Abstract
Photonic crystal is one of the most important materials in optoelectronics technology, it is made up of periodic dielectrics and imitates the arrangement of atoms in the crystal. The basic optical property of a photonic crystal is band gap, it is necessary to use the numerical method when computing the band gap. The propagation behavior of electromagnetic waves in photonic crystals is governed by Maxwell's equations, applying the time harmonic assumption, the governing equations will become a time-independent frequency domain problem. There are many numerical methods, after comparison, we believe that Yee's scheme is the best discrete method. So far, this method can only be applied to simple cubic lattice and face-centered cubic lattice, but there are 14 Bravais lattices in three-dimensional space. The first important work of this dissertation is to extend this method to all of the Bravais lattices. Using the Yee's scheme to discretize the Maxwell's equations, we can get a general eigenvalue problem, and we will analyze this general eigenvalue problem. The second important task of this article is to find the eigen-decomposition of the discrete curl operator, after a series of complicated calculations, we find that all the lattices can be summed up into two kinds of decomposition. We are interested in finding the several few smallest real eigenvalues, but the large dimension of null space is 1/3 of all, this seriously affected the convergence of calculation, we use a technique which is called nullspace-free method to avoid this trouble. But this technique transforms the sparse matrix in our problem to a dense matrix, fortunately, the eigenvectors we found before are related to discrete Fourier transformation. The efficiency of calculation has been significantly improved by using the fast Fourier transformation. Finally, we calculate the band structures of photonic crystals on various lattices, and implement high performance calculations on the GPU.