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Introduction to the Optical Transfer Function
Author(s): Charles S. Williams; Orville A. Becklund
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Book Description

Originally published by Wiley in 1989, this timeless classic provides a well-illustrated treatment of the fundamental concepts of spatial frequency, spread function, wave aberration, and transfer function--and how these concepts are related in an optical system, how they are measured and calculated, and how they may be useful.

Book Details

Date Published: 15 September 2002
Pages: 412
ISBN: 9780819443366
Volume: PM112

Table of Contents
SHOW Table of Contents | HIDE Table of Contents
1. OTF Historical Background / 1
Introduction / 1
The Early History of Optical Design and Image Evaluation / 2
Laying the Foundation for OTF-1850 to 1940 / 6
The Appearance of Some Important Mathematics / 8
Growing Awareness of OTF-The 1940s / 9
Inventive OTF Instrumentation-The 1950s / 10
Adjustment to Practive-The 1960s / 13
Acceptance-The 1970s / 15
The 1980s / 16
Perspective / 18
References / 19
2. Concepts / 23
Introduction / 23
Spatial Frequency / 23
Flux Density and Distributions / 26
Frequency Spectrum / 28
Three-Bar Pattern Spectrum / 30
Even and Odd Harmonics and Functions / 32
A Stepladder Bar Pattern / 33
Spectrum for a General Distribution / 35
Extension to Two Dimensions / 37
Contrast and Contrast Transfer / 38
Distributions of Physical Quantities / 40
Point Source / 41
Stops and Pupils / 42
Point Spread Functions / 43
Spread Functions for Small Aberrations / 50
Line Spread Functions / 55
The Edge Trace / 57
Isoplanatism / 60
Linear Superposition / 60
Coherence / 61
References / 62
3. Notation and Coordinates / 64
Introduction / 64
Sign and Nomenclature Conventions / 66
Cardinal Points / 66
Paraxial Notation / 67
Need for Special Coordinates / 69
Wave-Front Aberration / 70
Nonparaxial Notation / 73
Transfer Equations / 78
Pupil Variables / 80
Reduced Coordinates / 81
Shifting the Image Plane / 84
Magnification with Distortion / 89
References / 91
4. Diffraction Integral and Wave-Front Aberration Function / 92
Introduction / 92
Wave-Front Expressions and the Diffraction Integral / 93
The Strehl Ratio / 100
Anamorphotic Stretching / 101
The Pupil Function / 102
The Wave Aberration Function / 103
Spherical Aberration / 108
Coma / 115
Astigmatism / 119
Curvature of Field / 124
Distortion / 124
Expansion fo the Wave Aberration Function in Zernike Polynomials / 126
References / 131
5. Mathematical Theory of OTF / 134
Introduction / 134
Definitions, Nomenclature, and Conventions / 135
Linearity and Isoplanatism / 142
Image of a General Distribution / 144
One-Dimensional Analysis / 146
Optical Transfer Function / 149
The Perfect OTF / 152
Perfect OTF from Spread Function / 158
Effects of Certain Aberrations on the Optical Transfer Function / 162
Apodization / 170
The Geometrical Optics OTF Approximation / 177
The Polychromatic OTF / 178
References / 179
6. Optical Design and Image Criteria / 181
The Nature of Optical Design / 181
Automatic Lens Design / 188
Selected Features of Design Programs / 192
Manufacturing Tolerances / 195
Assessment of Image Quality / 196
Resolving Power versus Acutance / 199
The Phase Transfer Function / 204
References / 208
7. Merit Functions and Aberration Balancing / 211
Introduction / 211
Single MTF Values and Certain Graphical Areas as Criteria of Performance / 213
A Merit Function Based on the Low-Frequency End of the MTF / 216
Other OTF-Related Merit Functions / 217
Merit Evaluations Based on the Aberration Function / 218
Mean Square Value of the Aberration Function as a Merit Function / 218
Variance of the Aberration Function as a Merit Function / 219
Variance of the Aberration Difference Function as a Merit Function / 221
Aberration Balancing Based on the Power Series Expansion of the Wave
Aberration Function / 224
Aberration Balancing with Zernike Polynomials / 234
Comparisons of Optimizing and Balancing Procedures / 237
The Effect of Optical Parameter Variations on the Optical Transfer Function / 240
References / 244
8. Measurement / 246
Introduction / 246
Components of a Measuring System / 249
Requirements of the Components / 249
Direct Methods / 255
Effect of Finite Grating Length / 258
Changing Spatial Frequency / 261
The Area Grating / 263
Effect of Slit Width / 268
Square Wave Gratings / 270
Indirect Methods / 272
Interferometric Methods / 274
The Interferometer / 275
An Interferometric Measuring Equipment / 282
Other Interferomteric Equipment / 285
References / 288
9. Calculation of the OTF: Analytical Methods / 291
Introduction / 291
The OTF Calculated for Defocusing / 293
The OTF Calculated for Astigmatism / 300
References / 316
10. Calculation of the OTF: Numerical Methods / 317
Introduction / 317
Optical Path Difference Data by Interferometry / 320
Calculation of the Aberration Polynomial / 323
Extension to More Than One Independent Variable / 325
Choice of Orthogonal Polynomial / 326
Gauss Quadrature / 329
References / 335
Appendix A. Calculated Otpical Transfer Functions / 337
Introduction / 337
Defocusing / 337
Primary Spherical Aberration / 341
Primary with Seconday Spherical Aberration / 341
Primary and Secondary Coma with Defocusing / 345
Spherical Aberraton with Color / 348
Optimum Balanced Fifth-Order Spherical Aberration / 349
Primary Coma at Different Azimuths / 354
Nonrotationally Symmetric Systems / 357
References / 360
Appendix B: Some Mathematics / 362
The Fourier Transform / 362
The Delta Function / 365
The Convolution Integral / 367
Convolution Identities / 369
Convolution Integral When One Function is Sinusoidal / 370
Significance of the Convolution Integral / 372
Convolution and Spread Functions / 378
Other Convolution Integrals / 379
The Correlation Function / 380
Examples / 381
References / 386
Appendix C: Diffraction Intregral Fundamentals / 387
Introduction / 387
The Traveling Wave Equation / 387
Spherical Wave-Fronts / 391
Application of the Huygens-Fresnel Principle to a Spherical Wave-Front / 395
Application of the Huygens-Fresnel Principle to Chapter 4 / 398
References / 400
Appendix D: Updated Calculations / 401
Index / 403

