TABLE OF CONTENTS The geometrical theory of diffraction (GTD) was conceived by Keller in the mid-1950s (1953-57), and published only some time later [1, 2, 3, 4]. He showed that diffraction phenomena can be incorporated into a geometrical strategy and phrased in geometrical terms by introducing diffracted rays. These rays have paths determined by a generalisation of Fermat's principle.
The concept of diffracted rays was developed by Keller from the asymptotic evaluation (as the wave number k tends to infinity) of the known exact solution to scattering from simple shapes, referred to as the canonical problems of GTD. There exists a direct relationship between ray representations and the asymptotics of the solution of the Helmholtz equation ? u + k 2 u = 0 (or a system of Maxwell's equations) first outlined by Sommerfeld and Runge [5] in 1911 for the geometrical optics (GO) rays. It has given the basis to a formal technique called the ray method for constructing asymptotic expansions with respect to the inverse power of k of GO solutions of diffraction problems by smooth objects [6]. Construction of a high-frequency asymptotic solution by the ray method is possible only if the field of rays is regular. This condition, which will be mathematically specified in Section 1.1.4, is normally satisfied away from caustics and shadow boundaries.
In the Russian literature (see Reference 7), Keller's technique is known as the...
