## 1 Introduction
More importantly, the machining time can be reduced considerably to improve machining efficiency. A few studies have been made on machinable material via conventional diamond sculpturing. [PERSON] et al. [17] conducted a series of sculpturing experiments and found that the slope angle of the tool path played a key role in the determination of the material spring back and the burr formation, then proposed a theory to explain the tool indentation effect in diamond cutting of optical micro-structured surfaces. [PERSON] et al. [18] conducted the sculpturing of a microlens array on an intrinsic silicon substrate. Although the surface roughness of the machined structure was in the nanometer range, form accuracy was poor, which suggested that it might be difficult for conventional sculpturing to accurately machine hard materials. Currently, little research has been found to provide new insight into the diamond sculpturing of microlens arrays with sharp edges on steel materials. Reduced tool wear [19], optimal surface finish [20] and fine form accuracy are the concerns in the machining process. To improve machinability and reduce tool wear in the machining of microlens array on steel, the high-frequency ultrasonic vibration-assisted sculpturing was applied. As a result, the purpose of this research is to conduct a theoretical and experimental investigation of high-frequency ultrasonic vibration-assisted sculpturing of optical microstructures on steel workpieces. The machining principle of diamond sculpturing, as well as the cutting dynamics of ultrasonic vibration on the theoretical form error in diamond sculpturing, was analyzed and determined. Form accuracy, the surface morphology of the machined structures, and the tool wear were measured and analyzed. Furthermore, conventional sculpturing was also conducted for comparison. Principle and Theoretical Form Error of Ultrasonic Vibration-Assisted Sculpturing of Quadrilateral Microlens Array  
Quadrilateral microlens array can be described by Equation (1):  
\[z(x,y)=\frac{\left[x-p_{1}(i-0.5)\right]^{2}+\left[y-p_{2}(j-0.5)\right]^{2}}{r+ \sqrt{r^{2}-\left[x-p_{1}(i-0.5)\right]^{2}-\left[y-p_{2}(j-0.5)\right]^{2}}} \tag{1}\]  
where \(i\) and \(j\) are the sequence number of the lenslet in the \(x\) and \(y\) direction, \(r\) is the microlens radius. \(p_{1}\) and \(p_{2}\) are pitches of the lenslet in the \(x\) and \(y\) direction, which were set as the same value in this work. Figure 1 presents the configuration of the cutting system and tool path in slow-slide-servo diamond turning and diamond sculpturing of microlens array. In slow-slide-servo diamond turning as shown in Figure 1a, C-axis, \(X\)-axis, and \(Z\)-axis are configurated and the tool path is generated along a spiral line. This tool path generation strategy could encounter difficulties in the machining of those structures with sharp edges, including linear interpolation deviation of tool path sampling, overshoot, and vibration of machine motion response at sharp edges. By contrast, in diamond sculpturing, as shown in Figure 1b, \(X\)-axis, \(Y\)-axis, and \(Z\)-axis are configurated and the tool path is generated along the symmetric line of the lenslets. In this technique, a round tool is used to machine lenslets by one tool path instead of multiple tool paths enveloping the structure profile as diamond turning. The tool path generation strategy becomes simple, and it could ensure that the sampling points of the tool path appear at the sharp edge and segment cutting can be performed to avoid overshoot and vibration of machine motion at the sharp edge caused by a quick acceleration of the \(Z\)-axis slide. Figure 1: Configuration of the cutting systems and tool paths of microlens array for: **(a)** diamond turning and **(b)** diamond sculpturing.