## 3 Results and Discussion
### Fabrication of High Precision Sinusoidal Microstructures  
The above investigations indicate that the optimized machining parameters for fabricating sinusoidal microstructures on a curved surface include a feed rate of 8 \(\upmu\)m/r, a number of sampling points of 7200 d/r and a minimal cutting depth of 4 \(\upmu\)m. With the optimized machining parameters, high precision sinusoidal microstructures are fabricated on a curved copper surface with a surface radius of 10 mm and a curvature radius of 500 mm. Before the fabrication of sinusoidal microstructures, a cleaning pass is performed to obtain an ultra-smooth curved surface with a curvature of 200 mm, similar to that shown in Figure 4. Figure 8 presents the finished part with sinusoidal modulation microstructures. Note that the figure shows some variations in surface quality, which are caused by the contaminations attached to the surface. Furthermore, the oxidation of the machined surface leads to the \"debris\" in black shown on the surface.  
Figure 9 presents the profile of the as-fabricated sinusoidal microstructures by a white light interferometer, which demonstrates that the sinusoidal grid microstructures on a curved surface are arranged regularly, and the shape of the sinusoidal grid has a high degree of consistency. Furthermore, the joints between neighboring sinusoidal girds are flat, and the surface is basically free of impurities. Figure 9 also indicates that there is no burr formed on the microstructures. Furthermore, there are basically no scratches observed on the workpiece surface, demonstrating the effectiveness of applying the rotary \(B\)-axis in suppressing the over-cutting phenomenon. The surface roughness of the microstructured surface is 18.9 nm.  
Figure 7: Effect of the number of sampling points on the profile of sinusoidal microstructures. Number of sampling points: (a) 7200 d/r; (b) 5700 d/r.  
The two-dimensional profile of the sinusoidal microstructures fabricated on a curved surface is extracted, as shown in Figure 10. The amplitude and wavelength of the sinusoidal grid structures are measured. The wavelength and the peak-to-valley of the sinusoidal grid structure are 700.6 and 18.7 \(\upmu\)m, respectively.  
Figure 8: Fabricated sinusoidal modulation microstructures on a curved copper surface.  
Figure 10: Two-dimensional profile of sinusoidal grid structures on a curved surface.  
Figure 9: (**a**) Profile and (**b**) 3D contour of fabricated sinusoidal microstructures characterized by white light interferometer by jointly using the FTS system and rotary \(B\)-axis under the optimized machining parameters.  
The contour deviation, \(\mathrm{S_{T}}\), is defined as the evaluation parameter for the contour error of the curved, sinusoidal mesh. According to the principle of surface reconstruction, the calculation of the \(\mathrm{S_{T}}\) is performed by comparing the actual machined surface with the corresponding ideal machined surface through frequency spectrum conversion simulation. Firstly, the ideal workpiece surface profile is derived by using the Butterworth low-pass filter in the white light interferometer. Then the spectrogram of the ideal machined surface is obtained using the Fourier transform. Next, the frequency corresponding to the ideal machined surface is removed by using spectral symmetry. Finally, the contour error of the corresponding machined area is obtained by performing an inverse Fourier transform, as shown in Figure 11, which indicates a contour deviation ST of 4.326 \(\upmu\)m. While the current work demonstrates the effectiveness of applying the rotary \(B\)-axis together with FTS in the fabrication of high precision microstructures on a curved surface, systematical investigations on the influence of machining parameters, the accuracy of the rotary \(B\)-axis, tool path generation, coupling degree between the rotary \(B\)-axis and FTS are needed in further investigations.