Year

2013

Degree Name

Doctor of Philosophy

Department

School of Mathematics and Applied Statistics

Abstract

A nonlinear medium that displays promise in all-optical communications is a nematic liquid crystal. A nematic liquid crystal exhibits a “huge” nonlinearity, so that nonlinear effects can be observed over millimetre distances for relative low powered input beams (milliwatt power). Spatial optical solitons, termed nematicons, are supported in nematic liquid crystals. A further property of nematic liquid crystals is that there optical response is nonlocal, in that the elastic response of the nematic extends beyond the optical perturbing beam. This nonlocal response allows two dimensional beams, such as nematicons and optical vortices, to be stable.

The equations governing nonlinear optical beam propagation in nematic liquid crystals form a non-integrable, coupled system of an nonlinear Schr¨odingertype equation for the beam and a Poisson’s equation for the medium response. This system has no known, general solutions. In this thesis, an approximate variational technique, termed modulation theory, and numerical solutions will be used to analyse the evolution and propagation of nematicons, both circular and elliptical in cross section, and optical vortices in a finite liquid crystal cell. Particular attention is paid to the effect of boundaries on the beam trajectory and stability. Modulation theory has the advantage that the coupled partial differential equations governing the beam are reduced to a finite dimensional dynamical system, which yields insights into the underlying physical mechanisms. In addition, modulation theory can be easily extended to account for the effect of the diffractive radiation shed as a beam evolves.

Two methods are used to solve the equation for the medium response, Fourier series and the method of images, with the latter found to give a much more efficient solution. It is found that the cell boundaries act as a repulsive force on a beam, so that a beam has a spiral path down a cell. It is also found that interaction with cell walls can destabilise an optical vortex. A linearised stability analysis is used to determine the minimum distance of approach to a cell boundary before instability sets in. This minimum distance is found to be in excellent agreement with numerical solutions. Finally, the propagation of an elliptic nematicon with orbital angular momentum in a finite-sized cell is analysed. It is found that the inclusion of angular momentum loss to radiation is vital for the accurate description of this beam. This loss is included for the first time.

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