Periodic Optical Waveguides: Exact Floquet Theory and Spectral Properties

We consider the steady propagation of a light beam in a planar waveguide whose width and depth are periodically modulated in the propagation direction. Using methods of soliton theory, a class of periodic potentials is presented for which the complete set of Floquet solutions of the linear Schrödinger equation can be found exactly at a particular optical frequency. For potentials in this class, there are exactly two bound Floquet solutions at this frequency, and they are degenerate, having the same Floquet multiplier. We study analytically the behavior of the waveguide under small changes in the frequency and observe a breaking of the degeneracy in the Floquet multiplier at first order. We predict, and observe numerically, the disappearance of both bound states at second order. These results suggest applications to spectral filtering.