacement of a Shack–Hartmann wavefront sensor in the y (vertical) direction

∆z

Range or change along the optical axis, such as depth of focus, axial distance between pupil planes, or in the context of optical coherence tomography, the translational change of the reference mirror

e0

Permittivity

f

In the context of the radially averaged OTF metric, the orientation variable for the integral over all orientations

f

In the context of a periodic function, such as a sinusoid, phase

h

Quantum efficiency

ϕc (x, y, t)

Wavefront profi le of correction, as a function of position and time

ϕi (x, y, t)

Wavefront profi le, uncompensated, as a function of position and time

ϕm

Wavefront profi le at actuator m

ϕr (x, y, t)

Wavefront profi le of residual aberration, as a function of position and time

l

Wavelength of light

n

Frequency of light

SYMBOL TABLE

Symbol

555

Represents

q

In the context of polar coordinates, the angular component of a polar coordinate

q

In the context of optical wavefronts, the slope of the wavefront

q max

Maximum wavefront slope that can still include one spot within each virtual subaperture

q min

Measurement sensitivity: the minimum wavefront slope that the wavefront sensor can measure

qdl

Angle subtended by the peak of the Airy disk and the fi rst minimum (width of the point spread function)

qvis

Visual angle

Θ(u, v)

Wiener fi lter in the spatial frequency domain

ς

Integration variable, shifted from x, in the defi nition of convolution

s

Standard deviation

s2

Variance

sCALIB

Calibration error

sn

Standard deviation of noise distribution

t

Delay time constant or time lag

tc

Computational delay

u

Integration variable, shifted from y, in the defi nition of convolution

wo

Resolution at the focal plane in microns

Ψ

Complex field of the corrected wavefront

Ψref

Complex wavefront from the reference arm of an optical coherence tomography system

556

SYMBOL TABLE

Symbol

Represents

Ψretina

Complex wavefront from the retina measured by an optical coherence tomography system

a

Length of stimulus along an axis orthogonal to the direction of viewing

abim

Bimorph thickness

agap

The distance between the electrodes and the membrane in a membrane mirror

akm

An element of the actuator influence matrix A (see below). Each element is obtained by applying known voltage to each actuator element m and obtaining the resulting slope measurement k in x and y directions for each lenslet.

amn

Zernike coefficients expressed according to Gram– Schmidt orthogonalization method for the cornea surface. The coefficients are indexed by angular frequency, m, and radial order, n

A

Actuator influence matrix that is used to compute the average slope of the wavefront produced by a given actuator voltage vector as vA = s

A†

Pseudo-inverse of the actuator influence matrix A. A† is used to compute the desired actuator voltage from the measured slope vector as v = A†s

bave

Average blur strength

cj

Element of the Zernike coefficient vector at mode index j

cJ

Maximum Zernike coefficient (mode) included

cmn

Zernike coefficient for the wave aberrations for angular frequency, m, and radial order, n

C1 or C 2

Principal curvature map

Cb (x, y)

Blur mapping function

C Gauss

Gaussian curvature map

SYMBOL TABLE

Symbol

557

Represents

CJ

Astigmatism curvature map

Cmean

Mean curvature

C3

Zernike coefficient value at the third mode (astigmatism term)

C4

Zernike coefficient value at the fourth mode (defocus term)

C5

Zernike coefficient value at the fi fth mode (astigmatism term)

c

Zernike coefficient vector

d

In the context of optical design, the diameter of a lenslet, pupil, or subaperture. May also represent lenslet or pixel spacing

d[n]

Residual wavefront as measured by the diagnostics

d′

In the context of signal detection theory, sensitivity, also called “d-prime”

dpix

Pixel diameter

D(f)

Residual mirror commands obtained in the diagnostics (i.e., input to compensator or “control computer”)

D(p)

Power spectrum of the diagnostics for frame p where p = 1 to P

D

Diopters

e−sT

Exponential function for converting the complex z-transform variable z to the Laplace domain using frequency variable s

exp(x)

Exponential function of variable x

E

Error metric of an optical system

f

Frequency; such as i2pf or spatial frequency

558

SYMBOL TABLE

Symbol

Represents

fc

Cutoff frequency

fs

Sampling frequency

F

Focal length

FS

Sagittal focal length

FT

Tangential focal length

f/#

f-number

gA (x, y)

Gaussian response used to model actuator surface

gN (x, y)

Gaussian neural weighting function used to compute neural sharpness

h

Plank’s constant (where the energy of a photon is hn)

H(f)

Transfer function for spatial frequency f

H(s)

Transfer function for frequency s, where s = i2pf

H(u, v)

Transfer function for two-dimensional spatial frequency u, v

i

Square-root of negative 1

i(x, y)

Image irradiance in the spatial domain

I(u, v)

Fourier transform of the image irradiance

I

Intensity

I(zref) retina

Intensity of a reflection from a slice of retina

Iref

Intensity at the reference arm of an optical coherence tomography system

Iretina

Intensity from the retina in an optical coherence tomography system

j

Index of Zernike mode (coefficient) in the range 1 to J

SYMBOL TABLE

Symbol

559

Represents

J

Number of Zernike modes (coefficients)

k

The index of a lenslet in a Shack–Hartmann wavefront sensor in the range 1 to K.

K

A constant in an expression, such as constant gain or unit conversion factor

KG

Coefficient of the controller (proportional to the system gain, K)

Km

Membrane stress constant

Kp

Piezoelectric constant of a bimorph mirror

KW

Weber fraction

K

Number of lenslets in a Shack–Hartmann wavefront sensor

l

Distance, as in viewing distance

lo

Length of a characteristic feature size

lc

Coherence length

Ll

Radiance at a given wavelength

Lback

Background luminance in a stimulus image

L max

Maximum luminance in a stimulus image

L min

Minimum luminance in a stimulus image

L stim

Stimulus luminance in a stimulus image

Lv

Luminance

m

In the context of Zernike coefficients, angular frequency index, used as a superscript

m

In the context of the adaptive optics control systems, the index of the deformable mirror actuator in the range 1 to M

560

SYMBOL TABLE

Symbol

Represents

m(x, y)

Mirror actuator signal magnitudes at actuator position (x, y)

M(f) or M(s)

Mirror position signal from a linear system

m0,N

Magnification between the eye and the plane where the lens is placed

ma..z

Product of the magnifications of the telescopes between conjugate planes a and z in a reaster scanning optical system

M

Number of deformable mirror actuators

M

Influence matrix of actuators

n

In the context of signal detection theory, a noise signal

n

In the context of Zernike coefficients, the index of the radial order

n¯

Mean of noise n

N(f) or N(s)

Additive noise signal input to a closed-loop linear system

N(u, v)

Spatial frequency (Fourier) domain expression for noise influence, on the Wiener filter

Ncsf(x, y)

Neural weighting function based on the inverse Fourier transform of the contrast sensitivity function

n

In the context of control systems at discrete time intervals, the time index

n or n′

Refractive index

ncore

Refractive index of the core of an optical fiber

nret

Refractive index of the retina

N

Number of Zernike radial orders

NA

Number of resolution elements in an A-scan

SYMBOL TABLE

Symbol

561

Represents

o(x, y)

True object irradiance in the spatial domain

ô(x, y)

Estimate of object irradiance in the spatial domain

O(u, v)

Fourier transform of the true object irradiance

p

Index of a diagnostic frame in a closed-loop control system, in the range 1 to P

pk (x, y)

Subpupil function used to simulate the spot pattern for each lenslet in a Shack–Hartmann wavefront sensor

P(u, v)

Fourier transform of the point spread function of an optical system

PA

Oblique astigmatism in diopters

Pcyl

Optical power of a cylindrical lens

PJ0 or PJ45

Optical power of a Jackson crossed cylinder

Pretina

Optical power of a retinal reflection

Psph

Optical power of a spherical lens (diopters)

PSE

Spherical-equivalent power

P

Number of diagnostic frames in a closed-loop control system

psf(x, y)

Point spread function in the spatial domain

psf N (r, q)

Normalized point spread function in polar coordinates

qA (x, y)

Binary function used to compute the correlation width of light distribution as a function of the autocorrelation of the PSF

qH (x, y)

Binary function that is 1 where the PSF is greater than one half the maximum and 0 elsewhere, used to compute the half-width at half-height metric

r

Radial distance as a variable; radius of beam of light at the eye

562

SYMBOL TABLE

Symbol

Represents

R

Radius of curvature

R(f ) or R(s)

Residual aberration in a closed-loop system, where s = i2pf

r

Radius

R

Reconstruction matrix

s

In the context of modeling the dynamic behavior of a system, complex frequency variable, where s = i2pf

sk

In the context of slope vectors, a single element, at lenslet measurement k, of the slope vector s

¯sx

Global tip-tilt from the horizontal component slope vectors

¯sy

Global tip-tilt from the vertical component slope vectors

(s + n)

Mean of signal plus noise

S

Strehl ratio

S(l)

Weighting function for computing polychromatic metrics

S CALIB

Strehl ratio corresponding to calibration error

S

Spot size

s

Slope vector or vector of centroid measurements

s[n]

Set of many residual centroid measurements at time n

s˜

Uncorrectable residual of centroid measurements

s′

Slope vector corrected for tip-tilt

s¯

Average of set s[n] of centroid measurements over many frames

TN ( f )

Contrast threshold function as a function of spatial frequency

T

Exposure time or sampling period. Used in the context of wavefront sensing or optical coherence tomography A-scan

SYMBOL TABLE

Symbol

563

Represents

u

Normalized axial unit

u[n]

Input to wavefront compensator at time n (unit step)

v(x, y)

Mirror voltage profi le at position (x, y)

v¯

Average of the actuator control voltage vector. To remove the piston component from the control voltage vector, subtract this value from each element of the vector v.

vm

The mth deformable mirror actuator control voltage

V′l

Scotopic luminous efficiency function at wavelength l

Vl

Photopic luminous efficiency function at wavelength l

v

Deformable mirror command vector

v′

Deformable mirror command vector with the piston component removed.

