## 1 Introduction
According to Equations (2) and (3), it is evident that the phase function response to atmospheric turbulence at a certain point on the receiving cross-section is related to the shock response function \(\delta_{n}(\overrightarrow{r},z)\) and the wave number \(k=2\pi/\lambda\). Therefore, the wavefront phase aberration of beams with different wavelengths after turbulence will differ. This dispersion non-equal halo error between the phases of beams of different wavelengths is represented by \(\Delta\phi\) and can be expressed as:  
\[\Delta\phi\left(\overrightarrow{r}\right) =\phi_{\lambda_{1}}-\phi_{\lambda_{2}} \tag{4}\] \[=k_{1}\int_{0}^{L}\delta_{n}(\overrightarrow{r},z)dz-k_{2}\int_{ 0}^{L}\delta_{n}(\overrightarrow{r},z)dz\]  
Based on Equation (4), it can be observed that \(\Delta\phi\) is affected by the wavelength difference, transmission distance between the two points in the received cross-section, and atmospheric turbulence. In regions of strong turbulence, the wavefront phase changes rapidly, which in turn leads to rapid and random changes in the aberration difference between the two wavefronts. These changes do not satisfy the conditions of a strictly smooth stochastic process [16]. The average value can only be maintained constant over a fairly short period. To solve this problem, instead of studying the difference value \(\Delta\phi\left(\overrightarrow{r}\right)\) directly, we examine the rate of change in the difference value, denoted as \(\Delta\phi\left(\overrightarrow{r}\right)-\Delta\phi\left(\overrightarrow{r },\Delta t\right)\). We consider the relative rate of change to satisfy the conditions of a smooth stochastic process, and the values that change slowly can be described by the structure function \(D_{\Delta\phi}\left(\overrightarrow{\rho}\right)\) of the phase difference as follows:  
\[D_{\Delta\phi}\left(\overrightarrow{\rho}\right)=\left\langle\left[\Delta \phi\left(\overrightarrow{\rho}+\overrightarrow{r}\right)-\Delta\phi\left( \overrightarrow{r}\right)\right]^{2}\right\rangle \tag{5}\]  
where \(\overrightarrow{\rho}\) denotes the position vector between two points on the receiving cross-section and \(\left\langle\cdot\right\rangle\) represents ensemble average. From Equation (5), it can be observed that the total amount of the phase structure function is composed of a stabilizing quantity and small portion that exhibits slight variations. The spatial distribution of these variations in the receiving cross-section can be expressed as follows:  
\[ \begin{array}{ll}D_{\Delta\phi}\left(\overrightarrow{\rho}\right)&=\left\langle \left[\Delta\phi\left(\overrightarrow{\rho}+\overrightarrow{r}\right)- \Delta\phi\left(\overrightarrow{r}\right)\right]^{2}\right\rangle\\ &=2\left[B_{n}\left(\overrightarrow{\rho}\right)-B_{n}\left(\overrightarrow {r}\right)\right]\end{array}  \tag{6}\]  
where \(B_{n}\) denotes the mutual coherence function. The phase response function is defined as the response to a unit impulse that is uniformly distributed across the receiving surface. At this point, the impact of atmospheric turbulence on the optical phase front can be described by the distribution of refractive index variations, which establishes a relationship between the structure function of phase difference and structure function for the random distribution of refractive index. This relationship is denoted as follows:  
\[ \begin{array}{ll}D_{\Delta\phi}\left(\overrightarrow{r}\right)&=\left\langle \left[\left(k_{1}\int_{0}^{L}\delta_{n}\left(\overrightarrow{r}_{1},z\right) dz\right)-\left(k_{2}\int_{0}^{L}\delta_{n}\left(\overrightarrow{r}_{2},z \right)dz\right)\right]^{2}\right\rangle\\ &=(k_{1}-k_{2})^{2}\left\langle\left[\left(\int_{0}^{L}\delta_{n}\left( \overrightarrow{r}_{1},z\right)dz\right)-\left(\int_{0}^{L}\delta_{n}\left( \overrightarrow{r}_{2},z\right)dz\right)\right]^{2}\right\rangle\\ &=(k_{1}-k_{2})^{2}D_{n}\left(\overrightarrow{r}\right)\end{array}  \tag{7}\]  
where \(D_{n}\left(\overrightarrow{r}\right)\) denotes the phase structure function of atmospheric turbulence.