Section: New Results

Spectral theory and modal approaches for waveguides

Scattering solutions in an unbounded strip governed by a plate model

Participants : Laurent Bourgeois, Sonia Fliss.

Together with Lucas Chesnel (EPI DEFI), we have initiated a new work on a particular waveguide which consists of a thin strip governed by a Kirchhoff-Love bilaplacian model. The aim is to build some radiation conditions and prove well-posedness of scattering problems for that simple model and for two kinds of boundary conditions: the strip is either simply supported or clamped. In the first case, we have shown that using a Dirichlet-to-Neumann operator enables us to prove fredholmness. Such approach is not possible in the second case, for which a completely different angle of attack is chosen: a Kondratiev approach involving weighted Sobolev spaces and detached asymptotics.

Modal analysis of electromagnetic dispersive media

Participants : Christophe Hazard, Sandrine Paolantoni.

We investigate the spectral effects of an interface between vacuum and a negative material (NM), that is, a dispersive material whose electric permittivity and/or magnetic permeability become negative in some frequency range. Our first work in this context concerns an elementary situation, namely, a two-dimensional scalar model (derived from the complete Maxwell's equations) which involves the simplest existing model of NM, referred to as the non-dissipative Drude model (for which negativity occurs at low frequencies). By considering a polygonal cavity, we have shown that the presence of the Drude material gives rise to various components of an essential spectrum corresponding to various unusual resonance phenomena: first, a low frequency bulk resonance (accumulation at the zero frequency of positive eigenvalues whose associated eigenvectors are confined in the Drude material); then, a surface resonance for one particular critical frequency (at which the so-called surface plasmons occurs, that is, localized highly oscillating vibrations at the interface between the Drude material and the vacuum); finally, corner resonances in a critical frequency interval (here, localized highly oscillating vibrations occur near any corner of the interface, interpreted as a "black hole" phenomenon). An article which presents these results has been submitted. Most recent works were devoted to the numerical simulation of these resonance phenomena in the context of the code XLiFE++ developped in the lab.

Formulation of invisibility in waveguides as an eigenvalue problem

Participant : Anne-Sophie Bonnet-Ben Dhia.

This work is done in collaboration with Lucas Chesnel from EPI DEFI and Vincent Pagneux from Laboratoire d'Acoustique de l'Université du Maine.

We consider an infinite acoustic waveguide (with a bounded cross-section) which is locally perturbed. At some exceptional frequencies and for particular incident waves, it may occur that all the energy of the incident wave is transmitted, the only effect in reflection being a superposition of evanescent modes in the vicinity of the perturbation. We have proposed an approach for which these reflection-less frequencies appear directly as eigenvalues of a new problem. This problem is very similar to the formulation of the scattering problem using Perfectly Matched Layers, except a slight modification in the PML. Precisely, we use two conjugated dilation parameters, $\alpha$ in the outlet and $\overline{\alpha }$ in the inlet, in order to select outgoing waves in the outlet and ingoing waves in the inlet. In fact, we show that the real eigenfrequencies that are obtained correspond either to trapped modes or to reflection-less modes. In addition to this real spectrum, we find intrinsic complex frequencies, which also contain information about the quality of the transmission through the waveguide. Mathematically, the non-selfadjoint eigenvalue problem with conjugated PMLs has strange properties: the discreteness of the point spectrum is not stable by compact perturbations and pathological examples can be exhibited.