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

Tsung Ming Huang, Tiexiang Li*, Jia Wei Lin, Wen Wei Lin, Heng Tian

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

2 Citations (Scopus)


This work is devoted to the numerical computation of complex band structure k= k(ω) ∈ C3, with ω 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 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 ⊤-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.

Original languageEnglish
Article number35
JournalJournal of Scientific Computing
Issue number2
Publication statusPublished - 2020 May 1


  • Complex band structure
  • Dispersive permittivity
  • FFT
  • Gyroscopic quadratic eigenvalue problem
  • ⊤-palindromic quadratic eigenvalue problem

ASJC Scopus subject areas

  • Software
  • Theoretical Computer Science
  • Numerical Analysis
  • General Engineering
  • Computational Theory and Mathematics
  • Computational Mathematics
  • Applied Mathematics


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