Étude mathématique et numérique de structures plasmoniques avec coins

Camille Carvalho 1
1 POEMS - Propagation des Ondes : Étude Mathématique et Simulation
Inria Saclay - Ile de France, UMA - Unité de Mathématiques Appliquées, CNRS - Centre National de la Recherche Scientifique : UMR7231
Abstract : In this thesis, we investigate the propagation of electromagnetic waves in plasmonic structures, made of a dielectric and a metal. At optical frequencies metals exhibit unusual electromagnetic properties like a negative dielectric permittivity whereas dielectrics have a positive one. This change of sign allows the propagation of surface waves (called surface plasmons) at the metal-dielectric interface. This thesis is focused on the case where the interface presents corners. In the past decade theoretical studies have been carried out, combining results of the T-coercivity approach and the analysis of corner singularities. In particular it has been shown the existence of two states, depending on the parameters of the problem (frequency, material, geometry). The goal of this thesis is to develop, for the 2D case, a stable numerical method adapted to each state, with a specific treatment at the corners. In the first state (where the solutions belong to the function space of "classical energy") we develop meshing rules adapted to the geometry to guarantee the convergence’s optimality of the approximation with the finite element method: we say that the meshes are T-conforming. For the second state (where the solutions are no longer of finite energy), we propose an original numerical method using Perfectly Matched Layers (PMLs) at corners to capture the singularities, called black-hole waves as they carry energy absorded by the corners. These techniques are applied to two physical problems: the diffraction by a plane wave of a polygonal metalic inclusion, and the search of guided modes in a plasmonic waveguide with a polygonal cross-section. For the scattering problem, we prove that the corners of the metalic inclusion can trap some energy carried by the black-hole waves, and we quantify numerically the amount of trapped energy by each corner. Concerning the study of guided modes of a plasmonic waveguide, the problem is a non standard spectral problem. In the presence of black-hole waves, the eigenvalues associated to the guided modes are embedded in the essential spectrum. We use again PMLs at the corners to reveal them. This consists in computing the eigenvalues of an extended operator with a discrete spectrum.
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Camille Carvalho. Étude mathématique et numérique de structures plasmoniques avec coins. Mathématiques [math]. ENSTA ParisTech, 2015. Français. ⟨tel-01240904⟩

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