## 2 Measurement Principle  
The optical configuration of the proposed dual-axis level is shown in Figure 1a. The reference laser emitted from the laser diode (LD) passes through the transparent liquid in a transparent container and is incident on an autocollimator unit composed of a focus lens (FL) and a quadrant photodetector (QPD). The mirrors M1 and M2 are used to reflect the reference laser to reduce the size. When the sensor is inclined, the liquid surface remains level. The reference laser emitted from the transparent liquid into the air is refracted. The angle change of the refracted reference laser is measured by the autocollimator unit. According to [PERSON]'s law, the relationship between the inclined angle and the angle change of the reference laser can be expressed as:  
\[\begin{array}{l}n_{1}\sin(\varepsilon_{y})=n_{2}\sin(\varepsilon_{ym})\\ n_{1}\sin(\varepsilon_{x})=n_{2}\sin(\varepsilon_{xm})\end{array}, \tag{1}\]  
where \(\varepsilon_{y}\) is the inclined angle around the Y axis, as shown in Figure 1b, and \(\varepsilon_{x}\) is the inclined angle around  
According to the auto-collimation principle, the change in the reference laser's incident angle from \(\varepsilon_{ym}\) to \(\varepsilon_{xm}\) will cause the change in light spot's position from \(\delta_{yQPD}\) to \(\delta_{xQPD}\) on the sensitive surface of the QPD:  
\[\varepsilon_{xm}=\frac{\delta_{xQPD}}{f};\varepsilon_{ym}=\frac{\delta_{yQPD}}{ f}, \tag{3}\]  
where \(f\) is the focal length of FL, the change in the light spot's position from \(\delta_{yQPD}\) to \(\delta_{xQPD}\) would lead to a change in the light intensity received by each photodiode of the QPD sensor, resulting in a change in the photocurrent output by the QPD. The photocurrents of the QPD are converted to voltages by an I/V conversion circuit. Based on the working principle of QPD and Equations (2) and (3), the relationship between the output voltages of the QPD (V1, V2, V3, V4) and the inclined angle \(\varepsilon_{y}\) and \(\varepsilon_{x}\) can be expressed by:  
\[\begin{array}{l}\varepsilon_{x}=\frac{m_{2}}{m_{1}}\varepsilon_{xm}=k_{ex} \frac{(V_{1}+V_{4})-(V_{2}+V_{3})}{(V_{1}+V_{2}+V_{3}+V_{4})}\\ \varepsilon_{y}=\frac{m_{2}}{m_{1}}\varepsilon_{ym}=k_{cy}\frac{(V_{1}+V_{2}) -(V_{3}+V_{4})}{(V_{1}+V_{2}+V_{3}+V_{4})}\end{array}, \tag{4}\]  
where \(k_{ex}\) and \(k_{xy}\) are the constants obtained by the calibration process, which will be mentioned in Section 4. The proposed measurement principle is modified from the author's previous work [20] to avoid the influence of the reflected light from the liquid surface on the QPD, which would cause a random error. In addition, compared with the measurement principle using the reflected light from the liquid surface [21], the intensity of the reference laser is lower and the signal of QPD is higher in our proposed measurement principle, leading to the better resolution performance of the proposed system.  
It should be noticed that, although the bottom wall of the transparent container will produce the additional refraction of the laser beam, this refraction is always the same regardless of the angle of the level, because the LD, container, and autocollimator are tilted together. The principle of angle measurement by the autocollimator is to detect only the angle change in the incident laser beam, and this angle change is purely from the change in the refraction angle between the liquid surface and the laser beam. In addition, the autocollimator is not sensitive to the beam shift.