# Theoretical Investigation for Angle Measurement Based on Femtosecond Maker Fringe
## 1 Introduction
On the other hand, the high peak intensity of the femtosecond laser generates a strong electrical field, which can produce various nonlinear optical phenomena, such as the second harmonic generation (SHG) [25]. It is well-known that the SHG process is closely related to propagation direction in the nonlinear crystals, and the brilliant angle dependence of the output intensity of the second harmonic wave was found in 1962 [26]. In our previous works, the angle dependence of SHG was utilized to design sensors for the detection of angular displacement. There are two types of angular measurement solutions based on SHG that were designed and verified, including the intensity-based angular measurement and the optical frequency domain angular measurement [27; 28]. The intensity-based measurement is based on the angle dependence of the generated second harmonic wave (SHW). The output intensity of SHW is maximized only when the femtosecond laser propagates along the phase-matching angle in nonlinear materials [29]. As the laser gradually rotates away from the phase-matching angle, the generated SHW degrades rapidly, and then the angular displacement can be determined according to the intensity change. The basic principle of optical frequency domain angle measurement is based on the dispersion effects of the nonlinear crystal. Due to the refractive index dispersion, the phase-matching angle of each mode in the femtosecond laser pulse is different. Therefore, the different optical frequency modes reach their intensity peaks as the femtosecond laser rotates. Intuitively, the intensity peak shifts in the optical frequency domain as the angle changes, and thus the angular displacement can be measured according to the peak shift of SHW in the frequency domain. However, the angle dependence of SHG not only exists in the phase-matching condition but also can be observed in the phase-mismatching area [30; 31]. The angle dependence in the phase-mismatching area is called the \"Maker fringe\" because the experimental results were first measured by [PERSON] [26]. The incident angle dependence of the Maker fringe can even reach the range of \(\pm 80^{\circ}\) in some materials [32], and thus it has a huge potential to expand the angular measurement range and improve measurement sensitivity. Differing from our previous work, the analytical model for the Maker fringe is more complex because the effects of the crystal boundary, the cut way of the crystal, and the azimuth of the measurement should be taken into account [30; 33]. Therefore, the theoretical investigation of angular measurement based on the Maker fringe should be done before the experimental setup design. This study investigates the angle dependence of the Maker fringe for both the intensity-based and the optical frequency domain angle measurement. The fundamental measure ment principles and the calculation model are briefly introduced for understanding. Compared with our previous research work, the Maker fringe could have multiple visible peaks as the incident angle changes. Therefore, the measurement range can be primarily expanded in expectation. In the calculation analysis, several z-cut negative uniaxial nonlinear materials and different polarizations of Maker fringe have been considered. For the intensity-based method, we discussed the effects of dispersion and refractive index on angle measurement and calculated the visibility and sensitivity. The simulation also verified several peak shifts in the optical frequency domain, and we used the central wavelength to denote the peak shift. Then, we showed the angle dependence of the central wavelength, and the measurement sensitivity was also calculated for the investigation. The contour map of the refractive index was introduced, and the results show that the refractive index is closely related to the measurement sensitivity.