# Theoretical Investigation for Angle Measurement Based on Femtosecond Maker Fringe
## 3 Simulation Results and Discussion
### The Investigation of Angle Measurement in the Optical Frequency Domain  
Figure 9 takes the p-s polarization of LiNbO\({}_{3}\) as a typical example to show the Maker fringe shift in the optical domain. The azimuthal position and the length are set as 0\({}^{\circ}\) and 1 mm for the calculation. The 2-D result can be seen Figure 9a, where the color bar represents the normalized intensity. The incident angle range is from 0\({}^{\circ}\) to 20\({}^{\circ}\), and the spectral range is from 740 nm to around 820 nm, corresponding to half the wavelength of the FW's spectrum. There are several peaks of maker fringe shift shown in Figure 9a as the white arrow shown in the white dotted square. It can be seen that all the peaks move towards longer wavelengths as the incident angle increase, and the higher intensity areas are 780 nm to around 800 nm in the optical frequency domain, which is correlated with the higher intensity light combs of FW. The 3-D plot of peak shift in the range of 5 to around 20\({}^{\circ}\) is illustrated in Figure 9b, where the modes are from 780 nm to 800 nm and stepped by 2 nm. It is visible that the six peaks of Maker fringe change their peak position in the optical domain, and all the peak shifts could be used to measure the angular displacement. In Figure 9a, these peaks cover the range of 0\({}^{\circ}\) to around 20\({}^{\circ}\) in 780 nm to around 800 nm. Therefore, the maker fringe can realize an angular measurement range of \(\pm\)20\({}^{\circ}\) only with a bandwidth of 20 nm (only the result of 0\({}^{\circ}\) to around 20\({}^{\circ}\) shown in Figure 9, and a significant savings of the femtosecond laser bandwidth can achieve this, compared with our previous study [27]). Figure 8: The angular dependence of the refractive indices difference \(\Delta n\). (**a**) BBO; (**b**) KDP; (**c**) LiNbO\({}_{3}\); (**d**) Ag\({}_{3}\)AsS\({}_{3}\). The red solid lines, pink dotted lines, and blue dotted lines in each subplot represent the p-s polarization, p-p polarization, and the s-s polarization, respectively. For further analysis, the materials in Table 1 are both investigated. The \(\varphi\) and \(L\) are set as 25\({}^{\circ}\) and 1 mm, and the calculation results are shown in Figure 10, where the (a), (b), (c), (d) indicate the nonlinear materials BBO, KDP, LiNbO\({}_{3}\), and Ag\({}_{3}\)AsS\({}_{3}\) in order, the numbers 1, 2, 3 denote the polarization combinations p-s, p-p, and s-s, respectively. In each subplot, the discussed incidence angle range is from 0\({}^{\circ}\) to 20\({}^{\circ}\), and the spectral range is from 780 nm to 800 nm because of the higher intensity. The s-s polarization of KDP is not given in Figure 10 because of the corresponding \(d_{\text{eff}}=0\) in Table 1. We notice that all the subplots have at least one peak shift, and the peaks almost cover the whole measurement range of interest. Besides this, it is easy to find that the shape of different peaks in the same subplot is similar. Thus, we only cut a single peak from every subplot to discuss its angle dependence in the frequency domain. The single peaks shift corresponding to Figure 10 is given in Figure 11. Similar to our previous work, the central wavelengths \(\lambda_{\text{P}}\) is used to illustrate the peak's wavelength in the frequency domain [27]. The calculation is given as the following formula:  
\[\lambda_{\text{P}}(\theta_{\text{i}})=\frac{\sum\limits_{ \begin{subarray}{c} \lambda_{2}=780\text{nm}\\ \delta 00\text{nm}\\ \sum\limits_{\lambda_{2}=780\text{nm}}\end{subarray} }^{800\text{nm}}\lambda_{ 2}I_{2}(\lambda_{2},\theta_{\text{i}})}{\sum\limits_{ \begin{subarray}{c} \lambda_{2}=780\text{nm}\\ \delta 0\text{nm}\end{subarray} }^{800\text{nm}}I_{2}(\lambda_{2},\theta_{\text{i}})} \tag{13}\]  
The angle dependence of \(\lambda_{\text{P}}\) of Figure 11 is shown in Figure 12. In this figure, the (a), (b), and (c) denote the subplot p-s polarization, p-p polarization, and s-s polarization, respectively, and the lines in different colors represent the corresponding material.