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Contents
Preface of the Book Series xxi
Preface of the First Volume xxiii
Acknowledgements xxv
1 Introduction 1
1.1 Modelling and Goal of Lens Design 2
1.2 Optical System Types and Aperture Field Classification 3
1.2.1 Selected Classes of Systems 7
References 10
2 Optical Materials 13
2.1 Introduction 13
2.2 Dispersion 14
2.2.1 Definition 14
2.2.2 Wavelengths 15
2.2.3 Abbe Number 16
2.2.4 Dispersion of Crown and Flint Glasses 17
2.2.5 Glass Diagram 18
2.2.6 Interpolation of the Refractive Index 18
2.2.7 Relative Partial Dispersion 20
2.2.7.1 Definition 20
2.2.7.2 Line of Normal Dispersion 22
2.2.7.3 Anomalous Partial Dispersion 25
2.2.7.4 Hoogland Diagram 26
2.2.7.5 The Anomalous Dispersion Representation of Münz 26
2.2.8 Generalized Mathematical Descriptions of Dispersion 29
2.2.8.1 Introduction 29
2.2.8.2 Derivative Based Dispersion Description 31
2.2.8.3 Buchdahls Chromatical Coordinates 32
2.2.8.4 Adaptive Dispersion Formula of Sasian 33
2.2.9 Miscellaneous 35
2.2.9.1 Dispersion of Diffractive Surfaces 35
2.2.9.2 Dispersion of Gradient-Index Materials 36
2.3 Group Velocity Dispersion and Short Pulses 37
2.3.1 Introduction 37
2.3.2 Dispersion Coefficients 38
2.3.3 Pulse Changes due to Dispersion 38
2.3.4 Pulse Dispersion Data of Glasses 39
2.3.4.1 Introduction 39
2.3.4.2 Numerical Calculation of Index Derivatives 39
2.3.4.3 GVD Properties of Glasses 40
2.4 Absorption and Transmission 42
2.4.1 Complex Index of Refraction 42
2.4.2 Lambert-Beers Law 44
2.4.3 Transmission of a Component 44
2.4.4 Transmission of Glasses 45
2.5 Thermal Properties 46
2.5.1 Thermal Expansion 46
2.5.2 Index Change 47
2.5.3 Thermo-Optical Coefficient 48
2.5.3.1 Introduction 48
2.5.3.2 Temperature-Dependence of Dispersion 49
2.6 Other Optical Materials 49
2.6.1 Crystal Materials for IR and UV 49
2.6.2 Plastics 53
2.6.3 Water 54
2.6.4 Technical Liquids 54
2.6.5 Immersion Liquids 56
2.6.6 Optical Cements 57
References 57
3 Geometrical Optics 61
3.1 Introduction 61
3.2 Law of Refraction 61
3.2.1 Introduction 61
3.2.2 Ray Bending 62
3.2.3 Description of Refraction in the k-Space 64
3.3 Fresnel Formulas 65
3.3.1 Amplitude Coefficients 65
3.3.2 Reflectivity and Transmittivity 67
3.3.3 Transmission of Systems 68
3.3.4 Total Internal Reflection 70
3.3.5 Reflection at Metals 73
3.4 Raytrace 74
3.4.1 Introduction 74
3.4.2 Paraxial Formulas 76
3.4.3 Meridional Q-U Method 77
3.4.4 Three-dimensional Case 78
3.4.5 Pitfalls and Special Cases 80
3.4.6 Modelling of Mirrors 81
3.4.7 Ray Intersection with Surfaces of Higher Order 82
3.4.8 Gradient Index Media 83
3.4.9 Differential Raytrace, Parabasal Rays and Ray Tubes 84
3.4.10 Diffractive Surfaces 86
3.4.10.1 Introduction 86
3.4.10.2 Plane linear gratings 87
3.4.10.3 Generalized Diffractive Surfaces 88
3.4.11 Non-Sequential Raytrace 88
3.4.12 Special Ray Types 89
3.4.13 Ray Aiming 91
