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Under the help of extended Huygens–Fresnel integral, we acquire the analytic expressions of second-order moments of the Wigner distribution function (WDF) for a twisted electromagnetic Gaussian Schell-model (TEGSM) beam propagating in a uniaxial crystal. Moreover, we obtain the formulae for propagation factor, the effective radius of curvature and Rayleigh range in which we studied the properties of the TEGSM beam numerically. It is found that the initial beam parameters and the uniaxial crystal parameters play a decisive role in the TEGSM beam propagating in a uniaxial crystal. The TEGSM beam with smaller absolute deviation of coherence width δxx and δyy is less affected by anisotropic diffraction in a uniaxial crystal, which is much different from the evolution properties of an EGSM beam in a uniaxial crystal. The Rayleigh range will decreases while the crystal parameter e increases. Thus, the propagation properties of a TEGSM beam could be controlled in a uniaxial crystal by changing the initial beam parameters and the uniaxial crystal parameters where it is necessary.

We derived analytical formula for the cross-spectral density matrix of a radially polarized partially coherent twist (RPPCT) beam in a uniaxial crystal orthogonal to the optical axis with the help of the extended Huygens-Fresnel integral formula and unified theory of coherence and polarization. Furthermore, we explore the evolution properties of a RPPCT beam in a uniaxial crystal. It is found that the RPPCT beam becomes an anisotropic RPPCT beam and the beam is rotating with beam propagation in uniaxial crystal. Its evolution properties change with twist factor and the ratio of extraordinary index to ordinary refractive index. The twist factor affects their rotation orientation angles, and the ratio of extraordinary index to ordinary refractive index has an effect on their twisted levels.

In this paper, analytical expressions for elements of the cross-spectral density matrix of Square Multi-Gaussian Schell-Model (SMGSM) beam, propagating in non-Kolmogorov turbulence, are derived based on the extended Huygens Fresnel integral. Polarization properties of the SMGSM beam are studied in detail based on the derived expressions. It is interesting to find that, apart from keeping the square flat shape of intensity distribution, polarization properties of the SMGSM beam, such as degree of polarization, the orientation angle and degree of ellipticity for the polarization ellipse also form square plateau distributions in the same areas as that of intensity distribution, and can preserve relatively long distance in atmospheric turbulence. This novel characteristic can be used for coherent light free space communications, polarization remote sensing and imaging in atmosphere.