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Mathematical Methods for Optical Physics and Engineering

Book Description

The first textbook on mathematical methods focusing on techniques for optical science and engineering, this text is ideal for upper division undergraduate and graduate students in optical physics. Containing detailed sections on the basic theory, the textbook places strong emphasis on connecting the abstract mathematical concepts to the optical systems to which they are applied. It covers many topics which usually only appear in more specialized books, such as Zernike polynomials, wavelet and fractional Fourier transforms, vector spherical harmonics, the z-transform, and the angular spectrum representation. Most chapters end by showing how the techniques covered can be used to solve an optical problem. Essay problems based on research publications and numerous exercises help to further strengthen the connection between the theory and its applications.

Table of Contents

  1. Cover
  2. Half Title
  3. Title Page
  4. Copyright
  5. Contents
  6. Preface
  7. 1. Vector algebra
    1. 1.1 Preliminaries
    2. 1.2 Coordinate system invariance
    3. 1.3 Vector multiplication
    4. 1.4 Useful products of vectors
    5. 1.5 Linear vector spaces
    6. 1.6 Focus: periodic media and reciprocal lattice vectors
    7. 1.7 Additional reading
    8. 1.8 Exercises
  8. 2. Vector calculus
    1. 2.1 Introduction
    2. 2.2 Vector integration
    3. 2.3 The gradient, ∇
    4. 2.4 Divergence, ∇·
    5. 2.5 The curl, ∇×
    6. 2.6 Further applications of ∇
    7. 2.7 Gauss’ theorem (divergence theorem)
    8. 2.8 Stokes’ theorem
    9. 2.9 Potential theory
    10. 2.10 Focus: Maxwell’s equations in integral and differential form
    11. 2.11 Focus: gauge freedom in Maxwell’s equations
    12. 2.12 Additional reading
    13. 2.13 Exercises
  9. 3. Vector calculus in curvilinear coordinate systems
    1. 3.1 Introduction: systems with different symmetries
    2. 3.2 General orthogonal coordinate systems
    3. 3.3 Vector operators in curvilinear coordinates
    4. 3.4 Cylindrical coordinates
    5. 3.5 Spherical coordinates
    6. 3.6 Exercises
  10. 4. Matrices and linear algebra
    1. 4.1 Introduction: Polarization and Jones vectors
    2. 4.2 Matrix algebra
    3. 4.3 Systems of equations, determinants, and inverses
    4. 4.4 Orthogonal matrices
    5. 4.5 Hermitian matrices and unitary matrices
    6. 4.6 Diagonalization of matrices, eigenvectors, and eigenvalues
    7. 4.7 Gram–Schmidt orthonormalization
    8. 4.8 Orthonormal vectors and basis vectors
    9. 4.9 Functions of matrices
    10. 4.10 Focus: matrix methods for geometrical optics
    11. 4.11 Additional reading
    12. 4.12 Exercises
  11. 5. Advanced matrix techniques and tensors
    1. 5.1 Introduction: Foldy–Lax scattering theory
    2. 5.2 Advanced matrix terminology
    3. 5.3 Left–right eigenvalues and biorthogonality
    4. 5.4 Singular value decomposition
    5. 5.5 Other matrix manipulations
    6. 5.6 Tensors
    7. 5.7 Additional reading
    8. 5.8 Exercises
  12. 6. Distributions
    1. 6.1 Introduction: Gauss’ law and the Poisson equation
    2. 6.2 Introduction to delta functions
    3. 6.3 Calculus of delta functions
    4. 6.4 Other representations of the delta function
    5. 6.5 Heaviside step function
    6. 6.6 Delta functions of more than one variable
    7. 6.7 Additional reading
    8. 6.8 Exercises
  13. 7. Infinite series
    1. 7.1 Introduction: the Fabry–Perot interferometer
    2. 7.2 Sequences and series
    3. 7.3 Series convergence
    4. 7.4 Series of functions
    5. 7.5 Taylor series
    6. 7.6 Taylor series in more than one variable
    7. 7.7 Power series
    8. 7.8 Focus: convergence of the Born series
