Abstract
The problem of resonant mode conversion in multiple spatial dimensions is considered. Using phase space methods, a complete theory is developed for constructing matched asymptotic expansions that fit incoming and outgoing WKB solutions. These results provide, for the first time, a complete and practical method for including multidimensional conversion in ray tracing algorithms. The paper provides a self-contained description of the following topics: (1) how to use eikonal (also known as ray tracing or WKB) methods to solve vector wave equations and how to detect conversion regions while following rays; (2) once conversion is detected, how to fit to a generic saddle structure in ray phase space associated with the most common type of conversion; (3) given the saddle structure, how to carry out a local projection of the full vector wave equation onto a local two-component normal form that governs the two resonantly interacting waves. This determines both the uncoupled dispersion functions and the coupling constant, which in turn determine the uncoupled WKB solutions; (4) given the normal form of the local two-component wave equation, how to find the particular solution that matches the amplitude, phase, and polarization of the incoming ray, to the amplitude, phase, and polarization of the two outgoing rays: the transmitted and converted rays.
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