Wannier functions and discrete NLS equations for nematicons

Abstract

We derive nonlocal discrete nonlinear Schrödinger (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 Schrödinger 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 Schrödinger 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.

Keywords

• nonlinear optics
• waveguide arrays
• nematic liquid crystals
• periodic media
• nonlocal media