Section: New Results

Spectral theory and modal approaches for waveguides

Guided modes in ladder-like open periodic waveguides

Participants : Sonia Fliss, Patrick Joly, Elizaveta Vasilevskaya.

This work is done in the context of the PhD of Elizaveta Vasilevskaya, in collaboration with Bérangère Delourme, from Paris 13 University. We consider the theoretical and numerical aspects of the wave propagation in ladder-like periodic structures. We exhibit situations where the introduction of a lineic defect into the geometry of the domain leads to the appearance of guided modes and we provide numerical simulations to illustrate the results. 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. Numerical computations are based on specific transparent conditions for periodic media.

Absence of trapped modes for a class of unbounded propagative media

Participants : Anne-Sophie Bonnet-Ben Dhia, Christophe Hazard, Sonia Fliss, Antoine Tonnoir.

We have proposed a new approach to prove that there does not exist square-integrable solutions to the two-dimensional Helmholtz equation in a homogeneous conical domain with a vertex angle greater than $\pi$. This shows that for a medium filling the whole plane, there can be no trapped modes if all the inhomogeneities (penetrable or not) are concentrated in a conical domain with a vertex angle less than $\pi$. The proof uses the compatibility of Fourier representations of the field in different half-spaces. One interesting consequence of our result concerns the case of curved open waveguides (e.g., bended optical fibers). Unlike closed waveguides for which trapped modes confined near the bend may occur, our result implies that trapped modes cannot exist if the core of the waveguide is located in a cone with vertex angle less than $\pi$. Our results can be extended to higher space dimensions, and to some Y-junctions of open waveguides (using a generalized Fourier transform instead of the usual one).

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 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 is 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 quantum graph in which, along each edge, one is reduced to compute the electrical potential and current by solving 1D wave equations (the telegrapher's model) coupled by vertex conditions. Using the method of matched asymptotics, we have derived and justified improved Kirchhoff conditions.

Geometrical transformations for waveguides of complex shapes

Participant : Jean-François Mercier.

In collaboration with Agnès Maurel from the Langevin Institut and Simon Felix from the LAUM, we have developed multimodal methods to describe the acoustic propagation in rigid waveguides of general shapes, with varying curvature and cross section. A key feature is the use of a flexible geometrical transformation to a virtual space in which the waveguide is straight but associated to Robin boundary conditions. We have revisited an efficient method developed earlier which consists in adding two extra non-physical modes to the usual modal expansion of the field on the Neumann guided modes, in order to obtain a better convergence of the modal series.

This method has been extended to a half guide with an end wall of general shape, transformed into a flat surface by a geometrical transformation, thus avoiding to question the Rayleigh hypothesis. The transformation only affects a bounded inner region that naturally matches the outer region, which allows to easily select the ingoing and outgoing waves.