3.4.13.1 Introduction 91
3.4.13.2 Paraxial Aiming Approach 94
3.4.13.3 Real Ray Aiming 95
3.4.13.4 General Formulation of Greynolds 97
3.4.13.5 Examples 97
3.4.14 Complex Rays 98
3.4.14.1 Introduction 98
3.4.14.2 Gaussian Beams with Complex Source Point 99
3.4.14.3 Tracing Complex Rays 100
3.5 Paraxial Approximation 101
3.5.1 Introduction 101
3.5.2 Abbe Invariant 104
3.5.2.1 Paraxial Definition 104
3.5.2.2 Generalized Abbe Invariant 105
3.5.3 Superposition of Rays 105
3.5.4 Generalized Paraxiality 106
3.5.5 Collinear Transform 107
3.6 Matrix Calculus 108
3.6.1 Introduction 108
3.6.2 Properties of the Matrices 110
3.6.3 Matrices of Simple Components 111
3.6.4 Decompositions of an ABCD Matrix 112
3.6.4.1 Iwasawa Decomposition 112
3.6.4.2 Alternative Decompositions 113
3.6.5 Matrices for Generalized Geometries 115
3.6.5.1 Two-dimensional 3 x 3 Matrices for Plane-Symmetric Systems 115
3.6.5.2 Centered 4 x 4 Matrices 116
3.6.5.3 Special 4 x 4 Matrices 117
3.6.5.4 Decomposition of a 4 x4 Matrix 118
3.6.5.5 General 5 x 5 Matrices 119
3.7 Helmholtz-Lagrange Invariant 120
3.7.1 Introduction 120
3.7.2 Lagrange Invariant for Arbitrary z-Positions 122
3.7.3 Generalized Etendue Definitions 122
3.7.4 Smith's Cosine Invariant 124
3.8 Delano Diagram 124
3.8.1 Definition 124
3.8.2 Properties of the Delano Diagram 125
3.8.3 Vignetting 127
3.8.4 Examples 128
3.9 Gaussian Brackets 131
3.9.1 Introduction 131
3.9.2 Alternative Representations 132
3.9.3 Relation to ABCD Formalism 132
3.9.4 Applications 133
3.9.4.1 Tolerancing 133
3.9.4.2 Achromatization 133
3.9.4.3 Zoom Systems 133
References 134
4 Optical Components 137
4.1 Overview 137
4.2 Single Refractive Spherical Surface 137
4.3 Plane Plates 138
4.3.1 Introduction 138
4.3.2 Perpendicular Plates 139
4.3.2.1 Beam Displacement 139
4.3.2.2 Aberrations 140
4.3.3 Tilted Plates 140
4.3.4 Non-Orthogonal Prisms 141
4.3.5 Plane-parallel Plate in a Convergent Beam 141
4.4 Lenses 143
4.4.1 Notations 143
4.4.2 Focal Length 144
4.4.3 Principal Planes and Surfaces 146
4.4.4 Lens Shape and Bending 147
4.4.5 Generalized Bending 151
4.4.6 Modified Bending Definition 152
4.4.7 Thick Lenses 152
4.4.8 The Ideal Lens Dilemma 154
4.5 Mirrors 157
4.6 Aspheres 158
4.6.1 Introduction 158
4.6.2 Conic Surfaces 159
4.6.2.1 Introduction 159
4.6.2.2 Parabolic Mirrors 161
4.6.2.3 Ellipsoidal Mirror 162
4.6.2.4 Hyperboloidal Mirror 163
4.6.2.5 Grazing Incidence Telescopes 163
4.6.3 Cartesian Ovaloids 164
4.6.3.1 Introduction 164
4.6.3.2 Special Case Infinity 165
4.6.3.3 Special Case Aplanatism 167
4.6.4 Asphere with Polynomial Expansion 167
4.6.4.1 Traditional Taylor Expansion 167
4.6.4.2 Strong Forbes Aspheres 168
4.6.4.3 Mild Forbes Aspheres 170
4.6.4.4 Superconical Surfaces 171
4.7 Freeform Surfaces 172
4.7.1 Introduction 172
4.7.2 Basic Shape 174
4.7.3 Boundary and Projection Factor 175
4.7.4 Polynomial Functions 176
