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Abstract
We report on a quantitative quasi-distributed vibration sensing (DVS) system enabled by phase-sensitive optical frequency domain reflectometry (φ-OFDR), which allows for multiple vibration events over consecutive spatial resolutions. To achieve effective crosstalk suppression and mitigation of the instability during the phase extraction, fiber with embedded ultra-weak grating arrays has been adopted as the sensing fiber. It exhibits a particularly customized low spatial duty cycle, that is, high ratio between the size of the gratings and their spacing and the spacing is additionally designed to match the integer multiple of the theoretical spatial resolution. In combination with a rectified frequency-modulated continuous-wave optical probe enabled by the optical phase-locked loop, it allows to achieve quantitative quasi-DVS for multiple events over consecutive sensing spatial resolution as high as ∼2.5 cm along the distance over ∼2200 m. The ability to simultaneously retrieve arbitrary multi-point vibration events over spatially consecutive sensing spatial resolutions with consistently linear response and sensitivity up to a few nano-strain level even at long distances has shown great potentials for the application of φ-OFDR from a practical point of view.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
An optical frequency domain reflectometer (OFDR), which relies on the interrogation of Rayleigh backscattering (RBS) occurring along optical fibers [1] using frequency-modulated continuous-wave (FMCW) optical probes that exhibit high sensitivity, spatial resolution, and dynamic range with moderate detection bandwidth requirements, has long been regarded as an attractive tool in applications such as the characterization of fiber-optic networks [2,3], distributed fiber-optic sensing (DFOS) [4–8], and structural health monitoring [9]. With the advent of coherent detection [7,10–13], direct access to the information imprinted on the phase of RBS signals has been made possible, expediting the implementation of the phase-sensitive OFDR (φ-OFDR). It is considered beneficial because of its potentially high sensitivity and, in particular, its linear relation with respect to the external disturbances of interest rather than other aspects such as the amplitude [4,8], enabling practical prosperity in distributed sensing for a wide range of different physical quantities, such as vibration, acoustic, and temperature [6,7,10,12].
Concerning the interferometric nature that links the change in physical quantities to that of the relative phase for the localized coherent superposition of the RBS within a certain spatial resolution, in addition to the relatively low intensity of the RBS in silica optical fibers [14], to date, several critical issues that not only deteriorate the linearity and sensitivity but also hamper phase demodulation in a distributed manner remain problematic.
In principle, the phase error including the sweep nonlinearity and laser phase noise of the FMCW probe, deteriorates drastically when approaching or exceeding the coherence length [24] and, in most cases, eventually constitutes a lower bound for the phase noise [25]. Thus, a highly rectified FMCW optical probe with high coherence, linearity, and a broad sweep range is required for high-resolution and high-precision measurement at long distances.
In addition, as a result of the random refractive index variation [15], the well-known coherent Rayleigh fading arising from distractive interference results in drastically low signal-to-noise ratio (SNR) [16] points located stochastically on the RBS traces. This, inevitably, leads to abrupt phase variations, which hinder the demodulation process. This common issue for coherent detection based RBS reflectometers [17] can be, to some extent, addressed by averaging using independent RBS measurements [12,18] or spatial redundancies [11,33] at the expense of compromised spatial resolution or processing consumption.
Concerning the nature of the frequency domain analysis, the demodulated phase is the average within an interrogating period. Undesired time-variant phase changes, such as laser frequency drift or external perturbations will be imprinted on the probing signal, leading to an obvious phase drift and can even be manifested as equivalent phase modulation extending over adjacent sensing units if sufficiently large [7]. The former can be coped with via proper compensation, whereas the latter is, most of the time, directly regarded as an important cause of crosstalk [18]. Systematic optimization with a hybrid time-frequency approach [19,20] has been applied to deal with potential non-ignorable overlaps, and thus the resulting indistinguishable dummy perturbations. In addition, the unavoidable sidelobes stemming from phase extraction such as matched filtering [12] or Fourier transform (FT) for finite-length signals [19], technically complicate this issue and can be alleviated by temporal windowing with a sacrificed spatial resolution.
Moreover, owing to the abovementioned cumulative effect, the phases at adjacent sensing units are altered, particularly if the perturbation induced phase modulation varies during a single measurement period [20]. This results in unpredictable phase errors once the two spatial resolutions that are involved in the phase differential are disturbed, namely, their phases are modulated or unstable during the differential process within a single measurement period. This phenomenon is closely associated with another critical impairment, referred to as instability of the phase differential [21]. To guarantee an accurate and reliable quantitative measurement, a pair of undisturbed differential regions is required [22], which can be translated into the condition that a single spatial resolution, that is, sensing unit, must be at least wider than the perturbation region [23]. Otherwise, the resulting irregularly responses will make it difficult to distinguish between different events in adjacent sensing units.
