## 1 Introduction  
The demand for ultraprecision machining of micro-structured functional surfaces, such as microlens array, pyramid array, retroreflective array, and diffraction grating, has been increasing in many advanced optical products and devices. Ultra-precision machining technology, including slow slide servo [1], fast tool servo [2], and ultra-precision milling [3] is applicable to the machining of these functional structures with complex geometry. To enhance the widespread use of these structured surfaces, high temperature forming processes such as injection molding or compression molding is considered as an effective and efficient method. The premise of this technology is to fabricate a mold with difficult-to-machine materials such as steel. However, direct diamond machining of steel with the super finish is not amenable due to extreme chemical tool wear [4]. As a result, UVAC has been applied due to reduced tool wear [5, 6] and better surface finish [7, 8]. There are several UVAC methods to fabricate complex structured surfaces on difficult-to-machine materials. [PERSON] and [PERSON] developed a novel structure sculpturing method by modifying vibration amplitude according to workpiece geometry in an ultrasonic elliptical vibration cutting process so complex micro/nanostructures were successfully machined on hardened steel without moving the \(Z\)-axis slide towards and backward the workpiece [9; 10]. To enhance geometric complexity as well as improve machining efficiency, a fast tool servo was integrated with ultrasonic vibration-assisted cutting. Consequently, the double-frequency vibration cutting method has been proposed, in which the system consists of one-dimension (1D) low-frequency vibration used to generate the complex tool path for the workpiece geometry acting as a fast tool servo, and a high-frequency vibration responsible for improving the machinability of the workpiece materials [11; 12]. [PERSON] et al. [13] added ultrasonic elliptical vibration cutting to a low-frequency setup for texturing micro-dimple patterns on steel. Regarding slow-slide-servo machining, a few scholars applied the state-of-art and high-frequency ultrasonic tooling system (UTS2) to machine structures with smooth tool paths such as sinusoidal grids on hard-to-machine materials including steel [14] and high entropy alloy [15], achieving sub-micrometer form accuracy and surface finish with nanometer roughness. Slow-slide-servo based machining in which the \(Z\)-axis slide moves out and in according to the workpiece geometry to generate the structured surface is a more convenient processing method compared with the structure sculpturing method or the double-frequency vibration cutting method, with no need for extra signal control operation or fast tool servo equipment. However, as the \(Z\)-axis slide has a large volume and inertia, the tool path is expected to be smooth enough to avoid overshoot and vibration induced by the quick acceleration of the \(Z\)-axis slide. Unfortunately, it is hard or impossible to ensure a smooth tool path for slow-slide-servo diamond turning of complex structures with sharp edges such as quadrilateral microlens array since the tool path is generated randomly along a spiral line. [PERSON] et al. [16] proposed a slow-slide-servo segment turning method to cut hexagonal microlens arrays by multiple groups so that a smooth approaching zone for each lenslet could be established to avoid the sudden change in direction of the tool path. However, it requires reconduction of the 3D structure to add the smooth approaching zone and it is time-consuming due to the need for multiple times of machining. In contrast, in diamond sculpturing, the cutting tool moves linearly to machine the lenslets by one tool path instead of multiple tool paths enveloping the lenslets as diamond turning. As a result, the target profile is simpler and more regular so that a smooth tool path can be generated by data processing to avoid overshoot or vibration of the cutting tool.