## 4 Experimental Results and Discussion  
Freeform surface requires an extremely high precision form and nanometric surface finish [32] and single crystal germanium is a typical brittle material, ultra-precision turning can be used to realize the processing requirements of the system. In this freeform system, the surface accuracy and assembly tolerance should be considered comprehensively; one compensator and routine tolerances were used during the design. They include the irregularity tolerance (CYN), thickness tolerance (DLT), displacement tolerance (DLZ), and surface tilt tolerance (DLA), and are presented in Table 4. The assembly method and tolerance curve are presented in Figure 9. The thickness, tilt, and displacement tolerances are all within 20 \(\upmu\)m. The radius and irregularity tolerances are less than 3 \(\upmu\)m. The overall size of the thermal imager is less than 45 mm. These indicators are the basic requirements of the installation process, proving the feasibility of the implementation. If it is assumed that the errors can only take extreme values of the tolerances, the 97.7% probable change in the MTF is \(-0.0462\) in the central FOV and \(-0.0555\) in the marginal FOV. The particular tolerance results show that the performance of final system thermal image can maintain the design simulation value very well. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Type (unit)** & **Minimum** & **Maximum** \\ \hline CYN(fringes) & 0.5 & 3 \\ DLT(mm) & 0.02 & 0.05 \\ DLZ(mm) & 0.02 & 0.05 \\ DLA(radians) & 0.02 & 0.05 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Tolerance types. Figure 8: **(a)** Ge freeform surface sag map with machining tool path and **(b)** distribution of fluctuation in the sagittal direction \(\Delta d\). Moreover, for the thermal imager systems, a combination of both spatial resolution and thermal sensitivity determines the final system performance. The minimum resolvable temperature difference (MRTD) combines the spatial resolution and thermal sensitivity. Spatial resolution represents the resolution of the shape of the target space by the thermal imager. A small spatial resolution refers to a high-resolution ability in the thermal imager. The testing result of the MRTD of this IR imager is 0.3 K at the spatial frequency of 0.44 cycles/mrad. Figure 12a-d present the imaging results of the thermal imager at different distances. The images are very clear at each different distances and the observer cannot feel the distortion of the imaging edge of the large field system. These results exemplify that the freeform TI exhibits an excellent image quality. The final optical system works as an un-cooled long-wave infrared thermal imager is one of the important development directions of infrared thermal imaging technology. Compared with the cooled thermal imager, the outstanding advantage of un-cooled infrared thermal imager is miniaturization and low power consumption, because it leaves out the complex and unstable cooling system. Notably, its thermal sensitivity has been reduced. So we tried to maximize the performance of this thermal imager by optimizing the performance parameters of the optical system. First, the F number of the system ensures a certain radiant power. Second, [PERSON]'s criteria [PERSON] (1968) describes both spatial domain and frequency domain approaches to analyze the ability of observers to perform visual tasks using image intensifier technology. The pitch size of the detector will affect the spatial resolution of the thermal imager, and the smaller the pixel size corresponds to the higher spatial resolution of the system. According to [PERSON]'s criteria, we adopted detector with 20 \(\upmu\)m pitch size for this un-cooled thermal imager to ensure its spatial resolution. Third, the view field of the system guarantees a large scope of observation. In the final test data of the thermal image performance,  
Figure 11: Measurement graph of MTF vs. frequency.