## 1 Introduction  
Infrared (IR) imaging systems have been utilized in an increasing number of applications. There is a demand for a new generation of IR thermal imager with not only a high resolution and farther detection range, but also a smaller size. Unfortunately, these two requirements are contradictory. If the system volume is fixed, increasing the focal length and field-of-view (FOV) will introduce higher-order aberrations, which are difficult to correct. The object field is transformed to an image within the FOV of the thermal imaging. The FOV is the angular extent of the observable object field [1]. Freeform surface can provide more design degrees to optical designers to satisfy the different optical systems than traditional optical elements such as spheres and aspheric lenses [2]. Especially when the deviation of FOV increases from the rotational symmetric configuration, it becomes interesting to find how non-rotational symmetric optics can help to provide tailor-made solutions with increased overall imaging performance [3]. For each real image point, it can be customized for aberration correction by freeform surface. Thus, a trial to change aspheric lens into freeform optics at the same location in the coaxial infrared system is in store to achieve a good performance imager. With the development of optical manufacturing technology, aspheric optical elements have been applied maturely in different optical systems and have developed rapidly during the 1950-1960s. Although the aspheric optical element can be thought of as a special kind of freeform surface, it cannot entirely replace the freeform surface. If the high order aberrations in the system are too much, they cannot be corrected effectively by using the aspheric surface [4]. Various papers have reported on the use of aspheric technology to eliminate distortion for large FOV systems [5; 6; 7]. Multiple aspheric lenses are often required to minimize the comprehensive aberrations, but they can be balanced by using only one freeform lens. This is especially important for IR optical systems because the optical components number was reduced by using freeform lens in the complex system to increase the system's transmittance. This is especially important for IR optical systems because this simplification can greatly improve the transmittance of the system. Certainly, the complexity and difficulty of the processing of the freeform surface are greater than those of the optical aspheric surface. In recent years, with the development of the ultra-precision process and requirements of higher-performance IR imaging systems, freeform surfaces have been applied primarily in the following three components. They are the conformal dome [8; 9; 10; 11], the miniaturization of the IR imaging system [12; 13; 14] and the panoramic imaging optical system [15; 16; 17; 18]. It was found that the freeform lens was also used for helmet display applications [19; 20]. With regard to the application of freeform surface, there have been too many articles about the advantages of freeform surface in the off-axis reflection system [21]. It was reported that freeform cylindrical lenses were used to correct the astigmatism in the off-axis reflected system by using user defined error function [22]. There are also some reports on the application of a freeform lens in coaxial refractive system. Typical freeform lenses, Alvarez lenses, perform focal length change in conventional infrared optical system. It is a good example of freeform surfaces that are used to imprint a deformation into the propagating wavefront to change the focal of the system [23]. Freeform optics can be also used as a kind of field correction element in the overall telescope system to get ideal results [24]. It illustrates the validity of freeform surface to correct specific aberration for different systems. There are two main reasons for the difficulty of a freeform surface in IR refractive imaging systems.