# Coupling Efficiency of a Partially Coherent Radially Polarized Vortex Beam into a Single-Mode Fiber
## 3 Results and Discussion  
In this section, with the help of the formulas obtained in Section 2, we will study the coupling efficiency of PCRP vortex beams for different topological charges, coherence length, and wavelength. First, we carefully chose the values of \(\omega_{j}(j=x,y)\) in Equation (3) to match with the core radius of the fiber because the theoretical model is only valid for the single-mode fiber. Let us recall the definition of the normalized frequency of the fiber [33]  
\[V=\frac{2\pi a}{\lambda}\sqrt{n_{1}^{2}-n_{2}^{2}} \tag{10}\]  
where \(\alpha\) is the radius of the fiber core, \(\lambda\) is the wavelength of the incident light. \(n_{1}\) and \(n_{2}\) are the peak values of the refractive index of the fiber core and the refractive index of the fiber cladding, respectively. It is known that when the value of \(V\) is smaller than 2.405, the fiber belongs to the single-mode fiber. Assuming that the fiber is weakly guiding with the refractive index profile, the last term in Equation (9) can be written as \(\sqrt{n_{1}^{2}-n_{2}^{2}}\approx n_{1}\sqrt{2\Delta}\) with \(\Delta=(n_{1}-n_{2})/n_{1}\) as a good approximation. To ensure the single-mode fiber, the parameter is chosen to be \(\alpha=5\upmu\)m, \(n_{1}=1.44\), \(\Delta=0.0034\), and \(\lambda=1550\,\)nm. On substituting these parameters into Equation (9), we obtained a value of \(V\) of 2.4, smaller than 2.405. Under this condition, we found the radius of the mode field of the fundamental propagating mode in the fiber by using the empirical formula to be \(\omega_{j}=5.15\upmu\,(j=x,y)\). Therefore, in the following numerical examples, the value of \(\omega_{j}\) was kept fixed at 5.15 \(\upmu\)m throughout the paper. If the focal length of the lens in plane A is 10 cm, the radius \(\omega_{jA}\,(j=x,y)\) of the mode field was calculated to be 9.6 mm. Figure 2 shows the intensity distribution of PCRP vortex beams in plane B with different spatial coherence width and topological charge. The propagation properties of PCRP vortex beams through a paraxial ABCD optical system have been studied in Ref. [28]. Figure 2 shows that the focusing properties of a PCRP vortex beam are closely affected by its initial spatial coherence and topological charge. The beam spot got larger with increasing topological charge from \(m=1\) to 3, but not \(m=0\). This is because the PCRP vortex beam became the PCRP beam for \(m=0\), which has a ring-shaped beam profile. The PCRP vortex beam had a dark hollow beam spot (see Figure 2h) with large coherence. When the spatial coherence was decreased, a flat-topped beam spot was formed (see Figure 2d). With the further decrease of the spatial coherence, a focused Gaussian beam spot was formed. Figure 2: Distribution of the intensity of PCRP vortex beams in plane B for different coherence length and topological charge. Figure 3 shows that the coupling efficiency varied with NA for different values of \(\delta_{0}\) under different topological charges. In Figure 3, one can find that the coupling efficiency increased at first and then decreased with increasing NA, and there was a peak value of the coupling efficiency, which means there was a best match between the incident beam and the distribution of the mode field. We also confirm that the position of the peak (i.e., the value of NA) was different for different coherence length and topological charge. We could adjust the spatial coherence to improve the coupling efficiency, and an optimum value was found. Obviously, the result of Figure 3b is different to other results. We found that when PCRP vortex beams had \(m=1\), the coupling efficiency increased with the increasing of the spatial coherence width of the beam. We can explain this phenomenon using the property of PCRP vortex beams (see Figure 2), where the PCRP vortex beams with \(m=1\) have a similar Gauss distribution beam profile. Based the analysis above, we know that there is a best choice (peak value of the coupling efficiency) of the value of coherence and NA for different topological charge.