4.7.4.1 Introduction 176
4.7.4.2 Monomials 176
4.7.4.3 Zernike Polynomials 176
4.7.4.4 Chebyshev Polynomials 178
4.7.4.5 Legendre Polynomials 178
4.7.4.6 Forbes Freeform Surfaces 178
4.7.5 Functional Systems with Local Support 179
4.7.5.1 Introduction 179
4.7.5.2 Radial Basis Functions 179
4.7.5.3 Splines 182
4.7.6 Technological Constraints and Manufacturability 185
4.8 Special Component Types 185
4.8.1 Cylindrical Lenses 185
4.8.1.1 Introduction 185
4.8.1.2 Aspherical Cylindrical Lenses 186
4.8.1.3 Toroidal Lenses 188
4.8.1.4 Toroidal Ring Lens 188
4.8.1.5 Combinations of Cylindrical Lenses 189
4.8.2 Fresnel Lenses 191
4.8.2.1 Introduction 191
4.8.2.2 Basic Equations 193
4.8.2.3 Properties of Fresnel Lenses 193
4.8.3 Axicons 194
4.8.3.1 Geometrical Concept 194
4.8.3.2 Physical Description 195
4.8.4 Variable Components 197
4.8.4.1 Introduction 197
4.8.4.2 Electrowetting Liquid Lenses 198
4.8.4.3 Electrophoretic Liquid Lenses 199
4.8.4.4 Hydraulic Membrane Liquid Lenses 200
4.8.4.5 Liquid Crystal Lenses 200
4.8.4.6 Deformable Mirrors 200
4.8.4.7 Alvarez Lenses 202
4.8.5 Lenslet Arrays 202
4.8.5.1 Introduction 203
4.8.5.2 Special Lens Arrays 203
4.8.5.3 Matrix Calculation of Arrays 204
4.8.5.4 Applications 205
4.8.6 Digital Mirror Device 208
4.9 Gradient Index Lenses 210
4.9.1 Introduction 210
4.9.2 Quadratic Radial Index Lenses 211
4.9.3 Axial Gradient Index Lenses 212
4.9.4 Stigmatic Imaging Gradient Index-Lenses 212
4.10 Prisms 215
4.10.1 Reflecting Prisms 215
4.10.2 Roof Prisms 216
4.10.2.1 Description of the Geometry 216
4.10.2.2 Dimensioning of the Size 217
4.10.2.3 Image Formation Problems Related to Roof Edges 217
4.10.3 Dispersion Prisms 218
4.10.3.1 Introduction 218
4.10.3.2 Achromatic Prism Pairs 219
4.10.3.3 Anamorphic Prism Pair 220
4.10.3.4 Fery Prisms 221
4.10.4 Risley Prisms 221
4.11 Diffractive Elements 223
4.11.1 Linear Gratings 223
4.11.1.1 Introduction 223
4.11.1.2 Blazed Grating 224
4.11.2 General Diffractive Elements 226
4.11.2.1 Introduction 226
4.11.2.2 Discretization and Quantization 230
4.11.2.3 Types of DOE 231
4.11.2.4 Fresnel Zone Plates 232
4.11.2.5 Classical Diffractive Lens 232
4.11.2.6 Decomposition into Orders or Zones 233
4.11.2.7 Problems of Real Diffractive Elements 234
4.11.3 Modelling DOEs by Raytrace 235
4.12 Diffusor Plates 235
4.12.1 Types of Diffusers 236
4.12.2 Properties of Diffusers 236
4.12.3 Angle Characteristic 238
References 239
5 Imaging Systems 243
5.1 Introduction 243
5.1.1 Mathematical Description of Optical Systems 243
5.1.2 System Notations 243
5.1.3 System Layout 244
5.1.4 Terms of Performance Evaluation 246
5.2 Geometrical Imaging 247
5.2.1 Introduction 247
5.2.2 Lens Makers' Formula 248
5.2.3 Paraxial Formulas 251
5.2.4 Imaging Transfer Length 251
5.2.5 Principal Planes 252
5.2.5.1 Principal Planes of Compound Systems 252
5.2.5.2 Principal Planes of Two-Lens Systems 253
5.2.6 Multi-Lens Systems 255
