A periodic FMM for Maxwell’s equations in 3D and its applications to problems related to photonic crystals

This paper presents an FMM (Fast Multipole Method) for periodic boundary value problems for Maxwell’s equations in 3D. The effect of periodicity is taken into account with the help of the periodised moment to local expansion (M2L) transformation formula, which includes lattice sums. We verify the proposed method by comparing the obtained numerical results with analytic solutions for models of the multi-layered dielectric slab. We then apply the proposed method to scattering problems for periodic two-dimensional arrays of dielectric spheres and compare the obtained energy transmittances with those from the previous studies. We also consider scattering problems for woodpile crystals, where we find a passband related to a localised mode. Through these numerical tests we conclude that the proposed method is efficient and accurate.