# Theoretical Investigation for Angle Measurement Based on Femtosecond Maker Fringe
## 3 Simulation Results and Discussion  
### The Simulation Setup  
Figure 5 shows the typical spectrum of femtosecond laser from our previous work, and it is used as the fundamental wave for the angle dependence investigation. The spectrum range of the fundamental wave is from 1480 nm to 1640 nm, meaning that the generated SHW will have a distribution from 740 nm to 820 nm. In the calculation process, the spectrum of the femtosecond laser is divided into a number of laser combs. For each comb, we assume it produces a corresponding mode of transmitted SHW, and the intensity of each mode can be calculated by Equation (4). Figure 5: The spectrum of a femtosecond laser used for the angular measurement investigation; the spectrum range of high intensity is from 1560 nm to around 1600 nm. The investigated materials include barium borate (BBO), lithium niobate (LiNbO\({}_{3}\)), potassium dihydrogen phosphate (KDP), and proustite (Ag\({}_{3}\)As\({}_{3}\)), which are both negative uniaxial crystals. The Sellmeier equations given in Equation (8) are used to empirically describe the dispersion relation of nonlinear materials, where the parameters \(A\), \(B\), \(C\), \(D\), \(E\) are related to the specific materials. The coefficients for refractive indices calculation are shown in Table 1[35]. \[ \begin{split} N_{\text{e}}(\lambda),N_{\text{o}}(\lambda)=\sqrt{ A+\frac{B\lambda^{2}}{\lambda^{2}-\text{C}}-D\lambda^{2}}\text{BBO LiNbO}_{3}\\ N_{\text{e}}(\lambda),N_{\text{o}}(\lambda)=\sqrt{A+\frac{B\lambda^{ 2}}{\lambda^{2}-\text{C}}-\frac{D}{\lambda^{2}-E}}\text{KDP}\\ N_{\text{e}}(\lambda),N_{\text{o}}(\lambda)=\sqrt{A+\frac{B}{\lambda^ {2}-\text{C}}-\frac{D}{\lambda^{2}-E}}\text{Ag}_{3}\text{AsS}_{3}\end{split}  \tag{8}\]  
The angle dependence of the transmitted second harmonic wave also should consider the effective nonlinear coefficients \(d_{\text{eff}}\). For the SHG process, the \(d_{\text{eff}}\) of the crystals are decided based on their point groups and the polarization combinations. The derivation of \(d_{\text{eff}}\) could refer to the literature [28], and \(d_{\text{eff}}\) are directly given in Table 2, where the dispersion and the walk-off effects of \(d_{\text{eff}}\) are ignored. There are three polarization combinations, the p-s case, p-p case, and s-s case, which will be discussed, with the exclusion of the s-p case. This is because the s-p case denotes that the s-polarization fundamental wave generates the p-polarization, and it means that the ordinary fundament wave generates the extraordinary second harmonic wave, which is the phase-matching case discussed in our previous works [26]. Therefore, the s-p case will not be investigated in this paper. The BBO, LiNbO\({}_{3}\), and Ag\({}_{3}\)AsS\({}_{3}\) have a 3 m point group structure, except that the KDP belongs to the \(\overline{4}\)2 m point group. Thus, the KDP is the only one with a different \(d_{\text{eff}}\) distinguished from the others. Table 2 lists the nonlinear coefficients for all materials according to the references [35], and it is important to note that the nonlinear coefficients are different even for the same point group structure materials.