5.3 Magnification, Field of View, Aperture and Vergence 257
5.3.1 Introduction 257
5.3.2 Numerical Aperture and F-Number 257
5.3.3 Generalized F-number 259
5.3.4 Effective Aperture 259
5.3.5 Magnification 259
5.3.5.1 Finite Systems 259
5.3.5.2 Influence of the Pupil Upon the Magnification 260
5.3.5.3 Magnification of Afocal Systems 263
5.3.5.4 Depth Magnification 263
5.3.6 Vergence 263
5.4 Pupil 264
5.4.1 Introduction 264
5.4.2 Pupil Matching in Composed Systems 266
5.4.3 Field Lenses 266
5.4.4 Pupil and Perspective 269
5.5 Vignetting 269
5.5.1 Introduction 269
5.5.2 Photometric Natural Vignetting 271
5.5.3 Artificial Geometrical Vignetting 272
5.5.3.1 Introduction 272
5.5.3.2 Approximation of the Pupil by an Ellipse 274
5.5.3.3 Analytical Approach in the Infinity case 274
5.5.3.4 Definition of Vignetting Coefficients 276
5.5.3.5 Paraxial Vignetting Calculation 277
5.5.3.6 Vignetting for Complicated Pupil Shapes 279
5.5.3.7 Vignetting Inside a Gradient index Lens 279
5.5.3.8 Vignetting in Phase Space 279
5.6 Infinity Cases for Field and Pupil 281
5.6.1 Overview 281
5.6.2 Afocal Imaging 282
5.6.2.1 Introduction 282
5.6.2.2 Relay Imaging with Afocals 283
5.6.2.3 Applications and Examples 284
5.6.3 Telecentricity 284
5.6.3.1 Introduction 284
5.6.3.2 Examples 286
5.7 Imaging with Mirrors 287
5.7.1 Introduction 287
5.7.2 Centred Two-Mirror Setups 288
5.7.3 Obscuration-Free Two-Mirror Systems 290
5.7.4 Three-Mirror Setups 291
5.8 Imaging Without Rotational Symmetry 292
5.8.1 Anamorphic Systems 292
5.8.1.1 Introduction 292
5.8.1.2 Classes of Anamorphic Systems 294
5.8.1.3 Anamorphic Imaging Systems 295
5.8.1.4 Refractive Anamorphic Systems 296
5.8.1.5 Reflective Anamorphic Systems 297
5.8.2 Scheimpflug Imaging 297
5.8.2.1 Introduction 297
5.8.2.2 Keystone Distortion 299
5.8.2.3 Generalized Scheimpflug Setup 299
5.8.3 General Skew Imaging 301
5.8.4 Image Plane Tilt 302
5.8.4.1 Introduction 302
5.8.4.2 Plane Symmetric Systems and the Scheimpflug Condition 302
5.8.4.3 Tilt of a Skew Spherical Surface 303
5.8.4.4 Tilt of a Skew Conical Mirror Surface 303
5.8.4.5 Prism Systems 305
5.9 Miscellaneous 305
5.9.1 Herzberger's Diapoints 305
5.9.2 Canonical Coordinates 305
5.9.2.1 Axis Case 305
5.9.2.2 Off-Axis Case 307
5.9.3 Imaging of Curved Objects 308
References 309
6 Diffraction and Point Spread Function 311
6.1 Diffraction Phenomena 311
6.1.1 Introduction 311
6.1.2 Fresnel Number 311
6.1.3 Huygens Principle 315
6.2 Calculation of Diffraction Effects 316
6.2.1 Introduction 316
6.2.2 Levels of Modelling 317
6.2.3 Wave Equation 318
6.2.4 Green's Theorem 318
6.2.5 Alternative Formulations of Diffraction Integrals 319
6.2.5.1 Introduction 319
6.2.5.2 Kirchhoff Integral 320
6.2.5.3 Rayleigh-Sommerfeld Diffraction Integrals 321
6.2.5.4 Angular Spectrum of PlaneWaves 322
6.2.5.5 Convolution Formulation 322
6.2.5.6 Fresnel Approximation 323
