# Nonparaxial Propagation Properties of Specially Correlated Radially Polarized Beams in Free Space
## 3 Statistical Properties of a Nonparaxial SCRP Beam
We calculate in Figures 9 and 10 the degrees of polarization of nonparaxial SCRP and PCRP beams  
Figure 8: Normalized intensity (cross line at \(y=x\)) \(\mathrm{I_{p}}/\mathrm{I_{Pmax}}\), \(\mathrm{I}/\mathrm{I_{max}}\), \((\mathrm{I_{x}}+\mathrm{I_{y}})/\mathrm{I_{max}}\), and \(\mathrm{I_{z}}/\mathrm{I_{max}}\) of a nonparaxial PCRP beam at \(z=10z_{\mathrm{R}}\) with \(\rho=\sqrt{x^{2}+y^{2}}\) and \(\sigma_{0}=\lambda\) for different values of the coherence widths. \(\mathrm{I_{p}}\) and \(\mathrm{I}\) denote the total intensity for the paraxial and nonparaxial SCRP beam, respectively. \(\mathrm{I_{x}}\) + \(\mathrm{I_{y}}\) denotes the transverse intensity, while \(\mathrm{I_{z}}\) denotes the longitudinal intensity for the nonparaxial SCRP beam. The coherence widths are marked in the figure. Figure 7: Normalized intensity distributions (cross line at \(y=x\)) \(\mathrm{I_{p}}/\mathrm{I_{Pmax}}\), \(\mathrm{I}/\mathrm{I_{max}}\), \((\mathrm{I_{x}}+\mathrm{I_{y}})/\mathrm{I_{max}}\), and \(\mathrm{I_{z}}/\mathrm{I_{max}}\) of a SCRP beam at \(z=10z_{\mathrm{R}}\) with \(\rho=\sqrt{x^{2}+y^{2}}\) and \(\sigma_{0}=\lambda\) for different values of the coherence widths. \(\mathrm{I_{p}}\) and \(\mathrm{I}\) denote the total intensity for the paraxial and nonparaxial SCRP beams, respectively. \(\mathrm{I_{x}}+\mathrm{I_{y}}\) denotes the transverse intensity, while \(\mathrm{I_{z}}\) denotes the longitudinal intensity for the nonparaxial SCRP beam. The coherence widths are marked in the figure. at \(z=10z_{\rm R}\) for different \(\delta_{0}\), respectively. The beam waist width in Figures 9a and 10a is set to be \(\sigma_{0}\)\(=10\lambda\), which indicates the paraxial propagation, while, in Figures 9b and 10b is selected as \(\sigma_{0}=\lambda\), which indicates the nonparaxial propagation. One finds that the degree polarization of both SCRP and PCRP beams, whether for paraxial or nonparaxial propagation, form an \"Inverse Gaussian\" shape during propagation, i.e., the SCRP and PCRP beams were depolarized, and the degree of polarization of increased as the transverse coordinate increased. The depolarization was attributed to the unneglectable \(z\) component and its limited spatial coherence length. What is interesting is that the evolution properties of the degree of polarization of the SCRP and PCRP beams are strikingly different, i.e., the degree of polarization of an SCRP beam increases as the initial coherence width decreases, while that of the PCRP beam decreases as the initial coherence width decreases. The difference arises from the different correlation structures (correlation matrix) of the SCRP and PCRP beams. A PCRP beam with a conventional Gaussian correlation structure will become less and less polarized with the increase of the propagation distance or the decrease of the initial coherence [74]. By contrast, a SCRP beam will become more and more polarized with the increase of the propagation distance or the decrease of the initial coherence. The physics behind this phenomenon has been discussed in detail in reference [21]. To learn about the SDOC of nonparaxial SCRP and PCRP beams on propagation, we comparatively calculate the modulus of the SDOC of nonparaxial SCRP and PCRP beams between two transverse points \((x,y)\) and \((-x,-y)\) at several propagation distances \(z\) with \(\delta_{0}\)\(=2\lambda\) for different values of \(\sigma_{0}\) in Figures 11 and 12, respectively. The beam waist widths in Figures 11a and 12a are both chosen as \(\sigma_{0}\)\(=10\lambda\) and indicate the paraxial propagation of the SCRP and PCRP beams, respectively. One sees clearly in Figure 11 that the evolution properties of SDOC of nonparaxial and paraxial SCRP beams are substantially different, i.e., with the increase of \(z\), the distributidegenerates from the initial non-Gaussian distribution with two sidelobes around the central bright spot into Gaussian distribution in the far field. However, the degeneration speed is higher with the decrease of \(\sigma_{0}\).