# Nonparaxial Propagation Properties of Specially Correlated Radially Polarized Beams in Free Space
## 3 Statistical Properties of a Nonparaxial SCRP Beam
Fourth column: the longitudinal intensity \({\rm I}_{z}\) for the nonparaxial SCRP beam. Fifth column: the corresponding cross lines \((y=x)\) with \(\rho=\sqrt{x^{2}+y^{2}}\). Here, the beam width \(\sigma_{0}=10\lambda\), and the intensities are normalized with respect to their maximums. Figure 3: Intensity distributions of the SCRP beam and the nonparaxial SCRP 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. Fourth column: the longitudinal intensity \({\rm I}_{z}\) for the nonparaxial SCRP beam. Fifth column: the corresponding cross lines \((y=x)\) with \(\rho=\sqrt{x^{2}+y^{2}}\). Here, the beam width \(\sigma_{0}=0.1\lambda\), and the intensities are normalized with respect to their maximums. Now, we can make some conclusions for the evolution of the intensity from Figures 1-8. The nonparaxiality of the SCRP and PCRP beams is closely determined by both their initial beam waist sizes and their coherence widths. When both the beam waist sizes and the coherence widths are larger than the wavelength, the difference between the results calculated by the nonparaxial propagation  
Figure 5: Intensity distributions of the 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. Fourth column: the longitudinal intensity \({\rm I}_{z}\) 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 maximums. Figure 6: Intensity distributions of the 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. Fourth column: the longitudinal intensity \({\rm I}_{z}\) for the nonparaxial SCRP beam. Fifth column: the corresponding cross lines \((y=x)\) with \(\rho=\sqrt{x^{2}+y^{2}}\). Here, the beam width \(\sigma_{0}=0.1\lambda\), and the intensities are normalized with respect to their maximums. formulas and those calculated by the paraxial propagation formulas can be negligible. If either the beam waist size or the coherence width is smaller than the wavelength, a significant difference appears, and nonparaxial propagation formulas are necessary for treating the propagation of SCRP and PCRP beams. Moreover, the modulation of the beam waist widths and the coherence widths on the profiles of the SCRP and PCRP beams in the far field is different. Next, we turn to study the polarization properties of the nonparaxial SCRP beam in free space.