Some numerical investigations in nonlinear optics

The problem of the self-focusing of a light beam in nonlinear media is the central problem in nonlinear optics. The powerful laser beam propagation through a real medium under certain conditions is accompanied with such a phenomenon. Mathematically the problem deals with the investigation of the asymptotic behaviour of the solution of the parabolic equation 2i ∂u ∂z= ∂²u ∂r²+ 1 r ∂u ∂r+f{hook}({divides}u{divides}²)u with given initial distribution u(r,0) and boundary condition u(∞, z) = 0 where u is the electromagnetic field amplitude, f is a function which describes the refractive index deviation from its constant value in the linear medium. It is complex in the case of nonconservative media. In our investigation we combine analytical and numerical methods. The computational study of the self-focusing problem is complicated due to the boundary condition at infinity and the abrupt light amplitude behaviour in the paraxial region. We managed to overcome these difficulties by introducing the socalled quasi-uniform grid for the radial variable and by using the special technique of the correct transfer of the boundary condition from infinity. The main physical results are: (1) the conditions for the light self-trapping and waveguide creation are found, (2) the self-focusing mechanism and the law of increasing beam amplitude when approaching the collapse point are discovered; (3) the influence of the different kinds of absorption is investigated and the process of light "turbulence" is explained;. All the analytical and numerical results are comparable with the experimental situation as well as with treatments by other authors.