### Spie Press Book

Optical Imaging and Aberrations, Part II. Wave Diffraction Optics, Second EditionFormat | Member Price | Non-Member Price |
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Ten years have passed since the publication of the first edition of this classic text in April 2001. Considerable new material amounting to 100 pages has been added in this second edition. Each chapter now contains a Summary section at the end. The new material in Chapter 4 consists of a detailed comparison of Gaussian apodization with a corresponding beam, determination of the optimum value of the Gaussian radius relative to that of the pupil to yield maximum focal-point irradiance, detailed discussion of standard deviation, aberration balancing, and Strehl ratio for primary aberrations, derivation of the aberration-free and defocused OTF, discussion of an aberrated beam yielding higher axial irradiance in a certain defocused region than its aberration-free focal-point value, illustration that aberrated PSFs lose the advantage of Gaussian apodizaton in reducing the secondary maxima of a PSF, and a brief description of the characterization of the width of a multimode beam. In Chapter 5, the effect of random longitudinal defocus on a PSF is included. The coherence length of atmospheric turbulence is calculated for looking both up and down through the atmosphere. Also discussed are the angle of arrival of a light wave propagating through turbulence, and lucky imaging where better-quality short-exposure images are selected, aligned, and added to obtain a high-quality image.

- 1. IMAGE FORMATION /1
- 1.1 Introduction /3
- 1.2 Rayleigh-Sommerfeld Theory of Diffraction and the Huygens-Fresnel Principle /5
- 1.2.1 Rayleigh-Sommerfeld Formula /5
- 1.2.2 Fresnel and Fraunhofer Approximations /9
- 1.2.3 Transfer Function of Free Space /12
- 1.3 Gaussian Image /12
- 1.4 Diffraction Image /14
- 1.4.1 Pupil Function /14
- 1.4.2 Diffracted Wave /17
- 1.4.3 Incoherent PSF and Shift-Invariant Imaging of an Incoherent
- Object /22
- 1.5 Physical Significance of Incoherent PSF /24
- 1.6 Optical Transfer Function (OTF) /27
- 1.6.1 General Relations /27
- 1.6.2 Physical Significance of OTF /31
- 1.6.3 Properties of OTF /33
- 1.6.4 OTF Slope at the Origin /35
- 1.6.5 OTF in the Limit of Zero Wavelength /40
- 1.6.6 Geometrical OTF /41
- 1.6.7 Comparison of Diffraction and Geometrical OTFs /44
- 1.6.8 Determination of OTF /45
- 1.6.9 Significance of PTF /45
- 1.7 Asymptotic Behavior of PSF /45
- 1.7.1 Point-Spread Function /46
- 1.7.2 Encircled Power /47
- 1.8 PSF Centroid /50
- 1.8.1 Centroid in Terms of OTF Slope /50
- 1.8.2 Centroid Related to Wavefront Slope /51
- 1.8.3 Centroid Related to Wavefront Perimeter /52
- 1.9 Strehl Ratio /53
- 1.9.1 General Relations /53
- 1.9.2 Approximate Expressions for Strehl Ratio /56
- 1.9.3 Determination of Strehl Ratio /58
- 1.10 Hopkins Ratio /59
- 1.11 Line- and Edge-Spread Functions (LSF and ESF) /61
- 1.11.1 Line-Spread Function /61
- 1.11.2 Edge-Spread Function /64
- 1.11.3 LSF and ESF in Terms of OTF /65
- 1.12 Shift-Invariant Imaging of a Coherent Object /67
- 1.12.1 Coherent Point-Spread Function /67
- 1.12.2 Coherent Transfer Function /70
- 1.13 Summary of Theorems /71
- Appendix A: Fourier Transform Definitions /74
- Appendix B: Some Frequently Used Integrals /75
- References /76
- Problems /78
- 2. OPTICAL SYSTEMS WITH CIRCULAR PUPILS /79
- 2.1 Introduction /81
- 2.2 Aberration-Free System /82
- 2.2.1 Point-Spread Function /82
- 2.2.2 Encircled Power /87
- 2.2.3 Ensquared Power /88
- 2.2.4 Excluded Power /90
- 2.2.5 Optical Transfer Function /93
- 2.2.6 PSF and Encircled Power from OTF /96
- 2.3 Strehl Ratio and Aberration Tolerance / 97
- 2.3.1 Strehl Ratio /97
- 2.3.2 Primary Aberrations /98
- 2.3.3 Balanced Primary Aberrations /99
- 2.3.4 Comparison of Approximate and Exact Results /101
- 2.3.5 Rayleigh's λ/4 Rule /102
- 2.3.6 Strehl Ratio for Nonoptimally Balanced Aberrations /103
- 2.4 Balanced Aberrations and Zernike Circle Polynomials /105
- 2.5 Defocused System /110
