## 4 Results and Discussion  
Figure 5 shows numerically simulated and experimentally observed beam profiles at the two detection positions for incident LP OVs with \(\ell=4\). We recorded bright bar-like spots at both detection positions. As described in the theory section, the two detection positions correspond to the LCP and RCP components of the incident beam. From the number of dark lines, we found that the LCP and RCP components of the incident beam both have TC of \(\ell=4\). From this information, we are able to characterize the incident beam as an OV with \(\ell=4\). In addition, the total energy at the two detection positions was measured to 70.2% As described in the Section 2, our system can realize 100% optical throughput in ideal. We can consider that this deterioration between the theory and experiment was caused by the imperfection and the Fresnel reflection of the SPG and PPGs, so that we can improve the optical throughput by adjusting the retardation of SPG and PPGs and, introducing anti-reflection coating. If we only use a single PPG for LP OV characterization, the optical throughput is limited to 50% because one of the circularly polarized components is cylindrically spread after passing through the PPG [Figure 2b,c]. The result shown in Figure 5 proves that both circular polarization components are cylindrically focused by the setup with one SPG and two PPGs. Therefore, even if the incident OV has an arbitrary SoP, our system can characterize the TC value of the incident OV with high optical throughput. In addition, if we measure the ratio of the beam intensities at the two detection positions, we can reconstruct the ellipticity of the incident OV.  
Figure 4: Schematic showing the optical setup for the demonstration of (**a**) OV and (**b**) PV characterizations using one SPG and two PPGs.  
We can find the fading or twisting bright spots, especially at the edge of the experimentally obtained results shown in Figures 5 and 6. We consider this distortion to be caused by the initial amplitude distribution of OV before the incident on the proposed system. Due to the accuracy of our experimental setup for generating OV, the measured OV does not have a doughnut-shaped amplitude distribution which is completely matched with the pure Laguerre Gaussian beam. By measuring the number of dark lines along the centerline perpendicular to the dark line orientation, we would mitigate this impact.  
Since the SPG and PPG is made of polymer liquid crystal, its birefringence and alignment pattern is almost immune to the temperature and vibration within the room temperature. As a result, the observed beam profiles in the experiment are stable for environmental fluctuation.  
Figure 5: Numerically simulated and experimentally observed beam profiles obtained at the two detection positions of the mode characterizer for an incident OV with \(\ell=4\).  
Figure 6: Numerically simulated and experimentally observed beam profiles at the two detection positions of the mode characterizer for an incident PV with (**a**) \(p=1\) and (**b**) \(p=2\).  
From numerically simulated and experimentally obtained beam profiles at the two detection positions for an incident PV with (a) \(p=1\) and (b) \(p=2\) [Figure 6a,b], we found that the LCP and RCP components of the incident beam have TC values of Figure 6a (\(\ell_{L}=1\), \(\ell_{R}=-1\)) and Figure 6b (\(\ell_{L}=2\), \(\ell_{R}=-2\)), respectively. From this information, we characterize the incident beam as PV with (a) \(p=1\) and (b) \(p=2\). These results indicate that our system is highly efficient not only as an OV characterizer but also as PV characterizer. Furthermore, our system can characterize vector vortex beams, which have both phase and polarization singularity. Because these beams can be decomposed into LCP and RCP OVs with \(|\ell_{L}|\
eq|\ell_{R}|\), the two detection positions show bright bar-like spots that differ in their number of dark lines.