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Abstract
Phase-sensitive optical time-domain reflectometry (Φ-OTDR) has been proposed for distributed vibration sensing purpose over recent years. Emerging applications, including seismic and hydroacoustic wave detection, demand accurate low-frequency vibration reconstruction capability. We propose to use the direct-detection Φ-OTDR configuration to achieve quantitative demodulation of external low-frequency vibrations by phase-shifted dual-pulse probes. Simultaneous pulsing and phase shifting modulation is realized with a single acousto-optic modulator to generate such probes, relaxing the need for an additional optical phase modulator. In the experiments, vibrations with frequency as low as 0.5 Hz are successfully reconstructed with 10 m spatial resolution and 35 dB signal-to-noise ratio. Excellent linearity and repeatability are demonstrated between the optical phase demodulation results and the applied vibration amplitudes. The proposed method is capable of quantitative demodulation of low-frequency vibrations with a cost-effective system configuration and high computation efficiency, showing potential for commercial applications of distributed seismic or hydroacoustic wave acquisition.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Phase-sensitive optical time-domain reflectometry ($\Phi$-OTDR) has been extensively studied as a promising solution to distributed vibration sensing for various applications over recent years, including perimeter security [1], pipeline protection [2], powerline monitoring [3,4] and transportation tracking [5].$\Phi$-OTDR is capable of simultaneously probing thousands of spatial sensing channels at meter-level resolution over standard optical fiber with length of several tens of kilometers. Recent research advancements have greatly enhanced the overall sensing performance of $\Phi$-OTDR, making it commercially ready for intrusion detection purposes [6–8]. Meanwhile, new application scenarios are emerging, e.g. hydroacoustic sensing [9], and seismic wave detection [10]. The dominant seismic or hydroacoustic wave are in the low-frequency region (below 20 Hz), which demands better performance of $\Phi$-OTDR in the low-frequency region.
$\Phi$-OTDR utilizes the Rayleigh backscattered (RBS) signal generated from a forward propagating probe pulse for distributed vibration sensing. By measuring the RBS amplitude variations along the fiber under test (FUT), external vibrations can be located accurately [11–13]. However, because RBS amplitude change is not proportional to the dynamic strain of FUT induced by vibration, amplitude demodulation $\Phi$-OTDR is not capable of recovering the actual vibration waveform. Hence, the optical phase information of the RBS signal is exploited to reconstruct the vibration waveform for advanced vibration characteristics analysis, since phase evolution along the perturbed sections of FUT varies linearly with the applied dynamic strain [14–30]. Two general approaches are taken to extract the phase of RBS signal. One approach is to introduce a local oscillator (LO) light and perform coherent detection by mixing the RBS with LO at the receiver side [14–21]. The LO light improves the signal-to-noise ratio (SNR), but deteriorates the phase demodulation results, especially in the low-frequency region, because the phase noise induced by laser source frequency drift becomes significant as a result of the relatively large temporal delay between LO and signal [31].
An alternative approach for RBS phase extraction is to create a time-delayed version of the RBS signal and apply self-interference detection between them [22–29]. The resultant phase signal is the differential phase between two sensing channels separated by a fixed length along the FUT. In order to perform time-delayed RBS self-interference operation, an imbalanced Mach-Zehnder interferometer [22–24] or Michelson interferometer [25] has been adopted at the receiver side. But the imbalanced interferometer is subject to environmental disturbances, thus additional noise is introduced to the system. Besides, double probe pulse scheme can be employed for self-interferenced $\Phi$-OTDR. Two probe pulses are launched sequentially into the FUT. The two corresponding RBS traces are interfered with each other at the receiver side with a time delay [26–29]. Obviously, compared to the imbalanced interferometer scheme, the dual-pulse $\Phi$-OTDR approach has a higher SNR, because two pulses are utilized instead of one; and the system configuration is more robust, because simple direct detection is sufficient, relaxing the need of an imbalanced interferometer.
