## Abstract

For a distributed fiber-optic vibration sensor (DFVS), the vibration signal extracted from the phase of backscattering has a linear response to the applied vibration, and is more attractive than that from the intensity term. However, the large phase noise at a random weak-fading-point seriously limits the sensor's credibility. In this paper, a novel phase-detection DFVS is developed, which effectively eliminates the weak-fading-point. The relationship between phase noise and the intensity of backscattering is analyzed, and the inner-pulse frequency-division method and rotated-vector-sum method are introduced to effectively suppress phase noise. In experiments, two simultaneous vibrations along the 35-kilometer-long fiber are clearly detected by phase detection with the signal-to-noise ratio (SNR) over 26 dB. The spatial resolution approaches 5 m and the vibration response bandwidth is 1.25 kHz.

© 2017 Optical Society of America

## 1. Introduction

Distributed fiber-optic vibration sensor (DFVS) can detect and locate one or more vibration events occurring at any position of the fiber under test (FUT). It has promising applications in border intrusion monitoring, oil and gas pipeline security monitoring, structural health monitoring, etc [1]. At present, phase-sensitive optical time domain reflectometry (Φ-OTDR) based configurations are very popular and many significant researches have been done to improve the SNR and the sensitivity. For example, coherent detection method is proposed to increase the optical power of received Rayleigh backscattering [2]; polarization-maintain configuration [3] is presented to eliminate negative effects caused by polarization instability; Moving average algorithm [3] and wavelet denoising algorithm [4] are introduced to reduce noise; and so on.

The primitive vibration detection method is based on detecting the variance of reflection intensity traces. Its principle is briefly presented as follows. Since the probe light is highly coherent [1], Rayleigh backscattering from different reflection points within a pulse interfere with each other, which results in a jagged reflectivity trace [5]. This phenomenon is usually called interference fading. The vibration will change the optical path among reflection points [6], and the reflection intensity trace varies accordingly. This method is simple and effective, but the detected vibration waveform is distorted with high-order harmonics, because the intensity variance of backscattering has a nonlinear relationship with the vibration amplitude.

The Phase-detection DFVS based on coherent detection and I/Q demodulation is also developed [6]. The phase shift of the backscattering has a linear relationship with the vibration amplitude, so the high-order harmonics problem can be avoided. However, along the FUT, there are lots of random positions where the intensity of backscattering is extremely low (those positions are named *weak fading points* in this paper), which are mainly caused either by interference fading or by polarization fading. The former one has been discussed above, and the latter one is due to polarization mismatch between the local oscillator and the backscattering. Because of the poor SNR of backscattering, the extracted phases at those weak fading points are companied with large noise, making it difficult to locate vibrations correctly by phase detection method. Polarization-diversity-detection method has been proposed to eliminate polarization fading [7], which raises the complexity and cost. In order to eliminate weak fading points caused by interference fading, phase-shifted pulse pairs are injected into the FUT [8], but it sacrifices the vibration frequency response bandwidth; Φ-OTDR with a multi-frequency light source has also been reported [9], which is not suitable for long-distance sensing due to the lower limitation of the peak power of the pulse compared with the traditional Φ-OTDR.

Recently, we have developed a novel reflectometry named time-gated digital optical frequency domain reflectometry (TGD-OFDR) [10], which is characterized by the high spatial resolution over a long measurement range. The DFVS based on TGD-OFDR has been demonstrated [11], where the demo system successfully detected a weak vibration over a 40-km-long FUT with a high spatial resolution of 3.5 m. Distributed acoustic sensing (DAS) was also realized based on TGD-OFDR configuration, with frequency bandwidth of 21 kHz over 10-km-long fiber [12]. The polarization fading is well suppressed with this type of configurations [12], but the interference fading problem still exists.

