## 3 Simulation and Discussion
In the case where the topological charge of the incident beam is \(l\) = 2, on the circumference, the phase of the scattered field remains consistent. The helical phase of the scattered field also shows two topological charges where the beam is incident with \(l\) = 4. Figure 4: Weights of the OAM modes of the incident beam with different topological charges as sampling the field on the whole measurement plane: panels (**a**–**d**) are for the topological charges of 1–4. Panels (a)-(e) in Figure 6 show the one-dimensional intensity of the scattered field in the polarization direction. Figure 6f presents the phase sampling on the circumference with the location of maximum intensity as the radius. Comparing Figures 3 and 6, the one-dimensional intensity distribution of the scattered field and the incident field is different, not only in the amplitude of the intensity but also in the position with the maximum intensity. For the topological charge of 2, the one-dimensional intensity of the scattered field shows a peak instead of a valley at y = 0. This phenomenon is similar to the case in which the topological charge carried by the incident beam is zero, which can also explain the phase behavior in Figure 5f. It can also be seen that when the topological charge of the incident beam is zero, the positions of the intensity peak of the incident field and scattered field remain the same, with the largest intensity at the center. Figure 6f also shows that the sampling phase on the circumference with the maximum one-dimensional intensity cannot directly and accurately reflect the phase structure of the center of the scattered field. Figure 7 shows the spiral spectrum results of the entire scattered field with the measurement plane at 50 \(\upmu\)m away from the center of the particle. The topological charge marked in the panels is the OAM mode when the beam is incident. Comparing Figures 4 and 7, the spiral spectrum distribution of the scattered field is different from that of the incident field. For the incident beams with orbital angular momentums of 2, 3, and 4, the spiral spectrum of the scattered field does not show the mode of the field, but it carries the phase information of the whole scattered field. Combined with the above analysis of the sampling phase along the circle, it can be seen that sampling the whole field on the receiving plane can stably, accurately, and concretely provide the normalized weights of the OAM modes of the current beam. This can be considered as an important basis for identifying the transmission conditions. Figure 5: The intensity and phase distributions of the scattering field of the y-polarized Bessel–Gaussian beams with different topological charges. The beam is scattered by a uniform and isotropic particle and the measurement plane is located at z = 50 \(\upmu\)m: panels (**a–d**) show the intensity; and (**e–h**) the present phase. The corresponding topological charge from left to right is 1 to 4. We further consider using the difference in the spiral spectrum of the scattered field at the receiving end to inverse the physicochemical parameters of the particle scatterer. Therefore, it is necessary to eliminate the possibility that the normalized weights of the OAM modes change with the transmission distance due to the calculation algorithm of the field. Figure 8 focuses on the incident beam and shows the spiral spectrum of the incident field on the receiving plane that is 50 \(\upmu\)m away from the particle center. By comparing Figures 4 and 8, can be seen that using GLMT and ASDM to calculate the beam field does not lead to a significant difference in the intensity of different sampling planes. Taking the detection probability of the OAM modes at the source as the true value, Table 1 presents the relative error between the weights of the OAM modes in Figures 4 and 8. The maximum relative error is 5.99%, and the maximum magnitude of absolute error is \(10^{-3}\), which is very small for the probability range 0-1.