## 3 Simulation Results and Discussion
### The Focusing Parameter's Effects on Angle Measurement  
The influence of the focusing parameters on the THG angle measurement is investigated by changing the focal length \(f\) and the focused position. Here, we only put the results of calcite crystal because of its better sensitivity performance compared with rutile. The results are similar to those of \(\alpha\)-BBO. In the simulation, the crystal length \(L=2\) mm, \(d=3.6\) mm, \(\lambda_{1}=1850\) nm. The focused point is in the center of the crystal and the azimuthal angle \(\varphi=30^{\circ}\). Figure 9 shows the results of angle dependence \(I_{3}(\theta_{\rm i})\), where the lines in different colors correspond to different focal lengths. In Figure 9a, the range of incident angle \(\theta_{\rm i}\) is from \(-13^{\circ}\) to \(1^{\circ}\). The peak intensities corresponding to the focal lengths are noted in the figure. The peak intensities are not monotonically decreasing as the focal length  
Figure 8: (**a1**,**b1**,**c1**) The 3-D plot \(I_{3}\); (**a2**,**b2**,**c2**) The 2-D plot \(I_{3}\). The letters a, b, and c correspond to the materials of \(\alpha\)-BBO, calcite, and rutile in order. The green dotted arrows show the spectrum peak shifts in frequency domain. The white dotted lines denote the weighted wavelength \(\lambda_{\rm w}(\theta_{\rm i})\).  
increases, and the highest intensity is 1.00 a.u. appear as \(f\) = 70 mm. The overall trend of the peak intensity is that it increases from 0.74 a.u. (10 mm) to 1.00 a.u. (70 mm) then decreases from 1.00 a.u. (70 mm) to 0.51 a.u. (150 mm). We can also see the width of the angle dependence curve is reduced as the focal length becomes longer from \(f\) = 10 mm (11.9 deg.) to \(f\) = 150 mm (0.5 deg.). In Figure 9b, \(\theta_{\rm  
where the decrease of \(b_{1}\) will make the part of \(J_{3}^{(2)}\) to dominate \(J_{3}^{(1)}\). It will induce an extra phase in \(\left|J_{3}\right|^{2}\), which is harmful to the phase-matching when \(\Delta k(\theta_{\mathrm{m}})=0\). Therefore, there should be a trade-off between the beam waist \(w_{1}\) and confocal parameter \(b_{1}\). The focal length of 100 mm is the best choice, which has a Rayleigh length of 2.2 mm. In addition, as can be found in Equation (14), a longer focal length \(f\) makes the \(J_{3}^{(2)}\) smaller and \(\left|J_{3}\right|^{2}\) will degenerate to the \(\mathrm{sinc}^{2}(\Delta k(\theta)L/2)\) function form. Then, the side peaks of \(\mathrm{sinc}^{2}(\Delta k(\theta)L/2)\) in the Maker fringe area also can be seen in Figure 9c [33].  
In the optimized case of \(f=100\) mm, the effect of the beam waist position is considered in Figure 10. Lines and circles are used to denote the beam waist positions of \(\pm 0.2\) mm, \(\pm 0.4\) mm, \(\pm 0.6\) mm, \(\pm 0.8\) mm, and \(\pm 1\) mm relative to the crystal center. The maximum intensity is obtained in the center of the crystal at 1.00 a.u., which can be seen in the red line. This is because the central position makes the high-intensity area of the Gaussian beam to be fully used for the THG process. As the beam waist gradually moves away, the peak intensity falls from 0.95 a.u. (\(\pm 0.2\) mm) to 0.51 a.u. (\(\pm 1\) mm). The symmetric results for the positive/negative beam waist movements are the consequence of the Gaussian beam's symmetric shape. In addition, the angle dependence areas of \(I_{3}\) are both close to 0.63 deg, as shown in the figure. Thus, the best measurement sensitivity should be obtained when the beam waist is in the center associated with the highest intensity.