Stable vortices in the model of a two-dimensional amplifying medium

A ubiquitous model of bulk amplifying laser media is provided by two-dimensional complex Ginzburg-Landau (CGL) equations with the cubic-quintic (CQ) nonlinearity. Solutions to these equations of fundamental physical significance are solitons and solitary vortices. While the zero-vorticity solitons may easily be stable in this setting, a severe problem which impedes the stabilization of vortices and, thus, the observation of such modes in the experiment, is the lack of an effective diffusion term in the physically relevant CGL equations (light does not undergo diffusionin lasing cavities). The formal diffusion term is necessary for the stability of vortex solitons. Recently, it has been reported that the addition of a 2D periodic potential, which may be induced by a transverse grating in the laser cavity, readily stabilizes compound (four-peak) vortices. Nevertheless, the most fundamental axisymmetric "crater-shaped" vortices (CSVs), alias vortex rings, have not been found before in a stable form. In this chapter, we report families of stable compact CSVs with vorticity S = 1 in the CGL model with the CQ nonlinearity and an external potential of two different types: an axisymmetric harmonic-oscillator trap, and the periodic potential corresponding to a grating (in the latter case, the CSV must be essentially squeezed into a single cell of the grating). We identify stability regions for the CSVs, as well as for the fundamental solitons (S = 0). If CSVs are unstable in the harmonic-oscillator potential, the vortex ring breaks up into robust dipoles. All the vortices with S = 2 are found to be unstable, splitting into tripoles. Stability regions for the dipoles and tripoles are reported too. The periodic potential does not stabilize CSVs with S > 1 either; instead, in this case unstable vortices with S = 2 evolve evolve into families of stable square-shaped quadrupoles.