Le 12 septembre 2023 de 11h00 à 12h00 / Amphithéâtre François Canac, LMA
Résumé : An open-source Quasi Normal Mode (QNM)-based solver, able to handle various types of non-linear eigenvalue problems and relying on Finite Elements is presented. Modal analysis using Quasi-Normal Modes (QNM) is now an essential tool to in- terpret the behavior of open photonic devices based on their intrinsic resonances. When dealing with frequency dispersive media, which is unavoidable in optics, the eigenvalue problem to be considered becomes non-linear due to the dependence of the permittivity with respect to the eigenvalue. The same situation occurs when dealing with boundary conditions at infinity which are intrinsically dispersive: Perfectly Matched Layers (PMLs), Absorbing Boundary Conditions (ABC), or any higher order approximations of the Dirichlet-to-Neumann operator.
In this talk, we will present the various numerical tools recently introduced allowing to perform the QNM expansion with dispersive media in open geometries: (i) Starting with a general causal rational function as a permittivity model [1], (ii) we will review various linearization schemes [2] to tackle the non-linear eigenvalue problem using finite elements, (iii) to finally show a general frame based on the Keldysh theorem to expand [3] the solution of direct problems.
The talk will also focus on another major theoretical difficulty: the discretization of the continuous spectrum associated with open structures. When using finite elements, Perfectly Matched Layers (PML) represent both a theoretical and practical tool to reveal the complex resonances. In their traditional fashion in harmonic regime, PMLs parameters are considered frequency independent, leading to a eigenvalue-dependent damping of the eigenfields when moving away from the resonator. Other strategies, along with their effect on the reconstruction of scattered fields solutions of direct problems in presence of sources, will be discussed.
Finally, we will detail an efficient implementation of the method in a fully open-source Finite Element framework [4, 5, 6] and typical examples [7, 8] will be shown. The approach adopted allows to simply pass the various rational functions involved, as well as the various right-hand sides for which the QNM expansion should be performed. Other examples can be easily adapted from template models also available open-source.