Geometrical Theory of Diffraction
Details the ideas underlying geometrical theory of diffraction (GTD) along with its relationships with other EM theories.
Inspec keywords: optical waveguides; geometrical theory of diffraction; light reflection; light diffraction
Other keywords: asympotic diffraction theory; ray fields; plane wave diffraction; physical optics; optical waveguide; geometrical theory of diffraction; reflection problems
Subjects: Edge and boundary effects; optical reflection and refraction; Optical waveguides and couplers; Optical diffraction and scattering
 Book DOI: 10.1049/PBEW037E
 Chapter DOI: 10.1049/PBEW037E
 ISBN: 9780852968307
 eISBN: 9781849193931
 Page count: 408
 Format: PDF

Front Matter
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1 Introduction
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During the last three decades, close relationships have been established between geometrical optics and the mathematical theory of diffraction and brought about a hybrid formalism known as the geometrical theory of diffraction (GTD). What is the essence of this theory? By diffraction one usually means those features in the behaviour of wave fields that cannot be described by the laws of geometrical optics. In other words, diffraction is what lies beyond geometrical optics. Moreover, the shortwave asymptotic behaviour of the known rigorous solutions (diffraction by a wedge, cylinder, and sphere) reveals that the laws of geometrical optics break down only in narrow transition regions where there occur diffraction fields not envisaged by these laws. Further propagation of these fields, far from the places where they have occurred, is again described by geometrical optics.

2 Fundamendals of the geometrical theory of diffraction
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Fundamentals of the geometrical theory of diffraction is reported. GTD is an extension of geometrical optics (GO). They divide in two groups [11]: the GO laws in boundless media where they allow the construction of rays and wavefronts and the calculation of the field and its polarisation along the ray, and the laws of field transformation in reflection and refraction.

3 Ray fields and reflections from smooth bodies
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This chapter presents the rationale behind and the refinement of the laws of geometrical optics. Ray fields are considered, i.e. the solutions of the wave equation, that possess a special form of asymptotic expansions called ray expansions. The leading term of a ray expansion is the geometrical optics solution, and the following terms are corrections to this solution. The ray expansions for the general case were analysed and then their form for particular and some recurrent types of waves, namely, plane, cylindrical, spherical, and toroidal, the analogue of cylindrical waves for axially symmetric problems were examined. These results are used to refine the second group of geometrical optics laws and to solve simple boundaryvalue problems which do not evolve diffraction fields in the form of edge waves or creeping waves.

4 Caustic and focal expansions
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Geometrical optics and its refinement, ray expansions, are inapplicable near caustics. The error of the GO approximation and ray expansions increases when the point of observation approaches a caustic and at the caustic these approximations become infinite. Therefore, these ray expansions are termed nonuniform asymptotic expansions. In this chapter we explore another type of field asymptotics, called the caustic expansion, applicable to points of observation in the vicinity of caustics. The error of these asymptotic approximations remains bounded (and tends to zero as £ oo) however close the caustic is approached. Therefore, the caustic expansions are called uniform expansions of ray fields.

5 Diffracted waves
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This section considers the basic types of diffraction field postulated in the early chapters on the basis of heuristic considerations. These fields make up for the breaks of the GO solution on the lightshadow boundary where some terms of this solution, say reflected waves, vanish jumpwise. Therefore, a natural classification of diffracted fields should be associated with the structure of GO solutions near the boundary that occurs when the incident wave interacts with the body. Three main typical situations of the occurrence of diffracted fields are examined.

6 GTD or physical optics methods?
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All the model problems considered in the early chapters have been endowed with asymptotic solutions (edge waves, caustic and penumbra fields) accurate to the given orders of k. Either the shortwave asymptotic of the known solutions, say plane wave diffraction by a wedge, or a known rigorous asymptotic where an exact solution is unavailable, as with diffraction of an arbitrary ray field by a surface with edge, have been used. Unfortunately, many model problems of practical interest have neither rigorous solutions to extract shortwave asymptotics nor appropriate asymptotics. Under these circumstances, one has to resort to approximate methods. A widely used approximate method employed for large sizetowavelength ratios is the Kirchhoff approximation (KA). Its modification, the physical theory of diffraction (PTD), alternatively called the edge wave method, in addition takes into account the field perturbations at the edges of the body.

7 Diffraction by bodies of complex geometry
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This chapter demonstrates with examples how the findings of the early chapters are incorporated in the GTD to solve diffraction by bodies of intricate shape mainly by the successive diffraction (SD) and selfconsistent field (SCF) methods.

8 Diffraction at open end of waveguide
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Thus far only single diffraction of a penumbra field at a wedge has been discussed, tacitly assuming that at a subsequent stage of diffraction the fields incident on the edges will be again rayoptical. However, there are problems with successive diffractions where in each diffraction order the incident field is penumbra. In the circumstances, the scattered field is of the same order in ka (a is characteristic dimension) as the incident field and so one has to sum all successively arising diffraction waves. The simplest example of this variety is diffraction at the open end of a waveguide. It serves as a model for many related problems: diffraction at the open end of a waveguide with nonsymmetrical flanges or with impedance boundary conditions, a wide slot in a waveguide, an obliquecut waveguide, and a waveguiding cross (intersection of two waveguides). Also related to this geometry are the threedimensional problems of the open end of a circular pipe with conical flange, which arises in circular waveguidehorn junction analysis, and the allied problem of plane wave incidence on a sawtoothprofiled surface (Wood echelettes). Some of these problems are discussed in this chapter.

9 Methods and results of asympotic diffraction theory
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The chapter is about methods and results of asymptotic diffraction theory using Sommerfield integral method.

10 Appendix
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This chapter contains generalised Fresnel integral; expressions for radiation pattern of reflector antenna; and calculation of sums of integrals.

Back Matter
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