VM

Michelson contrast

VW

Weber contrast

w[n]

Windowing function used to avoid spectral leakage

W

Wavefront aberration in waves (relates to phase, via f = W2p /l)

W(r, q)

Wavefront or wave aberration function in polar coordinates

Wk (x, y)

Wavefront or wave aberration function, for the subaperture of a particular lenslet, k, expressed in Cartesian coordinates

W(x, y)

Wavefront or wave aberration function in Cartesian coordinates

∂W ( x, y ) ∂x

Wavefront slope in the x direction, expressed as the partial derivative of the wavefront

∂W ( x, y ) ∂y

Wavefront slope in the y direction, expressed as the partial derivative of the wavefront

564

SYMBOL TABLE

Symbol —

Wx —

Represents Mean of the spatial derivative in x of the wave aberration

Wy

Mean of the spatial derivative in y of the wave aberration

x

Variable in the horizontal direction, typically orthogonal to the optical axis of a system

X( f ) or X(s)

Aberrations (input signal to linear system)

y

Variable in the vertical direction, typically orthogonal to the optical axis of a system

y[n]

Compensator consists of an integral controller of the form y[n] = y[n − 1] + Ku[n]

z

In the context of Cartesian coordinate systems, the direction along the optical axis, where z is perpendicular to the (x, y) plane

z

In the context of the Laplace transform, the complex z-transform variable where z = e−s−T

z(r, q)

Corneal elevations, defi ned as the distance from each point of the corneal surface to a reference plane tangential to the vertex of the cornea

z′jk

Derivative of the jth mode of the Zernike representation of the wavefront for the kth lenslet

Z

Zernike polynomial

Zj (x, y)

Zernike polynomial at mode j, where j = 1 to J

Zmn (r, q)

Zernike polynomial, for angular frequency m and radial order, n

Z

Reconstructor matrix that computes Zernike coefficients from slope vectors

Zm

Zernike reconstruction matrix for individual actuator m

Z†

Pseudo-inverse of reconstructor matrix Z

INDEX

Abbe error, 162 Aberration(s): characterized, 36–37, 66, 90, 236, 239 chromatic, 9, 33–34, 43–45, 51, 79, 90, 107–108, 200, 268–271, 354–356, 540, 550 corneal, 35–37, 533 correction of, 307, 34, 92, 291 defi ned, 529 generator, 77–78, 127–128 internal, 537 lenticular, 239 map, 340–341, 357–358 measurement, 63, 297 monochromatic, xvii, 4, 6, 33–40, 238–239, 358–359 in normal eye, 34–35 ocular, defi ned, 542 off-axis, 46–51, 239–240, 426 polarization effects, 34, 53–55 population statistics: Indiana, 52, 97–99, 100–109 Murcia optics lab, 52 Rochester, 52 Rochester & Bausch & Lomb, 97–99, 100–109 principal components analysis of, 52 refractive, 332 reporting (OSA Standards), 518–522 scatter effects, 34, 55 statistics of aberrations, 34, 52–53 temporal properties, 40–43, 97

Aberrometry, 98, 358 Aberro-polariscope, 54 Aberroscope, crossed-cylinder, xviii Ablation, see Corneal ablation conventional, 325, 532 customized, 324, 533 ocular, 40–43 optical zone, 312, 529 rate, 323–324 Absorption/absorption spectra, 42, 218–220, 225, 284, 420 Accommodation, 7, 34, 40–43, 336 Achromatization process, 44–45 Acoustic impedance mismatch, 257–258 Acousto-optic modulator (AOM), 419, 529 Acquisition bandwidth, 263 Actuator, see specifi c types of wavefront correctors AOSLO, 422–423 configuration of, 122–124 deformable mirror, discrete, 86, 97, 534 DM-to-WS registration, 183 influence function, 122, 184 lead magnesium niobate (PMN), 397, 452 mirror, 266 Nyquist criterion, 192–193 Rochester Adaptive Optics Ophthalmoscope, 400–401 slope influence function and, 124–125 spacing of, 156 stroke, wavefront correctors, 99–100 voltages, 150, 175, 184, 196

Adaptive Optics for Vision Science, Edited by Porter, Queener, Lin, Thorn, and Awwal Copyright © 2006 John Wiley & Sons, Inc.

565

566

INDEX

Acuity, defi ned, 9 Adaptive optics scanning laser ophthalmoscope (AOSLO): axial resolution, 434–438 basic layout of, 249 calibration, 431–432 characterized, 17–18, 20, 23 compensation, 432–434 electronic hardware control of, 428 image acquisition, 426–429 imaging results, 438–440 light detection, 254 light path, 249–251 optical layout for, 425–426 performance strategies, 441–444 SLO system operation, 255 software interface for, 429–431 Adaptive optics sensing and correcting algorithm (AOSACA), 430 Adaptive optics (AO) system: aberration correction, 38 assembly of, 181–182, 491 benefits of, 63, 83–84 defi ned, 529 fi rst-order optics, 156–157 human factors, 272 imaging time, 276 light budget, 271–272 optical alignment, 157–174 optomechanical design, 155 performance: bandwidth error, 199–200 calibration error, 191–192 fitting error, 192–194 measurement error, 194–199 significance of, 163, 189 Strehl ratio, 189–191 testing procedures, 492 wavefront errors, 200–201 principal components of, 84–86 real-time, 42 refraction, 272–276 registration, 496–498 retinal imaging, 11–24 software, 139, 489 system integration, 174–186 transfer function of, 130–135 vision correction, 9–11 wavefront correctors, 86–111 Adaptive optics with optical coherence tomography (AO-OCT): basic layout of, 264–266 characterized, 23, 448 chromatic aberrations, 271

Afocal relay telescope, 157–158, 170–174 Aging eye/aging process: effect on aberrations, 34, 40–43 fundus, 220 light absorption, 42 retinal changes, 218, 224–225 Airy, George Biddell, Sir, 4 Airy disk, 33, 237, 254, 435, 441 Algorithm(s): centroid, 200 center-of-mass, 144, 483–484, 488, 531 control, 119–135 direct slope, 124–127, 534 iterative centroid, 144 least-squares, 127 mirror control, 421–423 phase retrieval, 192 phase-unwrapping, 176 reconstruction, 76–77 Aliasing, 5, 51, 127, 423, 507 Alignment, see Optical alignment AO system: aligning optics, 167–170 CAD applications, 159–160, 164 common practices, 163–170 detectors, 166–167 error budget and, 160–161 general tools, 166–167 layout, 164 mirrors, flat and interchangeable, 167 offl ine alignment, 170–174 optical axis establishment, 164–166 sources, 166–167 laser, positioning of, 165 Rochester Adaptive Optics Ophthalmoscope, 402 telescope, 164 wavefront measurement methods, OSA Standards, 515–517 American National Standards Institute (ANSI), 419 ANSI standards Z80.28, 334 Ametropia, 292–293, 305, 529 Amplifiers, high-voltage (HVAs), 132 Analog/digital (A/D) conversion, 255 Angiograms, 210–211, 439 Angle targets, optical axis, 165–166 Annexin-5, 24 AO loop, 139, 530 Aperture: annular, 222 AOSLO, 422 confocal, 207, 224 entrance, 334

INDEX light scattering techniques, 222–224 numerical, 241, 259, 451 photoreceptors, 237 Rochester Adaptive Optics Ophthalmoscope, 403 size of, 278 stop, defi ned, 530 subaperture, 71–72, 75, 156, 177, 180, 183–184, 478, 481, 490–491, 500 wavefront correctors and, 110 Aphakic, defi ned, 530 AreaMTF, 350–351 AreaOTF, 351 Arcades, 209 Arteries, retinal, 206–207, 209–211, 222, 228–229 Artificial eye, 166, 182, 185 A-scan, 257, 263–264, 268, 270, 450, 465, 469, 530 Astigmatism: AO-OCT experiments, 454 AO system assembly, 181 characterized, 4, 36, 46, 83, 293, 306, 401 conventional imaging, 237, 240 correction of, 6, 50–51, 84, 273, 294, 335, 413, 431, 457, 474, 478, 503 custom-correcting contact lenses, 296, 301–302, 333–334 far point and, 335 high-resolution retinal imaging, 264 image quality and, 11 oblique, 47–48, 51 ocular adaptive devices, 9 off-axis aberrations, 47–48 peripheral refraction, 47–48 refractive surgery, 313 Shack-Hartmann wavefront sensing, 73, 75 on shear plate, 170 statistics of aberrations, 52 surgical correction of, 306–307, 313 wavefront correctors and, 101, 105, 107 Astrocytes, 215, 218 Atmospheric turbulence, impact of, 7, 95 Automated refraction, 9–10 Avalanche photodiodes (APDs), 5, 131, 255 Axial: point spread function (PSF), 465, 530 resolution, 5, 23, 256, 265, 267, 270, 434–438, 441, 467 sectioning, closed-loop AO system, 423–424 slicing, 441 Azimuthal frequency, 295

567

Babcock, Horace, 7 Backscattered light, 23, 222, 227–230 Bandwidth: acquisition, 263 closed-loop, 8, 133–135 error, 194, 199–200, 530 error transfer function, 133, 135 narrow, 242 open-loop, 133–134 spectral, 23, 90, 270 temporal, 97, 112 Basal lamina, 214 Bayesian statistics, applications of, 280, 379 Beam path, slope of, 165 Beam size, in scanning laser imaging, 253 Beamsplitter(s), 78, 240–241, 266, 272, 303, 399, 418, 425, 452, 483 Beer’s law, 212 Bias, SLM, 495 Bimorph mirrors, 88, 91–92, 96–97, 530 Bimorph technology, 86 Bipolar cells, 217 Birefringence, 227, 229, 443 Bi-sinc function, 177 Bit stealing, 387 Blackman-Harris window, 197 Bleaching, 15, 225 Blind spot, 217 Blood: -brain barrier, 209 flow: high-resolution imaging of, 21–22 measurement of, 439 monitoring, 23 light absorption, 218–220, 225 retinal barrier, 210, 214 Blur/blurring: conventional imaging, 243 convolution and, 278 eye movement, 5 neural adaptation to, 11 optical, 51 retinal image, 7 sources of, 326 strength, 334 strength map, 342–343 Bode analysis, 409 Boresight, 167, 175, 182, 530 Brain: blur/blurring and, 11 central nervous system, 209 cortical neural processing, 335 blood-brain barrier, 209 eye movement and, 17

568

INDEX

Bruch’s membrane, 214, 220 B-scan, 448–450, 465, 467, 471–472, 531 Calibration: AOSLO, 431–432 OSA Standards, 523–527 chromatic aberration, 79 defocus and, 431–432 error, 191–192, 531 hardware, 75–76 lenslet arrays, 77–78 liquid crystal AO phoropter, 492–502 performance errors, 191–192 reconstruction algorithm, 76–77, 127 reference centroids, 185–186 Shack-Hartmann wavefront sensing, 75–79, 158–159 Camera(s): charge-coupled device (CCD), 65–66, 75, 79, 130–131, 141, 400, 404, 406, 413, 418, 452, 479, 482, 506–507 high-resolution, 140 retinal, 83, 95, 110, 450 science, 85, 241–242, 246 stare, 195 Cane toad, photoreceptor cells, 5 Cannulation, 22 Capillaries, 206, 209–210 Capsulorhexis, 304 Cataract(s): characterized, 63, 304, 306, 308 formation of, 55 surgery, 39 Cathode ray tube (CRT): defi ned, 531 monitors, 381–385, 388, 405, 492 projectors, 386, 533 Catoptric image, 331 Center for Adaptive Optics, xviii Center-of-mass algorithm, 144, 483–484, 488, 531 Central nervous system, blood-brain barrier, 209 Central-pupil method, 343–344 Centroid algorithm: bounding box, 488 implications of, 71, 144, 200, 406, 459 pyramidal, technique, 488, 490 RMS wavefront error, 489 Centroids: AOSLO, 423, 430 characterized, 143, 484 conventional imaging, 240 defi ned, 531

estimates, 8 image preparation, 143–144 liquid crystal AO phoropter, 487–488 measurement of, 177–180 reference, 185–186 standard deviations of, 488 system performance and, 192 wavefront reconstruction process, 180–181 Charge-coupled device (CCD): applications, 239, 245–246, 256, 268, 449 camera: characterized, 141 retinal images, 452 Indiana University AO-OCT System, 452 liquid crystal adaptive optics, 479, 482, 506–507 Rochester Adaptive Optics Ophthalmoscope, 400, 404, 406, 413 scanning laser ophthalmoscope design, 418 wavefront correctors, 130–131 wavefront sensing, 65–66, 75, 79 detector, 473 plane: optical alignment, 156, 169, 177–179, 182 registration of, 183 Chief ray, 157, 531 Choriocapillaris, 210 Choroid: functions of, 213–214 photoreceptors, 261 Choroidal neovascular membrane, 210, 219–220 Chromatic: aberrations: axial, 354 calibration of, 79 characterized, 43–44, 243 conventional imaging, 241 intrinsic, 90 longitudinal (LCA), 44–45, 51, 107–108, 270–271, 540 monochromatic aberration interaction, 45 OCT ophthalmoscopes and, 268–271 ocular adaptive optics, 9 peripheral image quality and, 51 significance of, 33 transverse (TCA), 44–45, 51, 354–355, 550 wavefront errors, 200 compensation, 44

