## 3 Calibration of the Measurement System  
### Calibration of the Measuring Probes  
To implement the measurement of the inner profile of the high aspect ratio aspheric workpiece, the lateral displacement gauge and the axial displacement gauge must have high measurement accuracy and good linearity of tens of nanometers, especially for the lateral displacement gauge due to the nonlinearity of its inner structure and its output circuit. The calibration procedure for the lateral displacement gauge is shown in Figure 5.  
Figure 4: Schematic of the misalignment, roundness errors and motion errors associated with the measurement system. The lateral displacement gauge is mounted on the \(Z\)-slide and a standard gauge block is mounted on the \(Y\)-slide for calibration. To calibrate the horizontal measurement of the lateral displacement gauge, as shown in Figure 5a, the measurement output of the lateral displacement gauge varies from \(S_{i}\) to \(S_{i+1}\) while the \(Y\)-slide moves from \(Y_{i}\) to \(Y_{i+1}\), and the measurement error of the lateral displacement gauge can be evaluated as:  
\[e_{i}=(S_{i+1}-S_{i})-(Y_{i+1}-Y_{i}) \tag{5}\]  
Taking a series of samples within the measurement range of the lateral displacement gauge, the measurement error of each measurement output can be evaluated. It should be noted that the resolution of the \(Y\)-slide is 20 nm, and the repetition error of the \(Y\)-slide is \(\pm\)5 nm. The errors of \(\pm\)25 nm of the \(Y\)-slide are much smaller than the errors of the lateral gauge, which is \(\pm\)293 nm before calibration, according to results shown in Section 4.2. Furthermore, the calibration procedure should be carried out several times to achieve a stable calibration accuracy to reduce the effect of the errors of the \(Y\)-slide. In addition, the structure of the lateral gauge looks similar to a simple pendulum. During the measurement, there are crosstalk displacements of the probing sphere in the \(Y\)-direction and \(Z\)-direction with respect to the gauge body. The displacement in \(Y\)-direction is measured by the lateral gauge, while the displacement in the \(Z\)-direction is ignored. Thus, the crosstalk displacement in the \(Z\)-direction should be calibrated. As shown in Figure 5b, the crosstalk output of the lateral displacement gauge with respect to the movement along the \(Z\)-slide is calibrated by linking the gauge probe with the \(Y\)-slide by a spring. The little spring pushes the probe of the lateral gauge to the right side and keeps the probe in contact with the standard gauge block. When the \(Z\)-slide moves down from \(Z_{j}\) to \(Z_{j+1}\), the output correspondingly varies from \(S_{j}\) to \(S_{j+1}\). Thus, the relationship between the lateral gauge output and its vertical movement can be obtained for compensation:  
\[\Delta Z_{j}=\frac{Z_{j+1}-Z_{j}}{S_{j+1}-S_{j}}\Delta S_{j} \tag{6}\]  
Moreover, the lateral displacement gauge and the axial displacement gauge should be carefully mounted vertically to reduce the cosine error [25]. To evaluate the misalignment error of the two displacement gauges, a reference sphere with a radius of \(R\) is used, as shown in Figure 6. Tilt alignment of the lateral displacement gauge with a roll angle in the \(YZ\)-plane of \(\alpha_{A}\) is shown in Figure 6a. Firstly, the lateral probe is moved along the \(X\)-axis to find the minimum measurement output position and the \(YZ\)-plane of this position is the \(YZ\) central cross-section plane of the reference sphere. Figure 5: Schematic of calibration for the measurement of the lateral displacement gauge. (**a**) Horizontal calibration for the lateral probe; (**b**) vertical calibration for the lateral probe. The lateral displacement gauge, which is moved along the \(Z\) direction, is employed to measure the reference sphere from the position \(Z_{A0}\), which has the minimum output of \(S_{A0}\) to the position \(Z_{An}\), with the output of \(S_{An}\) reaching the limited value of the measurement range.