## 3 Numerical Calculation Results  
Considering an apertured BGCSM beam generating from the source plane (\(z=0\)) focused by a thin lens optical system with a focal length \(f\), the distance from the source plane to the thin lens is \(f\) and the distance from the thin lens to the receiver plane is \(z\). Thus, the transfer matrix for the optical system can be expressed as:  
\[\left( \begin{array}{cc}A&B\\ C&D\end{array} \right)=\left( \begin{array}{cc}1&z\\ 0&1\end{array} \right)\left( \begin{array}{cc}1&0\\ -1/f&1\end{array} \right)\left( \begin{array}{cc}1&f\\ 0&1\end{array} \right)=\left( \begin{array}{cc}1-\frac{z}{f}&f\\ -1/f&1\end{array} \right). \tag{20}\]  
We studied numerically focusing characteristic of the intensity, the effective beam width, and the SDOC of an apertured BGCSM beam focused by a thin lens focusing system by applying the formulae derived in Section 2. For the following numerical examples, the parameters were chosen as \(f=150\) mm, \(\lambda=632.8\) mm, \(\sigma_{0}=1\) mm, \(a=1.5\) mm, \(\delta_{0}=0.2\) mm, and \(\beta=2.5\), unless other values are indicated in the figures. To study the intensity properties of an apertured BGCSM beam focused by a thin lens focusing system, in Figures 1-3 we calculated the normalized intensity distribution of an apertured BGCSMbeam in the \(\rho\)-\(z\) plane for different values of the aperture radius \(a\), the transverse coherence width \(\delta_{0}\), and the parameter \(\beta\), respectively. One can see from Figure 1 that a partially coherent optical cage was formed with or without an aperture separately. The shape of the optical cage became much more uniform and the size of the cage became larger with the decrease of the aperture radius. The generated optical cage is useful for trapping low-refractive index particles. As shown in Figure 2, with the small value of \(\delta_{0}\), a more uniform optical cage was formed. With the increase of the coherence width \(\delta_{0}\), both transverse and longitudinal sizes, as well as the depth of the optical cage, decreased. As shown in Figure 3, with the increase of the parameter \(\beta\), the focal intensity profile gradually evolved from a peak-centered shape into an optical cage, and both transverse and longitudinal sizes, as well as the depth of the optical cage increased. However, it was also found that, with the increase of the parameter \(\beta\), the transverse intensity surrounding the optical cage increased while the longitudinal intensity surrounding the optical cage decreased, so the shape of the optical cage became less uniform. Hence, a \(J_{0}\)-correlated beam (i.e., a BGCSM beam with \(\beta\)\(\rightarrow\)\(\infty\)) is not appropriate for producing an optical cage. From Figures 2 and 3, one can explain that the effect of the special correlation function can modulate the optical cage and will be enhanced with a small \(\delta_{0}\) or a moderate \(\beta\). Furthermore, we conclude that one can conveniently obtain an adjustable optical cage by focusing an aperture BGCSM beam through choosing appropriate aperture radius, transverse coherence width, and correlation function (i.e., the Bessel-Gaussian correlation function). However, the conventional Gaussian Shell-model beams do not have those characteristics. Figure 1: Normalized intensity distribution of an aperture Bessel–Gaussian correlated Schell-model (BGCSM) beam focused by a thin lens in the \(\rho\)–\(z\) plane for different values of aperture radius \(a\), with \(\delta_{0}\) = 0.2 mm and \(\beta\) = 2.5. (**a**) \(a\) = 0.5 mm; (**b**) \(a\) = 1 mm; (**c**) \(a\) = 1.5 mm; and (**d**) without aperture. Next, let us study the effective beam width of an apertured BGCSM beam.