  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.
