Simulation of light beam propagation in nonlinear media

The present paper concerns the simulation of nonlinear wave propagation in the ray regime, i.e., in the limit of geometrical optics. The medium involved is nonlinear. Linear ray propagation is conventionally computed by using Hamilton´s ray equations, whose inhomogeneous terms are derived from the dispersion equation. The formalism used to solve such a set of equations is the Runge-Kutta algorithm. In the present case of nonlinear propagation, the inhomogeneous terms depend on field amplitudes which are heuristically determined by the convergence (or divergence) of the rays in the beam. However, in the present case the varying convergence depends on the solution of the Hamilton equations, and it is therefore necessary to modify the original Runge-Kutta scheme, by building into it some iteration mechanism, such that the process converges to values which take into account the amplitude effect. As expected the results display self-focusing effects characteristic of nonlinear optics problems.