## 4 Coaxiality Error Analysis  
The influence of assembly error on beam-pointing accuracy and the influence of the CCR geometrical error (planeness error and dihedral angle error) on the spot energy distribution are analyzed separately. However, the variation characteristics of the CE under the coupling effect of the two factors are unclear. Therefore, further research is necessary. In a typical APT system, the CE measurement method based on CCR calculates the deviation between the centroid of the retro-reflected laser spot and the detector's center. The determination of the retro-reflected beam-spot centroid is usually solved by the centroid method. \[x_{0}=\left(\sum\limits_{i=1}^{m}\sum\limits_{j=1}^{n}iI(x,y)\bigg{/}\sum \limits_{i=1}^{m}\sum\limits_{j=1}^{n}I(x,y)\right),y_{0}=\left(\sum\limits_{ i=1}^{m}\sum\limits_{j=1}^{n}jI(x,y)\bigg{/}\sum\limits_{i=1}^{m}\sum\limits_{j=1}^{n}I(x,y)\right) \tag{16}\]  
Then, the CE can be calculated based on the angle measurement principle. The CE caused by the CCR and assembly error simulation is performed using Zemax software. We assume that the convergence lens is ignored in the system and that all the optical elements have no deformation. In each simulation group, we suppose the CCR geometrical error is fixed and gradually increases the assembly error of different APT elements. The distance of the beam's route from where it is being emitted to where it is being reflected by the CCR and, lastly, where it is being received by the detector is 60 mm, the size of the detector is 6 mm \(\times\) 6 mm, the pixel number is 350 \(\times\) 350, and the size of each pixel is approximately 17 um in the simulation. We record the centroid position of the facula on the detector surface with or without CCR, and then calculate the CE based on the two facula centroid positions and the length of the optical path. The results can be viewed in Tables 2-7. Compared to the data in Tables 2 and 6, the planeness error has a great influence on CE, and the error increases by one order of magnitude when other conditions are equal, compared to Figures 2 and 3. Figure 5: Influence of assembly error on beam-pointing accuracy analysis: (**a**–**c**) the influence on outgoing beam; (**d**–**f**) the influence on incoming beam analysis. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \multicolumn{4}{c}{**Assembly Elements**} \\ \cline{2-5}
**Assembly Error (\({}^{\circ}\))** & \(M_{1}\) & \(M_{2}\) & \(Az\) FM & \(EI\) FM \\ \cline{2-5}
**Assembly Error (\({}^{\circ}\))** & \(M_{1}\) & \(M_{2}\) & \(Az\) FM & \(EI\) FM \\ \hline \multicolumn{5}{c}{**Coaxiality Error (Urad)**} \\ \hline
0.001 & 18 & 18 & 23 & 10 \\
0.002 & 37 & 17 & 36 & 9 \\
0.003 & 61 & 13 & 53 & 10 \\
0.004 & 83 & 19 & 61 & 10 \\
0.005 & 102 & 21 & 71 & 13 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Spot centroid error at \(\delta 1=\delta 2=\delta 3=0.001^{\circ}\) and N1 = N2 = N3 = 0. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \multicolumn{4}{c}{**Assembly Elements**} \\ \cline{2-5}
**Assembly Error (\({}^{\circ}\))** & \(M_{1}\) & \(M_{2}\) & \(Az\) FM & \(EI\) FM \\ \cline{2-5}
0.001 & 39 & 29 & 38 & 19 \\
0.002 & 73 & 31 & 40 & 31 \\
0.003 & 77 & 29 & 61 & 33 \\
0.004 & 107 & 30 & 75 & 43 \\
0.005 & 133 & 25 & 86 & 34 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Spot centroid error at \(\delta 1=\delta 2=\delta 3=0.003^{\circ}\) and N1 = N2 = N3 = 0. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \multicolumn{4}{c}{**Assembly Elements**} \\ \cline{2-5}
**Assembly Error (\({}^{\circ}\))** & \(M_{1}\) & \(M_{2}\) & \(Az\) FM & \(EI\) FM \\ \cline{2-5}
0.001 & 112 & 86 & 91 & 79 \\
0.002 & 138 & 85 & 98 & 81 \\
0.003 & 165 & 90 & 112 & 79 \\
0.004 & 192 & 84 & 111 & 83 \\
0.005 & 232 & 88 & 130 & 82 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Spot centroid error at \(\delta 1=\delta 2=\delta 3=0^{\circ}\) and N1 = N2 = N3 = 0.01. In contrast, each element assembly error of the APT has an inconspicuous different effect on coaxiality.