Numerical computation of plasmonic resonances in dispersive media: Application to metallic gratings

We present several methods for the direct computation of the resonances associated with electromagnetic structures involving highly frequency dispersive permittivities, such as metals and/or semi-conductors in the visible or... [ view full abstract ]

We present several methods for the direct computation of the resonances associated with electromagnetic structures involving highly frequency dispersive permittivities, such as metals and/or semi-conductors in the visible or infrared range of frequencies - this is a fundamental problem for plasmonic applications.
Computing the eigenfrequencies corresponding to source free solutions of an  electromagnetic problem (e.g. the harmonic wave equation for the electric field E) is a spectral problem. In presence of materials with flat dispersion, the  discretization of such problems using the Finite Element Method (FEM) in the harmonic case classically leads to linear (matrix) eigenvalue problems giving pairs
of resonant frequencies (eigenvalues) together with associated eigen-fields (modes).

Now, we consider relative permittivity functions heavily dependent on the very frequency that we aretrying to determine: We are facing a non-linear eigenvalue  problem. We are also interested in the case where several dispersive media are present in the structure as it often occurs in practice. We benchmark various FEM formulations of this non-linear eigenvalue problem, from auxiliary fields to brute numerical linearization, and we compare their performances. Very recent advances in linear algebra algorithms have provided efficient libraries able to directly tackle
such problems, such as the SLEPc library \cite{slepc-users-manual}. Direct calls to this library have been implemented in the open source FEM software GetDP.

The method is extended to open problems using Perfectly Matched Layers (PMLs) to determine the Quasi Normal Modes (QNMs). Note that the eigenfrequencies are then necessarily complex valued for the twofoldreason that the dispersive media are dissipative and that the geometry is unbounded. As an example, we present the modal analysis of a gold diffraction grating in the visible range. The structure is both open and periodic. The Floquet-Bloch theory is applied to tackle the  periodicity combined with PMLs to obtain the QNMs. A complex dispersion relation of the structure is obtained. It is shown that the physical behavior of this system can be efficiently described using a small number of QNMs - that can even be reduced to a single one.