Vortex solitons in quasi-phase-matched photonic crystals with the third harmonic generation

We report stable composite vortex solitons in the model of a three-dimensional photonic crystal with the third-harmonic (TH) generation provided by the quasi-phase-matched quadratic nonlinearity. The photonic crystal is designed with a checkerboard structure in the $(x\text{,}y)$ plane, while the second-order nonlinear susceptibility, $d(z)$, is modulated along the propagation direction as a chain of rectangles with two different periods. This structure can be fabricated by means of available technologies. The composite vortex solitons are built of fundamental-frequency (FF), second-harmonic (SH), and TH components, exhibiting spatial patterns which correspond, respectively, to a vortex with topological charges $s=1$, a quadrupole emulating $s=2$, and an antivortex with $s=\ensuremath{-}1$ in these components (due to the system's symmetry, the latter value is actually tantamount to $s=3$). Soliton profiles feature rhombic or square patterns, corresponding to phase-matching conditions ${\ensuremath{\varphi}}_{d}(x,y)=0$ or $\ensuremath{\pi}$, respectively, the rhombic solitons possessing a broader stability region. From the perspective of the experimental feasibility, we show that both the rhombic and square-shaped composite vortex solitons may stably propagate in the photonic crystals over distances of up to 1 m. The TH component of the soliton with $s=\ensuremath{\mp}1$ is produced by cascaded nonlinear interactions, starting from the FF vortex component with $s=\ifmmode\pm\else\textpm\fi{}1$ and proceeding through the quadrupole SH one. These findings offer a previously unexplored approach to the creation of stable vortex solitons in nonlinear optics.