    9. 7.9 Additional reading
    10. 7.10 Exercises
  14. 8. Fourier series
    1. 8.1 Introduction: diffraction gratings
    2. 8.2 Real-valued Fourier series
    3. 8.3 Examples
    4. 8.4 Integration range of the Fourier series
    5. 8.5 Complex-valued Fourier series
    6. 8.6 Properties of Fourier series
    7. 8.7 Gibbs phenomenon and convergence in the mean
    8. 8.8 Focus: X-ray diffraction from crystals
    9. 8.9 Additional reading
    10. 8.10 Exercises
  15. 9. Complex analysis
    1. 9.1 Introduction: electric potential in an infinite cylinder
    2. 9.2 Complex algebra
    3. 9.3 Functions of a complex variable
    4. 9.4 Complex derivatives and analyticity
    5. 9.5 Complex integration and Cauchy’s integral theorem
    6. 9.6 Cauchy’s integral formula
    7. 9.7 Taylor series
    8. 9.8 Laurent series
    9. 9.9 Classification of isolated singularities
    10. 9.10 Branch points and Riemann surfaces
    11. 9.11 Residue theorem
    12. 9.12 Evaluation of definite integrals
    13. 9.13 Cauchy principal value
    14. 9.14 Focus: Kramers–Kronig relations
    15. 9.15 Focus: optical vortices
    16. 9.16 Additional reading
    17. 9.17 Exercises
  16. 10. Advanced complex analysis
    1. 10.1 Introduction
    2. 10.2 Analytic continuation
    3. 10.3 Stereographic projection
    4. 10.4 Conformal mapping
    5. 10.5 Significant theorems in complex analysis
    6. 10.6 Focus: analytic properties of wavefields
    7. 10.7 Focus: optical cloaking and transformation optics
    8. 10.8 Exercises
  17. 11. Fourier transforms
    1. 11.1 Introduction: Fraunhofer diffraction
    2. 11.2 The Fourier transform and its inverse
    3. 11.3 Examples of Fourier transforms
    4. 11.4 Mathematical properties of the Fourier transform
    5. 11.5 Physical properties of the Fourier transform
    6. 11.6 Eigenfunctions of the Fourier operator
    7. 11.7 Higher-dimensional transforms
    8. 11.8 Focus: spatial filtering
    9. 11.9 Focus: angular spectrum representation
    10. 11.10 Additional reading
    11. 11.11 Exercises
  18. 12. Other integral transforms
    1. 12.1 Introduction: the Fresnel transform
    2. 12.2 Linear canonical transforms
    3. 12.3 The Laplace transform
    4. 12.4 Fractional Fourier transform
    5. 12.5 Mixed domain transforms
    6. 12.6 The wavelet transform
    7. 12.7 The Wigner transform
    8. 12.8 Focus: the Radon transform and computed axial tomography (CAT)
    9. 12.9 Additional reading
    10. 12.10 Exercises
  19. 13. Discrete transforms
    1. 13.1 Introduction: the sampling theorem
    2. 13.2 Sampling and the Poisson sum formula
    3. 13.3 The discrete Fourier transform
    4. 13.4 Properties of the DFT
    5. 13.5 Convolution
    6. 13.6 Fast Fourier transform
    7. 13.7 The z-transform
    8. 13.8 Focus: z-transforms in the numerical solution of Maxwell’s equations
    9. 13.9 Focus: the Talbot effect
    10. 13.10 Exercises
  20. 14. Ordinary differential equations
    1. 14.1 Introduction: the classic ODEs
    2. 14.2 Classification of ODEs
    3. 14.3 Ordinary differential equations and phase space
    4. 14.4 First-order ODEs
    5. 14.5 Second-order ODEs with constant coefficients
    6. 14.6 The Wronskian and associated strategies
    7. 14.7 Variation of parameters
    8. 14.8 Series solutions
    9. 14.9 Singularities, complex analysis, and general Frobenius solutions
    10. 14.10 Integral transform solutions
    11. 14.11 Systems of differential equations
    12. 14.12 Numerical analysis of differential equations
    13. 14.13 Additional reading
    14. 14.14 Exercises
  21. 15. Partial differential equations
    1. 15.1 Introduction: propagation in a rectangular waveguide
    2. 15.2 Classification of second-order linear PDEs
    3. 15.3 Separation of variables
