Solving three dimensional Maxwell eigenvalue problem with fourteen Bravais lattices

Tsung-Ming Huang Tiexiang Li Wei-De Li Jia-Wei Lin Wen-Wei Lin Heng Tian

Numerical Analysis and Scientific Computing mathscidoc:1912.43743

arXiv preprint arXiv:1806.10782, 2018.6
Calculation of band structure of three dimensional photonic crystals amounts to solving large-scale Maxwell eigenvalue problems, which are notoriously challenging due to high multiplicity of zero eigenvalue. In this paper, we try to address this problem in such a broad context that band structure of three dimensional isotropic photonic crystals with all 14 Bravais lattices can be efficiently computed in a unified framework. We uncover the delicate machinery behind several key results of our work and on the basis of this new understanding we drastically simplify the derivations, proofs and arguments in our framework. In this work particular effort is made on reformulating the Bloch boundary condition for all 14 Bravais lattices in the redefined orthogonal coordinate system, and establishing eigen-decomposition of discrete partial derivative operators by systematic use of commutativity among them, which has been overlooked previously, and reducing eigen-decomposition of double-curl operator to the canonical form of a 3x3 complex skew-symmetric matrix under unitary congruence. 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.
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@inproceedings{tsung-ming2018solving,
  title={Solving three dimensional Maxwell eigenvalue problem with fourteen Bravais lattices},
  author={Tsung-Ming Huang, Tiexiang Li, Wei-De Li, Jia-Wei Lin, Wen-Wei Lin, and Heng Tian},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191224205437760219307},
  booktitle={arXiv preprint arXiv:1806.10782},
  year={2018},
}
Tsung-Ming Huang, Tiexiang Li, Wei-De Li, Jia-Wei Lin, Wen-Wei Lin, and Heng Tian. Solving three dimensional Maxwell eigenvalue problem with fourteen Bravais lattices. 2018. In arXiv preprint arXiv:1806.10782. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191224205437760219307.
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