Four-dimensional eikonal theory of linear mode conversion

The problem of linear mode conversion in nonuniform media is solved for general geometry, in that the dispersion functions ${\mathrm{D}}_{\mathrm{a}}$(k,x),${\mathrm{D}}_{\mathrm{b}}$(k,x) of the two, orthogonally polarized, coupled modes a,b have spatial gradients ∂${\mathrm{D}}_{\mathrm{a}}$/∂${\mathrm{x}}^{\mathrm{\mu }}$,∂${\mathrm{D}}_{\mathrm{b}}$/∂${\mathrm{x}}^{\mathrm{\mu }}$ which need not be parallel. The transmission ratio is found to be exp(-2π‖η${\mathrm{\Vert }}^2$/‖B‖), where η is the coupling coefficient, and B is the Poisson Bracket {${\mathrm{D}}_{\mathrm{a}}$,${\mathrm{D}}_{\mathrm{b}}$}=(∂${\mathrm{D}}_{\mathrm{a}}$/${\mathrm{\partial }}_{\mathrm{X}}^{\mathrm{\mu }}$)(∂${\mathrm{D}}_{\mathrm{b}}$/∂${\mathrm{k}}_{\mathrm{\mu }}$)${\mathit{D}}_{\mathit{b}}$) -(∂${\mathit{D}}_{\mathit{a}}$/∂${\mathit{K}}_{\mathrm{\mu }}$)(∂${\mathit{D}}_{\mathit{b}}$/${\mathrm{\partial }}_{\mathit{X}}^{\mathrm{\mu }}$). /∂${\mathrm{x}}^{\mathrm{\mu }}$). The further generalization to weak dissipation and ray divergence is included.