# Precision Auto-Focusing Apparatus Based on Long-Working-Distance, Wide-Spectrum Microscope
## 2 Materials and Methods
### System Design
It can be seen that the MTF corresponding to a 1 mm half-field still basically coincides with the ideal MTF. When the field of view increases further, the MTF of the mentioned mirror-based objective is lower than that for an unobscured objective in the low and medium spatial frequencies. It should be noted in particular that the optical system has been carefully optimized, therefore, the different curves (T, S. T DIFF. Lim, etc.,) in the MTF curve (Figure 2a) in the small field of view are close to coincidence. Therefore, it cannot be completely independently reflected in the figure. Figure 2: Comparison of modulation transfer function (MTF) curves for the mirror-based objective with the ideal MTF. **(a)** MTF curve for smaller fields; **(b)** MTF curves for larger fields. In this study, the Monte Carlo method tolerance analysis was performed using ZEMAX; the tolerance settings are shown in Table 2 and the the simulation times were 10,000. The results of the tolerance analysis are shown in Table 3. The RMS spot size for the spatial image was used as the evaluation criterion. The primary and secondary mirrors of the aspheric Schwarzschild objective are made of fused quartz by a traditional optical processing method. The diameters of the two mirrors are 210 and 29 mm. Protected aluminum film is used to cover the machined surfaces. The results of the optical surface (without coating) are measured by a Fizeau interferometer, which indicates the deviation between the actual surface and the ideal surface, as shown in Figure 3. It can be seen that the peak-to-valley (PV) value of the surface was 0.135 and 0.08 \(\lambda\) (\(\lambda\) = 632.8 nm), respectively, for the two mirrors. According to [PERSON]'s judgment, as the surface PV value is less than \(\lambda\)/4, the surface error will have very little effect on the image quality. A diagram and photograph of the microscopic measurement integrated device are shown in Figure 4. In order to ensure the positioning accuracy and straightness of the motion mechanism of the focusing system, an air-bearing guide rail and roller screw are used as the driving mechanism, a brushless DC servo motor is used as the driving motor, and a grating encoder is used as the feedback sensor. Motion positioning accuracy better than 0.5 \(\upmu\)m can be achieved with this motion control system. All of the devices were placed on an active self-leveling isolation system in order to eliminate, as much as possible, the effects of vibrations on measurement results. \begin{table}
\begin{tabular}{c c} \hline \hline
**Description** & **Values** \\ \hline Radius & \(\pm\)0.05 mm \\ Thickness & \(\pm\)0.05 mm \\ Element decenter & \(\pm\)0.03 mm \\ Element tilt & \(\pm\)0.01 mm \\ Surface irregularity & \(\pm\)0.2 fringes \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Tolerances of the mirror-based objective. Figure 3: Surface measurement results for (**a**) primary mirror and (**b**) secondary mirror. \begin{table}
\begin{tabular}{c c} \hline \hline
**Proportion** & **RMS Spot Size** \\ \hline
90\(\%\leq\) & 32.6 \(\upmu\)m \\
50\(\%\leq\) & 26 \(\upmu\)m \\
10\(\%\leq\) & 5.2 \(\upmu\)m \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Monte Carlo tolerance analysis results at the relay image.