Thermal effects on nonlinear optical beam propagation in nematic liquid crystals

This thesis examines in detail the study of nonlinear wave phenomena arising in different contexts in applied nonlinear optics. In particular, it explores the intricate nonlinear interplay between light and matter using tools from nonlinear PDEs—from the modulation solution for the nonlinear Schr¨odinger equation to classical approaches, such as Newton’s method, to numerical techniques from nonlinear dynamical systems. A framework is presented to study the dynamics of nonlinear optical beams propagating in nematic liquid crystals (NLCs), a substance similar in its fluidity to ordinary (isotropic) liquids, but whose refractive index and other properties change in the presence of optical beams. NLCs are unique in that they exist in a state between a crystal and an isotropic liquid. This intermediate state gives rise to molecular long-range orientational order, which in turn can be controlled by weak electromagnetic fields. Thermo-optical solitary waves form as a result of the NLC’s response to the light beam, consisting of a combination of focussing electric field-induced molecular rotation and defocussing temperature effects due to optical absorption by the medium. This nonlinear medium can support an optical solitary wave, termed a ’nematicon’, which can travel through this self-focusing medium as a diffractionless beam, able to both direct itself and guide other optical signals, rendering these waves useful for all optically reconfigurable integrated circuits and light-controlled switching. Modulation theory and an averaged Lagrangian analysis is used to model light beam evolution in NLCs. With modulation theory, a (2 + 1)-dimensional model is developed, based on approximations of the full nonlinear system, to describe the interplay between light beams and the thermo-optical and reorientational responses they induce in NLCs. This model describes the trajectories that spatial optical solitary waves (nematicons) follow in the anisotropic (strongly directional) NLC environment. Introducing several approximations based on the nonlocal physics of the material enables the prediction of the effect of temperature on nematicon trajectories and their angular steering, which establishes the analytical structure of the energy exchange between the input beam and the medium through one-photon absorption. The theoretical results are compared with existing experimental data, showing excellent agreement. In an NLC, nematicons exhibit competing nonlinear responses: extraordinarily polarized light waves are self-focusing due to reorientation and self-defocusing due to thermal effects. ithhen the beam power is strong enough, these opposing effects can lead to the formation of two-humped and ring-shaped solitary waves possessing a cross-sectional volcano profile. The formation of two-humped nematicons in (1 + 1) dimensions and volcano shaped nematicons in (2 + 1) dimensions are analysed and numerically modelled by calculating numerical solutions of their full governing equations and variational approximations. These full equations governing nonlinear optical beam propagation in NLC consist of an NLS-type equation for the light beam and elliptic equations for both the reorientational and thermal responses, in contrast to the simplified molecular and thermal responses of previous work. The simplified variational solutions for these localized ring waves are in remarkably good agreement with numerical results