### Spie Press Book

Introduction to the Optical Transfer FunctionFormat | Member Price | Non-Member Price |
---|---|---|

- 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

### Preface

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.

CHARLES S. WILLIAMS

ORVILLE A. BECKLUND

Dallas, Texas

May 1988

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