- 2.5.1 Point-Spread Function /111
- 2.5.2 Focused Beam /113
- 2.5.3 Collimated Beam /119
- 2.6 PSFs for Rotationally Symmetric Aberrations /121
- 2.6.1 Theory /122
- 2.6.2 Numerical Results /124
- 2.6.3 Gaussian Approximation /134
- 2.6.4 Summary of Results /135
- 2.7 Symmetry Properties of an Aberrated PSF /136
- 2.7.1 General Theory /137
- 2.7.2 Symmetry About the Gaussian Image Plane /138
- 2.7.3 Symmetry of Axial Irradiance /141
- 2.7.4 Symmetry in Sign of Aberration Coefficient /141
- 2.8 PSFs for Primary Aberrations /142
- 2.8.1 Defocus /142
- 2.8.2 Spherical Aberration Combined with Defocus /142
- 2.8.3 Astigmatism Combined with Defocus /144
- 2.8.4 Coma /148
- 2.8.5 2D PSFs /150
- 2.8.6 Comparison of Diffraction and Geometrical PSFs /157
- 2.9 Line of Sight of an Aberrated System /159
- 2.9.1 PSF and Its Centroid /159
- 2.9.2 Numerical Results /162
- 2.9.2.1 Wavefront Tilt /162
- 2.9.2.2 Primary Coma / 162
- 2.9.2.3 Secondary Coma /165
- 2.9.3 Comments /168
- 2.10 Diffraction OTF for Primary Aberrations /169
- 2.10.1 General Relations /169
- 2.10.2 Defocus /172
- 2.10.3 Spherical Aberration /174
- 2.10.4 Astigmatism /174
- 2.10.5 Coma /176
- 2.11 Hopkins Ratio /182
- 2.11.1 Tolerance for Primary Aberrations /182
- 2.11.2 Defocus /182
- 2.11.3 Hopkins Ratio in Terms of Variance of Aberration Difference
- Function /185
- 2.11.4 Variance of Aberration Difference Function for Primary
- Aberrations /186
- 2.12 Geometrical OTF /187
- 2.12.1 General Relations /88
- 2.12.2 Radially Symmetric Aberration /189
- 2.12.3 Defocus /189
- 2.12.4 Spherical Aberration Combined with Defocus /190
- 2.12.5 Astigmatism Combined with Defocus /190
- 2.12.6 Coma /191
- 2.13 Incoherent Line- and Edge-Spread Functions /191
- 2.13.1 Theory /192
- 2.13.1.1 LSF From PSF /192
- 2.13.1.2 LSF From Pupil Function /192
- 2.13.1.3 Struve Ratio and Aberration Tolerances /193
- 2.13.1.3.1 Defocus /194
- 2.13.1.3.2 Astigmatism Combined with Defocus /195
- 2.13.1.4 LSF From OTF /196
- 2.13.1.5 ESF From OTF /198
- 2.13.2 Numerical Results /199
- 2.14 Miscellaneous Topics /205
- 2.14.1 Polychromatic PSF /205
- 2.14.2 Polychromatic OTF /208
- 2.14.3 Image of an Incoherent Disc /209
- 2.14.3.1 Gaussian Image /210
- 2.14.3.2 Diffraction Image /210
- 2.14.3.3 Numerical Results /213
- 2.14.4 Pinhole Camera /218
- 2.15 Coherent Imaging /222
- 2.15.1 Coherent Spread Function /222
- 2.15.2 Coherent Transfer Function /223
- 2.15.3 Coherent LSF /224
- 2.15.4 Coherent ESF /229
- 2.15.5 Image of a Coherent Disc /234
- 2.15.5.1 Diffraction Image /234
- 2.15.5.2 Numerical Results /235
- 2.15.6 Use of a Lens for Obtaining Fourier Transforms /238
- 2.15.7 Comparison of Coherent and Incoherent Imaging /241
- 2.15.7.1 Frequency Spectra of Images /241
- 2.15.7.2 Two-Point Resolution /245
- 2.16 Summary /253
- References /258
- Problems /262
- 3. OPTICAL SYSTEMS WITH ANNULAR PUPILS /265
- 3.1 Introduction /267
- 3.2 Aberration-Free System /267
- 3.2.1 Point-Spread Function /267
- 3.2.2 Encircled Power /271
- 3.2.3 Ensquared Power /271
- 3.2.4 Excluded Power /272
- 3.2.5 Numerical Results /273
- 3.2.6 Optical Transfer Function /278
- 3.3 Strehl Ratio and Aberration Tolerance /287
- 3.3.1 Strehl Ratio /288
- 3.3.2 Primary Aberrations /289
- 3.3.3 Balanced Primary Aberrations /289
- 3.3.4 Comparison of Approximate and Exact Results /290
- 3.4 Balanced Aberrations and Zernike Annular Polynomials /297
- 3.5 Defocused System /304
- 3.5.1 Point-Spread Function /304
- 3.5.2 Focused Beam /305
- 3.5.3 Collimated Beam /309
- 3.6 Symmetry Properties of an Aberrated PSF /311
- 3.7 PSFs and Axial Irradiance for Primary Aberrations /314
- 3.8 2D PSFs /317
- 3.9 Line of Sight of an Aberrated System /328
- 3.9.1 PSF and Its Centroid /328
- 3.9.2 Numerical Results /329
- 3.9.2.1 Wavefront Tilt /329
- 3.9.2.2 Primary Coma /330
- 3.9.2.3 Secondary Coma /333
- 3.10 Summary /336
- References /339
- Problems /340
- 4. OPTICAL SYSTEMS WITH GAUSSIAN PUPILS /341
- 4.1 Introduction /343
- 4.2 General Theory /344
- 4.3 Systems with Circular Pupils /346
