## 3 Simulation and Experimental Results
### Effect of the Filter Surface Roughness on the Combined Beam Quality
#### 3.3.2 The Multimode Gaussian Beam  
Generally, the multimode lasers that emerge from the resonator can be completely described as superpositions of the Hermite-Gaussian or Laguerre-Gaussian modes [25; 26; 27]. However, the accuracy of the modal decomposition technology is limited, and the expression of the complex amplitude is rather complex, especially for the beamlets that are achieved by the power combination of several laser oscillators. In this case, a large amount of work is theoretically needed when using the FSBC system, to obtain a precise expression for the multimode lasers. Fortunately, the PHASICS SID4 wave-front sensors, which are able to monitor the complex amplitudes of the beamlets in real time, provide a simple and reliable method for obtaining the complex amplitudes of the multimode lasers. These measurement results are shown in Figure 11.  
\(I_{0}\) and \(I_{1}\) are the normalized intensities of the incident lasers, and are expressed as:  
\[I_{1}=\frac{I_{1}}{\max(I_{1})} \tag{26}\]  
\[I_{0}=\frac{I_{0}}{\max(I_{1})} \tag{27}\]  
where \(\varphi_{0}\) and \(\varphi_{1}\) are the phases of the incident lasers. Hence, the complex amplitudes of the incident lasers can be expressed as:  
\[E_{0}=\sqrt{I_{0}}\exp(-i\varphi_{0}) \tag{28}\]  
Figure 10: (**a**) The encircled power versus the propagation distance, with a surface roughness of 2 nm. The intensity profiles of the combined beam at different propagation distances \(Z\); (**b**) \(Z\) = 0 mm; (**c**) \(Z\) = 80 mm; (**d**) \(Z\) = 160 mm; (**e**) \(Z\) = 240 mm; (**f**) \(Z\) = 320 mm; (**g**) \(Z\) = 500 mm.  
Figure 9: (**a**) The encircled power versus surface roughness at a distance of 500 mm. The intensity profile of: (**b**) the ideal laser (surface roughness \(\sigma\) = 0 nm); (**c**) the reflected laser (surface roughness \(\sigma\) = 2 nm); (**d**) the transmitted laser (surface roughness \(\sigma\) = 2 nm), and (**e**) the combined laser (surface roughness \(\sigma\) = 2 nm), all at a distance of 500 mm.  
\[E_{1}=\sqrt{I_{1}}\exp(-i\varphi_{1}) \tag{29}\]  
When Equations (28) and (29) are substituted into Equations (19) and (20), the relationship between the beam quality and the surface roughness at a distance of 500 mm can be potentially developed. The resulted encircled power distributions are shown in Figure 12, which shows a trend similar to that of an ideal Gaussian beam. The insets show the intensity profiles of the combined laser under different surface roughness values. As expected, the ratio of the scattered beam is gradually enhanced as the surface roughness increases, which leads to a deterioration in beam quality.  
On the other hand, the effect of the propagation distance on the PIB was also calculated. As shown in Figure 13, the encircled power displays a decreasing trend, with an increase in the propagation distance, and it is similar to that in the case of the single-mode lasers. The insets display the intensity profiles of the combined beam under different propagation distances, from which the enhancement of the scattering could be clearly observed. In this case, it is necessary to maintain the smoothness of the edge filter, which is not only beneficial for achieving combined beams with high power and good quality, but also helpful for enhancing the effective working distance of the combining system.  
Figure 11: The measured complex amplitudes of the incident lasers.  
Figure 12: The encircled power versus surface roughness at a distance of 500 mm. The insets are the intensity profiles of the combined laser under different surface roughness values.