Wannier functions and discrete NLS equations for nematicons$^\dagger$

Abstract

We derive nonlocal discrete nonlinear Schrodinger (DNLS) equations for laser beam propagation in optical waveguide arrays that use a nematic liquid crystal substrate. We start with an NLS-elliptic model for the problem and propose a simplified version that incorporates periodicity in one of the directions transverse to the propagation of the beam. We use Wannier basis functions for an associated Schrodinger operator with periodic potential to derive discrete equations for Wannier modes and propose some possible simplified systems for interactions of modes within the first energy band of the periodic Schrodinger operator. In particular, we present the simplest generalization of a model proposed by Fratalocchi and Assanto by including a linear nonlocal term, and see evidence for parameter regimes where nonlinearity is more pronounced.