## 2 Theoretical Analysis  
### Relationship between the Initial Phase and the Phase Sensitivity  
The structure of the optical fiber Sagnac AE sensor is shown in Figure 1. It is composed of a light source, a sensing probe, a 3 dB coupler, a polarization controller, a fiber delay coil, a photo-detector (PD), and a data acquisition card (DAQ). The light beam from the source is coupled into a single-mode fiber and is then split into clockwise (CW) beam 3 and counter-clockwise (CCW) beam 4 at port 1 by a 3 dB coupler. The two reverse propagating light beams will arrive at the PD after passing through the sensing probe, polarization controller, and delay coil. Ideally, there is no interference if the sensing probe is not affected by external signals. When the sensing probe bonded on the surface of the structure is affected by the ultrasonic waves generated by AE, the phase of the two light beams passing through the sensing probe will be modulated, which will result in a non-reciprocal phase shift. If there is no loss during the propagation of light and the power-coupling coefficient of the coupler is the ideal value 0.5, then the two optical fields denoted by \(E_{\text{CW}}\) and \(E_{\text{CCW}}\) respectively, at the PD, can be expressed as  
\[E_{\text{CW}}=0.5E_{0}\exp\{\text{j}[\omega t+\phi_{\text{CW}}-\phi_{\text{s}} (t-t_{1})]\} \tag{1}\]  
\[E_{\text{CCW}}=0.5E_{0}\exp\{\text{j}[\omega t+\phi_{\text{CCW}}-\phi_{\text{s}} (t-t_{2})]\} \tag{2}\]  
where \(E_{0}\) is the amplitude of the light field produced by the light source, \(\omega\) is the angular frequency of the light waves, and \(\phi_{\text{CW}}\) and \(\phi_{\text{CCW}}\) are the initial phases of CW and CCW light beams at the coupler, respectively; \(\phi_{\text{s}}(t-t_{1})\) and \(\phi_{\text{s}}(t-t_{2})\) are the phase changes of CW and CCW light beams at the sensing probe caused by ultrasonic waves, respectively; \(t_{1}\) and \(t_{2}\) are the times needed for the two light beams to return from the sensing probe to the PD through CW and CCW, respectively. The synthetic intensity of the two light beams at the PD can be expressed as  
\[I_{D}=(E_{\text{CCW}}+E_{\text{CCW}})\cdot(E_{\text{CW}}+E_{\text{CCW}})^{*}=0. 5I_{0}[1+\cos(\phi_{0}+\Delta\phi_{\text{s}})] \tag{3}\]  
where \(\phi_{0}=\phi_{\text{CW}}-\phi_{\text{CCW}}\) is the initial phase difference of the sensor, \(\Delta\phi_{\text{s}}=\phi_{\text{s}}(t-t_{1})-\phi_{\text{s}}(t-t_{2})\) represents the non-reciprocal phase difference caused by ultrasonic waves, and \(I_{0}=E_{0}{}^{2}\) is the light intensity of the light source. Assuming the phase change by ultrasonic waves at the sensing probe is given by  
\[\phi_{\text{s}}(t)=\phi_{\text{s0}}\cos\omega_{\text{u}}t \tag{4}\]  
where \(\phi_{\text{s0}}\) is the amplitude of phase change, \(\omega_{\text{u}}\) is the angular frequency of ultrasonic waves, \(\Delta\phi_{\text{s}}\) can be expressed as  
\[\Delta\phi_{\text{s}}=\phi_{\text{s0}}\cos\omega_{\text{u}}(t-t_{1})-\phi_{ \text{s0}}\cos\omega_{\text{u}}(t-t_{2})=2\phi_{\text{s0}}\sin\omega_{\text{ u}}(t-\frac{t_{1}+t_{2}}{2})\sin\omega_{\text{u}}(\frac{t_{1}-t_{2}}{2}) \tag{5}\]  
Because of the existence of the delay coil, therein \(t_{1}>>t_{2}\)  
\[t_{1}+t_{2}\approx t_{1}-t_{2}=L_{\text{d}}/c^{\prime}=\Delta\tau \tag{6}\]  
where \(L_{\text{d}}\) is the length of fiber delay coil, \(c^{\prime}\) is the propagation velocity of the light in the optical fiber. Figure 1: Principle of the optical fiber Sagnac acoustic emission (AE) sensor. Equation (5) can be expressed as  
\[\Delta\phi_{\rm s}=2\phi_{\rm s0}\sin\omega_{\rm u}(t-\frac{\Delta\tau}{2})\sin \omega_{\rm u}(\frac{\Delta\tau}{2}) \tag{7}\]  
As shown in Figure 1, the propagation path of the CW light beam is \(\rm Port\ 1\to Port\ 3\to fiber\) optic ring \(\to\rm Port\ 4\to Port\ 2\), and there is no coupling in this progress.