## 3 Results
From the PD spectra and the concrete values shown in Figure 6 and Table 4 respectively, when the value of \(C_{n}^{2}\) is no more than 1\(\times\)10\({}^{-14}\)m\({}^{-2/3}\), no matter how many the simulated random phase screens inserted, the pre-compensation of the GS algorithm can effectively reduce the crosstalk, and the OAM mode value of the transmitted multiplexing LG beam can be effectively judged according to the concrete spectrum distribution. However, when the value of \(C_{n}^{2}\) increase to 1\(\times\)10\({}^{-13}\)m\({}^{-2/3}\), which reaches a strong turbulence strength level, the introduced crosstalk is relatively larger and exceeds the compensation limit range of the GS algorithm; thus, the OAM mode cannot be accurately detected. Figure 6: The intensity and phase distributions of multiplexing OAM beams (\(l=2\) and \(l=6\)) and their PD spectra without and with the pre-compensation of GS algorithm under simulated atmospheric turbulence with different refractive-index structure parameter of \(C_{n}^{2}\) valued as \(1\times 10^{-15}\)m\({}^{-2/3}\), \(1\times 10^{-14}\)m\({}^{-2/3}\) and \(1\times 10^{-13}\)m\({}^{-2/3}\) (corresponding to the first, second and third horizontal line). The parts framed in green, blue and red dashes represent random phase screens number \(N=20\), 50 and 100 respectively. (a1, a4, a7), (c1, c4, c7) and (e1, e4, e7) are intensity profiles without pre-compensation, (a2, a5, a8), (c2, c5, c8) and (e2, e5, e8) demonstrate their corresponding phase distributions, (a3, a6, a9), (c3, c6, c9) and (e3, e6, e9) show their corresponding PD spectra; (b1, b4, b7), (d1, d4, d7) and (f1, f4, f7) are intensity profiles with the pre-compensation of GS algorithm; (b2, b5, b8), (d2, d5, d8) and (f2, f5, f8) depict their corresponding phase distributions, (b3, b6, b9), (d3, d6, d9) and (f3, f6, f9) correspond to their PD spectra. In addition, by comparing the above case of transmitting single OAM beam with that of multiplexing LG beam, we can find that, for the transmission of single OAM beam, under the same turbulence strength level, the PD of the target OAM mode after retrieving by the GS algorithm is significantly higher than that of the crosstalk channel. And when the turbulence strength does not exceed the level corresponding to the refractive index structure parameter of \(C_{n}^{2}=1\times 10^{-13}\)m\({}^{-2/3}\) with the number of simulated random phase screens \(N=20\), the GS algorithm can effectively compensate the distorted LG beam and the transmitted target OAM mode can be precisely determined according to the PD spectra distributions. However, for the transmission of multiplexing LG beam, since multiple channels will introduce more internecine crosstalk, especially when the interval between different OAM modes is not very large [18], when the \(C_{n}^{2}\) only increases to \(1\times 10^{-14}\)m\({}^{-2/3}\) with \(N=100\), the limit range that can be corrected by the GS algorithm is reached. This indicates that the retrieving performance of transmitting single LG beam is better than that of transmitting multiplexing LG beam. Simultaneously, we can conclude that for the transmission system in simulation, the maximum turbulence strength level that can be corrected by the GS algorithm is the level corresponding to the refractive index structure parameter valued as \(C_{n}^{2}=1\times 10^{-14}\)m\({}^{-2/3}\) with the random phase screen number valued as \(N=100\). To further investigate the limitation of the phase screen number, we have simulated the case of turbulence channel with phase screen number of \(N=150\). The corresponding performances of LG beams (\(l=5\)) with and without pre-compensation of GS algorithm under simulated atmospheric turbulence with \(C_{n}^{2}=1\times 10^{-15}\)m\({}^{-2/3}\) and \(C_{n}^{2}=1\times 10^{-14}\)m\({}^{-2/3}\) are shown in Figure 7. This shows that the PD spectra of distorted LG beams are 54.80% and 14.66% respectively for \(C_{n}^{2}=1\times 10^{-15}\)m\({}^{-2/3}\) and \(C_{n}^{2}=1\times 10^{-14}\)m\({}^{-2/3}\).