From a practical viewpoint, these requirements have seriously hindered distributed vibration sensing (DVS) for multiple events over consecutive sensing spatial resolutions. In this context, it is particularly interesting and urgently required to simultaneously address the aforementioned issues in φ-OFDR. Recently, discrete reflector arrays consist of such as weak gratings [26] have been employed to improve the backscattering strength in phase-sensitive optical time-domain reflectometry [27,28] in a quasi-distributed fashion or in connection with multi-core fibers for vectorial sensing [29]. Attempts have also been made to enhance the SNR and thus the sensitivity in OFDR [11,30] by sacrificing either the range or spatial resolution. In this study, we present a φ-OFDR based quasi-DVS that supports the quantitative detection of multiple vibration events over spatially consecutive sensing spatial resolutions. To achieve effective crosstalk suppression and mitigation of instability during the phase differential process, a specially customized optical fiber embedded with low duty-cycle ultra-weak grating arrays served as the sensing fiber. It exhibits a large length ratio between the grating size and spacing, while the spacing is additionally devoted to matching a suitable integer multiple of the theoretical spatial resolution. Along with a highly coherent FMCW optical probe controlled by the optical phase-locked loop (OPLL), we demonstrate quantitative quasi-DVS for multi-point events with arbitrary vibration waveforms over spatially consecutive sensing spatial resolution of ∼2.5 cm over a distance of ∼2200 m. The resulting linear responses with a sensitivity reaching up to a few tens of nanometers exhibit high consistency over spatially consecutive sensing cells, verifying the high resolution distributed sensing capability at long distances potentially for various scenarios.
2. Operation principle and system configuration
2.1 Principle
Considering a typical φ-OFDR, a highly coherent FMCW optical probe with an initial frequency ${\nu _0}$, sweep rate $\gamma $, sweep period ${T_\textrm{s}}$, and intrinsic laser phase noise $\varphi (t )$, is injected into the sensing fiber to interrogate the RBS from a large number of scattering elements along the fiber. Coherent detection is employed in terms of interferometric beating between the reflected signals and local reference, where the latter is one portion of the original FMCW optical probe. Assuming perfect polarization alignment, the resulting electrical field for a single RBS from any position ${L_x}$ of the sensing fiber can be expressed as:
Accordingly, the photocurrent when the coherent superposition of the RBS elements within the ${k^{\textrm{th}}}\textrm{}$ theoretical spatial resolution $\mathrm{\Delta }L = c/2n\gamma {T_\textrm{s}}$ starting at distance ${L_k} = \mathrm{\Delta }L \cdot ({k - 1} )$ can be written as:
Here $\mathrm{\Delta }\varphi ({t,{\tau_x}} )= \varphi ({t - {\tau_x}} )- \varphi (t )$ is responsible for the phase fluctuations owing to the intrinsic phase noise of the laser. Although constituting a fundamental lower noise floor for the phase measurement, it can be reasonably neglected, provided the distance is shorter compared to the laser coherence length.
According to interferometric demodulation in the optical domain, the distance information is linearly mapped to the frequency components, providing ideal sweep linearity. Thus, the phases associated with each spatial resolution can be directly extracted at the corresponding frequency component in the form of a phase spectrum along the distance. The initial phase of the ${k^{\textrm{th}}}\; $ theoretical spatial resolution is as follows:
Note that, owing to the limited frequency resolution in the Fourier domain, such phases are the coherent superposition of the RBS within a single spatial resolution. Fading occurs during this process, nevertheless, as manifested in a time-invariant fashion, supposing that such a section temporally unaffected.
When the ${k^{\textrm{th}}}$ theoretical spatial resolution experiences a disturbance-induced phase variation $\mathrm{\Delta }{\theta _k}({[{{L_k},\mathrm{\Delta }L} ]} )$, owing to the accumulative effect of the phases, as guaranteed by the spatial continuity along the fiber, the actual phase sensed at each $\mathrm{\Delta }L$ can be written as:
Accordingly, in the absence of disturbance, the above condition is smoothly translated into a stable relative phase between the spatial resolutions of the corresponding section of the sensing fiber. Nonetheless, as the probe is phase modulated once it enters the spatial resolution where perturbation occurs [21], the coherent superposition of the disturbed RBSs within such spatial resolution inevitably leads to randomness and thus errors in the associated phases. For an accurate quantitative vibration measurement, it is strictly required that the two spatial resolutions between which phase differential occurs must remain undisturbed [21,22]. Such criteria necessitate the differential to be taken between any pairs of spatial resolutions with a perturbation to be measured between them, namely, their spacing must be at least wider than the region where perturbation exists. Hence, this not only limits the actual sensing spatial resolution, that is, the size of a sensing unit, but also stringently hinders the quantitative DVS along consecutive sensing spatial resolutions for which such phase differential probably fails from a practical viewpoint, as discussed above.
