## 2 Propagation Analysis of MGSM Array Beams
### Propagation Theory
 \tag{10}\]  
where \(W_{ND}(x,z)\) and \(W_{ND}(y,z)\) are given as follows:  
\[W_{ND}(j,z)=\frac{\pi}{\sqrt{ab}}\exp[-\frac{\left(h_{1}\Lambda_{0}\right)^{2 }}{w_{0}^{2}}]\exp[-\frac{\left(h_{2}\Lambda_{0}\right)^{2}}{w_{0}^{2}}]\exp \left[\frac{1}{a}\left(\frac{h_{1}\Lambda_{0}}{w_{0}^{2}}+\frac{ik}{2z}j_{1}- \frac{j_{1}-j_{2}}{2\rho_{0}^{2}}\right)^{2}]\exp\left(\frac{c_{j}^{2}}{b}\right), \tag{11}\]  
with  
\[a=\frac{1}{w_{0}^{2}}+\frac{1}{2m\alpha^{2}}+\frac{1}{\rho_{0}^{2}}+\frac{ik}{ 2z}, \tag{12}\]  
\[b=\frac{1}{w_{0}^{2}}+\frac{1}{2m\alpha^{2}}+\frac{1}{\rho_{0}^{2}}-\frac{ik}{ 2z}-\frac{1}{a}\left(\frac{1}{\rho_{0}^{2}}+\frac{1}{2m\alpha^{2}}\right)^{2}, \tag{13}\]  
\[c_{j}=\frac{h_{2}\Lambda_{0}}{w_{0}^{2}}-\frac{ik}{2z}j_{2}+\frac{j_{1}-j_{2}}{ 2\rho_{0}^{2}}+\frac{1}{a}\left(\frac{1}{2m\alpha^{2}}+\frac{1}{\rho_{0}^{2}} \right)\left(\frac{h_{1}\Lambda_{0}}{w_{0}^{2}}+\frac{ik}{2z}j_{1}-\frac{j_{1}- j_{2}}{2\rho_{0}^{2}}\right) \tag{14}\]where \(j=x\) or \(y\) in all locations of equations. When \(j=x\), the symbols \(h\) and \(\Lambda\) represent \(n\) and \(X\) in all equations, respectively. When \(j=y\), the symbols \(h\) and \(\Lambda\) represent \(d\) and \(Y\) in all equations, respectively. The intensity [50] and effective beam widths [51] along the \(x\)- and \(y\)-axes of \(N\times D\) MGSM array beams propagating through non-Kolmogorov turbulent atmosphere can be written as follows:  
\[I(\mathbf{r},z)=W(\mathbf{r},\mathbf{r},z), \tag{15}\]  
\[W_{jz}=\sqrt{2\frac{\iint dxdy^{2}\langle I(\mathbf{r},z)\rangle}{\iint dxdy \langle I(\mathbf{r},z)\rangle}}, \tag{16}\]  
Substituting Equations (10)-(14) into Equation (16), the effective beam widths of a \(N\times D\) MGSM array beam propagating through non-Kolmogorov turbulent atmosphere can be obtained as follows:  
\[W_{xz}=\sqrt{2\frac{\sum\limits_{m=1}^{M}\left( \begin{array}{c}M\\ m\end{array} \right)\frac{(-1)^{m-1}}{m}\sum\limits_{n_{1}=-\frac{N-1}{2}}^{- \frac{N-1}{2}}\sum\limits_{n_{2}=-\frac{N-1}{2}}^{-\frac{N-1}{2}}\sum\limits_ {i=-\frac{D-1}{2}}^{\frac{D-1}{2}}\frac{\sum\limits_{i=-\frac{D-1}{2}}^{D-1}} {2}I(x,2)I(y,0)}}, \tag{17}\]  
\[W_{yz}=\sqrt{2\frac{\sum\limits_{m=1}^{M}\left( \begin{array}{c}M\\ m\end{array} \right)\frac{(-1)^{m-1}}{m}\sum\limits_{n_{1}=-\frac{N-1}{2}}^{- \frac{N-1}{2}}\sum\limits_{n_{2}=-\frac{N-1}{2}}^{-\frac{N-1}{2}}\sum\limits_ {i=-\frac{D-1}{2}}^{\frac{D-1}{2}}\sum\limits_{i=-\frac{D-1}{2}}^{\frac{D-1}{ 2}}I(x,0)I(y,2)}}}, \tag{18}\]  
where  
\[ \begin{array}{l}I(j,0)=\frac{\pi}{\sqrt{ab}}\exp\!\left[-\frac{(h_{1}\Lambda _{0})^{2}}{w_{0}^{2}}\right]\exp\!\left[-\frac{(h_{2}\Lambda_{0})^{2}}{w_{0}^ {2}}\right]\\ \exp\!\left[\frac{1}{a}\!\left[\frac{h_{1}\Lambda_{0}}{w_{0}^{2}}\right]^{2} \right]\exp\!\left[\frac{1}{b}\!\left[\frac{h_{2}\Lambda_{0}}{w_{0}^{2}}+\frac {1}{a}\!\left(\frac{1}{2m\alpha^{2}}+\frac{1}{p_{0}^{2}}\right)\!\frac{h_{1} \Lambda_{0}}{w_{0}^{2}}\right]^{2}\right]\,\\ \sqrt{\frac{\pi}{L}}\exp\!\left(\frac{G_{L}^{2}}{L}\right)\end{array}  \tag{19}\]  
\[ \begin{array}{l}I(j,2)=\frac{\pi}{\sqrt{ab}}\exp\!\left[-\frac{(h_{1} \Lambda_{0})^{2}}{w_{0}^{2}}\right]\exp\!\left[-\frac{(h_{2}\Lambda_{0})^{2}}{ w_{0}^{2}}\right]\exp\!\left[\frac{1}{a}\!\left(\frac{h_{1}\Lambda_{0}}{w_{0}^{2}} \right)^{2}\right]\\ \exp\!\left[\frac{1}{b}\!\left(\frac{h_{2}\Lambda_{0}}{w_{0}^{2}}+\frac{1}{a} \!\left(\frac{1}{2m\alpha^{2}}+\frac{1}{p_{0}^{2}}\right)\!\frac{h_{1} \Lambda_{0}}{w_{0}^{2}}\right)^{2}\right]\!\left[\frac{1}{2L}+\frac{G_{L}^{2} }{L^{2}}\right]\sqrt{\frac{\pi}{L}}\exp\!\left(\frac{G_{L}^{2}}{L}\right)\end{array}