## 3 Results
### M\({}^{2}\)-Factor  
In this section, the analytical expression Equation (21) is used to numerically simulate the M\({}^{2}\) factor of the partially coherent flat-topped beam rectangular arrays propagating in the atmospheric turbulence and plasma link. The beam parameters are set as \(\sigma=2\) cm, \(\lambda=632.8\) nm, and w\({}_{0}=4\) cm. Figure 7 shows the normalized M\({}^{2}\) factors of the partially coherent flat-topped beam rectangular arrays for the different beam orders propagating in the turbulent atmosphere and plasma link. As the beam order N decreases to 1, the normalized M\({}^{2}\) factors are reduced to express the partially coherent beam rectangular arrays. From Figure 7, it becomes evident that the normalized M\({}^{2}\) factors increase with propagation distance for these beams when they traverse through both a turbulent atmosphere and plasma. Specifically, in the near field, the normalized M\({}^{2}\) factors are nearly identical, regardless of the beam order. However, in the far field, beams with higher beam orders exhibit smaller normalized M\({}^{2}\) factors compared to those with lower beam orders, suggesting better quality and coherence preservation for higher-order beams over longer distances. Additionally, at the same propagation distance, the normalized M\({}^{2}\) factors reach their maximum value for the partially coherent beam  
Figure 5: Intensity distribution of the partially coherent flat-topped beam rectangular arrays during the forward transmission with the different propagation distances and beam widths (H = P = 3). Figure 6: Intensity distribution of the partially coherent flat-topped beam rectangular arrays during the reverse transmission with the different propagation distances and beam widths (H = P = 3). rectangular arrays. Upon examining the forward and reverse transmissions, it is observed that the beams in the forward direction exhibit smaller normalized M\({}^{2}\) factors. Here, we demonstrate that by varying the beam order N, the beam quality of the flat-topped array beams significantly outperforms that of the Gaussian rectangular array beams after long-distance transmission, regardless of whether the transmission is forward or backward. Figure 8 shows the transverse spatial coherence width effect on the normalized M\({}^{2}\) factors of the partially coherent flat-topped beam rectangular arrays propagating in the turbulent atmosphere and plasma link. From Figure 8, the partially coherent flat-topped beam rectangular arrays with the small transverse spatial coherence width display the small normalized M\({}^{2}\) factors when N = 4, i.e., the low coherence characteristic of these beams can be used to mitigate the turbulent atmosphere and plasma disturbance. Furthermore, at a given transverse spatial coherence width and over a specified propagation distance, the normalized M\({}^{2}\) factors associated with these beams during forward transmission are consistently smaller than those for the reverse transmission in the far field. This result indicates that forward transmission may be less susceptible to degradation compared to reverse transmission in certain scenarios. These findings align with our expectations and further confirm the substantial advantages of partially coherent array beams over fully coherent array beams during transmission in complex environments. Figure 9 illustrates the normalized M\({}^{2}\) factors of partially coherent flat-topped beam rectangular arrays as they propagate through turbulent atmospheric and plasma conditions at various beam widths. From Figure 9, the results clearly show that arrays with larger beam widths have a notable advantage in mitigating disturbances caused by both atmospheric turbulence and plasma during both forward and reverse transmissions. This  
Figure 8: Normalized M\({}^{2}\) factors of the partially coherent flat-topped beam rectangular arrays during the (**a**) forward and (**b**) reverse transmission with the different transverse spatial coherence widths (H = P = 3).