To accurately demodulate the differential phase from the sensing signal of dual-pulse $\Phi$-OTDR, several methods have been proposed. X. He et al. proposed a dual-pulse heterodyne $\Phi$-OTDR system in 2017, where two AOMs were employed to generate two pulses with different frequency offsets [26]. Later in 2020, Z. Ju et al. utilized a single AOM to realize a dual-pulse heterodyne $\Phi$-OTDR system [27]. The phase information is extracted from the self-interferenced RBS traces along the slow axis with digital IQ demodulation, but the maximum fiber sensing length and minimum detectable vibration frequency are limited as a result. For dual-pulse homodyne $\Phi$-OTDR system, one AOM is sufficient to generate the dual-pulse probe, but rather complex algorithm is needed to extract the phase information from the self-homodyne RBS traces. A. Alekseev et al. adopted a phase shifted dual-pulse $\Phi$-OTDR scheme by adding an electro-optic phase modulator [28]. The relative phase differences within the probe pulse pairs are shifted with a 2$\pi$/3 step. One FUT interrogation is completed by applying the differential-and-cross-multiplication method to three consecutive signal traces. Z. Zhou et al. proposed a sum-and-difference method to construct two orthogonal components from the self-homodyne RBS traces and perform IQ demodulation for phase extraction [29]. However, the outcomes of sum-and-difference method is badly deteriorated when SNR is low.
The ability to accurately detect and reconstruct low-frequency vibrations is paramount for $\Phi$-OTDR in emerging applications. Several attempts have been made to enhance the system response against low-frequency vibrations [32–34]. D. Wang et al. analyzed the cross-correlation spectrum between the sensing signal and reference signal of coherent $\Phi$-OTDR system [32]. The spectrum of the low-frequency vibration can be obtained, but the vibration waveform is not reconstructed using this approach. Q. Yuan et al. subtracted the differential phase of a reference FUT section from the differential phase of the perturbed FUT section to partially suppress the low-frequency laser phase noise in coherent $\Phi$-OTDR [33]. But high sampling rate and bandwidth is still required in order to detect the heterodyne signal.
In our previous conference paper [35], a direct-detection based $\Phi$-OTDR scheme was proposed to achieve differential RBS phase demodulation by employing phase-shifted dual-pulse probes. Simultaneous double pulse generation and phase shifting is achieved by acousto-optic modulation only, without additional hardware requirement. The differential phase is calculated using simple trigonometric relations from the self-homodyne signal of the two RBS traces generated by the phase-shifted dual-pulse probes, improving efficiency in the signal processing stage, while also reducing the system requirement for detection bandwidth and sampling rate. In this paper, we further analyze the theoretical principle of the proposed method, and continue to explore its distributed low-frequency sensing performances through experiments. The phase demodulation results demonstrate a good agreement with the applied vibrations of frequency as low as 0.5 Hz. The proposed method is shown to be a promising solution for applications including intrusion detection, infrastructure health monitoring and geological exploration.
2. Operation principle
2.1 Dual-pulse $\Phi$-OTDR
For conventional single-pulse $\Phi$-OTDR, the RBS signal is the superimposed Rayleigh backscattered light generated by the forward-propagating probe pulse from fiber scatter centers. Under the one-dimension fiber scatterer model, the RBS signal can be expressed by [36,37]:
Figure 1 illustrates the working principle of dual-pulse $\Phi$-OTDR [24,29]. The two probe pulses, pulse A and pulse B, are launched into the FUT sequentially, with a temporal delay of $\Delta t$. The RBS traces generated by the two pulses are interfered with each other and detected by the photodiode. The resultant self-interferenced intensity signal $I(t)$ can be expressed by:
Fig. 1. Operation principle of the dual-pulse $\Phi$-OTDR scheme. The probe pulse pair is consisted of Pulse A (green) and B (red), with a temporal separation equal to $\Delta t$ between them. The RBS traces corresponding to pulse A and B are drawn at the bottom of the figure.
