# Experimental Study on Phase Scintillation of Optical Transmission in Atmospheric Turbulence
## 3 Experimental Measurements
### Mean Square Error of Wavefront Distortion  
Wavefront distortion can be decomposed using annular Zernike polynomials, where the annular Zernike coefficients represent the contribution of each annular Zernike polynomial to the wavefront distortion. The mean square error (MSE) of wavefront distortion is directly related to these annular Zernike coefficients [17]. Specifically, the MSE of wavefront distortion is proportional to the sum of the squares of the annular Zernike polynomial coefficients with the proportionality constant \(1/\pi(1-\varepsilon^{2})\). The proof is as follows. The MSE of wavefront distortion \(\sigma^{2}\) is defined as the average of the square integral of the wavefront distortion \(\phi(r)\) over the entire annular region. It is specifically expressed as:  
\[\sigma^{2}=\frac{1}{A}\int_{A}\phi^{2}(r)dA \tag{19}\]  
Here, \(A\) is the area of the annular region. The area \(A\) of the annular region can be expressed as:  
\[A=\pi(1-\varepsilon^{2}) \tag{20}\]  
Here, \(\varepsilon\) is the obscuration ratio of the annular Zernike polynomial. Substituting Equations (6) and (20) into Equation (19):  
\[\sigma^{2}=\frac{1}{\pi(1-\varepsilon^{2})}\int_{\varepsilon}^{1}\int_{0}^{2 \pi}\left(\sum_{i=1}^{N}a_{i}Z_{i}(r,\theta;\varepsilon)\right)^{2}rdrd\theta \tag{21}\]  
Here, \(a_{i}\) represents the coefficient of the \(i\)th-order annular Zernike polynomial, \(Z_{i}(r,\theta;\varepsilon)\) represents the \(i\)th-order annular Zernike polynomial, and \(N\) is the highest order of the selected annular Zernike polynomials. Since annular Zernike polynomials are orthogonal and normalized over the annular region, they satisfy:  
\[\int_{\varepsilon}^{1}\int_{0}^{2\pi}Z_{i}(r,\theta;\varepsilon)Z_{j}(r, \theta;\varepsilon)rdrd\theta=\delta_{ij} \tag{22}\]  
Here, \(\delta_{ij}\) is the Kronecker delta function, which is one when \(i=j\) and zero when \(i\
eq j\). Therefore, by expanding and simplifying Equation (19), we obtain:  
\[ \begin{array}{ll}\sigma^{2}&=\frac{1}{\pi(1-\varepsilon^{2})}\sum\limits_{i= 1}^{N}\sum\limits_{j=1}^{N}a_{i}a_{j}\int_{\varepsilon}^{1}\int_{0}^{2\pi}Z_{ i}(r,\theta;\varepsilon)Z_{j}(r,\theta;\varepsilon)rdrd\theta\\ &=\frac{1}{\pi(1-\varepsilon^{2})}\sum\limits_{i=1}^{N}\sum\limits_{j=1}^{N}a _{i}a_{j}\delta_{ij}\end{array}  \tag{23}\]Since \(\delta_{ij}\) is one only when \(i=j\) and zero otherwise, the above expression can be simplified to:  
\[\sigma^{2}=\frac{1}{\pi(1-\varepsilon^{2})}\sum_{i=1}^{N}a_{i}^{2} \tag{24}\]  
The proof process from Equation (19) to Equation (24) demonstrates the relationship between the mean square error (MSE) of wavefront distortion and the annular Zernike coefficients, indicating that the MSE of wavefront distortion is proportional to the sum of the squares of the annular Zernike polynomial coefficients, with the proportionality constant \(1/\pi(1-\varepsilon^{2})\). When studying wavefront distortion, the magnitude of the MSE can reflect the extent of atmospheric turbulence's impact on wavefront quality. A larger MSE indicates more severe wavefront distortion. Figure 16 shows the results of the MSE of wavefront distortion for the first 30 orders, calculated over a 300 s measurement period using Equation (24). Based on Figure 16 and Tables 9-11, in link 1, the MSE of the wavefront distortion fluctuates relatively little under sunny and cloudy conditions, indicating more stable wavefront quality. In contrast, the MSE fluctuates more under light rain, heavy rain, dust storm, and snowy conditions, with the mean values for dust storm and snowy conditions being close, but the standard deviation is higher under dust storm conditions, indicating greater variability than under snowy conditions. Particularly under heavy rain, the MSE of wavefront distortion reaches a maximum of 101.67 \(\upmu\)m\({}^{2}\), a minimum of 43.17 \(\upmu\)m\({}^{2}\), and a standard deviation of 13.13 \(\upmu\)m\({}^{2}\), which are significantly higher than under other weather conditions, indicating that severe weather has a substantial impact on wavefront quality.