In this paper, we discuss, both analytically and numerically, the paraxial propagation of the radially polarized Laguerre–Gaussian-correlated Schell-model (LGCSM) beams orthogonal to the optical axis in uniaxial crystals. The analytical expression for the cross-spectral density function and the second-order moments of the radially polarized LGCSM beams are derived, and the evolution properties of the normalized intensity distribution, the spectral degree of the coherence (SDOC), and the spectral degree of the polarization (SDOP) in uniaxial crystals are elucidated by numerical examples. It is found that the intensity distribution of the radially polarized LGCSM beams evolves from a doughnut shape into a solid shape and finally converts into an elliptical symmetric hollow-ring profile in uniaxial crystals due to the combined effect of special correlation functions and the anisotropy effect of the uniaxial crystals. The evolution of the SDOC and SDOP for the radially polarized LGCSM beams is quite different from that of the radially polarized Gaussian–Schell-model beams. In addition, the propagation properties of the radially polarized LGCSM beams are closely related to the spatial coherence length, the mode order, and the ratio of extraordinary and ordinary reflective indices. The results show that the uniaxial crystals could modulate the evolution properties of the radially polarized LGCSM beams.

A class of random, wide-sense stationary optical beams with uniform correlations, named the partially coherent quasi-rectangular beam, is introduced theoretically. Based on the extended Huygens-Fresnel principle, the analytical expressions for the cross-spectral density (CSD), effective radius of curvature, and beam wander of the beam in the non-Kolmogorov turbulence are derived. It is found that the position of maximum intensity of the partially coherent quasi-rectangular beams shifts farther from the axis at intermediate distance, the shift in the turbulence is depressed compared to that in free space. As the effective radius of curvature decreases from infinity to a constant with the increase of the coherence length, it always takes a higher value than that in free-space propagation when the other parameters are fixed. In addition, the beam wander can be reduced by picking a relative small initial beam width, short coherence length, or long wavelength. These results are of importance for optical systems operating through long-range turbulent channels in which a beam must have a range-dependent tilt, e.g. on travelling around an obstacle on the axis.

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