Analysis of a Generalized Dispersive Model Coupled to a DGTD Method with Application to Nanophotonics

International audienceIn 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 semidiscrete convergence and prove the stability (and to 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

ANALYSIS OF A GENERALIZED DISPERSIVE MODEL COUPLED TO A DGTD METHOD WITH APPLICATION TO NANOPHOTONICS

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 ε(ω), 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 demonstrate

Development of DGTD solver for nanophotonics applications

International audienceDuring the last ten years, the discontinuous Galerkin time-domain (DGTD) method has progressively emerged as a viable alternative to well established finite-difference time-domain (FDTD) and finite-element time-domain (FETD) methods for the numerical simulation of electromagnetic wave propagation problems in the time-domain. We discuss here about the development and application of such a DTGD method for solving the system of time-domain Maxwell equations coupled to material models relevant to nanophotonics. Our efforts aim at improving the accuracy, flexibility and efficiency of the method in view of the numerical treatment of realistic problems

Hybrid MIMD/SIMD High Order DGTD Solver for the Numerical Modeling of Light/Matter Interaction on the Nanoscale

This paper is concerned with the development of a scalable high order finite element type solver for the numerical modeling of light interaction with nanometer scale structures. From the mathematical modeling point of view, one has to deal with the differential system of Maxwell equations in the time domain, coupled to an appropriate differential model of the behavior of the underlying material (which can be a dielectric and/or a metal) at optical frequencies. For the numerical solution of the resulting system of differential equations, we have designed a high order DGTD (Discontinuous Galerkin Time-Domain) solver that has been adapted to hybrid MIMD/SIMD computing. Here we discuss about this later aspect and report on preliminary performance results on the Curie system of the PRACE research infrastructure

Simulation of three-dimensional nanoscale light interaction with spatially dispersive metals using a high order curvilinear DGTD method

In this work, we present and study a flexible and accurate numerical solver in the context of three-dimensional computational nanophotonics. More precisely, we focus on the propagation of electromagnetic waves through metallic media described by a non-local dispersive model. For this model, we propose a discretization based on a high-order Discontinuous Galerkin time-domain method, along with a low-storage Runge-Kutta time scheme of order four. The semi-discrete stability of the scheme is analyzed for classical numerical fluxes, i.e. centered and upwind.Furthermore, the numerical treatment is enriched with an enhanced approximation of the geometry based on isoparametric curvilinear meshes. We finally assess our approach on several test cases, from academic to more physical ones

Simulation de la propagation d'ondes électromagnétiques en nano-optique par une méthode Galerkine discontinue d'ordre élevé

The goal of this thesis is to develop a discontinuous Galerkin time-domain method to be able to handle realistic nanophotonics computations. During the last decades, the evolution of lithography techniques allowed the creation of geometrical structures at the nanometer scale, thus unveiling a variety of new phenomena arising from light-matter interactions at such levels. These effects usually occur when the device is of comparable size or (much) smaller than the wavelength of the incident field. This work relies on the development and implementation of appropriate models for dispersive materials (mostly metals), as well as on a large panel of classical computational techniques. Two major methodological developments are presented and studied in details: (i) curvilinear elements, and (ii) local order of approximation. This work is complemented with several physical studies of real-life nanophotonics applications.L’objectif de cette thèse est de développer une méthode Galerkine discontinue d’ordre élevé capable de prendre en considération des simulations réalistes liées à la nanophotonique. Au cours des dernières décennies, l’évolution des techniques de lithographie a permis la création de structure géométriques de tailles nanométriques, révélant ainsi une large gamme de phénomènes nouveaux nés de l’interaction lumière-matière à ces échelles. Ces effets apparaissent généralement pour des objets de taille égale ou (très) inférieure à la longueur d’onde du champ incident. Ce travail repose sur le développement et l’implémentation de modèles de dispersion appropriés (principalement pour les métaux), ainsi que sur un large éventail de méthodes computationnelles classiques. Deux développements méthodologiques majeurs sont présentés et étudiés en détails: (i) les éléments courbes, et (ii) l’ordre d’approximation local. Ces études sont accompagnées de plusieurs cas-tests réalistes tirés de la nanophotonique

Simulation of the interaction of light with 3‐D metallic nanostructures using a proper orthogonal decomposition‐Galerkin reduced‐order discontinuous Galerkin time‐domain method

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Hybrid MIMD/SIMD High Order DGTD Solver for the Numerical Modeling of Light/Matter Interaction on the Nanoscale

International audienceThis paper is concerned with the development of a scalable high order finite element type solver for the numerical modeling of light interaction with nanometer scale structures. From the mathematical modeling point of view, one has to deal with the differential system of Maxwell equations in the time domain, coupled to an appropriate differential model of the behavior of the underlying material (which can be a dielectric and/or a metal) at optical frequencies. For the numerical solution of the resulting system of differential equations, we have designed a high order DGTD (Discontinuous Galerkin Time-Domain) solver that has been adapted to hybrid MIMD/SIMD computing. Here we discuss about this later aspect and report on preliminary performance results on the Curie system of the PRACE research infrastructure

Fitting experimental dispersion data with a simulated annealing method for nano-optics applications

International audienceA considerable amount of materials in nanophotonics are dispersive, enabling the propagation of so called surface plasmons at their interfaces with dielectrics. Hence, a reliable fit of frequency-dependent permittivity functions with an appropriate model is a first-order necessity for the accurate design of nano-optics devices with time-domain numerical methods, such as finite-difference time-domain (FDTD) or discontinuous Galerkin time-domain (DGTD). In this paper, we present the necessary ingredients to fit experimental permittivity functions using the simulated annealing (SA) method with a generalized second-order dispersion model implemented in the Diogenes software suite (see https://diogenes. inria.fr/). By scanning through different classes of materials, we come up with effective rules of thumb to make the fitting process fast and accurate

Simulation of near-field plasmonic interactions with a local approximation order discontinuous Galerkin time-domain method

International audienceDuring the last ten years, the discontinuous Galerkin time-domain (DGTD) method has progressively emerged as a viable alternative to well established finite-difference time-domain (FDTD) and finite-element time-domain (FETD) methods for the numerical simulation of electromagnetic wave propagation problems in the time-domain. The method is now actively studied for various application contexts including those requiring to model light/matter interactions on the nanoscale. In this paper we further demonstrate the capabilities of the method for the simulation of near-field plasmonic interactions by considering more particularly the possibility of combining the use of a locally refined conforming tetrahedral mesh with a local adaptation of the approximation order