## 3 Optical Design
### Designed Optical System  
The specifications of the LW 640 \(\times\) 512 uncooled IR detector, which has a 20-\(\upmu\)m pitch, are listed in Table 1, and its physical models are illustrated in Figure 4. For extended sources, the Noise Equivalent Temperature Difference (NETD) is the smallest measurable signal produced and it is the temperature difference when signal-to-noise (SNR) equals to 1. The derivation of the SNR and NETD equations shall be presented and the in particular the impact the optical system design has on these characteristics shall be emphasized. FPA usually plays a role of the imaging focal plane of infrared detector. Figure 3: Sampling points diagram. (**a**) Rotating axis-symmetric system and (**b**) XYP orthogonal system. The characteristics of the IR refractive optics are presented in Table 2. The initial structure of the optical system has a great influence on the design results and optimization cycle because the damping least-squares method depends on the initial structure. The traditional spherical refractive optical system has been selected as the initial model. We should apply the XYP surface to an appropriate surface to optimize the aberration. In the optimization process, the sag of the freeform surface \(\Delta Z\) and the largest fluctuation in the sagittal direction \(\Delta d\) are added to the evaluation function with other aberrations. According to our sample method, we realized the freeform refractive optical system. In order to verify the validity of the distortion correction by the freeform surface, we replaced the freeform surface with the aspheric surface at the same position and kept all other settings same in these two optical systems. The grid distortion map and the Modulation Transfer Function (MTF) chart of system are shown in Figure 5.  
From the comparison of the different simulation pictures, we can find that freeform lens has stronger aberration correction capability (Figure 5a,b). On one hand, it can make the imaging effect \begin{table}
\begin{tabular}{c c} \hline \hline
**Item** & **Value** \\ \hline Spectral range & 8–12 μm \\ Pitch size & 20 \(\times\) 20 μm \\ Number of elements & 640 \(\times\) 512 \\ Pixel NETD, average & 60 mK \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Detector characteristics. Figure 4: LW 640 \(\times\) 512 μm-cooled IR detector. Figure 5: Grid distortion map in (a) aspherical system and (b) freeform surface system. match the detector's shape. The relative distortion value \(q^{\prime}\) in freeform refractive optical system is 1.9% and \(q^{\prime}\) in aspheric refractive optical system is 6.1%. Although the actual ray height at the margin of the field on the image-plane is optimized from \(P_{i}\)(4.45, 6.08) to \(P_{i}\)(4.66, 6.34), sure to use only traditional aspheric surfaces are not enough, the use freeform surface can achieve better results. On the other hand, the MTF is a frequency domain representation of a spatial domain signal and it contains complete information about the spatial resolution. Because of aberration, not including distortion, the MTF of actual optical system will decrease with an increasing spatial frequency until the cut-off frequency. This is reason why MTF can be used to evaluate the quality of optical systems imaging. When spatial frequency is 26 cycles/mm, it can be inferred from Figures 6 and 7 that MTF in freeform refractive optical system is 48% and MTF in aspheric refractive optical system is 30% at the same margin of the field. From the comparison of two aspects above, it proves that the freeform lens has advantages of aberration correction comparing with the aspheric surface, even in the application of coaxial refractive optical system. The designed freeform surface for Ge meets the requirements of the machining. There are 45 coefficients in this XYP extension polynomial. The machining path is presented in Figure 8a the sag of the freeform surface satisfies the property of NRFS.