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Abstract
We propose a broadband chirped-pulse conversion algorithm (BCPCA) to convert a finite-step scanning probe pulse into an equivalent broadband chirped probe pulse by convolving a chirp factor on the received signal in coherent phase-sensitive optical time domain reflectometry (Φ-OTDR). Combined with Rayleigh interference pattern (RIP) demodulation in chirped-pulse Φ-OTDR (CP-ΦOTDR), environmental perturbations, such as strain and temperature along the sensing fiber, can be quantitatively measured. The equivalent broadband chirped pulse is generated by digital processing, and its bandwidth can be increased by changing the composition of the scanning pulse. Thus, the measurable perturbation range of the system can be expanded. As a proof-of-concept experiment, a high-performance distributed strain measurement was realized on a 10 km fiber, the frequency response was 5 kHz, which is only limited by the fiber length, and the strain resolution was 8.04 $\mathrm {p \varepsilon } / \sqrt {\mathrm{Hz}}$. The proposed method of generating equivalent broadband chirped pulse through the digital domain can be used as a supplement to CP-ΦOTDR.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Distributed optical fiber sensing based on Rayleigh backscattering (RBS) has attracted extensive attention in recent years, due to its fast response, high sensitivity, and long sensing distance. It has broad application prospects in many significant fields [1–4], such as seismic wave detection, oil and gas exploration, train status monitoring, and large structure health monitoring. Environmental perturbations along the sensing fiber will change the optical path of RBS light, thus affecting its phase and intensity. The perturbation can be demodulated by measuring the phase and intensity of the RBS trace. There are two major quantitative demodulation schemes: phase demodulation and Rayleigh interference pattern (RIP) demodulation.
The phase demodulation is based on the linear relationship between the phase change of the RBS trace and the environmental perturbation. The phase demodulation scheme is simple in principle and has been widely studied [5–10]. However, due to the non-uniqueness between demodulated phase and the perturbation caused by the periodicity of the trigonometric function, this scheme can only be used for dynamic perturbation measurement. In addition, it suffers from phase fading due to the low visibility of interference fringes in RBS traces [11–13]. These shortcomings limit application scenarios of the phase demodulation scheme.
Since the RBS trace is the interference result of the scattering signals within the pulse, it is directly related to the frequency of incident coherent light. When the fiber is disturbed, the RIP will change, which can be compensated by changing the frequency of the probe pulse. Thus, the RIP demodulation scheme is derived [14–17]. However, this scheme requires frequency scanning to obtain a complete RIP, which seriously limits the frequency response bandwidth of the system and is not conducive to dynamic perturbation measurement.
To solve this problem, J. Pastor-Graells et al. proposed a chirped-pulse phase-sensitive optical time domain reflectometer (CP-$\Phi$OTDR), which quantitatively demodulates environmental perturbations by measuring the time delay between chirped pulse RBS traces [18]. In the sensing process, a linear chirped pulse with a large frequency range was injected into the sensing fiber as the probe pulse. Due to the frequency and time of the linear chirped pulse being proportional, the frequency shift caused by the perturbation is directly converted into the time delay of the RBS trace. Like conventional $\Phi$-OTDR, the frequency response bandwidth of CP-$\Phi$OTDR is only limited by the length of the sensing fiber. In addition, CP-$\Phi$OTDR has the advantage of high sensitivity. At present, the strain sensitivity can reach $10^{-12}$ $\mathrm { \varepsilon } / \sqrt {\mathrm{Hz}}$ for km-length distributed sensors in conventional single-mode fibers [19], which can be used to measure micro-perturbations.
