A Trigonometric Galerkin Method for Volume Integral Equations Arising in TM Grating Scattering

TL;DR

This paper introduces an FFT-based trigonometric Galerkin method for solving volume integral equations in TM grating scattering, offering an efficient and easy-to-implement numerical approach without requiring Green's functions.

Contribution

It develops a novel FFT-based Galerkin discretization for periodized volume integral equations in TM scattering, proving optimal convergence without Green's function evaluation.

Findings

Converges with optimal order to the scattering solution.

Avoids the need for quasiperiodic Green's functions.

Efficiently evaluates integral operators using FFT.

Abstract

Transverse magnetic (TM) scattering of an electromagnetic wave from a periodic dielectric diffraction grating can mathematically be described by a volume integral equation. This volume integral equation, however, in general fails to feature a weakly singular integral operator. Nevertheless, after a suitable periodization, the involved integral operator can be efficiently evaluated on trigonometric polynomials using the fast Fourier transform (FFT) and iterative methods can be used to solve the integral equation. Using Fredholm theory, we prove that a trigonometric Galerkin discretization applied to the periodized integral equation converges with optimal order to the solution of the scattering problem. The main advantage of this FFT-based discretization scheme is that the resulting numerical method is particularly easy to implement, avoiding for instance the need to evaluate quasiperiodic Green's functions.