# Experimental Study on Phase Scintillation of Optical Transmission in Atmospheric Turbulence
## 2 Theoretical Analysis
The wavefront phase \(\phi(r)\) can be expanded using annular Zernike polynomials as:  
\[\phi(r)=\sum_{i=1}^{N}a_{i}Z_{i}(r,\theta;\varepsilon) \tag{6}\]  
In the Equation (6), \(Z_{i}(r,\theta;\varepsilon)\) represents the \(i\)th-order annular Zernike polynomial, \(r\) represents the radius of the circle, \(\varepsilon\) is the obscuration ratio of the annular Zernike polynomial, defined as the ratio of the radius of the central obscured region to the radius of the unit circle, \(a_{i}\) represents the coefficient of the \(i\)th-order annular Zernike polynomial, and \(N\) is the highest order of the selected annular Zernike polynomials. Therefore, in this paper, annular Zernike polynomials are used for description, as shown in Equation (7) [11]. \[Z_{i}(r,\theta;\varepsilon)=Z_{n,m}(\rho,\theta;\varepsilon)=R_{n,m}(\rho; \varepsilon)e^{im\theta} \tag{7}\]  
Here, \(R_{n,m}(\rho;\varepsilon)\) represents the annular radial polynomial, \(\rho\) is the normalized radial coordinate, \(i\) denotes the imaginary unit, \(n\) is the order of the annular Zernike polynomial, and \(m\) represents the angular frequency. In Equation (3), \(R_{n,m}(\rho;\varepsilon)\) is defined as [12]. \[R_{n,m}(\rho;\varepsilon)=\frac{n-|m|}{\sum_{s=0}^{2}}\ (-1)^{s}\frac{(n-k)!(1- \varepsilon^{2})\frac{|m|}{2}}{s!(\frac{n+|m|}{2}-s)!(\frac{n-|m|}{2}-s)!}( \rho)^{n-2s} \tag{8}\]  
Here, \(n\) and \(m\) are non-negative integers that satisfy \(n\geq|m|\), with \(n-|m|\) being an even number. The annular Zernike polynomial defined by Equation (6) is used to characterize the wavefront phase. Excluding the first term, which represents the translation coefficient, the annular Zernike coefficients can be expressed as:  
Figure 1.0  
The covariance matrix of \(A\) is expressed as:  
\[C_{a}=E\Big{[}a\cdot a^{T}\Big{]}= \begin{bmatrix}E(a_{2},a_{2})&E(a_{2},a_{3} )&\cdots&E(a_{2},a_{i})\\ E(a_{3},a_{2})&E(a_{3},a_{3})&\cdots&E(a_{3},a_{i})\\ \vdots&\vdots&\ddots&\vdots\\ E(a_{i},a_{2})&E(a_{i},a_{3})&\cdots&E(a_{i},a_{i})\end{bmatrix}  \tag{10}\]  
In Equation (10), \(C_{a}\) is an \((i-1)\times(i-1)\) matrix, \(E\big{(}a_{i},a_{j}\big{)}\) represents the covariance of the \(i\) and \(j\) order of the annular Zernike polynomial coefficients. The covariance expression for any two of the annular Zernike polynomials \(Z_{i}\) and \(Z_{i^{\prime}}\) is:  
\[E(a_{i},a_{i^{\prime}})=\frac{K_{zz^{\prime}}\delta_{z}\Gamma\Big{[}\frac{1}{ 2}\big{(}n+n^{\prime}-\frac{5}{3}\big{)}\Big{]}\Big{(}\frac{p}{r_{0}}\Big{)}^ {\frac{5}{3}}}{\Gamma\Big{[}\frac{1}{2}\Big{(}n-n^{\prime}+\frac{17}{3}\Big{)} \Big{]}\Gamma\Big{[}\frac{1}{2}\Big{(}n^{\prime}-n+\frac{17}{3}\Big{)}\Big{]} \Gamma\Big{[}\frac{1}{2}\Big{(}n+n^{\prime}+\frac{23}{3}\Big{)}\Big{]}} \tag{11}\]  
Here, \(n\) and \(m\) are the radial and azimuthal orders of \(Z_{i}(r,\theta;\varepsilon)\), \(n^{\prime}\) and \(m^{\prime}\) are the radial and azimuthal orders of \(Z_{i^{\prime}}(r,\theta;\varepsilon)\). \(\Gamma\) is the Gamma function, \(\delta_{z}\) is the Kronecker delta, \(D\) represents the aperture of the optical antenna, \(r_{0}\) represents the atmospheric coherence