## 2 Theory
### M\({}^{2}\)-Factor
The Wigner distribution function of the partially coherent flat-topped beam rectangular array propagating in the turbulent atmosphere and plasma link can be expressed as follows:  
\[h(r_{sx},r_{sy},\theta,z)=\frac{k^{2}}{4\pi^{2}}\int\limits_{-\infty}^{\infty} \int\limits_{-\infty}^{\infty}W(r_{dx},r_{sx},r_{dy},r_{sy},z)\exp(-ik\theta\cdot r _{d})dr_{dx}dr_{dy} \tag{12}\]where \(\mathbf{\theta}=(\theta_{x},\theta_{y})\) is the angle between the transmission direction (z direction) and the correlation vector, and \(k\theta_{x}\) and \(k\theta_{y}\) are the wave vector components in the x and y directions, respectively. Substituting Equation (11) into (12), the expression of the Wigner distribution function can be obtained as follows:  
\[ \begin{split} h(r_{sx},r_{sy},\theta_{x},\theta_{y},z)=\frac{k^{2 }w_{0}^{2}}{16A^{2}\pi^{3}(l_{1}+l_{2})}\iiint\int\limits_{-\infty}^{\infty} \int\limits_{l_{1}=1}^{N}\sum\limits_{l_{2}=1}^{N}\sum\limits_{h_{1}=-\frac{H- 1}{2}}^{\frac{H-1}{2}}\sum\limits_{h_{2}=-\frac{H-1}{2}}^{\frac{H-1}{2}}\sum \limits_{p_{1}=-\frac{p_{1}}{2}}^{\frac{p_{1}-1}{2}}\sum\limits_{p_{2}=-\frac {p_{1}}{2}}^{\frac{p_{1}-1}{2}}\\ \times\frac{(-1)^{l_{1}+l_{2}}}{N^{2}}\Bigg{(}\begin{array}{c}N \\ l_{1}\end{array}\Bigg{)}\Bigg{(}\begin{array}{c}N\\ l_{2}\end{array}\Bigg{)}\exp\bigl{(}\alpha_{1}r_{dx}^{2}+\alpha_{2}r_{dx}\kappa _{dx}+\alpha_{3}\kappa_{dx}^{2}+\beta_{1}r_{dx}\\ +\beta_{3}\kappa_{dx}+i\alpha_{4}r_{dx}\kappa_{xx}+:\frac{1}{A}r_{sx}\kappa_{ dx}-ik\theta_{x}r_{dx}+\alpha_{1}r_{dy}^{2}+\alpha_{2}r_{dy}\kappa_{dy}+\alpha_{3} \kappa_{dy}{}^{2}\\ +i\frac{1}{A}r_{sy}\kappa_{dy}-ik\theta_{y}r_{dy}+\beta_{2}r_{dy}+\beta_{4} \kappa_{dy}+i\alpha_{4}r_{dy}r_{sy}\Bigr{)}\\ \times\exp\biggl{\{}-H\biggl{[}r_{dx},r_{dy},\frac{1}{A}\biggl{(}r_{dx}+\frac {B}{k}\kappa_{dx}\biggr{)},\frac{1}{A}\biggl{(}r_{dy}+\frac{B}{k}\kappa_{dy} \biggr{)},z\biggr{]}\biggr{\}}\alpha_{5}dx_{dx}dx_{dy}dr_{dx}dr_{dy}\\ \end{split}  \tag{13}\]  
where  
