# A Micro-Coordinate Measurement Machine (CMM) for Large-Scale Dimensional Measurement of Micro-Slits
## 3 Experimental Result and Discussion
### Evaluation of Alignment Errors of the Measurement System
Figure 8 shows a schematic of the cosine error between the laser interferometer and \(X\)-directional moving axis. \(L\) is the distance measured by the laser interferometer. \(L^{\prime}\) is the actual displacement of the probe tip along the \(X\)-direction. \(L^{\prime}/L\) can be expressed by the following equation. \[\frac{L^{\prime}}{L}=\frac{1}{\cos\theta_{y\_mirror}\cos\theta_{z\_mirror}} \tag{6}\]  
In order to evaluate the alignment errors of the reflective mirror with respect to the laser interferometer, the tilt errors of the reflective mirror were measured by using the optical fiber displacement probe. Figure 9 shows a schematic diagram for evaluation of the alignment errors of the reflective mirror. One plane of the right-angle prism mirror is arranged perpendicular to the \(X\)-axis of the PZT stage. The difference of the tilting angles between the prism mirror and the reflective mirror is measured. Table 1 shows a summary of the evaluated angular alignment errors of the developed micro-CMM. Figure 6: Schematic for evaluation of the angular alignment errors: (**a**) \(X\)-directional stages; (**b**) \(Z\)-directional stage. Based on the evaluated alignment errors of the positioning stages, the geometric measurement errors are investigated. Figure 10 shows a geometrical model in the \(X\)-\(Y\) plane during the probing on the calibrated artifact. \(L_{gb}\) is the distance between the probing points on the calibrated artifact. \(h\) is the distance between the moving axes of the \(X\)-directional PZT stage on the both sides of the gauge block. \(D_{s}\) is the \(X\)-directional displacement of the servo motor stage. The thickness of the calibrated artifact \(w_{b}\) can be expressed by the following equation. \[w_{b}=\left(L_{gb}-L^{\prime}\right)\cos\left(\theta_{z\_X-PZT}-\theta_{gb}\right) \tag{7}\]  
\(L^{\prime}\) can be defined as follows. \[L^{\prime}=h\ \tan\left(\theta_{z\_X-PZT}-\theta_{gb}\right) \tag{8}\]  
\(h\) can be calculated by  
\[h=D_{s}\ \sin\left(\theta_{z\_X-Servo}-\theta_{z\_X-PZT}\right) \tag{9}\]  
Therefore, the measurement error \(\Delta w_{b}\) of the calibrated artifact width, which is also extended in the \(Y\)-\(Z\) plane, can be calculated by the following equation. \[ \begin{array}{ll}\Delta w_{b}&=\left(L_{gb}-D_{s}\right)-w_{b}\\ &=\left(\frac{w_{b}+D_{s}\ \tan(\theta_{z\_X-servo}-\theta_{y\_X-PZT}) \tan(\theta_{y\_X-PZT}-\theta_{gb})}{\cos(\theta_{z\_X-PZT}-\theta_{gb})}-w_{ b}\right)\\ &+\left(\frac{w_{b}+D_{s}\ \tan(\theta_{z\_X-servo}-\theta_{z\_X-PZT}) \tan(\theta_{z\_X-PZT}-\theta_{gb})}{\cos(\theta_{z\_X-PZT}-\theta_{gb})}-w_{ b}\right)\\ \end{array}  \tag{10}\]  
Figure 11 shows a geometrical model in the \(X\)-\(Y\) plane. The measurement error \(\Delta w_{s}\) of the slit gap width can be calculated by the following equation. \[\Delta w_{s}=\frac{w_{s}}{\cos\left(\theta_{y\_X-PZT}-\theta_{y\_sd}\right) \cos\left(\theta_{z\_X-PZT}-\cos\theta_{z\_sd}\right)}-w_{s} \tag{11}\]  
Figure 10: Geometrical model of the alignment error of the probing points during the qualification procedure. Since the gap width measurement is carried out by the probe displacement with the laser interferometer in the \(X\)-direction, the Abbe errors for the laser interferometer have to be considered for the evaluation of the combined standard uncertainty. When the center of the probe tip is not located on the optical axis of the laser interferometer, an Abbe error component will be caused in the measurement result of the probe displacement due to the tilt errors of the \(X\)-directional stages. Figure 12 shows the Abbe error between the probe tip and the laser interferometer. As shown in Figure 12, \(a_{1}\) indicates the \(X\)-directional distance between the probe tip and the reflective mirror. \(a_{2}\) is the \(Z\)-directional difference between the center of the probe tip and that of the laser spot on the reflective mirror. \(\theta_{Abbe}\) is the tilt error of the \(X\)-directional stages. The Abbe errors \(L_{y\_Abbe}\) and \(L_{z\_Abbe}\) can be expressed as follows.