6.2.5.7 Fraunhofer Far-field 324
6.2.5.8 Debye Diffraction Integral 325
6.2.5.9 Collins Integral 325
6.2.6 Numerical Computation of Diffraction Integrals 327
6.2.6.1 Introduction 327
6.2.6.2 The Phase Oscillation Problem 327
6.2.6.3 Sampling in Diffraction Calculations 328
6.2.6.4 Removal of Parabolic Curvature 330
6.2.6.5 Sinc-based Algorithm 333
6.2.6.6 Boundary and Sampling Problems 335
6.2.6.7 Phase Masks: TEA-, LPI- and LSI Approximations 336
6.3 Point Spread Function in Optical Systems 337
6.3.1 Hybrid Model of the Diffraction Calculation in Optical Systems 337
6.3.2 Ideal point spread function 338
6.3.2.1 Introduction 338
6.3.2.2 Airy Pattern 340
6.3.2.3 Axial Distribution 341
6.4 PSF in Case of Apodization 343
6.4.1 Apodization Effects 343
6.4.2 Super-Gaussian Profile 345
6.5 Focusing at Low Fresnel Numbers 347
6.5.1 Introduction 347
6.5.2 Focal Shift 349
6.6 Focusing at High Numerical Aperture 350
6.6.1 Introduction 350
6.6.2 Scalar Apodization Effects 351
6.6.3 Natural Mapping Function 354
6.6.4 Vectorial Diffraction Integral 355
6.6.4.1 Introduction 355
6.6.4.2 Richards-Wolf Integral 356
6.6.4.3 Cartesian Debye Integral Formulation 357
6.6.4.4 Formulation of Mansuripur 358
6.6.4.5 Relative Size of Vectorial Effects 358
6.6.4.6 Example Calculation 359
6.7 PSF for Compound Systems 359
6.7.1 Introduction 361
6.7.2 Coherent Field Propagation Through a Single Thin Lens 361
6.7.3 Coherent Propagation Through a Cascaded System 362
6.8 Cascaded Diffraction in Optical Systems 365
6.8.1 Introduction 365
6.8.2 System Model 366
6.8.3 Examples 369
6.8.3.1 Diffraction of a Gaussian Beam 369
6.8.3.2 Double Gauss Camera Lens 372
6.9 Miscellaneous 374
6.9.1 Polychromatic PSF 374
6.9.2 Line of Sight 375
6.9.3 Extended Zernike Approach 377
6.9.3.1 Introduction 377
6.9.3.2 Properties of the Extended Zernike Approach 378
6.9.4 Encircled Energy 379
6.9.5 Fresnel Edge Diffraction 380
6.9.6 Line Spread Function 381
6.9.7 Slit Diffraction 383
6.10 Field in a Tilted Plane 384
6.10.1 Introduction 384
6.10.2 Analysis of the Geometry 385
6.10.3 z-Rotation by Shear Transform 385
6.10.4 Calculation of the x-Rotation 386
6.10.5 Run Time Performance 388
6.10.6 Examples 388
References 390
7 Optical Transfer Function 395
7.1 Spatial Frequency Concept 395
7.2 Optical Transfer Function 398
7.2.1 Introduction 398
7.2.2 Duffieux Representation of the OTF 400
7.2.3 Contrast and Resolution 401
7.2.4 Ideal MTF 402
7.2.5 Phase Transfer Function 405
7.2.6 Coherent Transfer Function (CTF) 405
7.2.7 Geometrical Approximated Transfer Function (GTF) 407
7.2.8 Connection Between OTF and PSF 407
7.2.9 Three-Dimensional Coherent Transfer Function 409
7.3 Miscellaneous 410
7.3.1 Non-Isotropic Point Spread Function 410
7.3.2 Properties of MTF Curves 410
7.3.3 MTF for Defocused Systems 412
7.3.3.1 Introduction 412
7.3.3.2 Approximations of the MTF 413
7.3.3.3 The Ambiguity Function 413
