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Spie Press Book

Wavefront Optics for Vision Correction
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Book Description

This book addresses some of the issues in visual optics with a functional analysis of ocular aberrations, especially for the purpose of vision correction. The basis is the analytical representation of ocular aberrations with a set of orthonormal polynomials, such as Zernike polynomials or the Fourier series.

Although the aim of this book is the application of wavefront optics to laser vision correction, most of the theories discussed are equally applicable to other methods of vision correction, such as contact lenses and intraocular lenses.

Book Details

Date Published: 11 February 2008
Pages: 366
ISBN: 9780819469663
Volume: PM179

Table of Contents
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Symbols, Notations, and Abbreviations
1 Introduction
1.1 Wavefront Optics and Vision Correction
1.2 Purpose and Structure of the Book
2 Fundamentals of Ocular Wavefront Correction
2.1 Principle of Phase Conjugation
2.1.1 Refractive Index of the Cornea
2.1.2 Wavefront and Optical Path Difference
2.1.3 Phase Conjugation
2.2 Munnerlyn Equation
2.2.1 Myopic Correction
2.2.2 Hyperopic Correction
2.2.3 Astigmatic Correction
2.3 Principle of Customized Laser Vision Correction
2.3.1 Wavefront-Driven Correction
2.3.2 Ablation Equation
2.3.3 Profile Fitting with Simulated Annealing
2.4 Principle of Excimer Laser Ablation of the Cornea
2.4.1 Principle of Corneal Tissue Ablation
2.4.2 Key Parameters of Refractive Laser Systems
2.5 Fine-Tuning Ablation Profiles
2.5.1 Variable Index of Refraction of the Cornea
2.5.2 Adjusted Ablation Profile Due to Curved Cornea
2.5.3 Variable Ablation Rate Due to Hydration
2.5.4 Variable Ablation Rate Due to Energy Fluctuation
2.5.5 Compensation of Corneal Biomechanics and Healing
Appendix 2.A Derivation of the Munnerlyn Equation
Appendix 2.B Derivation of Laser Energy Loss due to Reflection
3 Ocular Wavefront Representation
3.1 Orthonormal Polynomials and Their Merits
3.2 Geometrical Aberrations and Power Series
3.2.1 Primary (Seidel) Aberrations
3.2.2 Secondary (Schwarzschild) Aberrations
3.2.3 Seidel Series
3.3 Zernike Polynomials
3.3.1 Definition
3.3.2 Aberration Balancing of Zernike Polynomials
3.3.3 Properties of Zernike Polynomials
3.3.4 Wavefront Refractions and Zernike Coefficients
3.3.5 Munnerlyn Equation and Zernike Defocus
3.4 Other Basis Functions for Ocular Aberrations
3.4.1 Fourier Series
3.4.2 Taylor Monomials
3.4.3 Elliptical Polynomials
3.5 Refractive Laser Profiles
3.5.1 Gaussian Beam
3.5.2 Top Hat Beam
3.5.3 A Comparison of Profile Fitting
Appendix 3.A Orthonormal Polynomials and Related Properties
Appendix 3.B Determination of Orthonormal Polynomials
Appendix 3.C Properties of the Inner Product of Polynomials
Appendix 3.D Zernike Polynomials up to the 10th Order
Appendix 3.E Aberration Balancing of Orthonormal Polynomials
Appendix 3.F Derivation of the Fourier Transform of Zernike Polynomials
Appendix 3.G Examination of the Munnerlyn Equation
4 Ocular Wavefront Sensing and Reconstruction