INDEX difference of magnification (CDM), 354, 531 dispersion: AO-OCT ophthalomoscopes, 266 implications of, 354 OCT ophthalmoscopes, 266 Chromaticity, 357, 382, 389 CIE standard observer, 368–370 Circle of least confusion, 47–48 Clinical trials, laser surgery, 317–318 Closed-loop, in AO systems: bandwidth, 8, 133–134 control, 129–130, 241, 432, 532 correction, 411, 491 implications of, 85, 241, 423–424, 431, 459–460 operation, 499–502 power spectrum, 198 response, AO, 131 tracking system, 322 transfer function, 132–133 transfer system, 408 Computer numerical control (CNC) contact lens lathe, 299–300 Coherence/coherent: AO parallel SD-OCT imaging, 469 laser source, 399 lengths, 398, 451, 469, 532 light, generally, 5–6, 390–391 Collagen, 212, 324 Collimation, 157, 163, 173, 178, 182, 185, 419 Color: appearance, adaptive optics, 15 blindness, 19 -channel independence, 384 fundus photography, 219 lookup tables (CLUTs), 387 vision, 15, 215 Colorimeter, 389–390, 532 Coma: aberration structure, 35, 41, 46–47, 49–50 correction, 477 corneal ablation, 322 customized vision correction devices, 291, 293–294, 296, 301, 306–307 Command vector, 125 Commission Internationale de l’Eclairage (CIE), 368–370, 382 Common path aberrations, 158 Compaq MP1600, 405 Compensation: AO-OCT experiments, 454 AOSLO, 432–434 characterized, 241, 252

569

chromatic, 44 liquid crystal AO phoropter, 504 scanning laser imaging, 252, 255 scanning laser ophthalmoscope (SLO), 421 spatial light modulator (SLM) response, 493 wavefront, 418, 421 zonal, 504 Compensator gain, 195 Complementary metal-oxide-semiconductor (CMOS), 92, 246 Computer-aided design (CAD) software, 159–160, 164–165 Computer software applications, see Software Computer technology, impact of, 6 Concave lenses, 4 Cone(s): absorption measurements, 284 angular tuning properties, 12, 14 AO parallel SD-OCT imaging, 474–475 characterized, 215–217, 410, 439 conventional imaging, 243–244, 246 daylight vision, 215 density, 17, 21, 215 diameter, measurement of, 21 directional sensitivity of, 13–14 long wavelength sensitive (L), 15, 16, 19, 215, 217, 226, 284 middle wavelength sensitive (M), 15, 16, 19, 215, 217, 226, 284 mosaic: AO-OCT experiments, 454, 462 AO parallel SD-OCT imaging, 470–472 color blindness and, 19 implications of, 5–6, 8, 367 primate, 24 retinal imaging, 12 trichromatic, 16 photoreceptor mosaic, 16, 20, 24, 110 photoreceptors, 11, 14, 225–227, 237 reflectance of, 14 short wavelength sensitive (S), 15, 16, 215, 217, 226 spacing, 6, 17 trichomatic, 14–15 Cone-rod dystrophy, mosaic in, 20–21 Confidence ellipse, 358 Confocal: aperture, 207, 224 imaging, AOSLO, 435 light imaging, 223 pinhole: AOSLO, 434, 437, 441 size of, 418, 441–443

570

INDEX

scanning laser imaging, resolution limits, 249 scanning laser ophthalmoscopes (cSLO), 23–24, 83, 89, 223, 236, 261 Contact lenses: advantages of, 291 benefits/beneficiaries of, 45, 83, 301–304, 332 cast molding process, 299 customized, 34, 63, 298–299 design considerations, 295–297 disposable, 298–299 fabrication for, 6 flexible, 305 hydrophilic lenses, 295, 300 lathing parameters, 299–300, 303 manufacturing issues, 300–301, 304 measurement for, 297–298 rigid gas-permeable (RGP), 293–295, 300 rotational stability, 295–296 silicone, 294, 305 soft, 293–295, 297–298, 302–304 spherical aberration, 291–293 tolerance/benefit ratio, 295 trial, 297–298 Contrast: agents, 21–22, 210–211 AO-OCT, 448 AOSLO, 443–444 attenuation, 335 improvement strategies for, 23–24 Michelson, 365, 377 resolution, 387–388 sensitivity: characterized, 8–9, 366, 410–411 function (CSF), 55, 364–367, 505–506, 532 Weber, 365 Control algorithms: actuators, configuration of, 119–122 implications of, 184–185 influence function measurement, 122–124 lenslets, configuration of, 119–123 spatial control command of wavefront corrector, 119, 124–128 temporal control command of wavefront corrector, 119, 128–135 Rochester Adaptive Optics Ophthalmoscope, 405 Control loop: schematic of, 194 system gain of, 184–185

Control matrices, 124–127, 184 Conventional imaging: basic system design, 237–239 field size, 244–246 implications of, 441 light source, 242–244 optical components, 239–241 resolution limits, 237 retinal, 236–246 science camera, 241–242, 246 system operation, 246 wavefront sensing, 240–242 Convergence: characterized, 502 error, 487 iterations, 487 Convolution, characterized, 277–278. See also Deconvolution CoolSNAP HQ, 452 Cornea, see Corneal abnormal conditions, 74 aging process, 43–44 astigmatism research, 4 cross section diagram, 313 LASIK surgery, 308 light exposure level at, 450, 452 polarization impact, 53 refractive errors on, 66 soft contact lenses and, 297 transparency in, 55 Corneal: aberrations: calculation of, 35–36 characterized, 63, 239 coupling with intraocular lens, 40 customized, 34 data, 33, 36 defi ned, 533 measurement, 35 pathological conditions, 301 refractive surgery, 38–39 schematic representation of, 37 ablation: anatomical customization, 319–320 biomechanics, 322–324 conventional, 325 correction strategies, 311 customized, 317–321, 324, 325, 533 excimer laser treatment, 312, 321–323, 325–326 functional customization, 317, 319 laser refractive surgery, 312–317, 326 LASIK flap, 324 optical customization, 320–321

INDEX variable rate, 322–324 apex, 297 curvature, 305 curvature map, 340 reflection, 241 topography, xviii, 36, 319 transplantation, 69, 74, 301 Corrective lenses, 336, 477 Corrective methods, conventional, 83 Corrector: devices, types of, 34, 86–88 segmented, 86–87, 548 stability, 461 wavefront, 84–111, 119, 124, 135, 170, 297, 398–403 Correction bandwidth, 408 Correlation width (CW), 347 Cortical neural processing, 335 Coupling coefficient, 86–87, 531 Cross-coupling, 86, 127, 423 Cross-cylinder convention, 333 Crossed-cylinder aberroscope, 36 Crossover, Shack-Hartmann wavefront sensing, 71 Crossover frequency, 198 Crystalline lens, 33, 36, 40, 43, 63, 336 C-scan, 257 Curvature map, 340 Curvature sensing, xviii Customized ablation, 317–321, 324, 325, 533 Customized vision correction devices: contact lenses, 291–304 intraocular lenses (IOLs), 304–308 Cutoff frequency, 280, 350, 429 Cycles per degree (cpd), 533 Cpd, defi ned, 533 Cyclopegic: drugs, 336 refractions, 311, 533 Cylinder: AO-OCT experiments, 454 characterized, 293, 478 correction of, 294 defi ned, 533 Cylindrical: lens, 333 trial lenses, 180 Cysts, macular, 220, 224 Dalsa CA-D1, 482 Dark noise, 178, 246, 484 Data: logs, 461 storage systems, 245

571

Daylight vision, 215 Decentered optical system, 307, 321–322, 513 Deconvolution: applications of, 282–283 linear, 278–280, 284 multiframe blind (MFBD), 281–282 nonlinear, 280–282 Defocus: aberrations, 52, 192 accommodation of, 41 AO-OCT system, 454, 456 AOSLO, 430–432, 436 AO system assembly and integration, 181–182 calibration, 431–432 characterized, 35, 83 closed-loop AO system, 423–424, 431 coefficient, wavefront corrector case illustration, 99 contact lenses and, 296–297, 301–302 conventional imaging, 237 correction/correction efficiency, 4, 6, 44, 50, 335, 413, 457, 474, 477, 503 custom-correcting contact lenses, 301–302 far point and, 335 image quality/image quality metrics, 348 induction of, 306 lens prescriptions, 334 OCT ophthalmoscopes, 264 off-axis aberrations, 47–48 overcorrections, 306 peripheral, 48, 50 point spread function (PSF), 357 polychromatic images, 360 polychromatic light, 355 Rochester Adaptive Optics Ophthalmoscope, 401 scanning laser imaging, 255 Seidel, 333 Shack-Hartmann wavefront sensing, 73, 75 undercorrections, 306–307 wavefront correctors and, 100–101, 105, 107 wavefront error computation, 41 Zernike, 100, 333 Deformable mirror: aberration correction, 149 actuator influence function, 149, 193 actuators on, 120 adaptive optics applications, 34 adaptive optics system, 11 AO-OCT system, 452

572

INDEX

AOSLO, 418, 421, 423–424, 426 characterized, 7–8, 10, 238 components of, 239 conventional, 478 defi ned, 533 discrete actuator, 86, 97, 534 fitting error and, 192 Fried configuration, 120–121, 184 OCT ophthalmoscope, 264–265 optical alignment, 167–168, 182 parallel SD-OCT system, 449 qualification of, 175–181 refraction, 272–273 Rochester Adaptive Optics Ophthalmoscope, 398–400, 402, 408, 411–412 scanning laser imaging, 255 Shack-Hartmann wavefront sensing, 70 Southwell configuration, 120 technology, future directions for, 23 transfer function of, 132 waffle, 120, 184 wavefront correctors, 101–102, 106, 112, 126–127 Degree of polarization (DOP), 54–55, 534 Degrees of freedom (DOFs), 4, 92, 160–161, 174 Depolarization, impact of, 54–55, 228–229 Depth of field, 9 Depth of focus, 168, 240, 261, 336–337, 347, 360, 467–469, 534 Detection threshold, 374–375, 534 Detector(s): alignment process, 166–167 charge-coupled device (CCD), 473 quantum efficiency (QE), 246 Shack-Hartmann wavefront sensor, 461 size of, 245 two-dimensional, 236 Development of adaptive optics, see Historical perspectives Diabetics/diabetes: retinal disease, 438–439 retinopathy, 22 visual effects, 210 Diagnostic display, Rochester Adaptive Optics Ophthalmoscope, 401 Dichromacy, 19 Dielectric beamsplitters, 241, 452 Diffraction: AOSLO, 435 implications of, 6, 23, 83, 157, 179, 236, 331 grating, 451, 454

