## 3 Numerical Calculation Results
The effective beam width of an an apertured BGCSM beam in the focal plane for different values of the aperture radius \(a\), the transverse coherence width \(\delta_{0}\), and the parameter \(\beta\) are depicted in Figure 4. From Figure 4a, we can see that with the increase of the aperture radius \(a\), the effective beam width first quickly decreased and then tended toward a minimum value, which was the effective beam width of the unapertured BGCSM beam in the focal plane. Additionally, the effective beam width increased with the increase of \(\beta\). From Figure 4b, we can see that if the aperture radius \(a\) kept invariant, with the increase of the aperture radius \(a\), the effective beam width \(\delta_{0}\) decreased with the increase of the aperture radius \(a\), the effective beam width \(\delta_{0}\) decreased with the increase of the aperture radius \(a\). Figure 3: Normalized intensity distribution of an apertured BGCSM beam focused by a thin lens in the \(\rho\)–\(z\) plane for different values of \(\beta\) with \(a\) = 1.5 mm and \(\delta_{0}\) = 0.2 mm. (**a**) \(\beta\) = 0; (**b**) \(\beta\) = 1.5; (**c**) \(\beta\) = 2.5; and (**d**) \(\beta\) = 4. Figure 2: Normalized intensity distribution of an apertured BGCSM beam focused by a thin lens in the \(\rho\)–\(z\) plane for different values of transverse coherence width \(\delta_{0}\) with \(a\) = 1.5 mm and \(\beta\) = 2.5. (**a**) \(\delta_{0}\) = 0.1 mm; (**b**) \(\delta_{0}\) = 0.2 mm; (**c**) \(\delta_{0}\) = 0.5 mm; and (**d**) \(\delta_{0}\) = 1 mm. coherence width \(\delta_{0}\), the effective beam width first quickly decreased and then approached a common value, which was independent of the coherence width. This common value can be considered as the effective beam width of a coherent Gaussian beam. As seen from Figure 4c, the effective beam width increased as the parameter \(\beta\) increased, which is consistent with Figure 4a. For a large value of \(a\) (i.e., \(a>1.5\) mm), the effective beam width was nearly unchanged, so the effect of an aperture can be neglected. Finally, we discuss the SDOC of an apertured BGCSM beam focused by a thin lens. Figure 5 shows the modulus of the SDOC \(|\mu(\rho,0)|\) of an apertured BGCSM beam at two transverse spatial positions, \(\rho_{1}=\rho\) and \(\rho_{2}=\mathbf{0}\), propagating in the \(\rho\)-\(z\) plane for different values of the aperture radius \(a\). One sees clearly that the \(|\mu(\rho,0)|\) displayed a non-Gaussian distribution with two side robes around the central bright spot on propagation, and finally degenerated into a Gaussian distribution in the focal plane when the aperture disappeared. While there was an existing aperture, with the decrease of \(a\), \(|\mu(\rho,0)|\) diverged more quickly on propagation, and displayed non-Gaussian distribution with more side robes in the focal region. To further study the SDOC in the focal plane, in Figure 6 we calculated the modulus of the SDOC \(|\mu(\rho,0)|\) in the focal plane between \(\rho_{1}=\rho\) and \(\rho_{2}=\mathbf{0}\) for different values of the aperture radius \(a\)  
Figure 4: Effective beam waist width of an apertured BGCSM beams in the focal plane versus (**a**) aperture radius \(a\), (**b**) transverse coherence width \(\delta_{0}\), and (**c**) \(\beta\). The other parameters were set as \(\delta_{0}=0.2\) mm in (**a**), \(a=1.5\) mm in (**b**), and \(\delta_{0}=0.2\) mm in (**c**). Figure 5: Modulus of the spectral degree of coherence (SDOC) \(|\mu(\rho,0)|\) of an apertured BGCSM beam focused by a thin lens in the \(\rho\)–\(z\) plane for different \(a\), with \(\beta=2.5\) and \(\delta_{0}=0.2\) mm. (**a**) \(a=0.5\); (**b**) \(a=1\); (**c**) \(a=1.5\); and (**d**) without aperture. the transverse coherence width \(\delta_{0}\), and the parameter \(\beta\), respectively. One finds from Figure 6a that without aperture, \(|\mu(\rho,0)|\) exhibited a Gaussian distribution in the focal plane. However, with the decrease of \(a\), \(|\mu(\rho,0)|\) exhibited several zero points.