# Theoretical Investigation for Angle Measurement Based on Femtosecond Maker Fringe
## 2 The Theory and Principle of Angle Measurement Based on the Maker Fringe  
### The Principle of Angle Measurement Based on the Maker Fringe  
Figure 1a shows the measurement schematic of the angular sensor based on second harmonic generation. The mode-locked femtosecond laser beam, as the incident fundamental wave (FW), induces the second harmonic wave in the nonlinear crystal, and then the transmitted SHW is detected by the optical spectrum analyzer (OSA) with a photodiode array for the fast measurement. For clarity, the transmitted FW is not shown in this figure. The X-Y-Z is the laboratory coordinate system, and the X-Z plane is parallel to the incident plane. The measurement target is fixed with the nonlinear crystal. As the target rotates along the Y-axis, the transmitted intensity of the SHW changes accordingly because of the angle dependence of the Maker fringe. Figure 1b,c show the diagram of the spectrum of incident FW and transmitted SHW, where the \(\lambda_{1}\) and \(\lambda_{2}\) denote the wavelength of incident FW and the transmitted SHW. The FW and SHW contain a series of equally-spaced modes in the optical frequency domain, and the frequency difference between the discrete modes of FW and SHW are \(\
u_{\text{rep}}\) and \(2\
u_{\text{rep}}\). Both the FW and SHW have \(N\) modes, and the \(i\)th mode of FW corresponds to the \(i\)th mode of SHW. \(\lambda_{1}^{i}\) and \(\lambda_{2}^{i}\) denote the wavelength of the \(i\)th mode of FW and SHW, and \(\lambda_{1}^{i}/2=\lambda_{2}^{i}\), that is to say that the wavelength of the generated SHW is half of the corresponding mode of FW. The intensity \(I_{2}(\lambda_{2},\,\theta_{i})\) of transmitted SHW is both related to the wavelength \(\lambda_{2}\) and the incident angle \(\theta_{i}\) because of the dispersion and the angle dependence of SHG, and it can be calculated analytically by the Maker fringe theory introduced in the next section.  
For the intensity-based angular measurement of the Maker fringe, the total intensity of all the modes in the SHW spectrum is used as the measurement signal to improve the  
Figure 1: (**a**) The schematic of angle measurement based on SHG; (**b**) the spectrum of incident femtosecond laser; (**c**) the spectrum of transmitted SHW.  
signal-noise ratio [26]. The total intensity \(I_{\text{SHW}}\) denotes the intensity summation of the transmitted second harmonic wave of all the discrete modes as shown in Equation (1):  
\[I_{\text{SHW}}(\theta_{i})=\sum_{\lambda_{2}^{1}}^{\lambda_{2}^{N}}I_{2}(\lambda_ {2},\theta_{i}) \tag{1}\]  
where \(\lambda_{2}^{1}\) and \(\lambda_{2}^{N}\) represent the wavelength of the first and the final mode in the SHW spectrum. The typical angle dependences of the Maker fringe and the case nearing the phase-matching condition of our previous study are shown in Figure 2a,b, where the Maker fringe has more visible peaks, which means that a wider measurement range is used.  
On the other hand, the Maker fringe also can be applied to the angular measurement in the optical frequency domain [27]. Figure 3a,b illustrate the peak shift in the optical frequency domain of the Maker fringe and the phase-matching case. Because of the wavelength dependence of \(I_{2}(\lambda_{2},\,\theta_{i})\), the modes \(\lambda_{2}^{i}\) (\(i=1,\,2,\,\ldots,\,N\)) have different angle dependence relationships. Therefore, as the modes' wavelength gradually increases in the optical domain, the peaks shift in a certain speed and direction in the optical domain, and the peak shift can use to denote the angular displacement [27]. The Maker fringe has multiple visible peaks, and each peak has its shift in the frequency domain. Thus, the measurement range in the optical frequency domain can also be largely expanded.