Applying the Wave Catastrophe Theory to Solve of Problems of EM Waves Propagation, Diffraction and Focusing in Non-uniform Media

摘要

The review of results of mathematical and numerical modeling for processes of propagation, diffraction, and focusing of electromagnetic waves in non-uniform and dispersive media is resulted on the basis of applying of the wave catastrophe theory. The mathematical means of the solving of problems is developed by methods of the theory of the main, edge and corner catastrophes. As against traditional asymptotical methods (such as, a method of geometrical optics, the geometrical theory of diffraction, physical optics, the physical theory of diffraction, method of the parabolic equation, Gaussian beams summation, etc) the wave catastrophe theory allows to analyze singularities of complex focal and diffraction structures of wave fields. Such structures are not described by traditional asymptotical methods, or their description is extremely complicated or in general it is impossible because of formation of the complex focal areas. The wave catastrophe theory operates in terms of singularities of smooth mapping or catastrophe which physically correspond to steady focusings of wave fields. In essence, the type of wave catastrophe determines the diffraction structure of a field in general. In this work is considered the application of the given approach to the solving of applied problems in which singularities of caustic types such as the main, edge and corner catastrophe take place.The general rules of the wave catastrophe theory, including classification, indexes and types of focusings, and also methods of uniform asymptotic construction, used for the description of structure of wave fields in such areas, the analysis of amplitude and phase structure of a field are considered. Classes of applied problems in which application of wave catastrophe appeared productive are submitted. The general description of classes of the special functions used for construction uniform asymptotic solving for wave fields, properties of these functions and methods of calculation of one-dimensional, two-dimensional, three-dimensional (space-time) focusings of wave fields is given.