It is assumed that the local physical properties of the FUT remain unchanged for the time duration between it is probed by pulse A and pulse B. Hence, the RBS signal generated by pulse B can be treated as a delayed version of the RBS signal generated by pulse A. Eq. (3) can then be written as:
From Eq. (5), we can conclude that the differential phase is obtained directly in the optical domain with the dual-pulse $\Phi$-OTDR scheme, as opposed to the conventional method where a phase subtraction operation is needed in the digital domain [17,18]. As shown in Fig. 1, when pulse A travels past the perturbed FUT section, whilst pulse B is still before the perturbed section, the corresponding local differential phase is proportional to the dynamic strain induced by external vibrations. The spatial resolution $L$ of dual-pulse $\Phi$-OTDR can be easily obtained as [24,29]:
2.2 Differential phase demodulation based on the phase-shifted probe
To accurately demodulate the differential phase embedded in Eq. (5), phase shifted pulsing is adopted to produce relative phase shift between the two pulses of the probe pulse pair (PPP). The initial phase of the first pulse (pulse A in Fig. 1) is always set as 0, while the initial phase of the second pulse (pulse B) is shifted with a step of $\pi /2$ , resulting in a relative phase shift of $\Delta \theta _{i}=i\cdot (\pi /2)$ between pulse A and B for the $i$-th PPP. Figure 2 presents a schematic diagram of the $\Phi$-OTDR data matrix, showing four consecutive RBS traces obtained by the proposed phase shifted dual-pulse $\Phi$-OTDR system. The fast axis $t$ and slow axis $i$ correspond to the distributed spatial sensing channels along the FUT, and the RBS traces obtained from different measurements, respectively [6]. The green dash line represents the RBS signal recorded at a specific sensing channel across multiple measurements, i.e. $t_{0} = 40 \mu s$ in this case.
Fig. 2. Schematic diagram of the $\Phi$-OTDR data matrix converted from the raw signal. Green dash line represents the RBS signal obtained at 40-${\mu}$s sensing channel along the FUT.
Therefore, by adding an additional phase shift $\Delta \theta _{i}$ inside the cosine function in Eq. (5), the $i$-th self-interferenced RBS signal can be expressed by:
where the variations of $I_{DC}(t)$, $I_{AC}(t)$ and $\Delta \varphi (t)$ are assumed to be negligible between the adjacent $i$-th and ($i$+1)-th RBS traces. Note that the dynamic range of the proposed system is reduced in order for this assumption to hold (see Section 5.2 for detailed discussion). Substituting $\Delta \theta _{i+1}=\Delta \theta _{i}+\pi /2$ into Eq. (7), the expression for the ($i$+1)-th RBS trace is obtained as:From Eqs. (7) and (8), it is obvious that the differential phase of the $i$-th RBS trace can be demodulated using the following expression:
Eq. (9) is obtained under ideal condition without any consideration for system noise. Y. Fu et al. studied the negative impact of noise-induced channel amplitude imbalance in I/Q demodulation $\Phi$-OTDR system based on phase diversity receiver [17]. Eq. (9) could be regarded as a modified I/Q demodulation approach. However, instead of two orthogonal sensing channels as in [17], there is only one signal channel in our approach. Hence no channel amplitude mismatch problem exists in the proposed system. Instead, the dominating noise factor is the time-varying random detection noise, including laser frequency drift, AOM phase modulation instability, amplified spontaneous emission (ASE) noise, thermal and shot noise of the photodetector, and environmental disturbances [38]. In order to suppress the random additive noise, we modify Eq. (9) using simple trigonometric relations and get:
The noise variance for Eqs. (9) and (10) can be expressed as [23]:
For Eq. (9), $N^2_{k}(t)+D^2_{k}=I^2_{AC}(t)$, meanwhile for Eq. (10), $N^2_{k}(t)+D^2_{k}=4I^2_{AC}(t)$. Therefore the phase demodulation results obtained from Eq. (10) would show a four-fold suppression of the additive noise compared to those obtained by Eq. (9). Besides, the DC component of $I_{i}(t)$ is automatically removed in Eq. (10) when performing the subtracting operations between two measurements of $I_{i}(t)$, as opposed to the DC filtering operation in Eq. (9) which may leave DC residuals and deteriorate phase demodulation results. In this work, we employ Eq. (10) for phase demodulation to suppress random noise while maintaining a reasonable system dynamic range.
