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1 NACHOS - Numerical modeling and high performance computing for evolution problems in complex domains and heterogeneous media CRISAM - Inria Sophia Antipolis - Méditerranée , JAD - Laboratoire Jean Alexandre Dieudonné : UMR6621 2 JAD - Laboratoire Jean Alexandre Dieudonné

Abstract : In this paper, we are concerned with the numerical modelling of the propagation of electromagnetic waves in dispersive materials for nanophotonics applications. We focus on a generalized model that allows for the description of a wide range of dispersive media. The underlying differential equations are recast into a generic form and we establish an existence and uniqueness result. We then turn to the numerical treatment and propose an appropriate Discontinuous Galerkin Time Domain framework. We obtain the semi-discrete convergence and prove the stability and in a larger extent, convergence of a Runge Kutta 4 fully discrete scheme via a technique relying on energy principles. Finally, we validate our approach through two significant nanophotonics test cases. 1. Introduction. Among the numerous phenomena encountered in electromag-netics, many rely on the dispersive properties of materials, e.g. the fact that their phase velocity varies with frequency. Indeed, in specific ranges of wavelengths, biological tissues GGC96, noble JC72 and transition metals JC74, but also glass Fle78 and certain polymers CC41 exhibit non-negligible dispersive behaviors. From the mathematical modeling point of view, this phenomenon is modeled by a frequency-dependent permittivity function εω, often derived from physical considerations. Regarding nanophotonics applications, an accurate modeling of the permittivity function for metals in the visible spectrum is crucial. Indeed, the free electrons of metals are the key ingredient in the propagation of surface modes of particular interest, called surface plasmons NH07. The implementation of dispersion models in time-domain electromagnetics solvers can be achieved by different methods. The most common is certainly the Additional Differential Equation ADE technique, which consists in the addition of one or more ODEs to the Maxwell system, the coupling being made via source terms. A consequent literature on this topic exists in the context of Finite-Difference Time-Domain FDTD see e.g. VLDC11 and references therein. More recently, more papers are concerned with Finite Element or even Discontinuous Galerkin Time-Domain approaches DGTD see e.g. GYKR12 and BKN11 and references therein, aiming at overcoming the limitations of FDTD. In this context, some works are more precisely focused on the numerical analysis. Several proofs exist for the standard dispersive media models and the most classical time and space discretization schemes see e.g. all the papers of J. Li and co-authors such as JL06, Li07, LCE08, Li09. Let us also mention the approach of WXZ10 for the integro-differential version of the classical dispersive models. The latter reference propose to analyze a semi-discrete divergence free discontinuous Galerkin framework. Finally, in a previous work LS13, the authors analyzed, for the Debye model, a fully discrete scheme based on a centered fluxes nodal Discontinuous Galerkin formulation and Leap frog discretization in time. In this paper, we present a complete study of a generalized dispersive model that encapsulates a wide range of dispersive media, its higher efficiency being demonstrated

Keywords : Nanophotonics Maxwell-s equations dispersive media Discontinuous Galerkin Runge Kutta schemes





Autor: S Lanteri - Claire Scheid - J Viquerat -

Fuente: https://hal.archives-ouvertes.fr/



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