## 3 Optical Design
### Field Sampling  
It was discussed in Section 3.1. that the model of (XYP) extended polynomial can calculate the actual ray height of each sampling point \(P_{i}(X_{i},\,Y_{i})\) to control the system aberration to meet the error evaluation function. Specific sampling methods will be analyzed in this section. In the optical design process, the field sampling method is especially important. If there are too few sample points, the completeness and accuracy of the design results cannot be ensured. However, if there are too many sample points, the calculation load will increase, which will reduce the optimizing speed. In the traditional design process of rotationally symmetric optics, field sampling usually involves choosing five FOVs, 0, 0.5, 0.707, 0.85, and 1, in the entire FOV. It should be noted here that this kind of sampling method is no longer applied to freeform lens designs. The freeform surface loses its rotational symmetry property that requires more ray sampling in the process of optimization, which gives rise to a slower ray tracing structure and larger concurrency value than spherical and aspheric surfaces. However, it should be noted that the kind of freeform lens presented in this paper loses its rotationally symmetric character but retains its side-symmetric character. According to the fast too servo (FTS) assisted ultra-precision turning of the NRFS, when the spindle speed is relatively large, the stroke of FTS should be small enough (usually within \(\sim\)100 \(\upmu\)m) to track the contour of the freeform surface and maintain the synchronization with the rotating spindle. The largest fluctuation in the sagittal direction \(\Delta d\) should be strongly controlled. If there are insufficient sampling points, the fluctuations cannot be ensured within the scope of the machining ability and some mutations will occur in the freeform surface, which can cause surface discontinuity and disrupt the system performance. Even in some cases, the freeform surface cannot be formed. Therefore, in our field sampling, sufficient intensive sample points were chosen to ensure the minimum variation of the surface sag difference \(\Delta Z\). By utilizing the side-symmetric character of the NRFS, we should choose every sample point in the entire FOV from 0 to 1 with a step of 0.1. Considering the side-symmetric nature, only the first FOV quadrant needs to be sampled. Figure 3a illustrates the traditional sampling method of rotational axisymmetric system, such as aspheric surfaces. However, it does not cater for the situation of freeform surfaces, which need more different complex sampling FOV Figure 3b shows one of ten FOV zoom samples, which includes 11 FOV values in each FOV zoom in all the sampling FOVs, there is a total of 121 spots, which is only a quarter of the full FOV, and the optimization time can clearly be reduced using this approach. In view of a sufficient number of sampling points, the machining quality and efficiency of the freeform lens is ensured because the sag variable can be controlled within the range of manufacturability. The 121 FOV values are sampled in the design process. According to the sampling strategy, the optimization of the freeform surface is conducted in following steps:  
1. Equally distributed points are sampling along \(x\)-direction in 0-20\({}^{\circ}\) and \(y\)-direction in 0-15\({}^{\circ}\), respectively. The weight of the central field angle (0\({}^{\circ}\), 0\({}^{\circ}\)) is set to 1, and the weight of the adjacent field is increased by 0.5 increments to the edge field angle (15\({}^{\circ}\), 20\({}^{\circ}\)) which is set to 6. 2. The coordinates of ideal image (\(X_{i0}\), \(Y_{i0}\)) and the actual image (\(X_{i}^{\prime}\), \(Y_{i}^{\prime}\)) are derived from the principal ray of each sampling points (\(X_{i}\), \(Y_{i}\)). 3.