# Nonparaxial Propagation Properties of Specially Correlated Radially Polarized Beams in Free Space
## 3 Statistical Properties of a Nonparaxial SCRP Beam
One finds that the nonparaxiality of the SCRP and PCRP beams is also closely determined by their spatial coherence widths \(\delta_{0}\). When \(\delta_{0}\geq\)\(\lambda\), the longitudinal component intensity \(\mathrm{I}_{x}/\mathrm{I}_{\mathrm{max}}\) is negligible compared with \(\mathrm{I}/\mathrm{I}_{\mathrm{max}}\), or \((\mathrm{I}_{x}+\mathrm{I}_{y})/\mathrm{I}_{\mathrm{max}}\), thus the paraxial approximation is allowable. However, with a decrease of \(\delta_{0}\), the ratio of \(\mathrm{I}_{x}/\mathrm{I}_{\mathrm{max}}\) increases quickly, for a very small value of \(\sigma_{0}=0.1\lambda\) or \(0.5\lambda\), the longitudinal component intensity \(\mathrm{I}_{z}/\mathrm{I}_{\mathrm{max}}\) becomes extremely noticeable, and a notable discrepancy between the nonparaxial and paraxial results appears. A possible explanation is that a very small coherence width leads to a larger divergence angle, thus the beam becomes nonparaxial. In this case, the vectorial nonparaxiality of the SCRP and PCRP beams has to be taken into consideration. One can also find from Figures 7 and 8 that the far-field intensity profiles of the SCRP and PCRP beams are also closely determined by their coherence length \(\delta_{0}\). With the decrease of \(\delta_{0}\), the beam profile of the SCRP field changes from Gaussian shape, to central-dark, flat-topped, and finally hollow beam spot, while the beam profile of the PCRP beam changes from hollow beam spot, to half-dark hollow beam spot, flat-topped, and finally Gaussian beam spot in the far field. Thus, one can shape the intensity distribution of nonparaxial SCRP and PCRP fields by modulating their initial spatial coherence, which is useful in material thermal processing and particle trapping. Figure 1: Intensity distributions of the paraxial specially correlated radially polarized (SCRP) beam and the nonparaxial SCRP beam at the propagation distances \(z=0\) (first row), \(z=0.1z_{\text{R}}\) (second row), \(z=0.5z_{\text{R}}\) (third row), \(z=10z_{\text{R}}\) (fourth row) in free space, where \(z_{\text{R}}\) is the Rayleigh distance. First column: the total intensity \(\text{I}_{p}\) for the paraxial SCRP beam. Second column: the total intensity \(\text{I}_{p}\) for the paraxial SCRP beam. Third column: the transverse intensity \(\text{I}_{x}+\text{I}_{y}\) for the nonparaxial SCRP beam. Fifth column: the corresponding cross lines (\(y=x\)) with \(\rho=\sqrt{x^{2}+y^{2}}\). Here, the beam width \(\sigma_{0}=10\lambda\), and the intensities are normalized with respect to their maxima. Figure 2: Intensity distributions of the SCRP beam and the nonparaxial SCRP beam at the propagation distances \(z=0\)(first row), \(z=0.1z_{\text{R}}\) (second row), \(z=0.5z_{\text{R}}\) (third row), \(z=10z_{\text{R}}\) (fourth row) in free space, where \(z_{\text{R}}\) is the Rayleigh distance. First column: the total intensity \(\text{I}_{p}\) for the paraxial SCRP beam. Second column: the total intensity \(\text{I}_{x}+\text{I}_{y}\) for the nonparaxial SCRP beam. Third column: the transverse intensity \(\text{I}_{x}+\text{I}_{y}\) for the nonparaxial SCRP beam. Fourth column: the longitudinal intensity \(\text{I}_{z}\) for the nonparaxial SCRP beam. Fifth column: the corresponding cross lines (\(y=x\)) with \(\rho=\sqrt{x^{2}+y^{2}}\). Here, the beam width \(\sigma_{0}=\lambda\), and the intensities are normalized with respect to their maxima. Figure 4: Intensity distributions of the paraxial partially coherent radially polarized (PCRP) beam and the nonparaxial PCRP beam at the propagation distances \(z=0\)(first row), \(z=0.1z_{\rm R}\) (second row), \(z=0.5z_{\rm R}\) (third row), \(z=10z_{\rm R}\) (fourth row) in free space, where \(z_{\rm R}\) is the Rayleigh distance. First column: the total intensity \({\rm I}_{p}\) for the paraxial SCRP beam. Second column: the total intensity \({\rm I}\) for the nonparaxial SCRP beam. Third column: the transverse intensity \({\rm I}_{x}+{\rm I}_{y}\) for the nonparaxial SCRP beam.