Geometrical theories of wave propagation: a contemporary review

The paper aims at an informal and elementary introduction to the subject of modern geometrical techniques in the study of wave propagation problems which may arise in practical contexts including reflector antennas, integrated optics and radiowave propagation. The term `modern¿ is to be taken to mean `co-ordinate-free¿, wherein the use of co-ordinate systems is eschewed in favour of intrinsic geometrical constructions, and Morse critical point theory (i.e. catastrophe theory) is used to introduce canonical co-ordinates only where necessary for integration. The term `geometrical¿ refers to the structure of rays and wavefronts. We concentrate particularly on the construction of uniform transition functions to repair the singularities of the Kline-Luneberg form of geometrical optics, which forms a major concern of current investigations in the subject. We examine diffraction from the points of view of GTD, UAT, Kirchhoff-Huyghens theory and spectral synthesis, and waveguides from those of the Poisson sum formula and intrinsic mode theory. The review begins with a brief historical overview to place a perspective on current work, and ends with an assessment of future directions for geometrical techniques.