However, the maximum measurable perturbation of CP-$\Phi$OTDR is limited by the bandwidth of the linear chirped pulse, which requires that the frequency shift corresponding to the time delay of the RBS spectrum caused by the perturbation should not exceed 3%−5% of the bandwidth [20]. A larger measurable perturbation range can be realized by constantly updating reference trace [21]. Still, the maximum measurable perturbation in unit time does not change, so it can only be used for measuring the low-frequency perturbation. Further, the noise accumulation in the integral process will deteriorate the quality of the sensing signal [22]. Although the noise accumulation can be reduced by setting a frequency shift threshold to reduce the number of reference trace updates, the threshold is still limited to 3%−5% of the chirped pulse bandwidth [23]. Therefore, CP-$\Phi$OTDR requires large bandwidth chirped-pulse modulation to achieve accurate measurement of large perturbations.
The conventional CP-$\Phi$OTDR generates chirped pulses based on simple laser harmonic current modulation, and its bandwidth can easily reach several GHz [18]. However, the variations of the chirped pulse profile from the ideal linear chirp will introduce errors in the perturbation measurement. Another promising method is to modulate the frequency of the laser with an external modulator [24,25]. It uses chirped electrical signal to modulate optical signal, and the chirp linearity is better, resulting in a smaller measurement error. Because the arbitrary waveform generator (AWG) can easily generate diverse types of electrical signals, this method is extremely flexible.
H. Qian et al. proposed a chirped-pulse conversion algorithm (CPCA) to generate chirped pulses through the digital domain, which can be used as a supplement to the chirp modulation method of CP-$\Phi$OTDR [26]. By convoluting the coherent $\Phi$-OTDR received signal with a chirp factor, a single-frequency narrow-pulse probe was converted into an equivalent chirped probe pulse with a bandwidth inversely proportional to the original single-frequency probe pulse width, and the RIP demodulation in CP-$\Phi$OTDR was used. This scheme does not require complex chirped-pulse modulation and greatly reduces the complexity of system. However, the bandwidth and the measurable perturbation range of the converted chirped pulse are limited.
In this work, we propose a broadband chirped-pulse conversion algorithm (BCPCA), which uses a finite-step scanning optical pulse as the probe pulse in coherent $\Phi$-OTDR and convolves a chirp factor on the received signals. The algorithm solves the bandwidth limitation of the chirped pulse modulation method that based on the digital domain. By changing the composition of the scanning optical pulse, the bandwidth and spatial resolution of the system can be changed. Therefore, the algorithm can expand the measurable perturbation range of the system with controlled spatial resolution.
2. Theory analysis
In coherent $\Phi$-OTDR, a highly coherent optical pulse is injected into the sensing fiber. The RBS light returns to the receiver mixing with the local oscillator (LO), and is transformed into an electrical beat signal. Assuming that the modulation signal of the probe pulse is $m(t)$, the probe pulse $s(t)$ can be expressed as:
where $k_1$ is the amplitude of the probe pulse and $\omega$ is the angular frequency of the light source.The electric field of the LO $E_{LO}(t)$ with amplitude $k_2$ can be expressed as:
According to the 1-D backscatter impulse model, the electric field of RBS light $E_{RBS}(t)$ is given by the convolution of the impulse response $h(t)$ [27] with the probe pulse $s(t)$, which becomes:
Then, a digital filter of the chirp factor $c(t)=\exp \left (j \pi k t^2 \right )$ with a chirp rate $k$ is established to convolve on $I(t)$ , and obtain:
The modulation signal $m(t)$ is changed to $m_c(t)$ after the convolution operation, and $m_c(t)$ can be expressed as:
The equivalent modulation signal $m_c(t)$ is also a chirped pulse. Its amplitude spectrum is the same as the spectrum of $m(t)$, which determined its frequency range, and its bandwidth B is equal to that of $m(t)$.
For a single-frequency narrow pulse, its 3 dB bandwidth $B$ is inversely proportional to its pulse width $T_m$ ($B=1/T_m$). The approximate condition can be represented as $\left |k T_m^2\right | \ll 1$ at this time, and a single-frequency narrow probe pulse can be converted into an equivalent chirped pulse [26]. However, the width of the probe pulse is normally more than 10 ns in $\Phi$-OTDR, so both the converted chirped pulse bandwidth $B$ and the measurable perturbation range are limited by the width of the original single-frequency probe pulse.
