# Research on Nanometer Precision Measurement Method of High Order Even Aspheres
## 3 Experiments and Results
### Results and Discussion
Three feature points are marked on the test surface and pictures are taken using the CCD of the interferometer. Feature points are identified by setting the gray scale thresholds and their coordinates are obtained in Matlab, as shown in Figure 9. Each time the test surface undergoes a translational transformation, feature points need to be set and their coordinates are recorded, as shown in Table 2. The effect on the absolute testing results is within 0.1 nm when the translation error is controlled within one pixel, which is shown in our previous work [31]. In the experiment of the shift-rotation method, \(\Delta x\) is 18.78 pixels, \(\Delta y\) is \(-17.70\) pixels, and \(\Delta\theta\) is \(90^{\circ}\). In the experiment of the N-position method, \(\varphi\) is \(30^{\circ}\) and the test surface is measured 12 times. Figure 10 illustrates the results of the absolute testing of the optical asphere and the system error. Since the first 200 Zernike polynomials are used to process the measurements of the shift-rotation method, the results in Figure 10b obtained in Matlab lose some mid-frequency and high-frequency information compared to Figure 8. The results obtained by the method proposed in this paper guide IBF for deterministic material removal and the RMS of the surface shape error of the optical asphere converges to 4.85 nm, as shown in Figure 11.  
Figure 8: The results of the four positions of the 4-inch TF. (**a**) Origin position, (**b**) rotation position, (**c**) X-position, and (**d**) Y-position. In summary, the proposed method consists of four stages: non-contact profilometer measurement, null compensation interferometry with a CGH, mapping distortion correction, and absolute testing. The measurement results guided deterministic material removal and successfully converged the RMS of the surface shape error of the optical asphere from 76.69 to 4.85 nm, realizing the nano-precision manufacturing of optical aspheres. We have tried to keep the effect of each error on the results within 0.1 nm, such as noise, translation error, and rotation error. A detailed error analysis of the measurement process can be found in our previous work [31]. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Unit: Pixels** & \(\mathbf{X_{1}}\) & \(\mathbf{Y_{1}}\) & \(\mathbf{X_{2}}\) & \(\mathbf{Y_{2}}\) & \(\mathbf{X_{3}}\) & \(\mathbf{Y_{3}}\) & **Average** \\ \hline Origin & \multirow{2}{*}{347.48} & \multirow{2}{*}{463.04} & \multirow{2}{*}{628.34} & \multirow{2}{*}{375.25} & \multirow{2}{*}{533.10} & \multirow{2}{*}{657.63} & \multirow{2}{*}{\multirow{2}{*}{}} \\ Position & & & & & & \\ X-Position & 328.50 & 463.12 & 609.75 & 375.17 & 514.30 & 657.65 & \\ \(\Delta x\) & \(-18.98\) & & \(-18.59\) & & \(-18.80\) & & \\ X–Y-Position & 380.99 & 385.94 & 666.65 & 539.39 & 437.19 & 722.84 & \\ Y-Position & 381.09 & 368.29 & 666.55 & 521.90 & 437.13 & 704.89 & \\ \(\Delta y\) & & \(-17.65\) & & \(-17.49\) & & \(-17.95\) & \(-17.70\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Coordinates of the feature points at translation positions. Figure 10: The results of the absolute testing of the optical asphere and the system error. (**a**) The result of the optical asphere with the N-position method, \(W_{aver}\); (**b**) the result of the system error with the shift-rotation method, \(W_{s}\). Figure 9: Image recognition of the coordinates of feature points. (**a**) Image of CCD; (**b**) coordinates of the feature points.