## 3 Simulation and Experimental Results
### Effect of the Filter Surface Roughness on the Combined Beam Quality
#### 3.3.1 The Ideal Gaussian Beam  
We first assume that two laser individual channels operating at 1070 nm and 1090 nm, respectively, are combined via an edge filter, and that their far-field intensity profiles have an ideal Gaussian  
Figure 8: **(a)** The combining efficiency as a function of the thickness error. **(b)** The combining efficiency as a function of the incident angle. The blue line represents the \(s\)-polarized light, and the red line represents the \(p\)-polarized light.  
Figure 7: **(a)** The measured normalized spectra of the 1070 nm laser (blue line) and the 1090 nm laser (red line). **(b)** The combining efficiency as a function of the surface roughness.  
distribution. According to Equations (14) and (15), the complex amplitudes of the reflected laser and transmitted laser can be expressed as:  
\[E_{r}(x_{r},y_{r},0) =r*E_{0}(x_{0},y_{0},0) \tag{19}\] \[E_{t}(x_{t},y_{t},0) =t*E_{1}(x_{1},y_{1},0) \tag{20}\]  
where \(E_{0}\) and \(E_{1}\) are the amplitudes of the incident lasers. Based on the angular spectrum theory, the complex amplitudes of the beams at a propagation distance \(z\) are obtained as:  
\[Ez_{0}(x,y,z) = \tag{21}\] \[*\exp\{j2\pi\big{[}f_{x}(x-x_{0})+f_{y}(y-y_{0})\big{]}\big{]}d _{f}x_{d}f_{d}x_{0}dy_{0}\] (22) \[Ez_{1}(x,y,z) =\] (23) \[*\exp\{j2\pi\big{[}f_{x}(x-x_{t})+f_{y}(y-y_{t})\big{]}\big{]}d _{f}x_{d}f_{d}y_{d}x_{t}dy_{t}\] (24) \[Ez_{t}(x,y,z) =\] (25) \[*\exp\{j2\pi\big{[}f_{x}(x-x_{t})+f_{y}(y-y_{t})\big{]}\big{]}d _{f}x_{d}f_{d}x_{t}dy_{t} \tag{26}\]  
where \(f_{x}\) and \(f_{y}\) are the spatial frequencies in the \(x\) and \(y\) directions, respectively, and the small incident angle (\(2.5^{\circ}\)) allows us to ignore the effects of this angle on the complex amplitude. The purpose of this laser beam combination is to overcome the limits that are set by nonlinear optical processes and to scale up the output power. In this case, the encircled power in the bucket (PIB), which is defined as the ratio of the actual energy within a circle, as determined by the \"diffraction-limited bucket\" as being the ideal energy level within the same bucket, is treated as a classical parameter for the evaluation of the laser beam quality. The PIB expression is:  
\[Ez_{t}(x,y,z) = \tag{27}\] \[*\exp\{j2\pi\big{[}f_{x}(x-x_{t})+f_{y}(y-y_{t})\big{]}\big{]}d _{f}x_{d}f_{d}x_{t}dy_{t}\]  
The numerical simulations of the PIB versus the surface roughness at a distance of 500 mm, based on Equations (21)-(25), are shown in Figure 9a. As expected, the surface roughness of the filter has a negative influence on the combined beam quality, and the PIB gradually decreases as the surface roughness increases, which is especially obvious in the case of the transmitted laser. When the surface of the filter is smooth, both the reflected laser and the transmitted laser show a typical Gaussian distribution (Figure 9b). However, with the increase in surface roughness, the ratio of scattered beam gradually increases in the combined laser beams, which reduces the power focusability for both cases. Worse still, due to the influence of scattering, the far-field intensity profiles of the reflected and transmitted lasers (Figure 9c,d) typically have a non-Gaussian distribution, and this could result in the degradation of the combined beam quality (Figure 9e).  
Meanwhile, based on the simulation results shown in Figure 10a, the propagation distance Z will enhance the effects of scattering, and this further reduces PIB. Figure 10b-f shows the far-field intensity profiles of the combined beam under different propagation distances. With an increase in the propagation distance, the intensity distribution of the combined laser beams gradually drift off from the Gaussian distribution, and a significant deterioration in the combined beam quality is observed. Hence, for a combining element with a large surface roughness, the laser beam combination system is present only within a limited available irradiation range.