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.

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.

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 short-wave 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 short-wave asymptotics nor appropriate asymptotics. Under these circumstances, one has to resort to approximate methods. A widely used approximate method employed for large size-to-wavelength 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.

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 ray-optical. 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 oblique-cut waveguide, and a waveguiding cross (intersection of two waveguides). Also related to this geometry are the three-dimensional problems of the open end of a circular pipe with conical flange, which arises in circular waveguide-horn junction analysis, and the allied problem of plane wave incidence on a sawtooth-profiled surface (Wood echelettes). Some of these problems are discussed in this chapter.

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