## Abstract

This paper proposes a distributed acoustic sensor (DAS) scheme, which is immune to the fading problem and can overcome the trade-off existing in traditional *ϕ*-OTDR between spatial resolution and sensing distance. An optical chirped pulse and non-matched filter method are used, and hence the spatial resolution is mainly determined by the bandwidth of the chirped pulse and non-matched ratio, rather than pulse duration. The Rayleigh interference pattern method is adopted here to quantitatively demodulate strain distribution along the whole sensing fiber, so there is no fading problem, which is a serious problem in the Rayleigh phase method. In proof-of-concept experiments, a DAS with 2-m spatial resolution and 10-km distance range is demonstrated. The response bandwidth of strain is 5 kHz, only limited by the fiber length. A *nε*-scale strain signal is detected at the far end of fiber with a high SNR of 35 dB.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Fiber-optic distributed acoustic sensor (DAS) can quantitatively measure the full distribution of strain along the whole fiber link. Owing to its unique ability of high-dense observation of strain information, it is promising in many significant applications [1–4], such as structural health monitoring, natural hazard prediction, underground resource exploration. Hence, DAS has become a hot topic in optical fiber sensing field and attracted lots of researches and investments in the past few years.

Apart from Rayleigh phase demodulation method [5–7], Rayleigh interference pattern (RIP) demodulation method [8–10] also attracts related researchers’ attentions. One of advantages of RIP demodulation method is that it is immune to interference fading problem [11,12], which is a serious problem in Rayleigh phase demodulation method [13]. However, since RIP demodulation method requires independent RIPs with different laser wavelengths and shifting laser wavelengths is time-consuming, it has limited response bandwidth of strain [8, 9]. In order to solve this problem, chirped pulse phase-sensitive optical time domain reflectometry (CP *ϕ*-OTDR) [10] is proposed. CP *ϕ*-OTDR uses optical chirped pulses as probes, so RIPs with different wavelengths can be obtained directly via one measurement, and dynamic sensing is realized. But these systems [9,10] still have the inherent trade-off that exists in traditional OTDR. Its spatial resolution is decided by the duration of optical pulse. Reducing the pulse duration can improve the spatial resolution, but it decreases the averaging power of optical pulses and, as a result, reduces the distance range or strain resolution.

In 2014, our group proposed a novel reflectometry configuration, called time-gated digital optical frequency domain reflectometry (TGD-OFDR) [14], and in 2015 we proposed the DAS based on it [15]. TGD-OFDR uses optical chirped pulses as probes too, but the purpose is for decoupling the relationship between the spatial resolution and the pulse duration [16]. Via matched filter [17], the chirped pulse with a long duration is compressed into a sinusoidal pulse with a short duration, and the effective spatial resolution is decided by the short duration, which is determined by the chirping range, rather than pulse duration. Therefore, TGD-OFDR can improve the sensing range by lengthen the pulse duration and, independently, improve the spatial resolution by broaden the chirping range. After matched filter, Rayleigh phase demodulation method is used to realize DAS. In order to solve fading problem, we propose many methods, but they make setup and algorithm more complicated [18,19]. Therefore, RIP method is a good alternative.

In this paper, we propose a DAS based on TGD-OFDR, but we use non-matched filter to process the signal and use RIP method to obtain the strain distribution, rather than matched filter algorithm and Rayleigh phase method. The proposed DAS is immune to fading problem, which is similar to CP *ϕ*-OTDR, and it can obtain high spatial resolution, long sensing distance and dynamic sensing at the same time, just like TGD-OFDR. In Section 2, RIP demodulation method is introduced. In Section 3, non-matched filter algorithm is proposed. In Section 4, the effective spatial resolution of the proposed DAS is analyzed. In Section 5, the proof-of-concept experiment demonstrates the scheme is feasible.

## 2. Rayleigh interference pattern method

Figure 1 is the schematic setup of direct-detection *ϕ*-OTDR. The laser is highly coherent and its center frequency *f _{c}* and peak power are constant. Arbitrary waveform generator (AWG) and frequency shifter (FS) are used to cut the continuous lightwave into optical pulse sequence and change the center frequency

*f*to a fixed frequency

_{c}*f*. Generally, FS is an optical modulator, such as acousto-optic modulator (AOM) and electro-optic modulator (EOM). Intensity of Rayleigh backscattering (RBS) from the fiber under test (FUT) is jagged due to interference [20]. Rayleigh intensity at the position

