Étude mathématique et numérique des résonances dans une micro-cavité optique

Abstract : This thesis is devoted to the study of resonance frequencies of bidimensional optical cavities. More specifically, we are interested in whispering-gallery modes (modes localized along the cavity boundary with a large number of oscillations). The first part deals with the numerical computation of resonances by the finite element method using perfectly matched layers, and with a sensibility analysis in the three following situations: an unidimensional problem, a reduction of the rotationally invariant bidimensional case, and the general case. The second part focuses on the construction of asymptotic expansions of whispering-gallery modes as the number of oscillations along of boundary goes to infinity. We start by considering the case of a rotationally invariant problem for which the number of oscillations can be interpreted as a semiclassical parameter by means of an angular Fourier transform. Next, for the general case, the construction uses a phase-amplitude ansatz of WKB type which leads to a generalized Schrödinger operator. Finally, the numerically computed resonances obtained in the first part are compared to the asymptotic expansions made explicit by the use of a computer algebra software.
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Zoïs Moitier. Étude mathématique et numérique des résonances dans une micro-cavité optique. Analyse numérique [cs.NA]. Université Rennes 1, 2019. Français. ⟨NNT : 2019REN1S053⟩. ⟨tel-02308978v2⟩

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