In this paper, we developed a novel DFVS based on TGD-OFDR, which can effectively overcome the interference fading problem, without sacrificing the vibration response bandwidth or measurement range. A mathematic model is built to explain the reason that the phase shift of Rayleigh backscattering has a linear relationship with the vibration in the proposed sensor. Then inner-pulse frequency-division method and rotated-vector-sum method are introduced to overcome the phase extraction noise due to weak fading points. In demo experiments, the phase noise caused by weak fading points are well suppressed along the whole 35-kilometer-long FUT. Two simultaneous vibrations are located correctly by phase detection method with a high SNR better than 26 dB. The vibration response bandwidth is 1.25 kHz, and the spatial resolution is 5 m.

## 2. System configuration

The system setup is schematically illustrated in Fig. 1. An arbitrary waveform generator (AWG) generates a linear frequency modulated (LFM) pulse sequence to drive an acousto-optic modulator (AOM). The time period *T* of the pulse sequence must be larger than the maximum round-trip time of lightwave traveling through the total FUT. The continuous-wave lightwave from a narrow-linewidth laser is split into two parts by a fiber-optic coupler. The upper one is probe light and is chopped into optical pulses with frequency chirping after passing the AOM. The other one acts as the local oscillator. Rayleigh backscattering from the FUT beats with local oscillator in the other fiber-optic coupler. The beat light is converted into photocurrent signal by a balanced photodetector (BPD) and the photocurrent signal is digitalized by an analog-to-digital converter (ADC). Finally, the raw data is transmitted to a processor for further process. The FUT is commercial single mode fiber and a segment of fiber is wrapped tightly around a cylinder piezoelectric transducer (PZT). A signal generator drives the PZT shrinking and expanding to simulate the mechanical vibration. The system injects the optical pulse sequence into the FUT. Each pulse travels through the FUT, and the vibration information along the fiber is recorded by the Rayleigh backscattering.

## 3. Principle of vibration measurement

One pulse in the LFM pulse sequence generated by the AWG is expressed as

*t*/

*τ*) is the rectangular window function in the range of

_{p}*t*∈ [0,

*τ*],

_{p}*τ*is the pulse width,

_{p}*f*

_{0}is the initial frequency, and

*κ*is the frequency chirp rate. The optical pulse and the local oscillator are respectively expressed as

*ω*is the lightwave frequency. The summation of Rayleigh backscattering from all reflection points in the FUT is expressed as

_{c}*N*is the number of reflection points in the total FUT,

*i*represents the

*i*-th reflection point,

*τ*is the round-trip time of lightwave traveling from the near end to the

_{i}*i*-th reflection point,

*a*is the attenuation coefficient, and ${r}_{i}=\left|{r}_{i}\right|{e}^{j{\theta}_{i}}$ is the complex reflection coefficient. |

_{i}*r*| is the amplitude coefficient that obeys Rayleigh distribution and

_{i}*θ*is the phase shift that obeys uniform distribution from −

_{i}*π*to +

*π*. After coherent detection and photoelectric conversion, the photocurrent signal is expressed as

*N*is extremely massive, the summation above can be rewritten as an integral,

*h*(

*t*) is the impulse response of the FUT and is expressed as

*s*

^{∗}(−

*t*) is generated in digital domain to process

*i*(

*t*) and the output signal is

*R*(

*t*) is a spike-like function and is expressed as

*R*(

*t*) is regarded as the effective probe pulse and the spatial resolution of TGD-OFDR is determined by the full width at half maximum (FWHM) of the main lobe of

*R*(

*t*) [10,11], where

*v*is light group velocity in optical fiber, and Δ

_{g}*F*=

*κτ*is the frequency chirp scale.

_{p}*r*_{c} (*t*) in Eq. (8) can be regarded as the compositive reflection coefficient trace of the FUT. If *R*(*t*) is an impulse, i.e., the width of the probe pulse tends towards zero, the spatial resolution is considered to be infinitely great and *r*_{c} (*t*) can reveal the condition of every reflection point directly. The impulse response *h*(*t*) can be rewritten as the function of the distance *z* over the FUT

