# Research on Nanometer Precision Measurement Method of High Order Even Aspheres
## 3 Experiments and Results
### Results and Discussion  
In the initial stage of processing optical aspheres, turning and grinding are applied to complete the rough machining of the blank. After that, Single-Point Diamond Turning (SPDT), ultra-precision grinding, and other technologies are used to complete the finishing processing. At this stage, the RMS of the surface shape error of the optical asphere can converge from several wavelengths to within one wavelength. When interferometry is performed, the fringes can be very dense and make it difficult to analyze, resulting in incomplete measurement results. A non-contact profiler is a good choice to solve this problem, which can quickly and accurately measure the error of the surface shape of complex optical components. The non-contact profiler from DUI is employed to measure the surface shape of the optical asphere during the stage of MRF. The result is shown in Figure 6 and the uncertainty of measurement is within 15 nm. Figure 4: The setup of the optical asphere using the N-position method. (**a**) The optical layout. The red line represents the test beam and the arrow points in the direction of the original position; (**b**) the actual experimental setup. Figure 5: The setup of the high-precision flat using the shift-rotation method. (**a**) The optical layout. The red line represents the test beam and the arrow points in the direction of the original position; (**b**) the actual experimental setup. With the gradual convergence of the RMS of the surface shape error, the optical asphere has to enter the stage of IBF. IBF accomplishes the accurate removal of material from a defined location on the surface. The accuracy of non-contact profilers is no longer sufficient to meet the requirements. The null test with CGHs is desired now, which is carried out in a 6-inch Fizeau interferometer that works at 632.8 nm. Mapping distortion can also affect the accuracy of material removal. The results before and after mapping distortion correction are shown in Figure 7, and mapping distortion correction is completed in Matlab (v.R2020b). It is clear that the PV and RMS of the surface shapes before and after correction have changed because the transverse coordinates of the imaging plane have been corrected and new data points have been generated by linear interpolation. The mapping profile looks similar before and after the distortion correction. That is because the test optical asphere is rotationally symmetric and the test hologram of the CGH used in the experiment is circular. No serious mapping distortion occurs when a plane wavefront passing through a test hologram becomes an aspherical wavefront. The surface shape error of a TF PV is about \(\lambda/20\), where \(\lambda\) is 632.8 nm. The RMS of the wavefront distortion caused by the CGH substrate used in this experiment is around 6 nm. When the surface shape error of the optical asphere converges to about 10 nm, the wavefront distortion caused by the TF and CGH can no longer be ignored. Absolute testing should be applied to separate errors. Two sets of absolute testing experiments have been carried out using the method described in Section 2.3. A 4-inch TF is applied as a high-precision optical flat to calibrate the error of the CGH and the 6-inch TF and measurement results of four positions (origin position, X-position, Y-position, and rotation position) are shown in Figure 8. It is clear that the results of the four positions do not vary  
Figure 6: The measurement result of non-contact profiler. Figure 7: The results of the null test with a CGH before and after mapping distortion correction. (**a**) Before mapping distortion correction; (**b**) after mapping distortion correction. much. That is because the object that undergoes the position change is a 4-inch TF with a PV of about \(\lambda/40\), which is more accurate than the system error. Image recognition is applied to determine the amount of translation of the 4-inch TF along either the \(x\)-axis direction or the \(y\)-axis direction.