- 4.3.1 Pupil Function and Transmitted Power /346
- 4.3.1.1 Gaussian Illumination on a Uniformly Transmitting
- Pupil /346
- 4.3.1.2 Uniformly Illuminated Pupil with a Gaussian
- Transmission /348
- 4.3.2 Aberration-Free System / 349
- 4.3.2.1 PSF /349
- 4.3.2.2 Focal-Point Irradiance and Optimum Beam Radius /353
- 4.3.2.3 OTF /354
- 4.3.3 Strehl Ratio, Aberration Balancing, and Zernike-Gauss
- Polynomials /355
- 4.3.3.1 Primary Aberrations: Standard Deviation and
- Tolerance /355
- 4.3.3.2 Balanced Primary Aberrations /357
- 4.3.3.3 Strehl Ratio for Primary Aberrations /359
- 4.3.3.4 Zernike-Gauss Circle Polynomials /370
- 4.3.4 Defocused System /372
- 4.3.4.1 Pupil Function /372
- 4.3.4.2 PSF /373
- 4.3.4.3 Axial Irradiance /374
- 4.3.4.3.1 Focused Beam /374
- 4.3.4.3.2 Depth of Focus /379
- 4.3.4.3.3 Diffraction Focus /379
- 4.3.4.3.4 Collimated Beam /382
- 4.3.4.4 OTF /385
- 4.3.5 Balancing of Defocus Aberration with Spherical Aberration or
- Astigmatism /387
- 4.3.5.1 Focused Beam /387
- 4.3.5.2 Collimated Beam /392
- 4.3.6 Aberrated System /395
- 4.3.6.1 PSF for Spherical Aberration /395
- 4.3.6.2 Symmetry Properties of an Aberrated PSF /398
- 4.3.7 Weakly Truncated Gaussian Circular Beams /401
- 4.3.7.1 Pupil Function /401
- 4.3.7.2 PSF /402
- 4.3.7.3 Radius of Curvature of the Propagating
- Wavefront /407
- 4.3.7.4 Collimated beam /408
- 4.3.7.5 Beam Focusing and Waist Imaging by a Lens /410
- 4.3.7.6 OTF /414
- 4.3.7.7 Strehl Ratio, Aberration Balancing, and Zernike-Gauss
- Circle Polynomials /415
- 4.3.7.8 Beam Characterization and Measurement /419
- 4.4 Systems with Annular Pupils /421
- 4.4.1 Pupil Irradiance /422
- 4.4.2 Aberration-Free System /423
- 4.4.3 Strehl Ratio and Aberration Tolerance /425
- 4.4.4 Balanced Aberrations and Zernike-Gauss Annular Polynomials /425
- 4.4.5 Defocused System /429
- 4.4.5.1 PSF /429
- 4.4.5.2 Axial Irradiance /429
- 4.4.5.3 Collimated Beam /432
- 4.4.6 Symmetry Properties of an Aberrated PSF /433
- 4.5 Line of Sight of an Aberrated System /434
- 4.5.1 PSF and Its Centroid /434
- 4.5.2 Numerical Results /435
- 4.5.2.1 Wavefront Tilt /435
- 4.5.2.2 Primary Coma /435
- 4.5.2.3 Secondary Coma /436
- 4.6 Summary /438
- References /441
- Problems /444
- 5. RANDOM ABERRATIONS /445
- 5.1 Introduction /447
- 5.2 Random Image Motion /447
- 5.2.1 Introduction /447
- 5.2.2 Transverse Image Motion /448
- 5.2.2.1 General Theory /448
- 5.2.2.2 Application to Circular Pupils /449
- 5.2.2.2.1 Theory /449
- 5.2.2.2.2 Gaussian Approximation /450
- 5.2.2.2.3 Numerical Results /451
- 5.2.2.3 Application to Annular Pupils /455
- 5.2.2.3.1 Theory /455
- 5.2.2.3.2 Numerical Results /456
- 5.2.3 Longitudinal Image Motion /459
- 5.2.3.1 Theory /459
- 5.2.3.2 Numerical Results /460
- 5.3 Imaging through Atmospheric Turbulence /467
- 5.3.1 Introduction /467
- 5.3.2 Kolmogorov Turbulence /468
- 5.3.3 Mutual Coherence and Wave Structure Functions /470
- 5.3.4 Atmospheric Coherence Length /473
- 5.3.5 Phase Structure Function and Power Spectrum of Phase
- Fluctuations /477
- 5.3.6 Long-Exposure Image /478
- 5.3.6.1 Theory /478
- 5.3.6.2 Application to Circular Pupils /482
- 5.3.6.3 Application to Annular Pupils /486
- 5.3.7 Phase Aberration in Terms of Zernike Circle
- Polynomials /493
- 5.3.7.1 Zernike Circle Polynomials /493
- 5.3.7.2 Covariance and Variance of Zernike Expansion
- Coefficients /494
- 5.3.7.3 Aberration Variance and Approximate Strehl
- Ratio /497
- 5.3.7.4 Modal Correction of Aberration Function /500
- 5.3.8 Short Exposure Image /502
- 5.3.8.1 Angle-of-Arrival Fluctuations /502
- 5.3.8.1.1 Zernike Tilt Fluctuations /502
- 5.3.8.1.2 Centroid Fluctuations /503
- 5.3.8.2 Near-Field Imaging /507
- 5.3.8.2.1 Circular Pupils /507
- 5.3.8.2.2 Annular Pupils /514
- 5.3.8.3 Far-Field Imaging /518
- 5.3.8.4 Simulated Star Images /521
- 5.3.9 Lucky Imaging and Adaptive Optics /525
- 5.4 Summary /527
- Appendix: Fourier Transform of Zernike Circle Polynomials /530
- References /532
- Problems /535
- BIBLIOGRAPHY /537
- REFERENCES FOR ADDITIONAL READING /539
- INDEX /547

