The Wigner function (WDF) has been used to study the second-order moments for a radially polarized partially coherent twisted (RPPCT) beam (i.e., a RPPC beam with a twist phase) propagating in a uniaxial crystal. With the help of the extended Huygens–Fresnel integral formula and definition of the WDF, the analytical formulas for propagation factor (-factor), effective radius of curvature (ERC), and Rayleigh range of a RPPCT beam propagating in a uniaxial crystal have been derived. Our numerical results show that the -factor of a RPPCT beam with larger absolute value of the twist factor or lower coherence is less affected by anisotropic diffraction in a uniaxial crystal. The dependence of the ERC and Rayleigh range of a RPPCT beam propagating in a uniaxial crystal on the parameter along the direction is much different from that along the direction due to anisotropic diffraction. We can judge how much a RPPCT beam carries the twist factor by measuring the deviation percentage of the -factor of a RPPCT beam propagating in a uniaxial crystal and even can modulate the properties of a RPPCT beam by varying the beam parameters, which will be useful in some applications where a RPPCT beam is required.
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