Mathematical Nonlinear Optics.

Abstract : The principal investigator, together with two post-doctoral fellows, several graduate students, and colleagues, has applied the modern mathematical theory of nonlinear waves to problems in nonlinear optics. Projects included (1) the interaction of laser light with nematic liquid crystals, (2) propagation through random nonlinear media, (3) cross polarization instabilities and optical shocks for propagation along nonlinear optical fibers, and (4) the dynamics of bistable optical switches coupled through both diffusion and diffraction. In project (1) the extremely strong nonlinear response of a cw laser beam in a nematic liquid crystal medium produced striking undulation and filamentation of the cw beam which was observed experimentally and explained theoretically. In project (2) the interaction of randomness with nonlinearity was investigated, as well as an effective randomness due to the simultaneous presence of many nonlinear instabilities. In the polarization problems of project (3) theoretical hyperbolic structure (instabilities and homoclinic orbits) in the coupled pdes was identified and used to explain cross polarization instabilities in both the focusing and defocusing cases, as well as to describe optical shocking phenomena. For the coupled bistable optical switches of project (4), a numerical code was carefully developed in two spatial and one temporal dimensions. The code was used to study the decay of temporal transients to 'on-off' steady states in a geometry which includes forward and backward longitudinal propagation, together with one dimensional transverse coupling of both electromagnetic diffraction and carrier diffusion.