## 2 Propagation of an Apertural BGCSM Beam through a Paraxial ABCD Optical System  
The second-order correlation properties of a scalar partially coherent beam are generally characterized by the cross-spectral density (CSD) in the spatial-frequency domain [44]. The CSD of a BGCSM beam at two arbitrary points in the source plane is expressed as [27; 40]:  
\[W(r_{1},r_{2})=\exp\left(-\frac{r_{1}^{2}+r_{2}^{2}}{4\sigma_{0}^{2}}\right) \exp\left[-\frac{(r_{1}-r_{2})^{2}}{2\delta_{0}^{2}}\right]J_{0}\left(\beta \frac{|r_{1}-r_{2}|}{\delta_{0}}\right), \tag{1}\]  
where \(r_{j}=(r_{jx}r_{jy})\) (\(j=1,2\)) denotes an arbitrary transverse position vector, \(\sigma_{0}\) is the transverse beam waist width of the corresponding Gaussian beam, \(\delta_{0}\) is the transverse coherence width, \(\beta\) is a real constant, and \(J_{0}\) (\(x\)) is the zeroth-order Bessel function of the first kind. Equation (1) shows that the BGCSM beam reduces to a conventional Gaussian Schell-model beam when \(\beta\) equals 0, and to a \(J_{0}\)-correlated beam when \(\beta\) trends towards infinity. In a practical optical system, an aperture is usually used to modulate the size of the beam spot of a partially coherent beam. Therefore, it was necessary to investigate the effect of the aperture radius on the propagation properties of a partially coherent beam. We assumed that a circular aperture with radius \(a\) is located in front of the BGCSM beam, so the CSD of the apertured BGCSM beam in the source plane is expressed as  
\[W(r_{1},r_{2})=\exp\left(-\frac{r_{1}^{2}+r_{2}^{2}}{4\sigma_{0}^{2}}\right) \exp\left[-\frac{(r_{1}-r_{2})^{2}}{2\delta_{0}^{2}}\right]J_{0}\left(\beta \frac{|r_{1}-r_{2}|}{\delta_{0}}\right)H(r_{1})H^{*}(r_{2}), \tag{2}\]  
where \(H(r_{i})\) is the transmission function of the circular aperture. This can be expanded as the following finite sum of complex Gaussian functions [45; 46]:  
\[H(r_{i})=\sum_{m=1}^{M}A_{m}\exp\left(-\frac{B_{m}r_{i}^{2}}{a^{2}}\right). \tag{3}\]  
Here, \(A_{m}\) and \(B_{m}\) are the expansion and Gaussian coefficients, respectively, which can be obtained through numerical optimization [45]. This expansion method has proved to be reliable and efficient,and the simulation accuracy improves as \(M\) increases. For a circular hard aperture, \(M=10\) already ensures a very good description of the propagation of the diffracted beam [45; 46]. The propagation of the CSD of an apertured BGCSM beam through a paraxial ABCD optical system can be studied by using the following extended Collins formula [47; 48]:  
\[ \begin{array}{l}W(\rho_{1},\rho_{2})=\left(\frac{k}{2\pi B}\right)^{2}\exp \left[-\frac{ikD}{2B}\big{(}\rho_{1}^{2}-\rho_{2}^{2}\big{)}\right]\int_{- \infty}^{\infty}\int_{-\infty}^{\infty}\int_{\infty}^{\infty}\int_{-\infty}^{ \infty}W(\mathbf{r}_{1},\mathbf{r}_{2})\\ \times\exp\left[-\frac{ikA}{2B}\left(\mathbf{r}_{1}^{2}-\mathbf{r}_{2}^{2} \right)+\frac{ik}{B}(\mathbf{r}_{1}\cdot\rho_{1}-\mathbf{r}_{2}\cdot\rho_{2}) \right]d^{2}\mathbf{r}_{1}d^{2}\mathbf{r}_{2},\end{array}  \tag{4}\]  
where \(\rho_{j}=(\rho_{jx}\rho_{jy})\) (\(j=1,2\)) denotes an arbitrary transverse position vector at the receiver plane and \(k=2\pi/\lambda\) is the wave number related to the wavelength \(\lambda\). For the convenience of integration, the following \"sum\" and \"difference\" coordinates were adopted:  
\[ \begin{array}{l}\mathbf{r}_{s}=\frac{\mathbf{r}_{1}+\mathbf{r}_{2}}{2}, \Delta\mathbf{r}=\mathbf{r}_{1}-\mathbf{r}_{2},\rho_{s}=\frac{\rho_{1}+\rho_{2 }}{2},\Delta\rho=\rho_{1}-\rho_{2}.\end{array}  \tag{5}\]  
Substituting Equations (2) and (5) into Equation (4), we obtained the following alternative expression of Equation (4):  
\[