## Materials and Methods
### Experimental Interferometer System with Phase Feedback Control Scheme  
The experimental interferometer system with a phase feedback control scheme is shown in Figure 3. As described in Figure 1, the interferometer structure keeps the same configuration, apart from adding a phase feedback control scheme. As a hardware platform to implement the feedback control algorithm, a data acquisition card (NI PCIe-6343) working on a desktop computer is used. The data acquisition card has a 32-channels analog input with a 16-bit resolution and a maximum sampling frequency of 500 kHz. In addition, it has a 4-channels analog output with a 16-bit resolution and a maximum update frequency of 900 kHz. The outputs of the 3 PDs are digitized by a multichannel data acquisition card and then processed in a feedback control algorithm. The output signal of the algorithm is exploited to control the fiber phase shifter in an interference arm. As far as is known, the interference intensity of two channels of light can be described by:  
\[I=I_{1}+I_{2}+2\sqrt{I_{1}I_{2}}\cos(\Delta\phi)=(I_{1}+I_{2})(1+\
u\cos( \Delta\phi)), \tag{7}\]  
where \(I_{1}\) and \(I_{2}\) are the power of the two channels of light, respectively; \(\Delta\phi\) is the phase difference of the two channels of light; and \(\
u=2\sqrt{I_{1}I_{2}}/(I_{1}+I_{2})\), which denotes the fringe visibility and is defined by the following formula:  
\[\
u=\frac{I_{\max}-I_{\min}}{I_{\max}+I_{\min}}, \tag{8}\]  
where \(I_{\max}\) and \(I_{\min}\) are the maximum and minimum of the interference intensity, respectively. The fringe visibility may also be degraded by a difference in the polarization of the two lights. The effect of  
environment should be considered and eliminated. A fast and accurate feedback control scheme to compensate for the phase fading is proposed, and the process of the direction and magnitude of the phase compensation is considered at the same time. First, the dc-offset of the interference signal is calculated as:  
\[I_{1}+I_{2}=(P_{1}+P_{2}+P_{3})/3. \tag{13}\]  
When the interference system is constructed and the power of the laser is set, the power of the light from the reference arm (\(I_{1}\)) will be determined and can be detected before the experiment. However, the power of the light from the sensing arm (\(I_{2}\)) may vary for some reason in the part of the reflected light received from the surface of the object. The vibration of the self-designed probe caused by the external environment and the different reflection on the surface of the object both give rise to variance when receiving the reflected light, but we can get \(I_{2}\) according to Equation (13). Consequently, the magnitude of \(\phi_{0}+\phi_{n}\) can be calculated by taking the arcsine in Equation (11) (supposing the displacement \(x\) is small enough), and can be expressed as:  
\[\phi_{0}+\phi_{n}=\arcsin(\frac{P_{32}}{-2\sqrt{3I_{1}I_{2}}}). \tag{14}\]  
We can get the magnitude of \(\phi_{0}+\phi_{n}\), but the quadrant is not determined, so the direction of the phase compensation is uncertain. In order to solve this problem, a method for using the orthogonal signal is proposed here. Supposing the displacement \(x\) is small enough, we can get the cosine form of \(\phi_{0}+\phi_{n}\) from Equations (10) and (13):  
\[P_{4}=P_{1}-\frac{P_{1}+P_{2}+P_{3}}{3}=2\sqrt{I_{1}I_{2}}\cos(\phi_{0}+\phi_ {n}). \tag{15}\]  
In order to describe the proposed method more clearly, a schematic diagram is shown in Figure 4. When the phase difference \(\phi_{0}+\phi_{n}\) is located in the phase range of \([-\pi/2,\pi/2)\), it should be stabilized in a phase of 0. At this time, the positive and negative of the phase difference correspond to the increase and decrease of the phase compensation, and the sign of \(P_{4}\) is positive. When the phase difference \(\phi_{0}+\phi_{n}\) is located in the phase range of \([\pi/2,3\pi/2)\), it should be stabilized in the phase of \(\pi\).