4.1 Wavefront Slopes
4.2 Ocular Wavefront Sensing Methods
4.2.1 Hartmann-Shack Aberrometry
4.2.2 Tscherning Aberrometry
4.2.3 Ray Tracing Aberrometry
4.3 Wavefront Reconstruction Methods
4.3.1 Zonal Reconstruction
4.3.2 Modal Reconstruction
4.4 Non-Fourier-Based Modal Reconstruction
4.4.1 Taylor Reconstruction
4.4.2 Zernike Reconstruction
4.5 Fourier-Based Modal Reconstruction
4.5.1 Fourier Reconstruction
4.5.2 Iterative Fourier Reconstruction
4.5.3 Comparison of Zernike and Fourier Reconstructions
Appendix 4.A Wavefront Tilts and Image Displacement
Appendix 4.B Matlab Code for Zonal Reconstruction
Appendix 4.C Derivation of Eq. (4.28)
5 Ocular Wavefront Conversion
5.1 General Discussion of Wavefront Conversion
5.1.1 Completeness of Basis Functions
5.1.2 Conversions from the Coefficients of A Complete Set of Basis Functions to the Coefficients of Orthonormal Basis Functions
5.1.3 Conversions between the Coeficients of A Complete Set of Basis Functions and Fourier Coefficients
5.2 Conversions of Zernike Polynomials and Seidel Series
5.2.1 Seidel Coefficients to Zernike Coefficients
5.2.2 Zernike Coefficients to Seidel Coefficients
5.3 Conversions of Zernike Polynomials and Fourier Series
5.3.1 Zernike Coefficients to Fourier Coefficients
5.3.2 Fourier Coefficients to Zernike Coefficients
5.4 Conversions of Taylor Monomials and Zernike Polynomials
5.4.1 Taylor Coefficients to Zernike Coefficients
5.4.2 Zernike Coefficients to Taylor Coefficients
5.5 Conversions of Fourier Series and Taylor Monomials
5.5.1 Taylor Coefficients to Fourier Coefficients
5.5.2 Fourier Coefficients to Taylor Coefficients
Appendix 5.A Derivation of Eq. (5.3)
Appendix 5.B Derivation of Eqs. (5.6) and (5.7)
Appendix 5.C Derivation of Conversion Matrices Cs2z and Cz2s
Appendix 5.D Proof of Eq. (5.15)
Appendix 5.E Derivation of Conversion Matrices Ct2z and Cz2t
Appendix 5.F Derivation of Qqp(k; Á)
6 Ocular Wavefront Transformation
6.1 Wavefront Transformation and Iris Registration
6.1.1 Review of Some Useful Definitions
6.1.2 Iris Registration
6.2 Wavefront Representation for Pupil Resizing
6.2.1 General Consideration
6.2.2 Pupil Resizing Polynomials
6.2.3 Taylor Resizing Monomials
6.2.4 Zernike Resizing Polynomials
6.2.5 Effective Power and Correction of Presbyopia
6.2.6 Pupil Resizing with Seidel Series
6.3 Wavefront Representation for Cyclorotation
6.3.1 Wavefront Rotation with Taylor Monomials
6.3.2 Wavefront Rotation with Zernike Polynomials
6.4 Wavefront Representation for Decentration
6.4.1 Wavefront Extrapolation
6.4.2 Wavefront Decentration with Taylor Monomials
6.4.3 Wavefront Decentration with Zernike Polynomials
6.4.4 Wavefront Refraction of Decentered Aberrations
6.5 Wavefront Representation for Resizing, Rotation, and Decentration
6.5.1 Wavefront Transformation with Zernike Polynomials
6.5.2 Wavefront Refraction after Transformation
Appendix 6.A Derivation of Eq. (6.19)
Appendix 6.B Zernike Resizing Polynomials
Appendix 6.C Derivation of Eq. (6.27)
Appendix 6.D Derivation of Eq. (6.28)
Appendix 6.E Derivation of Eq. (6.32)
Appendix 6.F Matlab Code for Geometrical Transformations
7 Ocular Wavefront Propagation
