Fast calculations of band diagrams of irregularly shaped scatterers in periodic structures using the multiple scattering theory and broadband Green's function

A fast method is developed utilizing the multiple scattering theory with broadband Green’s function (MST-BBGF) to calculate band diagrams for vector electromagnetic waves in 3D periodic structures composed of irregularly shaped scatterers. The method uses the T-matrix of irregular-shaped scatterers to formulate eigenvalue problems for the KKR (Korringa-Kohn, Rostoker) equations. The single scatterer T-matrix is derived from far-field scattering amplitudes using vector spherical harmonics and is re-usable since it is independent of lattice structures or Bloch vectors. The method is applied to a scatterer consisting of a two-layer triangular prism structure developed for 3D photonic topological insulators. Various comparisons are made to test the accuracies of the numerically derived T-matrix. The computations of the fast MST-BBGF method for the first five bands require only a KKR eigen equation dimension of 6, utilizing 6 vector spherical waves. The simulation time for the MST-BBGF method for two-layer triangular prism on a standard laptop is 4.63 seconds for band diagram calculations for 30 Bloch vectors given the pre-calculated T-matrix elements. The CPU is 3 orders of magnitude less than the CPU of 3.5 hours with COMSOL’s finite element method on a relatively coarse grid.