GUIDED MODES OF INTEGRATED OPTICAL GUIDES - A MATHEMATICAL STUDY

A waveguide in integrated optics is defined by its refractive index. T he guide is assumed to be invariant in the propagation direction while in the transverse direction it is supposed to be a compact perturbati on of an unbounded stratified medium. We are interested in the modes g uided by this device, which are waves with a transverse energy confine d in a neighbourhood of the perturbation. Our goal is to analyse the e xistence of such guided modes. Under the assumptions of weak guidance the problem reduces to a two-dimensional eigenvalue problem for a scal ar field. The associated operator is unbounded, selfadjoint, and bound ed from below. Its spectrum consists of the discrete spectrum correspo nding to the guided modes and of the essential spectrum corresponding to the radiation modes. We present existence results of guided modes a nd an asymptotic study at high frequencies, which shows that contraril y to the case of optical fibers, the number of guided modes can remain bounded. The major tools are the min-max principle and comparison of results between different eigenvalue problems. The originality of the present study lies in the stratified character of the unbounded refere nce medium.