# A Differential Measurement System for Surface Topography Based on a Modular Design
## 3 Motion Error Compensation Based on a Film Interferometer  
As described in the section above, the motion error caused by the linear guide represented by \(h_{2}\) was measured by an in-house developed film interferometer. The principle is shown in Figure 4.  
Figure 3: Geometric error factors: (**a**) Straightness and flatness, (**b**) Tilting angle. The key specifications of the components used in this system are listed in Table 1.  
As shown in Figure 4a, the film interferometer included two optical components, the optical flat and the right-angle prism. The optical flat moved with the stage and the right-angle prism was stationary. The two optical surfaces formed an air gap between themselves with a gradually changing thickness. The incident laser beam was split and reflected by the two optical surfaces. When the angle of the air gap was very small, the reflected laser beams propagated along the same path and generated an interferogram that was detected by the camera. By analyzing the phase shift of the interferogram, the distance variation between the optical flat and the right-angle prism could be measured. This distance variation represented the motion error caused by the imperfect straightness of the linear guide. The root cause of the interferogram was the optical path difference of the reflected beams from the two optical surfaces. As shown in Figure 4a, the optical path difference \(D\) could be expressed as:  
\[D=n_{1}(FG+GH+KN)+n_{2}\text{:HK}-n_{2}(FI+IL+LM), \tag{3}\]  
where \(n_{1}\) and \(n_{2}\) are the relative refractive indexes of the air and glass, respectively, and \(\lambda/2\) represents the half-wave rectification. Based on geometrically analyzing the optical path, it could be concluded that:  
(1) Since the angle of the air gap was very small, FG was approximately equal to GH:  
\[FG=GH. \tag{4}\]  
(2) Since HK and IL could be seen as the same light beam in the same medium, HK was equal to IL:  
\[HK=IL. \tag{5}\]  
(3) Since LM and KN could be seen as the same light beam in different media, they had the same length of the optical path:  
\[n_{2}\text{:}LM=n_{1}\text{:}KN \tag{6}\]  
It should be highlighted that Equations (4)-(6) were approximately true because the optical flat was not ideally parallel with the bottom of the right-angle prism. When the angle between the two \begin{table}
\begin{tabular}{c c} \hline
**Components** & **Specifications** \\ \hline Camera & Basler acA2000-165 um, frame rate 165 fps, resolution 2048 \(\times\) 1088 (2 MP) \\ Lens & Moritrex ML - MC50 HR, 0.8\(\times\), focal length 50 mm \\ Right-angle prism & 25.4 mm \(\times\) 25.4 mm \(\times\) 25.4 mm, K9 glass, refractive index 1.5163 \\ \hline \end{tabular}
\end{table} 
Table 1: Component specifications in the film interferometer. Figure 4: Principle of the film interferometer: (**a**) Principle, (**b**) Off-line testing setup. reflective surfaces was very small, the optical path differences \(FG-GH\), \(HK-IL\), and \(n_{2}\cdot LM-n_{1}\cdot KN\) could be considered as a higher-order infinitesimal of the angle. With Equations (5)-(7) substituted into Equation (4), the optical path difference could be expressed as:  
\[\mathrm{D}=2n_{1}\cdot FG-n_{2}\cdot FI. \tag{7}\]  
By assuming that the current distance between the two optical surfaces was GR = \(d\), \(D\) could be modified as:  
\[\mathrm{D}=\frac{2n_{1}d}{\cos\theta_{4}}-\frac{2n_{2}d\sin\theta_{3}}{\tan \theta_{4}}. \tag{8}\]  
Based on Shell's law,  
\[\left\{ \begin{array}{l}n_{2}sin\theta_{3}=n_{1}sin\theta_{4}\\ n_{1}sin\theta_{1}=n_{2}sin\theta_{2}\end{array}