Discontinuous Galerkin Time-Domain method for nanophotonics

The numerical study of electromagnetic wave propagation in nanophotonic devices requires among others the integration of various types of dispersion models, such as the Drude one, in numerical methodologies. Appropriate approaches have been extensively developed in the context of the Finite Differences Time-Domain (FDTD) method, such as in [1] for example. For the discontinuous Galerkin time-domain (DGTD), stability and convergence studies have been recently realized for some dispersion models, such as the Debye model [2]. The present study focuses on a DGTD formulation for the solution of Maxwell’s equations coupled to (i) a Drude model and (ii) a generalized dispersive model. Stability and convergence have been proved in case (i), and are under study in case (ii). Numerical experiments have been made on classical situations, such as (i) plane wave diffraction by a gold sphere and (ii) plane wave reflection by a silver slab. 1 Drude model The Drude model describes the response of certain dispersive media to an electromagnetic wave propagating in a certain range of frequencies. The considered model permits to establish a dependency between the permittivity of the material and the angular frequency of the electromagnetic wave in the following form: εr,d(ω) = ε∞ − ω2 d ω2 + iωγd , where ωd, γd and ω are respectively the plasma frequency and the damping constant of the medium, and the angular frequency. Adding a Drude dispersion model therefore implies a coupling, in the time domain, between the electric field E and an additional field, the dipolar current (Jp), through an ODE whose solution we choose is here approximated in a DG framework. A centered fluxes DG method has been chosen to develop a numerical approximation of the problem, given the geometry and the inhomogeneous media to be considered. It is associated with a second-order Leap-Frog scheme in time, therefore inducing a non-dissipative scheme. A theoretical study of the latter has been made, demonstrating an error convergence in O ( hmin(s,p) + ∆t2 ) , where p is the spatial order of approximation, and s is related to the regularity assumptions made on the electromagnetic field. 2 Generalized dispersive model Recently, several arbitrary dispersive models have been proposed, such as the Critical Points (CP) [3] and the Complex-Conjugate Pole-Residue Pairs (CCPRP) [1]. Here, another formulation is considered: in accordance with the fundamental theorem of algebra, the permittivity function is written as a decomposition of a constant, one zero-order pole (ZOP), a set of first-order generalized poles (FOGP), and a set of second-order generalized poles (SOGP). This leads to the following expression in the frequency domain: εr,g(ω) = ε∞− σ jω − ∑