Propagation properties of the partially coherent radially polarized beam with the cross phase

With the help of the generalized Huygens–Fresnel integral, an analytical expression of a partially coherent radially polarized beam with the cross phase (PCRPCP) in the paraxial propagation is derived. The statistical properties of the PCRPCP are studied through the numerical examples. It is shown that the PCRPCP rotates during propagation. However, the rotation directions of its two orthogonal components are opposite. The influence of the cross strength factor and the source coherence length on beam’s intensity, polarization, and coherence are investigated, respectively. The results show that the PCRPCP will first focus and then diverge until it reaches the far-field. In this process, the proportion of the fully polarized part of the PCRPCP decreases at first and increases then. The existence and increase of the cross strength factor lead the beam to depolarize weaker. Moreover, as long as the cross strength factor is large enough, the PCRPCP could have a higher coherence in the far-field than that in the source plane. Our results will be helpful to the fields of optical tweezers, microscopy, optical communication, and so on.