# Nonparaxial Propagation Properties of Specially Correlated Radially Polarized Beams in Free Space
## 2 Nonparaxial Propagation Theory of an SCRP Beam
\begin{array}{ll}\Delta=\frac{k^{2}}{4 MNr_{1}^{2}r_{2}^{2}}\exp[-ik(r_{1}-r _{2})]exp\bigg{[}\frac{k^{2}\left(b_{v}{}^{2}+b_{v}^{2}\right)}{4M}-\frac{k^{2 }}{4 Nr_{2}}\left(x_{2}^{2}+y_{2}{}^{2}\right)\bigg{]},\\ N=\frac{1}{4\delta_{0}^{2}}+\frac{1}{2\delta_{0}^{2}}-\frac{ik}{2r_{2}},\ M=\frac{1}{4\delta_{0}^{2}}+\frac{ik}{2\delta_{0}^{2}}+ \frac{ik}{2r_{1}}-\frac{1}{4N\delta_{0}^{4}},\\ O=\frac{1}{2\delta_{0}^{2}}-\frac{1}{\delta_{0}^{2}}-\frac{1}{4\delta_{0}^{2}},\ C=\frac{1}{2N\delta_{0}^{2}}-1,\ b_{a}=\frac{ika_{1}}{r_{1}}-\frac{ika_{2}}{2 Nr_{2}\delta_{0}^{2}}\\ Q_{a1}=\frac{ika_{0}}{2M},\ Q_{a2}=\frac{1}{2\delta_{0}^{2}}+\frac{ika_{1}^{2}}{4M^ {2}},\ Q_{a3}=\frac{3 kb_{h}}{4M^{2}}+\frac{k^{2}b_{h}}{8M^{3}},\ Q_{a4}=\frac{3 }{4M^{2}}+\frac{3k^{2}b_{h}}{4M^{3}}+\frac{k^{4}b_{h}}{16M^{4}}\\ A_{a0}=\left(-C+\frac{ik}{2 Nr_{2}}-\frac{3 ik}{4N^{2}\delta_{0}^{2}r_{2}}\right) \pm\left(2+\frac{1}{ikr_{2}}\right)\frac{k^{2}a_{3}^{2}}{4N^{2}\delta_{0}^{2}r _{2}^{2}},\\ A_{a1}=\frac{3}{4N^{2}\delta_{0}^{2}}-\frac{3}{2N\delta_{0}^{2}}+\left(\frac{ i}{2N\delta_{0}^{2}}-i+\frac{k}{2 Nr_{2}}-\frac{3k}{8N^{2}r_{2}\delta_{0}^{2}} \right)\frac{kx_{2}}{Nr_{2}\delta_{0}^{2}},\\ A_{a2}=-\left(1-\frac{1}{N\delta_{0}^{2}}+\frac{1}{4N^{2}\delta_{0}^{2}} \right)\frac{a_{2}}{\delta_{0}^{2}}-\left(\frac{1}{2}-\frac{1}{N\delta_{0}^{2} }+\frac{3}{8N^{2}\delta_{0}^{2}}\right)\frac{ika_{2}}{Nr_{2}\delta_{0}^{2}},\\ A_{a3}=\frac{1}{2N\delta_{0}^{4}}\left(1-\frac{1}{N\delta_{0}^{2}}+\frac{1}{4N ^{2}\delta_{0}^{4}}\right),\\ B_{a0}=\frac{1}{2r_{2}}+\left(i-\frac{k}{2n_{2}r_{2}}\right)\frac{kx_{2}}{2n _{2}r_{2}},\ B_{a1}=\left[-C+\left(1-\frac{1}{N\delta_{0}^{2}}\right)\frac{ ik}{2 Nr_{2}}\right]\alpha_{2},\ B_{a2}=\frac{C}{2N\delta_{0}^{2}},\\ C_{a1}=-\frac{ika_{1}a_{2}}{2 Nr_{2}},\ \ C_{a2}=\alpha_{1}C+\frac{ika_{2}}{2 Nr_{2}},\\ D_{a0}=\left(\frac{k}{2 Nr_{2}}-i\right)\frac{kx_{2}}{2 Nr_{2}}-\frac{1}{2N},D_{ a1}=\left[C-\left(1-\frac{1}{N\delta_{0}^{2}}\right)\frac{ik}{2 Nr_{2}}\right] \alpha_{2},\ D_{a2}=-\frac{C}{2N\delta_{0}^{2}}.\end{array}  \tag{24}\]  
Under the paraxial approximation, i.e., \(r_{j}\approx z+(x_{j}^{2}+y_{j}^{2})/2z,\ (j=1,2)\) the longitudinal component of the optical field can be usually neglected. Equations (16)-(18) can be reduced to the propagation formulae for the elements of the \(2\times 2\) CSD matrix of a paraxial SCRP beam in free space, which are consistent with the Equations (19)-(25) in [21]. The intensity of a nonparaxial partially coherent vector beam in the half space z \(>0\) is given by [70]:  
\[ \begin{array}{ll}I(x,y,z)&=I_{x}(x,y,z)+I_{y}(x,y,z)+I_{z}(x,y,z)\\ &=W_{xx}(x,y,x,y,z)+W_{yy}(x,y,x,y,z)+W_{zz}(x,y,x,y,z)\end{array}  \tag{25}\]  
The degree of polarization of a nonparaxial partially coherent vector beam in the half space \(z>0\) can be defined by the formula [71]:  
\[P(x,y,z)=\frac{p_{1}(x,y,z)-p_{2}(x,y,z)}{p_{1}(x,y,z)+p_{2}(x,y,z)+p_{3}(x,y,z)}, \tag{26}\]  
where \(p_{1}(x,y,z)\), \(p_{2}(x,y,z)\), \(p_{3}(x,y,z)\) are the three eigenvalues of the CSD matrix of a nonparaxial partially coherent vector field and satisfy the relation \(p_{1}(x,y,z)\geq p_{2}(x,y,z)\geq p_{3}(x,y,z)\). Another definition of the three-dimensional degree of polarization is claimed in reference [72]. Both definitions of the degree of polarization can be applied to study the polarization properties of a nonparaxial partially coherent vector beam. The SDOC of a nonparaxial partially coherent vector beam between two arbitrary points \((x_{1},y_{1},z)\) and \((x_{2},y_{2},z)\) in the half space z \(>0\) is defined by the formula [73]:  
\[\mu(x_{1},y_{1},x_{2},y_{2},z)=\frac{Tr\overset{\leftrightarrow}{W}(x_{1},y_{1},x _{2},y_{2},z)}{\sqrt{Tr\overset{\leftrightarrow}{W}(x_{1},y_{1},x_{1},y_{1},z) \sqrt{Tr\overset{\leftrightarrow}{W}(x_{2},y_{2},x_{2},y_{2},z)}}}, \tag{27}\]  
where \(Tr\) stands for the trace of the CSD matrix.