# Theoretical Investigation for Angle Measurement Based on Femtosecond Maker Fringe
## 2 The Theory and Principle of Angle Measurement Based on the Maker Fringe
### The Calculation Theory of the Maker Fringe
Once the refractive index has been determined, the propagation direction of all waves can be obtained using [PERSON]'s law in Equation (2), and then the intensity \(I_{2}\) of transmitted SHW can be known  
Figure 4: (**a**) The propagation model in the incident plane; (**b**) the refractive index ellipsoid. by solving the Maxwell equation and considering the boundary condition. With the assumptions of plane wave and the pump nondepletion, \(I_{2}\) is proportional to the production of Equation (4) [29], where \(t_{1}\), \(t_{2}\) are the transmission factors of FW and SHW, \(I_{1}\) is the intensity of FW, \(d_{\text{eff}}\) is the effective nonlinear coefficient, and \(\psi\) is the phase difference between bound SHW and free SHW in Equation (5). \[I_{2}\propto d_{\text{eff}}^{2}t_{1}{}^{4}t_{2}I_{1}^{2}(\lambda_{1})\big{[}1 /(n_{\text{b}}^{2}-n_{\text{f}}^{2})^{2}\big{]}\sin^{2}\psi \tag{4}\]  
\[\psi=\big{[}2\pi L/\lambda_{1}(n_{\text{b}}\cos\theta_{\text{b}}-n_{\text{f}} \cos\theta_{\text{f}})\big{]} \tag{5}\]  
\[t_{1}=\left\{ \begin{array}{l}2\cos\theta/\left(n_{\text{b}}\cos\theta_{ \text{b}}+\cos\theta\right)\text{s-wave}\\ 2\cos\theta/\left(n_{\text{b}}\cos\theta+\cos\theta_{\text{b}}\right)\text{p- wave}\end{array} \right. \tag{6}\]  
\[t_{2}=\left\{ \begin{array}{l}2n_{\text{f}}\cos\theta_{\text{f}}\frac{(n_{ \text{b}}\cos\theta_{\text{b}}+\cos\theta_{\text{b}})\left(n_{\text{f}}\cos \theta_{\text{b}}+n_{\text{b}}\cos\theta_{\text{f}}\right)}{\left(n_{\text{f}} \cos\theta_{\text{f}}+\cos\theta\right)^{3}}\text{s-wave}\\ 2n_{\text{f}}\cos\theta_{\text{f}}\frac{(\cos\theta+n_{\text{b}}\cos\theta_{ \text{b}})\left(n_{\text{b}}\cos\theta_{\text{b}}+n_{\text{f}}\cos\theta_{ \text{f}}\right)}{\left(n_{\text{f}}\cos\theta_{\text{f}}+\cos\theta\right)^{3 }}\text{p-wave}\end{array} \right. \tag{7}\]  
The transmission factors \(t_{1}\) and \(t_{2}\) represent the effects caused by the boundary of the nonlinear crystal. Thus, they are similar to Fresnel factors, and their values are related to the polarization of the fundamental wave and second harmonic wave as Equations (6) and (7). There are four different polarization combinations of SHG, including s-s case, s-p case, p-s case, and p-p case. For example, the s-p case implies that the incident s-polarization FW induces the p-polarization transmitted SHW, which means that the second harmonic wave may have a different polarization from the fundamental wave. The transmission factor has slowly varying angle dependence, which should be considered in the expanded angle measurement range calculation. Besides this, the effective nonlinear coefficient \(d_{\text{eff}}\) also has slowly varying angle dependence, and the calculation should refer to the point groups of crystal and polarization combination of SHG [28]. In Section 3, the paper gives extensive calculation results based on this model for two different measurement principles, including the intensity-based angular measurement and the optical frequency domain measurement.