# A Second Harmonic Wave Angle Sensor with a Collimated Beam of Femtosecond Laser
## 3 Design of the Second Harmonic Wave Angle Sensor and Experiments
The evaluation of this collimated beam is tabulated in Table 1. It should be noted that the beamwidth was to be smaller than the value estimated by Equation (1), 1058.5 um in the \(X\)-direction and 1097.5 um in the \(Y\)-direction. This discrepancy was due to the initial profile of the Fs laser beam; as the profiler was positioned farther than the first plano-convex lens, the beam at this position could experience dispersion, resulting in the detection of the profile with a larger diameter. Figure 6 shows the spectral characteristics of the initial Fs laser output and the Fs laser output after the reduction in the beam diameter. The initial Fs laser output had a power density of 0.35 W/cm\({}^{2}\). In contrast, a smaller beam width produced a high-power intensity \begin{table}
\begin{tabular}{c c c} \hline \hline
**Profile Direction** & **Diameter** & **Angle of Dispersion** \\ \hline \(X\) & (921.83 \(\pm\) 3.00) μm & 2.53 arc-seconds \\ \(Y\) & (948.70 \(\pm\) 13.47) μm & 44.79 arc-seconds \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The generated collimated laser beam. Figure 5: The initial beam profile of the Fs laser: **(a)** Light intensity distribution; **(b)** Cross-sectional profiles; **(c)** The change in the beam width as the increase in observation distance. with a power density of 19.17 W/cm\({}^{2}\). This spectral characteristic in Figure 6 was later used for the calculation. Figure 7a shows the intensity of the second harmonic wave to be generated by the FW estimated based on Equation (5). The angular displacement calculated based on the change in the spectral peak is presented in Figure 7b; this result indicates the measurement trend. As can be seen in Figure 7c, a sensitivity of 0.00725 arc-second/nm was observed in the most extended measurement range, while a sensitivity of 0.00226 arc-second/nm was observed in the least measurement range, as shown in Figure 7d. Recalling that the cut-off angle of MgO:LiNbO\({}_{3}\) crystal is 47.5\({}^{\circ}\) and Fs wavelength range is 1480 nm to 1640 nm, the measurement trend in Figure 7b theoretically occurs at an angle of incidence between 1.00\({}^{\circ}\) to 3.87\({}^{\circ}\). Although the calculated yield trendline was not as smooth as the calculated MgO:LiNbO\({}_{3}\) phase-matching angle shown in Figure 1, it still showed a nearly similar angle-wavelength dependence tendency. A close observation of the results shows the presence of the split peaks that causes the localized SHG intensities and discrepancy of the measurement trend line. For instance, Figure 8 shows one of the split peaks that appeared at the matching angle of 50.18\({}^{\circ}\). The split peak caused two peaks of the SHG spectrum to detect a specific angle position simultaneously. Figure 6: The comparison spectrum of initial Fs laser output and the Fs laser output after reduction in the beamwidth. Figure 7: Calculation results of angle measurement based on the birefringence equation: (**a**) The contour graph of calculated SHG; (**b**) The measurement trend based on the peak of the calculated SHG; (**c**) The measurement trend in the most extended measurement range area; (**d**) The measurement trend in the least measurement range area. Further analysis of the calculations was carried out to explore the measurement dynamics. Figure 9a shows the dynamics of phase mismatch for a certain angle and Figure 9b shows the dynamics of the sinc quadratic function. As shown in Figure 9a, a certain angular phase mismatch for a given wavelength established a pattern that affected the sinc quadratic function in Figure 9b. The peaks of sinc quadratic function gradually narrowed with the increase in angle. Since the sinc quadratic function was applied to the FW spectrum in Figure 6 consisting of peaks and valleys, the second harmonic wave spectrum was dynamically changed, and the split peaks were unavoidable. Figure 10 shows the performance of the proposed angle sensor.