Accurate determination of band structures of two-dimensional dispersive anisotropic photonic crystals by the spectral element method.

The spectral element method (SEM) is used to calculate band structures of two-dimensional photonic crystals (PCs) consisting of dispersive anisotropic materials. As in the conventional finite element method, for a dispersive PC, the resulting eigenvalue problem in the SEM is nonlinear and the eigenvalues are in general complex frequencies. We develop an efficient way of incorporating the dispersion in the system matrices. The band structures of a PC with a square lattice of dispersive cylindrical rods are first analyzed. The imaginary part of the complex frequency is the time-domain decay rate of the eigenmode, which is very useful for tracing a band from discrete numerical data. Modification of the band structure of TE mode by an external static magnetic field in the out-of-plane direction is explored for this square lattice. A plasmon resonance mode is found near the plasmon frequency when the magnetic field is nonzero. The band structure of a PC with a triangular lattice is also calculated with the SEM. Other types of lattices can also be treated readily by the SEM.