## 5 Results and Discussion  
### Performance Comparison Based on Aberration-Free OV Beams  
To compare the above five closed path determination methods, we first evaluated the performance under the aberration-free OV beam condition. In the experiments, we displayed various computer-generated holograms (CGHs) with a spiral phase structure on the LCOS-SLM to generate an OV beam with TC values ranging from \(\pm\) 1 to \(\pm\) 20. In order to enrich the data amount as well as eliminate the randomness, for each TC value, we repeatedly recorded the Hartmanngrams at 15 different SH-WFS positions by laterally moving the mechanical platform. Hence, we obtained a total of 600 Hartmanngrams (40 TC values and 15 SH-WFS positions). Figure 4 shows an example of the spiral phase pattern displayed on LCOS-SLM, and the corresponding Hartmanngram as well as the intensity profile image respectively recorded by SH-WFS and CMOS camera. Figure 3: Schematic diagram of the experimental setup. As described in Section 2, for each recorded Hartmanngram, we first performed the necessary preprocessing, and then summed all the pixel values in each lenslet region to obtain the intensity sum map and measured the phase slopes at the lenslet positions according to the SH-WFS working principle. After that, we used each of the studied five methods to determine the closed path, and then utilized the generalized counter-sum method to calculate the TC value. Considering that the theoretical TC value is an integer, for TC measurement, if the absolute difference between the measured TC value and the theoretical TC value is lower than 0.5, the measured TC value will be identical to the theoretical TC value after rounding--we define this as a well-determined TC measurement. Based on this definition, we introduced a parameter, the well-determined TC measurement ratio (WTCMR), which is the percentage of well-determined TC measurements within a given set of TC measurements, to quantitatively compare the performance of the diverse closed path determination methods. The performance evaluation results of the individual closed path determination methods are given in Figures 5 and 6. In Figure 5, each value is the WTCMR of a group of 30 measurements (15 positions for both positive and negative TC values), while in Figure 6, the WTCMR is the statistical average of all 600 measurements. Figure 6 also shows the corresponding processing speed of the individual methods in terms of frames per second, which was measured on an Intel Core i7 computer with a CPU frequency of 3.7 GHz and 16 GB of RAM. As shown in Figures 5 and 6, the WT, MAIC, and WT-MAIC methods performed perfectly, since the WTCMR was shown to be 100% for all tested TC values. However, the WTCMR of the FC-PRCM and the SC-PRCM was not always 100%, even when measuring aberration-free OV beams. Moreover, the performance of these methods deteriorated with the increase of the TC value. The SC-PRCM was the worst method in terms of both the WTCMR and the computing speed. Figure 4: Example of (**a**) spiral phase pattern, (**b**) Hartmanngram, and (**c**) intensity profile recorded by the complementary metal-oxide semiconductor (CMOS) camera. The bar indicates 1 mm. Figure 5: Well-determined topological charge (TC) measurement ratio (WTCMR) distributions of the five closed path determination methods when the TC value changes from \(\pm 1\) to \(\pm 20\). Considering that the TC determination progress is the same apart from the closed path being determined by diverse closed path determination methods, the reason for the differences in performance was exclusively due to the differences in the determined closed paths.