_{i}*z*is expressed as

*I*(

*f*,

_{i}*z*). Keeping FUT undisturbed and changing

*f*at a fixed frequency interval multiple times, we can obtain RIP of FUT {

_{i}*I*(

*f*,

_{i}*z*),

*i*= 0, 1, 2...}. When there is a strain event Δɛ occurring, we repeat the above operation and obtain a new RIP {

*I′*(

*f*,

_{i}*z*),

*i*= 0, 1, 2...}. According to the theory [8], {

*I*(

*f*,

_{i}*z*),

*i*= 0, 1, 2...} is similar to {

*I′*(

*f*,

_{i}*z*),

*i*= 0, 1, 2...} at the position where the strain occurs, but there is a frequency shift Δ

*f*between two RIPs, which can be determined by cross-correlation. Finally, according to the relationship [8]

*ε*can be determined quantitatively. Since

*f*is changed step by step, which is time-consuming, this method isn’t suitable for dynamic strain sensing.

_{i}When AWG and FS generate optical chirped pulse sequence, the system in Fig. 1 becomes CP *ϕ*-OTDR [10]. Since the optical pulse is chirped, Rayleigh intensity trace *I*(*z*) is taken as RIP directly and the system doesn’t need to change laser wavelength step by step. When FUT is kept undisturbed, we obtain *I*(*z*). When there is a strain event Δ*ε*, we obtain *I′*(*z*). According to the analysis [10], *I′*(*z*) is similar to *I*(*z*), but there is a horizontal position shift Δ*z* between two RIPs at the position where the strain occurs, which can be determined by cross-correlation too.

According to the relationship between Δ*z* and Δ*ε* [10],

*κ*is the chirping rate and

*c*is light speed in optical fiber and is about 2 ×10

^{8}m/s, Δ

*ε*can be determined quantitatively. Since we use cross-correlation to calculate the shift between RIPs and we don’t need to calculate Rayleigh phase, RIP method doesn’t have interference fading problem.

But the spatial resolution of OTDR is only determined by the duration of chirped pulse,

where*T*is the pulse duration. In order to obtain a high spatial resolution,

*T*is small and, as a result, the average power of chirped pulse is low. RIP method is sensitive to the SNR of Rayleigh intensity. When Rayleigh signal is dominated by additive white noise, it is hard to correctly determine the position shift Δ

*z*between RIPs via cross-correlation, and RIP method fails. In order to solve this problem, we propose non-matched filter method to decouple the relationship between the spatial resolution and the pulse duration. We can inject a long optical pulse with a large frequency range to realize high spatial resolution and long sensing range at the same time.

## 3. Non-matched filter

Figure 2 is the schematic setup of TGD-OFDR. The laser is highly coherent too, and its center frequency *f _{c}* and peak power are also constant. AWG generates electrical chirped pulse

*s*(

*t*) to drive FS to generate optical chirped pulse, which is usually

*μ*s-scale. Photocurrent signal

*i*(

*t*) from balanced photodetector (BPD) is the summation of lots of

*s*(

*t*) with different time delays and amplitude attenuations [19]. A digital chirped pulse

*h*(

*t*) is generated in digital domain to process

*i*(

*t*). Processed by

*h*(

*t*), each

*s*(

*t*) in

*i*(

*t*) is compressed into a new pulse

*R*(

*t*) and

*i*(

*t*) becomes Rayleigh interference pattern

*I*(

*t*). The full width at half maximum (FWHM) of |

*R*(

*t*)| is regarded as the effective pulse duration of

*R*(

*t*) and determines the effective spatial resolution of TGD-OFDR. Since

*R*(

*t*) is a transcendental function when

*h*(

*t*) isn’t equal to

*s*(

*t*), we use numerical simulation to discuss the characteristics of

*R*(

*t*) below.

In numerical simulation, the sampling rate is 2.5 GSa/s, which is identical to the sampling rate of ADC in the experiments. *s*(*t*) has the frequency chirping from 0.1 GHz to 1.1 GHz and its pulse duration is 4 *μ*s, which means the spatial resolution is 400 m. When *h*(*t*) is equal to *s*(*t*), *h*(*t*) is matched filter and *R*(*t*) is a sinusoidal pulse. As shown in Fig. 3(a), the effective pulse duration of *R*(*t*) is about 1 ns, which means the spatial resolution is improved to 0.1 m. Short time Fourier transform (STFT) of *R*(*t*) is shown in Fig. 3(b). The center frequency of *R*(*t*) is 0.6 GHz, and the bandwidth is 1 GHz. The simulation result is identical with previous theoretical analysis [17].