## Preface to the Second Edition

Ten years have passed since the publication of the first edition in April 2001. Many of the typographical errors were corrected in the Second Printing that took place after 3 years in 2004. Only a small amount of new material, approximately 11 pages, was added at that time. It included Appendix B in Chapter 1, Gaussian OTF in Chapter 4, and an extension of the discussion of the short-exposure image in Chapter 5. Additional corrections were made in the e-version of the book in 2009, and the discussion of Zernike circle, annular, and Gauss polynomials was streamlined utilizing an abbreviated notation with emphasis on their orthonormal form. However, a considerable amount of new material amounting to another 88 pages has been added in this Second Edition. Besides correction of some residual typographical errors, a Summary section has been included in Chapters 2, 3, and 5 for consistency with Chapters 1 and 4. Any compound references have been split into single ones. The new material is primarily in Chapters 4 and 5.

In Chapter 4, a Gaussian pupil obtained by apodization is compared with that of a Gaussian beam. The optimum value of the Gaussian radius relative to that of the pupil to yield the maximum focal-point irradiance is derived. The discussion of the standard deviation, aberration balancing, and Strehl ratio for primary aberrations for different values of the ratio of the beam and pupil radii has been expanded. It is shown that the approximate expression for Strehl ratio in terms of the aberration variance is not suitable for very narrow Gaussian beams. The aberration-free OTF has been extended to that of a defocused system. The problem of balancing defocus aberration with spherical aberration or astigmatism is discussed, illustrating that an aberrated beam can yield a higher axial irradiance in a certain defocused region than its aberration-free focal-point value. The PSFs aberrated by spherical aberration are considered to illustrate the loss of advantage of the Gaussian pupil in reducing the secondary maxima when the aberration is present. The characterization of the width of a multimode beam compared to that of a Gaussian beam is discussed briefly.

In Chapter 5, the effect of random transverse image motion on the PSF has been supplanted by a discussion of the effect of random longitudinal defocus. The coherence length of atmospheric turbulence is calculated for the Hufnagel-Valley model of the refractive index structure parameter, for both looking up and down through the atmosphere. The angle of arrival of the light wave propagating through turbulence is discussed for both the Zernike tilt as well as the centroid of the PSF, showing that they are nearly equal. A brief discussion of lucky imaging is also given, where better quality short-exposure images are selected, aligned, and added to obtain a high-quality image.

**Virendra N. Mahajan**

El Segundo, California

April 2011

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