Mathematical Methods for Optical Physics and Engineering

Cover......Page 1
Half-title......Page 3
Title......Page 5
Copyright......Page 6
Dedication......Page 7
Contents......Page 9
Preface......Page 17
1.1 Preliminaries......Page 21
1.2 Coordinate system invariance......Page 24
1.3.2 Scalar or dot product......Page 29
1.3.3 Vector or cross product......Page 30
1.4.1 Triple scalar product......Page 32
1.5 Linear vector spaces......Page 33
1.6 Focus: periodic media and reciprocal lattice vectors......Page 37
1.8 Exercises......Page 44
2.1 Introduction......Page 48
2.2.1 Path integrals......Page 49
2.2.2 Surface integrals......Page 51
2.3 The gradient, V......Page 55
2.4 Divergence, V......Page 57
2.5 The curl, Vx......Page 61
2.6.2…......Page 63
2.6.4 Laplacian,…......Page 64
2.7 Gauss\' theorem (divergence theorem)......Page 65
2.8 Stokes\' theorem......Page 67
2.9 Potential theory......Page 68
2.10 Focus: Maxwell\'s equations in integral and differential form......Page 71
2.10.1 Gauss’ law and “no magnetic monopoles”......Page 72
2.10.2 Faraday’s law and Ampère’s law......Page 73
2.11 Focus: gauge freedom in Maxwell\'s equations......Page 77
2.13 Exercises......Page 80
3.1 Introduction: systems with different symmetries......Page 84
3.2 General orthogonal coordinate systems......Page 85
3.3.1 The gradient......Page 89
3.3.2 The divergence......Page 90
3.3.3 The curl......Page 91
3.4 Cylindrical coordinates......Page 93
3.5 Spherical coordinates......Page 96
3.6 Exercises......Page 99
4.1 Introduction: Polarization and Jones vectors......Page 103
4.1.3 Optical attenuator......Page 107
4.2 Matrix algebra......Page 108
4.3 Systems of equations, determinants, and inverses......Page 113
4.3.1 Gaussian elimination......Page 119
4.3.2 Gauss–Jordan matrix inversion......Page 120
4.4 Orthogonal matrices......Page 122
4.5 Hermitian matrices and unitary matrices......Page 125
4.6 Diagonalization of matrices, eigenvectors, and eigenvalues......Page 127
4.6.1 Hermitian matrices, normal matrices, and eigenvalues......Page 132
4.7 Gram--Schmidt orthonormalization......Page 135
4.8 Orthonormal vectors and basis vectors......Page 138
4.10.1 Introduction......Page 140
4.10.2 Translation and refraction in matrix form......Page 143
4.10.3 Lenses......Page 145
4.10.4 Conjugate matrix......Page 148
4.10.5 Imaging by a thin lens......Page 150
4.10.6 Lens with a dielectric plate......Page 151
4.10.7 Telescope......Page 152
4.12 Exercises......Page 153
5.1 Introduction: Foldy–Lax scattering theory......Page 159
5.2 Advanced matrix terminology......Page 162
5.3 Left--right eigenvalues and biorthogonality......Page 163
5.4 Singular value decomposition......Page 166
5.5.1 Gaussian elimination with pivoting......Page 173
5.5.2 LU decomposition......Page 175
5.6.1 Einstein summation convention......Page 179
5.6.2 Covariant and contravariant vectors and tensors......Page 180
5.6.3 Tensor addition and tensor products......Page 182
5.6.4 The metric tensor......Page 184
5.6.5 Pseudotensors......Page 188
5.6.6 Tensors in curvilinear coordinates......Page 189
5.6.7 The covariant derivative......Page 192
5.8 Exercises......Page 194
6.1 Introduction: Gauss\' law and the Poisson equation......Page 197
6.2 Introduction to delta functions......Page 201
6.3.3 Composition......Page 204
6.4 Other representations of the delta function......Page 205
6.5 Heaviside step function......Page 207
6.6 Delta functions of more than one variable......Page 208
6.8 Exercises......Page 212
7.1 Introduction: the Fabry--Perot interferometer......Page 215
7.2 Sequences and series......Page 218
