1. Introduction

Recently, distributed dynamic strain measuring techniques attract more and more attentions [1–5]. As one of such techniques, phase sensitive optical time domain reflectometry (Φ-OTDR) utilizes coherent effects of the Rayleigh backscattering in the fiber to detect perturbations and locate their positions [6,7]. Its superior properties include high sensitivity, fast response speed, long sensing distance, etc. Thus it has wide-ranging applications in civil structure health, mining safety, geological prospecting, pipeline safety, perimeter intrusion and so on [8–13].

Φ-OTDR detects perturbations on fiber by demodulating the intensity or phases of the interference pattern of the backscattering signal. A probe lightwave with stable frequency is critical to Φ-OTDR because frequency variation will change phase difference between two points, resulting in the change of the interference pattern [14]. F. Zhu et al. has shown that a commercial laser always has slow frequency drift which induces significant noise in low frequency domain, restricting the capacity of Φ-OTDR measuring low frequency events [15]. Actually, quasi-static measurement on low-frequency events is essential to certain applications such as earthquake wavefield monitoring, galloping monitoring of power transmission line and suspension bridge, whose vibrational frequency ranges from Hz to sub-Hz level [16]. In order to reduce the influence of laser frequency drift (LFD), a few methods have been proposed. Differential method is a simple and common method. By calculating the difference of demodulated phases between two points on the fiber, most noises accumulated before the points can be eliminated [17,18]. F. Zhu has proposed an active compensation method for Φ-OTDR which can reduce the influence of the LFD. In this method, the backscattering signal at different laser frequency is pre-measured at first. Then the LFD is tracked and compensated from the cross-correlation between the real-time signals the pre-measured signal [15]. However, the change of the whole environment (e.g. the change of environmental temperature) may invalidate the pre-measurement results. M. Fernández-Ruiz et al. have used the correlation between two measurement results of a reference section to reduce the impact of the laser phase noise on the sensing signal in chirped-pulse distributed acoustic sensor. Up to 17 dB increase in signal-to-noise ratio was achieved [19,20]. N. Xue et al. have proposed a compensation method through time-shifting and multiplying a phase-shift term to successfully reduce the time-skew and phase mismatch noises in the detection scheme of Φ-OTDR [21]. M. wu et al. et al. have also successfully reduced the laser phase noises in Φ-OTDR with weak reflector array by using an auxiliary interferometer to monitoring the LFD and compensating for the relative noises [22]. However, the effect of the method may be nonuniform along the sensing distance when using normal single mode fiber.

In this paper, in order to further reduce the influence of the LFD for Φ-OTDR with phase demodulation, a twice differential method is proposed. The twice differential operation only requires a single measurement result, which is simple and can compensate for the LFD in real time. The principle of the method is given in detail. Its effect and restriction are estimated with several experiments. With the proposed method, the system can respond to slow and tiny external perturbations. Slow vibration and quasi-static strain are detected successfully in experiment.

2. Principle

The phase value of the backscattering signal in Φ-OTDR is an equivalent phase, because the signal is a comprehensive result of massive individual back scattered pulses. So we use a symbol ${\theta }_R(f,z)$ to represent the equivalent phase where $f$ is the frequency of lightwave, $z$ is the position under consideration. Heterodyne detection is a main method to receive the signal in order to obtain the phase of signal. In the scheme of heterodyne detection, the backscattering signal is mixed with a local reference signal. By introducing a frequency difference between the reference signal and the backscattering signal with an acoustic optical modulator (AOM), the detector will catch a beat signal which can be expressed as:

Several methods which can successfully extract the phase term of Eq. (1) have been proposed [10,23,24]. If there is no LFD, then $f\left(t\right)$ keeps constant. So one can accurately deduce the change of $\phi \left(t\right)$ which is quantitatively related to the external perturbation. Unfortunately, due to the influence of LFD, $f\left(t\right)$ changes with time, introducing an extra time-varying term $4\pi n/cf\left(t\right)z$ + ${\theta }_R(f(t),z)$ in the demodulated phase results. This induces a noise floor in the low frequency region, which limits the performance of Φ-OTDR in measuring small and low frequency perturbations.

In order to demodulate the phase changes induced by a local perturbation, a differential operation is commonly executed. In this operation, the phase of one position A is subtracted from the phase of another position B, so the phase difference of these two positions is obtained. When A and B are located on each side of the perturbation, the phase change induced by the perturbation can be obtained. It is noteworthy that the length $D_{AB}$ between the two positions A and B should be larger than the length of fiber under perturbation and the spatial resolution of system, so the phase change induced by the perturbation can be extracted. With the differential operation, the phase at a certain moment can be expressed as:

Apparently, the differential operation greatly reduces common noises which arise from LFD and other ambient perturbations before the position A. However, the LFD still has non-negligible impact on the section $D_{AB}$. So there is still a low frequency noise in the measurement result after this differential operation.

