## 5 Results and Discussion
### Performance Comparison Based on Distorted OV Beams  
Many practical applications of OV beams, e.g., free-space optical communication and optical remote metrology, require the determination of the TC value of the OV beam after its propagation over a certain distance in the atmosphere. Consequently, it is vital to evaluate and compare these methods for OV beams distorted by atmospheric turbulence. Generally, a turbulent atmosphere can be treated as an inhomogeneous refractive index media, featured by the structure parameter \(C_{n}^{2}\)[33], and its impact on the light beam propagating through it can be accumulated as phase screens [34]. Supposing the turbulence satisfies the Kolmogorov model, the phase screen can be simulated through the Zernike polynomial [35]. The coefficients of the Zernike polynomial are related to the normalized correlation length \(r_{0}/D\), where \(r_{0}\) is the Fried parameter and \(D\) is the beam size [36].  
In the experiments, considering that the beam size used in our optical system was 4 mm, we chose \(r_{0}\) to be 1, 2, 3, 4, and 6 mm, and the normalized correlation length \(r_{0}/D\) was chosen to be 0.25, 0.5, 0.75, 1, and 1.5. Under each \(r_{0}/D\) value, we performed 50 random realizations of phase screens, which were then separately superposed with the spiral phase pattern (SPPs) whose TC values were 1, 5, and 10. Therefore, we generated 750 phase patterns altogether (three TC values, five \(r_{0}/D\) values, and 50 phase screens). Afterwards, we displayed the phase patterns one by one on the LCOS-SLM and recorded the corresponding Hartmanngrams. The SH-WFS did not move herein, because we had already generated 50 diverse phase patterns under each condition for repeated testing. For each Hartmanngram, we respectively used WTM, MAICM, and WT-MAICM as the closed path determination method, and determined the TC value by the generalized contour-sum method.  
The experimental results evaluated by the WTCMR are given in Figure 9, where Figure 9a-c corresponds to a set of 50 measurements for the individual TC, and Figure 9d is the mean of these measurements.  
Figure 9: TC determination performance of three methods under different atmospheric turbulence conditions, where (**a**) is under \(TC=1\), (**b**) is under \(TC=5\), (**c**) is under \(TC=10\), and (**d**) is the average of (**a**), (**b**), and (**c**).  
Based on the experimental results, we found that the general tendency of TC measurement accuracy increased along with the increase of \(r_{0}\)/\(D\), which means the atmospheric turbulence becomes gentler. Moreover, when changing the closed path determination method from WTM to MAICM to WT-MAICM, the TC determination accuracy improved, especially when the atmospheric turbulence was severe. Supposing the accredited WTCMR to be over 0.9, we found that the limitations of \(r_{0}\)/\(D\) for WTM, MAICM, and WT-MAICM were around approximately 0.75-1, 0.5-0.75, and 0.25-0.5, respectively.  
Figure 10 is an example specifically illustrating why the WT-MAICM has a better performance than the WTM and MAICM. Here, the OV beam to be measured has a TC value of 10 and a turbulence strength parameter \(r_{0}\)/\(D\) of 0.25. The top, middle, and bottom rows show the closed path determination processes and corresponding results when utilizing WTM, MAICM, and WT-MAICM separately. From the significantly different determined closed paths, we can see that when the turbulence is severe, the distorted OV beam can have extremely nonuniform regions. These nonuniform regions can trap the MAICM searching process into a local loop (middle row), or lead the WTM-determined closed path to be more than one part (top row), which conflicts with the aim of extracting only one closed path; these results eventually deteriorate the performance of the WTM and MAICM.