When the chirping rate of *h*(*t*) isn’t equal to that of *s*(*t*), *h*(*t*) is non-matched filter. For example, when the duration of *h*(*t*) is 4.02 *μ*s and the frequency range is still from 0.1 GHz to 1.1 GHz, |*R*(*t*)| is shown in Fig. 3(c) and its STFT is in Fig. 3(d). The results shows that *R*(*t*) is still much shorter than *s*(*t*) and its effective duration is about 16 ns, which means the spatial resolution is improved to 1.6 m. Besides, *R*(*t*) is still a chirped pulse and its frequency is from 0.1 GHz to 1.1 GHz. When the duration of *h*(*t*) is 4.04 *μ*s and the frequency range is still from 0.1 GHz to 1.1 GHz, |*R*(*t*)| is shown in Fig. 3(e) and its STFT is in Fig. 3(f). The effective duration of *R*(*t*) is about 31 ns, which means the spatial resolution is about 3.1 m. *R*(*t*) is also a chirped pulse and the chirping range is still from 0.1 GHz to 1.1 GHz.

The results of numerical simulation indicates that via non-matched filter, the chirped pulse with a long duration is compressed to a new chirped pulse with a short duration, which determines the effective spatial resolution. Therefore, we can use non-matched filter to process the raw photocurrent signal from TGD-OFDR and obtain the Rayleigh intensity trace, rather than Rayleigh phase trace. Then, we use Rayleigh interference pattern method discussed above to demodulate the strain distribution. On the one hand, since its spatial resolution isn’t limited by the raw pulse duration, we can use a long chirped pulse with high power to realize a long sensing range. On the other hand, since RIP method is used, the proposed DAS will be immune to interference fading problem.

## 4. Spatial resolution

According to our previous theoretical analysis [17], when matched filter is used, the spatial resolution of TGD-OFDR is determined by the bandwidth of the chirped pulse,

where*B*is the bandwidth of

*s*(

*t*). According to Fig. 3, we find when the difference between the duration of

*h*(

*t*) and that of

*s*(

*t*) is bigger, the effective duration of

*R*(

*t*) is larger and the effective spatial resolution deteriorates. Keeping the frequency range fixed, we define the non-matched ratio Γ,

*T*is the duration of

*s*(

*t*) and

*T′*is the duration of

*h*(

*t*);

*κ*is the chirping rate of

*s*(

*t*) and

*κ′*is the chirping rate of

*h*(

*t*). Besides, we define the deterioration factor of spatial resolution Λ, where

*Z*is the spatial resolution by matched filter, and

*Z′*is the spatial resolution by non-matched filter.

In the first situation, *s*(*t*) has the frequency chirping from 0.1 GHz to 1.1 GHz, which means the bandwidth is 1 GHz, and its pulse duration is 4 *μ*s. When matched filter is used, the spatial resolution is about 0.1 m according to Eq. (4). We change the duration of *h*(*t*) from 4 *μ*s to 4.04 *μ*s, which means Γ is from 0 to 0.01. The relationship between Λ and Γ is shown in Fig. 4(a). We can see the relationship is linear and the slope is about 3000. In the second situation, the frequency of *s*(*t*) is still from 0.1 GHz to 1.1 GHz, but the duration is reduced to 2 *μ*s. The relationship between Λ and Γ, displayed in Fig. 4(b), is still linear, but the slope is the half and is about 1500. In the third situation, the frequency of *s*(*t*) is from 0.1 GHz to 0.6 GHz, and the duration is 8 *μ*s. The relationship curve between Λ and Γ in Fig. 4(c) shows the slope is the same with that in the first situation and is about 3000.

There is an important parameter of chirped pulse called time-bandwidth product *B* · *T*. In the first and the third situation, although the frequency ranges and durations are different, their time-bandwidth products are the same, and the slope are the same. Time-bandwidth product in the second situation is the half of those in the first and third situation, and the slope is the half too. Therefore, we can conclude that the slope has a linear relationship with *B* · *T* as

*K*is the slope. Finally, we can obtain the spatial resolution

*Z′*when non-matched filter is used, According to the requirements of applications, the spatial resolution can be adjusted flexibly by changing the non-matched ratio. For example, when the frequency range of

*s*(

*t*) is from 0.1 GHz to 1.1 GHz, and the duration is 4

*μ*s, we can obtain 1-m spatial resolution in theory, if we set non-matched ratio Γ as 0.003. It is noted that the pulse shapes of

*s*(

*t*) and

*h*(

*t*) are rectangular in this manuscript. Changing pulse shapes, such as using Hanning window, only affects the spatial resolution by a constant ratio, and doesn’t affect other characteristics.

