An FMM for orthotropic periodic boundary value problems for Maxwell's equations

This paper presents a Fast Multipole Method (FMM) for orthotropic periodic problems for Maxwell's equations in 3D. We consider scattering problems where two-dimensional arrays of dielectric scatterers are arranged periodically with different periodic lengths in two orthogonal directions. The multipole and local expansions of the periodic Green function are used for accelerating the solution of the integral equations. We derive Fourier integral expressions for the periodic Green function and its derivatives, which are essential ingredients in our formulation. The cells used in the proposed FMM are not cubic because we identify the unit cell of the periodic structure with the 'level 0 cell' in the FMM tree. We show modifications required to the FMM algorithm for introducing non-cubic cells. Through numerical tests we conclude that the proposed method is efficient and accurate.