7.3 Series convergence......Page 221
7.4 Series of functions......Page 230
7.5 Taylor series......Page 233
7.6 Taylor series in more than one variable......Page 238
7.7 Power series......Page 240
7.8 Focus: convergence of the Born series......Page 241
7.10 Exercises......Page 246
8.1 Introduction: diffraction gratings......Page 250
8.2 Real-valued Fourier series......Page 253
8.3 Examples......Page 256
8.5 Complex-valued Fourier series......Page 259
8.6.1 Symmetric series......Page 260
8.6.2 Existence, completeness, and closure relations......Page 262
8.7 Gibbs phenomenon and convergence in the mean......Page 263
8.8 Focus: X-ray diffraction from crystals......Page 266
8.10 Exercises......Page 269
9.1 Introduction: electric potential in an infinite cylinder......Page 272
9.2 Complex algebra......Page 274
9.3 Functions of a complex variable......Page 278
9.4 Complex derivatives and analyticity......Page 281
9.5 Complex integration and Cauchy\'s integral theorem......Page 285
9.6 Cauchy\'s integral formula......Page 289
9.7 Taylor series......Page 291
9.8 Laurent series......Page 293
9.9 Classification of isolated singularities......Page 296
9.10 Branch points and Riemann surfaces......Page 298
9.11 Residue theorem......Page 305
9.12.1 Definite integrals…......Page 308
9.12.2 Definite integrals…......Page 309
9.12.3 Definite integrals…......Page 312
9.12.4 Other integrals......Page 314
9.13 Cauchy principal value......Page 317
9.14 Focus: Kramers--Kronig relations......Page 319
9.15 Focus: optical vortices......Page 322
9.17 Exercises......Page 328
10.2 Analytic continuation......Page 332
10.3 Stereographic projection......Page 336
10.4 Conformal mapping......Page 345
10.5 Significant theorems in complex analysis......Page 352
10.5.2 Morera’s theorem......Page 353
10.5.3 Maximum modulus principle......Page 355
10.5.4 Arguments, zeros, and Rouché’s theorem......Page 356
10.6 Focus: analytic properties of wavefields......Page 360
10.7 Focus: optical cloaking and transformation optics......Page 365
10.8 Exercises......Page 368
11.1 Introduction: Fraunhofer diffraction......Page 370
11.2 The Fourier transform and its inverse......Page 372
11.3 Examples of Fourier transforms......Page 374
11.4 Mathematical properties of the Fourier transform......Page 378
11.4.1 Existence conditions for Fourier transforms......Page 379
11.4.2 The Fourier inverse and delta functions......Page 381
11.4.3 Other forms of the Fourier transform......Page 382
11.4.6 Time shift......Page 383
11.4.8 Scaling property......Page 384
11.5 Physical properties of the Fourier transform......Page 385
11.5.1 The convolution theorem......Page 386
11.5.2 Parseval’s theorem and Plancherel’s identity......Page 388
11.5.3 Uncertainty relations......Page 389
11.6 Eigenfunctions of the Fourier operator......Page 392
11.7 Higher-dimensional transforms......Page 393
11.8 Focus: spatial filtering......Page 395
11.9 Focus: angular spectrum representation......Page 397
11.9.1 Angular spectrum representation in a half-space......Page 398
11.9.2 The Weyl representation of a spherical wave......Page 400
11.10 Additional reading......Page 402
11.11 Exercises......Page 403
12.1 Introduction: the Fresnel transform......Page 406
12.1.2 Scaling property......Page 407
12.1.4 Inverse Fresnel transform......Page 408
12.1.5 Convolution......Page 409
12.1.7 Observations......Page 410
12.2 Linear canonical transforms......Page 411
12.2.3 Composition......Page 413
12.2.4 Identity limit......Page 414
12.3 The Laplace transform......Page 415
12.3.1 Existence of the Laplace transform......Page 416
12.3.5 Differentiation in the time domain......Page 418
