# Research on Nanometer Precision Measurement Method of High Order Even Aspheres
## 2 Methods  
### Null Compensation Interferometry with a CGH  
The null test is the preferred method of optical measurement. For optical aspheres, null optics are required to achieve the null test. CGH is a commonly used compensator, which works like a diffraction grating with variable period and has curves in the pattern [23]. The phase of the incident wavefront passing through the CGH is modulated and transformed into a test wavefront that is consistent with the ideal surface shape of the test optical asphere, thus accomplishing the null test of the asphere. Therefore, the design of the phase function is crucial for a CGH to work properly and determines the diffraction pattern on the CGH. The diffraction pattern is a contour plot of the phase function intercepted according to a certain duty cycle. A single etched period on the diffraction pattern allows a phase modulation of the incident wavefront by 2 m\(\pi\) where m is the order of diffraction. CGHs are categorized into amplitude-type and phase-type. Diffraction patterns on the substrate are created using raster scanning beam writing. A phase-type CGH is used in this paper and its phase distribution of the test hologram is shown in Figure 1. The design parameters of the test hologram are shown in Table 1.  
A CGH often uses its \(+1\) order diffraction for wavefront transform. However, other diffraction orders, such as 0 order, \(-1\) order, etc., can have an effect on \(+1\) order diffraction, creating ghost images in the interferometer and interfering with the measurement. Adding a carrier to the phase function of CGH to separate the other diffraction orders can effectively eliminate the effect of ghost images. Tilt carriers and power carriers are common choices [24; 25]. A tilt carrier of \(1.45^{\circ}\) is applied to the CGH used in this paper to eliminate the effects of ghost images. Moreover, a pinhole with a radius of 3 mm is used to examine the ghost image in the Multi-Configuration of Zemax (v.2020), which shows that the applied tilt carrier eliminates the effect of other interfering diffraction orders.  
Figure 2 shows the optical layout for the null test of an optical asphere with a CGH in this paper. In addition to the hologram used for the null test, alignment hologram and positioning hologram are also designed on the CGH [26; 27]. The alignment hologram is used to adjust the relative position of the CGH to TF or TS. The positioning hologram is used to adjust the relative position of the test surface to the CGH. Although the diffractive structure period of CGH is larger at spherical wavefront incidence, the optical cavity length is too long in this case, which is about 500 mm. Consequently, a 6-inch TF is applied to produce parallel light incident on the CGH. The substrate of CGH is made of quartz with a certain thickness. When parallel light enters the substrate from the air, there is some distortion in the incident wavefront due to the change in refractive index. In order to eliminate the effects of substrate thickness and refractive index variations, the diffraction pattern was fabricated on the lower surface of the substrate. The optical layout of measurement and the CGH are designed in Zemax (v.2020).  
\begin{table}
\begin{tabular}{c c c c} \hline
**Design Parameter** & **Setup** & **Design Parameter** & **Setup** \\ \hline Type & Phase-type & Average period & 0.0073 mm \\ Size & 89.2 mm & Carrier frequency & Tilt with \(1.45^{\circ}\) \\ Minimum period & 0.0018 mm & Etching depth & 0.5 μm \\ Maximum period & 2.7045 mm & Duty cycle & 0.5 \\ \hline \end{tabular}
\end{table}
Table 1: The design parameters of the test hologram.  
Figure 1: The phase distribution of the test hologram on the CGH.  
Parallel light selected as the incident wavefront is also intended to facilitate the subsequent calibration of the substrate error of the CGH by absolute testing using the same optical layout [28].