The principle of the twice differential method, which can reduce the influence of LFD, is shown in Fig. 1. Assuming the differential operation is between points A and B which are located each side of the perturbation, the first differential result can be expanded as Eq. (2). Then in order to eliminate the phase noise on the length$D_{AB}$, we select another two reference points C and D on the non-vibration region. Because the backscattering signals generated on the whole fiber by a probe pulse have the same frequency, the differential result between the phases of C and D also contains the phase noise induced by the LFD. Analogously, the first differential result between points C and D can be expressed as:

 

Fig. 1 The principle of twice differential algorithm.

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Thus if we make $D_{CD}$ be equal to$D_{AB}$, then we can deduce the following result:

In Eq. (4), it can be seen that the phase noise term $4\pi n/cf\left(t\right)D_{AB}$ in Eq. (2) induced by the LFD is eliminated. However, another noise term $\Delta {\theta }_R$ related to the LFD emerges. Apparently, if $\Delta {\theta }_R<<4\pi n/cf\left(t\right)D_{AB}$, then the proposed method can suppress the phase noise induced by LFD effectively. Because of the random characteristic of the term ${\theta }_R(f,z)$, we need to study the influence of $\Delta {\theta }_R$ in experiment.

3. Experiment and results

Several experiments were carried out to verify the effects of proposed method. The setup of the Φ-OTDR system is shown in Fig. 2. The narrow linewidth laser source (∆ν = 100 Hz) worked on 1550.12 nm. Its output lightwave was split into two paths with a 90:10 coupler. The 90% lightwave in the first path was modulated into a probe pulse by an AOM which also introduced 40 MHz frequency shift to the lightwave. The pulse width was 100 ns, corresponding to a spatial resolution of 10 m. The 10% lightwave in the second path served as the reference signal. The probe pulse was launched into the sensing fiber through a circulator after it was amplified by an EDFA. The backscattering signal was mixed with the reference signal via a 3dB coupler and their beat signal was received by a balanced photodetector (BPD). Then the signal of the sensing path was converted into digital form by a dual-channel data acquisition (DAQ) card with a sampling rate of 500 MSa/s. The length of the FUT was 6 km and a vibration was applied on the fiber with a PZT (OPTIPHASE PZ2) at about 5 km.

 

Fig. 2 The setup of the Φ-OTDR system.

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The PZT was driven by a 0.5 Hz sinusoidal signal and the repetition rate of the probe pulse was 1 kHz. One of the temporal signal traces from 4 km to 6 km is shown in Fig. 3. In the inset of Fig. 3, the 40 MHz beat signal can be seen clearly. In experiments, we used the IQ demodulation method to demodulate the phase of the beat signal [24].

 

Fig. 3 Intensity of coherent light along fiber.

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The variation of phase with time for the signal at 5.1 km is shown in Fig. 4. Figure 4(a) shows the results extracted directly from the phase-distance traces. However, it hardly indicates the driving signal, which we attribute to the influences of the LFD and ambient noises along the fiber. In Fig. 4(b), the results were obtained via the differential operation, in which the differential length $D_{AB}$ was setting as 80 m (the fiber wounded on the PZT is 60 m). It shows that the 0.5 Hz driving signal can be obtained clearly. However, a slowly downward component is also contained in Fig. 4(b). This component was not in the driving signal, so it was a low frequency noise which was mainly induced by the LFD.

 

Fig. 4 The variation of phase with time at 5.1 km (a) before and (b) after differential operation.

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In order to observe the noise induced by the LFD more obviously and optimize the effect of the twice differential method, we made several measurements without external perturbation on the fiber and each measurement lasted for hundreds of seconds. Due to the limitation of the DAQ card, the repetition rate of the probe pulse was set to 5 Hz. The signal for the fiber section between 2 km and 3.6 km was digitized and the measurement duration lasted for 800 s.

Figure 5(a) shows the differential signals for four fiber sections locating at 2240~2320 m, 2320~2400 m, 3440~3520 m and 3520~3600 m respectively. It can be seen that the phase noises induced by the LFD are similar for close fiber sections, whereas the difference of the phase noises is large for the fiber sections far away from each other. Figure 5(b) shows the compensation result by using the twice differential method. The blue curve is the result between two close fiber sections and the red curve is the result between two fiber sections far away from each other. For the blue curve, the amplitude of the phase variation induced by the LFD is decreased by more than 95%. It also implies that we cannot compensate for all the phase noises along a long fiber through a single reference section. We believe that this is because the measured phase of Φ-OTDR signal is an equivalent result of massive individual back scattered pulses instead of a pure reflected signal. So a good strategy is to choose the reference section near the measurement section to compensate for the influence of the LFD when using the twice differential method.

