# Research on Nanometer Precision Measurement Method of High Order Even Aspheres
## 2 Methods
### Absolute Testing of Optical Asbjres  
Absolute testing can improve measurement accuracy down to the nanometer scale. An absolute testing method suitable for rotationally symmetric aspherical surfaces is used in this paper, which is a combination of the shift-rotation method and the N-position method. Details can be found in previous work [31].  
The measurement result of the null test on an optical asphere with a CGH can be expressed as follows:  
\[W(x,y)=W_{R}(x,y)+W_{CGH}(x,y)+W_{T}(x,y) \tag{4}\]  
where \(W_{R}(x,y)\) is the deviation of the reference surface, \(W_{CGH}(x,y)\) is the wavefront aberration due to the CGH, and \(W_{T}(x,y)\) is the surface shape of the test surface.  
\[\left\{\begin{aligned} W_{1}(\rho,\theta)&=W_{R}( \rho,\theta)+W_{CGH}(\rho,\theta)+W_{T}(\rho,\theta)\\ W_{2}(\rho,\theta+\varphi)&=W_{R}(\rho,\theta)+W_{CGH}( \rho,\theta)+W_{T}(\rho,\theta+\varphi)\\ &\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad \qquad\qquad\vdots\\ W_{N}(\rho,\theta+(N-1)\varphi)&=W_{R}(\rho, \theta)+W_{CGH}(\rho,\theta)+W_{T}(\rho,\theta+(N-1)\varphi)\end{aligned}\right. \tag{5}\]  
where \(N=\frac{2\pi}{\varphi}\) and \(\varphi\) is the angle at which the test surface rotates around the optical axis each time.  
Figure 3: Tracing rays in mapping distortion correction.  
A reverse rotation operation is performed on Equation (5), after which all the results are summed and averaged to obtain the following:  
\[W_{aver}(x,y)=W_{R}^{sym}(x,y)+W_{CGH}^{sym}+W_{T}(x,y) \tag{6}\]  
where \(W^{sym}\) is the rotationally symmetric component of the wavefront distortion.  
Subsequently, the test asphere is replaced with a high-precision optical flat. Measurements at 4 positions are taken to calibrate the errors of the CGH and the reference surface. The results obtained are denoted as follows:  
\[\left\{\begin{aligned} & W_{1}(x,y)=W_{R}(x,y)+W_{CGH}(x,y)+W_{F}(x,y)\\ & W_{2}(\rho,\theta+\Delta\theta)=W_{R}(\rho,\theta)+W_{CGH}(\rho, \theta)+W_{F}(\rho,\theta+\Delta\theta)\\ & W_{3}(x+\Delta x,y)=W_{R}(x,y)+W_{CGH}(x,y)+W_{F}(x+\Delta x,y) \\ & W_{4}(x,y+\Delta y)=W_{R}(x,y)+W_{CGH}(x,y)+W_{F}(x,y+\Delta y) \end{aligned}\right. \tag{7}\]  
where \(W_{F}(x,y)\) is the surface shape error of the test optical flat, \(\Delta x\) is the distance of the test surface translated along the \(x\)-axis direction, \(\Delta y\) is the distance of the test surface translated along the \(y\)-axis direction, and \(\Delta\theta\) is the angle of the test surface rotated about the optical axis.  
Reverse the results of Equation (7) and subtract the last three terms from the first to obtain the following:  
\[\left\{\begin{aligned} & W_{2}^{\prime}(\rho,\theta)-W_{1}(\rho,\theta)=W_{R} (\rho,\theta-\Delta\theta)-W_{R}(\rho,\theta)+W_{CGH}(\rho,\theta-\Delta\theta )-W_{CGH}(\rho,\theta)\\ & W_{3}^{\prime}(x,y)-W_{1}(x,y)=W_{R}(x-\Delta x,y)-W_{R}(x,y)+W_ {CGH}(x-\Delta x,y)-W_{CGH}(x,y)\\ & W_{4}^{\prime}(x,y)-W_{1}(x,y)=W_{R}(x,y-\Delta y)-W_{R}(x,y)+W _{CGH}(x,y-\Delta y)-W_{CGH}(x,y)\end{aligned}\right. \tag{8}\]  
The surface shape is reconstructed by Zernike polynomials and the rotationally symmetric components are extracted as follows:  
\[W_{s}^{sym}(x,y)=W_{R}^{sym}(x,y)+W_{CGH}^{sym}(x,y) \tag{9}\]  
The absolute surface shape of the test asphere \(W_{T}(x,y)\) can be obtained by subtracting Equation (6) from Equation (9).