-limited imaging, 41, 95, 111–113, 245, 254, 264, 442 point-, 265–266, 453, 455 scalar theory, 103 Shack-Hartmann wavefront sensing, 75 Digital current (DC), 427 Digital light projector, 386, 404, 405, 414, 534 Digital micromirror devices (DMDs), 386, 404–405, 534 Digital numbers (DN), 481–482, 534 Digital Video Interface (DVI), 388 Digital-to-analog conversion (DAC), 131–132, 388–389 Dilation methods, 23, 98, 167, 453, 459 Diopters, 273, 331, 334, 336, 343, 431, 454 Dioptric value, 46–47 Direct backscattered light, 222, 227–230 Direct slope control algorithm/matrix, 124–127, 406, 534 Discrete actuator deformable mirrors: applications, 96 characterized, 86, 111–112 defi ned, 534 Gaussian, 97 macroscopic, 89–90 wavefront corrector illustration, 102–104, 109 Discrete Fourier transform (DFT), 195, 534 Discrimination threshold, 374–375, 535 Dispersion balancing, 259, 265, 270 Displays for psychophysics: cathode ray tube (CRT), 381–385, 388, 405, 531 characterization, 383–384, 388–390 contrast resolution of, 387–388 digital light projector (DLP), 386, 404, 405, 414, 534 gamma function, 389 gamut, 382–383, 536 liquid crystal display (LCD), 381, 384–385, 539 plasma displays, 381, 385, 545 projector systems, 381, 384–386 Display stimuli, 150–151 Disposable contact lenses, 298–299 Distance vision, 41, 317 DM-WS: geometries, 178 registration, 178, 183–184 Doppler OCT techniques, 21, 259 Double-pass (DP) retinal images, 53 d-prime (d′), 374. See also Signal detection theory (SDT) Drusen, 214, 219

INDEX Dry eyes, 320 Dual Purkinje Eye Tracker, 17–18 Dynamic behavior: camera stare, 195 computational delay, 194 zero-order, 195–196 Dynamic corrections, 41 Dynamic focusing, 168–169 Dynamic range: characterized, 480–481 conventional imaging, 246 defi ned, 535 liquid crystal AO phoropter, 507 measuring spherical lenses, 481 microelectromechanical systems, 92 Shack-Hartmann wavefront sensor (SHWS), 481 wavefront sensing research, 67–68, 71–75 Elderly, light absorption in, 42. See also Aging eye/Aging process Electrostatics, 88, 95 Emmetropia, 336, 535 En face: AO-OCT retinal camera, 450 imaging, 448 scanning, 256–257, 265 Entropy (ENT), image quality metrics, 347 Epi-LASIK, 320 Epiretinal membranes, removal of, 22 Equivalent quadratic, 339, 358, 535 Equivalent width (EW), image quality metrics, 346 Error(s), types of: Abbe, 162 bandwidth, 194, 199–200, 530 budget, 160–161 calibration, 191–192, 531 convergence, 487 fitting, 192–194 measurement, 194–199 noncommon path, 409 performance, 191–192 phase, 409 refractive, 66, 104–105, 240, 331, 338, 546 transfer function, 133, 135 wavefront, 41, 158, 175, 180, 190, 200–201, 302–303, 324, 326–327, 332, 334, 338, 340–342, 397, 402–403, 406–409, 423, 432–433, 441, 449, 455–457, 459, 461, 474, 491, 505, 505–506 Euler’s theorem, 342

573

Excimer laser systems, corneal ablation: characterized, 9, 312, 321–322 clinical results, 325–326 surgery, biomechanical response, 323 Extrinsic markers, retinal imaging, 24 Eye, see Cornea; Corneal; Pupil aging process and, 40 artificial, 166 basic components, 206 biomechanical changes, 9, 322–323, 325 diffraction-limited, 442 -lens system, 336 light sensitivity, 12 movement, 40, 311, 321–322, 433, 443 peripheral optics of, 51 position, tracking with adaptive optics, 15–18 refraction, 11 tracking (in surgery), 322 wave aberration measurement, 10 wavefront sensors for, 63–79 Far point, 334–335, 339 Farsightedness, see Hyperopia Far vision, perfect correction for, 41–42 Fast Fourier transform (FFT), 190, 535 FC fiber, 165 Feedback closed-loop control system, 128 Femtosecond laser, 319 Ferroelectric technology, 91 Fiber-based OCT systems, 266 Field: angle, 50–51, 245 curvature, 46–47 scan angle, 253–254 size, significance of, 244–246, 267–268, 432 of views (FOVs), 175, 182, 469 Fifth-order: aberrations, 50 Zernike polynomials, 98 Fill factor(s), 87–88, 95, 106, 246 Finite-element analysis, 101–102, 112–113 First-order optics, 156–157, 535 Fitting errors, 192–194, 536 Fixation: implications of, 255, 272 target, 156, 246 Flap cut, effect of, 324 Flash lamp, 242–243, 399, 414 Flat-fielding, 200 Flat fi le, 176 Flexible intraocular lenses (IOLs), 305 Floaters, 218

574

INDEX

Flood illumination: AO-OCT system, 266, 447–448, 452, 461–463 characterized, 182, 238 conventional, 236, 256, 263, 266–269, 449, 451, 454, 456, 461–464, 474 OCT ophthalmoscope, 256–257, 259 retinal cameras, 95 Rochester Adaptive Optics Ophthalmoscope, 403, 413 SD-OCT system, 450 source of, 156 Fluorescein, 21, 211, 439 Fluorescent imaging, vascular structure and blood flow, 21 Flourescent markers, 24 Flying-spot lasers, 312 F-number, 168 Focal: length, 73–74 plane, 263 Focus/focusing: conventional imaging and, 242 depth of, 336–337, 347, 360, 467–469 optical alignment, 168–169 scanning laser imaging, 255 Fold mirrors, installation of, 181–182 Foucault knife-edge technique, 36 Fourier optics, 277 Fourier transform: Discrete Fourier transform (DFT), 195, 534 Fast Fourier transform (FFT), 190, 535 implications of, 104, 259, 348, 450, 487, 493 inverse, 473 linear deconvolution, 278 nonlinear deconvolution, 280 Fourth-order aberrations, 6, 46 Fovea/foveal: anatomical view, 514 avascular zone, 206 blood flow, 439 characterized, 34, 206, 215, 217, 225, 410–411 cones, human, 15 crest, 207–208 off-axis aberrations, 51 defi ned, 536 photoreceptors, 227, 334 pit, 207–208, 270, 467 refraction, 47–48 vision, 47 Frame: grabbing, 255, 426, 428–429 rate, 140, 200

Free run mode, 140–141 Free-space OCT systems, 266 Frequency domain, 351, 357 Fresnel/lenticular screens, 385 Fresnel microlenses, 75 Fried: configuration, 120–121 geometry, 178, 183–184 Full width at half height (FWHH), 259–260, 455–456, 465 Full width at half maximum (FWHM), 179, 191, 441, 443, 536 Functional customization, 317, 319 Fundus, 18, 220, 223–225, 229, 536 Gabor function, 366 Gabor pattern, 364–366, 536 Gain/gain factor, 184–185, 406, 536 Galileo, 4 Galvanometric scanner, 253, 419 Gamma function, 389 Gamut, 382–383, 536 Ganglion cells/cell layer: AO parallel SD-OCT imaging, 467–468, 470–471 characterized, 20, 23–24, 218, 261 of retina, 216 Gaussian: beam, 470 curvature, 342, 492 fit algorithm, 484 moment, 346 optics, 535 probability density function, 372 statistics, 281 Genesis-LC, 426 Geometrical optics, 205, 207 Glaucoma, 23, 224 Glial cells, 215, 218 Gram-Schmidt orthogonalization method, 35 Graphical user interface (GUI), 153 Ground-based telescopes, 7, 97 Half width at half height (HWHH), 346–347, 359 Hamming window, 197 Hanning window, 197 Haptics, 304–305 Hardware calibration, Shack-Hartmann wavefront sensing, 75–76 Hartmann spot, 169, 178. See also ShackHartmann wavefront sensor Helmholtz, 4, 236 Heterochromatic fl icker photometry, 369

INDEX High-contrast visual acuity, 303 Higher order aberrations: correction strategies, 4, 6–9, 11, 44, 49–50, 291, 294, 296–297, 302–303, 305–308, 312, 322, 325–327, 336, 358–359, 401, 462, 478, 492 surgical correction of, 307–308 High-frequency scanning, 253 High-quality eye, 345 High-resolution: camera, 140 imaging, 23–24 OCT images, 467 retinal, see High-resolution retinal imaging wavefront sensor, 7 High-resolution retinal imaging: AOSLO, 433 characterized, 63, 79 common issues of, 271–276 conventional imaging, 236–246 image postprocessing, 276–284 OCT ophthalmoscope, 236, 256–271 overview of, 235–236 scanning laser imaging, 247–255 High-speed imaging, 243 High-voltage amplifiers (HVAs), 132 Historical perspectives: aberration correction in human eye, 3–9 ocular adaptive optics, 9–24 Horizontal cells, 217 Horizontal scanner, AOSLO, 427–428, 432 Horizontal scanning mirror, 419–420 Horizontal synchronization pulse (hsync), 420, 537 hsync signal, 426, 428–429, 537 Hubble space telescope, 7 Hue, 355 Human eye, see Eye Human subjects, 502–503 Hydrophilic lenses, 295 Hyperfocal: point, 336, 537 refraction, 336–337, 358, 360 Hyperopia: characterized 4, 209, 317, 321, 323, 332 corneal laser surgery, 324 defi ned, 537 latent, 337 positive spherical lenses, 336 refractive surgery, 313 Identification threshold, 374–375, 537 Illumination beam, 223 Image, see Image acquisition; Image postprocessing; Image quality

575

flux, 190 optical alignment, 169 recording, 418 sharpening, 191–192, 457 stabilization, 18 Image acquisition, AO software: frame rate, 140 pupil imaging, 141–142 rates, 245 synchronization, 140–141 Image postprocessing: convolution, 277–278 deconvolution applications, 282–283 linear deconvolution, 278–280 nonlinear deconvolution, 280–282 overview of, 276–277 Image quality: loss of, 336 metrics, 11 Area of visibility for rMTF (AreaMTF), 350–351 Area of visibility for rOTF (AreaOTF), 351 Entropy (ENT), 347 Equivalent width (EW), 346 Light-in-the-bucket (LIB), 347 Neural contrast threshold function (CSF N), 351–352 Neural sharpness (NS), 348 Radially averaged modulation transfer function (rMTF), 350 Radially averaged optical transfer function (rOTF), 350 Square root of second moment (SM), 346 Strehl ratio, frequency domain, MTF Method (SRMTF), 351 Strehl ratio, frequency domain, OTF Method (SROTF), 351 Standard deviation (STD), 347, 374, 377, 379 Visual Strehl ratio, frequency domain, MTF Method (VSMTF), 352 Visual Strehl ratio, frequency domain, OTF Method (VSOTF), 352, 359 Visual Strehl ratio, spatial domain (VSX), 348 Volume under neurally weighted OTF (VNOTF), 352 Volume under OTF (VOTF), 352, 359 neural mechanisms of, 11 retinal, 296 significance of, 143, 200, 335, 357, 359–360 Imaging, see Conventional imaging pipeline, 140

576

INDEX

time, 276 wavelength, 210 Impedance, 257–258, 485 Incoherent: light, retinal imaging research, 6 subsystem, SD-OCT system, 450–451 Independent influence, 101–102 Index of refraction, 230, 438, 469 Indiana AO ophthalmoscope, 239, 241–243, 246, 266, 460 Indiana Eye model, 354–355 Indiana schematic eye, 270 Indiana University AO-OCT system: AO performance, 455–461 conventional flood-illuminated imaging, 461–463 description of, 448–453 experimental procedures, 453–455 parallel SD-OCT imaging, 463–474 significance of, 447–448 Indocyanine green dye, 210–211 Influence: functions, 101–102, 122, 149–150 matrix, 126, 184 In-focus reflection, 241 Infrared beams, 166 Infrared (IR) light, 44 Inner and outer photoreceptor segments (IS/OS), 215–216, 467–468, 470–472, 537, 543 Inner limiting membrane (ILM), 216, 466 Inner layers: nuclear (INL), 216, 218, 467–468, 471 plexiform (IPL), 216, 218, 467–468, 470–472 Integrator, defi ned, 184 Interchangeable mirrors, alignment and, 167 Interference: AO parallel SD-OCT imaging, 473 effects, 340 fi lter (IF), 242–243 signature, 259 Interferogram, 94, 300 Interferometer: AO system integration, 176 in optical alignment, 170–171, 173–174 Shack-Hartmann wavefront sensing, 78–79 Interferometry, 5, 85, 256 Intermediate image location, 186 Internal aberration: defi ned, 537 measurement of, 37 Internal ocular optics, 35–38

