Quasioptical modeling of wave beams with and without mode conversion: I. Basic theory

This work opens a series of papers where we develop a general quasioptical theory for mode-converting electromagnetic beams in plasma and implement it in a numerical algorithm. Here, the basic theory is introduced. We consider a general quasimonochromatic multi-component wave in a weakly inhomogeneous linear medium with no sources. For any given dispersion operator that governs the wave field, we explicitly calculate the approximate operator that governs the wave envelope $\psi$ to the second order in the geometrical-optics parameter. Then, we further simplify this envelope operator by assuming that the gradient of $\psi$ transverse to the local group velocity is much larger than the corresponding parallel gradient. This leads to a parabolic differential equation for $\psi$ ("quasioptical equation") in the basis of the geometrical-optics polarization vectors. Scalar and mode-converting vector beams are described on the same footing. We also explain how to apply this model to electromagnetic waves in general. In the next papers of this series, we report successful quasioptical modeling of radiofrequency wave beams in magnetized plasma based on this theory.