## 4 Results and Discussions  
### Form Accuracy Analysis of the Machined Microlens array  
Figure 7a shows a figure of the microlens array machined by CS and HFUVAS. It can be seen that CS produced the microlens array with a poor and blurry structure surface while HFUVAS achieved a smooth and mirror surface finish. Figure 7b,c presents the measured 3D structure of the machined microlens array. The 2D profiles across the centers  
Figure 5: Theoretical form error in HFUVAS of microlens array induced: (**a**) by overcut of vibration, (**b**,**c**) by overcut of tool edge, (**d**) smooth tool path generation of the target line.  
Figure 6: (**a**) A photo of the experimental setup, (**b**) a schematic of tool motion in HFUVAS.  
of lenslets along the feed direction (P1) and the cutting direction (P2) were measured and analyzed for further study.  
Figure 8 shows the results of the form deviation between the designed profile and the measured profile for the microlens array machined by CS and HFUVAS. The results show that the measured profile of the microlens array machined by CS had a large deviation from the corresponding designed profile. The measured profiles were not smooth with surface variation as compared with the designed one. In contrast, for the form deviation between the target profile and the measured profile of the microlens array machined by HFUVAS, the measured profiles were smoother as compared with those machined by CS, and the form deviation between the measured profile and the designed profile is significantly reduced. As a result, it is found that HFUVAS can improve surface finish and reduce form deviation as compared with CS.  
To further identify the form error of the machined profile. Figure 9 shows the form error between the designed profile and the measured profile for the microlens array machined by CS and HFUVAS. The form error included two parts: namely E1 and E2. E1 represents the form error induced by the machining of the profile apart from the sharp edge between the lenslets, and E2 represents the form error induced by the machining of the sharp edge between the lenslets (as marked with a red box). For form error of microlens array machined by CS, as shown in Figure 9a. E1 and E2 reached 0.5 and 0.3, respectively, making the total form error of the machined microlens array intolerable for ultraprecision applications. For form error of microlens array machined by HFUVAS as presented in Figure 9b. E1 and E2 were close to 0.2 and 0.2, respectively. The total form error of the machined microlens array was about 0.4, which was reduced significantly compared with the result of CS. It seemed that the form error induced by the machining of the sharp edge between lenslets occupied quite a portion of the total form error and it could be associated with generation burrs or plastic deformation which will be analyzed in the SEM image of the machined microlens array surface morphology.  
Figure 8: Form deviation analysis (Blue: designed profile; red: measured profile): (**a**) P1, (**b**) P2, for the microlens array machined by CS, (**c**) P1, (**d**) P2 for the microlens array machined by HFUVAS.  
Figure 7: (**a**) A photo of the microlens array machined by CS and HFUVAS, (**b,c**) measured 3D structure of the machined microlens array.