## 3 Simulation and Discussion
Compared with the results of the incident beam shown in Figures 3 and 4, the maximum one-dimensional intensity of the external field after scattering is significantly changed. The phase corresponding to the sampling angle along the circumference is different. However, the phase along the circle within the whole sampling range still shows the OAM mode carried by the incident beam. This is the case even if the phase structure of the external field becomes extremely irregular. There is a weight difference between the spiral spectrum of the external field and that of the incident field. It can be seen that the larger the topological charge of the incident beam, the smaller the difference. This is because the beam is incident on the axis, so the lower the energy irradiates on the particle when the topological charge is larger, the weaker the scattering effect and the smaller the proportion of the scattered field in the total external field. The above analysis shows that the weights of OAM modes in both scattered and external fields at the receiving end are different from that of the incident field. This enables significant application opportunities, where the spiral spectrum of the field on the measurement plane can be used as a reference for retrieving the physicochemical parameters of the scatterer particle. In addition, sampling the phase along the circumference can only show the phase performance of the intensity on a certain sampling radius, hence unable to reflect the deformation of the OAM states of the whole field. may lead to the same spiral spectrum simulation results, it might be much more accurate to use both the circumferential sampling phase and the spiral spectrum to inverse the particle parameters. Figure 12: The information of the external field in the far region after the beam is scattered by a uniform and isotropic spherical particle: (**a**) lists the position of the maximum intensity of the corresponding external field in the polarization direction where the beam is incident with a different topological charge; (**b**) shows the sampling phase along the circle where the maximum lies; and (**c–f**) are the spiral spectra of the entire external fields. The measurement plane is located at z = 50 \(\upmu\)m.  
Figure 13: The same as in Figure 12, but here the scattered field is simulated. The radius of the illuminated particle is changed to 1.0 \(\upmu\)m and other parameters are unchanged. Similarly, the particle parameters can also be retrieved by using the circumferential sampling phase and the spiral spectrum of the total external field at the receiving end. We further change the properties of the scatterer and replace it with a charged spherical particle. The surface electrostatic potential is 50 V and the temperature is 30 \({}^{\circ}\)C, reflecting the amount of charge carried on the particle. The simulation results are shown in Figure 15. Comparing Figure 15 with Figures 6 and 7, we can see that the results of one-dimensional intensity, sampling phase along a circle, and spiral spectrum of the scattered field of a charged spherical particle are consistent with those of a neutral dielectric sphere. The reason is that when the particle size is small enough, the scattering effect of a charged spherical particle is different from that of a neutral one, and the difference is more obvious in the near region than in the far region. Therefore, the scattered field in the far region can hardly identify whether the scatterer particle carries charges or not. Similarly, the external field in the far region is unable to identify this. Finally, we discuss whether other types of vector Bessel-Gaussian vortex beams can show the differences in the weights of OAM modes for various characteristic parameters of particles. Here, we consider the right circular polarization and the radial polarization.