## 3 Numerical Examples  
In this section, we numerically investigate the evolutions of \(N\times D\) MGSM array beams propagating through free space and non-Kolmogorov turbulent atmosphere by applying the derived formulae. In the numerical simulation, unless other values are given, the following parameters are used: \(w_{0}=5\) mm,\(\lambda=1550\) nm, \(X_{0}=Y_{0}=5\) cm, \(N=D=3\), \(\sigma=3\) mm, \(M=40\), \(C_{n}^{2}=10^{-12}\) m\({}^{3-\alpha}\), \(\alpha=3.5\), \(L_{0}=20\) m, and \(l_{0}=1\) mm. The normalized intensity of \(N\times D\) MGSM array beams propagating through free space (\(C_{n}^{2}=0\)) and non-Kolmogorov turbulent atmosphere are illustrated in Figures 2 and 3, respectively. According to Figure 2, we can see that the \(N\times D\) MGSM array beams in free space will maintain the rectangular distribution of beamlets in the short propagation distance. As \(z\) increases, the individual beamlets will gradually translate into a flat-topped beam caused by the influence of the MGSM sources. As \(z\) further increases, the individual flat-topped beams will overlap with each other. Eventually, the MGSM array beam will translate a solid beam; at last, the MGSM array beams will involve into a beam with flat-topped profile. While \(N\times D\) MGSM array beams propagating through non-Kolmogorov turbulent atmosphere will have similar evolution properties with array beams in free space, the individual beamlets in non-Kolmogorov turbulent atmosphere will overlap with each other more rapidly than the array beams in free space at the same distance, \(z=200\) (Figures 2b and 3a). As \(z\) further increases, the array beams in non-Kolmogorov turbulent atmosphere will evolve into a Gaussian beam. The phenomenon in Figures 2 and 3 can be explained as the general beams propagation in atmosphere can translate into Gaussian beams [52]. To view the effects of non-Kolmogorov turbulent atmosphere, the cross sections of \(N\times D\) MGSM array beams with \(N=D=3\) through non-Kolmogorov turbulent atmosphere for the different \(C_{n}^{2}\) are shown in Figure 4. It can be seen that the MGSM array beams propagating through non-Kolmogorov turbulent atmosphere with a larger \(C_{n}^{2}\) will translate into a larger Gaussian-like beam spot more rapidly than in free space (\(C_{n}^{2}=0\)) at the same distance \(z\). Figure 2: The contour graphs of \(N\times D\) MGSM array beams propagating through free space with \(N=D=3\). (**a**) \(z=100\) m, (**b**) \(z=200\) m, (**c**) \(z=600\) m, and (**d**) \(z=5000\) m.  
The effects of coherence length, \(\sigma\), on \(N\times D\) MGSM array beams with \(N=D=3\) propagating through non-Kolmogorov turbulent atmosphere are illustrated in Figure 5. As can be found, the MGSM array beams with smaller \(\sigma\) will evolve into a Gaussian beam more rapidly with increasing propagation distance \(z\), and will have a larger spot. To investigate the effective beam size of a Gaussian beam translated by the MGSM array beam, Figure 6 gives the effective beam size \(W_{x}\) of the Gaussian beam for different \(C_{n}^{2}\) and \(\sigma\). One can see that \(W_{x}\) of a Gaussian beam obtained by a MGSM array beam will increase with the increase of \(C_{n}^{2}\) or the decrease of \(\sigma\). Figure 4: The cross sections of \(N\times D\) MGSM array beams with \(N=D=3\) propagating through non-Kolmogorov turbulent atmosphere for \(C_{n}^{2}\). (**a**) \(z=600\) m and (**b**) \(z=5000\) m.  
Figure 3: The contour graphs of \(N\times D\) MGSM array beams propagating through non- Kolmogorov turbulent atmosphere with \(N=D=3\). (**a**) \(z=100\) m, (**b**) \(z=600\) m, (**c**) \(z=2000\) m, and (**d**) \(z=5000\) m.  
By changing the parameters \(M\) and \(N=D\), the effective beam size for MGSM array beams propagating through non- Kolmogorov turbulent atmosphere is shown in Figure 7. As can be found, when the propagation distance \(z\) remains the same, the Gaussian beam translated by the MGSM array beams with larger \(M\) can have a larger effective beam size.