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Structure-Preserving Methods for Computing Complex Band Structures of Three Dimensional Photonic Crystals

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Abstract

This work is devoted to the numerical computation of complex band structure \(\mathbf {k}=\mathbf {k}(\omega )\in {\mathbb {C}}^3\), with \(\omega \) being positive frequencies, of three dimensional isotropic dispersive or non-dispersive photonic crystals from the perspective of structured quadratic eigenvalue problems (QEPs). Our basic strategy is to fix two degrees of freedom in \(\mathbf {k}\) and to view the remaining one as the eigenvalue of a complex gyroscopic QEP which stems from Maxwell’s equations discretized by Yee’s scheme. We reformulate this gyroscopic QEP into a \(\top \)-palindromic QEP, which is further transformed into a structured generalized eigenvalue problem for which we have established a structure-preserving shift-and-invert Arnoldi algorithm. Moreover, to accelerate the inner iterations of the shift-and-invert Arnoldi algorithm, we propose an efficient preconditioner which makes most of the fast Fourier transforms. The advantage of our method is discussed in detail and corroborated by several numerical results.

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Acknowledgements

The authors were partially supported by the ST Yau Center in National Chiao Tung University and the Shing-Tung Yau Center of Southeast University. T.-M. Huang was partially supported by the Ministry of Science and Technology (MoST) 108-2115-M-003-012-MY2 National Center for Theoretical Sciences (NCTS) in Taiwan. T. Li was supported in parts by the National Natural Science Foundation of China (NSFC) 11971105. W.-W. Lin was partially supported by MoST 106-2628-M-009-004-. H. Tian was supported by MoST 107-2811-M-009-002-.

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Authors and Affiliations

  1. Department of Mathematics, National Taiwan Normal University, Taipei, 116, Taiwan

    Tsung-Ming Huang

  2. School of Mathematics, Southeast University, Nanjing, 211189, People’s Republic of China

    Tiexiang Li

  3. Department of Applied Mathematics, National Chiao Tung University, Hsinchu, 300, Taiwan

    Jia-Wei Lin, Wen-Wei Lin & Heng Tian

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Correspondence to Tiexiang Li.

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Huang, TM., Li, T., Lin, JW. et al. Structure-Preserving Methods for Computing Complex Band Structures of Three Dimensional Photonic Crystals. J Sci Comput 83, 35 (2020). https://doi.org/10.1007/s10915-020-01220-1

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  • DOI: https://doi.org/10.1007/s10915-020-01220-1

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