Claire Scheid

A parallel non-conforming multi-element DGTD method for the simulation of electromagnetic wave interaction with metallic nanoparticles

The present work is about the development of a parallel non-conforming multi-element discontinuous Galerkin time-domain (DGTD) method for the simulation of the scattering of electromagnetic waves by metallic nanoparticles. Such nanoparticles most often have curvilinear shapes, therefore we propose a numerical modeling strategy which combines the use of an unstructured tetrahedral mesh for the discretization of the scattering structures with a structured (uniform cartesian) mesh for treating efficiently the rest of the domain. The overall goal is to increase the flexibility in the meshing process while decreasing the needs in computational resources for the target applications. The latter are here modeled by the system of 3D time-domain Maxwell equations coupled to a Drude dispersion model for taking into account the material properties of nanoparticles at optical frequencies. We propose an auxiliary differential equation (ADE) based DGTD method for solving the resulting system and present numerical results demonstrating the benefits of using non-conforming multi-element meshes in this particular application context.

Travelling Waves for the Nonlinear Schrödinger Equation with General Nonlinearity in Dimension Two

We investigate numerically the two-dimensional travelling waves of the nonlinear Schrödinger equation for a general nonlinearity and with nonzero condition at infinity. In particular, we are interested in the energy–momentum diagrams. We propose a numerical strategy based on the variational structure of the equation. The key point is to characterize the saddle points of the action as minimizers of another functional that allows us to use a gradient flow. We combine this approach with a continuation method in speed in order to obtain the full range of velocities. Through various examples, we show that even though the nonlinearity has the same behaviour as the well-known Gross–Pitaevskii nonlinearity, the qualitative properties of the travelling waves may be extremely different. For instance, we observe cusps, a modified KP-I asymptotic in the transonic limit, various multiplicity results and “one-dimensional spreading” phenomena.

A 3D curvilinear discontinuous Galerkin time-domain solver for nanoscale light-matter interactions

Classical finite element methods rely on tessellations composed of straight-edged elements mapped linearly from a reference element, on domains which physical boundaries and interfaces are indifferently straight or curved. This approximation represents a serious hindrance for high-order methods, since they limit the accuracy of the spatial discretization to second order. Thus, exploiting an enhanced representation of physical geometries is in agreement with the natural procedure of high-order methods, such as the discontinuous Galerkin method. In this framework, we propose and validate an implementation of a high-order mapping for tetrahedra, and then focus on specific photonics and plasmonics setups to assess the gains of the method in terms of memory and performances.

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 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.

A discontinuous finite element time-domain solver for nanophotonic applications

We present a discontinuous finite element time-domain solver for the computer simulation of the interaction of light with nanometer scale structures. The method relies on a compact stencil high order interpolation of the electromagnetic field components within each cell of an unstructured tetrahedral mesh. This piecewise polynomial numerical approximation is allowed to be discontinuous from one mesh cell to another, and the consistency of the global approximation is obtained thanks to the definition of appropriate numerical traces of the fields on a face shared by two neighboring cells. Time integration is achieved using an explicit scheme and no global mass matrix inversion is required to advance the solution at each time step. Moreover, the resulting time-domain solver is particularly well adapted to parallel computing. In this paper, we discuss about recent contributions for improving the accuracy, flexibility and efficiency of the method, as well as its adaptation to physical models relevant to nanophotonic applications.