# Theoretical Investigation for Angle Measurement Based on Femtosecond Maker Fringe
## 4 Conclusions  
In this paper, we propose to use the femtosecond Maker fringe for angular measurement and present various calculation results of different z-cut nonlinear crystals and polarization combinations. The calculation investigation has been done for both the intensity-based angular measurement and the optical frequency domain angular measurement. For the intensity-based method, the calculation results show that p-p polarization and s-s polarization only has the slow varying angle dependence in the Maker fringe, which means low sensitivity and visibility. In contrast, the Maker fringe of p-s polarization has a relatively higher sensitivity and visibility because of the oscillation in the angle dependence. To explain the oscillation of the Maker fringe, the refractive index difference \(\Delta n\) was introduced, and the results show that the \(\Delta n\)'s angle dependence of p-s polarization is significantly faster than the others, thus confirming the relationship between the oscillation and the \(\Delta n\). Besides this, we also calculate the Maker fringe in different modes to explore the effect of dispersion and calculate the sensitivity and visibility of BBO and KDP. As a typical case, the multiple peaks shift of Maker fringe in the optical frequency domain can be clearly seen in the p-s polarization of LiNbO\({}_{3}\), and it can cover a measurement range of \(\pm 20^{\circ}\) only with a bandwidth of 20 nm. Moreover, we show other cases of the Maker fringe shift, and the multiple fringes can also be seen in some materials. We cut the single peak from the calculation results and used the central wavelength to denote the peak's angle dependence. The measurement sensitivity of central wavelength was calculated, and the p-s polarization of BBO has the best sensitivity of 1.43 \(\times\) 10\({}^{-3}\) nm/arcsecond. In addition, the contour map of refractive index difference \(\Delta n\) was used to explore the determining factor of measurement sensitivity, and the results confirmed that the slope and number of contour lines are closely related to the measurement sensitivity and the peaks' number. It should be noted that the theoretical investigation in this paper has been mainly focused on visibility and sensitivity. However, there are many other aspects that are also important for designing a sensor in the actual application, such as measurement selectivity and stability. Taking the selectivity as an example, the Maker fringe should not only be  
Figure 13: The contour map of \(\Delta n\) of the investigated materials. (**a**) BBO; (**b**) KDP; (**c**) LiNbO\({}_{3}\); (**d**) Ag\({}_{3}\)As\({}_{3}\). In every subplot, the lines in different colors denote the different constants of \(\Delta n\), and the step of \(\Delta n\) between different lines is 0.001.  
sensitive to incident angle \(\theta_{\rm i}\) but also should be insensitive to other factors that may affect the measurement, such as the azimuthal angle \(\varphi\). Therefore, a comprehensive analysis will be done in future works by considering both selectivity and stability. Then, a measurement scheme will be considered to verify the feasibility of angle measurement based on the Maker fringe. Conceptualization, W.G. ; methodology, W.G., H.M. and K.L. ; software, K.L. and W.D.A. ; formal analysis, K.L. ; investigation, W.D.A. and R.S. ; resources, W.G. ; data curation, K.L. and H.M.; writing--original draft preparation, K.L. ; writing--review and editing, H.M. and W.G. ; visualization, K.L. ; supervision, W.G. ; funding acquisition, W.G. and H.M. All authors have read and agreed to the published version of the manuscript. This work is supported by the Japan Society for the Promotion of Science (JSPS) 15H05759 and 20H00211. Not applicable. The data presented in this study are available on request from the corresponding author. [PERSON] would like to thank the Graduate Program for Integration of Mechanical Systems (GP-Mech) of Tohoku University, Japan. The authors declare no conflict of interest.