Integrable and chaotic polarization dynamics in nonlinear optical beams

Abstract

The problem of two counterpropagating optical laser beams in a nonlinear medium is investigated as a Hamiltonian system. The phase space for travelling-wave solutions is the manifold C2 × C2, coordinated by two complex two-compnent electric field amplitudes, one for each beam. Invariance of the Hamiltonian function under various actions of the rotation group allows for reduction of the phase space to the two-sphere S2, on which the reduced Hamiltonian system, being two-dimensional, is completly integrable. We determine all the fixed points of the system and describe the bifurcations of the phase portrait which occur as parameters are varied. Among the various orbits in the reduced phase space, those connecting the hyperbolic fixed points are special and correspond to soliton-like and kink-like travelling-wave solutions. We also investigate how chaos, in particular horseshoe chaos and Arnold diffusion, arises when the system is subjected to certain types of perturbations.