## 1 Introduction
### Theory  
Zernike polynomials are a set of mutually orthogonal polynomials defined within a unit circle. Their coefficients, which are linear combinations of basis functions, can accurately describe different types of wavefront aberrations. Wavefront aberration, \(\phi(\rho,\theta)\), can be expressed as a linear combination of Zernike polynomials [17]:  
\[\phi(\rho)=\sum_{i=1}^{N}a_{i}z_{i}(\rho,\theta) \tag{16}\]  
where \(N\) denotes the total aberration order, \(a_{i}\) denotes the coefficient of the Zernike polynomial of order \(i\), \((\rho,\theta)\) denotes the polar coordinate, and \(Z_{i}(\rho,\theta)\) denotes the expression of the Zernike polynomial of order \(i\) in polar coordinates as follows:  
\[Z_{i}=\left\{ \begin{array}{ll}\sqrt{n+1}R_{n}^{m}(\rho)\sqrt{2}\cos(m\theta )&m\
eq 0\text{ and }m\text{ {\it is even}}\\ \sqrt{n+1}R_{n}^{m}(\rho)\sqrt{2}\sin(m\theta)&m\
eq 0\text{ and }m\text{ {\it is odd}}\\ \sqrt{n+1}R_{n}^{m}(\rho)&m=0\end{array} \right. \tag{17}\]  
Figure 3: The difference in beam wavefront phase distortion between different points on the receiving cross-section. **(a)** Different wavelengths. **(b)** Difference between different wavelengths. where  
\[R_{n}^{m}(\rho)=\sum_{s=0}^{\dfrac{n-m}{2}}\ \ \ \dfrac{(-1)^{s}(n-s)!}{s![(n+m/2)-s]![(n-m/2)-s ]!}\rho^{n-2s} \tag{18}\]  
where \(m\) and \(n\) denote the angular and radial numbers in polar coordinates of the Zernike polynomials, respectively. The essence of employing Zernike polynomials to model wavefront aberrations lies in determining both the order of the polynomials and the coefficients associated with each order. Essentially, higher-order Zernike polynomials encompass a broader spectrum of aberrations, thereby yielding greater fitting accuracy. Following the fitting of Zernike polynomials to the wavefront aberration phase for signal and beacon lights of varying wavelengths, discrepancies emerge in the coefficient matrices of the Zernike polynomials corresponding to the two beams within the same Zernike mode order. The coefficients of the Zernike polynomials, which represent wavefront aberrations for signal light at wavelength \(\lambda_{1}\) and for beacon light at wavelength \(\lambda_{2}\), can be expressed as follows [17]:  
\[ \begin{array}{l}a_{1}(\lambda_{1}),a_{2}(\lambda_{1}),a_{3}(\lambda_{1}), \ldots\ldots,a_{j}(\lambda_{1})\\ a_{1}(\lambda_{2}),a_{2}(\lambda_{2}),a_{3}(\lambda_{2}),\ldots\ldots,a_{j}( \lambda_{2})\end{array}  \tag{19}\]  
The numerical relationship between the coefficients of Zernike polynomials of the same order can be established through the wavelength function, which can be expressed as:  
\[ \begin{array}{l}a_{1}(\lambda_{1})=f_{1}[a_{1}(\lambda_{2}),\lambda_{1}]\\ a_{2}(\lambda_{1})=f_{2}[a_{2}(\lambda_{2}),\lambda_{1}]\\ \vdots\\ a_{j}(\lambda_{1})=f_{j}[a_{j}(\lambda_{2}),\lambda_{1}]\end{array}  \tag{20}\]  
Based on the actual transmission process of laser communication, we established a mode analysis model to determine the functional relationship among polynomial coefficients. We selected an infinitesimal region, denoted as \(\Delta\delta\), on the wavefront phase plane and directed the signal light and beacon light into a common optical path. Assuming perfect alignment, the corresponding region \(\Delta\delta\) experiences the same transmission path through the atmospheric channel and the same variations in refractive index. Based on \"Taylor's\" frozen-flow hypothesis [18], we selected a particular instant as the time for analysis. At this moment, the combined fluctuations in the atmospheric refractive index along the path of the beam can be approximated by a lens with an inhomogeneous refractive index distribution. This approximation can be analyzed using the formula for refractive index dispersion, which is commonly used for engineering materials.