## 6 Conclusions  
In this study, we theoretically and experimentally analyzed the CSD distributions of a PCLG vortex beam transmitted through two cylindrical lenses at varying angles. When the cylindrical lenses were placed vertically, the orientation of the CSD magnitude can  
Figure 8: Experimental results for the CSD amplitude and phase distribution with medium coherence width (\(\mathrm{\sigma_{g}}=0.4\omega_{0}\)) and different TCs. Included angles of cylindrical lenses was \(30^{\circ}\). CSD amplitude distributions for TC equal to (**a**) +1, and (**b**) +2. The arrows mark the positions of the stripes in the CSD amplitude. CSD phase distributions for TC equal to (**c**) +1, and (**d**) +2. The arrows mark the positions of the “dumbbell pieces and handles” in the CSD phase.  
be used to distinguish the TC sign. However, the TC magnitude can only be determined for extremely low-coherence cases using the relationship between the number of CSD stripes and the TC, i.e., N = 2 1 11 +1. As the coherence increased, the stripes evolved into singularities, the number of which did not correlate with the TC. However, when the cylindrical lens's angle was changed gradually from 90\({}^{\circ}\) to 0\({}^{\circ}\), stripes reappeared in the CSD amplitude for the case of medium and high coherences. Meanwhile, the magnitude of the TC in the PCLG vortex beams can be determined by the number of bright stripes in the amplitude or the dumbbell structure in the phase patterns. The quantitative relationship was N = 2 1 11 +1. Compared with the findings regarding the CSD amplitude, this conclusion was more evident in the CSD phase patterns. Our findings can facilitate the development of applications in the field of free-space, turbulence-atmosphere optical communication, and information encryption.  
Conceptualization, [PERSON] ([PERSON]) and [PERSON] ([PERSON]); methodology, [PERSON] ([PERSON]) and Z.W.; formal analysis, [PERSON] ([PERSON]) and Z.W.; investigation, [PERSON] ([PERSON]); writing--original draft preparation, [PERSON] ([PERSON]) and [PERSON] ([PERSON]); writing--review and editing, [PERSON] ([PERSON]) and C.Z.; supervision, [PERSON] ([PERSON]), C.Z. and Y.C.; project administration, [PERSON] ([PERSON]); funding acquisition, [PERSON] ([PERSON]), C.Z. and Y.C. All authors have read and agreed to the published version of the manuscript.  
This work was supported by the National Key Research and Development Program of China (No. 2022 YFA1404800 and No. 2019 YFA0705000), National Natural Science Foundation of China (No. 12204340, No. 12174280, No. 12192254, No. 11974218, and No. 92250304), China Postdoctoral Science Foundation (2022M722325), Priority Academic Program Development of Jiangsu Higher Education Institutions, Tang Scholar, Local Science and Technology Development Project of the Central Government (No. YDZX20203700001766), and Key Lab of Modern Optical Technologies of Jiangsu Province (KJS2138). [PERSON] and [PERSON] Chinese Undergraduate Research Endowment (CURE).  
Not applicable.  
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.  
The authors declare no conflict of interest.