## 2 Theory and Method
The expression of light intensity distribution on the final optical axis is  
\[I(0,z)=|u(0,z)|^{2}=\frac{2\pi k}{(z-\gamma)z}(|C|\gamma z)^{\frac{2}{2-\gamma }}e^{-\frac{2}{\omega_{0}^{2}}(|C|\gamma z)^{\frac{2}{2-\gamma}}}. \tag{10}\]  
To find the intensity maximum, we compute the first derivative:  
\[u(0,z)=\frac{k}{iz}e^{ikz}\int_{0}^{\infty}e^{-\frac{\rho^{2}}{\epsilon_{0}}}e^{ ik(|C\frac{\rho}{\epsilon_{0}}\gamma+\frac{\rho^{2}}{\epsilon_{0}}|)}pd\rho. \tag{2}\]0Because \(z\
eq 0\), \(z_{peak\;intensity}\) denotes the position of the peak intensity on the optical axis, which is  
\[u(0,z)=\frac{k}{iz}e^{ikz}\int_{0}^{\infty}e^{-\frac{\rho^{2}}{\epsilon_{0}}}e^{ ik(|C\frac{\rho}{\epsilon_{0}}\gamma+\frac{\rho^{2}}{\epsilon_{0}}|)}pd\rho. \tag{2}\]1  
Equation (12) shows that the peak intensity position of the OPB depends on the shape factor \(\gamma\), waist radius \(\omega_{0}\), and modulation depth \(C\). This enables generating different OPB with identical peak intensity positions by adjusting \(\gamma\), \(\omega_{0}\), and \(C\). The analytical model (Equations (1)-(12)) characterizes OPB propagation in vacuum. In the context of weak fluctuation conditions that satisfy the [PERSON] approximation, atmospheric turbulence is incorporated via the stochastic phase perturbation \(\psi(r,z)\) in \(u(r,z)=u_{0}(r,z)e^{\psi(r,z)}\). Here, \(u_{0}(r,z)\) denotes the field in the absence of turbulence, and \(\psi(r,z)\) represents the perturbation to the turbulent phase. While turbulence distorts the wavefront and shifts the focal position, it does not redefine the fundamental focusing mechanism governed by \(\gamma\) and \(C\). The scintillation index \(\sigma_{I}^{2}\) is defined as the normalized variance of irradiance fluctuations [25], which is given by  
\[u(0,z)=\frac{k}{iz}e^{ikz}\int_{0}^{\infty}e^{-\frac{\rho^{2}}{\epsilon_{0}}}e^{ ik(|C\frac{\rho}{\epsilon_{0}}\gamma+\frac{\rho^{2}}{\epsilon_{0}}|)}pd\rho. \tag{2}\]2  
Here, the symbol \(I_{S}\) denotes the received power in the receiving aperture. OPB has the capability for stable energy over transmission distances, so it inherently resists atmospheric turbulence. These characteristics hold significant research value for FSOC. This study investigates the scintillation behavior of OPB under varying turbulence conditions and optimizes the system design for practical applications. The experiment utilized a rectangular sampling screen with a length of 559.8 mm, and the number of sampling points was set to \(N=1024\). The 1550 Nm Gaussian beam with a waist radius \(\omega_{0}=15\) mm is generated and collimated. The beam has been modulated in OPB phase using either a spatial light modulator or a phase mask. Following modulation, the OPB is expanded by a factor of 9.33 through an expansion system before transmitting into space. At the receiver, an aperture-restricted screen captures the far-field spot to analyze scintillation effects (Figure 1). The experiments compare the scintillation performance of OPB and conventional Gaussian beams (GB) under the same turbulence conditions. By simulating atmospheric turbulence and calculating the ARP and SI, this work proposed a method to design the OPB to suppress light intensity scintillation. The results demonstrate that OPB exhibits superior stability in scenarios with turbulence, offering a viable solution for enhancing the reliability of FSOC systems. \[u(0,z)=\frac{k}{iz}e^{ikz}\int_{0}^{\infty}e^{-\frac{\rho^{2}}{\epsilon_{0}}}e^{ ik(|C\frac{\rho}{\epsilon_{0}}\gamma+\frac{\rho^{2}}{\epsilon_{0}}|)}pd\rho. \tag{2}\]3 Schematic of the OPB spatial transmission methodology under atmospheric turbulence.