## 1 Introduction
The wavefront sensor fails to capture the continuous phase distribution within the strong turbulence region, rendering the phase relationship unusable for correcting wavefront aberrations. Additionally, conventional geometric optics analysis proves insufficient for analyzing wavefront phase across different wavelengths. Consequently, we opted for the statistical optics method, also termed as the phase structure function method. In statistical optics, the coefficient average, also referred to as the ensemble average, can be considered equal to the time average of a single stochastic process under the preconditions of a generalized stochastic smooth process. Leveraging the definitions of the coefficient average and root mean square (RMS), we derive the RMS of the error between beams of varying wavelengths by squaring the mean square value of Equation (12), which can be expressed as:  
\[\text{RMS}=\sqrt{D_{\Delta\rho}\left(\overrightarrow{r}\right)} \tag{15}\]  
It is assumed that the values of the outer scale of turbulence \(L_{0}\) correspond to 20 m, 30 m, 50 m, 60 m, or 80 m, the turbulence inner scale parameter \(l_{0}\) is 0.1 m, and the atmospheric refractive index structural parameter value is \(C_{n}^{2}=1\times 10^{-12}\) m\({}^{-2/3}\). When signal light with a wavelength of 1550 nm and beacon light with a wavelength of 632.8 nm are transmitted in the atmospheric channel in a co-optical path with \(L=\) 10 km, the variation curves of the difference RMS in the wavefront phase aberration between different points on the received cross-section are shown in Figure 1. Figure 1 illustrates the beam wavefront aberration difference at the receiving end cross-section with the augmentation of the outer scale of turbulence when the distance between the two points is held constant. Notably, beyond an outer scale of 40 m, the rate of change in wavefront aberration diminishes, suggesting a gradual reduction in the impact of the turbulence outer scale on beam modulation. Furthermore, when the outer scale of turbulence is fixed, an increase in the distance between the two points at the receiver does lead to a higher wavefront aberration at the receiver's cross-section; however, this overall variation magnitude remains relatively small. This observation underscores that, while different distances between two points influence the wavefront aberration at the receiver's cross-section to some extent, the effect is not substantial. It is assumed that the turbulence inner scale \(l_{0}\) values range from 0.01 m to 0.15 m and from 0.2 m to 0.4 m, with a turbulence outer scale parameter of \(L_{0}\) = 40 m and an atmospheric refractive index structural constant value of \(C_{n}^{2}=1\times 10^{-12}\) m\({}^{-2/3}\). When signal light with a wavelength of 1550 nm and beacon light with a wavelength of 632.8 nm are transmitted through the atmospheric channel on a co-optical path with \(L\) = 10 km, the influence of the turbulence inner scale on the RMS difference in wavefront phase aberration across various points on the received cross-section is shown in Figure 2. Upon comparing Figure 2a,b, it becomes apparent that when the separation between the two points is smaller than the inner scale of turbulence, there is a discernible effect on the wavefront phase aberration difference. As the distance between the points increases and approaches the inner scale of turbulence, this effect gradually diminishes, and the curves begin to overlap significantly. Subsequently, when the distance between the points surpasses the inner scale of turbulence, the wavefront phase aberration differences stabilize and the curves' slopes exhibit no significant changes. This outcome may be attributed to beam diffraction within small-scale turbulence. However, as the distance between the points extends beyond the inner scale of turbulence, the light undergoes refraction once more, leading to a diminished influence of small-scale turbulence on the beam.