Beam-tracing and profile evolution for localised beams in inhomogeneous plasmas

TL;DR

This work develops a complete beam-tracing framework for localised beams in inhomogeneous cold plasmas, deriving a second-order PDE for the beam profile in addition to the standard envelope equations. By introducing ladder operators that commute with the profile-evolution operator, the authors construct two biorthogonal families of exact solutions and show that Gauss-Hermite beams generally evolve into superpositions of rotated Hermite modes in inhomogeneous media. They demonstrate consistency with Gaussian profiles, connect to Hermite tensors, and provide exact Hermite-polynomial solutions, including explicit boundary embeddings for 'elegant' and 'traditional' Gauss-Hermite bases. The theory accommodates weak dissipation and offers a path toward real-time diagnostic analyses for microwave plasma diagnostics. This framework thus enables flexible, exact treatment of arbitrary initial beam profiles and clarifies how inhomogeneity alters modal content during propagation.

Abstract

We derive the beam tracing and profile evolution for the propagation of any localised beam with arbitrary profile through an inhomogeneous cold plasma. We recover standard Gaussian beam-tracing, with an additional PDE describing the evolution of the beam's profile as it propagates through the plasma. We then solve for generic families of solutions to the PDE using ladder operators, which can be chosen to reduce to Gauss-Hermite beams in homogeneous media. We importantly obtain an exact expression for the resulting beam profile, demonstrating that Hermite modes will generally evolve into a superposition of different modes during propagation through inhomogeneous plasmas, contrary to prior work on the subject. Importantly, this approach facilitates future analysis of the diagnostic signal received from arbitrary beams.