## 3 Simulation Results and Discussion
### The Sensitivity Investigation of Angle Measurement Based on Direct THG  
As illustrated before, the intensity-dependent and wavelength-dependent methods are based on the dispersion-less and dispersive properties of angle dependence. Here, we use the phase-matching angle \(\theta_{\text{m}}(\lambda_{1})\) to denote the dispersion speed in a certain wavelength range because \(\theta_{\text{m}}\) represents the angle of the highest point in the angle dependence curve. We first consider a wavelength range from 900 nm to 2500 nm, which covers a commonly used range of femtosecond laser, to find a suitable wavelength range for intensity-dependent angle measurement. The simulation results are given in Figure 6a, where the lines correspond to \(\alpha\)-BBO, calcite, and rutile. In the investigated wavelength range, the \(\theta_{\text{m}}(\lambda_{1})\) of rutile is monotonically decreasing from 79.8\({}^{\circ}\) to 34.2\({}^{\circ}\), which indicates the angle dependence of rutile has a very significant dispersion in this range. Therefore, rutile is not an ideal crystal for intensity-dependent measurement in the investigated range. Compared with rutile, both the \(\theta_{\text{m}}(\lambda_{1})\) curves of \(\alpha\)-BBO and calcite go down firstly from 900 nm to around 1700 nm, then keep stable from 1700 nm to 2000 nm, and finally slowly rise from 1700 nm to 2000 nm. We can see that the trends of \(\alpha\)-BBO and calcite are very similar, and the range 1700 nm to 2000 nm is the ideal range for intensity-dependent angle measurement because of the low dispersion. In the dispersion-less area, the \(\theta_{\text{m}}(\lambda_{1})\) of \(\alpha\)-BBO and calcite are around 33.0\({}^{\circ}\) and 27.1\({}^{\circ}\). It is easy to understand that the derivative of \(\theta_{\text{m}}(\lambda_{1})\) in the dispersion-less area should be close to zero. Figure 6b shows the calculated phase-matching angle derivative in a unit of deg/nm. In this figure, the \(\alpha\)-BBO and calcite correspond to the left vertical axis. It can be seen that the dispersion speed of \(\theta_{\text{m}}(\lambda_{1})\) is in the order of 10\({}^{-3}\) deg/nm. Here, the red circles represent the positions having zero derivatives, which are 1803 nm and 1858 nm for calcite and \(\alpha\)-BBO, respectively. As a \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Nonlinear Crystal** & \(A\) & \(B\) & \(C\) & \(D\) & \(E\) \\ \hline \(\alpha\)-BBO & & & & \\ \(N_{\text{o}}\) & 2.7471 & 0.01878 \(\upmu\)m\({}^{2}\) & 0.01822 \(\upmu\)m\({}^{2}\) & 0.01354 \(\upmu\)m\({}^{-2}\) & \\ \(N_{\text{e}}\) & 2.3715 & 0.01224 \(\upmu\)m\({}^{2}\) & 0.01667 \(\upmu\)m\({}^{2}\) & 0.01516 \(\upmu\)m\({}^{-2}\) & \\ Calcite & & & & \\ \(N_{\text{o}}\) & 1.7335 & 0.96464 \(\upmu\)m\({}^{-2}\) & 1.94325 \(\upmu\)m\({}^{2}\) & 1.8283 \(\upmu\)m\({}^{-2}\) & 120 \(\upmu\)m\({}^{2}\) \\ \(N_{\text{e}}\) & 1.3585 & 0.82427 \(\upmu\)m\({}^{-2}\) & 1.06689 \(\upmu\)m\({}^{2}\) & 0.1442 \(\upmu\)m\({}^{-2}\) & 120 \(\upmu\)m\({}^{2}\) \\ Rutile & & & & \\ \(N_{\text{o}}\) & 5.913 & \(2.441\times 10^{5}\)\(\upmu\)m\({}^{-2}\) & \(0.803\times 10^{5}\)\(\upmu\)m\({}^{2}\) & \\ \(N_{\text{e}}\) & 7.197 & \(3.322\times 10^{5}\upmu\)m\({}^{-2}\) & \(0.843\times 10^{5}\upmu\)m\({}^{2}\) & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The coefficients of Sellmeier Equations [26; 31; 32]. comparison, the derivative result of rutile is also given in the figure. It corresponds to the right vertical axis. We see the dispersion speed of \(\theta_{\mathrm{m}}(\lambda_{1})\) is in the order of \(10^{-2}\) deg/nm, which is much higher than the dispersion-less cases. To evaluate the angle measurement sensitivity of the intensity-dependent method, we calculate the angle dependence of \(I_{\mathrm{THW}}(\theta_{\mathrm{i}})\).