## 2 Proposed Sub-Meter Spatial Resolution \(\varphi\)-OTDR: Theoretical Model  
The working principle of the proposed \(\varphi\)-OTDR system is to detect the phase change induced by the coherent RS caused by multiple scattering centers within the injected pulse duration. A narrow-linewidth laser is modulated by an acousto-optic modulator (AOM) to produce a short optical pulse. The short optical pulse is launched into the sensing fiber and the returned RS light is sent to the imbalanced MIs. Then the photodetector (PD) collects the interfering RS light after MI; the interference light from a certain position, \(L\), with a time delay, which is caused by the optical path difference, \(L_{MI}\), in MI, the interference light intensity from the position \(L\) can be given as [7]:  
\[E(t)=E_{L}(t)+E_{L-L_{MI}/2}(t)=A+B\exp[j\beta L_{MI}+\Delta\varphi_{L}(t)] \tag{1}\]  
where \(\Delta\varphi(t)=\varphi_{L-L_{MI}/2}(t)-\varphi_{L}(t)\) is the phase change carrying the sensing information related to external disturbance, and \(A\) and \(B\) are the simplified coefficients. The intensity of the interference light from MI is given by [7]:  
\[I(t)=|E(t)|^{2}=I_{D}+I_{c}\cos[\beta L_{MI}+\Delta\varphi] \tag{2}\]  
where \(I_{D}=A^{2}+B^{2}\) and \(I_{c}=2 AB\). After using a PGC algorithm and arctangent operation, phase information can be recovered by arctangent operation as follows [7]:  
\[\varphi(t)=\arctan[I(t)/Q(t)] \tag{3}\]  
where \(I(t)\) and \(Q(t)\) are the in-phase and quadrature components, respectively. The experimental system setup is illustrated in Figure 1. An arbitrary function generator (AFG) generated two different periodic signals; a narrow pulse signal (red line, 1) and a sinusoidal signal (purple line, 2). The pulse signal was used to drive an acousto-optic modulator (AOM) and control the optical switch on-off. The sinusoidal signal was intended for modulating a piezoelectric ceramic transducer and applied for phase-generated carrier demodulation. A continuous light from a narrow line-width laser with the wavelength of 1550.15 nm and linewidth of 3 kHz was modulated by an AOM (insertion loss 5 dB; extinction ratio 50 dB) as optical pulse, of which the duration time and repetition rate were 50 ns and \(f_{r}\) = 8 kHz. The optical pulse was amplified by an erbium-doped fiber amplifier (EDFA). An optical isolator was used to prevent any backward amplified spontaneous emission (ASE) generated in the EDFA from causing instabilities or damage to the laser. A fiber Bragg grating (FBG) with a center wavelength of 1550.35 nm, 3 dB bandwidth of 0.25 nm and reflectivity of 99.6% was utilized for eliminating forward ASE from the EDFA. The filtered light was launched into the sensing fiber through a circulator. After that, the backscattering light was, in turn, injected into the two unbalanced MIs (MI1 and MI2), comprised of a coupler and two Faraday rotation mirrors (FRMs) controlled by the optical switch that is synchronized with the pulse signal. The two MIs were almost the same, except that the path length difference of MI1 and MI2 were \(L_{MI1}\) = 10 m and \(L_{MI2}\) = 9.2 m, respectively. A sinusoidal electrical signal with oscillation frequency of \(f\) = 2 kHz was applied to the piezoelectric transducer (PZT) placed at one arm of the MIs for the PGC algorithm. Interfering signal was collected by a high sensitive balance photodetector (BPD, bandwidth 200 MHz, transimpedance gain 60 kV/A). The output of BPD and the sinusoidal signal were transferred to a data acquisition and signal processing unit to retrieve the phase information. An isolator was employed at the end of the sensing fiber to remove unwanted end reflection. The output of BDP for the two MIs-related signals can be expressed as:  
\[ \begin{array}{l}I_{1}(t)=I_{D1}+I_{c1}\cos[\beta L_{MI1}+\Delta\varphi_{1}] \\ I_{2}(t)=-I_{D2}-I_{c2}\cos[\beta L_{MI2}+\Delta\varphi_{2}]\end{array}  \tag{4}\]  
The above signals are in turns obtained in a short time (\(T=1/f_{r},f_{r}=8\) kHz).