Guided modes of integrated optical guides. A mathematical study

A waveguide in integrated optics is defined by its refractive index. The guide is assumed to be invariant in the propagation direction while in the transverse direction it is supposed to be a compact perturbation of an unbounded stratified medium. We are interested in the modes guided by this device, which are waves with a transverse energy confined in a neighbourhood of the perturbation. Our goal is to analyse the existence of such guided modes. Under the assumptions of weak guidance the problem reduces to a two-dimensional eigenvalue problem for a scalar field. The associated operator is unbounded, selfadjoint, and bounded from below. Its spectrum consists of the discrete spectrum corresponding to the guided modes and of the essential spectrum corresponding to the radiation modes. We present existence results of guided modes and an asymptotic study at high frequencies, which shows that contrarily 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 reference medium.