Fast Multipole Method for Maxwell's Equations in Layered Media

TL;DR

This paper develops a fast multipole method for efficiently solving Maxwell's equations in 3D layered media, enabling rapid computation of electromagnetic interactions with proven $\,\mathcal{O}(N\log N)$ complexity.

Contribution

It introduces a novel FMM framework for Maxwell's equations in layered media using layered Green's functions and Chebyshev polynomial expansions for improved efficiency and stability.

Findings

Achieves $\,\mathcal{O}(N\log N)$ computational complexity.

Demonstrates rapid convergence for low-frequency electromagnetic interactions.

Provides numerical validation of the method's efficiency and accuracy.

Abstract

We present a fast multipole method (FMM) for solving Maxwell's equations in three-dimensional (3-D) layered media, based on the magnetic vector potential $\boldsymbol A$ under the Lorenz gauge, to derive the layered dyadic Green's function. The dyadic Green's function is represented using three scalar Helmholtz layered Green's functions, with all interface-induced reaction field components expressed through a unified integral representation. By introducing equivalent polarization images for sources and effective locations for targets to reflect the actual transmission distance of different reaction field components, multiple expansions (MEs) and local expansions (LEs) are derived for the far-field governed by actual transmission distance. To further enhance computational efficiency and numerical stability, we employ a Chebyshev polynomial expansion of the associated Legendre functions to speed up the calculation of multipole-to-local (M2L) expansion translations. Finally, leveraging the FMM framework of the Helmholtz equation in 3-D layered media, we develop a FMM for the dyadic Green's function of Maxwell's equations in layered media. Numerical experiments demonstrate the $\mathcal O(N\log N)$-complexity of the resulting FMM method, and rapid convergence for interactions of low-frequency electromagnetic wave sources in 3-D layered media.