## 2 Theory
### Intensity Distribution
\begin{split} W(x_{1},y_{1},x_{2},y_{2},z)=\frac{1}{\lambda^{2}B^{2}} \iint\limits_{-\infty}^{\infty}\sum\limits_{l_{1}}^{N}\sum\limits_{l_{2}=1}^{ \frac{l_{2}-1}{2}}\sum\limits_{l_{1}=-\frac{l_{1}}{2}}^{\frac{l_{2}-1}{2}}\sum \limits_{l_{2}=-\frac{l_{2}}{2}}^{\frac{l_{2}-1}{2}}\sum\limits_{l_{2}=-\frac {l_{2}}{2}}^{\frac{l_{2}-1}{2}}\\ \times\frac{(-1)^{l_{1}-1}}{N}\frac{(-1)^{l_{2}-1}}{N}\left( \begin{array}{c}N\\ l_{1}\end{array}\right)\left(\begin{array}{c}N\\ l_{2}\end{array}\right)\exp\left[-\frac{l_{1}\left(x_{1}^{\prime}-h_{1}x_{0} \right)^{2}}{w_{0}^{2}}-\frac{l_{1}\left(y_{1}^{\prime}-p_{1}y_{0}\right)^{2}} {w_{0}^{2}}\right]\\ \times\exp\left[-\frac{l_{2}\left(x_{2}^{\prime}-h_{2}x_{0}\right)^{2}}{w_{0} ^{2}}-\frac{l_{2}\left(y_{2}^{\prime}-p_{2}y_{0}\right)^{2}}{w_{0}^{2}}\right] \exp\left[-\frac{\left(x_{1}^{\prime}-x_{2}^{\prime}\right)^{2}+\left(y_{1}^{ \prime}-y_{2}^{\prime}\right)^{2}}{2\sigma^{2}}\right]\\ \times\exp\left\{-\frac{ik}{2B}\left[A\left(x_{1}^{\prime 2}-x_{2}^{\prime 2}+y_{1}^{ \prime 2}-y_{2}^{\prime 2}\right)^{2}+D\left(x_{1}^{2}-x_{2}^{2}+y_{1}^{2}-y_{2}^{ 2}\right)\right.\right.\\ \times\left.\left.-2\left(x_{1}x_{1}^{\prime}-x_{2}x_{2}^{\prime}+y_{1}y_{1}^{ \prime}-y_{2}y_{2}^{\prime}\right)\right]\right\}\\ \times\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 dx_{1}^{\prime }dx_{2}^{\prime}dy_{1}^{\prime}dy_{2}^{\prime}\end{split}  \tag{3}\]  
where \((\kappa_{\rm L},y_{1})\) and \((x_{2},y_{2})\) are the arbitrary position vectors at the receiver plane, respectively. \(\rm k=2\pi/\lambda\) is the wave number, and \(\lambda\) is the wavelength. The symbol \(<>\) is the ensemble average. The term \(\exp(\mathbb{Y})\) describes the complex phases induced by the atmospheric turbulence [42], which can be expressed as  
\[ \begin{split}\left\langle\exp[\psi^{*}(x_{1},y_{1},x_{1}^{\prime },y_{1}^{\prime},z)+\psi(x_{2},y_{2},x_{2}^{\prime},y_{2}^{\prime},z)]\right\rangle \\ =\exp\left\{-\frac{1}{\rho_{0}^{2}}\left[\left(x_{1}^{\prime 2}+y_{1}^{ \prime 2}-2x_{1}^{\prime}x_{2}^{\prime}-2y_{1}^{\prime}y_{2}^{\prime}+x_{2}^{ \prime\,2}+y_{2}^{\prime\,2}\right)\right.\right.\\ \left.\left.+(x_{1}^{\prime}x_{1}+y_{1}^{\prime}y_{1}-x_{1}^{ \prime}x_{2}-y_{1}^{\prime}y_{2}-x_{2}^{\prime}x_{1}-y_{2}^{\prime}y_{1}+x_{2} ^{\prime}x_{2}+y_{2}^{\prime}y_{2}\right)\right.\\ \left.\left.+(x_{1}-x_{2})^{2}+(y_{1}-y_{2})^{2}\right]\right\} \end{split} \] (4a) where \(\rho_{0}=\left[3/(\pi^{2}k^{2}zT)\right]^{1/2}\) represents the transverse spatial coherence radius to describe the strength of the atmospheric turbulence and the parameter \[T=\int_{0}^{\infty}\Phi_{n}(\kappa)\kappa^{3}d\kappa\] (4b) where \(\Phi_{n}\) is the power spectrum for the refractive index fluctuation of the atmospheric turbulence with the spatial frequency \(\kappa\). Substituting Equation (4a) into Equation (3), the CSD function at the receiver plane can be obtained as  
\[\begin{split}& W(x_{1},y_{1},x_{2},y_{2},z)=\frac{1}{\lambda^{2}B^{2}} \sum\limits_{l_{1}=1}^{N}\sum\limits_{l_{2}=1}^{\frac{l_{2}-1}{2}}\sum\limits_{ l_{1}=-\frac{l_{1}}{2}}^{\frac{l_{2}-1}{2}}\sum\limits_{l_{2}=-\frac{l_{2}}{2}}^{ \frac{l_{2}-1}{2}}\sum\limits_{p_{1}=-\frac{l_{2}}{2}}^{\frac{p_{1}-1}{2}} \sum\limits_{p_{2}=-\frac{p_{2}-1}{2}}^{\frac{p_{2}-1}{2}}\\ &\frac{Equation (5) describes the CSD function of the partially coherent flat-topped beam rectangular array through a plasma and atmospheric turbulence link at the receiver plane. Equation (5) is used to describe the intensity distribution function when the arbitrary position coordinates \({\rm x_{1}}^{\prime}={\rm x_{2}}^{\prime}={\rm x}^{\prime}\), \({\rm y_{1}}^{\prime}={\rm y_{2}}^{\prime}={\rm y}^{\prime}\).