3. Experimental setup
3.1 Phase-shifted pulsing based on acousto-optic modulation
$\Phi$-OTDR system with phase-diversity-based probe pulse has been adopted by several previous works for RBS phase demodulation [28] or interference fading suppression [39,40] purposes. However, to the best of our knowledge, probe pulse phase diversity has been realized by adding an additional phase modulator at the transmitter side, e.g. an electro-optic modulator, or a fiber stretcher, resulting in an increase of system complexity and cost. In this paper, we propose to use an AOM for simultaneous probe pulse generation and pulse phase shifting. As AOM is a standard pulse generation device for conventional $\Phi$-OTDR, the proposed method requires no additional hardware components.
Figure 3(a) illustrates the basic working principle of an AOM. An acoustic transducer is driven by a radio-frequency (RF) signal to modulate the density of a transparent crystal, resulting in a traveling acoustic wave equivalent to a moving Bragg grating. The incident beam will be diffracted by this moving grating, if the acoustic wavelength matches with the incident light wavelength under Bragg condition. The diffracted beam is frequency up-shifted and phase modulated by the moving grating. Hence, by switching the on-and-off state of the acoustic wave, the optical intensity of the incident beam is modulated to generate optical pulses; whilst by modulating the initial phase of the acoustic wave, the initial optical phase of the probe pulse is also modulated accordingly. Simultaneous probe pulse generation and pulse phase shifting is achieved with a single AOM using the proposed method.
Fig. 3. (a) Schematic diagram of the basic working principle of an acousto-optic modulator. $\Omega$: acoustic wave. (b) Experimental setup for testing the phase shifted pulsing method based on AOM. NLL: narrow-linewidth laser; OC: optical coupler; AWG: arbitrary wave generator; RF: radio frequency; OSC: oscilloscope; AOM: acousto-optic modulator; BPD: balanced photodiode. (c) The waveforms of two pairs of RF double pulse signal, with frequency of 80 MHz, pulse width of 125 ns and pulse separation of 100 ns. The initial phases of the two pairs of RF double pulses $(\varphi _{\Omega,A},\varphi _{\Omega,B})$ are $(0,0)$ (blue) and $(0,\pi )$ (red) respectively. (d) The optical beating waveforms of the two PPPs with initial phase modulation of $(0,0)$ (blue) and $(0,\pi )$ (red) respectively.
To test the feasibility of the proposed phase shifted pulsing method, an experimental setup is constructed as shown in Fig. 3(b). The laser output of a narrow-linewidth laser (NLL) is split into two beams by an optical coupler (OC). The two beams form a Mach-Zehnder interferometer, where one beam is modulated by an AOM while the other beam serves as the reference. The AOM is driven by the intensity and phase modulated RF signal from an arbitrary wave generator (AWG), resulting in phase shifted pulsing of the signal arm. The two beams are mixed with each other to produce beating signal at the balanced photodiode (BPD). An example of the RF signal applied to the AOM and the detected beating signal are plotted in Figs. 3(c) and 3(d). The center frequency of the RF signal is equal to the acoustic frequency of the AOM, which is $\omega _{\Omega }=80$ MHz in this case. Two pairs of RF double pulses, with pulse width of 125 ns and pulse separation of 100 ns are detected by an oscilloscope and shown in Fig. 3(c). The initial phases $(\varphi _{\Omega,A},\varphi _{\Omega,B})$ of the twp pairs of double pulses are $(0,0)$ (blue) and $(0,\pi )$ (red) respectively. After applying the two RF pulse pairs to the AOM, two PPPs are generated and detected by the experimental setup as in Fig. 3(b). The optical beating waveforms of the two PPPs are depicted in Fig. 3(d). It is obvious that the initial phases $(\varphi _{0,A},\varphi _{0,B})$ of the optical pulses are correctly modulated by the RF pulses into $(0,0)$ (blue) and $(0,\pi )$ (red) respectively, proving the effectiveness of the proposed simultaneous optical pulse generation and phase shifting method. This allows the use of advanced pulse coding technique in conventional $\Phi$-OTDR systems, without the need for extra hardware components.
