## 3 Numerical Simulation and Analysis  
As demonstrated in Section 2, the theoretical analysis shows numerous OPB types at a given focal distance. This section aims to explore how adjusting beam parameters enables OPBs to achieve optimal scintillation suppression effectiveness. The numerical simulations will be employed to demonstrate and compare their scintillation suppression capabilities. Firstly, we systematically investigate the focal distance of the OPB under two distinct parametric configurations: (1) with a fixed beam modulation depth \(C=0.00005\), varying the shape factor \(\gamma\) (0.3-1.2, step 0.1) and beam waist radius \(\omega_{0}\)\(\in\)[40 mm, 80 mm, 120 mm]; and (2) with a fixed \(\omega_{0}\) = 70 mm, scanning \(\gamma\) over the same range while adjusting the modulation depth \(C\in\) {0.00006, 0.00008, 0.0001}. Locate the position of peak intensity using the spatial transmission simulation method without turbulence and compare it with the value calculated from Equation (12). The simulation results are presented in Figure 2. Comparative analysis of the focal positions calculated by both methods shows a maximum deviation of less than 200 m for the OPB. This consistency validates the formula-based approach as a reliable method for designing OPB in practical applications. In order to examine the distinct propagation patterns, the comparative analysis was conducted of OPB and GB under both turbulence and turbulence-free conditions. The OPB phase used is shown in Figure 3. Far-field spot distributions for all four configurations were  
Figure 2: Comparison of OPB focal distances calculated using the spatial transmission method and Equation (12). measured at 400 m intervals under turbulence conditions (\(D/r_{0}=5\)), with results illustrated in Figures 4 and 5. The oscillating white curve traces the on-axis normalized intensity \(lI_{0}\) versus propagation distance, capturing turbulence-induced signal fluctuations along the beam path. It is exhibited that atmospheric turbulence disrupts OPB self-focusing, inducing premature peak intensity extrema along the propagation axis (900 m earlier than transmission without turbulence conditions). In contrast, GB display distinct propagation dynamics. They sustain continuous divergence profiles and avoid forming focal points during transmission (Figure 4c). However, turbulent environments induce intensity fluctuations. This phenomenon directly accounts for the apparent quasi-focal behavior observed in the statistical ARP of GB, originating from turbulence-induced phase distortion effects rather than intrinsic self-focusing mechanisms. Figure 4: Comparative Schematic Analysis of OPB and GB Transmission Characteristics under Turbulent (\(D/r_{0}=5\)) vs. Non-Turbulent Conditions. Figure 3: Diagrams of phase profile for OPB. In order to investigate the selection of OPB with the same focal length in practical applications, a study was conducted in which five types of OPB with a focal length of 1500 m were designed and their far-field scintillation characteristics were studied. Firstly, the five designed OPB with a focal length of 1500 m are propagated together with a GB under four turbulence conditions: \(D/r_{0}=5\), \(D/r_{0}=10\), \(D/r_{0}=15\), and \(D/r_{0}=20\). Data are collected at 50 m intervals along the propagation path. The receiving aperture is 50 mm. For a more intuitive analysis of the SI values of the OPB and the GB at the focal point (specifically, at 1500 m), Figure 6 is presented. The outcomes reveal that the OPB, characterized by maximum modulation depth and shape factor, exhibits the least intense scintillation across various turbulence scenarios. Notably, as turbulence intensity increases, the scintillation index at the focal point does not continually escalate. This phenomenon is primarily due to turbulence impairing the OPB\({}^{\prime}\)s self-focusing property. Consequently, the focal point emerges earlier, and the entire self-focusing region shifts anteriorly.