Nonlinear Optical Vector Ampliude Equations. Polarization and Vortex Solutions

Abstract

We investigate two kind of polarization of localized optical waves in nonlinear Kerr type media, linear and combination of linear and circular. In the first case of linear polarized components we obtained the vector version of 3D+1 Nonlinear Schr\"{o}dinger Equation (VNSE). It is shown that these equations admit exact vortex solutions with spin l=1. We have determined the dispersion region and the medium parameters necessary for experimental observation of these vortices. In the second case we represent the electric and magnetic fields as sum of circular and linear components. We supouse also that our nonlinear media admit linear magnetic polarization. This allows us to reduce the Maxwell's equations to a set of amplitude Nonlinear Dirac Equations (NDE). WE find two representation of the NDE- spherical and spinor. In the spherical representation we obtain exact vortex solutions with spin l-1 and in the spinor representation, exact vortex solutions with spin j=1/2.