Square-shaping of sturdy optical vortex droplets in quasi-phase-matched photonic crystals
We elaborate a scheme for controllable shaping of self-trapped vortex states in a quasi-phase-matched three-dimensional photonic crystal with the combination of self-focusing quadratic and defocusing cubic material nonlinearities. The setting gives rise to sturdy droplet-like vortex modes, capable to adapt to externally imposed strong geometric constraints. The application of a square-shaped modulation in the transverse $\left(x,y\right) $ plane and periodic quasi-phase-matching to the quadratic nonlinear coefficient $d_{z}$ along the propagation direction $z$ leads to the formation of square vortex droplets (VDs) with fourfold rotational symmetry ($C_{4}$). These states preserve the vortical phase circulation and exhibit robust propagation in a broad parameter region. In the oversaturated regime dominated by the cubic self-defocusing, the square-shaped VDs obey the anti-Vakhitov--Kolokolov stability criterion. The results, which are produced, chiefly, for the VDs with topological charge $S=1$, and also, in a partial form, for $S=2$ and $4$, reveal an unexpected universality: the vortex robustness is not contingent upon the circular symmetry. Thus, the combination of the competing nonlinearities and geometric confinement provides not only an effective method for the formation of self-trapped vortex states, but also new insight into generality of the topological protection in nonlinear optical fields.