# Nonparaxial Propagation Properties of Specially Correlated Radially Polarized Beams in Free Space
## 2 Nonparaxial Propagation Theory of an SCRP Beam  
On the basis of the unified theory of coherence and polarization, the second-order correlation of a vector partially coherent beam can be described by a \(3\times 3\) electric cross-spectral density (CSD) matrix \(\overset{\leftrightarrow}{W}\). In Cartesian coordinate system, the elements of the \(3\times 3\) CSD matrix in the source plane \(\mathrm{z}=0\) are given by [66]:  
\[ \begin{array}{l}\overset{\leftrightarrow}{W}(x_{10},y_{10},x_{20},y_{20},0) \\ =\left(\begin{array}{cc}W_{xx}(x_{10},y_{10},x_{20},y_{20},0)&W_{xy}(x_{10},y_{ 10},x_{20},y_{20},0)&0\\ W_{yx}(x_{10},y_{10},x_{20},y_{20},0)&W_{yy}(x_{10},y_{10},x_{20},y_{20},0)&0 \\ 0&0&0\end{array}\right),\end{array}  \tag{1}\]  
where the matrix element \(W_{\alpha\beta}(x_{10},y_{10},x_{20},y_{20},0)=\left\langle E_{\alpha}^{*}(x_ {10},y_{10},0)E_{\beta}(x_{20},y_{20},0)\right\rangle\) denotes the coherence properties of the random electric field components \(E_{\alpha}\) and \(E_{\beta}\) along the \(x\) and \(y\) directions, respectively. The asterisk denotes the complex conjugate, and the angular brackets denote ensemble average. For an SCRP beam, the elements of the CSD matrix in the source plane \(\mathrm{z}=0\) take the form of: [21; 37]  
\[ \begin{array}{ll}W_{aa}(x_{10},y_{10},x_{20},y_{20},0)&=\exp\left(-\frac{x_{10 }^{2}+y_{10}^{2}+x_{20}^{2}+y_{20}^{2}}{4\sigma_{0}^{2}}\right)\left[1-\frac{(x _{20}-a_{10})^{2}}{\delta_{0}^{2}}\right]\\ &\times\exp\left[-\frac{(x_{10}-x_{20})^{2}+(y_{10}-y_{20})^{2}}{2\delta_{0}^{2 }}\right],\qquad\quad(\alpha,\beta=x,y)\end{array}  \tag{2}\]  
\[ \begin{array}{ll}W_{xy}(x_{10},y_{10},x_{20},y_{20},0)&=-\exp\left(-\frac{x_{1 0}^{2}+y_{10}^{2}+x_{20}^{2}+y_{20}^{2}}{4\sigma_{0}^{2}}\right)\frac{(x_{20} -x_{10})(y_{20}-y_{10})}{\delta_{0}^{2}}\\ &\times\exp\left[-\frac{(x_{10}-x_{20})^{2}+(y_{10}-y_{20})^{2}}{2\delta_{0}^{2 }}\right],\end{array}  \tag{3}\]  
\[W_{yx}(x_{10},y_{10},x_{20},y_{20},0)=W_{xy}^{*}(x_{20},y_{20},x_{10},y_{10},0), \tag{4}\]  
where \(\sigma_{0}\) is the beam waist width, and \(\delta_{0}\) is the correlation width. On the basis of the vectorial Rayleigh diffraction integral, the nonparaxial propagation of a fully coherent vector beam in the half space \(\mathrm{z}>0\) can be related to its electric field distribution in the plane \(\mathrm{z}=0\)[53]:  
\[E_{\alpha}(x,y,z)=-\frac{1}{2\pi}\int\limits_{-\infty}^{\infty}\int\limits_{- \infty}^{\infty}E_{\alpha}(x_{0},y_{0},0)\frac{\partial}{\partial z}\left[ \frac{\exp(ikR)}{R}\right]dx_{0}dy_{0},\qquad(\alpha=x,y) \tag{5}\]  
\[E_{z}(x,y,z)=\frac{1}{2\pi}\int\limits_{-\infty}^{\infty}\int\limits_{-\infty }^{\infty}\biggl{\{}E_{x}(x_{0},y_{0},0)\frac{\partial}{\partial x}\left[ \frac{\exp(ikR)}{R}\right]+E_{y}(x_{0},y_{0},0)\frac{\partial}{\partial y} \left[\frac{\exp(ikR)}{R}\right]\biggr{\}}dx_{0}dy_{0} \tag{6}\]  
where \(E_{\alpha}\) (\(x_{0}\), \(y_{0}\), \(0\)) and \(E_{\alpha,z}(x,y,z)\) are components of the electric field vector in the plane \(z=0\) and \(z>0\), \(R=\sqrt{\left(x-x_{0}\right)^{2}+\left(y-y_{0}\right)^{2}+z^{2}}\), respectively, and \(k=2\pi/\lambda\) is the wave number related to the wavelength \(\lambda\). When \(R\gg\lambda\), in Equations (5), (6), the partial derivatives of the function \(\exp(ikR)/R\) on the variables \(\alpha\) and \(z\) are usually approximated as [66]:  
\[\frac{\partial}{\partial\alpha}\left[\frac{\exp(ikR)}{R}\right]=\frac{ik\exp( ikR)}{R^{2}}(\alpha-\alpha_{0}), \tag{7}\]  
\[\frac{\partial}{\partial z}\left[\frac{\exp(ikR)}{R}\right]=\frac{ikz\exp( ikR)}{R^{2}}. \tag{8}\]  
Now, we extend the nonparaxial propagation theory of coherent vector beams to the partially coherent vector case. The \(3\times 3\) CSD matrix of a partially coherent vector beam in the half space \(\mathrm{z}>0\) in a Cartesian coordinate system are given by [66]:  
\[