An FFT-accelerated direct solver for electromagnetic scattering from penetrable axisymmetric objects

TL;DR

This paper introduces an FFT-accelerated solver for electromagnetic scattering from axisymmetric dielectric objects, offering high accuracy and efficiency in inverting integral equations for complex geometries.

Contribution

The paper presents a novel FFT-accelerated separation of variables method for efficiently solving Maxwell's equations for penetrable axisymmetric bodies, including edge geometries.

Findings

Rapid evaluation of modal Green's functions and derivatives

High accuracy with generalized Gaussian quadratures

Efficient handling of geometries with edges

Abstract

Fast, high-order accurate algorithms for electromagnetic scattering from axisymmetric objects are of great importance when modeling physical phenomena in optics, materials science (e.g. meta-materials), and many other fields of applied science. In this paper, we develop an FFT-accelerated separation of variables solver that can be used to efficiently invert integral equation formulations of Maxwell's equations for scattering from axisymmetric penetrable (dielectric) bodies. Using a standard variant of M\"uller's integral representation of the fields, our numerical solver rapidly and directly inverts the resulting second-kind integral equation. In particular, the algorithm of this work (1) rapidly evaluates the modal Green's functions, and their derivatives, via kernel splitting and the use of novel recursion formulas, (2) discretizes the underlying integral equation using generalized Gaussian quadratures on adaptive meshes, and (3) is applicable to geometries containing edges. Several numerical examples are provided to demonstrate the efficiency and accuracy of the aforementioned algorithm in various geometries.