*β*is the propagation constant in optical fiber. The magnitude term |

*h*(

*z*)| can provide macroscopic parameters of the FUT. Specifically,

*a*(

*z*) is the attenuation coefficient and |

*r*(

*z*)| shows the reflectivity. Besides, the phase term, angle{

*h*(

*z*)}, can reveal the vibration. We assume that there is a vibration occurring at the position between two reflection points, A and B. The vibration is regarded as an axial dynamic strain and can change the optical path between A and B due to photoelastic effect [13–15]. Hence the differential phase between A and B is proportional to the vibration amplitude [6] and is expressed as

*γ*is the effective photoelastic coefficient,

*L*is the distance between A and B,

*ε*is the vibration amplitude, and

_{z}*C*is a phase constant. Another benefit of the differential phase is that the phase noise from laser source can be suppressed well [16]. The spatial resolution of reflectometry is only determined by the frequency chirp scale according to Eq. (10), but the interval

*L*determines the practical spatial resolution of vibration, since the differential of phase signal is required. Generally, Δ

*Z*is chosen as the interval

*L*.

Practically, *R*(*t*) is not an impulse. However, it still can be proven that the phase term can reveal the vibration and has a linear relationship with the vibration amplitude. The compositive reflection coefficient of the *k*-th reflection point in Eq. (8) can be simplified as

*M*is the number of reflection points within the spatial resolution Δ

*Z*. We assume that there is a vibration affecting a segment of the FUT near this reflection point, and its compositive reflection coefficient in Eq. (13) can be rewritten as

*kε*is the phase variance of the

_{zi}*i*-th reflection point caused by the vibration according to Eq. (12). It is noted that the vibration affecting the segment of the FUT may not distribute uniformly. Equation. (14) can be expanded as

*ε*→ 0. From Eq. (15), we can find that the phase shift at the

_{zi}*k*-th reflection point is the weighted average of the phase shifts at nearby reflection points within the region Δ

*Z*,

*ε*. It indicates that the phase shift is still proportional to the vibration amplitude, which is also verified in the demo experiment below.

_{za}## 4. Principle of fading noise reduction

The compositive reflection coefficient trace *r _{c}* (

*t*) in Eq. (8) is complex, and the magnitude part, |

*r*(

_{c}*t*)|, represents the reflection intensity trace of the FUT. Practically,

*r*(

_{c}*t*) is contaminated by noise. Therefore, its phase term will be extracted with phase extraction noise [16]. Numerical simulation indicates that when

*r*(

_{c}*t*) is contaminated by white noise, the noise in extracted phase obeys Gaussian distribution as displayed in Fig. 2(a), and the variance is approximatively inverse proportional to the SNR of intensity as shown in Fig. 2(b). In the case that the SNR of intensity is greater than 0 dB, the variance of phase extraction noise can be approximately expressed as

Similar to Φ-OTDR [17], Eq. (8) indicates that *r _{c}* (

*t*) is associated with the impulse response of the FUT and the parameters of the probe pulse, namely, the pulse width

*τ*, the frequency chirp rate

_{p}*κ*, the lightwave frequency

*ω*and the initial chirping frequency

_{c}*f*

_{0}. Varying the frequency of probe light is an efficient method to reduce fading noise, which is generally utilized in coherent OFDR and coherent OTDR and is called frequency shift averaging method (FSAV) [18].

In order to obtain multiple independent Rayleigh backscattering with different lightwave frequencies, inner-pulse frequency-division method is introduced. The LFM pulse is a kind of broadband signal, and can be divided into multiple sections with different frequency ranges. *K* digital bandpass filters with different bandwidths are generated to separate the received photocurrent signal, coming from one optical LFM pulse, into *K* sections. These *K* sections are then demodulated individually by their corresponding matched filters and *K* compositive reflection coefficient traces are obtained. As discussed above, these *K* traces are different from each other. Since these *K* sections are separated from the same probe pulse, the proposed scheme does not sacrifice the vibration frequency response bandwidth, compared with the scheme in [8]. Moreover, the power of each section is still high due to the long pulse width, and hence the proposed system can detect a long FUT, compared with the scheme in [9].

