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

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.