I.V. Allin, A.S. Kryukovsky, D.S. Lukin, D.V. Rastyagaev
In work the processes of propagation and focusing of electromagnetic pulses in dispersive media by methods of the wave catastrophe theory are considered. The asymptotic methods based on the ray approach are applied to the description of processes of wave propagation in a short-wave case. However in conditions of a space focusing of a broadband signal caused by curvature of an initial wave field (or conditions of a signal propagation in medium) an electromagnetic field is not described by a linear superposition of rays (geometro-optical, edge, time), there are nonlinear effects connected to caustics and their singularities.
The modelling problem of propagation of an electromagnetic signal with the curved initial wave front in cold homogeneous plasma is considered. For the description of a field structure the integrated representation of the solution of the Klein-Gordon equation is used. It is well known, that at construction of the asymptotic decision the basic contribution is brought by saddle points of phase function of the integrated solution. For the analysis of types of saddle points in work the wave catastrophe theory allowing to execute classification of types of the singular points and the appropriate focal structures, arising in a task is used.
The influence of pulse final duration propagating in dispersive medium is investigated. Is shown, that the focusings of rectangular pulses in homogeneous plasma are spatial and correspond one-dimensional (cuspoid) and two-dimensional focusings. In work the classification of space – time focusings is carried out, the uniform asymptotic expressions for wave fields are constructed in the field of such focusings. In the work the new classes of special functions of wave catastrophes describing spatial focusing at signal propagation are constructed. The modeling which has allowed to describe wave effects in the field of signal focusings and to take into account edge effects connected to edge temporary rays is carried out.