## 2 The Principle of Angle Measurement Based on Direct THG
### The Angle Measurement Methods Based on Direct THG with a Femtosecond Laser  
Figure 3a shows the angle measurement schematic based on direct THG in centrosymmetric crystal. Because the magnitude of \(\chi^{(3)}\) for THG is very small, the femtosecond laser beam is needed to be focused into centrosymmetric crystal by a focused lens after passing through the polarizer 1. The generated third harmonic wave is collected by a detector. The polarizer 2 used here prevents the femtosecond laser beam from affecting the measured result. The crystal is mounted on a rotary stage. The angle \(\theta\) changes as the target rotates. In a THG process, the incident fundamental wave (FW) and the transmitted THW generally contain a certain bandwidth in the frequency domain, which can be seen in Figure 3a. For a mode-locked femtosecond laser, the FW has \(N\) equally spaced modes in the frequency domain. It is also the case for the THW. The spaces between the \(i^{\text{th}}\) mode and the \(i+1\)th mode are \(\mathbf{v}_{\text{rep}}\) and \(\mathbf{3}_{\text{rep}}\) for FW and THW, respectively. The carrier-envelope offsets are \(\mathbf{v}_{\text{CED}}\) and \(\mathbf{3}_{\text{VCED}}\) for FW and THW, respectively, because the THG process triples the frequency of FW in the crystal. The \(\lambda_{1,3}\) are used to denote the corresponding wavelengths of FW and THW, respectively, in Figure 3a. In this paper, we assume the focused beam has a typical Gaussian distribution in free space, and the characteristics are displayed in Figure 3b. Here, \(w\) and \(b\) are the beam waist and the confocal parameter of the Gaussian beam, respectively. The \(Z_{\text{L}}\) and \(Z_{\text{R}}\) are the coordinate value in the incident plane and the output plane, respectively. The crystal length \(L=Z_{\text{R}}-Z_{\text{L}}\). The \(d\) and \(f\) are the diameter of the incident laser beam and the focal length of the focused lens, respectively. Figure 2: (**a**) The refractive index surface of type I phase-matching case; (**b**) The type I phase-matching process in a crystal; (**c**) The refractive index surface of type I phase-mismatching case; (**d**) The type I phase-mismatching process in crystal. The angle dependence of transmitted THG \(I_{3}(\lambda_{3},\theta)\) is given in Equation (4) [27], where c and \(\epsilon_{0}\) are the light speed and dielectric constant of free space, respectively. \(I_{1}\) is the intensity of the fundamental wave. \(\chi_{\text{eff}}^{(3)}\) is the third-order effective nonlinear coefficient, which is related to the phase-matching type and the \(\mathbf{\chi}^{(3)}\) matrix of crystal. Here, the \(J_{3}\) function is crucial for the angle measurement. It is dependent on the phase-mismatching variation \(\Delta k(\lambda_{1},\theta)\). The equations for the calculation of \(\Delta k\), \(b_{1}\), and \(w_{1}\) are given as Equation (6) to Equation (8), respectively. The beam waist and the confocal parameter for THW are \(w_{3}=\sqrt{3}w_{1}\) and \(b_{3}=b_{1}\), respectively [28]. \(\chi_{\text{eff}}^{(3)}\) will be calculated in the next section corresponding to the investigated centrosymmetric crystals. \[I_{3}(\lambda_{3},\theta)=\frac{48(\chi_{\text{eff}}^{(3)})^{2}I_{3}^{2}}{n_{ 3}n_{1}^{3}\lambda_{1}^{2}\epsilon^{2}\epsilon_{0}^{2}w_{1}^{4}}|J_{3}(\Delta k (\lambda_{1},\theta),b_{1},Z_{L},Z_{R})|^{2},3\lambda_{3}=\lambda_{1} \tag{4}\]  
\[|J_{3}|^{2}=\left|\int_{z_{L}}^{z_{R}}\frac{e^{i\Delta k(\lambda_{1},\theta)z }dz}{(1+2 iz/b_{1})^{2}}\right|^{2} \tag{5}\]  
Figure 3: (**a**) The schematic of angle measurement based on direct THG in a centrosymmetric crystal; (**b**) The diagram of THG generated by focused beam; (**c**) The relationship between the incident angle \(\theta_{\text{i}}\) and \(\theta\).