# Finite Elements for Electrodynamics and Modal Analysis of Dispersive Structures

Thursday, December 15, 2016 - 9:00am - 10:00am

Keller 3-180

Andre Nicolet (Aix-Marseille Université)

Our purpose is to develop a numerical tool for the study of photonic devices. The electrodynamic behavior of these systems can be efficiently characterized by their resonances but realistic materials have a strong time dispersive permittivity at optical frequencies and it therefore depends on the very frequency we are trying to compute via an eigenvalue problem.

As an introduction, the interest of using the Finite Element Method (FEM) with edge elements in order to solve the vector eigenvalue problems in electrodynamics is sum up. The case of Quasi-Normal Modes (QNMs) for open resonators is also considered with the Perfectly Matched Layers (PMLs) taken to solve this problem. PMLs are considered here as material properties equivalent to a complex-valued change of coordinates just as in Transformation Optics (TO). All these techniques can be advantageously presented in the framework of Exterior Calculus.

A new method is then presented for the direct computation of the resonances associated with electromagnetic structures including media with highly dispersive permittivities. A classical way to introduce dispersion is to use Drude or Lorentz. All these models are in fact representations of a dispersive permittivity in the form of a rational function of the frequency f, i.e. N(f)/D(f) where N and D are polynomials in f. A direct way to obtain an eigenvalue problem is to multiply the equation by the denominator D(f). In this case, the discretization of the problem will lead to a generalized polynomial eigenvalue problem (of degree higher than 2). Very recent advances in linear algebra algorithms have provided efficient libraries able to directly tackle such problems (e.g. SLEPc). Nevertheless, the interesting cases may involve several dispersive media. In this case, D(f) is different in each medium and multiplying the equation by all the different D(f) leads to an eigenvalue problem of very high degree that is much more difficult to solve. Therefore, we multiply each region with the local D(f) (and D(f) can indeed be seen as a discontinuous function between the media). In this case, a special treatment has to be applied to the boundaries: the magnetic tangential field has to be introduced as a Lagrange multiplier on such boundaries and used to control the discontinuity introduced by the local D(f) factor. The practical implementation is performed using Onelab/Gmsh/GetDP open source packages. We present simple 2D models showing the validity of this approach.

As an introduction, the interest of using the Finite Element Method (FEM) with edge elements in order to solve the vector eigenvalue problems in electrodynamics is sum up. The case of Quasi-Normal Modes (QNMs) for open resonators is also considered with the Perfectly Matched Layers (PMLs) taken to solve this problem. PMLs are considered here as material properties equivalent to a complex-valued change of coordinates just as in Transformation Optics (TO). All these techniques can be advantageously presented in the framework of Exterior Calculus.

A new method is then presented for the direct computation of the resonances associated with electromagnetic structures including media with highly dispersive permittivities. A classical way to introduce dispersion is to use Drude or Lorentz. All these models are in fact representations of a dispersive permittivity in the form of a rational function of the frequency f, i.e. N(f)/D(f) where N and D are polynomials in f. A direct way to obtain an eigenvalue problem is to multiply the equation by the denominator D(f). In this case, the discretization of the problem will lead to a generalized polynomial eigenvalue problem (of degree higher than 2). Very recent advances in linear algebra algorithms have provided efficient libraries able to directly tackle such problems (e.g. SLEPc). Nevertheless, the interesting cases may involve several dispersive media. In this case, D(f) is different in each medium and multiplying the equation by all the different D(f) leads to an eigenvalue problem of very high degree that is much more difficult to solve. Therefore, we multiply each region with the local D(f) (and D(f) can indeed be seen as a discontinuous function between the media). In this case, a special treatment has to be applied to the boundaries: the magnetic tangential field has to be introduced as a Lagrange multiplier on such boundaries and used to control the discontinuity introduced by the local D(f) factor. The practical implementation is performed using Onelab/Gmsh/GetDP open source packages. We present simple 2D models showing the validity of this approach.