Construction of trapped modes for wave guides and diffraction gratings

This paper describes various methods for the construction of trapped-mode solutions within wave guides and along diffraction gratings; in the latter case the solutions are often called Rayleigh-Bloch waves. One of the main results is the explicit construction of a number of new trapped-mode solutions within a wave guide that correspond to eigenvalues that are embedded in the continuous spectrum of the relevant operators. The method of construction is to take 'trial' solutions of the field equation that satisfy the required conditions on the guide walls and have the correct decay at large distances and then to identify lines that can be interpreted as the boundary of a structure within the wave 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 approximate solution for more general geometries.