## 2 Theory  
Herein, the propagation of a PCLG vortex beam through a non-axisymmetric ABCD optical system, that is dual cylindrical lens system, is analyzed. For the convenience of understanding, a schematic diagram is presented in Figure 1. A PCLG vortex beam on the source plane passing through a dual cylindrical lens system (CL1 and CL2), the measurement plane is located on the defocus plane of CL2. The coordinates and symbols used in the following analysis are also marked. Figure 1: Schematic diagram of a PCLG vortex beam passing through a dual cylindrical lens system. CL1 and CL2, cylindrical lenses. \(\theta\) is the included angle between the cylindrical lens and \(y\)-axis. SLM2, spatial light modulator. CCD2, charge-coupled device. The labels of SLM2 and CCD2 are consistent with those in the diagram of experimental set up. For a monochromatic partially coherent beam, the second-order, two-point correlation of such a field \(E(\mathbf{r})\), i.e., the CSD is defined as \(W(\mathbf{r}_{1},\mathbf{r}_{2})=\langle E^{*}(\mathbf{r}_{1})E(\mathbf{r}_{2})\rangle\) in the space-frequency domain [16]. \(E(\mathbf{r})\) is the electric field with random fluctuations and \((\mathbf{r}_{1},\mathbf{r}_{2})\) are vector coordinates. Here, the asterisk \"*\" and the angle brackets \(\langle\ \rangle\) represent the complex conjugate and the ensemble average, respectively. The CSD of the PCLG vortex beam in the source plane can be expressed as [37]. \[ \begin{split} W_{0}(r_{1},\varphi_{1},r_{2},\varphi_{2})=\left( \frac{2r_{1}r_{2}}{\omega_{0}^{2}}\right)^{l}L_{p}^{l}\left(\frac{2r_{1}^{2}}{ \omega_{0}^{2}}\right)L_{p}^{l}\left(\frac{2r_{2}^{2}}{\omega_{0}^{2}}\right) \\ \times\exp[-\mathrm{i}l(\varphi_{1}-\varphi_{2})]\exp\left(- \frac{r_{1}^{2}+r_{2}^{2}}{4\omega_{0}^{2}}\right)\exp\left(-\frac{(r_{1}-r_{ 2})^{2}}{2\sigma_{8}^{2}}\right)\end{split}  \tag{1}\]  
where \((r_{1},\varphi_{1},r_{2},\varphi_{2})\) are the polar coordinates of \((\mathbf{r}_{1},\mathbf{r}_{2})\). \(\omega_{0}\) means the waist width on the source plane, \(\sigma_{8}\) denotes the initial coherence width, and \(L_{p}^{l}\) represents the Laguerre polynomial with a radial mode index \(p\) and a TC of \(l\). Subsequently, using the generalized Collins' formula [38], the CSD of the PCLG vortex beam through a non-axisymmetric ABCD optical system composed of two cylindrical lenses can be obtained as follows:  
\[ \begin{split} W(\rho_{x_{1}},\rho_{y_{1}},\rho_{x_{2}},\rho_{y_{2 }})=\frac{1/\lambda^{2}}{B_{1}B_{2}}\int_{-\infty}^{\infty}\int_{-\infty}^{ \infty}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}W_{0}(x_{1},y_{1},x_{2},y _{2})\\ \times\exp\left[-\frac{ik}{2B_{1}^{2}}\left(A_{1}^{*}x_{2}^{2}-2 x_{1}\rho_{x_{1}}+D_{1}^{*}\rho_{x_{1}}^{2}\right)-\frac{ik}{2B_{2}^{2}} \left(A_{2}^{*}y_{1}^{2}-2y_{1}\rho_{y_{1}}+D_{2}^{*}\rho_{y_{1}}^{2}\right) \right]\\ \times\exp\Bigl{[}\frac{ik}{2B_{1}}\left(A_{1}x_{2}^{2}-2x_{2} \rho_{x_{2}}+D_{1}\rho_{x_{2}}^{2}\right)+\frac{ik}{2B_{2}}\left(A_{2}y_{2}^{ 2}-2y_{2}\rho_{y_{2}}+D_{2}\rho_{y_{2}}^{2}\right)\Bigr{]}dx_{1}dx_{2}dy_{1}dy _{2}\end{split}  \tag{2}\]  
where \((x_{1},y_{1})\) and \((x_{2},y_{2})\) correspond to \(\mathbf{r}_{1}\) and \(\mathbf{r}_{2}\), respectively. \(\left(\rho_{x_{1}},\rho_{y_{1}},\rho_{x_{2}},\rho_{y_{2}}\right)\) are the coordinates after the dual-cylindrical-lens system; and \(A_{1}\), \(B_{1}\), \(C_{1}\), \(D_{1}\), \(A_{2}\), \(B_{2}\), \(C_{2}\), and \(D_{2}\) are the components of the transmission matrix of the dual cylindrical lens system. In fact, it can be obtained by multiplying the corresponding transmission matrices from the beginning to the end, i.e., in the order of free space, cylindrical lens, free space, cylindrical lens, and free space. The free space transmission matrix \(M_{F}\) and the cylindrical lens matrix \(M_{CL}\) can be expressed as:  
\[M_{F}= \begin{pmatrix}1&0&z&0\\ 0&1&0&z\\ 0&0&1&0\\ 0&0&0&1\end{pmatrix} \text{ and }M_{CL}= \begin{pmatrix}\mathbf{I}&0\\ \mathbf{\Phi}&\mathbf{I}\end{pmatrix}