Efficient Characterization of Topological Photonics Using the Broadband Green’s Function

Abstract

A novel method is developed in this paper to characterize the band diagram and modal fields of gyromagnetic photonic crystals that support topolgoical one-way edge states. We exploy an integral equation based method that utilizes the broadband Green’s function as the kernel. The broadband Green’s function is a hybrid representation of the periodic lattice Green’s function that includes an imaginary wavenumber component represented in exponentially decaying spatial series and a reminder in fast converging Floquet plane wave expansions. Special boundary conditions govern the fields across the interface of the gyromagnetic scatterers, leading to surface integral equations (SIEs) that involve three components including the pilot field, its normal derivative and its tangential derivative. To reduce the independent number of unknowns, roof-top basis functions and the Garlerkin’s method are used to discretize the SIEs into matrix equations. The broadband Green’s function allows converting the discretized SIEs into a linear eigenvalue problem of a small size. The eigenvalues and eigenvectors of the linear eigenvalue problem are directly related to the band solutions and modal fields of the photonic crystal. The proposed approach is an effective method to characterize wave interactions with periodic scatterers using integral equations. The solutions of the presented approach are compared against Comsol simulations for various cases to show its accuracy and efficiency.