Block-iterative frequency-domain methods for Maxwell's equations in a planewave basis

TLDR

The study introduces a fully‑vectorial, three‑dimensional algorithm that computes definite‑frequency eigenstates of Maxwell’s equations in arbitrary periodic dielectric structures—including anisotropic and magnetic materials—using preconditioned block‑iterative eigensolvers, and presents an effective dielectric tensor that achieves O(Δx²) convergence even with sharp discontinuities while allowing interior eigenvalue solutions without exhaustive state computation. The algorithm employs preconditioned block‑iterative eigensolvers in a plane‑wave basis, utilizing preconditioned conjugate‑gradient Rayleigh‑quotient minimization compared with the Davidson method, and explores multiple iteration variants and preconditioners to optimize eigensolution.

Abstract

We describe a fully-vectorial, three-dimensional algorithm to compute the definite-frequency eigenstates of Maxwell's equations in arbitrary periodic dielectric structures, including systems with anisotropy (birefringence) or magnetic materials, using preconditioned block-iterative eigensolvers in a planewave basis. Favorable scaling with the system size and the number of computed bands is exhibited. We propose a new effective dielectric tensor for anisotropic structures, and demonstrate that O Delta x;2 convergence can be achieved even in systems with sharp material discontinuities. We show how it is possible to solve for interior eigenvalues, such as localized defect modes, without computing the many underlying eigenstates. Preconditioned conjugate-gradient Rayleigh-quotient minimization is compared with the Davidson method for eigensolution, and a number of iteration variants and preconditioners are characterized. Our implementation is freely available on the Web.