3.2 Experimental setup of dual-pulse $\Phi$-OTDR
The experimental setup of the dual-pulse $\Phi$-OTDR system is shown in Fig. 4. An NLL with a linewidth of 5 kHz is adopted to produce highly coherent laser. An optical isolator is placed at the laser output to prevent reflection. An AOM driven by phase-shifted RF signal produced from an AWG is employed for simultaneous probe pulse generation and phase shifting, as explained in Section 3.1. Phase-shifted PPPs are periodically produced as a result. The relative phase shift within the PPP is linearly increased for each measurement, with a step of $\pi$/2. Before injecting into the FUT, the PPP is first amplified by an Erbium-doped fiber amplifier (EDFA). The FUT consists of 5.1-km-long standard simple-mode fiber. A piezoelectric transducer (PZT) is placed at a distance of 5 km from the injection port. A section of fiber with length of 1 meter is wrapped around the PZT. The PZT is driven by a signal generator to apply dynamic strain onto the FUT. The RBS signal travels back to the receiver side through a circulator. Another EDFA is employed to boost the energy of the RBS light. A fiber Bragg grating is used along with a circulator, so that the ASE noise from both EDFAs can be filtered out. Finally, the RBS signal is converted to electrical signal by a photodiode (PD) with 10 MHz bandwidth, and sampled by a data acquisition card with 100 MHz sampling rate. In this configuration, the repetition rate of the dual-pulse probe is set as 16 kHz. The pulse width of the two pulses are 100 ns, while the temporal delay $\Delta t$ between the two pulses is also set as 100 ns. According to Eq. (6), the system spatial resolution $L$ is approximately 10 meters.
Fig. 4. Experimental setup of the dual-pulse $\Phi$-OTDR scheme. NLL: narrow-linewidth laser; ISO: isolator; AOM: acousto-optic modulator; PPP: probe pulse pair; EDFA: Erbium-doped fiber amplifier; FUT: fiber under test; SG: signal generator; PZT: piezoelectric transducer; FBG: fiber Bragg grating; AWG: arbitrary wave generator; PD: photodiode; DAQ: data acquisition card.
4. Experimental results
To test the feasibility of the proposed method, vibrations with different frequencies and amplitudes are applied to the FUT by PZT. The differential phase variation at the perturbed FUT section is recovered with the proposed phase-shifted dual-pulse $\Phi$-OTDR scheme using Eq. (10).
First, the system’s ability to accurately locate the external vibration along the FUT is demonstrated. A 20-Hz sinusoidal signal is applied to the PZT placed about 5 km away from the FUT injection port, as described in Section 3.2. Eq. (10) is employed to acquire the differential phase information along the whole FUT over a 3-second sampling duration. Figure 5(a) illustrates the space-frequency diagram obtained by calculating the fast Fourier transform of the differential phase signal along the slow axis. Figure 5(b) represents the amplitude of the 20-Hz frequency component of the differential phase spectra along the whole FUT. The inlet of Fig. 5(b) shows the zoomed-in view at the perturbed FUT section. The length of the perturbed FUT section is measured as 12.3 meters.
Fig. 5. (a) Space-frequency diagram of the demodulated differential phase along the whole FUT during 3-second sampling period with 20-Hz vibration applied at 5 km position of the FUT. (b) Amplitude of the 20-Hz frequency component of the differential phase signal spectra along the FUT. The inlet is a zoomed-in view at the vibration location.
Next, multiple sinusoidal signals with peak-to-peak voltage of 7 V and different frequencies were applied to the PZT, respectively. By applying Eq. (10), the differential phase signal were successfully demodulated from the raw sampled data. Then we applied the method proposed in [32] to further suppress the laser instability induced phase drift by subtracting the differential phase signal at the vibration location with the differential phase signal at the reference position, which is the beginning of the FUT. As illustrated in Fig. 6(a)-(d), the temporal evolutions of external vibrations with frequency equal to 0.5 Hz, 1 Hz, 5 Hz and 20 Hz are well reconstructed by the demodulated differential phase results respectively. The corresponding power spectrum densities (PSDs) are shown in Fig. 6(e)-(h). SNR of more than 35 dB is achieved for all four experiments, proving that the proposed method is capable of accurately recovering the low-frequency vibrations.
Fig. 6. Differential phase demodulation results of external vibrations with frequencies of (a) 0.5 Hz, (b) 1 Hz, (c) 5 Hz and (d) 20 Hz respectively. (e)-(h) The corresponding power spectrum densities (PSD) of the demodulated differential phase waveforms.
