Mathematical methods for optical physics and engineering 1st Edition by Gregory J Gbur ISBN 1107022355 9780521516105 by Gbur G. 9780521516105, 0521516102 instant download after payment.

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ISBN 10: 1107022355 
ISBN 13: 9780521516105
Author: Gregory J Gbur

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.

Mathematical methods for optical physics and engineering 1st Table of contents:

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

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