## 3 Simulation and Discussion  
First, in order to verify the correctness of the algorithm we use, a validation example is used to compare the calculation results of the Mie scattering theory with those of the electromagnetic (EM) simulation software FEKO. Subject to the source of FEKO, a Bessel-Gaussian beam with x-polarization is considered to irradiate a uniform and homogeneous dielectric sphere of complex refractive index \(m=1.377+1.62\times 10^{-3}i\) and radius \(a=0.5\)\(\upmu\)m. The wavelength of the incident beam is \(\lambda=1.064\)\(\upmu\)m, the order is \(l=0\), the conic angle is \(\theta_{\rm{b}}=0^{\circ}\), and the beam waist radius \(w_{0}\rightarrow\infty\). The radar cross section (RCS) and the scattered field distribution calculated by the GLMT and FEKO are compared. The measurement plane of the scattered field is located at z = 50 \(\upmu\)m, and the results are shown in Figure 1. It is obvious that the results match very well and the relative errors are negligible. Figure 1: Comparison of the results calculated by the Mie scattering theory and FEKO: panel (**a**) is the radar cross section (RCS); panels (**b**) and (**c**) are the scattered field distributions obtained by the GLMT algorithm and FEKO simulation, respectively; the relative error between them is shown in panel (**d**). In the following simulations of the field sampling, we consider the OAM modes from \(-5\) to \(5\). We further set the radial, and angular sampling intervals to \(0.1~{}\upmu\)m, and \(1^{\circ}\), respectively. Considering the y-polarized Bessel-Gaussian beam as an example, the topological charges of the incident beam are \(1\), \(2\), \(3\), and \(4\), respectively. Unless otherwise specified, here the beam wavelength is \(1.064~{}\upmu\)m, the conic angle is \(25^{\circ}\) and the waist radius is \(0.5~{}\upmu\)m.  
Figure 2 shows the intensity and phase distributions of the y-polarized Bessel-Gaussian vortex beams at the source plane. The topological charges corresponding to the left panels to right are \(1\)-\(4\), respectively. Panels (a)-(d) present intensity and (e)-(h) show phase. The intensity distribution of the incident field of beam presents a hollow structure and the hollow shape is similar to an ellipse. It is not completely uniform and symmetrical in the angular space, but it follows the basic characteristics of the vortex beam. In other words, the greater the topological charge, the weaker the intensity and the larger the hollow size. Note that the panels at the bottom show the phase distributions, where the changes in the wavefront phase reflect the orbital angular momentum mode carried by the incident beam. As was seen above, there are two ways to detect the spiral spectrum according to the field distribution including sampling along the circumference at a certain radius and sampling the whole measurement plane. The OAM modes of the incident beam at the source plane are calculated according to these two sampling methods. Figure 3 provides the one-dimensional intensity of the incident field in the polarization direction (i.e., y-direction), and marks the specific coordinates with the maximum intensity. Taking the position as the determined radius, we then conduct phase sampling on the circle. A similar approach based on taking the intensity in the polarization direction as a sample was also considered in [28]. Figure 2: The intensity and phase distributions of the incident field of y-polarized Bessel–Gaussian beams with different topological charges at the source plane: panels (**a–d**); and (**e–h**) show intensity, and phase, respectively. The corresponding topological charge from left to right is \(1\)–\(4\). Panels (a)-(e) in Figure 3 show the one-dimensional intensity distribution of the incident field in the polarization direction for the topological charges of 0, 1, 2, 3 and 4, respectively. We also marked the specific coordinates with the maximum intensity. It can be seen that for the topological charges of 1 and 2, the central intensity is a valley with a non-zero value.