Vortex Solitons on Partially $\mathcal{PT}$-symmetric Azimuthal Lattices in a Medium with Quadratic Nonlinear Response

Abstract

Vortex solitons in parity-time ($\mathcal{PT}$) symmetric and partially $\mathcal{PT}$ (p$\mathcal{PT}$) symmetric azimuthal lattices are demonstrated for a media with quadratic nonlinear response. Stability properties of the vortices are investigated comprehensively by linear spectra and nonlinear evolution of the governing equations, and it is shown that, although the existence domain of the $\mathcal{PT}$-symmetric and p$\mathcal{PT}$-symmetric lattices are identical, the stability region of $\mathcal{PT}$-symmetric lattice is narrower than that of the p$\mathcal{PT}$-symmetric lattice. It is also observed that deeper real part in the azimuthal potentials supports stability of vortex solitons, whereas deeper imaginary part and strong quadratic electro-optic effects impoverish stability properties of the vortices. Moreover, it is shown that there are different stability properties of vortices in p$\mathcal{PT}$-symmetric azimuthal potentials for different vorticity values, while there is no such difference for vortices in $\mathcal{PT}$-symmetric potentials.