Tsung-Ming HuangDepartment of Mathematics, National Taiwan Normal UniversityTiexiang LiSchool of Mathematics, Southeast UniversityWei-De LiDepartment of Mathematics, National Tsing-Hua UniversityJia-Wei LinDepartment of Applied Mathematics, National Chiao Tung UniversityWen-Wei LinDepartment of Applied Mathematics, National Chiao Tung UniversityTian HengDepartment of Applied Mathematics, National Chiao Tung University
Numerical Analysis and Scientific ComputingNumerical Linear Algebramathscidoc:1810.07001
Calculation of band structures of three dimensional photonic crystals amounts to solving large-scale Maxwell eigenvalue problems, which are notoriously challenging due to high multiplicity of zero eigenvalues. In this paper, we try to address this problem in such a broad context that band structures of three dimensional isotropic photonic crystals in all $14$ Bravais lattices can be efficiently computed in a unified framework. In this work, we uncover the delicate machinery behind several key results of our framework and on the basis of this new understanding we drastically simplify the derivations, proofs and arguments. Particular effort is made on reformulating the Bloch condition for all $14$ Bravais lattices in the redefined orthogonal coordinate system, and establishing eigen-decomposition of discrete partial derivative operators by identifying the hierarchical structure of the underlying normal (block) companion matrix, and reducing the eigen-decomposition of the double-curl operator to a simple factorization of a $3$-by-$3$ complex skew-symmetric matrix. With the validity of the novel nullspace free method in the broad context, we perform some calculations on one benchmark system to demonstrate the accuracy and efficiency of our algorithm to solve Maxwell eigenvalue problems.