Here, the modulation signal $m(t)$ is designed as a phase-continuous finite-step scanning pulse with $n$ steps, and the scanning frequency interval is inversely proportional to the pulse width $T_s$ of the sub-signals ($\Delta B=1 / T_s$). According to the spectral broadening of narrow pulse, the bandwidth of $m(t)$ (equal to that of $m_c(t)$) is $B=n \Delta B$, and the total pulse width is $T_m=n T_s$. Then we can get the relationship as:
In this case, the approximate condition can be expressed as $\left |k T_s^2\right | \ll 1$, which has less restriction on the total pulse width of the modulated signal. Therefore, a finite-step scanning probe pulse can be converted into an equivalent broadband chirped pulse. The modulation signal with larger pulse width can be used to increase the energy of the probe pulse, thereby increasing the signal-to-noise ratio (SNR). According to Eq. (10), under the condition that the total pulse width $T$ related to the spatial resolution is controlled, the converted chirped pulse bandwidth $B$ and the measurable perturbation range can be expanded by reducing the pulse width $T_s$ of the sub-signals.
Figure 1 shows the process of broadband chirped-pulse conversion. In this simulation, the modulation signal was a 4-step phase-continuous scanning pulse with a total pulse width of 40 ns, and each sub-signal had a pulse width of 10 ns ($\Delta B=100\ \mathrm{MHz}$). The initial frequency of the modulation signal was 100 MHz, and the scanning frequency interval was set to 100 MHz to match the bandwidth of the sub-signal, so its total bandwidth was 400 MHz. The chirp rate of the chirp factor was $400\ {\mathrm{MHz}/ \mathrm{\mu} \mathrm{s}}$, and its frequency range was 50-450 MHz. After the convolution operation, the equivalent modulation signal was also a chirped pulse, and its bandwidth was equal to that of the original modulation signal. Here, the modulation signal was a 4-step scanning pulse, and the pulse width of each sub-signal was 10 ns. Reducing the pulse width of the sub-signal and increasing the number of the scanning step can increase its bandwidth.
Fig. 1. The process of broadband chirped-pulse conversion. (a), (c) and (e) Time-domain diagrams of the modulation signal, the chirp factor, and the equivalent modulation signal, respectively. (b), (d) and (f) Corresponding time-frequency diagrams.
According to the above analyses, a finite-step scanning probe pulse is converted to an equivalent broadband chirped pulse after the BCPCA processing, so the RIP demodulation can be used. The frequency shift $\Delta v$ caused by the perturbation is mapped to a local temporal delay $\Delta t$ in the RBS trace after the BCPCA process, which can be calculated by cross-correlation. The relationship between the temporal delay $\Delta t$ and strain change $\Delta \varepsilon$ is given by [18]:
where $v_0$ is the central frequency of the optical pulse.In Fig. 1(e), the amplitude of $m_c(t)$ is uneven, leading to the fluctuant amplitude of the converted equivalent broadband probe chirped pulse, which can reduce the correlation coefficient of RBS traces, but has little effect on the frequency shift corresponding to the maximum cross-correlation coefficient. The time-frequency diagram of $m_c(t)$ was obtained by short-time Fourier transform (STFT), and its chirp linearity is shown in Fig. 2. The black and red lines are the time-frequency and linear fitting curves, respectively. The residual spectrum represents the nonlinear frequency drift of $m_c(t)$, and the maximum drift is $\pm$2.5 MHz.
During demodulation process, the data are first intercepted by a sliding window and then processed by BCPCA. The width of the window finds its parallelism in the ’gauge length’ defined for the phase demodulation scheme (distance used to compute phase-difference between two fiber points to measure the applied strain, typically larger than the spatial resolution corresponding to the probe pulse width) [29]. It is set to be larger than the probe pulse width to obtain sufficient information. The spatial resolution of the system is determined by the convolution between the probe pulse width and the time window. The convolution results in an impulse response which presents a spatial FWHM of the same size of the time window (assuming a square-like optical pulse).
