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Spie Press Book

Optical Imaging and Aberrations, Part II. Wave Diffraction Optics
Author(s): Virendra N. Mahajan
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Book Description

This book discusses the characteristics of a diffraction image of an incoherent or a coherent object formed by an aberrated imaging system. Numerical results in aberrated imaging have been emphasized to maximize the practical use of the material.

This second printing includes a number of updates and corrections to the first printing.

Beginning with a description of the diffraction theory of image formation, the book describes both aberration-free and aberrated imaging by optical systems with circular, annular, or Gaussian pupils. As in part I, the primary aberrations are emphasized. Their effects on Strehl, Hopkins, and Struve ratios are discussed in detail. The balanced aberrations are identified with Zernike polynomials appropriate for each type of system. Imaging in the presence of random aberrations is also discussed that includes the effects of image motion and propagation through atmospheric turbulence. Each chapter ends with a set of practical problems.


Book Details

Date Published: 10 December 2001
Pages: 486
Volume: PM103

Table of Contents
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Preface xiii
Acknowledgments xv
Symbols and Notation xvii
CHAPTER 1: IMAGE FORMATION 1
1.1 Introduction 3
1.2 Rayleigh-Sommerfeld Theory of Diffraction and 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 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 64
1.12 Shift-Invariant Imaging of a Coherent Object 67
1.12.1 Coherent Point-Spread Function 67
1.12.2 Coherent Transfer Function 69
1.13 Summary of Theorems 71
Appendix: Fourier Transform Definitions 74
References 75
Problems 78
CHAPTER 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 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 121
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
2.8 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 2-D 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 164
2.9.3 Comments 168
2.10 OTFs for Primary Aberrations 169
2.10.1 General Relations 169
2.10.2 Defocus 172
2.10.3 Spherical Aberration 173
2.10.4 Astigmatism 173
2.10.5 Coma 175
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 188
2.12.2 Radially Symmetric Aberrations 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
2.13.1.4 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
2.14.4 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
2.15.6 Use of a Lens for Obtaining Fourier Transforms 238
2.15.7 Comparison of Coherent and Incoherent Imaging 241
References 253
Problems 257
CHAPTER 3: OPTICAL SYSTEMS WITH ANNULAR PUPILS 259
3.1 Introduction 261
3.2 Aberration-Free System 261
3.2.1 Point-Spread Function 261
3.2.2 Encircled Power 265
3.2.3 Ensquared Power 265
3.2.4 Excluded Power 266
3.2.5 Numerical Results 267
3.2.6 Optical Transfer Function 272
3.3 Strehl Ratio and Aberration Tolerance 281
3.3.1 Strehl Ratio 282
3.3.2 Primary Aberrations 283
3.3.3 Balanced Primary Aberrations 283
3.3.4 Comparison of Approximate and Exact Results 284
3.4 Balanced Aberrations and Zernike Annular Polynomials 291
3.5 Defocused System 298
3.5.1 Point-Spread Function 298
3.5.2 Focused Beam 299
3.5.3 Collimated Beam 303
3.6 Symmetry Properties of an Aberrated PSF 305
3.7 PSFs and Axial Irradiance for Primary Aberrations 308
3.8 2-D PSFs 311
3.9 Line of Sight of an Aberrated System 322
3.9.1 PSF and its Centroid 322
3.9.2 Numerical Results 323
3.9.2.1 Wavefront Tilt 323
3.9.2.2 Primary Coma 324
3.9.2.3 Secondary Coma 327
References 330
Problems 331
CHAPTER 4: OPTICAL SYSTEMS WITH GAUSSIAN PUPILS 333
4.1 Introduction 335
4.2 General Theory 336
4.3 Systems with Circular Pupils 337
4.3.1 Theory 337
4.3.2 Aberration-Free System 338
4.3.3 Strehl Ratio and Aberration Tolerance 343
4.3.4 Balanced Aberrations and Zernike-Gauss Circle Polynomials 344
4.3.5 Defocused System 348
4.3.5.1 Theory 348
4.3.5.2 Axial Irradiance 349
4.3.5.3 Defocused Distribution 350
4.3.5.4 Collimated Beam 352
4.3.6 Weakly Truncated Gaussian Circular Beams 353
4.3.6.1 Introduction 353
4.3.6.2 Irradiance Distribution and Beam Radius 354
4.3.6.3 Effect of Lenses on Gaussian Beam Propagation 359
4.3.6.4 Aberration Balancing 362
4.3.7 Symmetry Properties of an Aberrated PSF 365
4.4 Systems with Annular Pupils 366
4.4.1 Theory 367
4.4.2 Aberration-Free System 368
4.4.3 Strehl Ratio and Aberration Tolerance 370
4.4.4 Balanced Aberrations and Zernike-Gauss Annular Polynomials
371
4.4.5 Defocused System 374
4.4.5.1 Theory 374
4.4.5.2 Axial Irradiance 374
4.4.5.3 Defocused Distribution 376
4.4.5.4 Collimated Beam 376
4.4.6 Symmetry Properties of an Aberrated PSF 378
4.5 Line of Sight of an Aberrated System 379
4.5.1 PSF and its Centroid 380
4.5.2 Numerical Results 380
4.5.2.1 Wavefront Tilt 380
4.5.2.2 Primary Coma 381
4.5.2.3 Secondary Coma 382
4.6 Summary 382
References 385
Problems 386
CHAPTER 5: RANDOM ABERRATIONS 387
5.1 Introduction 389
5.2 Random Motion 389
5.2.1 General Theory 390
5.2.2 Circular Pupils 391
5.2.2.1 Theory 391
5.2.2.2 Gaussian Approximation 392
5.2.2.3 Numerical Results 393
5.2.3 Annular Pupils 393
5.2.3.1 Theory 393
5.2.3.2 Numerical results 397
5.3 Imaging Through Atmospheric Turbulence 401
5.3.1 Long-Exposure Image 402
5.3.2 Kolmogrov Turbulence 407
5.3.3 Circular Pupils 412
5.3.4 Annular Pupils 415
5.3.5 Phase Aberration in Terms of Zernike Polynomials 419
5.3.6 Short Exposure Image 425
Appendix: Fourier Transform of Zernike Polynomials 433
References 435
Problems 436
BIBLIOGRAPHY 437
REFERENCES FOR ADDITIONAL READING 439
INDEX 447

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