## 1 Introduction
Based on the relationship between the turbulent power spectrum and phase structure function, the following relationship for \(D_{n}\left(\overrightarrow{r}\right)\) can be obtained [17] as follows:  
\[ \begin{array}{ll}D_{n}\left(\overrightarrow{\rho}\right)&=2\left[B_{n} \left(\overrightarrow{r}+\overrightarrow{\rho}\right)-B_{n}\left( \overrightarrow{r}\right)\right]\\ &=8\pi\int_{0}^{\infty}\kappa^{2}\Phi(\kappa)\left[1-\frac{\sin(k\rho)}{k \rho}\right]dx\end{array}  \tag{8}\]  
where \(B_{n}\left(\overrightarrow{r}\right)\) represents the number of interrelationships between two points on the spot, \(\kappa\) denotes the number of spatial waves, and \(\Phi(\kappa)\) denotes the spatial power spectrum of the turbulence. The modified von-Karman power spectrum is used here, and it can be expressed as:  
\[\Phi(\kappa)=0.033C_{n}^{2}\frac{\exp\left(-\kappa^{2}/\kappa_{m}^{2}\right)}{ \left(\kappa^{2}+\kappa_{0}^{2}\right)^{11/6}},0\leq\kappa<\infty \tag{9}\]  
In the above equation, the relationship between the parameter \(\kappa_{0}^{2}\) and outer scale \(L_{0}\) of turbulence can be expressed as \(\kappa_{0}=c_{0}/L_{0}\), where \(c_{0}\) denotes a proportionality parameter. Typically, its value is \(8\pi\) in the region of strong turbulence. Furthermore, \(C_{n}^{2}\) denotes the atmospheric refractive index structure constant, which characterizes the strength of the turbulence and is a constant in horizontal transport. When \(C_{n}^{2}<10^{-17}\) m\({}^{-2/3}\), turbulence intensity is weak; \(C_{n}^{2}>10^{-13}\) m\({}^{-2/3}\) indicates strong turbulence intensity; and \(10^{-17}\) m\({}^{-2/3}<C_{n}^{2}<10^{-13}\) m\({}^{-2/3}\) corresponds to moderate turbulence intensity [17]. \(\kappa_{m}\) denotes a parameter that is related to \(l_{0}\) and can be expressed as \(\kappa_{m}=5.92/l_{0}\), where \(l_{0}\) denotes the internal scale of turbulence. The expansion \(\left[1-\frac{\sin(k\rho)}{k\rho}\right]\) in Equation (8) can be expressed using the Maclaurin series as [17]:  
\[\left[1-\frac{\sin(\kappa\rho)}{\kappa\rho}\right]=\sum_{n=1}^{\infty}\frac{ \left(-1\right)^{n-1}}{\left(2n+1\right)!}(\kappa\rho)^{2n} \tag{10}\]By substituting Equations (9) and (10) into Equation (8) and integrating, Equation (8) becomes [17]:  
\[ \begin{array}{l}D_{n}\left(\overrightarrow{\rho}\right)=1.685C_{n}^{2}\kappa_{ m}^{-2/3}\left[{}_{1}F_{1}\left(-\frac{1}{3};\frac{3}{2};-\frac{\kappa_{m}^{2} \rho^{2}}{4}\right)-1\right]\\ +1.050C_{n}^{2}\kappa_{0}^{-2/3}\left[1-{}_{0}F_{1}\left(-;\frac{2}{3};-\frac{ \kappa_{0}^{2}\rho^{2}}{4}\right)\right],\kappa_{0}\ll\kappa_{m}\end{array}  \tag{11}\]  
By substituting Equation (11) into Equation (7) and transforming, Equation (7) can be expressed as:  
\[ \begin{array}{l}D_{\Delta\rho}\left(\overrightarrow{r}\right)=1.685C_{n}^{2 }(k_{1}-k_{2})^{2}\kappa_{m}^{-2/3}\left[{}_{1}F_{1}\left(-\frac{1}{3};\frac{ 3}{2};-\frac{\kappa_{m}^{2}\rho^{2}}{4}\right)-1\right]\\ +1.050C_{n}^{2}(k_{1}-k_{2})^{2}\kappa_{0}^{-2/3}\left[1-{}_{1}F_{1}\left(1; \frac{2}{3};\frac{\kappa_{0}^{2}\rho^{2}}{4}\right)\right],\kappa_{0}\ll \kappa_{m}\end{array}  \tag{12}\]  
where function \({}_{1}F_{1}(a;c;x)\) is a type of confluent hypergeometric function. It can serve as an approximation for large parameters when the distance \(\rho\) between two points on the receiver's cross-section satisfies the condition \(l_{0}\ll\rho\ll L_{0}\). This approximation for large parameters is provided in [17]:  
\[{}_{1}F_{1}(a;c;-x)\sim\frac{\Gamma(c)}{\Gamma(c-a)}x^{-a} \tag{13}\]  
where \(\Gamma(\cdot)\) denotes the gamma function. When distance \(\rho\) between two points on the receiving cross-section satisfies the condition of \(\rho\ll l_{0}\). The following small parameter approximation can be used [17]  
\[{}_{1}F_{1}(a;c;-x)\sim 1-\frac{a}{c}x \tag{14}\]  
When the above-mentioned equation is combined with Equation (12), it can be observed that the beam wavefront aberration difference is not only related to the beam wavelength but also closely related to the inner and outer turbulence scales.