7.3.4 MTF for Special Pupil Shapes 414
7.3.4.1 MTF for a Square Pupil Shape 414
7.3.4.2 MTF of a Ring Pupil 415
7.3.4.3 MTF for Systems with Apodization 417
7.3.5 Relationships of System Response Functions 417
7.3.6 Imaging of Phase Structures 418
7.4 OTF of Cascaded Systems 420
7.4.1 Introduction 420
7.4.2 Coherent Cascaded Systems 420
7.4.3 Incoherent Cascaded Systems 421
7.4.3.1 Introduction 421
7.4.3.2 Simple Case of Two Gaussian PSFs 422
7.4.4 MTF of a Complete Signal Chain 422
References 423
8 Gaussian Beams 425
8.1 Introduction 425
8.1.1 Definition of Basic Properties 425
8.1.2 Amplitude and Intensity Distributions 427
8.1.3 Complex Representation and Collins Chart 427
8.2 Gaussian Beam Transformation 429
8.2.1 Transform By a Lens 429
8.2.2 Beam Transform Through Afocal Telescopes 430
8.2.3 Transform by an ABCD-Segment 432
8.2.4 Matching of Beams 433
8.2.4.1 Introduction 433
8.2.4.2 Given Incoming and Outgoing Gaussian Beams 433
8.2.4.3 Given Gaussian Beam, Lens and Size of the New Beam 434
8.2.4.4 Given Gaussian Beam, Lens and New Waist Position 434
8.2.4.5 Given Gaussian Beam, Lens Location and Size of the New Beam 434
8.2.4.6 LargestWaist Distance 434
8.2.4.7 Minimal Beam Size in a Certain Distance 435
8.2.4.8 Coupling of Two TEMoo-Beams 436
8.3 Astigmatic Beams 437
8.3.1 Orthogonal Astigmatic Gaussian Beams 437
8.3.2 General Astigmatic Gaussian Beams 438
8.4 Ray Equivalent of Gaussian Beams 440
8.4.1 Principle 440
8.4.2 Ray Equivalent in Three Dimensions 442
8.4.3 Selecting General Reconstruction Rays 443
8.4.4 Numerical Reconstruction of ABCD by Parabasal Rays 444
8.5 Truncated Gaussian Beams 445
8.5.1 One-dimensional Beam 445
8.5.2 Circular Symmetric Beam 446
8.5.2.1 Introduction 446
8.5.2.2 Exact Field Calculation 446
8.5.2.3 General Properties 447
8.6 Gaussian Beams Beyond the Paraxial Approximation 449
8.6.1 Introduction 449
8.6.2 Exact Solution of theWave Equation 449
8.6.3 Taylor Expansion of the Complex Field 450
8.6.4 Angle Expansion of Higher Order Modes 450
8.6.5 Off-axis-Focusing 451
8.7 Gaussian Beam with Spherical Aberration 452
8.7.1 Introduction 452
8.7.2 Intensity on Axis 452
8.7.3 Aberration Balancing 453
8.7.4 Numerical Example Calculations 453
8.8 Single Mode Fibre Coupling with Gaussian Beam 454
8.8.1 Basic Fibre Parameters 454
8.8.2 Fundamental Mode Shape 455
8.8.3 Calculation of Fibre Radiation Mode 455
8.9 Partial Coherent Gauss-Schell Beams 456
8.9.1 Introduction 456
8.9.2 Simple Gauss-Schell Beams 456
8.9.3 Propagation of Gauss-Schell Beams 458
8.9.3.1 Initial Values 458
8.9.3.2 Free-Space Propagation 458
8.9.3.3 ABCD-Propagation 459
References 459
9 Photometry and Radiometry 463
9.1 Introduction 463
9.1.1 General Remarks 463
9.1.2 Definition of the Radiometric Quantities 463
9.1.3 Photometric Quantities 464
9.1.4 Comparison of Quantities 464
9.1.5 Energy, Power and Photons 464
9.1.6 Solid Angle 466