## 5. Experiment and results

Based on the schematic setup in Fig. 2, the experimental DAS is designed as shown in Fig. 5. A fiber laser (FL) is used as light source, of which the center wavelength is about 1550.12 nm and the constant power is about 40 mW. The linewidth of FL is only 100 Hz, so the laser phase noise can be omitted. But the random drift of center frequency of FL can affect the low-frequency response performance of DAS, so compensating method [21] or stabilization method [22] should be used in future practical implementation. The lightwave is split into two parts by a 50/50 polarization-maintaining coupler. One part becomes probes and the other part acts as LO. FS in Fig. 2 is an intensity modulator (IM) here, since it has a wide modulation bandwidth. Monochromatic light from FL has multiple sidebands after passing the IM. Only one of the first order sidebands is needed and other sidebands have to be filtered by a fiber Bragg grating (FBG). A direct current source (DC) is used to adjust the bias of IM. The extinction ratio of IM is about 30 dB, and the suppression ratio of unneeded sidebands is about 15 dB.

The first channel of the AWG generates a chirped pulse sequence to drive the IM 1 to generate optical chirped pulses. An erbium-doped fiber amplifier (EDFA) is used to boost the peak power of optical chirped pulses. The maximum output power of EDFA is about 17 dBm. The period of the pulse sequence is 100 *μ*s, which means the maximum length of FUT is 10 km and the response bandwidth of dynamic strain is 5 kHz. The pulse duration is 4 *μ*s. The frequency range of each pulses is from 8.1 GHz to 9.1 GHz. The bandwidth of BPD is from 30 kHz to 1.6 GHz, so the second channel of the AWG has to generate a continuous sinusoidal signal to drive the IM 2 to shift the frequency of LO by 8 GHz. Finally, the frequency range of the photocurrent signal is from 0.1 GHz to 1.1 GHz. In the future work, IM 1 and FBG 1 can be replaced by single sideband modulator (SSBM), and IM 2 and FBG 2 can be removed. The system can be simplified.

In digital data processing, we generate non-matched filter *h*(*t*), of which the frequency range is also from 0.1 GHz to 1.1 GHz and the pulse duration is 4.02 *μ*s. Since non-matched ratio is 0.005, according to Eq. (8), the theoretical spatial resolution is about 1.6 m. According to Fig. 3(d), the chirping rate of *R*(*t*) is about 4.169×10^{16} Hz/s. The sampling rate of analog-to-digital converter (ADC) is 2.5 GSa/s, so the resolution of Δz is 0.04 m. Therefore, according to Eq. (2), the theoretical strain resolution is only about 55.23 n*ε*, or 781 *pε*/√Hz. In order to improve the theoretical strain resolution, the ADC with a higher sampling rate can be employed, but it will raise the cost of system. Alternatively, in this work, we use cubic spline interpolation [23] to improve the resolution of Δz by 100 times. Finally, the practical strain resolution is limited by noise.

The length of FUT is about 9.95 km and a PZT is placed at the far end of FUT to excite a 1-kHz sinusoidal strain. The DAS totally injects 400 optical chirped pulses to measure the dynamic strain. Via non-matched filter, we can obtain 400 Rayleigh intensity traces and take them as RIPs. We focus on two regions; one region has no strain as shown in Fig. 6(a) and the other one has the strain as shown in Fig. 6(b). RIPs at the region without strain are identical with each other, except for additive Gaussian noise. However, RIPs at the region with strain are similar to each other, but there are horizontal distance shifts between each other, which agrees with the discussion in Section II. Taking the first RIP as the reference, we use cross-correlation to calculate the horizontal distance shifts and use the relationship in Eq. (2) to calculate the strain. Changing the region of RIP and repeating the above operation, we can obtain the full distribution of dynamic strain along the whole fiber link as shown in Fig. 7. Since coherent-detection scheme is used, polarization has effects on the DAS. However, we calculate the cross-correlation of a region of RIP of FUT, rather than just one point, so RIP method still has a better resistance to polarization fading problem than Rayleigh phase method. In order to totally eliminate the effect of polarization, polarization diversity receiver can be adopted. The region of strain is clearly detected without fading problem. The rising and falling edges of the strain region are less than 2 m, which proves the practical spatial resolution of DAS is better than 2 m and approaches the theoretical spatial resolution.

