Chaotic properties of standing waves in a Kerr grating

Abstract

Helmholtz equations for a 1-d Kerr grating depend periodically on the propagation direction and their trajectories show irregular, 'chaotic' behavior. The usual Coupled Mode model for a shallow grating is derived from Helmholtz equations by an averaging procedure. They are autonomous and the (spatially) chaotic eects do not appear.
Here we compare the predictions for standing monochromatic waves at the Bragg frequency in a multi-layer structure on the basis of the CM model, with those based on the original Helmholtz equation. A scaling result for the dimensions of the chaotic region in the phase plane is obtained in terms of the linear index contrast. Waves with period of the envelope of the order of the inverse squared of the index contrast show irregular features, primarily in phase.