# A Micro-Coordinate Measurement Machine (CMM) for Large-Scale Dimensional Measurement of Micro-Slits
## 3 Experimental Result and Discussion
### Experiments of On-Line Qualification and Gap Width Measurement
\begin{table}
\begin{tabular}{c c c c c} \hline
**Uncertainty sources** & **Symbol** & **Value** & **Coverage Factor** & **Standard Uncertainty** \\ \hline Uncertainty in \(w_{5}\) & \(u_{\text{ass}}\) & - & - & 10.5 \\ Cosine error by the alignment of the gauge block and the probing axis & \(u_{\text{cos},\text{delta}}\) & 2.6 & \(\sqrt{3}\) & 1.7 \\ Cosine error by the alignment of the interferometer & \(u_{\text{cos},\text{laser}}\) & 0.6 & \(\sqrt{3}\) & 0.3 \\ axis and the probing axis & \(u_{\text{rel},\text{delta}}\) & 5.27 & \(\sqrt{3}\) & 3.0 \\ Absolute error of the X-PZT stage & \(u_{\text{laser},\text{revolution}}\) & 0.79 & \(\sqrt{3}\) & 0.5 \\ Resolution of the interferometer & \(u_{\text{laser},\text{linearity}}\) & 5.0 & \(\sqrt{3}\) & 2.9 \\ Thermal drift by temperature change & \(u_{\text{Thermal drift}}\) & 7.0 & \(\sqrt{3}\) & 4.0 \\ Repeatability of \(L_{34}\) & \(u_{\text{rep},\text{l3},\text{l4}}\) & 14.8 & \(\sqrt{3}\) & 8.5 \\ Uncertainty in \(D_{e}\) & - & - & 26.4 \\ Cosine error by the alignment of the gauge block and the probing axis & \(u_{\text{cos},\text{gauge}}\) & 11.0 & \(\sqrt{3}\) & 6.4 \\ Cosine error by the alignment of the interferometer & \(u_{\text{cos},\text{laser}}\) & 3.58 & \(\sqrt{3}\) & 2.1 \\ Axis and the probing axis & \(u_{\text{rel},\text{delta}}\) & 5.27 & \(\sqrt{3}\) & 3.0 \\ Abbe error of the X-perx stage & \(u_{\text{strong},\text{abs}}\) & 41.5 & \(\sqrt{3}\) & 24.0 \\ Resolution of the interferometer & \(u_{\text{laser},\text{revolution}}\) & 0.79 & \(\sqrt{3}\) & 0.5 \\ Linearity error of the interferometer & \(u_{\text{laser},\text{linearity}}\) & 5.0 & \(\sqrt{3}\) & 2.9 \\ Thermal drift by temperature change & \(u_{\text{Thermal drift}}\) & 7.0 & \(\sqrt{3}\) & 4.0 \\ Repeatability of \(L_{12}\) & \(u_{\text{rep},\text{l12}}\) & 11.5 & \(\sqrt{3}\) & 6.6 \\ Uncertainty in \(w_{b}\) & \(u_{\text{sub}}\) & - & - & 17.3 \\ Length tolerance(Calibrated gauge block) & \(u_{\text{all},\text{simulate}}\) & 20.0 & \(\sqrt{3}\) & 17.3 \\ Uncertainty in Ds & \(u_{\text{s}}\) & - & - & 1.2 \\ Uncertainty due to water layer & \(u_{\text{under}}\) & 2.0 & \(\sqrt{3}\) & 1.2 \\ Expanded uncertainty (with a coverage factor \(k=2\)) & \(U\) & - & - & 66.6 \\ \hline \end{tabular}
\end{table} 
Table 3: Uncertainty budget (unit nm). \(u_{De}\), which was the combined standard uncertainty of the on-line qualification of the effective diameter of the probe tip ball, was estimated based on Figure 10, Equations (10) and (14). As shown in Table 3, the cosine error caused by the misalignment of the gauge block and the probing axis was a relatively large uncertainty source because the \(X\)-directional servo stage was moved during the on-line qualification process. The total time for the on-line qualification of the effective diameter was approximately 250 s, during which the temperature was measured to be 22.677 \({}^{\circ}\)C \(\pm\) 0.008 \({}^{\circ}\)C. As a result, \(u_{De}\) was estimated to be 26.4 nm. \(u_{wb}\) was the length tolerance of the calibrated gauge block. According to the calibration certification of the gauge block used in the micro-CMM, \(u_{wb}\) was estimated to be 17.3 nm. \(u_{As}\) was the uncertainty source introduced by the water layer on the measured surface. It was estimated to be 1.2 nm, based on Figure 4. Consequently, the expanded uncertainty \(U\) of the gap width measurement was estimated to be 66.6 nm (\(k\) = 2), which was smaller than the allowed maximum expanded uncertainty of 100 nm. By changing the probing position along the \(Y\)- and \(Z\)-directions, the gap width uniformity was measured. Figure 16 shows the gap widths at measurement positions with an equal interval of 10 mm along the \(Y\)-direction. The measurement depth was set to be 50 \(\upmu\)m. The probing was repeated 3 times at each measurement position. The dot plot in the figure shows the mean value of the repeated probing results and the error bar represents the corresponding standard deviation at each measurement position.