Propagation of optical spatial solitary waves in bias-free nematic-liquid-crystal cells

The propagation of a bulk optical solitary wave in a rectangular cell filled with a nematic liquid crystal—a nematicon—is mathematically modelled. In order to overcome the Freédricksz threshold the cell walls are rubbed to pretilt the nematic. A modulation theory, based on a Lagrangian formulation, is developed for the (2+1)-dimensional propagation of the solitary wave beam down the cell. This modulation theory is based on two different formulations of the director distribution. The relative advantages and disadvantages of these two methods are discussed. A previously unexplored method based on images is found to possess significant advantages. Excellent agreement with full numerical solutions of the nematicon equations is found for both methods. Finally, the implications of the results obtained for some widely used approximations to the nematicon equations are discussed, particularly their use in comparisons with experimental results

Elliptical optical solitary waves in a finite nematic liquid crystal cell

2015 Elsevier B.V. The addition of orbital angular momentum has been previously shown to stabilise beams of elliptic cross-section. In this article the evolution of such elliptical beams is explored through the use of an approximate methodology based on modulation theory. An approximate method is used as the equations that govern the optical system have no known exact solitary wave solution. This study brings to light two distinct phases in the evolution of a beam carrying orbital angular momentum. The two phases are determined by the shedding of radiation in the form of mass loss and angular momentum loss. The first phase is dominated by the shedding of angular momentum loss through spiral waves. The second phase is dominated by diffractive radiation loss which drives the elliptical solitary wave to a steady state. In addition to modulation theory, the chirp variational method is also used to study this evolution. Due to the significant role radiation loss plays in the evolution of an elliptical solitary wave, an attempt is made to couple radiation loss to the chirp variational method. This attempt furthers understanding as to why radiation loss cannot be coupled to the chirp method. The basic reason for this is that there is no consistent manner to match the chirp trial function to the generated radiating waves which is uniformly valid in time. Finally, full numerical solutions of the governing equations are compared with solutions obtained using the various variational approximations, with the best agreement achieved with modulation theory due to its ability to include both mass and angular momentum losses to shed diffractive radiation

Propagation of nonlinear optical beams in finite liquid crystal cells

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

Optical vortex solitary wave in bounded nematic-liquid-crystal cell

Modulation theory, based on a Lagrangian formulation of the governing equations, is used to investigate the propagation of a nonlinear, nonlocal optical vortex solitary wave in a finite nematic-liquid-crystal cell. The nematic response to the vortex is calculated using the approach of themethod of images (MOI). It is demonstrated that the MOI is a reliable alternative to the usual Fourier series solution as it requires an order of magnitude fewer terms to obtain excellent agreement with numerical solutions. It is found that the cell walls, in addition to repelling the optical vortex solitary wave, as for an optical solitary wave, can destabilize it due to the fixed director orientation at the walls. A linearized stability analysis is used to explain and analyze this instability. In particular, the minimum distance of approach of a stable vortex to the wall is determined from the stability analysis. Good agreement is found with numerical minimum approach distances

Nonlinear optical beams in bounded nematic liquid crystal cells

Stable nonlinear beams, both solitary waves (nematicons) and optical vortices, can form in a nematic liquid crystal due to a balance between the nonlinear, nonlocal response of the nematic and the diffractive spreading of the light beam. The `huge\u27 nonlinearity of a nematic liquid crystal makes it ideal for the experimental development of photonic devices as nonlinear effects occur over millimetre distances. In this work, a simple and fast method to analyse the trajectory of a nonlinear beam within a finite liquid crystal cell, based on a classical method not explored in this context, the method of images, is developed. With the orientation of the nematic molecules modelled using images, the evolution of the beam is obtained by using both asymptotics and modulation theory. The efficiency of this new method is shown by comparisons with a standard Fourier series solution for the nematic response and full numerical solutions of the governing equations. It is found that only a small number of images is required compared with the usual Fourier series technique in order to obtain excellent agreement with full numerical solutions. Finally, the contrasting effect of the cell boundaries on a nematicon and a vortex is explored