Lumière dans les milieux atomiques désordonnés : théorie des matrices euclidiennes et lasers aléatoires

Abstract : This thesis is devoted to the study of the properties of light emitted by a collection of atomic scatterers distributed at random positions in Euclidean space and interacting with the electromagnetic field. In this respect, an ab initio analytic theory of random lasing is formulated in terms of the statistical properties of the so-called `Green's matrix'. The latter belongs to the family of Euclidean random matrices (ERM's), for which we develop an analytic theory giving access to their eigenvalue distribution. First, we derive quantum microscopic equations for the electric field and atomic operators, and show how the non-Hermitian Green's matrix (a matrix with elements equal to the Green's function of the Hemholtz equation between pairs of atoms in the system) emerges in the quantum formalism. We provide expressions for the intensity and the spectrum of light in terms of the properties of the Green's matrix, characterize quantum Langevin forces, and reveal how the semiclassical random laser threshold is washed out by quantum fluctuations (chapters 2 and 3). A mesoscopic and semiclassical description of light scattered by an arbitrary large number of pumped atoms randomly distributed in free space is the subject of chapter 4. After deriving a universal lasing threshold condition valid for any configuration of atoms, we provide a microscopic derivation of transport equation in the presence of gain, discuss various approximations of the latter (Bethe-Salpeter, Boltzmann, diffusion equations), reveal a mapping to ERM's, and analyze the lasing threshold condition inferred from the transport equation. Facing the problem of characterizing analytically the statistical properties of the Green's matrix, we develop in chapters 5 and 6 a theory for Hermitian and non-Hermitian ERM's in the limit of large matrix size. We obtain self-consistent equations for the resolvent and the eigenvector correlator of arbitrary ERM and apply our results to three different ERM's relevant to wave propagation in random media: the three-dimensionnal Green's matrix, its imaginary part and its real part. From a physical point of view, we are able to describe analytically with a fair precision the full probability distribution of decay rates of light emitted by a large number of atoms, as well as of the collective frequency shift induced by the light-matter interaction. In addition, we promote the idea that the eigenvalue distribution of the Green's matrix can serve as a map on which signatures of various regimes of disorder can be distinguished (ballistic, diffusive, localized, effective medium, and superradiance regimes). Finally, we combine microscopic equations of motion of light-matter interaction with our results for non-Hermitian ERM's to tackle the problem of random lasing. Lasing threshold and the intensity of laser emission are calculated analytically in the semiclassical approximation, and the spectrum of light below threshold is computed by taking into account quantum effects. Our theory applies all the way from low to high density of atoms.
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Arthur Goetschy. Lumière dans les milieux atomiques désordonnés : théorie des matrices euclidiennes et lasers aléatoires. Autre. Université de Grenoble, 2011. Français. ⟨NNT : 2011GRENY083⟩. ⟨tel-00676988v2⟩

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