## 3 Measuring Process and Results
### Measuring the System Construction  
A five-axis precision motion system has high precision motion shafting and stability and is widely used in ultra-precision measurement fields. The experiment carried out the measurements on an ultra-precision machine tool with a confocal probe and the measured DOE to verify the measurement method proposed in the paper. An ultra-precision single-point diamond machine tool was used as a multi-axis system platform. With the help of the stability of the ultra-precision machine tool, the systematic errors in the measured data were minimized, to ensure the measurement accuracy of the diffraction structure. Figure 9 shows a DOE measurement system built based on the [PERSON] (Swanzey, NH, USA) Nanotech 350 ultra-precision single point diamond lathe. The confocal probe was driven by the machine tool to measure the curve-based DOE, and to realize the three-dimensional topography measurement of the complete DOE with a constant contour scanning speed (0.5 mm/min). The system consisted of an aerostatic spindle and a vacuum chuck, three orthogonal hydrostatic guides (X-axis, Y-axis, and Z-axis), a precision platform y-axis parallel to the Y-axis and fixed on the B-axis, and a rotating A-axis mounted on the y-axis; the confocal probe was fixed on the A-axis, and the curved-base DOE was fixed on the vacuum chuck. The optical axis of the probe was aligned with the Z direction through the A-axis and B-axis adjustment, and the height of the probe was adjusted by the Y- and the y-axes. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Times** & **Height (\(\mu\)m)** & **Error (nm)** \\ \hline
1 & 10.0065 & +30.5 \\
2 & 10.0073 & +31.3 \\
3 & 10.0168 & +40.8 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The results of an accuracy verification of the probe. Figure 8: Accuracy verification of probe. Figure 9: The experimental setup for measuring the curved substrate DOE. As mentioned in Section 2, the optical axis of the probe needed to be calibrated before the measurements. The data recorded by the confocal probe were a point cloud consisting of tens of thousands of points with height information. That is to say, each point had only z-value coordinates, which needed to match the workpiece or the probe movement (x, y) coordinates to form 3D data. As shown in Figure 10, the red line indicates that the probe scan path is a truncated line passing through the diameter of the base vertex, and the gray line indicates that the probe scan path is a truncated line at a position \(10^{\circ}\) away from the base vertex. The data obtained after the probe scan were the measured data with structural information flattened along the base arc length. The data point cloud of two measurements was normalized to the same coordinate system, it was found that there was a certain deviation in the scanning measured data of the probe without crossing the base vertex. This was because the probe measured data were not the maximum arc length across the vertex. When the coordinate transformation was restored to the actual size, the pitch became shorter and the feature information of the curve-based DOE could not be correctly evaluated. Therefore, correcting the relative position of the probe to the measured DOE and making the probe measurement path go through the center vertex of the curved base were important steps in the measurement scheme. In the experiment, the standard ball was used to complete the alignment of the probe with the measured DOE center, as shown in Figure 11. After the standard sphere was corrected by an inductance meter, the system first controlled the probe to move along the X-axis (horizontal direction) and record the height measured data on a line. The data could be obtained by fitting a maximum value. The X-coordinate of the extreme value was the X-coordinate of the center of the standard sphere. Then, at the X-coordinate, the system controlled the probe to move along the Y-axis (vertical direction) and record the height measured data on another line.