 

Fig. 5 The variation of phase with time on undisturbed fiber (a) with differential operation and (b) with twice differential method.

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Then we used the twice differential method to restore the actual vibration signal in Fig. 4(b). The reference fiber section was chosen at 4900~4980 m. In order to compare the results conveniently, all the signals are shown in Fig. 6. We can see that the phase variation of the reference fiber section (black curve) is very consistent with the downward component of the original signal (blue curve). Then by subtracting the signal in the reference section from the original signal, the final result was obtained (red curve). We can see that the downward component is almost totally eliminated. Thus the twice differential method can reduce the low frequency noises and improve the performance of sensing.

 

Fig. 6 Waveform compensated by the twice differential method.

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Then several vibrations with different frequencies were measured. Figure 7 shows the results of using the twice differential method for three different vibrations. The frequencies of the driving signals were 1 Hz, 0.5 Hz and 0.2 Hz respectively, and the voltages of the driving signal were all 0.1 V, which corresponding to a strain of 11.8 nε. The red curve represents the original phase differential signal and the blue curve represents the compensated signal. It can be seen that the proposed method can reduce the influence of LFD on restoring the external perturbation.

 

Fig. 7 Waveform with different frequencies before and after compensation.

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To test the limit of the proposed method, many driving sinusoidal signals with different frequencies and amplitudes were applied to the PZT. Figure 8 shows the measurement results for the driving signal with frequency of 0.1 Hz and amplitude of 0.05 V, corresponding to a strain of 5.9$n\epsilon$. From the figure, it can be seen that the slow and small sinusoidal signal can still be detected properly by the twice differential method.

 

Fig. 8 Sinusoidal waveform with 0.1Hz frequency by twice differential method.

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Because the response of PZT to slower driving signal is nonlinear, we replaced the PZT with a stepper motor stretching the fiber with a speed of 0.4 μm/s and measured the stretching process with Φ-OTDR in order to verify the effect of the proposed method in measuring quasi-static perturbation. Meanwhile, we setup a Mach-Zehnder interferometer (MZI) to give a comparison to the measurement result. The two arms of the MZI had nearly equal length to avoid the influence of the LFD. In Fig. 9, by comparing with the reference signal (black curve), we can see that although the original signal (red curve) can show the phase change induced by the stretching, it gives misleading information for the stable states before and after the stretching. And from the downward slope before the stretching and upward slope after the stretching, we can judge that the original signal gives an overestimation to the strain induced by the stretching. On the other hand, the compensated signal (blue curve) is consistent well with the reference signal. It shows both the stable states before and after the stretching have the similar phase changes as the reference signal. The result also shows that the stretching process of the stepper motor has small steps, which may be concerned in the applications using stretching.

 

Fig. 9 Compensation result of ultra-low frequency signal generated by stepper motor.

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4. Discussion and conclusion

Since the reference section should be close to the measurement section, we propose to choose the reference section with a similarity matching method if one has no prior knowledge of the reference section. As shown in Fig. 5(a), the phase noises induced by LFD are similar when the sections are close to each other. So the reference section is not necessarily adjacent to the measurement section. In practical applications, one can choose a few fiber sections in a relative large area around the measurement section. Then the reference section is determined from one of these fiber sections whose phase noise are similar to each other. In the case that most of the fiber is under perturbation, it would be impossible to find a reference section for the compensation, which is a drawback of the proposed method.

Meanwhile, it should be note that the proposed method will also eliminate the signal on the whole fiber. The most common scenario may be the temperature drift around the fiber. On one hand, this could be an advantage when people are only interested in the individual perturbations, and treat the temperature drift as a background noise. On the other hand, this may be a drawback when people concern the whole temperature drift.

In conclusion, we propose a twice differential method in a traditional Φ-OTDR system to eliminate the influences of LFD. This method is capable of compensating for the phase variation induced by the LFD. It does not require any additional components, because the reference signal can be extracted directly from the sensing signal. In experiment, the amplitude of the low frequency noise induced by the LFD is reduced by more than 95% with the proposed method. So it is very beneficial in measuring low frequency and quasi-static perturbations. A vibration with 0.1 Hz frequency and 5.9$n\epsilon$ amplitude is successfully detected by this method. And the stretching process of a stepper motor is also measured successfully.