## 3 Measurement Principles  
Since the measuring and reference arms of RI or MI are perpendicular to each other, as shown in Figure 1, the unbalanced environmental disturbance introduces drifts in the optical path difference (OPD) between the measuring and reference arms of RI or MI. Benefitting from the design of DPMI, the reference and measuring arms of the two interferometers are almost in the same optical path when M\({}_{1}\) and M\({}_{2}\) are in the initial position \(p_{0}\), so the unbalanced disturbances are almost the same and can be converted into a common-mode disturbance between the two interferometers. The common-mode disturbance can be minimized by converting the measurement of OPD between the measuring and reference arms to the measurement of OPD between two measuring arms of RI and MI. When M\({}_{2}\) moves from the initial position \(p_{0}\) to the current position \(p_{1}\), the displacement of M\({}_{2}\) can be expressed as  
\[\Delta z=\Delta\mathrm{L}_{\text{M2-MI}}\!=\Delta\mathrm{L}_{\text{M2-MR}}\!- \!\Delta\mathrm{L}_{\text{MI-MR}}=\frac{\lambda}{4\pi}(\Delta\varphi_{2}\!-\! \Delta\varphi_{1}) \tag{1}\]  
where \(\Delta\mathrm{L}_{\text{M2-MI}}\), \(\Delta\mathrm{L}_{\text{M2-MR}}\), and \(\Delta\mathrm{L}_{\text{MI-MR}}\) are the OPD changes between M\({}_{2}\) and M\({}_{1}\), between M\({}_{2}\) and M\({}_{R}\), and between M\({}_{1}\) and M\({}_{R}\) before and after the movement, respectively; \(\lambda\) is the wavelength of the laser in the air, while \(\Delta\varphi_{1}\) and \(\Delta\varphi_{2}\) are the phase changes corresponding to OPDs \(\Delta\mathrm{L}_{\text{MI-MR}}\) and \(\Delta\mathrm{L}_{\text{M2-MR}}\). For phase demodulation, the phase generated carrier (PGC) demodulation with a combined sinusoidal and triangular modulation signal [20] was adopted. In DPMI, EOM is used to modulate the source beam, different from our previous study [16], in which EOM was used to modulate the reference beam. When a combined modulation signal is applied to EOM, the x-direction polarization component of the source beam is modulated with  
Figure 1: Configuration of DPMI: IO, isolator; EOM, electro-optic phase modulator; BE, beam expander; NPBS, non-polarization beam splitting cube; PBS, polarization beam splitting cube; M\({}_{R}\), reference plane mirror; M\({}_{1,2}\), fixed and moveable measuring plane mirrors; QWP, quarter-wave plate; P, polarizer; PD, photodetector; APD, integrated 2 \(\times\) 2 array photodetector. respect to the y-direction polarization component. Then, the interference signals detected by PD and APD can be expressed as  
\[S_{1}(t)=A_{1}+B_{1}\cos\big{[}\big{(}m_{x}-m_{y}\big{)}\cos\omega_{c}t+\varphi_{1 }+\varphi_{tri}\big{]} \tag{2}\]  
\[S_{2}^{n}(t)=A_{2}^{n}+B_{2}^{n}\cos\big{[}\big{(}m_{x}-m_{y}\big{)}\cos\omega_{ c}t+\varphi_{2}^{n}+\varphi_{tri}\big{]} \tag{3}\]  
where \(A_{1}\) and \(B_{1}\) are the amplitudes of the DC and AC components of the PD signal, respectively; \(A_{2}^{n}\) and \(B_{2}^{n}\) are the amplitudes of the DC and AC components of the n-th APD signal, respectively; \(m_{x}=V_{m}/V_{\rm{rx}}\) and \(m_{y}=V_{m}/V_{\rm{ry}}\) are the modulation depths of EOM corresponding to x-direction and y-direction, respectively; \(V_{\rm{m}}\) is the amplitude of the sinusoidal modulated signal, \(V_{\rm{rx}}\) and \(V_{\rm{ry}}\) are the half-wave voltage of EOM corresponding to x-direction and y-direction, respectively; \(\omega_{c}\) is the frequency of the sinusoidal modulated signal, \(\varphi_{1}\) is the phase corresponding to the optical path difference (OPD) between the reference and measuring beams on PD; \(\varphi_{2}^{n}\) is the phase corresponding to the OPD between the reference and measuring beams on the n-th detector of APD; \(\varphi_{tri}\) is the phase introduced by the triangular modulation signal. The interference signals \(S_{1}(t)\) and \(S_{2}^{n}(t)\) are processed in FPGA for phase modulation.