Finally, sinusoidal waveforms with varying peak-to-peak voltage are applied to the PZT. The peak-to-peak values of the corresponding differential phase response at the perturbed FUT section are recorded and linearly fitted to the applied voltages. As depicted in Fig. 7, four experiments with vibration frequencies of 0.5 Hz, 1 Hz, 5 Hz and 20 Hz are conducted. The applied voltages vary from 1 V to 10 V, with a step of 1 V, for each experiment. The measured phase responses are linearly fitted to the applied voltages. The slopes for all four linear fitting results are almost identical (approximately 3 rad/V), showing good repeatability of the proposed method. The $R^2$ values for 0.5 Hz, 1 Hz, 5 Hz and 20 Hz linear fitting results are 0.9966, 0.9987, 0.9997, and 0.9995 respectively. Based on the results presented in Fig. 7, it can be concluded that the proposed method demonstrates excellent linearity and stability for quantitative demodulation of the external vibrations in the low-frequency range.
Fig. 7. The actual measured data (points) and their corresponding linear fitting results (dashed lines) of the peak-to-peak differential phase responses versus the applied peak-to-peak voltage on PZT are depicted when vibration frequency is 0.5 Hz (red), 1 Hz (blue), 5 Hz (orange) and 20 Hz (green).
To study the noise characteristics of the proposed method, the phase demodulation results from undisturbed FUT were recorded for a duration of 1 second (1600 traces), and compared with results obtained by conventional heterodyne $\Phi$-OTDR configuration as in Ref. [11] using the same hardware components and sampling duration. The differential phase traces obtained by the proposed scheme (red) and conventional scheme (blue) during the 1 second period were shown in Figs. 8(a) and (b). Both schemes suffer from Rayleigh interference fading effect, resulting in abrupt phase changes at random spatial sampling channels. Compared to the heterodyne scheme, the proposed scheme appears to have fewer fading channels and smaller phase errors. This phenomenon may be resulted from better polarization state mismatch immunity of the proposed scheme. We further calculated the room mean square (RMS) level of differential phase results for all spatial sampling channels. Histograms showing the probability of RMS value for arbitrary spatial channels are presented in Figs. 8(c) (proposed) and (d) (conventional) with their median RMS values in red dotted line. It is obvious that the proposed scheme demonstrated statistically lower RMS value, proving that the proposed method is more immune to random noise than conventional heterodyne scheme. The median RMS value of 0.078 rad corresponds to a minimum detectable PZT voltage of 0.026 Vpp for the proposed method, while the conventional method reached a vibration sensing resolution of 0.041 Vpp.
Fig. 8. Time-domain differential phase demodulation results from undisturbed FUT acquired by the proposed method (a) and conventional heterodyne $\Phi$-OTDR (b) during 1-second sampling period. The corresponding histograms of spatial sampling channel RMS value probability for the proposed method (c) and conventional method(d) The red dotted lines correspond to the median RMS value across all channels.
5. Discussion
5.1 Laser source instability
As discussed earlier, laser source instability will induce additional phase noise to the RBS signal. Eq. (10) is capable of suppressing the Gaussian white noise embedded in the signal, but the laser phase noise is still present in the demodulated differential phase waveforms, e.g. the upward or downward trends in Fig. 6(a)-(d). This phenomenon can be explained by recalling Eq. (9). When laser instability is considered, a low-frequency time-varying term $\omega _{\textrm{drift}}(t)$ is added to the laser angular frequency, and $\theta _{0}$ in Eq. (9) becomes:
5.2 Dynamic range
