Patrick Joly

Maxence Cassier, Patrick Joly, Maryna Kachanovska

In this work, we investigate mathematical models for electromagnetic wave propagation in dispersive isotropic media. We emphasize the link between physical requirements and mathematical properties of the models. A particular attention is devoted to the notion of non-dissipativity and passivity. We consider successively the case of so-called local media and general passive media. The models are studied through energy techniques, spectral theory and dispersion analysis of plane waves. For making the article self-contained, we provide in appendix some useful mathematical background.

Maxence Cassier, Christophe Hazard, Patrick Joly

We explore the spectral properties of the time-dependent Maxwell's equations for a plane interface between a metamaterial represented by the Drude model and the vacuum, which fill respectively complementary half-spaces. We construct explicitly a generalized Fourier transform which diagonalizes the Hamiltonian that describes the propagation of transverse electric waves. This transform appears as an operator of decomposition on a family of generalized eigenfunctions of the problem. It will be used in a forthcoming paper to prove both limiting absorption and limiting amplitude principles.

Bérangère Delourme, Patrick Joly, Elizaveta Vasilevskaya

In this note, we exhibit a three dimensional structure that permits to guide waves. This structure is obtained by a geometrical perturbation of a 3D periodic domain that consists of a three dimensional grating of equi-spaced thin pipes oriented along three orthogonal directions. Homogeneous Neumann boundary conditions are imposed on the boundary of the domain. The diameter of the section of the pipes, of order $ε$ \textgreater{} 0, is supposed to be small. We prove that, for $ε$ small enough, shrinking the section of one line of the grating by a factor of $\sqrt$ $μ$ (0 \textless{} $μ$ \textless{} 1) creates guided modes that propagate along the perturbed line. Our result relies on the asymptotic analysis (with respect to $ε$) of the spectrum of the Laplace-Neumann operator in this structure. Indeed, as $ε$ tends to 0, the domain tends to a periodic graph, and the spectrum of the associated limit operator can be computed explicitly.