In physics, wave propagation is traditionally analyzed by means of regular solutions of wave
equations. However, solutions of wave equations in two and three dimensions often possess
singularities, that is, points or lines in space at which mathematical quantities that describe
physical properties of waves become infinite or change abruptly (Berry [2000]). For example,
at the point of phase singularity, the phase of the wave is undefined and wave intensity
vanishes. Phase singularities are recognized as important features common to all waves …

Optical vortices are phase singularities in electromagnetic waves that constitute a unique
and fascinating class of phenomena within the context of the physics of light. We present a
brief overview of the recent advances in the study of optical vortices propagating in
nonlinear media. We demonstrate that self-focusing nonlinearity leads, in general, to the
azimuthal instability of a vortex-carrying beam, but it can support novel types of stable (or
quasi-stable) self-trapped beams carrying nonzero angular momentum, such as vortex …