Internal optics, aging process, 43 Intraocular lenses (IOLs): aberrations, types of, 305–306 characterized, 34 defi ned, 304, 537 flexible, 305 higher order aberrations, 306–308 implanted, 39, 306 lens decentration, 307 manufacturing process, 308 phakic, 321 polymerization, 306 rigid, 305 Intraocular scatter, 42, 537 Inverse Fourier transformation, 473 Iris: custom-correcting contact lenses, 303 as point target, 165 Irradiance, 277 Irregular astigmatism, 4 Isoplanatism, 245 Jackson crossed cylinder, 333 Just noticeable difference (JND), 374 Keck telescope, 456 Kepler, 4 Kerataconus, 9, 69, 74, 300–301, 303, 538 Keratoplasty, 9, 301 Kinematic: placement techniques, 164 principles, 167 Knife edge, optical alignment technique, 169 Kolmogorov distribution, 52 Krypton flash lamp, 242–243, 414 Lambertian scatter, 212 Lamina cribrosa, 209, 228 Laminar flow, 211 Laplace transforms, 131, 194, 408 Laser-assisted epithelial keratoplasty (LASEK), 320, 325–327, 538 Laser(s): ANSI standards, 419 corneal ablations, 311–327 diodes, 243, 451, 484–485 illumination, 219 low coherence, 419 propagation, 157 ray tracing, xviii, 63, 65–66 surgical procedures, see Laser-assisted epithelial keratoplasty (LASEK); Laser in situ keratomileusis (LASIK); Laser refractive surgery

INDEX Laser in situ keratomileusis (LASIK): applications, 325, 327 corneal ablation, 311, 319, 323 defi ned, 538 higher order aberration correction, 307–308 hyperopic treatment, 316–317 myopic treatment, 314–315, 317–318 research, 314–316 wave aberration, 127 Laser refractive surgery: applications, 73, 79, 98, 311 basics of, 312–317 customized, 63 Lateral chromatic aberration, see Longitudinal chromatic aberration (LCA) Lateral misalignment, 442 Lateral resolution, 265, 270 Lawrence Livermore National Laboratory (LLNL), 9, 478, 494 Layout, 164 Lead-magnesium-niobate (PMN) actuators, 89–90 Least-squares: algorithm, 127 fitting, 339 method, 487 Lens: cataract surgery, 304 contact lenses and, 295–298 crystalline, 33, 36, 40, 43, 63, 336 displacement of, 44 intraocular lenses (IOLs), 304–305 natural, 305–306 optical power of, 306, 308 optical zone of, 299 polarization impact, 53 prescriptions, 332–334 schematic diagram, 206 transparency of, 55 Lenses, cylindrical, 180, 273 Lenslet(s), see Lenslet arrays AOSLO, 420–422 centroid measurement, 142, 179 characterized, 66 configuration of, 120 conventional imaging, 241 defi ned, 539 high-density configuration, 121–122 Rochester Adaptive Optics Ophthalmoscopes, 400 Shack-Hartmann wavefront sensing, 67–71, 75, 78–79

577

Lenslet arrays: characterized, 238 calibration process, 77–78 focal length of, 73–74 Lenticular aberrations, 239 Lesions: retinal, 208, 220 solar retinopathy and, 440 Leukocytes, 11 Life span, retinal image quality, 43 Light: absorption, 42 angular dependence of, 12 budget, 271–272 delivery optics, 418–419 detection, 262–264, 418 distribution, polarization effects, 53–54 scatter, see Light scatter/scattering sources, 242–244, 267 Light Adjustable Lens (LAL), 306 Light-emitting diodes (LEDs), 391 Light-in-the-bucket (LIB), 357 Light scatter/scattering, implications of, 55, 206–207, 420 Limbal junction, 295, 539 Linear deconvolution, 278–280, 284 Linear inversion fi lter, 279–280 Linear sum model (LSM), 102 Line of sight (LOS), optical alignment, 165, 167–168, 174, 181, 297 Lipperhey, Hans, 4 Liquid crystal AO phoropter: AO assembly, 491–492 beacon selection, 484–485 calibration, 492–502 human subject results, 502–506 integration, 491–492 software interface, 489–491 system performance, 492–502 testing procedures, 492–502 troubleshooting, 491–492 wavefront corrector selection, 485–486 wavefront reconstruction, 486–489 wavefront sensor selection, 478–484 Liquid crystal display (LCD): characterized, 75 defi ned, 539 monitors, 381, 384–385 PAL-SLM, 485–486 projectors, 386, 539 Liquid crystal spatial light modulators (LC-SLMs), 34, 88, 90–91, 99, 176, 477, 539 Liquid crystal technology, 86

578

INDEX

Littrow’s angle, 449 LogMAR: defi ned, 539 visual acuities, 302–303, 368, 539 Longitudinal chromatic aberration (LCA), 44–45, 51, 108, 270–271, 346, 354–355, 357, 404, 540 Long-range scatter, 225 Long wavelength sensitive (L) cones, 15, 16, 19, 215, 217, 226, 284 Lookup table, 493–494, 508 Loop gain, 195, 200 Low coherence laser sources, 419 Low-contrast visual acuity, 303 Lower-order aberrations, 191, 307, 320, 401, 477 L-square applications, 166 Luminance, 335, 355, 357, 365–366, 368–369, 381, 540 Macromers, photosensitive, 306 Macula/macular: anatomical view, 514 bull’s-eye lesion in, 20 characterized, 206, 209, 211 degeneration of, see Macular degeneration disease, 224 pigment, 217–219, 225 Macular degeneration, 210–211, 220, 224–225 Macwave, 456, 461 Magnetic resonance imaging (MRI), 209 Magnification telescope, 158 Manifest refraction, 311, 540 Manifest refractive spherical equivalent, 315, 540 Maréchal approximation/criterion, 158, 192, 200 Matched-fi lter correlation, 178 Material dispersion, 107 MATLAB, 151, 380, 405, 456–457, 461 Maximum a posteriori (MAP), 280 Maximum likelihood (ML), 280–281 Maximum permissible exposure (MPE), 23, 245, 398, 419 Maxwellian-view optical systems, 390 Measurement error, 194–199, 540 Measurement sensitivity: defi ned, 540 Shack-Hartmann wavefront sensing, 67, 71–75 Meiosis, 215 Melanin, 66, 214, 218–220 Melanosomes, 213

Membrane mirrors, 88, 96, 109–112, 540 Method of adjustment, 375–376, 540 Method of constant stimuli, 378–379, 541 Michelson: contrast, 377 interferometer, 258 Microaneurysms, 22, 438–439 Microelectromechanical system (MEMS) mirrors: AO software applications, 152 applications, 96 characterized, 92–95, 157, 478 deformable, 10 scanning laser imaging, 255 Microelectromechanical system (MEMS) technology, 86, 176, 541 Microflashes, detection of, 19 Microglia, 215, 218 Microkeratome laser, 319, 324 Middle wavelength sensitive (M) cones, 15, 16, 19, 215, 217, 226, 284 Midpoint, out-of-focus planes, 168 Mirror(s), see specifi c types of mirrors AOSLO, 425–426 bimorph, 88, 91–92, 96–97, 530 control of, see Mirror control curved, 240 diameter, wavefront correctors, 112 OCT ophthalmoscope, 258 raster scanning, 418 reflective wavefront correcter and, 85–86 scanning laser imaging, 253, 255 Mirror control: algorithm, 421–423 AOSLO, 430 significance of, 151 Misalignment: implications of, 158–159 pinhole, 442–443 sources of, 161 Misregistration, 497 Mitosis, 215 Modal correctors, 88 Modal wavefront control algorithm, 127, 541 Modulation transfer function (MTF): aging eyes, 42 characterized, 42, 277–278, 281, 349–350, 357, 367, 505–506 defi ned, 541 human eye, 367 implications of, 277–278, 281, 349, 357, 505–506 off-axis aberrations, 48, 50–51 polarization effects, 53

INDEX Monochromatic: aberrations: characterized, xvii, 4, 6, 35–40, 45, 238–239, 358–359 influential factors, 33–34 interaction with chromatic aberrations, 45 measurement of, 33 light, 9, 15, 411 metrics, 358 refraction, estimation from aberration map: equivalent quadratic, 339 methodology evaluation, 358–359 numerical example, 353–354 overview, 327, 337–338 virtual refraction, 339–353, 355 retinal image quality, 51 Monostable multivibrator, 428, 541 Monovision, 317 Müller cell matrix, 23, 215, 218 Multiply scattered light imaging: characterized, 222–223 contrast from, 227–230 Multiframe blind deconvolution (MFBD), 281–282, 284 Multimode laser diode, 451 Multimode step index optical fiber, 451 Multiphoton imaging, 24 Multiplexer, 428, 541 Murcia optics lab study, 52 Mutated genes, 19 Myopia: characterized, 98, 209, 293, 314–315, 317–318, 321, 323, 332, 356 correction of, 4, 294 defi ned, 541 monochromatic refraction, 358–359 negative spherical lenses, 336 polychromatic light, 356 refractive surgery, 313–314 treatment strategies, 323, 325 Myopic deconvolution, 284 N-alternative-forced-choice (NAFC) procedure, 376, 529, 542 Narrow bandwidth, 242 National Science Foundation Science and Technology Center, xviii National Television Subcommittee (NTSC), 426, 542 Near-infrared: illumination, 210, 212, 462 imaging, 241

579

source, 399 spectrum, 267 Nearsightedness, see Myopia Near-ultraviolet light, 306 Negative magnification, 158 Neovascular membrane, 221–222 Nerve fiber bundles, 227–229 Nerve fiber layer: AO parallel SD-OCT imaging, 467–469, 471, 473–474 AOSLO, 436 retinal, 219–220, 227, 229, 261 Nervous system, blur adaptation and, 11 Neural contrast threshold function (CSF N), 351–352 Neural sampling, 335 Neural sharpness (NS), image quality metrics, 348 Neural vision system, 334 Neurosensory retina, 220 Neutral density fi lters, 464 Night driving, 321 Noise: AO parallel SD-OCT imaging, 473 AO system performance, 196 AOSLO trace, 17–18 conventional imaging, 241, 246 dark, 178, 246, 484 error measurement, 198–199 linear deconvolution and, 284 Shack-Hartmann, 483–484 speckle, 270 Noise-alone distribution, 371–373, 377 Noncommon path: aberrations, 159, 185, 413, 457, 542 errors, 409 lengths, AO-OCT system, 453 Nonlinear deconvolution, 280–282 Nonnulling operation, 424, 436 Nuclear layer, of retina, 216, 218, 261 Numerical aperture (NA), 23, 241, 259, 451 Nyquist criterion, 192 Nyquist sampling: limit, human eye, 367 theorem, sampling rate, 487 Objective refraction, 360 Object irradiance, 278–282, 284 Oblique astigmatism, 46 Oblique: coma, 47 effect, 366

