Non-linear eigenvalue problems with GetDP and SLEPc: Eigenmode computations of frequency-dispersive photonic open structures

TL;DR

This paper develops a framework combining GetDP and SLEPc to solve complex non-linear eigenvalue problems in photonics, enabling accurate computation of electromagnetic modes in frequency-dispersive open structures with detailed analysis of numerical methods.

Contribution

It introduces a novel approach for solving non-linear eigenvalue problems in photonics using open-source tools, with detailed comparison of multiple solution strategies.

Findings

Different solution methods are compared for accuracy and efficiency.

Convergence behavior is analyzed with mesh refinement and polynomial order.

Numerical challenges due to sharp corners and sign-changing coefficients are addressed.

Abstract

We present a framework to solve non-linear eigenvalue problems suitable for a Finite Element discretization. The implementation is based on the open-source finite element software GetDP and the open-source library SLEPc. As template examples, we propose and compare in detail different ways to address the numerical computation of the electromagnetic modes of frequency-dispersive objects. This is a non-linear eigenvalue problem involving a non-Hermitian operator. A classical finite element formulation is derived for five different solutions and solved using algorithms adapted to the large size of the resulting discrete problem. The proposed solutions are applied to the computation of the dispersion relation of a diffraction grating made of a Drude material. The important numerical consequences linked to the presence of sharp corners and sign-changing coefficients are carefully examined. For each method, the convergence of the eigenvalues with respect to the mesh refinement and the shape function order, as well as computation time and memory requirements are investigated. The open-source template model used to obtain the numerical results is provided. Details of the implementation of polynomial and rational eigenvalue problems in GetDP are given in appendix.