# 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  
To quantificationally investigate the angle dependence of the Maker fringe, the calculation theory of the Maker fringe should be explained in detail. The theory of the Maker fringe is different from that of our previous works because the incident angle of the Maker fringe can vary in a wider range, where the internal reflection and refraction caused by the boundary should be considered. The propagation model proposed by the literature  
Figure 3: (**a**) The peak shifts of Maker fringe; (**b**) the peak shift nearing the phase-matching condition. Figure 2: (**a**) The typical angle dependence of the Maker fringe; (**b**) the angle dependence nearing the phase-matching condition. is referred to calculate \(I_{2}\) in this paper [29]. The model is illustrated in Figure 4a, where \(X^{\prime}\)-\(Y^{\prime}\)-\(Z^{\prime}\) and \(L\) denote the Cartesian coordinator and the length of the rotary crystal. In this model, the incident FW inputs on the cut plane from the Region I (air), and then induces two different SHW (the bound SHW and free SHW) inside the Region II (crystal). The two SHW waves can pass through the crystal as transmitted SHW in Region III (air), and they also result in the reflected SHW in the Region I (air). The \(k_{1}\), \(k_{\text{b}}\), \(k_{\text{f}}\), \(k_{\text{r}}\), and \(k_{\text{t}}\) are the wave vectors of the fundamental wave, bound SHW, free SHW, reflected SHW, and transmitted SHW, respectively. Correspondingly, the \(\theta_{\text{i}}\), \(\theta_{\text{b}}\), \(\theta_{\text{f}}\), \(\theta_{\text{r}}\), and \(\theta_{\text{t}}\) are the angles between the wave vectors and the normal. For calculation, the incident angle \(\theta_{\text{i}}\) is a known parameter, and the other angles in Figure 4a should be calculated. It can be derived by considering the conservation of the electrical field's tangential component and the result is shown as Equation (2) [34]:  
\[\theta_{\text{i}}=\theta_{\text{r}}=\theta_{\text{t}}=n_{\text{b}}\sin\theta_ {\text{b}}=n_{\text{f}}\sin\theta_{\text{f}} \tag{2}\]  
Equation (2) shows that \(\theta_{\text{b}}\) and \(\theta_{\text{f}}\) are related to the refractive indices of the bound SHW (\(n_{\text{b}}\)) and the free SHW (\(n_{\text{f}}\)). In this paper, we assume that the optic axis is parallel to the norm to simplify the analysis. That is to say that the z-cut nonlinear crystal is used for the investigation, and the coordinator of the refractive index ellipsoid is the same as \(X^{\prime}\) -\(Y^{\prime}\) -\(Z^{\prime}\). For the commonly used uniaxial crystal, the refractive index ellipsoid is shown in Figure 4b [35]. The \(N_{\text{o}}\) and \(N_{\text{e}}\) are the length of a refractive ellipsoid, and their dispersion could be described by empirical Sellmeier equations [36]. The \(\theta\) denotes the angle between the optical axis and the wave vector \(k\), and \(\varphi\) denotes the azimuthal position between the \(X^{\prime}\) axis and the projection of wave vector \(k\) in the \(X^{\prime}\) -\(Y^{\prime}\) plane. For the z-cut crystal, the s-wave or the o-wave represents the wave whose electrical field is perpendicular to the incident plane. The electrical field of the p-wave or the e-wave is parallel to the incident plane. According to their polarization, the refractive index of s-wave and p-wave can be calculated as Equation (3):  
\[n_{\text{b, f}}(\lambda,\theta_{\text{b, f}})=\frac{1}{\sqrt{ \left(\cos^{2}(\theta_{\text{b, f}})/N_{\text{o}}^{2}(\lambda_{1,2})+\sin^{2}( \theta_{\text{b, f}})/N_{\text{e}}^{2}(\lambda_{1,2})\right)}}\text{ P-wave} \tag{3}\] \[n_{\text{b, f}}=N_{\text{o}}(\lambda_{1,2})\text{ s-wave}\]  
where \(\lambda_{1}\) and \(\lambda_{2}\) correspond to the wavelength of the fundamental wave and second harmonic wave in a vacuum. Since the velocity of bound SHW is the same as the FW, the corresponding wavelength for \(n_{\text{b}}\) is \(\lambda_{1}\), and \(\lambda_{2}\) is for \(n_{\text{f}}\) of free SHW.