## 3 Optical Design
### Choosing Freeform Surface Description Model  
There are many ways of describing the freeform surface to more effectively match the rectangular characteristics of the refractive IR detector. The imaging freeform optical systems have been designed using non-uniform rational basis-spline (NURBS) surfaces in Reference [30]. It has a dense sampling grid close to the freeform surface and uses an optimization algorithm relying on point-by-point control, but the NURBS freeform design for the off-axis refraction system is not suitable for our optical system because the initial structure of this freeform lens does not need to be set up. In the whole coaxial refractive optical system, it still shares the same optical axis, combined with our field sampling method,  
Figure 1: The Schematic diagram of freeform lens changes the direction of the ray light.  
Figure 2: Configuration of the infrared (IR) thermal imager. **(a)** Three-dimensional drawing system and **(b)** structure illustration.  
so the system is still plane-symmetrical that the initial structure of the freeform lens is automatically set up. Herein, the XYP surface to describe the freeform surface system has been adopted. First, according to the way of sampling, XYP surface can calculate the actual ray height of each sampling point \(P_{i}(X_{i},\,Y_{i})\) to control the system aberration to meet the design requirement. Second, due to this freeform surface retaining its plane-symmetric character, the XYP surface can save optimizing time. This extended polynomial is a monomial extension, which increases the number of supreme power series based on the conic coefficient, as shown in Equation (2) [29].  
\[z=\frac{c(x^{2}+y^{2})}{1+(1-(1+k)c^{2}(x^{2}+y^{2}))^{1/2}}+\sum_{m,n}^{p}C_{( m,n)}x^{m}y^{n},\quad 1\leq m+n\leq p \tag{2}\]  
where \(c\) is the curvature, \(k\) is the conic constant, and \(C_{(m,n)}\) is a monomial coefficient and \(r=(x^{2}+y^{2})^{1/2}\). Equation (2) describes XYP surface. In traditional rotationally symmetric systems, aspheric surfaces that can be described by Equation (3) have been used to correct aberration [29].  
In traditional rotational symmetry, aspheric surfaces have been used to correct balance, as shown in Equation (3). In this formula, \(k\) is the quadric surface coefficient, \(r\) is the radial coordinate, and \(\alpha_{1}\), \(\alpha_{2}\), and \(\alpha_{3}\dots\) are the radial coordinate coefficients.  
\[z=\frac{cr^{2}}{1+[1-(1+k)c^{2}r^{2}]^{1/2}}+\alpha_{1}r^{2}+\alpha_{2}r^{4}+ \alpha_{3}r^{6}+\dots \tag{3}\]  
Compared with an aspheric surface, the XYP surface has considerably more parameter variables. In the two vertical \(x\)- and \(y\)-directions, although they also share the same root axis of symmetry, the coefficients in their respective directions are different, which can satisfy the design requirements of the IR imager according to the valuation function optimization.  
In the XYP extended polynomial, this freeform lens loses rotational symmetry because it has different variable parameters in the \(x\)- and \(y\)-directions. At the beginning of the design, the sphere system is used rather than the freeform surface. When the existing spheres cannot further improve the image quality, the freeform surface is added to this system.  
There are two main reasons for adopting this design method. The first is that the freeform surface in the refractive system plays a significant role in correcting the high-order aberrations. The second is that it can be machined with the processing method that is used to design a near-rotational freeform surface (NRFS) with a low non-rotational degree [31]. This constrains the variation of the traditional freeform optics to address the balance issue of the machining quality on the freeform surfaces and the brittle nature of IR materials. In the machining process of the freeform surface, the largest fluctuation in the sagittal direction \(\Delta d\) is small enough to satisfy the machining on IR materials.