## Section: New Results

### Spectral theory and modal approaches for waveguides

#### Modal analysis of electromagnetic dispersive media

Participants : Christophe Hazard, Sandrine Paolantoni.

We investigate the spectral effects of an interface between a usual dielectric and a negative-index material (NIM), that is, a dispersive material whose electric permittivity and magnetic permeability become negative in some frequency range. We consider here an elementary situation, namely, 1) the simplest existing model of NIM : the Drude model (for which negativity occurs at low frequencies); 2) a two-dimensional scalar model derived from the complete Maxwell's equations; 3) the case of a simple bounded cavity: a camembert-like domain partially

lled with a portion of non dissipative Drude material. Because of the frequency dispersion (the permittivity and permeability depend on the frequency), the spectral analysis of such a cavity is unusual since it yields a nonlinear eigenvalue problem. Thanks to the use of an additional unknown, we show how to linearize the problem and we present a complete description of the spectrum.

#### Formulation of invisibility in waveguides as an eigenvalue problem

Participants : Antoine Bera, 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.
A scatterer placed in an infinite waveguide may be *invisible* at particular discrete frequencies. We consider two different definitions of invisibility: no reflection (but possible conversion or phase shift in transmission) or perfect invisibility (the scattered field is exponentially decaying at infinity). Our objective is to show that the invisibility frequencies can be characterized as eigenvalues of some spectral problems. Two different approaches are used for the two different definitions of invisibility, leading to non-selfadjoint eigenvalue problems.

More precisely, for the case of no-reflection, we define a new complex spectrum which contains as real eigenvalues the frequencies where perfect transmission occurs and the frequencies corresponding to trapped modes. In addition, we also obtain complex eigenfrequencies which can be exploited to predict frequency ranges of good transmission. Our approach relies on a simple but powerful idea, which consists in using PMLs in an original manner: while in usual PMLs the same stretching parameter is used in the inlet and the outlet, here we take them as two complex conjugated parameters. As a result, they select ingoing waves in the inlet and outgoing waves in the outlet, which is exactly what arises when the transmission is perfect. This simple idea works very well, and provides useful information on the transmission qualities of the system, much faster than any traditional approach.

#### Transparent boundary conditions for general waveguide problems

Participants : Anne-Sophie Bonnet-Ben Dhia, Sonia Fliss.

In this work, done in collaboration with Antoine Tonnoir from INSA of Rouen, we propose a construction of transparent boundary conditions which can be used for quite general waveguide problems. Classical Dirichlet-to-Neumann maps used for homogeneous acoustic waveguides can be constructed using separation of variables and the orthogonality of the modes on one transverse section. These properties are also important for the mathematical and numerical analysis of problems involving DtN maps. However this framework does not extend directly to stratified, anisotropic or periodic waveguides and for Maxwell's or elastic equations. The difficulties are that (1) the separation of variables is not always possible and (2) the modes of the waveguides are not necessarily orthogonal on the transverse section. We propose an alternative to the DtN maps which uses two artificial boundaries and is constructed using a more general orthogonality property.