Existence of Optical Vortices

Optical vortices arise as phase singularities of the light fields and are of central interest in modern optical physics. In this paper, some existence theorems are established for stationary vortex wave solutions of a general class of nonlinear Schr\"{o}dinger equations. There are two types of results. The first type concerns the existence of positive-radial-profile solutions which are obtained through a constrained minimization approach. The second type addresses the existence of saddle-point solutions through a mountain-pass-theorem or min-max method so that the wave propagation constant may be arbitrarily prescribed in an open interval. Furthermore some explicit estimates for the lower bound and sign of the wave propagation constant with respect to the light beam power and vortex winding number are also derived for the first type solutions.