CONSTRUCTION OF TRAPPED MODES FOR WAVE-GUIDES AND DIFFRACTION GRATINGS

This paper describes various methods for the construction of trapped-m ode solutions within wave guides and along diffraction gratings; in th e latter case the solutions are often called Rayleigh-Bloch waves. One of the main results is the explicit construction of a number of new t rapped-mode solutions within a wave guide that correspond to eigenvalu es that are embedded in the continuous spectrum of the relevant operat ors. The method of construction is to take 'trial' solutions of the fi eld equation that satisfy the required conditions on the guide walls a nd have the correct decay at large distances and then to identify line s that can be interpreted as the boundary of a structure within the wa ve guide. The same idea is also investigated for Rayleigh-Bloch waves within the context of diffraction gratings. Sensible choices of trial function are made by reference to solutions for a grating of circular cylinders, obtained by a multipole expansion method, and an approximat e solution for more general geometries.