## 2 System and Principle
### Optimization of Measurement Configuration  
A charge-coupled device (CCD) camera (pixel number 2592 \(\times\) 1944, pixel size 2.2 \(\upmu\)m \(\times\) 2.2 \(\upmu\)m) was used as a laser receiver for the convenience of subsequent signal processing, in which the intensity of the spot image was represented by a pixel value (ranging from 0 to 255). However, the quality of the beam spot, which converged on the camera, was worse when it was formed by the rotating paraboloid rather than by a plane. If the measuring beam was reflected by a plane, the beam spot at the focal plane of the convergent lens converged to a point, as shown in Figure 4a. Unfortunately, when the measuring beam was reflected by the rotating paraboloid, the imaging light spot was dispersed, which caused difficulty in calculating the accurate position of the center point, as shown in Figure 4b. In order to clearly show the shape of the spot in Figure 4b, we showed the image near the spot, which caused the spot to seem bigger than its normal size. Figure 2: Schematic diagram of the slope sensor. The relationship between the size of the imaging spot and the diameter of the light spot on the specimen is depicted in Figure 5a. The following equation can be obtained by analysing the edge position of the beam:  
\[d_{2}=sd_{1} \tag{11}\]  
where \(d_{1}\) and \(d_{2}\) are the diameters of the measuring beam and imaging spot, respectively, and \(s\) is the sensitivity (\(s=f_{c}/f_{r}\)) of the measurement system. Equation (11) shows that the diameter of the imaging spot is linearly related to the diameter of the measuring beam. Generally, the accuracy of the system can be improved by increasing the sensitivity of the system appropriately, which means that the system sensitivity can be improved by increasing the focal length of the convergent lens or by reducing the focal length of the rotating paraboloid. However, the higher the sensitivity of the system, the larger the size of the imaging spot, which reduces the detecting accuracy of the spot center. In addition, the measuring range of the system will be reduced due to the reduction of the focal length of the rotating paraboloid. Therefore, there are several inadequacies when the collimated light beam is used as the measuring beam:  
Figure 4: The imaging light spot: (**a**) The imaging light spot reflected by a plane; (**b**) the imaging light spot reflected by a rotating paraboloid. Figure 3: Schematic diagram of relative measurement. (1) The poor quality of the imaging spot will cause a reduction of measurement accuracy;  
(2) Although the optimization of the imaging spot can be achieved by reducing the diameter of the measuring beam, the smaller the diameter of the measuring beam, the more sensitive the system is to interference factors, such as tiny scratches and dust on the surface of the rotating paraboloid;  
(3) A larger imaging spot can reduce the effective working range of the slope sensor. According to the above analysis, the quality of the imaging spot is poor when the collimated light beam is reflected by a rotating paraboloid. Therefore, the system has been optimized by using a focused light beam instead of a collimated light beam. The schematic of the optimized measurement system is shown in Figure 5b. First of all, according to the geometrical principle of reflection, the relationship between angles can be obtained as:  
\[\beta=2\alpha+\gamma \tag{12}\]  
where \(\alpha\) is the inclination angle at the measuring point on the rotating paraboloid, \(\beta\) is the angle between the reflected beam at the measuring point and the \(z\)-axis, and \(\gamma\) is the divergence angle of the measuring beam.