Fast Multipole Method for Maxwell’s Equations in Layered Media

We present a fast multipole method (FMM) for solving Maxwell’s equations in 3-D layered media, using the magnetic vector potential [Formula: see text] with the Lorenz gauge in deriving 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, multipole expansions and local expansions 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 expansion translations. Finally, leveraging the FMM framework of the Helmholtz equation in 3-D layered media [B. Wang, W. Zhang, and W. Cai, SIAM J. Sci. Comput., 41 (2019), pp. A3954–A3981], we develop a FMM for the dyadic Green’s function of Maxwell’s equations in layered media. Numerical experiments demonstrate the [Formula: see text]-complexity of the resulting FMM method, and rapid convergence for interactions of low-frequency electromagnetic wave sources in 3-D layered media.