However, we cannot calculate the average of these *K* traces directly just like FSVA [18], since they are complex. As illustrated in Fig. 3(a), if the included angle between two complex numbers, *r*_{1} and *r*_{2}, is obtuse angle, |*r*_{ave}| decreases instead. Only if the included angle is acute angle, |*r*_{ave}| will increase. When the included angle tends towards zero, |*r*_{ave}| can be maximized. Therefore, we propose rotated-vector-sum method. Before the average, we calculate their normalized conjugations,
${r}_{1}^{*}/\left|{r}_{1}\right|$ and
${r}_{2}^{*}/\left|{r}_{2}\right|$, and rotate the phase terms of *r*_{1} and *r*_{2} towards zero together by multiplication, as illustrated in Fig. 3(b). The normalized conjugations of *r _{c}* (

*t*) demodulated from the first optical pulse are taken as the references to rotate

*r*(

_{c}*t*) demodulated from the other probe pulses. After rotated-vector-sum method, the fading can be reduced, while the phase information relative to the vibration is reserved.

Figure 3(b) describes an ideal situation that the included angle between two complex numbers are zero after conjugate multiplication. However, there are phase extraction noise due to the intensity noise in practice, as illustrated in Fig. 3(c). Phase extraction noise after average is

*δθ*represents phase extraction noise. Equation (19) indicates that rotated-vector-sum method is the weighted average of phase extraction noise substantially and the weight is the intensity. Since

_{i}*δθ*is independent with each other, the variance of

_{i}*δθ*

_{ave}is

*r*|,

_{i}*i*= 1,…,

*K*}, is in descending order, and |

*r*| =

_{i}*α*|

_{i}*r*

_{1}|, where

*α*∈ (0, 1]. The reduction of

*δθ*after the average is

*G*(

*K*+ 1,

*A*

_{K}_{+1})

*< G*(

*K, A*) < 1, which requires

_{K}Equation (22) indicates that not all *r _{i}* should be taken into the summation and only if |

*r*| meets the condition above, phase extraction noise can be reduced after the summation.

_{i}In [18], K. Shimizu et al. proposed a mathematic model to describe FSAV quantitatively, which can guide the selection of *K* in this scheme. The square root of the variance of the fading noise after rotated-vector-sum method is

*F*is the frequency variance range, and

*K*> 1. Equation (23) indicates the number of correlative reflection points decreases and hence fading noise is reduced. Also, Δ

*F/F*determines the lower limit of

*σ*and

_{r}*σ*is inversely proportional to

_{r}*K*when Δ

*F/F*is fixed. However, the spatial resolution deteriorates as

*K*increases. Besides, the calculation also raises with the increase of

*K*, so

*K*should be properly chosen in practice.

## 5. Experimental results and analysis

The experimental system is shown in Fig. 4. The fiber laser (NKT, E15) with the narrow linewidth of 1 kHz is utilized as the light source. The AWG (Tektronix, AFG3252C) generates a LFM pulse sequence with the time interval of 120 *μs* to drive the AOM (Gooch & Housego). The frequency range of each pulse is from 150 MHz to 250 MHz and the pulse width is 4 *μs*. The bandwidth of the BPD (Thorlabs, PDB470C) is DC-400 MHz. The sampling rate and the resolution of the ADC (NI, PXIe-5185) are 1 GSa/s and 8 bit respectively. The total length of the FUT is about 10 km. Two PZTs are placed at the distance of 9.83 km and 9.93 km respectively and there is about 10-meter-long fiber wrapped on the PZT.

Δ*F* is 20 MHz, so the theoretical spatial resolution is 5 m according to Eq. (10). The bandwidth of the AOM is only 100 MHz, and hence *F* is 80 MHz at most. Considering the calculation, *K* is selected as four. Therefore, the processor generates four digital bandpass filters with different bandwidths, which are 150–170 MHz, 175–195 MHz, 200–220 MHz, and 225–245 MHz respectively, to separate the raw data into 4 sections (marked from #1 to #4). Four corresponding matched filters are also generated to demodulate these 4 sections. It is calculated that *σ _{r}* is 1.04 dB and 6

*σ*is 8.82 dB according to Eq. (23).

