Vectorial effect on the evolution of fractional-order vector vortex beams in a strongly nonlocal nonlinear medium

Propagation of a vector vortex optical field (VVOF) with both fractional order of polarization topological charge $m$m and fractional order of vortex topological charge $n$n with spatially variant states of polarization (SoP) in a strongly nonlocal nonlinear medium (SNNM) is studied. The optical field always evolves reciprocally with a cycle of stretch and shrink in a SNNM with dark stripes forming at $z=t\pi {z_p}$z=tπz_p ($t$t denotes an integer number, and ${{z}_p}$z_p is a parameter that depends on the initial power of the VVOF and the material constant associated with the response function), as a result from the coherent superposition of the vortices with different order of topological charges and weighting coefficients. In particular, the conversions between linear and circular polarization components occur during propagation, and the converted SoP distributions in different propagation distances depend closely on the topological charges and the initial powers. The evolutions of the Stokes parameters of the fractional-order VVOF (FO-VVOF) during propagation in a SNNM show that the spatial distributions of different polarization components are closely related to the topological charges, the initial powers and the propagation distances, implying that the FO-VVOF can be regarded as a superposition of two different fractional-order vortices with orthogonal circular polarization components. These results provide new strategies on tailoring polarization states in a structured optical field with fractional topological charges.