Airy-Gaussian vortex beams in the fractional nonlinear-Schr\"odinger medium

We address the propagation of vortex beams with the circular Airy-Gaussian shape in a (2+1)-dimensional optical waveguide modeled by the fractional nonlinear Schrodinger equation. Systematic analysis of autofocusing of the beams reveals a strongly non-monotonous dependence of the peak intensity in the focal plane on the corresponding Levy index, with a strong maximum at alpha =1.4. Effects of the nonlinearity strength, the ratio of widths of the Airy and Gaussian factors in the input, as well as the beam vorticity, on the autofocusing dynamics are explored. In particular, multiple autofocusing events occur if the nonlinearity is strong enough. Under the action of the azimuthal modulational instability, an axisymmetric beam may split into a set of separating bright spots. In the case of strong fractality (for alpha close to 1), the nonlinear beam self-traps, after the first instance of autofocusing, into a breathing vortical quasi-soliton. Radiation forces induced by the beam field are considered too, and a capture position for a probe nanoparticle is thus identified.