Two Methods for the Exact Solution of Diffraction Problems

by Frederick E. Alzofon

Overview

In analyses of radiation scattering, accurately assessing the shape of the scatterer and the wavelength of the incident radiation is a goal that has challenged researchers since the beginning of optical science. This innovative text presents two methods of calculating the electromagnetic fields due to radiation scattering by a single scatterer. Both methods yield valid results for all wavelengths of the incident radiation as well as a wide variety of scatterer configurations.

Product Details

ISBN-13:	9780819451415
Publisher:	SPIE Press
Publication date:	10/28/2003
Series:	SPIE Press Monograph Series
Pages:	142
Product dimensions:	6.90(w) x 9.90(h) x 0.40(d)

Table of Contents

	Preface	xi
Chapter 1	Introduction	1
1.1	Sommerfeld's Method	2
1.2	Generalizing Boundary Surfaces	3
	References	4
Chapter 2	Historical Background of the Sommerfeld Method	7
2.1	The Kelvin Image Method	7
2.2	The Sommerfeld Image Method	9
	References	12
Chapter 3	Two-Leaved Generalization of a Spherical Wave: One Branch Line	15
3.1	The Point Radiation Source in Physical Space	15
3.2	Complex Number Notation	16
3.3	Outline of the Construction of a Multiple-Valued Radiation Source	17
3.4	The Analytic Continuation of [phi]'	20
3.5	The Cauchy Integral for a Point Source: Definition of U[subscript 1]	22
3.6	Uniqueness of the Solution	29
3.7	Explicit Expressions for U[subscript 1]	30
3.8	Multiple-Valued Generalization of a Plane Wave	31
	References	33
Chapter 4	Fresnel Diffraction by a Semi-Infinite Plane	35
4.1	Scalar Theory	35
4.1.1	Reflection of a spherical wave by a perfectly reflecting semi-infinite plane: scalar theory	36
4.1.2	Diffraction of a spherical wave by a nonperfectly reflecting semi-infinite plane	38
4.2	The Electromagnetic Field Equations	39
4.3	Boundary Conditions	40
4.4	Poincare/Sommerfeld Solution	40
4.5	Solution Using Two Independent Scalar Solutions	43
	References	44
Chapter 5	Fresnel Diffraction by a Circular Disk	45
5.1	Coordinate-System Construction	45
5.2	Analytic Continuation of [theta]'	49
5.3	A Multiple-Valued Green's Function with a Circle as a Branch Curve	50
5.3.1	Plane wave approximation	52
5.3.2	Static solution and the harmonic measure of the two-leaved space	52
5.3.3	An alternative method of constructing a multiple-valued spherical wave	53
5.4	Diffraction of a Spherical Wave by a Perfectly Conducting Disk	58
5.5	Diffraction by a Perfectly Conducting Spherical Dome	59
5.6	Comments on the Foregoing Analysis	61
	References	61
Chapter 6	Fresnel Diffraction by a Flat Circular Annulus	63
6.1	Outline of the Generalized Sommerfeld Method	65
6.2	The Coordinate System	66
6.3	The Branch Points of D[subscript 2]	70
	References	74
Chapter 7	Fresnel Diffraction by a Slit between Perfectly Conducting Half-Planes	75
7.1	Coordinate Systems for Two Branch Lines	75
7.2	Analytic Continuation of [theta]'	79
7.3	Construction of U[subscript 1]	81
7.4	Diffraction of a Spherical Wave by a Slit between Two Perfectly Conducting Half-Planes	82
7.5	Some Remarks on the Sommerfeld Method	83
	References	84
Chapter 8	Coordinate Systems	85
8.1	Generalization of the Branch Curves	85
8.2	Cylinders of Arbitrary Shape	87
8.3	Closed Surfaces of Arbitrary Shape	89
8.4	Interpolated Coordinate Systems	89
	References	90
Chapter 9	Radiation Scattering by a Hexagonal Ice Cylinder: Coordinate System	91
9.1	Configuration	91
9.2	Unit Vectors	93
9.3	Inscribed Circle	95
	References	96
Chapter 10	Radiation Scattering by a Hexagonal Ice Cylinder: Boundary Conditions	97
10.1	Wave Propagation Equation and Elementary Solutions	97
10.2	Boundary Conditions	99
10.3	Field Continuity along the z-Axis	101
10.4	The Boundary Conditions on E[subscript gamma] and H[subscript gamma]	103
10.5	Simplifications by Use of Symmetry	105
10.6	Evaluation of the Fourier Transforms	106
10.6.1	Perturbation method	107
10.6.2	Fourier transforms for small radiation wavelength	108
10.6.3	Trigonometric interpolation	110
	References	111
Appendix A	Alternative Methods of Exact Diffraction Analyses	113
A.1	General Comments	113
A.2	Historical Development of Some Diffraction Solutions: Post-Sommerfeld	113
A.3	Modern Alternatives to Sommerfeld's Method	114
A.3.1	Finite-element method	114
A.3.2	Integral-equation method	115
A.3.3	The T-matrix	115
	References	115
Appendix B	Sommerfeld's Original Analyses	117
B.1	Static Fields	117
	References	121
Appendix C	Analytic Functions of a Complex Variable	123
C.1	Complex Numbers	123
C.2	Differential Properties	123
C.3	Integral Properties of Analytic Functions	124
C.4	Singularities	125
C.5	Contour Integration	125
C.6	Analytic Continuation	126
	References	126
Appendix D	Uniform Convergence	127
D.1	Definition of Uniform Convergence	127
	References	128