## 3 Simulation and Analysis
Comparing Figures 4c and 6c, when the transmission distance increases to 10 km, the mode of transmission is increased from 0.180 to 0.382, and at this point, the transmitted OAM mode is still the main component after adaptive compensation. For long distance transmission (5 km-10 km), the OAM mode stability becomes worse along with the order number increase of the topological charge \(l\). Therefore, the adaptive compensation can effectively increase the proportion of the main mode, and the crosstalk caused by turbulence can be reduced. Figure 7 displays the intensity and phase distribution of electromagnetic vortex waves with the transmission distance 1 km, 5 km and 10 km for multiplexing \(l=1\), 3, 5 under moderate-to-strong turbulence conditions. It can be seen from the two-dimensional intensity distribution diagram of OAM beam multiplexing that, in the process of turbulent transmission, the intensity of the electromagnetic vortex waves decreases and the intensity spot diffuses due to the influence of the turbulence effect. The intensity distribution deforms and the power is dispersed gradually with the increasing transmission distance. Due to atmospheric disturbance, the phase distortion of the OAM multiplexed beam is different, and the phase diffuses from the center to the periphery. The isophase lines gradually bend and even blur out of shape, and it is difficult for the receiving end to separate the effective information carried by the OAM mode. Figure 6: Variance of crosstalk through the weak-to-strong turbulence with transmission distance of (**a**) 1 km, (**b**) 5 km and (**c**) 10 km, with adaptive compensation. For the whole OAM multiplexing system, electromagnetic vortex waves transmitting into the atmosphere will be affected by atmospheric turbulence, so it needs to be compensated to improve the communication quality. With adaptive compensation, the intensity and phase distribution of electromagnetic vortex waves with the transmission distance 1 km, 5 km and 10 km under moderate turbulence conditions are shown in Figure 8. The intensity of multiplexed OAM modes is increased and the shape is restored. The distortion degree of isophase lines are improved, and they become smooth and tidy. Therefore, the intensity and phase effects of atmospheric turbulence on OAM multiplexing can be compensated well by the method of adaptive compensation. Figure 9 displays the crosstalk distribution of electromagnetic vortex waves with transmission distances of 1 km, 5 km and 10 km under moderate-to-strong turbulence conditions. The first line in Figure 9 is the crosstalk probability distribution of the OAM multiplexed beam transmitted in turbulence. After the transmission of the OAM multiplexed beam through atmospheric turbulence, its purity is damaged. At this time, all the OAM modes are doped with other modes with different topological charges to varying degrees. The singularity occurs when the topological charge is 0. Due to large mode crosstalk among different modes, each beam no longer maintains the original distribution state. Then, the dispersion between adjacent modes is serious, and the proportion of OAM modes with topological charges of 2, 4 and 6 have a high percentage. The second row of graphs in Figure 9 shows the crosstalk distribution of electromagnetic vortex waves with the increasing transmission distance after adaptive compensation. The purity of the OAM mode remains within its own transmission mode, and the crosstalk modes are of smaller proportions. At this moment, the mode information can be judged from the mode of the receiving end. Figure 7: Intensity and phase distribution of electromagnetic vortex waves with the transmission distance (**a**) 1 km, (**b**) 5 km and (**c**) 10 km for multiplexing \(l\) = 1, 3, 5 under moderate-to-strong turbulence conditions. We show the signal-to-crosstalk ratio with transmission path in Figure 10. The signal shows a weakening trend along with the transmission path, whereas the crosstalk shows the opposite trend.