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Analyse de quelques problèmes de conductivité avec changement de signe

Abstract : In this thesis, we study the behaviour of electromagnetic waves when interacting with a negative material. Such a material has a negative electric permittivity and/or magnetic permeability. Here we only focus on negative permittivity materials. In dimension two, Maxwell's equations in harmonic regime reduce to a couple of scalar, easier to tackle, sub-problems. One of these sub-problems allows surface waves to propagate along the interface between a negative material and a dielectric, which makes it very interesting for the applications. Such surface waves are called surface plasmons. Here, we focus on this sub-problem and more specifically on its main part which is a conductivity equation. Yet, as the permittivity sign changes between the negative material and the dielectric, it is not allowed to use the classical Lax-Milgram framework. In the first chapter, we introduce tools which are useful to understand the rest of this thesis. In particular, we describe how studying conductivity equation leads us to deal with the Poincar{'e}-Neumann operator. The spectrum of this operator encodes permittivity ratios that allow surface plasmons to propagate. We propose both the integral formulation and the variational formulation of this operator, and we explain the link existing in-between. In the second chapter of this thesis, we focus on the well-posedness property of the conductivity equation when permittivity sign changes. Using integral equation methods, we propose a sufficient well-posedness condition for this problem. In the third chapter, we deal with the numerical computation of the Poincaré-Neumann operator spectrum using finite element methods. We are interested in the convergence of numerically computed eigenvalues to the theoretical ones. In the last chapter, we study the electromagnetic wave transmission problem in a metallic layer with a negative permittivity from the Green's function point of view. In particular, we investigate the Green's function behaviour when the metallic layer thickness goes to zero.
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Submitted on : Friday, May 10, 2019 - 5:09:06 PM
Last modification on : Friday, March 25, 2022 - 9:44:28 AM


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Lionel Salesses. Analyse de quelques problèmes de conductivité avec changement de signe. Thermique [physics.class-ph]. Université Grenoble Alpes, 2018. Français. ⟨NNT : 2018GREAM087⟩. ⟨tel-02125956⟩



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