Self-focussed optical structures in a nematic liquid crystal

In recent experiments investigating the nonlinear interaction between light and nematic liquid crystals, Braun et al. (1993) observed complex optical beam structures that were generated by the strong self-focussing of laser light. For a simplified partial differential equation (PDE) model which captures the essential coupling between optical refraction and nematic deformation, we demonstrate two of the experimentally observed features - undulation and filamentation. For the mathematical analysis, we develop a novel asymptotic representation for this strongly coupled nonlinear system which exploits the natural separation of scales at which these optical structures are created by the self-focussing process. This approach uses geometrical optics, paraxial optics, and scale-separation to identify tractable outer and inner problems. The outer problem describes the undulation of the beam, and is given by a free-boundary problem for the distortion of the nematic crystal. The inner problem describes the filamentation of the beam, and is given by a nonlocal-nonlinear Schrödinger (NLS) equation for evolution of the light wave. For the outer problem, we demonstrate analytically the existence of small amplitude undulations of the beam. Large amplitude undulations are studied numerically. For the inner problem, waveguide modes are constructed. Simulations of the nonlocal NLS show that the interaction of these modes generates filamentary beam structures. Thus the PDE model, when reduced asymptotically into two decoupled systems at two distinct spatial scales, produces a theoretical corroboration of the unusual nonlinear optical behavior of undulation, as well as a nonlinear mechanism for filamentation.