3. Experimental setup
The experimental setup is shown in Fig. 3. A coherent external cavity laser (ECL) with a 3 dB bandwidth of 1.8 kHz and a central wavelength of 1550.12 nm was used as the optical source to generate continuous-wave (CW). A polarization-maintaining optical isolator (PM ISO) was used after the ECL to avoid laser instability caused by unexpected reflection. A 10:90 polarization-maintaining optical coupler divided the CW light into two paths to provide a signal and a LO. The continuous signal was frequency and pulse modulated by an IQ modulator and a semiconductor optical amplifier (SOA). The signal in Fig. 1(a) and another signal with a phase difference of $\pi /2$ were used as the two driving electrical signals for the IQ modulator, generated by an AWG module in the PXI system. The pulsed light was a 4-step scanning pulse with a frequency interval of 100 MHz, and the pulse width of each sub-signal was 10 ns. The pulse width and repetition rate were 40 ns and 10 kHz, respectively. The pulsed light was amplified by an erbium-doped fiber amplifier (EDFA) to get a suitable peak power. Then the amplified pulsed light was injected into the fiber under test (FUT) through a three-port optical circulator, propagated along the fiber, and generated the RBS light. The RBS light interfered with the LO in a 50:50 coupler, resulting in an interference light. A high-sensitivity balanced photodetector (BPD) with 1.2-GHz bandwidth was used to detect the interference light traces. The output of the BPD signal was sampled by a sampling module with a sampling rate of 2.5 GS/s. A piezoelectric ceramic transducer (PZT) cylinder with 3 m fiber wounded was placed at 5 km over 10 km sensing fiber as a vibration source.
4. Results
In the experiment, the PZT cylinder was driven by a 2 V sinusoidal signal with a frequency of 200 Hz. During data processing, the data were first intercepted with a window of 80 ns, which determined the spatial resolution of the system was 8 m. Comparisons of the experimental results with and without the above BCPCA are shown in Fig. 4. The RBS traces collecting directly are shown in Fig. 4(a) and Fig. 4(b), corresponding to the non-disturbed and disturbed regions, respectively. Since the probe signal did not meet the condition of being a chirp signal, it could not correctly reflect the expected vibration signal. Then, the BCPCA was carried out, the intercepted signal was convolved with a preset chirp factor with 250 MHz central frequency, 400 MHz bandwidth, and $400\ {\mathrm{MHz}/ \mathrm{\mu} \mathrm{s}}$ chirp rate. The converted RBS intensity traces remained stable in the non-disturbed region, as shown in Fig. 4(c). In the disturbed region, Fig. 4(d) shows that they had horizontal shifts that could be converted into temporal delay. The results are the same as the conventional CP-$\Phi$OTDR system.
Fig. 4. The RBS intensity traces. (a) and (b) The intercepted signals without BCPCA processing in the non-disturbed and disturbed regions, respectively. (c) and (d) The intercepted signals with BCPCA processing in the non-disturbed and disturbed regions, respectively.
Then, the RIP demodulation by cross-correlation was used to measure the vibration signal in the above disturbed region quantitatively. This algorithm has better anti-polarization fading ability by calculating the cross-correlation of the region rather than just one point. The demodulated strain depicted the desired sinusoidal signal in Fig. 5(a), and its peak-to-peak value was about 0.14 ${\mathrm{\mu}\mathrm{\varepsilon}}$. Figure 5(b) shows its power spectral density (PSD). The SNR reached 60.96 dB, and the second harmonics suppression ratio was 38.22 dB at 400 Hz, which may be relative to the demodulation distortion caused by the uneven amplitude of the converted equivalent broadband chirped pulse seen in Fig. 1(e). The strain noise was −101.90 $\mathrm {dB\ \mu \varepsilon ^2 /Hz}$, corresponding to the strain resolution of 8.04 $\mathrm {p \varepsilon } / \sqrt {\mathrm {Hz}}$ or 0.57 $\mathrm {n \varepsilon }$. According to Eq. (11), the strain resolution can be improved by reducing the chirp rate. When the chirp rate is reduced to about $400\ {\mathrm{MHz}/ \mathrm{\mu} \mathrm{s}}$, the resolution reaches saturation and approaches the performance limit for the given set of parameters.
