## 3 Simulation Results and Discussion
### The Sensitivity Investigation of Angle Measurement Based on Direct THG
In the simulation process, the parameters are set as: the focal length \(f\) = 70 mm, the Z positions of the crystal are \(Z_{\mathrm{L}}\) = \(-\)1 mm and \(Z_{\mathrm{R}}\) = 1 mm, respectively, the crystal length \(L\) = \(Z_{\mathrm{R}}\) - \(Z_{\mathrm{L}}\) = 2 mm, the diameter of incident femtosecond beam \(d\) = 3.6 mm. We assume the focused point (beam waist) is perfectly in the center of the crystal, and the azimuthal angle \(\varphi\) = \(30^{\circ}\). The wavelength range of the incident femtosecond laser is set from 1800 nm to 1900 nm, covering the dispersion-less area. The space between the modes is 1 nm. Figure 7 shows the simulated results, where the index (a), (b), and (c) of subplots correspond to \(\alpha\)-BBO, calcite, and rutile, respectively. In each subplot, the lines in different colors correspond to the angle dependence \(I_{3}(\theta_{\mathrm{i}})\) of different modes or the total intensity \(I_{\mathrm{THW}}(\theta_{\mathrm{i}})\). The horizontal axis denotes the angular displacement of the incident angle \(\theta_{\mathrm{i}}\). The left and right vertical axes denote the intensity of \(I_{3}\) and \(I_{\mathrm{THW}}\), respectively. It is clear that \(I_{3}(\theta_{\mathrm{i}})\) does not have an obvious change between the wavelength of 1800 nm, 1850 nm, and 1900 nm for \(\alpha\)-BBO and calcite, and the angle dependence of \(I_{\mathrm{THW}}(\theta_{\mathrm{i}})\) keeps the same shape as \(I_{3}(\theta_{\mathrm{i}})\). The maximum total intensities of \(I_{\mathrm{THW}}\) are strengthened to 96.8 a.u. (\(\alpha\)-BBO), 93.0 a.u. (calcite), respectively. Conversely, due to the dispersion, the rutile has a much lower maximum total intensity \(I_{\mathrm{THW}}\) of 2.7 a.u. The measurement sensitivity is calculated from about 80% to 20% of \(I_{\mathrm{THW}}\), and they are \(-\)0.087 a.u./arcsec, \(-\)0.108 a.u./arcsec, 0.002 a.u./arcsec, for \(\alpha\)-BBO, calcite, and rutile, respectively shown in Figure 7.  
Figure 6: (a) The calculated results of phase-matching angle \(\theta_{\mathrm{m}}\) dispersion from 900 nm to 2500 nm; (b) The phase-matching angle derivative in the dispersion-less area from 1700 nm to 2000 nm. For the wavelength-dependent method, the spectrum peak shift can be found except for the dispersion-less area (1700-2000 nm) of \(\alpha\)-BBO and calcite. In our previous work, a commercial femtosecond laser source was used for producing harmonic waves, and the central wavelength of the laser was around 1560 nm. Therefore, the wavelength-dependent method will be discussed near 1560 nm for future experimental work. In the simulation results, the parameters are the same as those of the intensity-dependent method. Figure 8 shows the calculated results for three crystals. In this figure, the letters a, b, and c correspond to the results of materials of \(\alpha\)-BBO, calcite, and rutile, respectively. The numbers 1 and 2 correspond to the 3-D plot and 2-D plot third harmonic intensity \(I_{3}\), respectively. For each subplot, the color bar is used to indicate the amount of intensity. Here, we fixed the range from \(-0.8^{\circ}\) to \(0.6^{\circ}\) to compare the peak shifts. The results in Figure 8(a,b,c,c) show that the peak shifts exist for all the investigated materials, and all the peak shifts toward the longer wavelength in the frequency domain as \(\theta_{\mathrm{i}}\) decreases. The peak shifts of \(\alpha\)-BBO and calcite both cover a range from around 500 nm to 540 nm. In contrast, the peak shift of rutile is relatively slow because it even cannot cover the wavelength range from 519 nm to 521 nm. The essence of peak shift is the point movement, which makes \(\Delta n(\lambda_{1},\,\theta)=\mathrm{const}\). This movement can be described by a contour map of \(\Delta n(\lambda_{1},\,\theta)\). It has been analyzed in our previous work on SHG. Therefore, it is not repeated here because of the similarity.