## 3 Error Modeling and Analysis
### Geometrical Error of CCR  
The CCR is a tetrahedral prism with three orthogonal surfaces. If the beam incidence is directed toward the CCR, it will be retro-reflected by three mirrors and produce six reflection sequences (123, 132, 213, 231, 312, and 321). Hence, the spot will be divided into six equally sized sectors (shown in Figure 2a), and the outgoing beam will be parallel to the income beam but in the opposite direction. As mentioned earlier, there are two kinds of machining errors along with the CCR: dihedral error angle and planeness error. Generally, as shown in Figure 2a, the three surfaces obey the following rule: OAB\(\perp\)OBC\(\perp\)OAC. However, if there is a dihedral angle error, the angle ceases to be \(90^{\circ}\) and will be \(90^{\circ}-8_{\rm ij}\) instead (where \(\delta_{\rm ij}\) represents the error angle between OAC and OBC, OAB and OAC, and OAB and OBC, and i and j = 1, 2, 3). Figure 1: Coaxiality measurement schematic based on CCR. Figure 2b shows the geometrical error of the CCR, where W(x, y) is the wavefront of the reflected beam when there is no geometrical error, and n is the normal. Accordingly, W(x', y') represents the distorted wavefront of the reflected beam from the CCR along with the dihedral angle error and planeness error, and n' is the normal. Thus, from Figure 2b, the optical path difference (OPD) induced by the CCR geometrical error can be described as  
\[\Delta d=n(\overline{BC}-\overline{AB}-\overline{AD}). \tag{1}\]  
Because the geometrical error has two parts, the optical path difference can therefore be written as  
\[\Delta d=ds+dt. \tag{2}\]  
In Equation (2), \(ds\) represents the OPD induced by the dihedral angle error, and \(dt\) represents the OPD induced by the planeness error. When a ray of light is reflected, the reflection process can be described as  
\[A^{\prime}=A-2(A\cdot N)N. \tag{3}\]  
In Equation (3), \(A\) = (\(Ax\), \(Ay\), \(Az\)) is the vector of the incoming beam, and \(A^{\prime}\) = (\(A^{\prime}x\), \(A^{\prime}y\), \(A^{\prime}z\)) is the vector of the reflected beam. Suppose that \(\delta_{12}\), \(\delta_{23}\), and \(\delta_{31}\) are the dihedral angle error of OAC and OBC, OAB and OAC, and OAB and OBC, respectively, then the normal of Surfaces I, II, and III can be obtained. \[ \begin{array}{l}N_{1}=(\cos(\delta_{12}),\,0,\,\sin(\delta_{12}))\,\times \,(\cos(\delta_{31}),\,\sin(\delta_{31}),\,0)\\ N_{2}=(\sin(\delta_{12}),\,0,\,\cos(\delta_{12}))\,\times\,(0,\,\sin(\delta_{ 23}),\,\cos(\delta_{23})).\\ N_{3}=(\sin(\delta_{31}),\,\cos(\delta_{31}),\,0)\,\times\,(0,\,\cos(\delta_{ 23}),\,\sin(\delta_{23}))\end{array}  \tag{4}\]  
Because \(\delta_{12}\), \(\delta_{23}\), \(\delta_{31}\) is very small, the normals \(N_{1}\), \(N_{2}\), and \(N_{3}\) can thus be rewritten as \(N_{1}\) = (\(\delta_{12}\), 0, \(\delta_{12}\)) \(\times\) (\(\delta_{31}\), \(\delta_{31}\)); \(N_{2}\) = (\(\delta_{12}\), 0, \(\delta_{12}\)) \(\times\) (0, \(\delta_{23}\), \(\delta_{23}\)); and \(N_{3}\) = (\(\delta_{31}\), \(\delta_{31}\), 0) \(\times\) (0, \(\delta_{23}\), \(\delta_{23}\)). In this way, the laser's vector going through the three reflectors can be written as follows:  
\[\text{Through Surface I}:\ A^{\prime}{}_{11}=A_{11}-2(A_{11}\cdot N_{1})N_{1}; \tag{5}\]  
\[\text{Through Surface II}:\ A^{\prime}{}_{12}=A_{12}-2(A_{12}\cdot N_{2})N_{2}= A^{\prime}{}_{11}-2\big{(}A^{\prime}{}_{11}\cdot N_{2}\big{)}N_{2}; \tag{6}\]  
\[\text{Through Surface III}:\ A^{\prime}{}_{23}=A_{23}-2(A_{23}\cdot N_{3})N_{3}= A^{\prime}{}_{12}-2\big{(}A^{\prime}{}_{12}\cdot N_{3}\big{)}N_{3}. \tag{7}\]  
Figure 2: The CCR schematic and its geometrical error. (**a**) CCR diagram; and (**b**) diagram of CCR geometrical error. The derived out-ray vectors need to convert into the coordinate in Figure 2b through the transformation described in [16] so that the wavefront solving model can be deduced. \[mx+ny+lz=0 \tag{8}\] \[z=ds=-(mx+ny)/l,\]  
where \(m\), \(n\), and \(l\) are the three components of the vector \(A^{\prime}{}_{23}\).