## 3 Simulation and Discussion
Therefore, it is reasonable to consider that the detection error of the spiral spectrum caused by the algorithm is negligible. We further investigate whether the distance between the receiving plane and the target particle affects the spiral spectrum of the scattered field of the beam. Considering the divergence of the beam, the intensity distribution of scattered field at different transmission distances is shown in Figure 9, in which the topological charge is \(l=3\). It can be seen that, as the transmission distance increases, the spot size of beam increases and the beam energy decreases. Therefore, when the measurement plane is set at the receiving end, the divergence of the beam needs to be fully considered, so as to avoid that the energy of field cannot be completely received due to the insufficient size of the measurement plane, and then the obtained OAM spectrum is not accurate. Figure 6: (**a**-**e**) present the one-dimensional intensity of the scattered field in the polarization direction, where the beam is incident with topological charges of 0-4; (**f**) is the phase sampling results along the circumference. The scatterer is a uniform and isotropic spherical particle, the measurement plane is located at z = 50 \(\upmu\)m.  
Figure 8: The same as in Figure 4, but the measurement plane is located at z = 50 \(\upmu\)m.  
Figure 7: Weights of the OAM modes of the scattered field on the whole measurement plane: panels (**a–d**) correspond with the cases that the topological charges are 1–4, where the beam is incident. The scatterer is a uniform and isotropic spherical particle, and the measurement plane is located at z = 50 \(\upmu\)m.  
Figure 10 shows the spiral spectrum of the scattered field for the receiving plane located at z = 1000 \(\upmu\)m. The scatterer is a homogeneous and isotropic dielectric spherical particle. Compared with Figure 7 and the relative error between them is shown in Table 2, where the detection probability of the OAM modes of the scattered field at z = 50 \(\upmu\)m is set to the true value. It is seen that the maximum relative error is 6.38%, and the maximum magnitude of absolute error is \(10^{-3}\). Therefore, the position of the receiving plane in the far region does not affect the spiral spectrum of the scattered fields. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**OAM Modes**} & \multicolumn{4}{c}{**Relative Error Caused by the Algorithm**} \\  & _I = 1_ & _I = 2_ & _I = 3_ & _I = 4_ \\ \hline \(-\)5 & - & - & - & - \\ \(-\)4 & - & - & - & - \\ \(-\)3 & - & - & - & - \\ \(-\)2 & - & - & - & - \\ \(-\)1 & 2.12\% & - & - & - \\ \(0\) & - & 3.72\% & - & - \\ \(1\) & \(-\)0.21\% & - & 3.03\% & - \\ \(2\) & - & \(-\)0.40\% & - & 3.53\% \\ \(3\) & 5.99\% & & \(-\)0.50\% & - \\ \(4\) & - & 5.31\% & - & \(-\)0.30\% \\ \(5\) & - & - & 4.88\% & - \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Relative error in the detection probability of the OAM modes at the source and at z = 50 \(\upmu\)m for the incident field with topological charge \(l\). Detection errors are caused by the calculation algorithm of the field. Figure 9: The intensity distribution of the scattered field at different transmission distances, where the topological charge is \(l\) = 3 and the other parameters are the same as Figure 5: panels (**a–d**) correspond to the transmission distances of z = 50 \(\upmu\)m, z = 200 \(\upmu\)m, z = 500 \(\upmu\)m and z = 1000 \(\upmu\)m, respectively. it can be seen that the energy of the incident beam has been substantially reduced if it is transmitted to the position at which the receiving plane is located. Figure 12 shows the position of the maximum intensity of the total external field in the polarization direction (Figure 12a), the sampling phase on the circumference where the maximum is located (Figure 12b), and the spiral spectrum results of the whole external field (Figure 12c-f).