# 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 performance was validated by the characteristic of the detected FW and second harmonic wave associated with angular change and the variation of the crystal position. The observed second harmonic wave in Figure 10b,d,f revealed a different contour profile compared to the calculation result in Figure 7a. Inconsistency in the FW intensity and second harmonic wave was found when the crystal position was changed. Such inconsistencies were frequently encountered in the use of lasers, especially when re-alignment of the experimental setup was carried out. A different start point and endpoint of the angle measurement were observed in the figures as the initial position of the crystal was manually adjusted. However, it did not affect the sensor performance. The localized SHG intensity appeared in all variations in crystal position. The augmented contour graphs give a better view of the lower wavelength range area where the localized intensities are also spotted. Compared to one another, those results have a resemblance in characteristics and tendencies. It indicates that variations in crystal position along the laser beam propagation direction had no significant impact on the FW and SHG spectra. Figure 8: Calculated SHG spectrum in the split peak region. Figure 9: (a) The dynamics of phase mismatch for a certain angle: (b) The sinc\({}^{2}\) function characteristic due to the dynamics of phase mismatch for a certain angle. Figure 11 shows the measurement trend of the proposed angle sensor, determined by estimating the SHG peak with respect to the angle transition. By considering the SHG peak only, the problem of intensity inconsistency at the crystal position variations mentioned earlier can be solved. The thorough measurement trend results in Figure 11a,c,e revealed different measurement trends compared to calculation results in Figure 7b. The calculation  
Figure 10: The contour graphs of the observed spectrum: (**a**) FW of 30 mm; (**b**) SHG of 30 mm; (**c**) FW of 50 mm; (**d**) SHG of 50 mm; (**e**) FW of 70 mm; (**f**) SHG of 70 mm. could not confirm the measurement trend in the least intensity of localized SHG due to the very low efficiency in this range. The measurement trends in Figure 11 are tabulated in Table 2. The most extended measurement range of all crystal positions changes showed identical sensitivity and resolution. In this area, the measurement target represented by the crystal position could be placed at any position along the collimated beam propagation direction. Meanwhile, the intensity was not steady throughout the experiment in areas with a lesser measurement range, resulting in inconsistent measurement sensitivity. The angle measurement range  
Figure 11: Angle measurement trend based on the peaks of SHG at several crystal positions: (**a**) 30 mm; (**b**) the most extended measurement area of 30 mm; (**c**) the middle range measurement area of 30 mm; (**d**) the least measurement area of 30 mm; (**e**) 50 mm; (**f**) the most extended measurement area of 50 mm; (**g**) the middle range measurement area of 50 mm; (**h**) the least measurement area of 30 mm; (**i**) 70 mm; (**j**) the most extended measurement area of 70 mm; (**k**) the middle range measurement area of 70 mm; (**l**) the least measurement area of 70 mm. of 18,998.31 arc-second, 19,948.44 arc-second, and 19,256.59 arc-second was achieved at the crystal position of 30 mm, 50 mm, and 70 mm, respectively. It is also noticed in Table 2 that the experiment results afforded lower measurement sensitivity but had a wider-angle range than that predicted in the calculation results. The main cause of the discrepancy was down to the optical factor in the calculations being ignored. For example, it is assumed that the laser propagated through the air before entering the crystal, where the refractive index of the crystal is greater than that of air. The direction of Fs laser propagation could thus change when it entered the crystal. However, the difference in the refractive index was not taken into account in the calculation.