The strain signal covers from 9935 m to 9945 m. We display the reconstructed waveform at 9940 m in Fig. 8(a) and its power spectrum density (PSD) in Fig. 8(b). The frequency of dynamic strain is 1 kHz and there is no harmonic observed, which proves the DAS has linear response to strain. The peak-to-peak value of dynamic strain is about 200 *nε*. The strain resolution is decided by the noise floor in PSD, which is about −75 dB *με*/√Hz, that is, 178 *pε*/√Hz. The SNR of reconstructed waveform is about 35 dB. In order to analyze the crosstalk, we display the reconstructed waveform at 9947 m, which is 2 meters away from the vibration zone, in Fig. 8(c), and show its PSD in Fig. 8(d). Ideally, there should be no signal out the vibration zone. However, due to the existence of side lobes of |*R*(*t*)| as shown in Fig. 3, the signal also appears near the vibration zone. According to Figs. 8(c) and 8(d), the power of leaked signal is reduced by 23 dB. In order to further reduce the crosstalk, the shape of chirped pulse can be set as Hanning window to reduce the side lobes. Next, a frequency-chirped dynamic strain is excited by the PZT. The frequency range of the dynamic strain is from 100 Hz to 4 kHz and the period is 20 ms. The reconstructed waveform and its STFT are displayed in Figs. 8(c) and 8(d) respectively. The results show the DAS can well measure the complicated strain signal.

Since the strain resolutions at different regions of FUT are different, we define the effective strain resolution as below. We calculate the probability density function (PDF) of strain resolution along the whole FUT, and then obtain its cumulative distribution function (CDF) by integration. We define the full width at 99 % of CDF as the effective strain resolution of DAS. PDF and CDF of strain resolution along 10-km FUT are displayed in Fig. 9. According to Fig. 9(a), the worst strain resolution is about 18.4 n*ε* or 260.2 *pε*/√Hz, which means there is no serious fading occurring. According to Fig. 9(b), the effective strain resolution of 10-km FUT is about 10.2 n*ε* or 144 *pε*/√Hz.

Next, we demonstrate that increasing the pulse duration can improve the effective strain resolution of the proposed DAS. We only change the pulse durations, namely 4 *μs*, 1 *μs*, and 0.4 *μs*, and keep the other parameters of optical pulse unchanged, such as peak power and frequency range. We generate non-matched filters with different non-matched ratios, namely 0.005, 0.02, and 0.05 respectively, so the spatial resolutions under three conditions are the same. When the pulse duration is only 0.4 *μ*s, SNR of Rayleigh intensity is too low at the far end of 10-km FUT, so we only calculate PDF and CDF of a 1-km FUT. PDFs and CDFs under three conditions are displayed in Fig. 10. Shorter pulse duration means lower optical power, worse SNR of Rayleigh intensity, and worse strain resolution. When the pulse duration is 4 *μ*s, the effective strain resolution of 1-km FUT is about 2.65 n*ε* or 37 *pε*/√Hz. When the pulse duration is 1 *μ*s, it is about 5.85 n*ε* or 82 *pε*/√Hz. When the pulse duration is 0.4 *μ*s, it is about 9.3 n*ε* or 131 *pε*/√Hz. The results in Fig. 10(c) indicates if the pulse duration is lengthened by *N* times, the effective strain resolution can be improved by $\sqrt{N}$ times. The proof-of-concept experiments in this manuscript well demonstrate the advantage of non-matched filter method.

## 6. Conclusion

Chirped pulse with a long pulse duration and a large bandwidth are used in the proposed DAS. Long pulse duration means high average power and can realize long sensing range. By non-matched filter algorithm, the effective spatial resolution is determined by the bandwidth of chirped pulse and non-matched ratio, so the spatial resolution can be flexibly adjusted by changing non-matched ratio according to the requirements of applications. Chirped pulse processed by non-matched filter is still chirped pulse, so Rayleigh interference pattern method can be used to demodulate the strain, and there is no fading problem. In the proof-of-concept experiment, DAS with 2-m spatial resolution and 10-km sensing range is realized, and the response bandwidth of strain is 5 kHz. A dynamic strain signal with the frequency of 1 kHz and peak-to-peak value of 200 *nε* is detected at the far end of 10-km FUT and with the SNR of 35 dB.

## Funding

National Key R&D Plan of China (2017YFB0405500); National Natural Science Foundation of China (61875121 and 61620106015).

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