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

Novel Methods for the Time-Dependent Maxwell’s Equations and their Applications

This dissertation investigates three different mathematical models based on the time domain Maxwell\u27s equations using three different numerical methods: a Yee scheme using a non-uniform grid, a nodal discontinuous Galerkin (nDG) method, and a newly developed discontinuous Galerkin method named the weak Galerkin (WG) method. The non-uniform Yee scheme is first applied to an electromagnetic metamaterial model. Stability and superconvergence error results are proved for the method, which are then confirmed through numerical results. Additionally, a numerical simulation of backwards wave propagation through a negative-index metamaterial is given using the presented method. Next, the nDG method is used to simulate signal propagation through a corrugated coaxial cable through the use of axisymmetric Maxwell\u27s equations. Stability and error analysis are performed for the semi-discrete method, and are verified through numerical results. The nDG method is then used to simulate signal propagation through coaxial cables with a number of different corrugations. Finally, the WG method is developed for the standard time-domain Maxwell\u27s equations. Similar to the other methods, stability and error analysis are performed on the method and are verified through a number of numerical experiments

Error analysis of implicit and exponential time integration of linear Maxwell\u27s equations

This thesis is concerned with the numerical analysis of some well-known time integration methods, such as implicit collocation methods and exponential integrators, for linear Maxwell\u27s equations in time-domain. The error analysis of time integrators is done both for continuous Maxwell\u27s equations in a semigroup theory framework and for space discrete problem obtained by discretizing Maxwell\u27s equations in space by using discontinuous Galerkin finite element method

Discontinuous Galerkin Finite Element Methods for Maxwell\u27s Equations in Dispersive and Metamaterials Media

Discontinuous Galerkin Finite Element Method (DG-FEM) has been further developed in this dissertation. We give a complete proof of stability and error estimate for the DG-FEM combined with Runge Kutta which is commonly used in different fields. The proved error estimate matches those numerical results seen in technical papers. Numerical simulations of metamaterials play a very important role in the design of invisibility cloak, and sub-wavelength imaging. We propose a leap-frog discontinuous Galerkin Finite Element Method to solve the time-dependent Maxwell\u27s equations in metamaterials. The stability and error estimate are proved for this scheme. The proposed algorithm is implemented and numerical results supporting the analysis are provided. The wave propagation simulation in the double negative index metamaterials supplemented with perfectly matched layer (PML) boundary is given with one discontinuous Galerkin time difference method (DGTD), of which the stability and error estimate are proved as well in this dissertation. To illustrate the effectiveness of this DGTD, we present some numerical result tables which show the consistent convergence rate and the simulation of PML in metamaterials is tested in this dissertation as well. Also the wave propagation simulation in metamaterals by this DGTD scheme is consistent with those seen in other papers. Several techniques have appeared for solving the time-dependent Maxwell\u27s equations with periodically varying coefficients. For the first time, I apply the discontinuous Galerkin (DG) method to this homogenization problem in dispersive media. For simplicity, my focus is on obtaining a solution in two-dimensions (2D) using 2D corrector equations. my numerical results show the DG method to be both convergent and efficient. Furthermore, the solution is consistent with previous treatments and theoretical expectations