Metaplectic geometrical optics for modeling caustics in uniform and non-uniform media

Abstract

The optimization of radiofrequency-wave (RF) systems for fusion experiments is often performed using ray-tracing codes, which rely on the geometrical-optics (GO) approximation. However, GO fails at caustics such as cutoffs and focal points, erroneously predicting the wave intensity to be infinite. This is a critical shortcoming of GO-based methods, as often the wave intensity at a caustic is precisely the quantity being optimized, for example, when a wave is focused on a resonance to provide plasma heating. Researchers often turn to full-wave modeling in such situations, which are computationally expensive and thereby limit the speed at which such optimizations can be performed. In previous work, we have developed a less expensive alternative called metaplectic geometrical optics (MGO). Instead of evolving the electric field ψ in the usual x (coordinate) or k (spectral) representation, MGO uses a mixed X = Ax + Bk representation. By continuously adjusting the matrix coefficients A and B along the rays, one can ensure that GO remains valid in the X variables, so ψ(X) can be calculated efficiently and without caustic singularities. The result is then mapped back onto the original x space using integrals (called metaplectic transforms) that can be efficiently computed using a numerical steepest-descent method. Here, we overview the theory of MGO and discuss recently developed algorithms that will aid the development of an MGO-based ray-tracing code. We also demonstrate the potential utility of MGO by numerically computing the spectrum of a wave bounded between two cutoffs in a quadratic plasma cavity.

Steepest-descent algorithm for simulating plasma-wave caustics via metaplectic geometrical optics

“…Here we provide a brief overview of MGO; for more details, see Refs. [31,32]. Let ψ(x) be a scalar stationary wavefield in a plasma described by an N -dimensional (N -D) Euclidean coordinate system.…”

“…( 16) requires B t to be invertible; this is done for simplicity, and the generalization to arbitrary B t is provided in Ref. [32]. Also, the prefactor N t (x) in Eq.…”

“…Along the steepest-descent contour that passes through a given saddlepoint at κ = κ 0 , denoted C 0 , Eq. ( 19) takes the general form (32) or equivalently, (33) where we have introduced κ(l) as a 1-D parameterization of C 0 with κ(0) = κ 0 , we have defined…”

“…In response to this need, a new reduced theory called metaplectic GO (MGO) has been recently developed that leads to solutions that are finite at caustics [31,32]. By default, MGO yields an integral representation of the wavefield, which can be approximated analytically to some extent but in general must be evaluated numerically.…”

“…As part of ongoing work on MGO algorithms [31,32,34,35], here we present a quadrature rule for calculating MGO integrals based on numerical steepest-descent integration [36]. This algorithm emerges naturally from the MGO framework in that MGO integrals always contain saddlepoints that correspond to the ray contributions to the wavefield.…”