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Spie Press Book

Two Methods for the Exact Solution of Diffraction Problems
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Book Description

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.

Book Details

Date Published: 26 November 2003
Pages: 142
ISBN: 9780819451415
Volume: PM127

Table of Contents
SHOW Table of Contents | HIDE Table of Contents
Preface / xi
Chapter 1 Introduction / 1
1.1 Sommerfeld's Method / 2
1.2 Generalizing Boundary Surfaces / 3
References / 5
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 / 16
3.2 Complex Number Notation / 16
3.3 Outline of the Construction of a Multiple-Valued Radiation Source /19
3.4 The Analytic Continuation of 20
3.5 The Cauchy Integral for a Point Source: Definition of U1 / 24
3.6 Uniqueness of the Solution / 30
3.7 Explicit Expressions for U1 / 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 an imperfectly 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 / 42
4.6 Generalization of (exp ikD)/D / 43
References / 44
Chapter 5 Fresnel Diffraction by a Circular Disc / 45
5.1 Coordinate-System Construction / 45
5.2 Analytic Continuation of / 49
5.3 Construction of a Spherical Wave Generalization: An Alternative Method / 50
5.3.1 Direct method for construction of a multiple-valued Green's function / 55
5.4 Applications / 57
5.5 Diffraction of a Spherical Wave by a Perfectly Conducting Disc / 57
5.6 Diffraction by a Perfectly Conducting Spherical Dome / 58
5.7 Comments on the Foregoing Analysis / 59
References / 60
Chapter 6 Fresnel Diffraction by a Flat Circular Annulus / 61
6.1 Outline of the Generalized Sommerfeld Method / 63
6.2 The Coordinate System / 64
6.3 The Branch Points of D2 / 68
6.4 Construction of W1 / 70
References / 71
Chapter 7 Fresnel Diffraction by a Slit between Perfectly Conducting Half-Planes /73
7.1 Coordinate Systems for Two Branch Lines / 73
7.2 Analytic Continuation of / 77
7.3 Construction of U1 / 79
7.4 Diffraction of a Spherical Wave by a Slit between Two Perfectly Conducting Half-Planes / 80
7.5 Some Remarks on the Sommerfeld Method / 81
References / 82
Chapter 8 Coordinate Systems / 83
8.1 Generalization of the Branch Curves / 83
8.2 Cylinders of Arbitrary Shape / 85
8.3 Closed Surfaces of Arbitrary Shape / 86
8.4 Interpolated Coordinate Systems / 87
References / 88
Chapter 9 Radiation Scattering by a Hexagonal Ice Cylinder: Coordinate System /89
9.1 Configuration / 89
9.2 Unit Vectors / 92
9.3 Inscribed Circle / 93
References / 94
Chapter 10 Radiation Scattering by a Hexagonal Ice Cylinder: Boundary Conditions / 95
10.1 Wave Propagation Equation and Elementary Solutions / 95
10.2 Boundary Conditions / 97
10.3 Field Continuity along the z-Axis / 99
10.4 The Boundary Conditions in E( and H( / 101
10.5 Simplifications by Use of Symmetry / 103
10.6 Evaluation of the Fourier Transforms / 104
10.6.1 Perturbation method / 104
10.6.2 Fourier transforms for small radiation wavelength / 106
10.6.3 Trigonometric interpolation / 107
References / 108
Appendix A Alternative Methods of Exact Diffraction Analyses / 109
A.1 General Comments / 109
A.2 Finite Element Method / 109
A.3 Integral Equation Method / 110
A.4 The T-Matrix / 110
References / 110
Appendix B Sommerfeld's Original Analyses / 111
B.1 Static Fields / 111
References / 115
Appendix C Analytic Functions of a Complex Variable / 117
C.1 Complex Numbers / 117
C.2 Differential Properties / 117
C.3 Integral Properties of Analytic Functions / 118
C.4 Singularities / 119
C.5 Contour Integration / 119
C.6 Analytic Continuation / 120
C.7 Branch Point / 120
References / 120
Appendix D Uniform Convergence / 121
D.1 Definition of Uniform Convergence / 121
References / 122
Index / 123


This monograph is about 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. Only the theory is discussed; numerical consequences of the theory are not presented.

Ruling out any changes of state owing to high-energy processes, two essential features of radiation scattering are: (1) the shape of the scatterer, and (2) the wavelength of the incident radiation. Taking these accurately into account in scattering analyses is a goal that has challenged researchers since the formulation of optical science. Many approximate methods have been devised to this end.

One of the methods to be generalized and discussed in this book was originated by Arnold Sommerfeld in two fundamental papers written in 1896 and 1897. The promising simplicity and accuracy of the expressions derived, well-verified by experiment, led to expectations of further progress in the use of Sommerfeld's method that were not realized in the years that followed. However, the need for exact solutions has been demonstrated by frequent references to the implications of Sommerfeld's work, even when the scatterers did not have the original configuration. The lack of success in generalizing Sommerfeld's analysis has led to a general belief that it cannot be done.

The author was fortunate to find an alternate formulation of Sommerfeld's method a logical continuation of research originated under the direction of Professor Griffith C. Evans. This research was directed toward a generalization of Sommerfeld's method of constructing multiple-valued potentials defined on a 3D multileaved space, i.e., solutions to Laplace's equation. Eventually, such a generalization was achieved and detailed in a published work. It was then evident that a similar generalization could be applied to construct multiple-valued Green's functions for the equation of wave propagation. The results constitute part of the work herein, indicating a variety of ways in which Green's functions can be constructed for a variety of configurations. For this introductory work, the method and the numerous consequences that follow from it render the theory more significant than the possible resulting calculations. However, it is hopeful that such calculations will soon be carried out. Although Sommerfeld's method delivers solutions in a convenient, closed, and analytic form, these solutions have the drawback of being applicable only to surfaces spanning given space curves; it is not applicable to solids bounded by closed surfaces. This limitation is remedied in a method of analysis originated by the author following the exposition of Sommerfeld's method. The new method provides a means of solving boundary-value problems for solids bounded by a large variety of surface configurations.

Although the analyses presented are primarily mathematical, they are not strictly rigorous. In this respect, a discussion follows the example set in the historical development of optics theory in which physical models and sufficient (rather than necessary) conditions have played an important role. The use of sufficient conditions, for example, is illustrated in a series of superpositions of elementary solutions to the wave equation based on a physical model; it is then assumed that the series will converge to the correct solution of the boundary-value problem. Further, if the physical model is defined well enough to serve as a guide to any approximations, even a divergent series can be used as a solution, i.e., asymptotic expansions. Indeed, Sommerfeld's original solution for diffraction by a semi- infinite plane was given as the first term of an asymptotic expansion. A similar expansion is used in the second method of this book, although it is not an essential part of the analysis.

It will be seen that the following work can be considerably extended.

Frederick E. Alzofon
July 2003

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