For conventional $\Phi$-OTDR technique, the vibration induced phase variation must not exceeds $\pi$ between two consecutive measurements at the disturbed fiber section, i.e. $|\Delta \varphi _{i+1}(t)-\Delta \varphi _{i}(t)|<\pi$, in order to avoid any phase unwrapping error, which is sometimes referred to as the dynamic range of the $\Phi$-OTDR system. For the proposed method, we have the assumption that $\Delta \varphi _{i}(t)=\Delta \varphi _{i+1}(t)$ in Eq. (8). As a result, this assumption would further reduce the dynamic range of the proposed $\Phi$-OTDR system. In order for this assumption to hold, the dynamic range of the proposed method would inevitably be reduced compared to that of conventional $\Phi$-OTDR. Figure 9 shows the numerical simulation results when sinusoidal dynamic strain is applied on the fiber with 16 kHz pulse repetition rate and no system noise. In Fig. 9(a) and (b), the vibration frequency is fixed at 20 Hz while the amplitude of the vibration induced differential phase variation at the disturbed fiber section is increased from 1 to 1000 rad. During the increment of differential phase amplitude, the maximum value of $|\Delta \varphi _{i+1}(t)-\Delta \varphi _{i}(t)|$ is also increased. The resultant SNR and root mean square error (RMSE) of the phase demodulation results by the proposed method are plotted against the phase amplitude as well as the maximum value of $|\Delta \varphi _{i+1}(t)-\Delta \varphi _{i}(t)|$. In Fig. 9(c) and (d), the differential phase amplitude is fixed at 10 rad, while the vibration frequency is increased from 1 to 4000 Hz. The increment of vibration frequency would also lead to a larger maximum value of $|\Delta \varphi _{i+1}(t)-\Delta \varphi _{i}(t)|$. The SNR and RMSE results are presented with respect to frequency change as well as the change in maximum value of $|\Delta \varphi _{i+1}(t)-\Delta \varphi _{i}(t)|$. As can be seen in Fig. 9, when the maximum value of $|\Delta \varphi _{i+1}(t)-\Delta \varphi _{i}(t)|$ exceeds $\pi /2$, the proposed solution suffers a significant drop in SNR, and a dramatic increase in phase demodulation error. Compared to conventional $\Phi$-OTDR methods, the proposed solution would reduce the system dynamic range by half to $\pi /2$. Note that the vibration frequency detection range would hardly be affected by the proposed solution with Eq. (10), considering a total number of $N$ FUT interrogations would yield $N-3$ RBS differential phase measurements.
Fig. 9. Numerical simulation results of SNR and RMSE with increased maximum value of $|\Delta \varphi _{i+1}(t)-\Delta \varphi _{i}(t)|$. (a) SNR and (b) RMSE results obtained by the proposed method when vibration frequency is fixed at 20 Hz with increasing vibration amplitude. (c) SNR and (d) RMSE results obtained by the proposed method when vibration amplitude is fixed at 10 rad with increasing vibration frequency.
6. Conclusions
We propose a phase-shifted dual-pulse $\Phi$-OTDR system with phase demodulation capability using direct detection scheme. A simultaneous probe pulse generation and phase shifting technique based on acousto-optic modulation is implemented to generate phase-shifted dual-pulse probes, without adding extra hardwares into the $\Phi$-OTDR system. The vibration induced optical differential phase variation along the FUT is quantitatively recovered with direct-detection receiver from the RBS self-homodyne signal, reducing system requirements for detection bandwidth and sampling rate. The proposed system demonstrates good performance for distributed low-frequency vibrations sensing purpose. More than 35 dB SNR for vibration sensing with frequencies of 0.5 Hz, 1 Hz, 5 Hz and 20 Hz at a distance of 5 km is experimentally achieved. High consistency of the demodulated phase response versus the applied vibration amplitude is obtained across four vibration frequencies with good repeatability. Compared to conventional heterodyne $\Phi$-OTDR scheme, the proposed solution is proved to be less subjective to phase noise induced by laser frequency drift owing to the double probe pulse scheme, demonstrating higher SNR for low-frequency vibration reconstruction. Compared to other direct-detection based $\Phi$-OTDR systems with phase demodulation capability, the proposed solution adopts a more robust and cost-effective system configuration, while also implements a phase extraction procedure with higher computation efficiency, making it suitable for future large-scale commercial applications that require the reconstruction of infrasonic waveforms, including seismic and hydroacoustic wave acquisition.
Funding
Future Greater-Bay Area Network Facilities for Large-scale Experiments and Applications (LCZ0019); The Verification Platform of Multi-tier Coverage Communication Network for Oceans (LZC0020); Guangdong Department of Science and Technology (2021A0505080002); Guangdong Department of Education (2021ZDZX1023); Shenzhen Science, Technology & Innovation Commission (20200925162216001).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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