## 2 Theory  
### Intensity Distribution  
Figure 1 shows the intensity distribution of the partially coherent flat-topped beam rectangular array at the source plane. The electric field can be expressed as follows [47]:  
\[ \begin{split} E(x^{\prime},y^{\prime},0)=\sum\limits_{l=1}^{N}& \frac{\frac{H-1}{2}}{\sum\limits_{h=-}^{\infty}}\frac{\frac{P-1}{2}}{ \sum\limits_{p=-}^{\infty}}\frac{\frac{P-1}{2}}{\frac{P-1}{2}}\frac{\left(-1 \right)^{l-1}}{N}\Bigg{(}\begin{array}{c}N\\ l\end{array}\Bigg{)}\\ &\times\exp\left[-\frac{l(x^{\prime}-hx_{0})^{2}}{w_{0}^{2}}- \frac{l(y^{\prime}-py_{0})^{2}}{w_{0}^{2}}\right]\end{split}  \tag{1}\]  
where (x\({}^{\prime}\), y\({}^{\prime}\)) are the position vectors, w\({}_{0}\) is the beam width, H and P are the number of beamlets for the transverse and longitudinal beam array, respectively. (N i)\({}^{\text{T}}\) = N!/[l! (N - l)!] describes the binomial coefficient, and N is the beam order. The parameters x\({}_{0}\) and y\({}_{0}\) are the transverse and longitudinal distance between two adjacent beamlets. Equation (1) reduces to the electric field function of a partially coherent beam rectangular array when the beam order N = 1. For the partially coherent beams, the evolution properties of the intensity distribution can be described by the cross spectral density (CSD) function. The CSD function of a partially coherent flat-topped beam rectangular array at the source plane can be expressed as  
\[ \begin{split}& W\big{(}x_{1}^{\prime},y_{1}^{\prime},x_{2}^{ \prime},y_{2}^{\prime},0\big{)}=\sum\limits_{l_{1}=1}^{N}\sum\limits_{l_{2}=1} ^{N}\frac{\frac{H-1}{2}}{\sum\limits_{h_{1}=-}^{\infty}}\frac{\frac{P-1}{2}}{ \sum\limits_{h_{2}=-}^{\infty}}\frac{\frac{P-1}{2}}{\sum\limits_{h_{2}=-}^{ \infty}}\frac{\frac{P-1}{2}}{\sum\limits_{p_{1}=-}^{\infty}}\frac{\left(-1 \right)^{l_{1}-1}}{N}\frac{\left(-1\right)^{l_{2}-1}}{N}\Bigg{(}\begin{array}[ ]{c}N\\ l_{1}\end{array}\Bigg{)}\Bigg{(}\begin{array}{c}N\\ l_{2}\end{array}\Bigg{)}\\ &\times\exp\left[-\frac{l_{1}\big{(}x_{1}^{\prime}-h_{1}x_{0}\big{)}^{2}}{w_{ 0}^{2}}-\frac{l_{1}\big{(}y_{1}^{\prime}-p_{1}y_{0}\big{)}^{2}}{w_{0}^{2}} \right]\exp\left[-\frac{l_{2}\big{(}x_{2}^{\prime}-h_{2}x_{0}\big{)}^{2}}{w_{ 0}^{2}}-\frac{l_{2}\big{(}y_{2}^{\prime}-p_{2}y_{0}\big{)}^{2}}{w_{0}^{2}} \right]\\ &\times\exp\left[-\frac{\big{(}x_{1}^{\prime}-x_{2}^{\prime}\big{)}^{2}+ \big{(}y_{1}^{\prime}-y_{2}^{\prime}\big{)}^{2}}{2\sigma^{2}}\right]\end{split}  \tag{2}\]  
where \((x_{1}^{\prime},y_{1}^{\prime})\) and \((x_{2}^{\prime},y_{2}^{\prime})\) are arbitrary position vectors at the source plane, respectively. \(\sigma\) denotes the transverse spatial coherence width. The intensity distribution is conforms to the extended Huygens-Fresnel principle when the partially coherent flat-topped beam rectangular array propagates in the atmospheric turbulence and plasma link. According to the Collins formula, the CSD function at the receiver plane of the partially coherent flat-topped beam rectangular array, propagated through the ABCD optical system with the matrix elements A, B, C, and D in the turbulent atmosphere and plasma link, can be obtained as  
Figure 1: Schematic diagram of a partially coherent flat-topped beam rectangular array. \[