    4. 15.4 Hyperbolic equations
    5. 15.5 Elliptic equations
    6. 15.6 Parabolic equations
    7. 15.7 Solutions by integral transforms
    8. 15.8 Inhomogeneous problems and eigenfunction solutions
    9. 15.9 Infinite domains; the d’Alembert solution
    10. 15.10 Method of images
    11. 15.11 Additional reading
    12. 15.12 Exercises
  22. 16. Bessel functions
    1. 16.1 Introduction: propagation in a circular waveguide
    2. 16.2 Bessel’s equation and series solutions
    3. 16.3 The generating function
    4. 16.4 Recurrence relations
    5. 16.5 Integral representations
    6. 16.6 Hankel functions
    7. 16.7 Modified Bessel functions
    8. 16.8 Asymptotic behavior of Bessel functions
    9. 16.9 Zeros of Bessel functions
    10. 16.10 Orthogonality relations
    11. 16.11 Bessel functions of fractional order
    12. 16.12 Addition theorems, sum theorems, and product relations
    13. 16.13 Focus: nondiffracting beams
    14. 16.14 Additional reading
    15. 16.15 Exercises
  23. 17. Legendre functions and spherical harmonics
    1. 17.1 Introduction: Laplace’s equation in spherical coordinates
    2. 17.2 Series solution of the Legendre equation
    3. 17.3 Generating function
    4. 17.4 Recurrence relations
    5. 17.5 Integral formulas
    6. 17.6 Orthogonality
    7. 17.7 Associated Legendre functions
    8. 17.8 Spherical harmonics
    9. 17.9 Spherical harmonic addition theorem
    10. 17.10 Solution of PDEs in spherical coordinates
    11. 17.11 Gegenbauer polynomials
    12. 17.12 Focus: multipole expansion for static electric fields
    13. 17.13 Focus: vector spherical harmonics and radiation fields
    14. 17.14 Exercises
  24. 18. Orthogonal functions
    1. 18.1 Introduction: Sturm–Liouville equations
    2. 18.2 Hermite polynomials
    3. 18.3 Laguerre functions
    4. 18.4 Chebyshev polynomials
    5. 18.5 Jacobi polynomials
    6. 18.6 Focus: Zernike polynomials
    7. 18.7 Additional reading
    8. 18.8 Exercises
  25. 19. Green’s functions
    1. 19.1 Introduction: the Huygens–Fresnel integral
    2. 19.2 Inhomogeneous Sturm–Liouville equations
    3. 19.3 Properties of Green’s functions
    4. 19.4 Green’s functions of second-order PDEs
    5. 19.5 Method of images
    6. 19.6 Modal expansion of Green’s functions
    7. 19.7 Integral equations
    8. 19.8 Focus: Rayleigh–Sommerfeld diffraction
    9. 19.9 Focus: dyadic Green’s function for Maxwell’s equations
    10. 19.10 Focus: scattering theory and the Born series
    11. 19.11 Exercises
  26. 20. The calculus of variations
    1. 20.1 Introduction: principle of Fermat
    2. 20.2 Extrema of functions and functionals
    3. 20.3 Euler’s equation
    4. 20.4 Second form of Euler’s equation
    5. 20.5 Calculus of variations with several dependent variables
    6. 20.6 Calculus of variations with several independent variables
    7. 20.7 Euler’s equation with auxiliary conditions: Lagrange multipliers
    8. 20.8 Hamiltonian dynamics
    9. 20.9 Focus: aperture apodization
    10. 20.10 Additional reading
    11. 20.11 Exercises
  27. 21. Asymptotic techniques
    1. 21.1 Introduction: foundations of geometrical optics
    2. 21.2 Definition of an asymptotic series
    3. 21.3 Asymptotic behavior of integrals
    4. 21.4 Method of stationary phase
    5. 21.5 Method of steepest descents
    6. 21.6 Method of stationary phase for double integrals
    7. 21.7 Additional reading
    8. 21.8 Exercises
  28. Appendix A The gamma function
    1. A.1 Definition
    2. A.2 Basic properties
    3. A.3 Stirling’s formula
    4. A.4 Beta function
    5. A.5 Useful integrals
  29. Appendix B Hypergeometric functions
    1. B.1 Hypergeometric function
    2. B.2 Confluent hypergeometric function
    3. B.3 Integral representations
  30. References
  31. Index