## 3 Methods  
### Machining Parameters  
Microlens array with \(10\times 10\) lenslets was machined. \(r\) and \(p\) were 1 mm and 199.75 \(\upmu\)m, resulting in a lenslet depth \(d\) of 5 \(\upmu\)m and a maximum cutting depth of 10 \(\upmu\)m of the microlens array. Table 1 shows the machining parameters of the cutting experiments. To avoid tool interference with the workpiece, the tool radius must not be larger than the workpiece contour in the feed direction and the tool clearance angle must be larger than the maximum slope angle in the cutting direction in consideration of the fact that the tool rake angle is \(0^{\circ}\) so the rake face would not interfere with the workpiece. In this research, the tool radius is equal to the workpiece contour. The maximum slope angle of the target profile of the microlens array is \(5.7^{\circ}\). As a result, a diamond tool with a rake angle of \(0^{\circ}\), clearance angle of \(15^{\circ}\), nose radius of 1 mm, tool edge radius of 50 nm, and are waviness of fewer than 0.25 \(\upmu\)m provided by Contour Fine tooling Ltd. was used in the experiment. Mirrax 40 steel was used as the workpiece material in this study. It is a remelted stainless steel pre-hardened to 40 HRC with excellent ductility, machinability, polishability, and toughness, which makes it appropriate for injection molds, flow molds and extrusion dies. The chemical composition of Mirrax 40 steel includes (wt.%): C: 0.21, Si: 0.9, Mn: 0.45, Cr: 13.5, Mo: 0.2, Ni: 0.6, V: 0.25.  
Figure 4: Definition of coordinate system for sculpturing of microlens array: (**a**) view in the \(y_{w}O_{w}z_{w}\) plane, (**b**) view in the \(x_{w}O_{w}y\) plane.  
Before the cutting experiment, it is necessary to analyze the theoretical form error of the microlens array to evaluate the machining feasibility. Figure 5a shows the theoretical form error induced by overcutting of vibration in HFUVAS of microlens array with 3 lenslets. The maximum form error arose at the edge of two adjacent lenslets and it is within 0.1 um. Figure 5b,c shows the form error induced by overcutting of tool edge in HFUVAS of microlens array with 3 \(\times\) 3 lenslets. The maximum form error arose at the intersection cusp of four adjacent lenslets and it is only 0.025 um. These two form errors are relatively small and insignificant so the microlens array is machinable theoretically. It should be noted that the theoretical form error between the designed microlens array and the machined microlens array by HFUVAS could increase to an intolerable value for ultraprecision machining with increasing lenslet depth and decreasing lenslet radius. In the cutting experiment, the cutting depth was 1 um larger than the structure depth to ensure the successful fabrication of the structure, which leads to a total cutting depth of 11 um. To protect the cutting tool from damage and achieve a good surface finish, the microlens array was cut with four depth layers of 5 um, 3 um, 2 um, and 1 um, respectively. It is important to note that to avoid overshoot and vibration of machine motion caused by a quick acceleration of the \(Z\)-axis slide arising at the sharp edge between lenslet, segment cutting was conducted, and each layer was machined by two steps for both CS and HFUVAS, as shown in Figure 5d. This tool path smoothing process would not affect the analyzing results of theoretical form error based on target lines as mentioned before because the cut lines left on the workpiece between the target line and the smoothed tool path are the same. After a list of lenslet was fabricated, the cutting tool was moved by a distance of \(p_{1}\) in the \(x\)-axis to fabricate another list of lenslets. The coolant was claisol 330/odorless kerosene and minimal quantity lubrication (MQL) method use applied. Quadrilateral microlens array was fabricated by means of conventional sculpturing and high-frequency vibration-assisted sculpturing, respectively, under the same tool path.