## 3 Simulation Results  
Based on Equation (2), the CSD characteristics of a PCLG vortex beam through a non-axisymmetric ABCD optical system composed of two cylindrical lenses were simulated. The radial mode index \(p\) was set to 0 in the following analysis. The focal lengths of the cylindrical lenses were \(f_{1}=f_{2}=200\) mm, the propagation distance between the source field and the first cylindrical lens was \(z_{0}=500\) mm. Due to the size of the lens holder, the cylindrical lenses cannot be too close, so the distance between two the cylindrical lenses was \(s=60\) mm. During the focusing/defocusing process, the rotation of the vortex beam is related to the sign of TC, as discussed in [20]. Therefore, the distance from the second cylindrical lens to the detection plane was \(z=300\) mm. In addition, the beam waist width \(\omega_{0}=0.35\) mm and the wavelength \(\lambda=532\) nm. Figure 2 shows the theoretical CSD amplitude distributions with varying TCs under different initial coherence widths, i.e., \(\sigma_{\rm g}=0.035\), \(0.14\), and \(0.35\) mm, which correspond to extremely low coherence, medium coherence, and high coherence, respectively. The two cylindrical lenses were placed vertically. Here, the amplitude is shown based on an on-axis reference point \(\left(\rho_{2x},\rho_{2y}\right)=\left[0,0\right]\)) and the pattern is in log scale, i.e., log \(\left[1+\left|W\left(\rho_{1x},\rho_{1y},0,0\right)\right|\right]\). The CSD amplitude exhibits anisotropy and presents a stripe distribution at extremely low coherence, i.e., \(\sigma_{\rm g}=0.1\omega_{0}\), as shown in Figure 2a-d. For example, the bright stripes are denoted by white arrows in Figure 2b,c. The relationship between the number of bright stripes and TC is N = 2 \(|l|\) + 1. Furthermore, a comparison between the cases of \(l=3\) and \(l=-3\) indicates that the sign of the TC can be determined by the orientation of the CSD amplitude. As the coherence increased, the CSD amplitude no longer exhibited a strip distribution, but singularities (i.e., dark dots) occurred [33]. However, the quantitative relationship between the number of singularities and the TC was not constant for different coherence cases. This is because the singularities exited from the amplitude envelope. Although the CSD phase can be experimentally measured (Section 4), the phase outside the amplitude envelope was noisy and difficult to distinguish. This implies that when the two cylindrical lenses are placed vertically, accurate TC measurements can only be achieved in the case of  
Figure 2: Theoretical simulation results of CSD amplitude of PCLG vortex beams in log scale, i.e., \(\log\left[1+\left|W\left(\rho_{x_{1}},\rho_{y_{1}},0,0\right)\right|\right]\). CSD amplitude distributions for (**a–d**) low coherence, (**e–h**) medium coherence and (**i–l**) high coherence cases with four different TCs. The arrows mark the positions of stripes in the CSD amplitude. extremely low coherence. Therefore, the CSD distributions for different cylindrical lenses angles were investigated. Currently, obtaining a final analytical solution for other included angles is difficult. Thus, in non-vertical cases, the CSD was numerically calculated using the random screen simulation method to avoid onerous integral operations [39]. Figure 3 presents the numerical simulation results of the CSD amplitude of the PCLG vortex beams; the parameters are the same as those used in Figure 2. These results agreed well with the theoretical calculations shown in Figure 2. A comparison of the analytical (Figure 2) and simulation results (Figure 3) for the case of \(90^{\circ}\) confirmed the accuracy and effectiveness of the random-screen simulation method presented in [39]. The speckle noise outside the central region was caused by the speckle superposition applied in numerical simulation. The positions of the singularities shown in Figures 2 and 3 mirrored each other, except for \(l=+3\) (Figure 3k) and \(l=-3\) (Figure 3l). This error was similarly caused by the speckles applied in the simulation.