“Exact” solutions for circularly polarized solitons and vortices in a Kerr medium

For the curl-curl type vector equation describing a monochromatic light wave in a Kerr medium, an exact substitution has been analyzed, which leads to a system of four first-order ordinary differential equations for functions of the transverse radial coordinate. This system includes the integer multiplicity m of a vortex in the longitudinal electric field component. In this case, the multiplicity of a vortex in a wave with the left and right circular polarizations is $m-1$ and $m + 1$, respectively. With $|m| = 1$, numerical solutions of this system with appropriate boundary conditions make it possible to obtain the full information on the internal structure of a strongly nonlinear circularly polarized optical beam in a focusing medium taking into account the longitudinal field component and a small fraction of the opposite polarization. For $m = 0$, a solution in the form of a left-handed vortex in a left circularly polarized wave exists for a defocusing medium, which differs qualitatively from the right-handed vortex in the left circularly polarized wave for $m = 2$.