## 3 Simulation and Discussion
In contrast, for the topological charges of 3 and 4, the intensity in the center is further reduced to 0. The position with the intensity peak indicates the size of the hollow. In Figure 3f we consider the location where the intensity has its maximum value as the radius and plot the phase samples along the circumference. Figure 3f shows that the phase sampling results along the circle can intuitively characterize the topological charge of the incident beam, i.e., through the relationship between the phase change and \(2\pi\) within the sampling angle range, regardless of whether the phase changes regularly or not. We also found that for different radii of the circle used for phase sampling, the specific phase corresponding to each sampling angle may change significantly. This is despite the same change periods of the phase within the sampling angle range. Therefore, obtaining the normalized weights of OAM modes based on the phase sampling results on a certain circle might be unstable. Furthermore, in practice, it is much easier to obtain the intensity distribution on a measurement plane, which also reflects the spatial information of the beam more comprehensively. This is the main reason that sampling is often applied on the whole measurement plane at the receiving end when the intensity and phase changes of vortex beam propagating in a turbulent environment are studied. In Figure 4, we consider sampling the entire incident field to obtain the weights of OAM modes. Figure 3: (**a–e**) show the one-dimensional intensity distribution of the incident beams with different topological charges in the polarization direction; (**f**) shows the phase sampling results along the circumference. The electric field of the vector Bessel-Gaussian vortex beam in this manuscript is calculated based on the GLMT and ASDM. Nevertheless, the intensity distribution is not as ideal and uniform as the scalar Bessel-Gaussian vortex beam. The spiral spectrum of the incident field is also shown in Figure 4. As can be seen, the decisively dominant OAM mode clearly shows the topological charge carried by the incident beam. In addition, the modes separated by one bit from the main mode account for a small proportion. The size of the measurement plane of the field that is used to calculate the OAM modes directly affects the calculation results. The sampling plane should be large enough to ensure full reception of the beam's energy. Here, we further calculate the intensity and phase distributions of the scattered field of the beam that is scattered by a small spherical particle. As an example, we consider a uniform and isotropic dielectric spherical particle and set its radius and complex refractive index as \(a\) = 0.5 \(\upmu\)m, and \(m\) = 1.377 + 1.62 \(\times\) 10\({}^{-3}\)\(i\), respectively. Assuming that the measurement plane is 50 \(\upmu\)m away from the particle center, the energy of the scattered field can be fully received if the diameter of the circular receiving plane is 2000 \(\upmu\)m.  
Figure 5 shows the intensity and phase distributions of the scattered field in the whole measurement plane. Here, the scatterer is a homogeneous and isotropic spherical particle. Panels in the first and second rows present the intensity and phase, respectively. The topological charges corresponding to the panels from left to right in each row are 1, 2, 3, and 4. The area with the concentrated intensity and the area with clear phases are small, hence, we show the results in a smaller region in the upper right corner of each panel, and the specific dimensions are marked in white. Figure 5 shows that the intensity distribution of the scattered field presents a scattered crescent shape for the topological charge 1 or 2. For all beams considered here, the scattered fields tend to rotate to the right relative to the incident fields. As for the phase distribution, for the incident beams with OAM of 1 and 3, the phases of their scattered fields at the receiving end are one topological charge, whereas the spiral direction is the opposite.