## 2 The Principle of Angle Measurement Based on Direct THG  
### The Basic Principle of Angle Dependence of Direct THG in Centrosymmetric Crystal  
Figure 1a shows a brief schematic of the THG process in centrosymmetric crystal. The fundamental wave (FW) passes through the crystal and produces a third harmonic wave (THW) by three-order susceptibility \(\chi^{(3)}\). The produced third harmonic wave has three times the frequency (\(\
u_{3}=3\
u_{1}\)) and the wave number (\(k_{3}=3k_{1}\)) of the fundament wave. Its wavelength becomes one-third of the FW accordingly (\(\lambda_{3}=\lambda_{1}/3\)). The angle dependence of THG is mainly related to the birefringence and dispersion of the crystal's refractive index, which makes the light propagate at different speeds in different directions and wavelengths. Therefore, we introduce the refractive index first. For the common uniaxial crystal, Figure 1b gives the refractive index ellipsoid of the uniaxial crystal, where \(X\)-\(Y\)-\(Z\) is the Cartesian coordinates of the crystal, and the \(Z\) axis is assumed to be parallel to the optical axis. \(\theta\) is the angle between the wave vector \(\mathbf{k}\) and the optical axis. \(\varphi\) denotes the azimuthal angle, which is the angle between the projection of \(\mathbf{k}\) in the \(X\)-\(Y\) plane and the \(X\)-axis. \(N_{\rm o}\) and \(N_{\rm e}\) are the length of the refractive ellipsoid. There are two different propagated waves in crystal, called the ordinary wave (o-wave) and the extraordinary wave (e-wave), of which the electrical fields are perpendicular and parallel to the incident plane, respectively. It should be noted that the effect of the walk-off is ignored in this paper. Thus, we assume that the e-wave is perpendicular to the \(\mathbf{k}\). The refractive index can be calculated by Equation (1), where the numbers 1 and 3 correspond to the FW and THW. We can see that the refractive index of the e-wave is dependent on the angle \(\theta\), and it is also related to the wavelength \(\lambda\). Because the refractive index of the e-wave is determined by both \(\theta\) and \(\lambda\), the angle \(\theta\) can be used for the compensation of refractive index dispersion between FW and THW. That is to say, the refractive index \(n_{3}\) of the third harmonic wave may be equal to that of the fundamental wave \(n_{1}\) by propagating at one special angle and choosing a suitable polarization combination, which is so-called birefringent phase-matching. \[n_{\rm 1e,\ 3e}(\lambda_{1,3},\theta)=\left(\frac{\cos^{2}\theta}{[N_{\rm o }(\lambda_{1,3})]^{2}}+\frac{\sin^{2}\theta}{[N_{\rm e}(\lambda_{1,3})]^{2}} \right)^{-1/2}\rm e-wave\ \ n_{\rm 1o,\ 3o}=N_{\rm o}(\lambda_{1,3})\ \rm o-wave \tag{1}\]  
Figure 2 illustrates the type I phase-matching of positive uniaxial crystal as an example. In this figure, FW and THW are assumed as e-wave and o-wave, respectively. \(X\)-\(Y\)-\(Z\)' and \(X\)-\(Y\)-\(Z\) denote the stationary laboratory's coordinate and the rotated crystal coordinate, respectively. The \(Z\)' axis is parallel to the FW propagation direction \(\hat{k}_{\rm 1e}\) (laser direction) in crystal. Figure 2a shows the refractive index surface of the phase-matching case. We can see \(\hat{k}_{\rm 1e}\) is in the direction of phase-matching angle \(\theta_{\rm m}\), which makes the phase mismatching \(\Delta k(\theta_{\rm m})=6\pi/\lambda_{1}(n_{\rm 3o}-n_{\rm 1e}(\lambda_{1}, \theta_{\rm m}))=0\) because of the refractive index difference \(\Delta n(\theta_{\rm m})=n_{\rm 3o}-n_{\rm 1e}(\lambda_{1},\ \theta_{\rm m})=0\). In the phase-matching case, as given in Figure 2b, FW and THW have the same propagated speed in the crystal because \(\Delta k=0\) and \(\Delta n=0\). The THW generated in different positions (the first, second, and third, pos  
can be strengthened by constructive interference. Therefore, the detectable THG intensity is generated.