submitted in the IMA Journal of Applied Mathematics
Anne-Sophie Bonnet-BenDhia
LSMP
ENSTA
Centre de l'Yvette
Chemin de la Hunière
91120 Palaiseau, France
Gabriel Caloz
IRMAR, Université de Rennes 1
Campus de Beaulieu
35042 Rennes Cedex, France
Fabrice Mahé
IRMAR, Université de Rennes 1
Campus de Beaulieu
35042 Rennes Cedex, France
A waveguide in integrated optics is defined by itsrefractive index. The guide is assumed to be invariant in thepropagation direction while in the transverse direction itis supposed to be a compact perturbation of an unbounded stratified medium. We are interested in the modes guidedby this device, which are waves with a transverseenergy confined in a neighborhood of the perturbation.Our goal is to analyze the existence of such guided modes. Under the assumptions of weak guidance the problem reduces toa two dimensional eigenvalue problem for a scalar field. Theassociated operator is unbounded, selfadjoint, and boundedfrom below. Its spectrum consists of the discrete spectrumcorresponding to the guided modes and of the essentialspectrum corresponding to the radiation modes. We present existence results of guided modes and an asymptoticstudy at high frequencies, which showsthat contrarily to the case of optical fibers, thenumber of guided modes can remain bounded. The major tools arethe min-max principle and comparison results between different eigenvalue problems. The originality ofthe present study lies in the stratified characterof the unbounded reference medium.