9.2 Lambertian Source 466
9.2.1 Classical Lambertian Radiator 466
9.2.2 Generalized Lambertian Radiator 467
9.3 Radiometric Transfer of Energy 467
9.3.1 Radiance and Irradiance 467
9.3.2 Radiometric Flux Transfer 468
9.3.3 Analytical Solutions for Simple Geometries 469
9.3.3.1 Surface Element Illuminated by a Point Source 469
9.3.3.2 Circular Plane Surface Illuminated by a Point Source 469
9.3.3.3 Parallel Surface Elements 470
9.3.3.4 Circular Lambertian Source and Point-Like Receiver 470
9.3.3.5 Circular Lambertian Source and Circular Receiver 470
9.3.4 Numerical Radiation Transfer 471
9.3.4.1 Introduction 471
9.3.4.2 Monte-Carlo Raytracing Approach 472
9.3.4.3 Source Modelling 473
9.3.4.4 Evaluation of the Detector Irradiance 474
9.3.4.5 Examples 474
9.3.5 Radiation Transport with Interaction 475
9.3.6 Ray Tube Model 476
9.4 Radiometry of Optical Systems 477
9.4.1 Introduction 477
9.4.2 Sine Condition and Photometry 478
9.4.3 Aplanatic Systems 479
9.4.4 Natural Vignetting 480
9.4.4.1 General Case Free of Vignetting 480
9.4.4.2 Formulation With Entrance and Exit Pupil 481
9.4.4.3 Monocentric and Telecentric 4-f -Systems 482
9.4.4.4 System With Rear Stop 482
9.4.4.5 System with Front Stop 484
9.5 Radiometry with Partial Coherent Light 485
9.5.1 Introduction 485
9.5.2 Partial Coherent Light 486
9.5.3 Generalized Radiance 486
9.5.4 Propagation of the Generalized Radiance 487
9.5.4.1 Introduction 487
9.5.4.2 Example Lambertian Source 487
References 488
10 Phase Space Representation 491
10.1 General Aspects 491
10.1.1 Motivation 491
10.1.2 Legendre Transform and Equation of Motion 492
10.1.3 Uncertainty Relation 494
10.1.4 Analogy to Mechanics 495
10.2 Geometrical Ray Model 495
10.2.1 Introduction 495
10.2.2 Phase Space Representation of Ray Bundle Transport 496
10.2.3 Ray Aberrations 499
10.3 Wigner Distribution Function 500
10.3.1 Introduction 501
10.3.2 Theory of the Wigner Function 502
10.3.3 Wigner Function of Gaussian Beams 503
10.3.4 Wigner Function of a Decomposed Field 503
10.3.5 Propagation of the Wigner Function 505
10.3.6 Transfer Through Thin Masks 506
10.3.7 Wigner Function of a Slit 506
10.3.8 Wigner Function for Gauss-Schell Beams 507
10.3.9 Examples 508
10.4 Photometry in Phase Space 508
10.4.1 Introduction 508
10.4.2 Conservation of Energy 510
10.4.3 Vignetting 511
10.4.4 Fibre Illumination 512
10.4.5 Further Examples 513
10.5 Miscellaneous 515
10.5.1 Caustic in Phase Space 515
10.5.2 Phase Space Discussion of Sampling 515
10.5.3 Phase Space Analyzer 517
10.5.4 Fractional Fourier Transform 518
10.5.4.1 Definition 518
10.5.4.2 Fresnel Integral and Fractional Fourier Transform 519
10.5.4.3 Gradient Index Lenses 520
10.5.5 Linear Canonical Transform 520
References 521
11 Computation and Digital Processing of Images 523
11.1 Introduction 523
11.2 Image Computation 523
11.2.1 Introduction 523
11.2.2 Ray Based Image Calculation 524
