# Coupling Efficiency of a Partially Coherent Radially Polarized Vortex Beam into a Single-Mode Fiber
## 2 Coupling Efficiency of Partially Coherent Radially Polarized Vortex Beams  
The schematic diagram of our coupling arrangement is shown in Figure 1. Suppose that a PCRP vortex beam is coupled into a single-mode fiber by a lens of focal length \(f\). Let the coordinate in the plane A be denoted by \(\rho\). Based on the theory of coherence and polarization, the second-order statistical properties of a PCRP beam can be characterized by the \(2\times 2\) CSD matrix \(\overset{\leftrightarrow}{W}(\rho_{1},\theta_{1},\rho_{2},\theta_{2})\) in the plane A, which are defined as ([PERSON] _et al._, 2012; [PERSON] _et al._, 2014):  
\[ \begin{split} W_{0 xx}(\rho_{1},\theta_{1},\rho_{2},\theta_{2})& =\frac{\rho_{1}\rho_{2}\cos\theta_{1}\cos\theta_{2}}{4w_{0}^{2}} \exp\left(-\frac{\rho_{1}^{2}+\rho_{2}^{2}}{4w_{0}^{2}}\right)\exp\left[- \frac{\rho_{1}^{2}+\rho_{2}^{2}-2\rho_{1}\rho_{2}\cos(\theta_{1}-\theta_{2})} {2\delta_{0}^{2}}\right]\\ W_{0 xy}(\rho_{1},\theta_{1},\rho_{2},\theta_{2})&= \frac{\rho_{1}\rho_{2}\cos\theta_{1}\sin\theta_{2}}{4w_{0}^{2}}\exp\left(- \frac{\rho_{1}^{2}+\rho_{2}^{2}}{4w_{0}^{2}}\right)\exp\left[-\frac{\rho_{1}^{ 2}+\rho_{2}^{2}-2\rho_{1}\rho_{2}\cos(\theta_{1}-\theta_{2})}{2\delta_{0}^{2} }\right]\\ W_{0 yx}(\rho_{1},\theta_{1},\rho_{2},\theta_{2})&= W_{0 xy}^{*}(\rho_{1},\theta_{1},\rho_{2},\theta_{2})\\ W_{0 yy}(\rho_{1},\theta_{1},\rho_{2},\theta_{2})& =\frac{\rho_{1}\rho_{2}\sin\theta_{1}\sin\theta_{2}}{4w_{0}^{2}}\exp\left(- \frac{\rho_{1}^{2}+\rho_{2}^{2}}{4w_{0}^{2}}\right)\exp\left[-\frac{\rho_{1}^{ 2}+\rho_{2}^{2}-2\rho_{1}\rho_{2}\cos(\theta_{1}-\theta_{2})}{2\delta_{0}^{2} }\right]\end{split}  \tag{1}\]  
where \(\omega_{0}\) is the beam width in the plane A and \(\delta_{0}\) is the spatial coherence length. PCRP beams can be embedded a vortex phase, and the new CSD of PCRP vortex beams can be rewritten as follows ([PERSON] _et al._, 2014)  
\[W_{\alpha\beta}(\rho_{1},\theta_{1},\rho_{2},\theta_{2})=W_{0\alpha\beta}( \rho_{1},\theta_{1},\rho_{2},\theta_{2})\exp(-im\theta_{1}+im\theta_{2}) \tag{2}\]  
where \(m\) is the topological charge. According to the coupling theory introduced in Ref. ([PERSON] _et al._, 2014), the field distribution of the fiber mode can be approximated by a Gaussian function ([PERSON] _et al._, 2014; [PERSON] _et al._, 2014):  
\[F_{j}=\sqrt{\frac{2}{\pi}}\omega_{j}^{-1}\exp\left(-\frac{\rho^{2}}{\omega_{j }^{2}}\right)\qquad(j=x,y) \tag{3}\]  
where \(\rho^{\prime}\) is a transverse position vector in the fiber plane B and \(\omega_{j}\) is a measure of the mode width. Furthermore, the field distribution of the mode at the aperture plane A is found by using the backpropagation technique from B to A and is given by:\[F_{jA}=\sqrt{\frac{2}{\pi}}\omega_{jA}^{-1}\exp\left(-\frac{\rho\prime^{2}}{\omega_{ jA}^{2}}\right) \tag{4}\]  
where \(\omega_{jA}=\lambda f/\pi\omega_{j}\) is the effective mode width at the aperture plane, \(\lambda\) is the wavelength of the incident, and \(f\) is the focal length of the coupling lens. We can get the coupling efficiency as:  
\[\eta_{c}=\frac{\langle P_{c}\rangle}{\langle P_{a}\rangle} \tag{5}\]  
where \(\langle P_{c}\rangle\) is the power coupled into the fiber and \(\langle P_{a}\rangle\) is the power in the aperture plane. They can be expressed as:  
\[ \begin{array}{l}P_{c}=\iint\limits_{D}W_{ij}(\rho_{1},\rho_{2},\omega)F_{ iA}^{*}(\rho_{1})F_{jA}(\rho_{2})\times\exp\left(-\frac{\rho_{1}^{2}+\rho_{2}^{2}}{W^ {2}}\right)\mathrm{d}^{2}\rho_{1}\mathrm{d}^{2}\rho_{2}\\ P_{a}=\iint\limits_{D}W_{jj}(\rho,\rho,\omega)\times\exp\left(-\frac{2\rho^{2 }}{W^{2}}\right)d^{2}\rho\end{array}  \tag{6}\]  
where \(D\) denotes a hard aperture of diameter, \(W^{2}=D^{2}/8\), and \(\mathrm{NA}=D/2f\)[32]. For PCRP vortex beams, we obtain the following expression for the power coupled into the single fiber with \(P_{c\_even}\) (\(m\) is even number) and \(P_{c\_odd}\) (\(m\) is odd number):  
\[\begin{array}{lcl}P_{c\_even}&=\frac{\pi}{2\omega_{0}^{2}w_{xA}^{2}}\left[