## 3 Experimental Phenomenon and Theoretical Analysis
### Asymmetric Etalon Effect Phenomenon and Theoretical Analysis
Based on the multi-beam interference theory, the effective reflection and transmission coefficients of mirror M1 are calculated respectively as Equation (5) and Equation (6):  
\[r_{eff}=(r_{1}+\phi r_{0}e^{-2i\delta_{0}})/(1+\phi r_{1}r_{0}e^{-2i\delta_{0}}) \tag{5}\]  
\[t_{eff}=(\sqrt{(1-{r_{1}}^{2})(1-\phi^{2}r_{0}^{2})}e^{-i\delta_{0}})/(1+\phi r _{1}r_{0}e^{-i2\delta_{0}}) \tag{6}\]  
where \(\delta_{0}=nd\cos(\theta)\cdot 2\pi/\lambda\), \(n\) is the reflective index of quartz glass (\(n=1.444\)), \(d\) is the thickness of M1 (d = 8 mm), \(r_{0}\), \(r_{1}\), \(r_{2}\), \(r_{3}\) and \(r_{4}\) represent the amplitude reflectivity of the mirror surface (\(r_{0}^{2}=0.0332,r_{1}^{2}=r_{4}^{2}=0.99999,r_{2}^{2}=r_{3}^{2}=0.99991\), calibration by cavity ring-down). The reflection coefficient \(r_{1}\) can be replaced by the effective reflection coefficient \(r_{eff}\), when the multi-beam interference occurs in the U-shaped cavity (as shown in Figure 3b). Consider that the resonant field of the U-shaped cavity is the TEM\({}_{00}\) fundamental mode. By the integral operation, the total coupling light of the transmitted light field at M1 is calculated as Equation (7):  
\[E_{\text{T1}}=A_{0}\sqrt{(1-r_{3}^{2})}r_{2}t_{eff}e^{-i(\omega t+kx)}e^{-i( \delta_{1}+\delta_{2})}\psi \tag{7}\]  
Figure 3: (**a**) shows the incident light A split into multiple beams A-n and B-n at the mirror M1; (**b**) shows the multi-beam interference of the U-shaped cavity by using a effective reflection coefficient. A total interference transmitted light field at M2 is calculated as Equation (8):  
\[E_{\rm T2}=A_{0}\sqrt{(1-r_{3}^{2})(1-r_{2}^{2})}r_{2}r_{eff}\cdot e^{-i(\omega t+kx )}e^{-i(2\delta_{1}+\delta_{2})}\psi \tag{8}\]  
Here, \(\psi=1/(1-r_{2}^{2}r_{3}^{2}r_{4}r_{eff}e^{-2i\delta})\), \(\delta_{0}=nd\cos(\theta)\cdot 2\pi/\lambda\), \(\delta_{1}=L_{1}\cdot 2\pi/\lambda\), \(\delta_{2}=L_{2}\cdot 2\pi/\lambda\), \(\delta=2\delta_{1}+\delta_{2}\), \(L_{1}\) and \(L_{2}\) is the length of each cavity arm (\(L_{1}=L_{3}=170\) mm, \(L_{2}=132\) mm). Considering that the transmission losses of mirrors M1 and M2 are different, the corresponding ripple amplitude of the transmission light intensity is different. According to the above mathematical model, the transmitted light intensity and ripple amplitude of the resonator with different loss mirrors are simulated. The transmission loss of mirror M1 (T\({}_{1}=1-R_{1}\)) is set to 10 ppm, and the transmission loss of mirror M2 (T\({}_{2}=1-R_{2}\)) varies from 10 ppm to 210 ppm. The simulation results are shown in Figure 4:  
As shown in Figure 4a, the transmission loss T1 of the end mirror M1 is unchanged. As the transmission loss of the folded mirror M2 gradually increases, the ripple amplitude of the transmission spectrum of M2 gradually decreases. When the transmission losses of M1 and M2 are 10 ppm and 100 ppm, respectively, the ripple amplitude of M1 (\(\sim\)0.2) is much larger than that of M2 (\(\sim\)6 \(\times\) 10\({}^{-3}\)). As shown in Figure 4b, the ripple amplitude of the transmission spectrum of folded mirror M2 is inversely proportional to its transmission loss value, that is, the larger the loss of M2, the smaller the ripple of the transmission spectrum. For the end mirror M1, as the loss of M2 increases, the relative loss of M1 (T1:T2) decreases and the M1 ripple amplitude increases. It can be inferred that the main reason for the phenomenon of asymmetric etalon effect is that the loss of the folded mirror M2 is much larger than that of the end mirror M1, and the weak ripple is easily drowned by system noise.