Section: New Results

Wave propagation in non classical media

Modal analysis of electromagnetic dispersive media

Participants : Anne-Sophie Bonnet-Ben Dhia, Christophe Hazard.

Except in vacuum, the velocity of electromagnetic waves generally depends on the frequency. This dispersion plays in particular a vital role in situations where the effective index takes values below unity or negative, which happens with metamaterials or plasmonic devices. However, most of the studies in this domain are considering only the time-harmonic regime, forgetting dispersion, which leads to apparent paradoxes. We have elaborated a project, in collaboration with the Institut Fresnel in Marseille. Our objective is to gather physical and mathematical points of view to explore a frequency-to-time approach for dispersive media. This approach is based on a general technique which allows to hide dispersion in an augmented formulation of Maxwell’s equations. Using this tool, our aim is first to carry the spectral analysis of dispersive systems, take advantage of this analysis to predict the time-dependent behaviour of dispersive systems, then design adapted numerical methods for their simulation and finally confirm predictions by real experiments. To begin with, during the internship of Bilal Yezza, a toy problem has been studied, where the presence of accumulation points in the spectrum is due to the dispersion. This project has been submitted to the ANR for the second year and has already led to preliminary common works and discussions, in particular during the workshop Leaky days organized by Christophe Hazard in Palaiseau in June 2015.

Perfectly Matched Layers in plasmas and metamaterials

Participants : Eliane Bécache, Patrick Joly, Maryna Kachanovska, Valentin Vinoles.

We work on the stability of Generalized Perfectly Matched Layers (GPMLs) in dispersive media for which classical PMLs are in general unstable. These new PMLs involve, in addition to the absorption parameter $\sigma \ge 0$, a real valued rational function of the frequency $\psi \left(\omega \right)$. We first worked on isotropic media and derived, using Fourier analysis methods, a necessary and sufficient condition on the function $\psi \left(\omega \right)$ for the stability of the PML model. This result has been presented in several conferences and used to design new stable PMLs for negative index metamaterials and uniaxial anisotropic plasmas (even though this last model is anisotropic, the anisotropy has a structure that permits a special decomposition of vector fields that give a new equivalent model adapted for our GPMLs).

We are currently working on the generalization of this analysis to a class of anisotropic dispersive models using a different approach based on Laplace transform in time.

However, this theory does not apply to more general cold plasma models that we wish to treat. Finding good PMLs in this case still remains a challenging open question. Several attempts, such as radial PMLs (which we discussed about with our visitor Martin Halla from TU Wien), have failed.