## 2 Effects of Freeform Lenses in Optical System  
For an ideal optical system, the lateral magnification \(\beta\) is constant in the conjugate plane. However, in the actual optical system, they are equal only when the FOV is very small. In aberration theory, the power series form is taken to represent the relationship between the aberrations and the maximum object value \(y\) (or viewing angle \(\omega\)) and beam aperture \(h\) (or angle \(u\)). The distortion will rapidly decrease with the decrease in FOV, and it is the reverse situation for large FOVs because of the third-power relation between the distortion value and FOV angle. The sides of the image are curved because the amount of distortion varies with the cube of the distance from the axis. Thus, the distortion of the marginal FOV is difficult to correct.  
Distortion is a kind of chief ray aberration and is denoted by the distance \(\delta y^{\prime}\) between the intersection of the chief ray and Gaussian image plane \({y_{z}}^{\prime}\) and the ideal image point \({y_{0}}^{\prime}\). In a common optical design, the relative distortion value \(q^{\prime}\) is used to express the relative error between the actual magnification and ideal magnification, which is denoted by the percentage of distortion \(\delta y^{\prime}\) relative to the maximum image point coordinate \({y_{0}}^{\prime}\). The relative distortion can be expressed as:  
\[qt=\frac{\delta yt}{{y^{\prime}}_{0}}\ \times\ 100\% \tag{1}\]  
This indicates that it is difficult to correct the distortion in the traditional optical design method including the aspheric application because other aberrations, such as spherical aberration, coma, astigmatism, and field, will be caused by the introduction of distortion correction in the optical system [28]. The freeform surface can help to reduce the incident angle and exit angle of marginal ray to more effectively balance the other aberrations. Herein, the main goal is to verify that a better performance will be achieved with the use of a freeform lens in a refractive IR optical system as compared to an aspheric lens.  
The height of main ray is controlled directly to make the distance \(\delta y^{\prime}\) equal to zero. The design method principle is illustrated in Figure 1. The following parameters are set as variables: \(c\) is the radius of curvature, \(k\) is the quadric coefficient, and \(C_{(m,n)}\) is the monomial coefficient. First, we calculate the maximum ideal imaging value according to the corresponding FOV. Second, we trace the actual rays by compiling the Macro command embedded in the CODE V optical optimization software [29]; then, we can calculate and control the difference between the maximum actual and maximum ideal imaging values with the evaluation function to optimize the freeform surface shape parameters. Finally, the shape parameters can be translated into the solution of a series of freeform surface sags \(\Delta Z\).  
In general, the freeform surface can correct the aberration in the x- and y-directions to match the rectangular focal plane shape better than an aspheric surface. Considering the rotation asymmetry of the freeform surface, there are two key points in the entire design process. The first issue involves choosing the surface description model; the second issue involves selecting the field sample points as optimization variables to ensure the imaging quality and machinability. The optimized optical lens data and simulation performance will be given. The above points will be presented in the next section.