580

INDEX

Ocular aberrations, see Aberrations Ocular adaptive optics, see Adaptive Optics (AO) system Ocular hazards, 271 Ocular media, 55, 83, 389 Ocular optics, aberration-free, 34 Off-axis: aberrations: characterized, 46–47, 239–240, 426 chromatic, 48–51 correction of, 51 monochromatic, 48–51 monochromatic image quality, 51 peripheral refraction, 47–48 field point, 157 SLD illumination, 399 Offl ine alignment, 170–174 On-axis: correction, 44 field point, 157 wavefront error, 190 Open-loop: AO control systems, 128–129, 424, 432 bandwidth, 133–134 control system, 128–129, 542 system transfer function, 408 Ophthalmoscope: conventional, 236 development of, 5, 8 OCT, see Optical Coherence Tomography (OCT) SLO, see Scanning Laser Ophthalmoscope (SLO) Ophthalmoscopy, conventional, 20 Optic: disk, 209 nerve: fiber layer, of retina, 216 head, 207–210, 218, 222, 226, 228–229, 542 schematic diagram, 206 Optical alignment: common practices, 163–170 components of, 157–158, 180–182 misalignment penalties, 158–159 offl ine alignment, sample procedure, 170–174 optomechanics, 159–163 Optical: axis: angle targets, 165–166 conventional imaging, 243 defi ned, 543 establishment of, 164

measurement of, 294 optic alignment onto, 167–170 point targets, 164–165 rough targets, 165–166 coherence tomography (OCT). See also Optical Coherence Tomography (OCT) density, 225, 389 focus, 334 path: difference (OPD), 172, 321 length, 35, 321, 464 OCT ophthalmoscopes, 265 quality metrics, 360 quality, off-axis, 48 slicing, 424 transfer function (OTF), 346, 349, 357, 359, 543 zone (OZ), 323 Optical Coherence Tomography (OCT): AO-OCT ophthalmoscopes, basic layout of, 264–266 resolution, 23 AO parallel spectral-domain OCT, 466–475 characterized, 5, 83–84, 95, 229–230, 236, 256–257, 447 chromatic aberrations, impact of, 268–271 dispersion balancing, 259 Doppler, 21, 259 fiber-based, 259 field size, 267–268 free-space, 259 imaging light source, 267 light detection, 262–264 optical components, 266 parallel spectral-domain (SD-OCT), 447, 452, 454, 463–465 phase-sensitive, 259 polarization sensitive, 259 principle of operation, 257–259 resolution limits, 5, 236, 259–263 speckle, impact of, 268–271 spectral-domain (SD-OCT), 89, 256–257, 259–260, 262–266, 268–270, 549 spectroscopic, 259 time-domain, 256–257, 259, 261–264, 266, 454 wavefront sensing, 266–267 Optical Society of America: functions of, 334 Standards for Reporting Optical Aberrations, 511–527

INDEX Optimal closed-loop feedback control, 128 Optomechanics, in AO system: adjustment fi xtures, 162 design considerations, 155 fundamentals of, 159–161 hardware selection, 161 immobilizing subjects, 162–163 mechanical isolation, 162–163 moving loads, 162 stray light, avoidance strategies, 163 thermal effects, 162 Ora serrata, 208 OSLO ray tracing software, 240 Outer limiting membrane, of retina, 216 Outer layers: nuclear (ONL), 216, 218, 467–468, 471– 472, 474 plexiform (OPL), 216, 218, 467–468, 471–472 Outer segment, 215, 218, 225, 543 Out-of-focus reflection, 241 Out-of-plane scatter, 224 Overrefraction, 275 Parallax, 156–157 Parallel aligned nematic liquid crystal spatial light modulator (PAL-SLM), 485, 543 Parallelism, 166 Paraxial: curvature matching, 339 optics, 535 rays, 333 Partial data, 145 Patient stability, 272 Peak sensitivity, 389 Peak-to-valley (PV): difference, 341, 499 errors, 101, 110, 180 wave aberrations, 97–101, 103, 106 wavefront error, 180 Penetrating keratoplasty, 301 Perceptual weighting, 335 Peripapillary region, 228–229 Peripheral light, 323 Phacoemulsification, 304 Phakic: defi ned, 543 intraocular lenses, 321 Phase: delay, spatial light modulator (SLM) response, 495 diversity, 192 errors, 350 function, testing SLM stroke, 176

581

modulation, 91, 485 plate, 293, 303 resolution, 106 retrieval, xviii, 192 -sensitive OCT, 259 transfer function (PTF), 349, 544 wrapping, 92, 100, 106–108, 341, 493–495, 500, 544 Phoropter, adaptive optics, 9–10, 544. See also Liquid crystal AO phoropters Photometers, 389, 544 Photomultiplier tube (PMT): characterized, 5, 255, 418, 426, 428–429, 432 defi ned, 544 Photons: conventional imaging, 245 flux, 166 multiphoton, 24 Photopigments: characterized, 15, 214, 217 defi ned, 544 distribution of, 217, 225 spectral sensitivities, 227 Photoreceptor(s): characterized, 5–6, 11, 206, 215 cones, see Cones defi ned, 544 degeneration, 19–20 distribution of, 225–226 inner segments, 215–216, 467–468, 470–472, 537 mosaic, 16, 20, 24, 220, 411 mosaic, cone-rod dystrophy, 21 optics, retinal imaging, 11–14 outer segments, 215, 218, 225, 467–468, 470–472, 543 rods, see Rods sampling, 51, 367 trichromatic mosaic, 16 Photorefractive keratectomy (PRK), 311, 320, 323, 325, 544 Piezoelectric: constant, 110 technology, 86 Pinhole pupil, 45 Pinning error, 102 Piston-only mirrors: characterized, 87, 111 segmented mirrors, 97–98, 106–107, 112, 544 Piston/tip/tilt mirror: functions of, 103, 107–109 segmented mirrors, 87, 94–95, 97, 111–112, 545

582

INDEX

Pixel(s): in AO system assembly and integration, 177–178 architecture, 246 CCD, 179 conventional imaging, 245 OCT ophthalmoscopes, 268 psychophysics, 383–384 sampling density, 245 SD-OCT system, 449–450 Shack-Hartmann wavefront sensor (SHWS), 481–482 Plane reflector, AOSLO, 437–438 Plasma display, 381, 385, 545 Plate scale, measurement of, 177, 179–180 Plexiform layer, of retina, 216, 218 Plexiglas contact lens, 300–301, 303 Point-diffraction: AO-OCT ophthalmoscopes, 266 interferometer, 265, 453, 455 Point source, 190 Point spread function (PSF): axial, 258–260, 465, 530 AO-OCT system, 456–458 AOSLO, 423, 431, 441–442 chromatic aberrations, 354–356 convolution and, 277 corneal aberrations, 37–38 deconvolution and, 280–282 defi ned, 545 diffraction-limited, 245 image quality metrics, 345–349 implications of, 11–12, 104, 189–190, 237, 301 monochromatic, 357 Rochester Adaptive Optics Ophthalmoscope, 407 scanning laser imaging, 255 selective correction, 151 wavefront correctors, 102 Point targets, optical axis: CCD, 165 defi ned, 164 on image plane, 166 iris, 165 machined targets, 164–165 optical fiber, 165 wire crosshair, 165 Polarimetry, 54, 228 Polarization: AOSLO, 443 imaging, 24 impact on ocular aberrations, 34, 53–55, 229–230

-sensitive: imaging, 91 OCT, 259 Polychromatic: light, 360 optical transfer function, 357 refraction: characterized, 354–356 clinical, 355 grating image metrics, 357 methodology evaluation, 359–360 point image metrics, 357 wavefront metrics, 356–357 Polygon scanner, 253 Postoperative healing process, 307 Postreceptoral sampling, 51 Postsurgical eyes, visual acuity, 303–304 Power: rejection curve, 459–460 spectra: AO parallel SD-OCT imaging, 471–473 defi ned, 277, 545 temporal, 196–198, 409, 458–459, 461 spectral density (PSD), 198–199, 545 vector, 334, 339, 341–342 Powered mirrors, installation of, 181 Precision optics, 34 Presbyopia, 305, 317, 545 Primates, retinal imaging, 24 Prisms, 331 Projector systems: characterized, 381, 384–386 CRT, 386 digital light (DLPs), 386, 404, 414 LCD, 386 Prototypes/prototyping, 34 Pseudo-code, 145–146 Pseudophakic eye, 304, 306, 546 Psychometric functions, 377–378, 546 Psychophysical criterion, 358 Psychophysical function, 364, 546 Psychophysical methods: constant stimuli, 378–379 forced-choice, 376 implications of, 156, 334–335 method of adjustment, 375 staircase, 379–380 threshold, 370–375 Psychophysical tests, liquid crystal AO phoropter, 507, 546 Psychophysics: characterized, 140, 363–364, 391 contrast sensitivity function (CSF), 364–367

INDEX defi ned, 546 displays, see Displays for psychophysics psychometric functions, 377–378, 546 psychophysical functions, 363–364 methodologies, see Psychophysical methods sensitivity, 371 stimulus display, Rochester Adaptive Optics Ophthalmoscope, 404–405 threshold, 370–371 yes/no procedure, 371–372 PsychToolbox software, 380, 388 Pupil: alignment, 255 artificial, 418 decentrations, 321 defi ned, 156, 546 diameter of, 243, 259 dilation of, 98, 453, 459 in the elderly eye, 42 entry point, 66 fraction, 343–344 fully dilated, 23 glow in, 5 imaging with AO software, 141–142 light passing through, 4–5, 65 magnification of, 74–75 measurement of, 143 off-axis aberrations, 48–50 optical alignment, 169–170 photoreceptor optics, 12–13 polarization effects, 55 prism differences, 45 reducing size of, 101 retinal imaging, 212 scanning laser imaging, 253 schematic diagram, 206 Shack-Hartmann wavefront sensing, 79 size of, 34, 83, 95, 98, 104–105, 107, 110, 112–113, 237, 271, 336 stabilization of, 443 tracking system, 152, 443 visual acuity measurement process, 303 wavefront, 182 wavefront sensor research, 63, 66–67 Purkinje, 5 Pyramid sensing, 36, 85 Quantitative imaging, 218 Quantix:57, 449 Quantum efficiency (QE), 241, 246, 263 QUEST staircase procedure, 379–380 Radially averaged modulation transfer function (rMTF), 350

583

Radial Zernike order, 7, 518–522 Radiation, energy density of, 425 Radially averaged optical transfer function (rOTF), 350 Raster scanning: characterized, 5, 253–254 mirrors, 418–420 Rayleigh: criterion, 341 range, 157, 469–470 resolution limit, 245 scatter, 212 Ray tracing software, 240, 266 Real-time AO, 441 Reconstruction: algorithm, calibration process, 76–77, 127 matrix, 124, 193, 423 wavefront, 180–181, 486–489, 504 Zernike matrix, 207 Zernike mode, 123–124, 128 Reconstructor, 193 Red blood cells, 23 Red, green, and blue (RGB) channels, 384– 386, 388–389 Red-green color blindness, 19 Redundancy, 160 Reference: axis selection, OSA Standards, 513–515 beacon, 156 beam, 263 centroids: AO system gain and, 184–185 calibration of, 185–186 characterized, 159, 181, 184, 192 Reflectance, 5, 66, 241–242 Reflectivity, 112, 259 Reflectometry, 213 Reflex tearing, 298 Refraction: automated, 9 clear vision range, 336–337 conventional, 84, 212, 272–276, 358, 360 far point, 334–335 goal of, 334–337 impact of, see Refractive methodology evaluation, 358–360 monochromatic, 339–354 off-axis aberrations, 51 peripheral, 47–48 polychromatic, 354–356 by successive elimination, 335 virtual, 339–353 Refractive: aberrations, 332

