Compact RBF meshless methods for photonic crystal modelling

Abstract

Meshless methods based on compact radial basis functions (RBFs) are proposed for modelling photonic crystals (PhCs). When modelling two-dimensional PhCs two generalised eigenvalue problems are formed, one for the transverse-electric (TE) mode and the other for the transverse-magnetic (TM) mode. Conventionally, the Band Diagrams for two-dimensional PhCs are calculated by either the plane wave expansion method (PWEM) or the finite element method (FEM). Here, the eigenvalue equations for the two-dimensional PhCs are solved using RBFs based meshless methods. For the TM mode a meshless local strong form method (RBF collocation) is used, while for the tricker TE mode a meshless local weak form method (RBF Galerkin) is used (so that the discontinuity of the dielectric function ϵ ( x ) can naturally be modelled). The results obtained from the meshless methods are found to be in good agreement with the standard PWEM. Thus, the meshless methods are proved to be a promising scheme for predicting photonic band gaps.