Full-vectorial Mixed Spectral Element Method For Band Structures Of Two-Dimensional Photonic Crystal

Photonic crystals(PCs)are controllable and perfect tools to control the electromagnetic wave according to our actual needs because of the photonic band gap,and the analysis of band structures of PCs is the key to the photonic band gap.Based on the shortcomings of some common numerical methods,we mainly propose mixed spectral element method(Mixed SEM)and full-vectorial mixed spectral element method(full-vectorial mixed SEM)for the band structures of two-dimensional(2D)PCs in complex media in different lattices.For 2D isotropic or dispersive PCs,Mixed SEM can accurately calculate the band structures.For many 2D PCs with complex media,such as 2D PCs with arbitrary three-dimensional(3D)anisotropy,the fullvectorial mixed SEM can accurately realize the complete band structures of PCs based on the advantages of Mixed SEM.Mixed SEM adopts Gauss-Lobatto-Legendre polynomials to form higher order basis functions and to discretize the Helmholtz equation and Gauss’ law.With the Gauss’law,Mixed SEM can suppress spurious zero eigenvalues and construct the accurate band structures of 2D PCs.Moreover,the Mixed SEM maintains the advantage of spectral element method(SEM),which is exponential convergence.That is,when the order of basis functions increases,the numerical error decreases exponentially and the eigenvalue results have good numerical convergence.The band structures of a 2D isotropic photonic crystal with square lattice and a 2D dispersive photonic crystal with square lattice are used to verify the effectiveness and advantages of Mixed SEM.Based on the Mixed SEM,the full vector eigenvalue equations of full-vectorial mixed SEM are established successfully by considering all the components of the electric or magnetic field.The full-vectorial mixed SEM combines the Helmholtz equation and Gauss’ law,utilizes the Gauss-Lobatto-Legendre polynomials to form higher order basis functions,and divides the variational equations into transverse and longitudinal part to take all the field components into consideration.Therefore,Inheriting the advantages of the Mixed SEM,the full-vectorial mixed SEM can smartly achieve complete band structures of 2D PCs under the situation of that transverseelectric and transverse-magnetic modes cannot be decoupled.The full-vectorial mixed SEM is verified by using a triangular photonic crystal filled with liquid crystal in Te material,and a complete the band structure of this photonic crystal is constructed.