584

INDEX

correction, 291, 331–332 errors, 66, 104–105, 240, 331, 546 index: aging process and, 42–43 corneal aberrations, 35 electronic control of, 88 statistics of aberrations, 52 surgery: applications, 34, 299, 326–327 defi ned, 547 intraocular lenses (IOLs), 307–308 wavefront-guided, 9, 38–39 Refractometer, xviii, 53, 63, 65 Regions of interest, 142 Registration: defi ned, 547 DM-to-WS, 183–184 spatial light modulator (SLM), 496–497, 499 Relay, 77–78, 156, 547 Relay telescope, 157–158, 170–174, 181–182, 449 Remainder lens, 35 Residual aberration, 133–134 Resolution: axial, 5, 23, 256, 267, 270, 434–438, 441, 463–467, 475 contrast, 387–388 conventional imaging, 237 defi ned, 191 high-: camera,140 imaging, 21–24, 467 retinal imaging, 63, 79, 235–284, 433 wavefront sensors, 7 improvement strategies for, 23–24 lateral, 265, 270 limits, 237, 245, 249, 259–262 OCT ophthalmoscopes, 259–262 in optical alignment, 160 phase, 106 spatial, 90, 129–130, 381, 383 spurious, 350–351 temporal, 130, 381, 383 transverse, 5, 21, 260–262, 448 ultra-high, 260, 263, 266 VGA, 90 volume, 418 XGA, 90 Resonant frequency, 99 Resonant scanner, 419 Retina, see Retinal angular tuning properties, 12 blood supplies, 209–210 contrast from scattered light, 227–230

cross section diagram, 212 defi ned, 205, 547 fundus, 210–218 images/imaging, see High-resolution retinal imaging; Retinal imaging; Resolution, high-, retinal imaging light: distributions on, 15 scattering, 220–227 main layers, diagram of, 216 neural, 212, 215 polarization, 227 shape of, 206–209 spectra, 218–220 stabilization of, 443 visual angle, 205 Retinal: cameras, 110 degeneration, 20–21 densitometry, 16 disease, 11, 18–21, 24 eccentricity, 33–34, 51 hazard, 245 illuminance, 156, 245, 370 image/imaging, see Retinal image pigment epithelium (RPE), 214–218, 467–471, 547 vein occlusions, 22 Retinal image: adaptive optics, 6–8, 11–24 AO-OCT experiments, 454–455 AO software applications, 140, 150 blood flow, 439 characterized, 205 conventional, 236–246 deconvolution of, 15 diffraction-limited, 41 double-pass, 36, 53 high-contrast, 225 high-resolution, see High-resolution retinal imaging; Resolution, high-, retinal imaging historical perspectives, 236 microscopic in vivo, 5–6, 11 one-shot, 150 peak-to-valley (PV) errors, 101 quality: influential factors, 33–34, 36, 42–44, 355–357 limitations on, 44 polarization state and, 53 Rochester Adaptive Optics Ophthalmoscope, 403–404 size, 354

INDEX Retinopathy, solar, 440 Retinoscopy, 207 Retroreflection, 214 Reversal, defi ned, 379 Rhodamine dextran, retinal imaging, 24 Rigid gas-permeable (RGP) lens, 293–295 Rigid intraocular lenses (IOLs), 305 Rochester Adaptive Optics Ophthalmoscope: characterized, 128, 239–244, 246, 397 control algorithm, 405–407 optical layout, 398–405 retinal image quality, improvement in, 409–410 schematic diagram of, 398 system limitations, 412–414 visual performance, improvement in, 410–412 wavefront correction performance, 406–409 Rods: characterized, 215–217 cone-rod dystrophy, 20–21 photoreceptors, 225–227 Root-mean-square (RMS): conventional imaging, 246 defi ned, 547 fitting errors, 192–194 phase errors, 190 wavefront, 68, 95, 110 wavefront correctors, 90, 102–104 wavefront error: AOSLO, 423, 432–433, 441 customized vision correction devices, 297, 302–303 customized corneal ablation, 324, 326–327 convergence of, 505–506 high-resolution retinal imaging, 238 history, trace of, 147–148 Indiana University AO-OCT System, 449, 455–457, 459, 461, 474 in normal young subjects, 34 liquid crystal AO phoropter, 489, 491, 502 misalignments, 158 refraction and, 332, 334, 338, 340–342 Rochester Adaptive Optics Ophthalmoscope, 397, 402–403, 406–409 wavefront sensor, 483–484, 499, 501–502 Saggital: focus, 46–48 plane, 46 Sampling frequency, 130, 198

585

Sandia National Laboratories, 9 Satellite-tracking telescopes, 8 Saturation, 355 Scalar: diffraction theory, 103 metrics, 340, 342 Scanning, see specifi c types of scanning AOSLO system, 425–426 lasers, 312 mirrors, AOSLO, 432 slit refractometer, xviii slit wavefront sensors, 321 Scanning laser imaging: basic layout of AOSLO, 249 compensation, 252 confocal, resolution limits, 249 frame grabbing, 255 light: delivery, 251–252 detection, 254–255 path, 249–251 overview, 247–248 raster scanning, 253–254 SLO system operation, 255 wavefront sensing, 252 Scanning laser ophthalmoscope (SLO): architecture, 256 characterized, 7, 95, 417–418, 447, 475 closed-loop AO system, axial sectioning, 423–424 custom, 92 development of, 5 high-magnification, 24 light delivery, 251–252, 419 mirror control algorithm, 421–423 raster scanning, 253, 419–420 retinal imaging, 211, 221–223, 229 system operation, 255 wavefront: compensation, 421 sensing, 252, 420–421 Scars, corneal, 63 Scatter/scattering: contrast and, 227–230 impact of, 34, 55, 206–207, 220–227, 245–246, 331, 420 intraocular, 55, 537 Lambertian, 212 long-range, 225 multiply, 222–223, 227–230 out-of-plane, 224 single, 222 Scheiner, Cristoph, 4 Science camera, 85, 241–242, 246

586

INDEX

Scintillation, 190, 239 Sclera: characterized, 212–213 defi ned, 547 RGP lenses, 295 Scripting, 151, 153 SD-OCT imaging, AO parallel: characterized, 466–474 defi ned, 447 sensitivity and axial resolution, 463–466, 475 Search box, 142, 548 Second-order: aberrations, 97, 99–101, 104, 106–108, 112, 337, 358 Zernike coefficients, 331, 334, 339 Zernike terms, 104 Segmented corrector, 86–87, 548 Segmented mirrors, 87–88, 97–98, 111, 548 Seidel: approximation, 46 power-series expansion, 332 Selective correction, 151 Senile miosis, 42 Sensitivity, defi ned, 365, 548 Sensor(s), functions of, see specifi c types of sensors Sensory response strength probability density function, 373 Shack-Hartmann wavefront sensor (SHWS): aberration detection, 291 aliasing, 507 AO-OCT system, 266, 452–453, 455–458, 460 AOSLO, 420, 422 characterized, xviiii, 6–8, 10, 36–37, 50, 64–68, 85, 89, 156, 237, 477–483 configuration of, 120 conventional imaging, 240–241 corneal ablation, 312, 321 crossover, 71 defi ned, 548 detector, 461 development of, 84 double spots, 507 dynamic range, 71–75, 480–481, 507 fitting errors, 193 hardware calibration, 75–76 limitations of, 71 measurement sensitivity, 71–75 microelectromechanical systems (MEMS) mirrors and, 92, 95 optimization strategies, 68–75 qualification of, 177

registration, 496–498 slope: displacement, 405–406 measurement, 420 spatial light modulation (SLM), 503, 507 spot pattern, 178, 241 spot image, 177, 179, 241, 400–401, 411, 484, 488–490, 504 time stamping of measurements, 460–461 Sharpness: image sharpening strategies, 191–192, 457 loss of, 335 neural, 348 Shear plate, in optical alignment, 170 Shift-and-add technique, 433 Short-burst images, 470, 472–473 Short wavelength sensitive (S) cones, 15, 16, 215, 217, 226 Shutter triggers, 151 Signal detection theory (SDT), 371–374, 377 Signal-plus-noise distribution, 371–373, 377 Signal-to-noise: fi lter, 279 ratio (SNR), 73, 135, 263, 420, 428, 441, 469, 473–474 Silicon nitride membrane, 110 Sine-wave gratings, 492 Single: cones, in vivo studies, 11–12 -image acquisition, 268 -mode fiber, 179–180, 182, 186 scattering, 222 Singular value decomposition (SVD), 68, 126, 145 Sinusoidal scanner/scanning, 253, 427 Sixth-order aberrations, 50 Skiagrams, 48 Slope: of beam path, 165 direct, 124–127, 406 influence function, 124–125 influence matrix, 184 map, 340 measurement, 184 Shack-Hartmann, 405–406, 420 vector, 123, 126, 145 wavefront: implications of, 64, 66, 71, 142–144, 149, 341–342 measurement, 142–145 Snake, photoreceptor cells, 5 Snellen visual acuity, 477 Snell’s law, 535 Soft contact lenses, 293–295, 297–298, 302–304

INDEX Software: applications: aberration correction, 149–150 aberration recovery, 144–149 application-dependent considerations, 150–151 AO loop, 139 AOSLO, 429–431 image acquisition, 140–142, 151, 245 liquid crystal AO phoropter, 488–491 overview, 139–142, 151–153 psychophysical experiments, 387–388 ray tracing, 240, 266 retinal imaging, 140, 150 wavefront slope measurement, 142–144 CAD, 159–160, 164–165 calibration, Shack-Hartmann wavefront sensing, 75–76 control, 489 design, 240, 425 diagnostic, 489 PsychToolbox, 380, 388 simulation, 490 Solar eclipse, 440 Southwell: geometry, 183 configuration, 120 Spatial: coherence/coherent light, 242, 269, 451 control command, wavefront correctors: control matrix for direct slope algorithm, 124–127 modal wavefront correction, 127 wave aberration generator, 127–128 fi ltering, 348 frequency, see Contrast sensitivity aberrations, 92 AO parallel SD-OCT imaging, 472 characterized, 278, 282, 350, 365, 367 cutoff (SFcMTF), 350–351 distribution, 280 spectrum, 335 homogeneity, 384, 389 independence, 384, 389 light modulator (SLM): characterized, 75, 175, 485–486 closed-loop operation, 499–502 defi ned, 478, 548 nonlinear characterization, 493 phase modulation, 493, 503 phase-response, 493 time delay, 505 wavelength-dependent, 506 phase, 349

587

resolution, 90, 129–130, 381, 383 vision, 366 Spatially resolved refractometer, 63, 65 Speckle: AO-OCT system, 451 AO parallel SD-OCT imaging, 469–470, 473, 475 defi ned, 548 high-resolution retinal imaging, 23, 242 noise, 241, 419, 448, 484–485 OCT ophthalmoscopes, 268–271 Spectacles, 3–4, 7–8, 45, 83, 294, 332 Spectral: absorption, 218 bandwidth, 23, 90, 270 constancy, 384 detection, 256 -domain OCT, 89, 256–257, 259–260, 262–266, 268–270, 447, 549 efficiency, human eye, 365, 368 leakage, 197 power distribution, 382 Spectralon, 431 Spectrometer, 259 Spectroradiometers, 390, 549 Spectroscopic OCT, 259 Specular reflector, 437 Sphere, characterized, 293–294, 454, 478, 549 Spherical: aberration, 35, 41, 291–293, 296–297, 302, 306–308, 322–324, 338, 477 -equivalent power, 333, 549 refraction, 46 trial lenses, 180 Spherocylindrical: correction, 105 defi ned, 549 lenses, 4, 294, 332–333 refractive correction, 338 refractive error, 335 Spot: array pattern, Shack-Hartmann wavefront sensing, 76–77 defi ned, 177, 549 displacement, 64–65, 71–73, 75, 144 position, 478 size, 240, 321 Spurious: aberrations, 303 resolution, 350–351 Square lenslets, 120 Square root (SR), image quality metrics, 346 SRMTF, 351 SROTF, 351