12.3.6 Inverse transform......Page 419
12.4 Fractional Fourier transform......Page 420
12.5 Mixed domain transforms......Page 422
12.6 The wavelet transform......Page 426
12.7 The Wigner transform......Page 429
12.8 Focus: the Radon transform and computed axial tomography (CAT)......Page 430
12.10 Exercises......Page 436
13.1 Introduction: the sampling theorem......Page 439
13.2 Sampling and the Poisson sum formula......Page 443
13.3 The discrete Fourier transform......Page 447
13.4.1 Nyquist frequency......Page 450
13.4.2 Shift in the n-domain......Page 451
13.5 Convolution......Page 452
13.6 Fast Fourier transform......Page 453
13.7 The z-transform......Page 457
13.7.1 Relation to Laplace and Fourier transform......Page 459
13.7.2 Convolution theorem......Page 462
13.7.4 Initial and final values......Page 463
13.8 Focus: z-transforms in the numerical solution of Maxwell\'s equations......Page 465
13.9 Focus: the Talbot effect......Page 469
13.10 Exercises......Page 476
14.1 Introduction: the classic ODEs......Page 478
14.2 Classification of ODEs......Page 479
14.3 Ordinary differential equations and phase space......Page 480
14.4.1 Separable equation......Page 489
14.4.2 Exact equation......Page 491
14.4.3 Linear equation......Page 493
14.5 Second-order ODEs with constant coefficients......Page 494
14.6 The Wronskian and associated strategies......Page 496
14.7 Variation of parameters......Page 498
14.8 Series solutions......Page 500
14.9 Singularities, complex analysis, and general Frobenius solutions......Page 501
14.10 Integral transform solutions......Page 505
14.11 Systems of differential equations......Page 506
14.12 Numerical analysis of differential equations......Page 508
14.12.1 Euler’s method......Page 510
14.12.2 Trapezoidal method and implicit methods......Page 512
14.12.3 Runge–Kutta methods......Page 514
14.12.4 Higher-order equations and stiff equations......Page 517
14.12.5 Observations......Page 520
14.14 Exercises......Page 521
15.1 Introduction: propagation in a rectangular waveguide......Page 525
15.2 Classification of second-order linear PDEs......Page 528
15.2.1 Classification of second-order PDEs in two variables with constant coefficients......Page 529
15.2.2 Classification of general second-order PDEs......Page 532
15.2.3 Auxiliary conditions for second-order PDEs, existence, and uniqueness......Page 535
15.3 Separation of variables......Page 537
15.4 Hyperbolic equations......Page 539
15.5 Elliptic equations......Page 545
15.6 Parabolic equations......Page 550
15.7 Solutions by integral transforms......Page 554
15.8 Inhomogeneous problems and eigenfunction solutions......Page 558
15.9 Infinite domains the d\'Alembert solution......Page 559
15.10 Method of images......Page 564
15.12 Exercises......Page 565
16.1 Introduction: propagation in a circular waveguide......Page 570
16.2 Bessel\'s equation and series solutions......Page 572
16.3 The generating function......Page 575
16.4 Recurrence relations......Page 577
16.5 Integral representations......Page 580
16.6 Hankel functions......Page 584
16.7 Modified Bessel functions......Page 585
16.8 Asymptotic behavior of Bessel functions......Page 586
16.9 Zeros of Bessel functions......Page 587
16.10 Orthogonality relations......Page 589
16.11 Bessel functions of fractional order......Page 592
16.12 Addition theorems, sum theorems, and product relations......Page 596
16.13 Focus: nondiffracting beams......Page 599
16.15 Exercises......Page 602
17.1 Introduction: Laplace\'s equation in spherical coordinates......Page 605
17.2 Series solution of the Legendre equation......Page 607
