## 3 Results
(a1, c1, e1) and (b1, d1, f1) are intensity profiles without pre-compensation for OAM beams with \(l=3\) and \(l=5\) respectively, (a3, c3, e3) and (b3, d3, f3) denote their corresponding phase distributions, (a5, c5, e5) and (b5, d5, f5) show their corresponding PD spectra; (a2, c2, e2) and (b2, d2, f2) are intensity profiles with the pre-compensation of GS algorithm for OAM beams with \(l=3\) and \(l=5\) respectively, (a4, c4, e4) and (b4, d4, f4) are their corresponding phases, (a6, c6, e6) and (b6, d6, f6) their corresponding PD spectra. We have explored the case of transmitting LG beam with single OAM state. In the actual OAM-based optical communication system, based on the orthogonal property, the multiplexing vortex beam with multiple OAM modes (topological charges) is usually adopted to represent more data bits. Here, we have also explored the case of transmitting coherent multiplexing LG beams under different turbulence channel by changing the refractive index structure parameter \(C_{n}^{2}\) from \(1\times 10^{-15}\)m\({}^{-2/3}\) to \(1\times 10^{-13}\)m\({}^{-2/3}\) and the simulated random phase screen number \(N\) from 20 to 100. In our simulations, two LG beams with topological charges of \(l=2\) and \(l=6\) are adopted for coherent multiplexing. The results under turbulence with refractive index structure parameter \(C_{n}^{2}=1\times 10^{-15}\)m\({}^{-2/3}\) are shown in Figure 6. \begin{table}
\begin{tabular}{c c c c c} \hline
**OAM Mode** & **W/o or W/ GS** & **N = 20** & **N = 50** & **N = 100** \\ \hline \(l=3\) & W/o GS & 27.73\% & 24.47\% & 15.16\% \\  & W/ GS & 43.43\% & 35.89\% & 35.08\% \\ \(l=5\) & W/o GS & 18.76\% & 10.60\% & 10.22\% \\  & W/ GS & 36.17\% & 23.31\% & 13.84\% \\ \hline \end{tabular}
\end{table} 
Table 3: The PD values of OAM beams without (W/o) and with (W) GS algorithm corresponding to Figure 5.  
Figure 5: The intensity and phase distributions of OAM beams and their PD spectra without and with the pre-compensation of GS algorithm under simulated atmospheric turbulence with refractive-index structure parameter \(C_{n}^{2}\) valued as \(1\times 10^{-13}\)m\({}^{-2/3}\). The parts framed in green, blue and red dashes represent random phase screens number \(N\) = 20, 50 and 100 respectively. (a1, c1, e1) and (b1, d1, f1) are intensity profiles without pre-compensation for topological charge \(l=3\) and \(l=5\) respectively, (a3, c3, e3) and (b3, d3, f3) are their corresponding phase, (a5, c5, e5) and (b5, d5, f5) are their corresponding PD spectra; (a2, c2, e2) and (b2, d2, f2) are intensity profiles with the pre-compensation of GS algorithm for \(l=3\) and \(l=5\) respectively, (a4, c4, e4) and (b4, d4, f4) demonstrate their corresponding phase distributions, (a6, c6, e6) and (b6, d6, f6) are their corresponding PD spectra accordingly. Sub-figures (a1, a4, a7), (c1, c4, c7) and (e1, e4, e7) of Figure 6 are their intensity distributions under the turbulence channel with three different simulated random phase screens without pre-compensation. Sub-figures (a2, a5, a8), (c2, c5, c8) and (e2, e5, e8) of Figure 6 are their corresponding phase distributions. Sub-figures (a3, a6, a9), (c3, c6, e9) and (e3, e6, e9) of Figure 6 are their corresponding PD spectra. Sub-figures (b1, b4, b7), (d1, d4, d7) and (f1, f4, f7), Sub-figures (b2, b5, b8), (d2, d5, d8) and (f2, f5, f8) and Sub-figures (b3, b6, b9), (d3, d6, d9) and (f3, f6, f9) of Figure 6 are the intensity and phase distributions and the PD spectra results with the pre-compensation of GS algorithm. And the PD values of OAM beams without (W/o) and with (W) GS algorithm corresponding to Figure 6 have demonstrated in Table 4. It can be seen that the distortion of multiplexing LG beam increases gradually when the refractive index structure parameter of \(C_{n}^{2}\) increases with the fixed phase screen number, which is similar to that of the single OAM beam. Meanwhile, with increasing phase screen number, the pre-compensation result is similar when the refractive structure parameter of \(C_{n}^{2}\) is fixed.