11.2.3 Physical Model of Image Formation 526
11.2.3.1 Introduction 526
11.2.3.2 Image Formation According to Fourier and Abbe 527
11.2.3.3 4-f -Fourier Imaging Model 528
11.2.3.4 Complete 6-f -Fourier Model 530
11.2.3.5 Coherent Image Formation 531
11.2.3.6 Incoherent Image Formation 532
11.2.3.7 Isoplanatic Condition 532
11.2.3.8 Multifocal Visual Perception 534
11.2.3.9 Image Computation Examples 535
11.3 Confocal Imaging 538
11.3.1 Introduction 538
11.3.2 Scanning Options 540
11.3.3 Image Model 540
11.3.4 Influence of Pinhole Size and Aberrations 542
11.4 Anisoplanatic Imaging 543
11.4.1 Introduction 543
11.4.2 Interpolation of Zernikes 546
11.4.3 Spatially Variant Convolution 547
11.4.4 Interpolation by Principal Components 549
11.4.5 Taylor Expansion of the PSF 551
11.4.6 Distortion 552
11.4.7 Special Varying Blur Types 552
11.5 Digital Imaging Processing 554
11.5.1 Introduction 554
11.5.2 Spatial Discretization and Resolution 555
11.5.3 Management of Image Colour 558
11.6 Image Quality Metrics 559
11.6.1 Introduction 559
11.6.2 MSE Criterion 560
11.6.3 SSim Criterion 561
11.6.3.1 Introduction 561
11.6.3.2 Definition and Computation 561
11.6.3.3 Examples 562
11.7 Digital Image Restoration 563
11.7.1 Introduction 563
11.7.2 Deconvolution 565
11.7.2.1 Wiener Method 565
11.7.2.2 Algebraic Deconvolution 565
11.7.2.3 Further Methods 567
11.7.2.4 Examples 567
11.7.3 Denoising 568
11.8 Plenoptical Imaging 569
11.8.1 Introduction 569
11.8.2 Model of the Plenoptical Camera 1.0 572
11.8.3 Phase Space Discussion 573
11.8.4 Digital Refocusing 575
11.8.5 Depth of Focus 575
11.8.6 Examples 578
11.9 Digital Phase Imaging 579
11.9.1 Introduction 579
11.9.2 Coherent Diffraction Imaging 580
11.9.3 Transport of Intensity Approach 581
11.9.4 Ptychography 582
11.10 Extended Depth of Focus 583
11.10.1 Introduction 583
11.10.2 Toraldo Phase Mask 586
11.10.3 Cubic Phase Plate 586
References 588
12 Mathematical Appendix 591
12.1 Fourier Transform and Related 591
12.1.1 Fourier Series 591
12.1.2 Fourier Transform 591
12.1.3 Discrete Fourier Transform 592
12.1.3.1 Definition 592
12.1.3.2 Sampling and Aliasing 593
12.1.3.3 Leakage 594
12.1.3.4 Zero padding and Balanced Sampling 596
12.1.4 Chirp-z-Transform 597
12.1.5 Semi-Analytical Fourier Transform 600
12.1.6 Non-Uniform Fourier Transform 601
12.1.7 Special Fourier Related Operations 603
12.1.7.1 Convolution Integral 603
12.1.7.2 Auto-Correlation 604
12.1.7.3 Cross Correlation and Covariance 604
12.1.8 Fourier Slice Theorem and Radon Transform 605
12.2 Miscellaneous 606
12.2.1 Singular Value Decomposition 606
12.2.2 Principal Component Analysis and Karhunen-Loeve Transform 607
12.2.3 Quasi-Random Sampling 608
12.2.4 Window Functions 609
12.2.5 Fresnel Integrals 610
12.2.6 Method of Stationary Phase 612
12.2.6.1 Introduction 612
12.2.6.2 One-Dimensional Case 613
12.2.6.3 Two-Dimensional Case 613
References 614
Index 615