Preface to the Reprinted Edition

When Orville Becklund and I began writing our book, a powerful and rapid computer was not available to us. The best we had was a hand-held programmable calculator. We used it to calculate solutions in which each solution consisted of a series of series. Many times, I had to program the calculator to run all night. I would turn it on to calculate until morning, and go to bed. Finally, I had a bunch of partial answers to put together. I think it should have been expected that we would always be uneasy about the accuracy of data that finally found its way into the text.

Computers, programs, and programmers have come a long way since then. One of the best for calculating problems relating to optics, is from the work of Dr. David F. Edwards of Tracy, California. Much of his work in optics programming was after his retirement from Lawrence Livermore National Laboratory as head of the Optical Sciences and Engineering Group. Our calculations are updated by Dr. Edwards's, and are found in Appendix D (p. 401).

Charles Williams
July 2002


An abundance of knowledge about the optical transfer function (OTF) has been published in many excellent articles during the past 35 years or so, but somehow a niche for this knowledge has never been found in the engineering and scientific structure. As a result, OTF publications are scattered throughout the archival literature of scientific and technical journals. Our book aims to bring together into one source much of this wealth of information.

Those concerned with grounding engineers and scientists in the procedures of optical evaluation have found that spatial frequency, wave-front distortion, and optical transfer function, though not particularly difficult concepts to understand, do not easily become part of one's thinking, and therefore practice, as the concepts of rays, ray tracing, and ray aberrations. The word ray (geometrical optics) is used so commonly in our language that it is no longer an esoteric term reserved for optics. Actually, there are advantages peculiar to each of the two viewpoints, and an optical analyst is handicapped by a lack of facility with either. We hope that our book is articulate enough in the art to bring practitioners up to speed in the realm of spatial frequency and the OTF.

Specifically, our text dwells on such fundamental concepts as spatial frequency, spread function, wave aberration, and transfer function-how these are related in an optical system, how they are measured and calculated, and how they may be useful. In the early chapters, we review the historical background for the OTF, and related concepts, and the necessary nomenclature and coordinate systems. We discuss in some detail the wave aberration function, which is a measure of an optical system's ability to produce an image that is a "reasonable facsimile" of the object and which, therefore, is a fundamental characterization of the system's excellence of performance. We derive the optical transfer function and related concepts mathematically, and we discuss some ways that the OTF can be used for assessing the quality of an optical system both during its design and during testing of the manufactured system.

We show how the OTF can be used: when specifications for the optical system are being drawn up, when the OTF is part of a merit function while the system is being designed by computer, and when the optical system is being tested to verify adherence to specifications. Finally, we show how the OTF can be calculated mathematically, both by analytical procedures and by numerical methods of integration. In the appendixes, some pertinent mathematical basics are reviewed, and we document a number of OTF calculations that other workers have made. Our book makes liberal use of illustrations. For the reader who wishes to pursue studies beyond the scope of our text, we provide a full complement of references at the end of each chapter.

The reader of our mathematical chapters should have had courses in calculus; a course in transform theory would be helpful but not necessary because the mathematics in the appendixes provide a review of all the Fourier transform theory that the reader will need. Besides the professional nonexpert in physical optics, the level of our text is intended to suit undergraduates with limited exposure to optics, such as juniors and seniors in science, mathematics, and engineering.

We have purposely avoided certain OTF topics: We do not treat the geometrical approximation of the OTF, the OTF of sampled images, or the polychromatic OTF, because we feel that the state of the art concerning each of these topics is not quite ready to be included in a tutorial book on the optical transfer function. We make no pretense that the ideas in this book are original with us. Our information has come through various paths and from many sources, and we have tried to give credit at the appropriate places in the text to the many whose work we have used.

Dallas, Texas
May 1988

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