## 1 Introduction
Firstly, machining freeform surfaces on IR optical materials is still a difficult task because of the brittle nature of IR materials [25]. Secondly, most IR refractive optical systems are still rotating axisymmetric systems, thus, some scholars think it is sufficient to just use aspheric surfaces to balance comprehensive aberrations. However, freeform lenses are considered to possess some advantages to match the rectangular FOV of the detectors. It was demonstrated that freeform optics could help provide solutions with a distinctly better overall imaging performance for on-axis imaging applications with high aspect ratios [26]. Although freeform lenses are used in rotationally symmetric optical systems, freeform surfaces can correct high-order aberrations in the x- and y-directions more effectively than high-order aspheric surfaces. Especially for large FOV thermal imaging (TI) to facilitate observation and reconnaissance, there will be a significant amount of distortion, which can reduce the image quality and must be corrected perfectly. Digital methods of the distortion correction have the disadvantage of resolution losses. In addition, there is a disadvantage of using software correction distortion which is that high definition image and video data are very lengthy, which means that the hardware requirements are very high. Another thing is that it takes some time for software to achieve image or video's distortion preprocessing. In other words, real-time comparison is difficult to guarantee. Therefore, a method of direct correction distortion is needed, which can be realized with pure optical hardware system [27]. If a freeform surface is adopted in this kind of IR imaging system, designers can directly control the chief ray to reach the ideal position, which not only greatly reduces the distortion, but can also improve the performance of imaging. At the same time, the ultra-precision turning process can ensure the surface shape of brittle IR materials and can be easier than freeform reflectors in the optical system alignment. These characteristics meet the demands of modern photoelectric imaging systems, so it has very extensive application prospects. In the design process, on the basis of aspheric system, the aspheric surface was given more degrees to be further optimized to be freeform surface lens. Although the system lost rotational symmetry, the plane symmetry is maintained. In the process of optimization, combined with the plane symmetry, optimizing the freeform surface can save time and the face shape constraints of ultra-precision cutting of infrared materials are also followed. The machinability and assembly tolerances of the freeform surface are considered roundly. This paper provides new ideas and strategies for coaxial optical systems. A high-performance, long-wave, un-cooled, refractive infrared (IR) thermal imager has been designed with one freeform lens. A design method is proposed to correct the distortion of large FOV refractive IR optical systems. The theory behind this corrected method is based on well-established results in optical aberration theory [28]. The final design system is verified by test results. Design method, analysis results, and test data are provided to prove the availability and excellence of freeform lenses. The effects of freeform lens in optical system is discussed and the design process is then presented. Testing results are illustrated to confirm the performance of the designed effects. It turns out that freeform surface can be used in the infrared coaxial system to perform better aberration correction. Conclusions are finally drawn to summarize the feasibility and qualities of the designed system.