The Homogeneous Problem

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Abstract

The title of this chapter may seem obscure to many physicists. What we intend to deal with consists of several curious phenomena, well known to experimentalists, or recently discovered. They are grating anomalies, usually referred to as Wood anomalies, total absorption of a plane wave by a metallic grating (bare or coated with a thin dielectric layer) as well as the coupling of a laser beam into an optical waveguide by means of a holographic thin film coupler. Despite their different appearances, all these phenomena have the same origin in common. They all are connected with the excitation of surface waves along the periodic structure. Such a surface wave carries energy parallel to the mean plane of the surface, but is also slightly attenuated in that direction. Thus, it is referred to by many authors as a “leaky wave”. From a mathematical point of view, it is a solution of Maxwell equations and the associated boundary conditions on the grating surface, without any wave impinging on the structure. It is the study of such a solution that we call the “homogeneous problem” and we wish to show how the resolution of the homogeneous problem can enlighten the study of the response of the structure to a given excitation.