## 3 Simulation and Analysis
In the joint atmospheric turbulence model, the turbulence intensity in the troposphere is greater than that in the stratosphere, so the influence on the phase distribution in the long distance transmission is smaller than that in the short distance transmission below 6 km. Next, we calculated the crosstalk for each of transmitted OAM modes. Figure 4a-c show histograms of the OAM crosstalk found for 1 km, 5 km and 10 km, respectively. In each of the three cases, we present examples that correspond to the crosstalk generated by transmitting on an isolated channel with \(l=1\), 3, or 5. Each subplot corresponds to a row in topological charge \(l\), such as \(l=1\), and the electromagnetic vortex wave transmits in moderate-to-strong turbulence, the proportion of the dominant mode gradually decreases from 0.898 to 0.180. When the transmission distance increases to 10 km, the dominant mode is submerged by the crosstalk mode. From Figure 4 we can see a trend that transmitted OAM mode transfers energy into the neighboring modes along with transmission distance, and in the case of the weak turbulence, mode crosstalk occurs mainly in adjacent modes, whereas in the case of moderate-to-strong turbulence, mode crosstalk gradually expands to other peripheral modes. Furthermore, for each of the sending OAM modes, the effect of turbulence on mode transmission in turbulence channels is gradually unstable. Under moderate-to-strong turbulence,  
Figure 2: Strength of atmospheric turbulence. (**a**) Weak. (**b**) Moderate. (**c**) Strong. because the electromagnetic wave is inevitably influenced by turbulence disturbance, the receiving end cannot correctly identify the mode of transmission. Therefore, the mode crosstalk caused by atmospheric turbulence needs to be compensated for by obtaining the right mode. Because the wavelength of microwave band is much larger than that of light band, the effect of weak turbulence on electromagnetic vortex waves is small. Therefore, only the results of compensation for the influence of moderate-to-strong turbulence on electromagnetic vortex waves are given. Figure 5a-c displays the intensity and phase distribution of electromagnetic vortex waves with adaptive compensation found for 1 km, 5 km and 10 km, respectively. After the adaptive compensation, the intensity of beam increased significantly and the distribution became uniform. The distortion degree of the isophase line was greatly improved, which indicates that the phase distortion is effectively compensated. This shows that the adaptive compensation scheme is very effective for electromagnetic vortex wave distortion correction with a single topological number. Figure 3: Intensity and phase distribution of electromagnetic vortex waves with the transmission distance (**a**) 1 km, (**b**) 5 km and (**c**) 10 km under moderate-to-strong turbulence conditions. Figure 4: Variance of crosstalk with the transmission distance (**a**) 1 km, (**b**) 5 km and (**c**) 10 km under moderate-to-strong turbulence conditions. Figure 5: Intensity and phase distribution of electromagnetic vortex waves with transmission distances of (**a**) 1 km, (**b**) 5 km and (**c**) 10 km, with adaptive compensation. Figure 6a-c show histograms of the OAM crosstalk within adaptive compensation found for 1 km, 5 km and 10 km, respectively. In each of the three cases, we present examples that correspond to the crosstalk generated by transmitting on an isolated channel with \(l=1\), 3, or 5. Each subplot corresponds to a row in topological charge \(l\), such as \(l=1\), and the electromagnetic vortex wave transmits in moderate-to-strong turbulence. The proportion of the dominant mode gradually decreases from 1 to 0.382. In Figure 6a, when a single OAM mode is compensated for the influence of turbulence by the adaptive compensation method, crosstalk among channels is negligible. In comparing Figure 4b with Figure 6b, when topological charge \(l=1\), the proportion of dominant mode before and after adaptive compensation is increased from 0.882 to 1.