# Contents. Gbur, Gregory J. Mathematical methods for optical physics and engineering digitalisiert durch: IDS Basel Bern

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1 Preface page xv 1 Vector algebra Preliminaries Coordinate System invariance Vector multiplication Useful products of vectors Linear vector Spaces Focus: periodic media and reciprocal lattice vectors Additional reading Exercises 24 2 Vector calculus Introduction Vector integration The gradient, V Divergence, V Thecurl, Vx Further applications of V Gauss' theorem (divergence theorem) Stokes' theorem Potential theory Focus: Maxwell 's equations in integral and differential form Focus: gauge freedom in Maxwell's equations Additional reading Exercises 60 3 Vector calculus in curvilinear coordinate Systems Introduction: Systems with different symmetries General orthogonal coordinate Systems Vector Operators in curvilinear coordinates Cylindrical coordinates 73 Gbur, Gregory J. Mathematical methods for optical physics and engineering 2011 digitalisiert durch: IDS Basel Bern

2 3.5 Spherical coordinates Exercises 79 Matrices and linear algebra Introduction: Polarization and Jones vectors Matrix algebra Systems of equations, determinants, and inverses Orthogonal matrices Hermitian matrices and unitary matrices Diagonalization of matrices, eigenvectors, and eigenvalues Gram-Schmidt orthonormalization Orthonormal vectors and basis vectors Functions of matrices Focus: matrix methods for geometrical optics Additional reading Exercises 133 Advanced matrix techniques and tensors Introduction: Foldy-Lax scattering theory Advanced matrix terminology Left-right eigenvalues and biorthogonality Singular value decomposition Other matrix manipulations Tensors Additional reading Exercises 174 Distributions Introduction: Gauss' law and the Poisson equation Introduction to delta functions Calculus of delta functions Other representations of the delta function Heaviside step function Delta functions of more than one variable Additional reading Exercises 192 Infinite series Introduction: the Fabry-Perot interferometer Sequences and series Series convergence Series of functions Taylor series Taylor series in more than one variable Power series Focus: convergence of the Born series 221

3 7.9 Additional reading Exercises Fourier series Introduction: diffraction gratings Real-valued Fourier series Examples Integration ränge of the Fourier series Complex-valued Fourier series Properties of Fourier series Gibbs phenomenon and convergence in the mean Focus: X-ray diffraction from crystals Additional reading Exercises Complex analysis Introduction: electric potential in an infinite cylinder Complex algebra Functions of a complex variable Complex derivatives and analyticity Complex integration and Cauchy's integral theorem Cauchy's integral formula Taylor series Laurent series Classification of isolated singularities Branch points and Riemann surfaces Residue theorem Evaluation of definite integrals Cauchy principal value Focus: Kramers-Kronig relations Focus: optical vortices Additional reading Exercises Advanced complex analysis Introduction Analytic continuation Stereographic projection Conformal mapping Significant theorems in complex analysis Focus: analytic properties of wavefields Focus: optical cloaking and transformation optics Exercises Fourier transforms Introduction: Fraunhofer diffraction 350 ix

4 x Contents 11.2 The Fourier transform and its inverse Examples of Fourier transforms Mathematical properties of the Fourier transform Physical properties of the Fourier transform Eigenfunctions of the Fourier Operator Higher-dimensional transforms Focus: spatial filtering Focus: angular spectrum representation Additional reading Exercises Other integral transforms Introduction: the Fresnel transform Linear canonical transforms The Laplace transform Fractional Fourier transform Mixed domain transforms The wavelet transform The Wigner transform Focus: the Radon transform and computed axial tomography (CAT) Additional reading Exercises Discrete transforms Introduction: the sampling theorem Sampling and the Poisson sum formula The discrete Fourier transform Properties of the DFT Convolution Fast Fourier transform The z-transform Focus: z-transforms in the numerical Solution of Maxwell's equations Focus: the Talbot effect Exercises Ordinary differential equations Introduction: the classic ODEs Classification of ODEs Ordinary differential equations and phase space First-order ODEs Second-order ODEs with constant coefficients The Wronskian and associated strategies Variation of parameters Series Solutions Singularities, complex analysis, and general Frobenius Solutions 481

5 14.10 Integral transform Solutions Systems of differential equations Numerical analysis of differential equations Additional reading Exercises Partial differential equations Introduction: propagation in a rectangular waveguide Classification of second-order linear PDEs Separation of variables Hyperbolic equations Elliptic equations Parabolic equations Solutions by integral transforms Inhomogeneous problems and eigenfunction Solutions Infinite domains; the d'alembert Solution Method of images Additional reading Exercises Bessel functions Introduction: propagation in a circular waveguide Bessel's equation and series Solutions The generating function Recurrence relations Integral representations Hankel functions Modified Bessel functions Asymptotic behavior of Bessel functions Zeros of Bessel functions Orthogonality relations Bessel functions of fractional order Addition theorems, sum theorems, and product relations Focus: nondiffracting beams Additional reading Exercises Legendre functions and spherical harmonics Introduction: Laplace's equation in spherical coordinates Series Solution of the Legendre equation Generating function Recurrence relations Integral formulas Orthogonality Associated Legendre functions 597 xi

6 xii Contents 17.8 Spherical harmonics Spherical harmonic addition theorem Solution of PDEs in spherical coordinates Gegenbauer polynomials Focus: multipole expansion for static electric fields Focus: vector spherical harmonics and radiation fields Exercises Orthogonal functions Introduction: Sturm-Liouville equations Hermite polynomials Laguerre functions Chebyshev polynomials Jacobi polynomials Focus: Zernike polynomials Additional reading Exercises Green's functions Introduction: the Huygens-Fresnel integral Inhomogeneous Sturm-Liouville equations Properties of Green's functions Green's functions of second-order PDEs Method of images Modal expansion of Green's functions Integral equations Focus: Rayleigh-Sommerfeld diffraction Focus: dyadic Green's function for Maxwell's equations Focus: scattering theory and the Born series Exercises The calculus of variations Introduction: principle of Fermat Extrema of functions and functionals Euler's equation Second form of Euler's equation Calculus of variations with several dependent variables Calculus of variations with several independent variables Euler's equation with auxiliary conditions: Lagrange multipliers Hamiltonian dynamics Focus: aperture apodization Additional reading Exercises Asymptotic techniques Introduction: foundations of geometrical optics 748

7 xiii 21.2 Definition of an asymptotic series Asymptotic behavior of integrals Method of stationary phase Method of steepest descents Method of stationary phase for double integrals Additional reading Exercises 773 Appendix A The gamma function 775 A.l Definition 775 A.2 Basic properties 776 A.3 Stirling's formula 778 A.4 Beta function 779 A.5 Useful integrals 780 Appendix B Hypergeometric functions 783 B.l Hypergeometric function 784 B.2 Confluent hypergeometric function 785 B.3 Integral representations 785 References 787 Index 793

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