_{r}Figure 5(a) shows the normalized reflection intensity trace demodulated from the section #1 of the 1-st probe pulse, and Fig. 5(b) shows 80 differential phase traces demodulated from the section #1 of 80 probe pulses. The reflection intensity trace appears to be very jagged due to interference fading and the intensities in weak fading points are extremely low. Phase extraction noise in the differential phase traces makes it impossible to locate the vibrations correctly. Figure 5(c) and 5(d) zoom around the position of the vibrations and show that the positions of phase errors in differential phase traces correspond to the positions of weak points in the intensity trace. Figure 6(a) and 6(c) display the normalized reflection intensity trace, processed by inner-pulse frequency-division method and rotated-vector-sum method. The intensity trace is still jagged, and 6*σ* is 9 dB approximately, which is identical to the theory. Extreme-weak fading points are all removed compared with that in Fig. 5(a). As a result, phase extraction noise is reduced effectively and two vibrations are clearly observed with very high SNR as shown in Fig. 6(b) and 6(d). As shown in Fig. 6(d), the perturbation area covers about 15 m. According to the average of the rise and fall time equivalent fiber length for the small section under vibration, the spatial resolution is estimated to be 5 m.

We calculated the root-mean value of 80 intensity traces and set the noise level to 0 dB, and the trace of the SNR of the intensity are obtained. The variance of *δθ* in theory can be calculated according to Eq. (18) and is displayed in Fig. 7 as the blue line. We also calculated the variance of the 80 corresponding differential phase traces, and the trace of variance is shown in Fig. 7 as the brown line. The two traces are very close to each other as shown in Fig. 7(b), indicating that the phase noise mainly depends on the SNR of intensity. In order to further increase the SNR, ADC with better resolution, the BPD and EDFA with lower noise level can be employed. Other denoising algorithms, such as wavelet denoising, can also be adopted in the further work.

To further test the performances of the system, the length of the FUT is extended to 35 km. Two PZTs are placed at the distance of 9.93 km and 34.6 km respectively. Their vibration frequencies are both 800 Hz. The rate of injecting probe pulses is reduced to 2.5 kHz and hence the vibration response bandwidth is about 1.25 kHz. The other parameters of the experimental system remain unchanged. The normalized intensity trace after average is shown in Fig. 8(a) and weak fading points are all removed. 50 differential phase traces are displayed in Fig. 8(b) and phase extraction noise caused by fading noise is reduced. The two simultaneous vibrations are clearly observed. Also, the waveforms of two vibrations are extracted and displayed in Fig. 8(c) and 8(d) respectively. Their power spectra are shown in Fig. 8(e) and 8(f). There is no high-order harmonic, which proves that the phase term has a linear relationship with the vibration amplitude. The SNR of vibration is still up to 26 dB at the distance of 34.6 km.

The demo system obtains the long measurement distance and high SNR of vibration, and the spatial resolution is 5m. This benefits from the intrinsic advantage of TGD-OFDR compared with Φ-OTDR. What’s more, the inner-pulse frequency-division method and rotated-vector-sum method proposed in this paper solve the fading problem effectively and further raise the SNR of vibration compared with the previous work.

## 6. Conclusion

In order to solve fading noise problem in phase detection method, the paper proposes a fading-noise-free DFVS based on TGD-OFDR. The system setup is compact and there are no optical filter and distributed optical amplifier required, which can reduce the complexity and the cost of the system. A mathematic model is built to introduce the principle of TGD-OFDR in detail and explain that the phase term can reveal the vibration and has a linear relationship of the vibration amplitude due to photoelastic effect. The inner-pulse frequency-division method and rotated-vector-sum method are introduced to remove the phase errors caused by fading noise and improve the SNR. In experiments, two simultaneous vibrations are located accurately by phase detection method along the 35-kilometer-long FUT and the SNR is over 26 dB. The vibration response bandwidth is up to 1.25 kHz and the spatial resolution approaches 5 m. The scheme proposed in this paper is promising in practical DFVS applications.

## Funding

This work was partly supported by the National Natural Science Foundation of China under Grant 61275097, 61307106, 61620106015, and 61327812.

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