# Nonparaxial Propagation Properties of Specially Correlated Radially Polarized Beams in Free Space
## 3 Statistical Properties of a Nonparaxial SCRP Beam  
In this section, we will numerically study the statistical properties, such as the intensity, the degree of polarization, and the SDOC of a nonparaxial SCRP by applying the formulae derived in the above section. The statistical properties of a nonparaxial PCRP beam in free space are studied comparatively as well. In the following numerical examples, the propagation distances are normalized to \(z/z_{R}\), where \(z_{R}=\pi\sigma_{0}^{2}/\lambda\) is the Rayleigh distance. Figures 1-3 show the normalized intensity distributions \(\mathrm{I}/\mathrm{I}_{\mathrm{max}}\), \((\mathrm{I}_{x}+\mathrm{I}_{y})/\mathrm{I}_{\mathrm{max}}\), and \(\mathrm{I}_{x}/\mathrm{I}_{\mathrm{max}}\) and their corresponding cross lines \((y=x)\) of a nonparaxial SCRP beam at different propagation distances for \(\sigma_{0}=10\lambda\), \(\lambda\), and \(0.1\lambda\), respectively. Figures 4-6 show the contour graphs of \(\mathrm{I}_{p}/\mathrm{I}_{\mathrm{max}}\), \(\mathrm{I}/\mathrm{I}_{\mathrm{max}}\), \((\mathrm{I}_{x}+\mathrm{I}_{y})/\mathrm{I}_{\mathrm{max}}\), and \(\mathrm{I}_{x}/\mathrm{I}_{\mathrm{max}}\) and the corresponding cross lines \((y=x)\) of a nonparaxial PCRP beam under the same parameter conditions as those in Figures 1-3. The coherence widths of both SCRP and PCRP beams are set to \(\delta_{0}=\lambda\). For the convenience of comparison, the normalized paraxial intensity distribution \(\mathrm{I}_{p}/\mathrm{I}_{\mathrm{max}}\) of SCRP and PCRP beams and their corresponding cross lines \((y=x)\) are also plotted in Figures 1-3 and Figures 4-6, respectively. Several interesting properties can be observed in these plots. First, from the comparison of Figures 1-3, we find that the intensity in the far field (fourth row of each figure) of the nonparaxial SCRP beam is closely determined by the initial beam waist width \(\sigma_{0}\). When \(\sigma_{0}\) is large, the far-field intensity is dark hollow (see Figure 1), while when \(\sigma_{0}\) is small, the far-field intensity is a quasi-Gaussian distribution (see Figure 3). Very similar conclusions can be drawn for the nonparaxial PCRP beam (see Figures 4-6). Second, the evolution properties of the intensity in the free space for the nonparaxial SCRP beam and the nonparaxial PCRP beam are totally different. From Figure 1, we find that for a nonparaxial SCRP beam, the total intensity changes gradually from a Gaussian shape to a dark hollow profile with the increase of the propagation distance. By contrast, from Figure 4, we find that for a nonparaxial PCRP beam, the total intensity changes gradually from a dark hollow shape to a Gaussian profile with the increase of the propagation distance. Furthermore, we find from Figure 1; Figure 4 that, in the case of \(\sigma_{0}=10\lambda\), there is a tiny discrepancy between nonparaxial and paraxial intensity distributions in the near and far fields due to the fact that \(\mathrm{I}_{z}/\mathrm{I}_{\mathrm{max}}\) is extremely small compared to \(\mathrm{I}/\mathrm{I}_{\mathrm{max}}\), or \((\mathrm{I}_{x}+\mathrm{I}_{y})/\mathrm{I}_{\mathrm{max}}\) and can be negligible. In this case, the paraxial approximation is valid. However, for a very small value of \(\sigma_{0}\)\(=0.1\lambda\), as displayed in Figures 3 and 6, the ratio of \(\mathrm{I}_{z}/\mathrm{I}_{\mathrm{max}}\) becomes extremely noticeable and results in an appreciable discrepancy between nonparaxial and paraxial results in both near and far fields. Therefore, nonparaxial propagation formulae should be adopted to describe the propagation of SCRP and PCRP beams with a beam waist width that is much smaller than \(\lambda\). Next, we will investigate the effect of the spatial coherence on the propagation properties of both SCRP and PCRP beams. We plot in Figures 7 and 8 the normalized intensity distribution of SCRP and PCRP beams at \(z=10z_{R}\), with \(\sigma_{0}=\lambda\) for different values of spatial coherence width \(\delta_{0}\). The corresponding paraxial results are also plotted together for comparison.