## 2 Propagation Analysis of MGSM Array Beams
### Propagation Theory  
Considering the extended Huygens-Fresnel integral, the CSD of \(N\times D\) MGSM array beams in non-Kolmogorov turbulent atmosphere at the plane \(z\) is written as follows [15; 16; 17; 18; 19; 20]:  
\[ \begin{array}{c}W_{ND}(\mathbf{r}_{1},\mathbf{r}_{2},z)=\frac{k^{2}}{4n^{2} \cdot 2^{2}}\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}\int_{-\infty}^{+ \infty}\int_{-\infty}^{+\infty}d\mathbf{r}_{10}d\mathbf{r}_{20}W_{ND}(\mathbf{ r}_{10},\mathbf{r}_{20},0)\\ \times\exp\left[-\frac{k}{2z}(\mathbf{r}_{1}-\mathbf{r}_{10})^{2}+\frac{k}{2z} (\mathbf{r}_{2}-\mathbf{r}_{20})^{2}\right](\exp[\psi(\mathbf{r}_{10},\mathbf{ r}_{1})+\psi^{*}(\mathbf{r}_{20},\mathbf{r}_{2})])\end{array} , \tag{4}\]  
Figure 1: The contour graphs of \(N\times D\) MGSM array beam at the plane \(z=0\). (**a**) \(N=3,D=1\), (**b**) \(N=3,D=1\), (**c**) \(N=5,D=2\), and (**d**) \(N=3,D=3\). where \(k=2\pi/\lambda\) is the wave number with \(\lambda\) as the wavelength. The definitions of \(\psi(\mathbf{r}_{0},\mathbf{r})\) can be found in reference [19], and as follows:  
\[\left\langle\exp[\psi(\mathbf{r}_{10},\mathbf{r}_{1})+\psi^{*}(\mathbf{r}_{20}, \mathbf{r}_{2})]\right\rangle=\exp[-\frac{(\mathbf{r}_{10}-\mathbf{r}_{20})^{2 }+(\mathbf{r}_{10}-\mathbf{r}_{20})(\mathbf{r}_{1}-\mathbf{r}_{2})+(\mathbf{r} _{1}-\mathbf{r}_{2})^{2}}{\rho_{0}^{2}}], \tag{5}\]  
where \(1/\rho_{0}^{2}=\pi^{2}k^{2}\underset{0}{\overset{\infty}{\underset{0}{\sum}}} \kappa^{3}\Phi(\kappa,\alpha)d\kappa/3\), \(\kappa\) is the magnitude of spatial frequency, and \(\alpha\) is the power law exponent. The non-Kolmogorov power spectrum \(\Phi(\kappa,\alpha)\) can be described by the following [15; 16; 17]:  
\[\Phi(\kappa,\alpha)=A(\alpha)C_{n}^{2}\frac{\exp(-\kappa^{2}/\kappa_{m}^{2})} {\left(\kappa^{2}+\kappa_{0}^{2}\right)^{\alpha/2}}\ 0\leq\kappa<\infty,3<\alpha<4, \tag{6}\]  
where \(C_{n}^{2}\) is the turbulent structure constant with units \(m^{3-\alpha}\); \(\kappa_{0}=2\pi/L_{0}\), where \(L_{0}\) is the outer scale; and \(\kappa_{m}=c(\alpha)/l_{0}\), where \(l_{0}\) is the inner scale; and  
\[A(\alpha)=\frac{1}{4\pi^{2}}\Gamma(\alpha-1)\cos(\frac{\alpha\pi}{2}), \tag{7}\]  
\[c(\alpha)=\left[\frac{2\pi}{3}\Gamma\left(\frac{5-\alpha}{2}\right)A(\alpha) \right]^{\frac{1}{\alpha-5}}, \tag{8}\]  
where \(\Gamma(x)\) denotes the Gamma function. Substituting Equations (6)-(8) into the definition of \(1/\rho_{0}^{2}\), we can obtain the following:  
\[\frac{1}{\rho_{0}^{2}}=\frac{\pi^{2}k^{2}z}{6(\alpha-2)}A(\alpha)C_{n}^{2} \left[\left(2\kappa_{0}^{2}-2\kappa_{m}^{2}+\alpha\kappa_{m}^{2}\right) \kappa_{m}^{2-\alpha}\exp\left(\frac{\kappa_{0}^{2}}{\kappa_{m}^{2}}\right) \Gamma\left(2-\frac{\alpha}{2},\frac{\kappa_{0}^{2}}{\kappa_{m}^{2}}\right)-2 \kappa_{0}^{4-\alpha}\right], \tag{9}\]  
where \(\Gamma(x,y)\) represents the incomplete Gamma function. Thus, the propagation CSD of \(N\times D\) MGSM array beams in non-Kolmogorov turbulent atmosphere are derived as follows:  
\[ \begin{array}{c}W_{ND}(\mathbf{r}_{1},\mathbf{r}_{2},z)=\frac{k^{2}}{4\pi^{ 2}z^{2}}\exp[-\frac{ik}{2z}(\mathbf{r}_{1}^{2}-\mathbf{r}_{2}^{2})]\exp[-\frac {(x_{1}-x_{2})^{2}+(y_{1}-y_{2})^{2}}{\rho_{0}^{2}}]\\ \frac{1}{C_{0}}\underset{m=1}{\overset{M}{\underset{m}{\sum}}}\left(\begin{array} []{c}M\\ m\end{array}\right)\frac{\left(-1\right)^{m-1}}{m}\underset{m=-\frac{N-1}{2}}{ \overset{-N-1}{2}}\underset{m=-\frac{N-1}{2}}{\overset{-N-1}{2}}\underset{m =-\frac{D-1}{2}}{\overset{-D-1}{2}}\overset{D-1}{\underset{d_{2}=-\frac{D-1 }{2}}}W_{ND}(x,z)W_{ND}(y,z),\end{array}