## Section: New Results

### Spectral theory and modal approaches for waveguides

#### Guided modes in ladder-like open periodic waveguides

Participants : Sonia Fliss, Patrick Joly, Khac Long Nguyen, Elizaveta Vasilevskaya.

The general objective is the study of localized modes in locally perturbed periodic media and of guided modes in periodic media with a lineic perturbation. We investigate the existence theory of such modes as well as their numerical computations.

The problem, that is investigated in the framework of the PhD thesis of E. Vasilevskaya, in collaboration with Bérangère Delourme (Paris 13 University), is the case where the propagation medium is a thin structure whose limit is a periodic graph. We exhibit situations where the introduction of a line defect into the geometry of the domain leads to the appearance of guided modes. From the theoretical point of view, the problem is studied by asymptotic analysis methods, the small parameter being the thickness of the domain, so that when the thickness of the structure is small enough, the domain approaches a graph. The spectral theory of the underlying limit operator defined in the graph plays a key role in the analysis. For 2D configurations, we have shown that for sufficiently thin structures, it suffices to reduce the width of one rung to make appear guided modes. Moreover, using matched asymptotic expansions, we have constructed asymptotic expansions at any order of the corresponding eigenvalues and guided modes. For 3D configurations, the spectral theory of the underlying limit operator was already studied. In a further step, one can expect, again by asymptotic analysis, to get corresponding existence results for the original problem, at least for sufficiently thin structures.

From a numerical point of view, the modes can be computed using non linear eigenvalue problems and specific transparent boundary conditions for periodic media. During his post-doc, Khac Long Nguyen has implemented an exact method based on Dirichlet-to-Neumann operators to compute localized modes in 2D locally perturbed periodic media or guided modes in 3D periodic media with a lineic perturbation. This was already done few years ago for waveguides configurations but here the construction of the transparent boundary conditions are much more involved.

#### Reduced graph models for networks of thin co-axial electromagnetic cables

Participants : Geoffrey Beck, Patrick Joly.

This work is the object of the PhD of Geoffrey Beck and is done in collaboration with Sébastien Imperiale (Inria, MEDISIM). The general context is the non destructive testing by reflectometry of electric networks of co-axial cables with heterogeneous cross section and lossy materials, which was the subject of the ANR project SODDA. We consider electromagnetic wave propagation in a network of thin coaxial cables (made of a dielectric material which surrounds a metallic inner-wire). The goal is to reduce 3D Maxwell’s equations to a 1D like model. During the past two years, we derived and justified generalized telegraphers model for a single cable. This year, we incorporated in our model the losses due to the skin effect induced by the non perfectly conducting nature of the metallic wire. Finally using the method of matched asymptotics, we have derived and justified improved Kirchhoff conditions.

#### Multimodal methods for the propagation of acoustic and electromagnetic waves

Participant : Jean-François Mercier.

In collaboration with Agnès Maurel from the Langevin Institut and Simon Felix from the LAUM, we have developed fast multimodal methods to describe the acoustic propagation in rigid waveguides or in periodic arrays. An incident wave is scattered by penetrable inclusions or by the succession of different penetrable media separated by interfaces of any shape. The difficulties are: to take into account the modes coupling and to get modes naturally decoupled at the entrance and at the exit of the computational domain. A weak formulation of the problem provides a modal formulation taking exactly into account the matching conditions at the interfaces. A consequence is that the obtained convergence is the best convergence expected, given the regularity of the solution. After the study of isotropic cases, we have generalized this approach to the case of anisotropic media, the difficulty being to take into account a tensor in the propagation equation.

#### Plasmonic waveguides

Participants : Anne-Sophie Bonnet-Ben Dhia, Camille Carvalho, Patrick Ciarlet.

This work, which is a part of the PhD of Camille Carvalho, is done in collaboration with Lucas Chesnel (Inria, Defi). A plasmonic waveguide is a cylindrical structure consisting of metal and dielectric parts. In a certain frequency range, the metal can be seen as a lossless material with a negative dielectric permittivity. The study of the modes of a plasmonic waveguide is then presented as an eigenvalue problem with a sign-change of coefficients in the main part of the operator. Depending on the values of the contrast of permittivities at the metal / dielectric interface, different situations may occur. In the "good" case, the problem is self-adjoint with compact resolvent and admits two sequences of eigenvalues tending to + and -$\infty $. But when the interface presents corners, for a particular contrast range, the problem is neither self-adjoint nor with compact resolvent. In this case, Kondratiev's theory of singularities allows to build extensions of the operator, with compact resolvent. Finally, we show that the eigenvalues for one of these extensions can be computed by combining finite elements and Perfectly Matched Layers at the corners. The paradox is that a specific treatment has to be done to capture the corners singularities, even to compute regular eigenmodes.