Abstract

Consider a reference homogeneous and isotropic electromagnetic waveguide with a simply connected cross-section embedded in a perfect conductor. In this setting, when the waveguide is straight, the spectrum of the associated self-adjoint Maxwell operator with a constant twist lies on the real line, is symmetric with respect to zero and exhibits a spectral gap around the origin. Moreover, the spectrum is purely essential, and contains 0 which is an eigenvalue of infinite multiplicity. In this talk, we present new results on the effects of geometric deformations of bending and twisting on the spectrum of the Maxwell operator. More precisely, we provide, on the one hand, sufficient conditions on the asymptotic behaviour of curvature and twist ensuring the preservation of the essential spectrum of the reference waveguide. Our approach relies on a Birman-Schwinger-type principle which has an interest of its own. On the other hand, we give sufficient conditions, involving in particular the shape of the crosssection of the waveguide, so that the geometrical deformation creates discrete spectrum within the gap of the essential spectrum. In addition, we give some results on the possible localization of these discrete eigenvalues. Finally, we show some theoretical and numerical results further investigating the sufficient condition involving the geometry of the cross-section. This is a joint work in collaboration with P. Briet, M. Cassier and T. Ourmières-Bonafos.