## 2 System and Principle  
### Measurement Principle and Configuration  
The structure of the optical freeform surface used in this paper is a rotating paraboloid. As shown in Figure 1, a 3D Cartesian coordinate system is established whose origin coincides with the center point of the rotating paraboloid. The equation of the rotating paraboloid can be expressed as:  
\[x^{2}+y^{2}=4f_{r}z \tag{1}\]  
where \(f_{r}\) is the focal length of the rotating paraboloid.  
The derivatives of Equation (1) in the direction of \(x\) and \(y\) are calculated as following, respectively:  
\[k_{x}(x)=\frac{dz}{dx}=\frac{x}{2f_{r}} \tag{2}\]  
\[k_{y}(y)=\frac{dz}{dy}=\frac{y}{2f_{r}} \tag{3}\]  
where \(k_{x}(x)\) and \(k_{y}(y)\) are the slopes of the point \(p(x,y,z)\) on the surface in the direction of the \(x\)-axis and \(y\)-axis, respectively. Equations (2) and (3) show that the slopes \((k_{x},k_{y})\) of a point on the rotating paraboloid are linearly related to the position \(x\) and \(y\) of that point, respectively. In other words, the displacement from the center point of the rotating paraboloid in the \(x\)-\(y\) plane can be obtained from Equations (2) and (3) when the slopes \((k_{x},k_{y})\) are detected by an optical slope sensor.  
The measurement configuration used for detecting the slopes of a tangent line of a point on the surface is based on the optical structure of the autocollimator, which is shown in Figure 2.  
Taking the angle measurement in the \(xoz\) plane as an example, \(\alpha\) is the angle between the interested plane and the reference plane, \(\beta\) is the angle between the incident beam and the specular beam reflected from the interested plane. The laser spot \(p_{0}(x_{0},y_{c0})\) in the focal plane of a convergent lens is formed from the measuring beam reflected by the reference plane, while the laser spot \(p_{1}(x_{c1},y_{c1})\) is generated from the measuring beam reflected by the interested plane. According to geometric optics, the following equations can be obtained by:  
\[\tan\beta=\frac{x_{c1}-x_{c0}}{f_{c}} \tag{4}\]  
\[\alpha=\frac{\beta}{2} \tag{5}\]  
where \(f_{c}\) is the focal length of the convergent lens. The equation can be simplified when \(\beta\) is particularly small:  
\[\alpha=\frac{\beta}{2}\approx\frac{x_{c1}-x_{c0}}{2f_{c}} \tag{6}\]  
Figure 1: Three-dimensional model of rotating paraboloid.  
Replacing the plane with the rotating paraboloid, the position of the beam spot on the surface can be obtained from Equations (7) and (8):  
\[x=\frac{f_{r}}{f_{c}}(x_{c1}-x_{c0}) \tag{7}\]  
\[y=\frac{f_{r}}{f_{c}}(y_{c1}-y_{c0}) \tag{8}\]  
where \(x\) and \(y\) are the positions in the coordinate system \((x,y,z)\), as shown in Figure 2.  
As can be seen in Figure 3, if the slope sensor moves in any direction, for instance, the measurement point on the reflecting surface shifts from \(p_{s1}\) to \(p_{s2}\), then the position change of the slope sensor can be expressed as:  
\[l_{x}=x_{1}-x_{2}=\frac{f_{r}}{f_{c}}(x_{c1}-x_{c0})-\frac{f_{r}}{f_{c}}(x_{c2 }-x_{c0})=\frac{f_{r}}{f_{c}}(x_{c1}-x_{c2}) \tag{9}\]  
\[l_{y}=y_{1}-y_{2}=\frac{f_{r}}{f_{c}}(y_{c1}-y_{c0})-\frac{f_{r}}{f_{c}}(x_{c2 }-x_{c0})=\frac{f_{r}}{f_{c}}(x_{c1}-x_{c2}) \tag{10}\]  
where \((x_{c1},~{}y_{c1})\) is the center position of the imaging spot while the measuring point \(p_{s1}\) and \((x_{c2},y_{c2})\) is the center position of the imaging spot, while measuring point \(p_{s2}\).