Towards an electromagnetic crystal Green function multiple scattering technique for arbitrary polarizations, lattices, and defects

An electromagnetic crystal (EC) Green function multiple scattering technique (MST) that permits the fast simulation of wave propagation in 2D EC devices was presented in D. Pissoort et al. (2005). The targeted devices were obtained by removing cylinders from doubly periodic, defect-less, and infinite ECs. Contrary to the conventional free-space Green function MST, which associates unknown currents with the surfaces of all physical cylinders that define the EC device and then subsequently describes their interactions using a free-space Green function, the EC Green function MST considers unknown currents on the surfaces of fictitious, removed cylinders and then models their interaction via a Green function innate to the surrounding infinite EC. For frequencies in the electromagnetic bandgap, this EC Green function decays exponentially with distance. Therefore, its pre-computation can be achieved using the free-space Green function MST by considering a centrally excited, finite, and small EC. For the same reason, the EC Green function MST´s system of equations comprises a sparse interaction matrix and a localized excitation that can be solved rapidly by multi-frontal methods. This paper details a generalized formulation of the scheme of D. Pissoort et al. (2005) that handles EC devices defined on square or triangular lattices supporting both TM_Z and TE _Z polarized fields, which are obtained by replacing cylinders from a defect-less EC by cylinders that do not conform to those of the EC background. Throughout the paper it is assumed that the angular frequency omega lies within the EC bandgap