Evanescent waves and complex rays

Evanescent wave theory (EWT) is an extension of the geometrical theory of diffraction (GTD) to the tracking of high-frequency plane wave fields with complex phase. The tracking takes place along phase paths perpendicular to the local phase fronts. The determination of the generally complicated configuration of phase paths constitutes a difficulty in the implementation of EWT. The viability of an approximate scheme is examined whereby the phase paths emanating from the initial surface, on which the field is prescribed, are taken locally as hyperbolas; these are known to represent exact global solutions of a particular canonical evanescent wave problem. Numerical comparisons to be presented elsewhere reveal, however, that the local hyperbolic phase path matching may generally be inadequate even for weakly evanescent fields, although this procedure is effective for certain types of initial conditions. To overcome this difficulty, the solution is obtained from an exact formulation involving field integration over the intital surface and subsequent asymptotic evaluation by the saddle point method. Since the saddle points are complex, the latter procedure is equivalent to an analysis in terms of complex rays, from which one may derive the phase paths of EWT. Therefore complex rays should be used as an algorithm for solving the phase path equations. The rigorous analysis also shows that complex ray theory, or EWT, cannot account for strongly evanescent fields, and that there may exist relevant ray contributions from deep within the complex space that cannot be found by the real-space tracking of EWT.