7.1 Review of Some Eye Models
7.1.1 Lens Equations and Cardinal Points
7.1.2 A Historic Note of Schematic Eyes
7.1.3 Reduced Schematic Eye
7.1.4 Gullstrand-LeGrand Schematic Eye
7.2 Classical Vertex Correction
7.2.1 Ignoring the Optics of the Eye
7.2.2 Considering the Optics of the Eye
7.3 Ray Tracing of the Ocular Wavefronts
7.3.1 Calculation of the Direction Factor
7.3.2 Calculation of the Boundary Factor
7.3.3 Boundary of Propagated Wavefront Determined by Zernike Polynomials
7.4 Ocular Wavefront Propagation
7.4.1 Taylor Monomials
7.4.2 Zernike Polynomials
7.5 Wavefront Propagation of Common Aberrations
7.5.1 Low order aberrations
7.5.2 Coma aberration
7.5.3 Trefoil aberration
7.5.4 Primary spherical aberration
7.5.5 Secondary spherical aberration
7.5.6 Secondary astigmatism aberration
7.5.7 Secondary coma aberration
7.5.8 Quadrafoil aberration
Appendix 7.A Proof of Eq. (7.24) for Refractions Between Any Two Planes
Appendix 7.B Derivation of Eq. (7.30)
Appendix 7.C Proof of Eq. (7.51) From Wavefront Propagation
8 Optical Metrics of Ocular Wavefronts
8.1 Pupil Plane Metrics for Ocular Aberrations
8.1.1 Root Mean Square Wavefront Error Zernike representation Fourier representation Taylor representation
8.1.2 Wavefront Refractions Zernike representation Fourier representation Taylor representation
8.1.3 Other Metrics
8.2 Image Plane Metrics for Ocular Aberrations
8.2.1 Strehl Ratio 256
8.2.2 Full Width at Half Maximum
8.2.3 Encircled Energy
8.2.4 Modulation Transfer Function
8.2.5 Compound Modulation Transfer Function
8.2.6 Volume under Modulation Transfer Function
8.2.7 Visual Strehl Ratio
8.2.8 Other Metrics
8.3 Visual Performance Metrics
8.3.1 Manifest Refractions
8.3.2 Visual Acuity
8.3.3 Contrast Sensitivity
8.3.4 Binocular Vision and Stereopsis
8.3.5 Neural Contrast Threshold
8.4 Simulation of Visual Outcomes
8.4.1 Analytic Point Spread Functions
8.4.2 Polychromatic Point Spread Functions
8.4.3 Calibration of the Point Spread Functions
8.4.4 Convolution of Point Spread Function and Vision Targets
8.4.5 Prediction of Visual Acuity from Ocular Aberrations
Appendix 8.A Derivation of Eq. (8.9)
Appendix 8.B Derivation of Eq. (8.28)
Appendix 8.C Matlab Code for Calculation of Point Spread Functions
9 Clinical Results of Wavefront-Driven Refractive Surgery
9.1 Statistics of Ocular Aberrations
9.1.1 Virgin Eyes
9.1.2 Post-Operative Eyes
9.1.3 Characteristics of Ocular Aberrations Age Corneal Keratometry Left Eye vs Right Eye High Order RMS Gender Race Eye Dominance Contact Lens Wearing
9.2 Treatment Validation
9.2.1 Generation of Phase Plates
9.2.2 Phase Plates for Calibrating Aberrometers
9.2.3 PreVUETM Lens for Treatment Validation
9.2.4 Scleral Lens for New Treatment Design
9.2.5 Adaptive Optics for Treatment Validation
9.3 Wavefront-Driven Myopic Correction
9.3.1 Photorefractive Keratectomy
9.3.2 Laser Assisted in situ Keratomileusis
9.3.3 Comparison of Conventional and Custom LASIK
9.4 Wavefront-Driven Hyperopic Correction
9.4.1 Photorefractive Keratectomy
9.4.2 Laser Assisted in situ Keratomileusis
9.4.3 Comparison of Conventional and Custom LASIK
Author Index
Subject Index


