## 3 Simulation and Discussion
Their field distributions in the x-direction at the receiving end are used to analyze the spiral spectrum characteristics in order to explore whether we can obtain the spiral spectrum varying with the parameters of the scatterer particle only, based on knowing the component field in a certain direction. Hence, the characteristic parameters of the scatterer can be retrieved according to the differences in the spiral spectrum. This may then be used for remote sensing applications in engineering. Figure 14: The same as in Figure 12, but simulating the scattered field. The complex refractive index of the scatterer particle is changed to \(m=1.75+0.44i\), and other parameters are unchanged. Figure 16 shows the spiral spectrum of the scattered field of the right circular polarized Bessel-Gaussian vortex beam. The topological charges 1-4 are considered, corresponding to the panels from left to right in each row. The radius \(a\), and the complex refractive index \(m\), of the irradiated particle in each row are different. In the first row, \(a=0.5\)\(\upmu\)m and \(m=1.377+1.62\times 10^{-3}i\), in the second, \(a=0.5\)\(\upmu\)m and \(m=1.75+0.44i\) and \(a=1.0\)\(\upmu\)m and \(m=1.377+1.62\times 10^{-3}i\) in the third row. The particle is homogeneous and isotropic. The spiral spectrum distribution of the scattered field of the right circular polarized Bessel-Gaussian vortex beam in the x-direction is highly sensitive to the change of the characteristic parameters of the illuminated particle. Instances of such parameters are the complex refractive index and size of the particle. In other words, for the right circular polarized Bessel-Gaussian vortex beam, the characteristic parameters of the scatterer particle can be retrieved by using the spiral spectrum of the scattered field in the x-direction. In addition, comparing the results of the first and the third rows, it can be seen that the particle size is larger, the weight of the primary mode becomes dominant, which visually shows the OAM state carried by the incident beam. Similar to Figure 16, Figure 17 discusses the simulation results where the incident beam is a radial polarized Bessel-Gaussian vortex beam. Refer to Figure 16 for specific contents and parameters of panels. As it is seen, variations in the characteristic parameters of the scatterer particle also affect the spiral spectrum distribution of the scattered field of radial polarized Bessel-Gaussian vortex beam. Figure 15: The same as in Figure 12, with the simulated scattered field. The scatterer is changed into a charged particle where other parameters remain unchanged. The surface electrostatic potential of the charged particle is \(\varphi=50\) V and the surface temperature is \(T=30\)\(\,\mathrm{\SIUnitSymbolCelsius}\). The above discussions show that changing the characteristic parameters of the illuminated particle significantly alters the spiral spectrum of the scattered field of the vector Bessel-Gaussian vortex beam. Therefore, for detecting parameters of the particle, one can select Bessel-Gaussian vortex beams with different polarization states for incidence. Fields in multiple directions can be then sampled and their spiral spectrum characteristics can be analyzed to build a multi-dimensional database. Similarly, a uniform and isotropic dielectric sphere can be used as an auxiliary objet. By changing the multiple characteristic parameters of this object, a multi-dimensional database can be then established to identify the parameters of the incident vector Bessel-Gaussian beam. The higher the number of such dimensions, the more accurate the inversion results of the beam and particle parameters. In our future research, we will investigate whether a similar method can be extended to other shaped beams and irregular scatterers. Figure 16: The spiral spectra of the scattering fields of right circular polarized Bessel–Gaussian vortex beams were scattered by a particle with different sizes and different complex refractive indices. The scatterer particle is a uniform and isotropic sphere.