588

INDEX

Staircase method, 379–380, 549 Standard deviation (STD), 347, 374, 377, 379 Starfi re Optical Range, 8 Static corrections, 34, 41 Stellar speckle interferometry, 5 Stiles-Crawford effect, 12, 14, 45, 217, 244, 403, 549 Stratus OCT3 images, 260, 454, 466–468 Stray light control, 179 Strehl ratio, 34, 102, 104–109, 158, 189–190, 293–294, 303, 347, 351, 402, 423, 549 Stroke: bimorph mirrors, 91–92 defi ned, 549 deformable mirrors, 90, 98, 158 microelectromechanical systems (MEMS) mirrors, 93–94 Rochester Adaptive Optics Ophthalmoscope, 397, 414 scanning laser imaging, 255 wavefront correctors, 113 Structured illumination, 23 Sturm interval, 46–47 Subaperture: DM-to-WS registration, 183–184 fi rst-order optics, 156 liquid crystal AO phoropters, 478, 481, 490–491, 500 monochromatic refraction, 344–345 Shack-Hartmann wavefront sensing, 71–72, 75 wavefront sensor, 177, 180 Superluminescent diodes (SLDs): characterized, 448 high-resolution retinal imaging, 237, 241–243 liquid crystal adaptive optics phoropter, 484–485, 504 OCT ophthalmoscope, 267–268, 270 Rochester Adaptive Optics Ophthalmoscope, 398–399, 401, 414 Supernormal vision, 478 Surgery, see specifi c surgical procedures Surgical microscope, 22 Symbol Table, 553–564 Symmetric aberrations, 297 Synaptic layer, 261 Synchronization, 140–142, 150–151 System integration, AO systems: assembly of AO system, 181–182 boresight FOVs, 182 control algorithms, 184–185 control matrices, generation of, 184

deformable mirror (DM), qualification of, 175–171 DM-to-WS registration, 178, 183–184 influence matrix, measurement of, 184 overview, 174–175 reference centroid calibration, 185–186 slope measurement, 184 system gain of the control loop, 184–185 wavefront: error measurement, 175 reconstruction, 180–181, 486–487 sensor qualification, 177–180 Tangential: focus, 46–48 plane, 46 refraction, 48 Tecnis lens, 307 Telescope applications, 4, 7, 404. See also specifi c types of telescopes Temporal: bandwidth, 97, 112 coherence, 242 detection, OCT signal, 256 frequencies, 278 power spectra, 196–198, 409, 458–459, 461 resolution, 130, 381, 383 2-alternative-forced-choice (t2AFC) procedure, 376–377 vision, 366 Terminology: lens prescriptions, 332–334 refractive correction, 331–332 refractive error, 331 Tessellation method, 344 Test bed, 494 Tests/testing: liquid crystal AO phoropters, 492–502 psychophysical, 507 SLM stroke, 176 Third-order aberrations, 6, 49 3D graphics, 149 Threshold: characterized, 370–371 contrast, 382 defi ned, 550 detection, 374–375, 534 discrimination, 374–375, 535 identification, 375 Thresholding, pyramidal, 488 Ti:sapphire lasers, 221, 267 Tilt: impact of, 46, 83, 296

INDEX piston/tip/tilt mirror, 87, 94–95, 97, 103, 107–109, 111–112 tip/tilt alignment, 171 wavefront, 179 Time-domain OCT systems, 256–257, 259, 261–264, 266, 454 Time lag, in wavefront sensor transfer function, 131, 134 Time of fl ight, 258 Timing signal, image acquisition, 426 Tip: customized vision correction devices, 296 piston/tip/tilt mirror, 87, 94–95, 97, 103, 107–109, 111–112 tip/tilt alignment, 171 Tomographic studies, 5, 23, 83–84, 95, 229–230, 236, 256, 264–266, 271, 447–448 Toric lenses, 299 Traction, retinal, 210, 222–223 Transfer function: implications of, 130–135, 195–196 optical (OTF), 346, 349, 357, 359, 543 modulation (MTF), 42, 277–278, 281, 349–350, 357, 367, 505–506, 541 phase (PTF), 349, 544 Transition zone, 312, 550 Translation, retinal imaging system, 17 Transmission grating, 449 Transplantation, corneal, 69, 74, 301 Transposition formulas, 333 Transverse: chromatic aberration (TCA), 44–45, 51, 354–355, 550 resolution, 5, 21, 260–262, 448 Trefoil, 35, 293–294, 296, 324 Trial lenses, 180–181, 240, 358, 401, 413, 454 Trichromatic cone mosaic, 14–15 Troland, 369, 550 Tscherning wavefront sensor, 65, 312, 321 20/20 vision, 9 2-alternative-forced-choice (2AFC) procedure, 376–378, 380 Two-dimensional (2D) detector, 236 Ultra-high-resolution: axial, 263 OCT, 260, 266 Ultrasonography, 257–258 Ultraviolet (UV) light, 385 Uncorrected visual acuity (UCVA), 325–326

589

U.S. Food and Drug Administration (FDA): contact lenses, 307 corneal ablations, 312, 317 University of Rochester, 9, 326 Vascular structure, high-resolution imaging of, 21–22 Veins: choroidal, 214 retinal, 206–207, 209–211, 220, 222, 228–229 Vernier: acuity task, 17 alignment, 36, 45 Vertical: scanner/scanning, 420, 427–428, 432 synchronization pulse (vsync), 420, 550 VGA resolution, 90 Video: imagery, real-time, 5 keratography, xviii signal, image acquisition, 426 Vienna AO tomographic scanning OCT, 265–266 Virtual refraction: defi ned, 550 experimental method evaluation, 357–358 image quality metrics: defi ned, 340 grating objects, 349–353 point objects, 345–349 wavefront, 340–345 Vision: correction: customized devices, see Customized vision correction devices historical perspectives, 3–8 ocular adaptive optics, 9–11 Vision Stimulus Generator (VSG) system, 388 Visual: acuity (VA), 10, 55, 303–304, 336, 358, 367–368, 411–412, 477, 539, 550–551 acuity, improvement strategies, 477–478 angle, 205, 365, 551 benefit, 301–304, 551 cycle, 215 postsurgical eyes, 303–304 stimuli: generation of, 380–391 types of, 335 Strehl ratio, spatial domain (VSX), 348. See also Strehl ratio

590

INDEX

Visual Strehl ratio, frequency domain, MTF Method (VSMTF), 352 Visual Strehl ratio, frequency domain, OTF Method (VSOTF), 352, 359 Visualization, 148 Vitreo-retinal: interface, 218, 220, 222, 224, 551 surgery, adaptive optics-assisted, 22 Vitreous humor, 206–208, 218, 551 Voice coil: stage, 464 translator, 448 Voltage: actuator, 150, 196 adaptive optics system, 85 AOSLO, 423 temporal control command and, 128 wavefront influence function and, 122–124 Volume resolution element, 418, 551 Volume under neurally weighted OTF (VNOTF), 352 Volume under OTF (VOTF), 352, 359 VSG Software Library, 388 VSIA standards, 418 vsync signal, 426, 428–429 Warping, retinal imaging system, 17–18 Wave aberration, see Aberration characterizing, 6, 293 correction of, 8, 307 defi ned, 33, 551 measurement techniques, 9–10, 301 Seidel power-series expansion, 332 temporal fluctuations in, 6–7 Wavefront: beacon, 166 compensation, 418, 421 correction of, see Wavefront corrector(s) defi ned, 551 error, 41, 158, 175, 180, 189–190, 200–201, 238, 297, 302–303, 324, 326–327, 332, 334, 338, 340–342, 397, 402–403, 423, 432–433, 441, 449, 455–457, 458, 461, 474, 489, 491, 502, 505–506 error measurement, 175 flatness, 340 -guided ablation, 324 quality metrics, 340 reconstruction, 64, 180–181, 486–487 sensing, see Wavefront sensing; Wavefront sensor slope, see Wavefront slope technology, 311 tilt, 179

Wavefront corrector(s): availability of, 86 bimorph mirrors, 88, 91–92, 109–111 characterized, 9, 111–113, 237–239 classes of, 86–88 deformable mirrors, discrete actuator, 86, 89–90, 109, 111 functions of, 85 historical perspectives, 84–85 key parameters of, 112 liquid crystal spatial light modulator, 90 measurement, 170, 297, 398–403 membrane mirrors, 88, 109–111 microelectromechanical systems (MEMS), 92 performance predictions, case illustrations, 95–111 segmented correctors, 86–88, 111 sensing, see Wavefront sensing spatial control command of, 119, 124–128 stroke requirements, 99–101, 111 temporal control command of, 119, 128–135 vision science applications, 88–95 Wavefront reconstruction, 504 Wavefront sensing: AOSLO, 418, 420–421 conventional imaging, 240–242 corneal aberration measurement, 37 current technology, 52 future directions for, 23 OCT ophthalmoscopes, 265–267 scanning laser imaging, 252 Wavefront sensor, see Shack-Hartmann wavefront sensor (SHWS) AOSLO, 418, 425, 430–432 calibration of, 75–79, 158–159 categories of, 63–64 computer delay, transfer function of, 131 contact lens measurement, 298, 302 conventional imaging, 241 corneal ablation, 312 defi ned, 552 functions of, 63, 85 historical, 4 laser ray tracing, 63–66 modern, 4 polarization and, 54 qualification of, 177–180 real-time, 40 reconstruction algorithm, 488, 495 Rochester Adaptive Optics Ophthalmoscope, 408 slope measurement, 180 spatially resolved refractometer, 63–65

INDEX transfer function of, 130–131 Tscherning, 65 verification, 495–496 Wavefront slope: implications of, 64, 66, 71 influence functions, 149 measurement: centroiding, 144 image coordinates issues, 143 image preparation, 143–144 image quality, 143 pupil measurement, 143 regions of interest, 142 monochromatic refraction, 341–342 Wavelength: in focus, 356 lambda, 237 spectrum, 335 Webb, Robert, 5 Weber’s law, 375 Weibull function, 377 Weighting, neural, 348 White blood cells, 23 White-light illumination, 9, 33, 411 Whole-pupil estimation method, 344 Wiener fi ltering, 279–280 Wire crosshair, point target technique, 165 XGA resolution, 90 Yes/no procedure, 371–372, 552 Young, Thomas, 4

колхоз 7/17/06

591

ZEMAX optical design software, 240, 425 Zernike: aberrations, 83, 331, 337 astigmatism, 457 azimuthal order, 518–522 coefficient, 11, 52, 68–71, 75–77, 79, 99–100, 144–145, 299, 302, 273, 334, 357, 402, 423, 433, 435, 437, 521 correction, contact lenses and, 300 defocus, 105, 107, 333, 456–457 expansion, 92, 332, 339, 423 mode reconstruction, 123–124, 128 polynomials: defi ned, 552 implications of, 35–36, 52, 68, 127, 144–145, 339, 406, 418, 422, 432–434 OSA Standards for Reporting Optical Aberrations, 517–522 radial order, 518–522 reconstruction matrix, 127 recovering from partial data, 145 spherical aberration, 353 terms, 291–293, 295–296, 300, 302, 304, 308, 423 wavefront analysis, 331 Zero-order-hold (ZOH) transfer function, 130, 132, 195 ZEST staircase procedure, 379–380 Zonal: compensation, 504 correctors, 87 z transform, 194