\[ \begin{split}\alpha_{1}&=\frac{l_{1}^{2}+l_{2}^{2}-2 l_{1}l_{2}}{4A^{2}w_{0}^{2}(l_{1}+l_{2})}-\frac{l_{1}+l_{2}}{4A^{2}w_{0}^{2}}- \frac{1}{2A^{2}\sigma^{2}},\alpha_{4}=\frac{kD}{B}\frac{kD}{B}\\ \alpha_{2}&=\frac{1}{2(l_{1}+l_{2})}\bigl{[}\frac{(l_{1} ^{2}+l_{2}^{2})^{2}-2 Bl_{1}l_{2}}{A^{2}w_{0}^{2}}+\frac{il_{1}-l_{2}}{A}\bigr{]} -\frac{(l_{1}+l_{2})B}{(2A^{2}w_{0}^{2}}-\frac{B}{A^{2}\kappa^{2}} \tag{14}\] \[\alpha_{3}&=\frac{1}{4(l_{1}+l_{2})}\bigl{[}\frac{B^ {2}(l_{1}^{2}+l_{2}^{2}-2l_{1}l_{2})}{A^{2}k^{2}w_{0}^{2}}-w_{0}^{2}+\frac{2 Bl_ {1}-2 Bl_{2}}{Ak}\bigr{]}-\frac{(l_{1}+l_{2})B^{2}}{4A^{2}w_{0}^{2}}-\frac{B^{2 }}{2A^{2}k^{2}\sigma^{2}}\\ \alpha_{5}&=\exp\bigl{[}\frac{c_{1}^{2}l_{1}+c_{3}^{2}l_{2}^{2}+ c_{1}l_{3}l_{2}+c_{2}l_{2}^{2}+c_{2}l_{4}l_{2}}{w_{0}^{2}(l_{1}+l_{2})}-\frac{l_{1}c_{1}^{2}l_{ 2}+l_{2}c_{3}^{2}+l_{1}c_{2}^{2}+2c_{4}l_{3}}{w_{0}^{2}}\bigr{]}\\ \beta_{1}&=\frac{c_{1}l_{1}+c_{1}l_{2}-c_{1}l_{2}^{2}-c_{1}l_{2}}{Aw _{0}^{2}(l_{1}+l_{2})}+\frac{c_{1}l_{1}-c_{1}l_{2}}{Aw_{0}^{2}}\\ \beta_{2}&=\frac{c_{1}l_{1}+c_{1}l_{2}-c_{1}l_{2}^{2}-c_{1}l_{2}}{Aw _{0}^{2}(l_{1}+l_{2})}+\frac{c_{2}l_{1}-c_{1}l_{2}}{Aw_{0}^{2}}\\ \beta_{3}&=\frac{1}{l_{1}+l_{2}}\bigl{(}\frac{Bc_{1}l_{1}-Bc_{1}l_{ 2}-Bc_{1}l_{2}-Bc_{1}l_{2}+Bc_{3}l_{2}}{Aw_{0}^{2}}-il_{1}c_{1}-i_{2}c_{3}\bigr{)} +\frac{B(l_{1}c_{1}-l_{2}c_{3})}{Akw_{0}^{2}}\\ \beta_{4}&=\frac{1}{l_{1}+l_{2}}\bigl{(}\frac{Bc_{2}l_{2}-c_{1}l_{ 1}-Bc_{2}l_{1}^{2}+Bc_{4}l_{2}}{Akw_{0}^{2}}-il_{1}c_{2}-i_{2}c_{4}\bigr{)}+ \frac{B(l_{1}c_{2}-l_{2}c_{4})}{Akw_{0}^{2}}\end{split}  \tag{15}\]  
At the receiver plane, the arbitrary order moments can be obtained from the Wigner distribution function as follows:  
\[ \begin{split}\bigl{\langle}r_{sx}^{n_{1}},r_{sy}^{n_{1}},\theta_{x }^{n_{1}},\theta_{y}^{n_{2}}\bigr{\rangle}=\frac{1}{P}\int\limits_{-\infty}^{ \infty}\int\limits_{-\infty}^{\infty}x^{n_{1}}y^{n_{2}}\theta_{x}^{n_{1}}\theta_ {y}^{m_{2}}h(r_{s},\theta,z)d^{2}r_{s}d^{2}\theta\end{split}  \tag{16}\]  
where P is the total power,  
\[