## 2 Theory
### M\({}^{2}\)-Factor  
The M\({}^{2}\)-factor is an important parameter for describing the laser beam propagation in a free space and random media. In order to obtain the M\({}^{2}\) of the partially coherent flat-topped beam rectangular array in a plasma and atmospheric turbulence link at the receiver plane, we use the following sum and difference notation:  
\[ \begin{array}{l}x_{d}={x_{1}}^{\prime}-{x_{2}}^{\prime},{x_{s}}^{\prime}= \frac{{x_{1}}^{\prime}+{x_{2}}^{\prime}}{2},y_{d}={y_{1}}^{\prime}-{y_{2}}^{ \prime},{y_{s}}^{\prime}=\frac{{y_{1}}^{\prime}+{y_{s}}^{\prime}}{2},\\ r_{dx}=x_{1}-x_{2},r_{sx}=\frac{{x_{1}}+{x_{2}}}{2},r_{dy}=y_{1}-y_{2},r_{sy}= \frac{{y_{1}}+{y_{2}}}{2}\end{array}  \tag{8}\]  
and  
\[c_{1}=h_{1}x_{0},c_{2}=p_{1}y_{0},c_{3}=h_{2}x_{0},c_{4}=p_{2}y_{0} \tag{9}\]  
The term in Equation (4), induced by the atmospheric turbulence, was used in the following form:  
\[ \begin{array}{l}\left\langle\exp[\psi^{*}(x_{1},y_{1},x_{1}^{\prime},y_{1}^ {\prime},z)+\psi(x_{2},y_{2},x_{2}^{\prime},y_{2}^{\prime},z)]\right\rangle\\ \qquad=-4\pi^{2}k^{2}\gamma_{1}^{0}d\xi_{0}^{y}\int_{0}^{\infty}\big{[}1-J_{0} (\kappa\big{|}x_{d}^{\prime}\xi+(1-\xi)x_{d}\big{|})\big{]}\Phi_{n}(\kappa) \kappa d\kappa\\ \qquad\qquad\times\Big{\{}-4\pi^{2}k^{2}z\int_{0}^{1}d\xi_{0}^{y}\Big{[}1-J_{0 }(\kappa\big{|}y_{d}^{\prime}\xi+(1-\xi)y_{d}\big{|})\Big{]}\Phi_{n}(\kappa) \kappa d\kappa\Big{\}}\end{array}  \tag{10}\]  
With the help of the equations \(x_{d}=\frac{1}{\lambda}(r_{dx}+\frac{B}{k}\kappa_{dx})\), \(y_{d}=\frac{1}{\lambda}(r_{dy}+\frac{B}{k}\kappa_{dy})\), Equation (3) can be rewritten as follows:  
\[ \begin{array}{l}W(r_{dx},r_{sx},r_{dy},r_{sy},z)=\bigg{(}\frac{1}{2\pi A} \bigg{)}^{2}\iiint\int\limits_{-\infty}^{\infty}\sum\limits_{l_{1}=1}^{N}\sum \limits_{l_{2}=1}^{\frac{ll_{1}-1}{l_{2}}}\sum\limits_{h_{2}=-\frac{H-1}{l_{2 }}}^{\frac{ll_{2}-1}{l_{2}}}\sum\limits_{h_{2}=-\frac{H-1}{l_{2}}}^{\frac{p-1} {2}}\sum\limits_{h_{2}=-\frac{p-1}{l_{2}}}^{\frac{p-1}{2}}\\ \frac{(-1)^{l_{1}-1}}{N}\frac{(-1)^{l_{2}-1}}{N}\Bigg{(}\begin{array}{c}N\\ l_{1}\end{array}\Bigg{)}\exp\Bigg{\{}-\frac{l_{1}}{w_{0}^{2}}\bigg{[}\frac{1 }{2A}(r_{dx}+\frac{B}{k}\kappa_{dx})+{x_{s}}^{\prime}-c_{1}\bigg{]}^{2}\Bigg{\}} \\ \exp\Bigg{\{}-\frac{l_{1}}{w_{0}^{2}}\bigg{[}\frac{1}{2A}(r_{dy}+\frac{B}{k} \kappa_{dy})+{y_{s}}^{\prime}-c_{2}\bigg{]}^{2}-\frac{l_{2}}{w_{0}^{2}}\bigg{[} {x_{s}}^{\prime}-\frac{1}{2A}(r_{dx}+\frac{B}{k}\kappa_{dx})-c_{3}\bigg{]}^{2} \Bigg{\}}\\ \exp\Bigg{\{}-\frac{l_{2}}{w_{0}^{2}}\bigg{[}{y_{s}}^{\prime}-\frac{1}{2A}(r_ {dy}+\frac{B}{k}\kappa_{dy})-c_{4}\bigg{]}^{2}-\frac{1}{2\sigma_{L}^{2}} \bigg{[}(\frac{r_{dx}}{A}+\frac{B_{dx}}{Ak})^{2}+(\frac{1}{A}r_{dy}+\frac{B \kappa_{dy}}{Ak})^{2}\Bigg{]}\Bigg{\}}\\ \exp\Bigg{\{}-\frac{ik}{2B}\bigg{[}(2r_{dx}+\frac{2B}{k}\kappa_{dx}){x_{s}}^{ \prime}+(2r_{dy}+\frac{2B}{k}\kappa_{dy}){y_{s}}^{\prime}+D(2r_{dx}r_{sx}+2 r_{dy}r_{sy})\\ \qquad-2\Big{(}\frac{r_{dx}}{2}+r_{sx}\Big{)}\bigg{(}\frac{1}{2A}(r_{dx}+\frac{ B}{k}\kappa_{dx})+{x_{s}}^{\prime}\Big{)}-\Big{(}r_{sx}-\frac{r_{dx}}{2}\Big{)} \bigg{(}{x_{s}}^{\prime}-\frac{1}{2A}(r_{dx}+\frac{B}{k}\kappa_{dx})\bigg{)} \\ \qquad+\bigg{(}\frac{r_{dy}}{2}+r_{sy}\bigg{)}\bigg{(}\frac{1}{2A}(r_{dy}+\frac {B}{k}\kappa_{dy})+{y_{s}}^{\prime}\bigg{)}-\Big{(}r_{sy}-\frac{r_{dy}}{2} \bigg{)}\bigg{(}{y_{s}}^{\prime}-\frac{1}{2A}(r_{dy}+\frac{B}{k}\kappa_{dy}) \bigg{)}\bigg{]}\Bigg{\}}\\ \exp\bigg{[}-H(r_{dx},r_{dy},\frac{1}{A}(r_{dx}+\frac{B}{k}\kappa_{dx}),\frac{ 1}{A}(r_{dy}+\frac{B}{k}\kappa_{dy}),z)\bigg{]}dx_{s}^{\prime}dy_{s}^{\prime} dx_{dx}d\kappa_{dy}\end{array}  \tag{11}\]  
The M\({}^{2}\)-factor can be obtained from the Wigner distribution function when the laser beam propagates through a random media.