Fig. 5. The demodulation result of single frequency vibration event by BCPCA. (a) Time domain result. (b) Frequency domain result.
Further, a sweep frequency strain from 0 to 5 kHz was loaded to investigate the frequency response of the system. The driving signal voltage was 1 V, and the period was 25 ms. The demodulation result and its STFT are shown in Fig. 6(a) and Fig. 6(b), respectively. In the range of the Nyquist sampling theorem, the frequency response is satisfactory, only limited by the fiber length.
Fig. 6. The demodulation result of chirped frequency vibration event by BCPCA. (a) Time domain result. (b) STFT result.
To test the relationship between the voltage applied to the PZT and the demodulated strain, and the maximum measurable strain of the system, the driving voltage was changed from 0 to 5 V at 200 Hz with a step of 0.5 V, and a comparison with the conventional CP-$\Phi$OTDR system was carried out. As mentioned above, the probe in our BCPCA-based system only included four independent frequency signals with a frequency interval of 100 MHz and a total width of 40 ns. In comparison, the probe in the conventional CP-$\Phi$OTDR system was a linear sweep signal from 50 MHz to 450 MHz with a width of 80 ns. Both demodulated strains are presented in Fig. 7, and the solid lines are the linear fitting curve. Two data results showed a high degree of consistency, which means they were equivalent. The amplitude response in the voltage range from 0 to 4 V was linear, and the coefficient of determination $R^2$ was 0.9998. With the increase of the voltage over 4 V, the larger strain would severely deform the RBS traces and reduce the correlation with the reference trace, which increased the possibility of demodulation failure. When the applied dynamic strain exceeded 0.28 $\mathrm {\mu \varepsilon }$, the demodulation results were distorted. The reason is that the maximum measurable strain of $\pm$0.14 $\mathrm {\mu \varepsilon }$ (corresponding to the frequency shift of $\pm$21 MHz) is $\pm$5.25% of the total spectral range, which confirms that the BCPCA-based system is comparable to the conventional CP-$\Phi$OTDR system in the measurable strain range. Further, the proposed BCPCA can expand the measurable strain range by reducing the sub-signals pulse width of the finite-step scanning optical pulse and increasing the number of sub-signals. The method of generating equivalent broadband chirped pulse through the digital domain can be used as a supplement to CP-$\Phi$OTDR.
Fig. 7. Test results of amplitude response capability of the BCPCA-based system and the conventional CP-$\Phi$OTDR system.
5. Conclusion
In conclusion, the proposed BCPCA combined with RIP demodulation can quantify environmental perturbations by using a finite-step scanning pulsed light as the probe light of coherent $\Phi$-OTDR. This scheme inherits the advantages of the conventional CP-$\Phi$OTDR system, such as anti-interference fading, and can increase the bandwidth of the converted equivalent probe chirped pulse by changing the composition of the scanning pulsed light, thereby expanding the measurable strain range. The experimental results show that the system can demodulate the dynamic strain with a sensing range of 10 km, a response bandwidth of 5 kHz, and a strain resolution of 8.04 $\mathrm {p \varepsilon } / \sqrt {\mathrm {Hz}}$.
Funding
Research Program of Sanya Yazhou Bay Science and Technology City (SKJC-2020-01-009); Strategic Priority Research Program of the Chinese Academy of Sciences (XDA22040105); National Natural Science Foundation of China (61875184).
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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