## 3 Results and Discussion
### Influence of Rotary \(B\)-Axis on the Microstructure Fabrication  
Preliminary cutting experiments are firstly performed to evaluate the influence of applying the rotary \(B\)-axis on the machining precision of sinusoidal microstructures on a curved surface. Firstly, the spherical datum surface with a surface radius of 2 mm and a curvature radius of 200 mm is obtained by diamond cutting, and the machining parameters are: the minimal cutting depth is 2 \(\mathrm{\SIUnitSymbolMicro m}\), the feed rate is 5 \(\mathrm{\SIUnitSymbolMicro m}\)/r and the spindle speed is 1200 r/s. Figure 4a presents the machined surface morphology of the curved surface characterized by a Zygo NewView 9000 white light interferometer and Vision 32 software, which demonstrates a surface roughness, \(\mathrm{S_{a}}\), of 10 nm. Figure 4b further plots the profile of the machined curve within a measurement range of 883.6 \(\mathrm{\SIUnitSymbolMicro m}\), as well as the fitting curve colored in red. Specifically, the sampling points are selected on the \(X\)-axis within a measurement range of 883.6 \(\mathrm{\SIUnitSymbolMicro m}\), and the values of the sampling points on the \(Y\)-axis within a range of \(-0.3\) to 0.2 \(\mathrm{\SIUnitSymbolMicro m}\) are utilized for fitting the curvature of the machined curved surface. Figure 4b indicates that the fitted curvature radius is 199.999 mm, which is very close to the designed value of 200 mm, which provides a good basis for the following fabrication of sinusoidal microstructures on the machined curved surface. The above  
Figure 3: Sinusoidal mesh surface and tool path. (**a**) Determining the starting position of the tool; (**b**) Dividing the tool nose arc radius; (**c**) Calculating the displacement in the \(Z\) direction; (**d**) Determining the minimal displacement in the \(Z\) direction. results indicate a high profile accuracy, as well as the ultra-low surface roughness of the machined curved surface, which falls into the regime of ultra-precision machining. Then, sinusoidal microstructures are fabricated on the spherical datum surface. The machining parameters for the microstructure fabrication are typical as a feed rate f of 10 \(\upmu\)m/r, a number of sampling point P of 7200 d/r and a minimal cutting depth of 3 \(\upmu\)m. Figure 5 presents the profiles of fabricated sinusoidal microstructures by different machining strategies. Note that the measurement range of Figure 5 is smaller than that of Figure 5, intending to present an enlarged view of the machining characteristics for the strategy of jointly using the FST system and the rotary \(B\)-axis. It is seen from Figure 5 that the surface quality in terms of surface roughness and surface error for the microstructures fabricated by solely using the FST system is lower than that by jointly using the FST and rotary \(B\)-axis. Firstly, the burr formation is significantly suppressed by using the rotary \(B\)-axis. Figure 5 shows that there are burrs formed in the transition section of the sinusoidal structure for the strategy of sol  
Figure 4: Surface quality and shape error of spherical datum surface characterized by white light interferometer. (**a**) Surface morphology; (**b**) Surface profile. Figure 5: Profile of fabricated sinusoidal microstructures by (**a**) the FTS system and (**b**) the FTS system and the rotary \(B\)-axis. In contrast, the burrs are rather limited for the strategy of jointly using the FST system and rotary \(B\)-axis. Second, the measured surface roughness for the strategy of jointly using the FST and rotary \(B\)-axis is 46.7 nm, which is lower than that of 59.6 nm for the strategy of solely using FST is 59.6 nm. Thirdly, Figure 5a demonstrates that most of the outer circle of the curved surface is not cut off completely, which is not observed for the strategy of jointly using the FST and rotary \(B\)-axis.