## 2 The Principle of Angle Measurement Based on Direct THG
### The Angle Measurement Methods Based on Direct THG with a Femtosecond Laser
\[\Delta k=\frac{6\pi}{\lambda_{1}}(n_{1\text{e}}(\lambda_{1},\theta)-n_{3\text{o}}) \text{ Type I }(\text{Positive crystal}) \tag{6}\] \[\Delta k=\frac{2\pi}{\lambda_{1}}(2n_{1\text{o}}+n_{1\text{e}}( \lambda_{1},\theta)-3n_{3\text{e}}(\lambda_{3},\theta))\text{ Type II }(\text{Negative crystal})\]  
\[b_{1}=\frac{8f^{2}\lambda_{1}}{\pi d^{2}n_{1}} \tag{7}\] \[2w_{1}=\frac{4f\lambda_{1}}{\pi dn_{1}} \tag{8}\]  
Because the actual rotated angle for the measurement target is the incident angle \(\theta_{\text{i}}\), the relationship between \(\theta_{\text{i}}\) and \(\theta\) should also be considered for the angle measurement. As shown in Figure 3c, \(\theta_{\text{r}}\) is the refractive angle, and \(\theta_{\text{spec}}\) is the angle between the optical axis and the normal direction, which is often around the phase-matching \(\theta_{\text{m}}\). This enables the incident laser to easily achieve phase-matching conditions at the normal incidence. The relationship can be described by [PERSON]'s law as Equation (9), where \(n_{1}\) can be calculated according to Equation (1), which is related to the dispersion and the angle \(\theta\). \[\sin\theta_{\text{i}}=n_{1}(\lambda_{1},\theta)\sin\theta_{\text{r}}=n_{1}( \lambda_{1},\theta)\sin(\theta_{\text{spec}}-\theta) \tag{9}\]  
As reported in our previous studies, there are two types of angle measurement methods based on the phase-matching principle of THG: the intensity-dependent method and the wavelength-dependent method. For the former, a single photodiode (PD) can be used as the detector (see Figure 3a), which converts the angle-related intensity change into an electrical current output. Then, the electrical current is amplified to a voltage by an I/V converter, and the voltage is displayed on an oscilloscope. The dispersion of angle dependence is very important because a single PD can only detect the total intensity \(I_{\text{THW}}\) of THW, which has a band from \(\lambda_{3}^{1}\) to \(\lambda_{3}^{N}\). Therefore, the characteristics in the frequency domain cannot be measured. The angle dependence of measured total intensity \(I_{\text{THW}}(\theta)\) can be evaluated by Equation (10) [12]. \[I_{\text{THW}}(\theta)=\sum_{i=1}^{N}I_{3}(\lambda_{3}^{i},\theta) \tag{10}\]  
Figure 4 is used to visually explain the effect of dispersion. Supposing the angle dependence \(I_{3}(\theta)\) is dispersion-less in the wavelength band as shown in Figure 4a, there is a less overlap area of angle dependence between the modes of \(\lambda_{3}^{1}\), \(\lambda_{3}^{i}\), \(\lambda_{3}^{i+1}\), and \(\lambda_{3}^{N}\). That means there is also less possibility for the angle sensitivity to be reduced between the modes. A relatively stronger total intensity \(I_{\text{THW}}(\theta)\) can then be expected because the summation of Equation (10) will be large. Conversely, as shown in Figure 4b, if the angle dependence is dispersive, the expected angle measurement sensitivity will be weakened because of the large overlap area of angle dependence. \[I_{3}(\lambda_{3},\theta)=\frac{48(\chi_{\text{eff}}^{(3)})^{2}I_{3}^{2}}{n_{ 3}n_{1}^{3}\lambda_{1}^{2}\epsilon^{2}\epsilon_{0}^{2}w_{1}^{4}}|J_{3}(\Delta k (\lambda_{1},\theta),b_{1},Z_{L},Z_{R})|^{2},3\lambda_{3}=\lambda_{1} \tag{4}\]0 **(a)** The diagram of dispersion-less angle dependence \(I_{3}(\theta)\); **(b)** The diagram of dispersive angle dependence \(I_{3}(\theta)\). Different from the intensity-dependent method, the wavelength-dependent method is based on the angle dependence dispersion. An optical spectrum analyzer (OSA) is used for THG detection in the frequency domain. As shown in Figure 5a, the different blue lines correspond to the angle dependence curve of different THG wavelengths \(\lambda_{3}^{1}\), \(\lambda_{3}^{i}\), \(\lambda_{3}^{i+1}\), and \(\lambda_{3}^{N}\). Due to the dispersion, the angle dependence curve for each wavelength is different. The corresponding phase-matching angles \(\theta_{\rm m}(\lambda_{3})\) also change in the THG band.