METAPLECTIC FORMULATION OF LINEAR-MODE CONVERSION

The metaplectic formulation of linear mode conversion is presented. We begin by discussing the connection between wave operators in weakly i nhomogeneous media, their symbols, and related pseudodifferential oper ators. A brief summary of WKB theory and Hamiltonian ray dynamics is g iven. In regions where mode conversion occurs, the WKB approximation b reaks down and must be replaced by an appropriate local approximation. This is done by expanding in a Taylor series about the degenerate reg ion and keeping only the leading-order terms. At leading order a linea r canonical transformation on the ray phase space can be performed whi ch brings the system into a simpler form. This linear canonical transf ormation induces a unitary transformation, called a metaplectic transf ormation, in the wave function's Hilbert space. This is a generalizati on of the Fourier transformation. The advantage of metaplectic techniq ues over Fourier techniques lies in the wider range of transformations available to simplify the problem. We show how to construct the S mat rix, relating incoming and outgoing waves, and the Wigner tensor. We e xamine the Wigner function in detail with particular attention to its asymptotic properties.