A Numerical Analysis of Stop Band Characteristics by Multilayered Dielectric Gratings with Sinusoidal Profile

An effective computational method based on a conventional modal expansion approach is presented for handling a multilayered dielectric grating whose profiles are multilayered and sinusoidally modulated. This structure fabricated by dielectric material is one of the useful photonic crystals. The method is based on Yasuura’s modal expansion, which is known as a least-squares boundary residual method or a modified Rayleigh method. In the extended method, each layer is divided into shallow horizontal layers. The Floquet modal functions and approximate solutions are defined in each shallow layer, and the latter are matched with boundary conditions in the least-squares sense. A huge-sized least-squares problem that appears in finding the modal coefficients is solved by the QR decomposition accompanied by sequential accumulation. This procedure makes it possible to treat the case where the groove depths are the same as or a little more than the grating period. As numerical example, we calculate a diffractive characteristic by a multilayered deep dielectric grating and confirm that a common band gap occurs for both polarizations.