Propagating modes in a periodic wave guide in the semi-classical limit
Résumé
It is well known that the number of propagating modes in a uniform wave guide is the transverse section divided by the wavelength λ (for a two-dimensional (2D) wave guide). In this paper we study the number of propagating modes Nmodes in the limit of small λ, in the case where the section is non-constant but periodic. Using results of a study done by Asch and Knauf (Asch J and Knauf A 1998 Nonlinearity 11 175–200), we show that for small λ, Nmodes grows like μb/λ where μb is the measure of the ballistic classical trajectories inside the guide. In the case of an ergodic wave guide, where there are no ballistic trajectories but only diffusive trajectories, we show that Nmodes grows like √D/√λ where D is the diffusion constant. These results are generalized for any Hamiltonian periodic in one direction, and numerical results with the kicked Harper model are given. Nmodes can be related to the Landauer conductance.