Determination of electromagnetic Bloch modes in a medium with frequency-dependent coefficients

TL;DR

This paper introduces a new numerical method to compute electromagnetic Bloch modes in media with frequency-dependent permittivity, using a quadratic eigenvalue formulation and finite element discretization.

Contribution

It develops a functional framework and a linear eigenvalue reformulation for Maxwell's equations with frequency-dependent coefficients, enabling efficient computation of Bloch modes.

Findings

Numerical scheme successfully applied to frequency-dependent media.

Reformulation as a linear eigenproblem simplifies computations.

Preliminary results demonstrate the method's effectiveness.

Abstract

We provide a functional framework and a numerical algorithm to compute the Bloch variety for Maxwell's equations when the electric permittivity is frequency dependent. We incorporate the idea of a mixed formulation for Maxwell's equations to obtain a quadratic eigenvalue for the wave-vector in terms of the frequency. We reformulate this problem as a larger linear eigenvalue problem and prove that this results in the need to compute eigenvalues of a compact operator. Using finite elements, we provide preliminary numerical examples of the scheme for both frequency independent and frequency dependent permittivity.