17.3 Generating function......Page 609
17.4 Recurrence relations......Page 610
17.5 Integral formulas......Page 612
17.6 Orthogonality......Page 614
17.7 Associated Legendre functions......Page 617
17.8 Spherical harmonics......Page 622
17.9 Spherical harmonic addition theorem......Page 625
17.10 Solution of PDEs in spherical coordinates......Page 628
17.11 Gegenbauer polynomials......Page 630
17.12 Focus: multipole expansion for static electric fields......Page 631
17.13 Focus: vector spherical harmonics and radiation fields......Page 634
17.14 Exercises......Page 638
18.1 Introduction: Sturm--Liouville equations......Page 642
18.2.1 Series solution......Page 647
18.2.3 Recurrence relations......Page 649
18.2.4 Orthogonality......Page 650
18.2.5 The quantum harmonic oscillator......Page 651
18.2.6 Hermite–Gauss laser beams......Page 653
18.3.1 Series solution......Page 661
18.3.2 Generating function......Page 662
18.3.3 Recurrence relations......Page 663
18.3.4 Associated Laguerre functions......Page 664
18.3.5 Wavefunction of the hydrogen atom......Page 665
18.3.6 Laguerre–Gauss laser beams......Page 667
18.4.1 Polynomials of the second kind......Page 670
18.4.2 Polynomials of the first kind......Page 672
18.5 Jacobi polynomials......Page 674
18.6 Focus: Zernike polynomials......Page 675
18.8 Exercises......Page 682
19.1 Introduction: the Huygens--Fresnel integral......Page 685
19.2 Inhomogeneous Sturm--Liouville equations......Page 689
19.3 Properties of Green\'s functions......Page 694
19.4.1 Elliptic equation: Poisson’s equation......Page 696
19.4.2 Elliptic equation: Helmholtz equation......Page 697
19.4.3 Hyperbolic equation: wave equation......Page 700
19.4.4 Parabolic equation: diffusion equation......Page 702
19.5 Method of images......Page 705
19.6 Modal expansion of Green\'s functions......Page 709
19.7.1 Types of linear integral equation......Page 713
19.7.2 Solution of basic linear integral equations......Page 715
19.7.3 Liouville–Neumann series......Page 716
19.7.4 Hermitian kernels and Mercer’s theorem......Page 719
19.8 Focus: Rayleigh--Sommerfeld diffraction......Page 721
19.9 Focus: dyadic Green\'s function for Maxwell\'s equations......Page 724
19.10 Focus: Scattering theory and the Born series......Page 729
19.11 Exercises......Page 732
20.1 Introduction: principle of Fermat......Page 735
20.2 Extrema of functions and functionals......Page 738
20.3 Euler\'s equation......Page 741
20.4 Second form of Euler\'s equation......Page 747
20.5 Calculus of variations with several dependent variables......Page 750
20.6 Calculus of variations with several independent variables......Page 752
20.7 Euler\'s equation with auxiliary conditions: Lagrange multipliers......Page 754
20.8 Hamiltonian dynamics......Page 759
20.9 Focus: aperture apodization......Page 762
20.11 Exercises......Page 765
21.1 Introduction: foundations of geometrical optics......Page 768
21.2 Definition of an asymptotic series......Page 773
21.3 Asymptotic behavior of integrals......Page 776
21.4 Method of stationary phase......Page 783
21.5 Method of steepest descents......Page 786
21.6 Method of stationary phase for double integrals......Page 791
21.7 Additional reading......Page 792
21.8 Exercises......Page 793
A.1 Definition......Page 795
A.2 Basic properties......Page 796
A.3 Stirling\'s formula......Page 798
A.4 Beta function......Page 799
A.5 Useful integrals......Page 800
Appendix B: Hypergeometric functions......Page 803
B.1 Hypergeometric function......Page 804
B.3 Integral representations......Page 805
References......Page 807
Index......Page 813