## 3 Numerical Simulation and Analysis
Within the self-focusing regime, the scintillation index of the OPB initially decreases and then increases. Therefore, the greater the turbulence, the greater the reduction in the scintillation index. In reality, as turbulence intensifies, the scintillation index increases overall along the entire propagation trajectory. Figure 5: Comparative Far-Field Beam Profiles of OPB and GB: Turbulent (\(D/r_{0}=5\)) vs. Non-Turbulent Conditions. In the entire propagation path, SI is calculated shown in Figure 7. The OPB with a focal length of 1500 m shows superior performance compared to the Gaussian beams within a reception range of 2000 m, as their scintillation factor is consistently lower in most cases. The SI along the propagation path shows a characteristic trend of increasing and then decreasing. In particular, the far-field SI of the OPB is minimized under conditions where the shape factor \(\gamma\) and the modulation depth \(C\) are maximized in the transmission path. Furthermore, the scintillation factor increases more rapidly with increasing turbulence intensity. Particularly, the minimum scintillation factor is observed after the beam passes through the focal point, as shown by the lowest SI at 2000 m. In addition to the SI, the ARP is a critical parameter for investigating the far-field characteristics of beams. Therefore, the ARP of the six beam types (five OPBs and one Gaussian beam) was calculated across the entire propagation path under the four turbulence conditions (\(D\)/\(r_{0}\) = 5, \(D\)/\(r_{0}\) = 10, \(D\)/\(r_{0}\) = 15, and \(D\)/\(r_{0}\) = 20). The results are presented in Figure 8. Compared to the Gaussian beam, the OPB with a focal length of 1500 m consistently shows a higher ARP within the 2000 m reception range. The ARP of the OPB shows a characteristic trend of increasing and then decreasing along the propagation path. In particular, the maximum ARP is observed between 200 and 600 m. Furthermore, as the turbulence intensity increases, the position of the maximum ARP shifts towards shorter distances. It is also observed that larger values of shape factor \(\gamma\) and modulation depth \(C\) result in the highest far-field ARP for the OPB. Therefore, it is essential to consider the shape factor \(\gamma\) and modulation depth \(C\) as critical parameters when designing OPB with the same focal length. In particular, larger values of shape factor \(\gamma\) and modulation depth \(C\) result in higher ARP over the entire propagation path and lower SI at the focal point. Figure 6: Comparison of SI between OPB and GB in the focal distance (1500 m) under Varying Turbulence Conditions (\(D\)/\(r_{0}\) = 5, \(D\)/\(r_{0}\) = 10, \(D\)/\(r_{0}\) = 15, and \(D\)/\(r_{0}\) = 20). Figure 8: Statistics diagrams of ARP at different transmission locations, GB and OPBs (**a**) S \(D/r_{0}=5\) (b) \(D/r_{0}=10\) (**c**) S \(D/r_{0}=15\) (**d**) \(D/r_{0}=20\). Figure 3.0 Statistics diagrams of SI at different transmission locations, GB and OPBs (**a**) S \(D/r_{0}=5\) (b) \(D/r_{0}=10\) (**c**) S \(D/r_{0}=15\) (**d**) \(D/r_{0}=20\). To further investigate the OPB's performance under long-range propagation conditions (2-10 km), we conducted additional simulations by adjusting the beam's focal parameters (\(\gamma\) = 0.8, \(C\) = 0.000049) while maintaining consistent system configurations. The result is in Figure 3.1  
The results demonstrate that the OPB can maintain relatively low scintillation levels over an extended range, with the scintillation index exhibiting a characteristic pattern of initial increase followed by reduction and subsequent rise at longer distances. So, OPB has great potential for applications in suppressing intensity scintillation over longer distances. To further evaluate OPB's performance in long-haul transmission, the link distance was extended to 15 km. With the transmitter aperture held constant, the receiver aperture was increased to 70 mm and the sampling screen expanded to 2099.25 mm, implementing a sampling grid of \(N\) = 1024 \(\times\) 1024 points.