Method of Fundamental Solutions for Maxwell's Equations in Bi-Periodic Multilayered Media

TL;DR

This paper introduces a highly accurate numerical method combining the Method of Fundamental Solutions with a periodization scheme to solve Maxwell's equations in bi-periodic multilayered media, demonstrating exponential convergence and potential for complex multilayer applications.

Contribution

The paper develops a novel, stable, and highly accurate numerical approach for Maxwell's equations in multilayered media using a combined fundamental solutions and periodization scheme.

Findings

Achieves exponential convergence close to 10^{-14} for single and multiple interfaces.

Successfully verifies the method with known solutions.

Demonstrates the method's efficiency with an example involving 39 interfaces.

Abstract

In this paper, we present an accurate numerical method for the time-harmonic Maxwell's equations for bi-periodic multilayered media with quasi-periodic incident waves using the Method of Fundamental Solutions in conjunction with a periodization scheme. Following an approach used in acoustic scattering problems, the electric and magnetic fields in each layer are expressed as a sum of near and distant interactions. The near interaction comprises interactions between the unit cell and its nearest neighboring copies, while the distant interaction is approximated by proxy source points placed on spheres surrounding the unit cell. Imposing continuity of tangential components at the layer interface, quasi-periodicity conditions on the walls of the unit cell, and Rayleigh-Bloch expansion for the radiation condition yields a system of equations for the unknown coefficients, which can be solved by Schur complement and a backward-stable solver. The scheme is verified with known solutions and exhibits exponential convergence close to $10^{-14}$ for both single and multiple interfaces. An example with 39 interfaces is presented to demonstrate the solver's performance. The paper provides promising results for extending this method to a fast and accurate boundary integral equation solver for many cutting-edge applications involving a large number of layers in electromagnetics and optics.