## 6 Experimental Verification of the Mirror Alignment Method
### Experimental Alignment of the Plane Unit
#### 6.2.2 Results and Discussion
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Metal Plane**} & \multirow{2}{*}{\begin{tabular}{c} **Before Alignment** \\ **(Measurement Value)** \\ \end{tabular} } & \multirow{2}{*}{\begin{tabular}{c} **OneT–Two Method** \\ **(Section 3.2) (Theoretical Value)** \\ \end{tabular} } & \multirow{2}{*}{\
\begin{tabular}{c} **Three–Rotation** \\ **Alignment Method** \\ **(theoretical Value)** \\ \end{tabular} } & \multirow{2}{*}{
\begin{tabular}{c} **After Alignment** \\ **(theoretical Value)** \\ \end{tabular} } \\ \hline \multirow{4}{*}{Plate 1} & \(\mathbf{N_{\text{II}}}=[0.0328,0.0041,1.0]\) & \(\mathbf{N_{\text{II}}}=[0.0333,0.0031,0.9994]\) & & \(\mathbf{N_{\text{A}\text{II}}}=[0.0026,-0.0056,1.0]\) \\ \cline{2-5}  & \(\theta=33.09\) mrad & \(\theta=33.47\) mrad & \(\theta=0.43\) mrad & \(\theta=6.16\) mrad \\ \cline{2-5}  & \(\mathbf{T_{\text{II}}}=[-457.90,286.00,-26.82]\) & \(\text{err}=1.09\) mrad & \(d_{\text{A}}=7.89\) mrad & \\ \cline{2-5}  & \(\mathbf{T_{\text{II}}}=[-180.80,289.94,-35.30]\) & & \(d_{\text{B}}=-1.45\) mrad & \\ \cline{2-5}  & \(\mathbf{T_{\text{II}}}=[-175.20,12.79,-35.12]\) & & \(d_{\text{C}}=-0.62\) mrad & \\ \hline \multirow{4}{*}{Plate 2} & \(\mathbf{N_{\text{II}}}=[-0.00201,0.0318,1.0]\) & \(\mathbf{N_{\text{II}}}=[-0.0145,0.0325,0.9994]\) & & \(\mathbf{N_{\text{A}\text{II}}}=[0,-0.0047,1.0]\) \\ \cline{2-5}  & \(\theta=37.59\)mrad & \(\theta=35.62\) mrad & \(\theta=0.13\) mrad & \(\theta=4.69\) mrad \\ \cline{2-5}  & \(\mathbf{T_{\text{II}}}=[-183.00,705.04,-31.53]\) & \(\theta\text{err}=5.67\) mrad & \(d_{\text{A}}=-8.70\) mrad & \\ \cline{2-5}  & \(\mathbf{T_{\text{II}}}=[-182.60,429.10,-23.60]\) & & \(d_{\text{B}}=0.40\) mrad & \\ \cline{2-5}  & \(\mathbf{T_{\text{II}}}=[-458.80,429.70,-28.88]\) & & \(d_{\text{C}}=-3.76\) mrad & \\ \hline \hline \end{tabular} Note: The angle \(\theta\) is the intersection angle between the measurement result (\(\mathbf{N_{\text{II}}}\) or \(\mathbf{N_{\text{A}\text{II}}}\)) or theoretical calculation result (\(\mathbf{N_{\text{I}}}\)) and ideal axis vector \(\mathbf{N_{\text{I}}}\) (\(\mathbf{N_{\text{II}}}\) = [0, 0, 1.0]); the angle \(\theta_{\text{err}}\) is the intersection angle between the measurement result (\(\mathbf{N_{\text{II}}}\)) and theoretical calculation result (\(\mathbf{N_{\text{I}}}\)).  
\end{table} 
Table 3: Experimental data from the facet alignment of the metal plane using the three-rotation alignment method. the three-rotation alignment method, the error angle \(\theta\) of the axis vector was decreased from 33.09 to 6.16 mrad and 37.59 to 4.69 mrad for plate 1 and 2, respectively (Table 3).Using the two-rotation alignment method, the error \(\theta\) was decreased from 26.75 to 6.26 mrad and 46.11 to 2.05 mrad for Plate 1 and 2, respectively (Table 4). The fitted plane for plate 1 and 2 before and after adjustment is shown in Figure 10 and their fitted plane equation as shown in Equations (28)-(31) respectively, where Equations (28) and (29) is fitted plane equation of plate 1 and 2, respectively, before adjustment, and Equations (30) and (31) is fitted plane equation of plate 1 and 2, respectively, after adjustment. These normal vectors of the fitted planes are used for determining the axis vector of the metal plane as shown in Table 3.  
\[0.0328x+0.0041y+z+46.21=0 \tag{28}\]  
\[-0.0201x+0.0318y+z+11.70=0 \tag{29}\]  
\[0.0026x-0.0056y+z+41.46=0 \tag{30}\]  
\[-0.0047y+z+31.13=0 \tag{31}\]  
For the facet alignment measurement of the metal plane, the intersection angle \(\theta_{\rm err}\) between the normal vector of the fitting plane \(\mathbf{N}_{\rm t1}\) and theoretically calculated \(\mathbf{N}_{\rm t}\) (using the mirror facet measurement method in Section 3) was 1.09 mrad in plate 1 (Table 3). Thus, the deviation between the normal vectors was very small. The results have shown that the mirror facet measurement method presented in this paper is valid.