## 3 Results and Discussion
From Figure 4, it can be gleaned that, at the same propagation distance, the signal OAM state of the beam with a long wavelength has a high detection probability in the receiver plane. The results indicate that the long wavelengths have slower phase accumulation; thus, the disturbance of the signal OAM state by the turbulent atmosphere is slight, and the purity of the signal OAM state is higher. One can see that the OAM state carried by the non-uniformly correlated high-order Bessel-Gaussian beams at the wavelength of \(\lambda\) = 1550 nm propagating in a turbulent atmosphere is superior to that of the beams at the shorter wavelengths, and the wavelength \(\lambda\) = 1550 nm is also the atmospheric window in FSO communications. Figure 2: Detection probability of the signal OAM state for the non-uniformly correlated high-order Bessel–Gaussian beams at the different receiving aperture diameters (D). Figure 5 shows the detection probability of the signal OAM state carried by the non-uniformly correlated high-order Bessel-Gaussian beams influenced by the beam width at the different propagation distances. As shown in Figure 5, the detection probability decreases when the propagation distance and beam width increase; this phenomenon can be attributed to the turbulent atmosphere having an accumulative effect as the propagation distance increases. Figure 4: Detection probability of the signal OAM state carried by the non-uniformly correlated high-order Bessel–Gaussian beams in a turbulent atmosphere for the different wavelengths studied. Figure 3: Detection probability of the different signal OAM states carried by the non-uniformly correlated high-order Bessel–Gaussian beams in a turbulent atmosphere. Figure 6 illustrates the relationship between the SNR of the signal OAM state and the beam width of the non-uniformly correlated high-order Bessel-Gaussian beams propagating in an atmospheric turbulence channel at the propagation distance of \(z=0.5\) km, for which the topological charges \(l_{0}\) of the signal OAM states are 1, 3, 5, and 7, respectively. As shown in Figure 6, the small-signal OAM state carried by the non-uniformly correlated high-order Bessel-Gaussian beam has a high SNR, and the non-uniformly correlated high-order Bessel-Gaussian beam carries a large topological charge \(l_{0}\); as a result, the spots of the beams become large, and the signal OAM state induced by the turbulent atmosphere becomes serious. Therefore, the purity of the signal OAM state decreases, and the SNR is large for the small topological charge under the same conditions; this result agrees well with Figure 3. The SNR of the signal OAM state is almost unchanged at a small beam width. As the beam width increases, the SNR of the signal OAM state decreases. These results agree well with the large spots of the beams influenced seriously by the turbulent atmosphere. The different signal OAM states have the same SNR when the non-uniformly correlated high-order Bessel-Gaussian beams have a very large beam width. Figure 7 plots the average BER of the different signal OAM states carried by the non-uniformly correlated high-order Bessel-Gaussian beams in a turbulent atmosphere at a propagation distance of \(z=0.5\) km. As shown in Figure 6, the variation trend of the average BER of the signal OAM state is opposite to that of the SNR of the signal OAM state shown in Figure 6. This result agrees well with the low BER of the signal OAM state at a high SNR. Figure 8 shows the SNR and average BER of the signal OAM state carried by the non-uniformly correlated high-order Bessel-Gaussian beams affected by the strength of a turbulent atmosphere. As shown in Figure 8, the SNR of the signal OAM state decreases when the strength of the turbulent atmosphere and the beam width increase (see Figure 8a); this phenomenon can be interpreted as the purity of the signal OAM state decreasing due to the strong effect of the turbulent atmosphere.