# Theoretical Investigation for Angle Measurement Based on Femtosecond Maker Fringe
## 3 Simulation Results and Discussion
### The Investigation of Intensity-Based Angle Measurement  
To investigate the intensity-based method, Figure 6 displays the results of the incident angle dependence of the transmitted second harmonic wave. The total intensity \(I_{\text{SHW}}\) denotes the summation of all the discrete light combs as Equation (1). The azimuthal angle \(\varphi\) and the crystal length \(L\) are set as 0\({}^{\circ}\) and 1 mm for the calculation. In each subplot, the incident angle range is in the range of \(\pm\)20\({}^{\circ}\), and all the curves are normalized. The subplot and the color represent the results of the crystals and the polarization combinations. It can be seen that all the curves in Figure 6 show an angle dependence in the range of \(\pm\)20\({}^{\circ}\). The p-s polarization of BBO and KDP show significant measurement visibility and sensitivity because of the oscillation of angle dependence. In comparison, the oscillation of the p-s polarization of the LiNbO\({}_{3}\) is subtle, and the Ag\({}_{3}\)AsS\({}_{3}\) does not appear to oscillate, which also means they are not suitable for intensity angle measurement. The visibility \(V\) and the averaged sensitivity \(S\) can be calculated according to the following equation:  
\[V=\frac{I_{\text{max}}-I_{\text{min}}}{I_{\text{max}}+I_{\text{min}}} \tag{9}\]  
\[S=\frac{I_{\text{max}}-I_{\text{min}}}{\Delta\theta} \tag{10}\]  
where \(I_{\text{max}}\) and \(I_{\text{min}}\) denote the maximum intensity and the minimum intensity in one oscillation period, and \(\Delta\theta\) denote the angle difference in one oscillation period [23]. The visibility and sensitivity are calculated for the BBO and KDP as shown in Figure 6. For the BBO, the visibility and measurement sensitivity in the shown oscillation period are about 25.1% and 0.08/deg, respectively. For the KDP, the visibility and sensitivity are 69.2% and 0.35/deg. The visibility and measurement sensitivity of p-s polarization are closely related to the dispersion of the material. To be intuitive, Figure 7 shows the angle dependence of different discrete light modes. In this figure, the light combs take the wavelengths of 1560 nm, 1570 nm, and 1580 nm, respectively, and the calculation results are given for all the crystals in each corresponding subplot. It can be seen that the KDP has the lowest dispersion. Thus, the oscillations of the different optical combs  
Figure 6: The angular dependence of the total intensity of the second harmonic wave. (**a**) BBO; (**b**) KDP; (**c**) LiNbO\({}_{3}\); (**d**) Ag\({}_{3}\)AsS\({}_{3}\). out, which is advantageous for improving both the sensitivity and the visibility of angular measurement. In contrast, the Ag\({}_{3}\)AsS\({}_{3}\) has a fast dispersion, and it causes the oscillation of angle dependence to disappear, as Figure 6d illustrated. Furthermore, it also can be noticed from Figure 6 that neither the p-p case nor the s-s case shows any oscillation in the angle dependence, which also means low measurement sensitivity and visibility. This is because the oscillation is related to the angle dependence of phase difference \(\psi\) in Equation (5). The more sensitive the angle dependence of the \(\psi\) is, the shorter the period of the oscillation will be, i.e., the better the sensitivity and visibility of the angular measurement. We note that the phase difference is proportional to the term as Equation (11), which is determined by the refractive index difference \(\Delta n\) as (12):  
\[\psi\propto(n_{\text{b}}\cos\theta_{\text{b}}-n_{\text{f}}\cos\theta_{\text{ f}}) \tag{11}\]  
\[\Delta n=n_{\text{b}}-n_{\text{f}} \tag{12}\]  
Figure 8 exhibits the angle dependence of \(\Delta n\) for the light comb of 1560 nm, where the different lines in each subplot denote the polarization combinations, and the subplots correspond to the materials.