During the past three to four decades, adaptive optics has evolved from a specific technology for atmospheric turbulence compensation in astronomy to a promising technology with ample applications in astronomy, the military, vision, and laser propagation. Wavefront sensing is a key component of an adaptive optics system that measures (in real time) the aberrations of an optical system. Thus, a wavefront is vital to research and development in adaptive optics and related fields.

The application of wavefront sensing technology in vision was pioneered in the early 1990s by Prof. Josef F. Bille and Dr. Junzhong Liang (then Bille's PhD student) at the University of Heidelberg; they built the world's first aberrometer for measuring both the low order and the high order ocular aberrations of human eyes. The application of wavefront sensing and adaptive optics in vision experienced explosive progress after Dr. Liang, together with colleagues in Prof. David R. Williams' lab, built the world's first adaptive optics system for retinal imaging at the University of Rochester. At the same time, laser vision correction enjoyed rapid growth. Early in the new century, several leading refractive laser manufacturers had obtained FDA approval for marketing the wavefront-guided LASIK procedures in the United States. Active research in ocular wavefronts for correction, interpretation, implication, and the like has spread from leading researchers to many ophthalmologists, optometrists, vision scientists, and refractive laser engineers.

During the research and development of wavefront-driven refractive surgical techniques, it becomes apparent that a new field of research is developing that deals with the optics of the ocular wavefront with a consistent mathematical treatment. The key to this mathematical approach is the use of orthonormal basis functions, namely Zernike polynomials, although another set of orthonormal basis functions—Fourier series and other sets of nonorthonormal basis functions, such as Taylor monomials and Seidel series—can also be used. With this in mind, I have attempted to arrange the content of this book so it will answer the following questions: How is the ocular wavefront represented? How is it obtained with wavefront sensing and reconstruction? How are the coefficients of different representing basis functions converted to each other? How do the coefficients change when an ocular wavefront undergoes a pupil constriction, a cyclorotation, or a decentration? How do the coefficients change when the wavefront propagates? How is the ocular wavefront evaluated? What is the clinical impact of the ocular wavefront?

Over the course of my research during the past 15 years or so, several individuals have influenced me in different ways that deserve special appreciation, including one I have never met. First, thanks goes to Prof. Arne Arderberg, who introduced me to the interesting field of astronomy and adaptive optics. Together with Prof. Mette Owner-Peterson, I enjoyed reading Robert J. Noll's famous paper on Zernike polynomials and atmospheric turbulence. His paper took me a long time to fully understand, but the result is my increased interest in mathematics and optics. I was lucky enough to work for both Prof. David R. Williams and Dr. Junzhong Liang—two important figures in the development of adaptive optics in vision that led to this book—at the University of Rochester and Visx Incorporated, respectively. Over the past couple of years, I have developed a very fruitful collaboration with Dr. Virendra N. Mahajan during our spare time. Our work on orthonormal polynomials lays the foundation for the representation of ocular aberrations.

This book would not have been completed without the generous support of my employer, Advanced Medical Optics, Inc. I am very grateful to Leonard Borrmann, PhD, Tom Shoup, PhD, and Carol Harner, PhD, for their support. During my past six-year employment with Visx/AMO, I enjoyed working with some leading surgeons: Drs. Noel Alpins, Eric Donnenfeld, Ken Greenberg, Jack Holladay, Bruce Jackson, Douglas Koch, Martin Mainster, Marguerite McDonald, Marc Odrich, Luis Ruiz, Steve Schallhorn, Kerry Solomon, Julian Stevens, Gustavo Tamayo, and Steve Trokel. I am indebted to Profs. David A. Atchison and Jim Schwiegerling for reviewing the entire manuscript on a tight schedule and for their valuable critique and suggestions. Drs. Linda Lundström and Sverker Norrby carefully read every chapter and provided me with detailed comments. I also received helpful comments and suggestions from some of the aforementioned surgeons and many of my colleagues in the research group—all of which helped polish the book. (Of course, I am responsible for any error or omissions still remaining in the book.) It has been a pleasure to work with the SPIE Press staff, in particular Scott Schrum and Tim Lamkins, for their professionalism and enjoyable cooperation.

Last but not least, I am extremely grateful to my lovely family—my wife Wendy and my sons Percy and Perry—for their understanding and physical and mental support, in particular during the nights and weekends over the past year while this book was being written. Finally, I thank God—my father in heaven—for His spiritual guidance while I finished this interesting work.

Guang-ming Dai
Fremont, California
November, 2007

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