In the proposed φ-OFDR, specially designed ultra-weak FBG arrays that exhibit a low duty-cycle between the length of each grating ${d_{\textrm{FBG}}}$ and the spacing between successive gratings ${D_{\textrm{FBG}}}$, that is, the length of each grating section, are embedded and act as the sensing fiber. To describe the principle of how this design can assist in addressing the above issues, a series of consecutive grating sections denoted as ${N^{\textrm{th}}}$, ${({N + 1} )^{\textrm{th}}}$, and so on sensing units, respectively, are considered, as depicted in Fig. 1.
Fig. 1. Fiber embedded with ultra-weak FBG. The gratings are plotted in red, and the green and blue dotted boxes represent the grating size and their spacing, respectively.
As has been demonstrated in previous literature, Rayleigh enhancement techniques are quite promising for the fading noise mitigation and for the SNR enhancement. Such ultra-weak FBGs here are essentially a popular implementation for Rayleigh enhancement, and thus intrinsically allowing for addressing these issues. This way, Eq. (2) can be rewritten as:
Accordingly, the phase differential originally taken between adjacent spatial resolutions can be regarded as equivalent to that between two adjacent gratings, namely, between ${d_N}$ and ${d_{N + 1}}$ instead of $\mathrm{\Delta }{L_\textrm{k}}$ and $\mathrm{\Delta }{L_{\textrm{k} + 1}}$. If the spatial duty-cycle between ${d_{\textrm{FBG}}}$ and ${D_{\textrm{FBG}}}$, that is, $\eta = {d_{\textrm{FBG}}}/{D_{\textrm{FBG}}}$, is sufficiently small, it essentially allows narrowing the region within which the phases are superposed, i.e. reducing the number of RBS involved, leading to the reduction of the randomness. As a result, the phase for the associated reflections of the gratings is expected to be more stable. It thus diminishes the phase errors during the phase demodulation, whereas the vibration acting on the non-grating sections can still be effectively retrieved owing to the accumulative effect. Thus, the achievable sensing spatial resolution can be the same as the theoretical transform-limited spatial resolution of the system.
In addition, to cope with the crosstalk due to the sidelobes in the Fourier transform, ${D_{\textrm{FBG}}}$ which represents the actual sensing spatial resolution or the size of the actual sensing unit, could be set coincident with the integral multiple N of the theoretical transform-limited spatial resolution $\mathrm{\Delta }L$. In this way, by exploiting the non-grating section as a buffering space, for instance, when $N = 2$ as illustrated in Fig. 2(b), the strong sidelobes that are probably considered one of the most impairing crosstalk can be made largely misaligned with the neighboring gratings, thus preventing unwanted interference. This, in connection with proper windowing, permits effective crosstalk suppression.
Fig. 2. Configuration for the actual sensing spatial resolution considering both the grating and sidelobes. ${D_{\textrm{FBG}}}$ equals to $\mathrm{\Delta }L$ with probably strong sidelobes acting on the neighboring gratings; ${D_{\textrm{FBG}}}$.is set twice $\mathrm{\Delta }L$ where sidelobes could be potentially mitigated.
Therefore, the exploitation of such customized low-duty cycle gratings with a spacing that is at least twice the integer multiple of the theoretical spatial resolution offers an opportunity to address the aforementioned issues during the phase demodulation, including the low reflectivity, fading, crosstalk, and in particular, the differential phase instability as a prerequisite for precise phase extraction, necessitating a reliable quantitative measurement. In addition, the limits due to laser phase noise and sweep nonlinearity can be largely alleviated by using optical phase-locking technique [25].
2.2 Experiment setup
As schematically shown in Fig. 3, the proposed system mainly consists of two parts that are in charge of FMCW optical probe generation and distributed sensing. The former is realized by a delay self-interferometry based OPLL, and the latter comprises a Mach-Zehnder interferometer along with the sensing fiber. A small fraction of the output from a narrow-linewidth fiber laser is injected into an unbalanced Mach-Zehnder interferometer (UMZI) with ${\tau _d}$ delay. According to the theoretical analysis in our previous work [31], ${\tau _d}$ is set to 2 km for an optimized phase-locking performance. It transforms the laser frequency fluctuation into the phase variation of the heterodyne beat note. The error signal is attained by discriminating the beat note with a reference from an arbitrary waveform generator, whereas the corresponding correction signal, after being processed by the loop filter, is fed to the piezo-tuned laser in combination with a pre-distorted ramp signal. Such OPLL structure in closed-loop case would allow the production of the required highly coherent FMCW optical probing signal with linearized frequency sweep and suppressed phase noise. This enables the highly sensitive phase detection required to address the above issues. In addition, this OPLL could enhance the stability of the probe laser, allowing measurement for slowly-varying vibrations, which has been regarded as challenging demands in many practical scenarios. The sweep rate $\gamma $ is dictated by ${\tau _d}$ and the frequency of the locking reference, to which the beat note is locked. The detailed operating principle is described in [32].
Fig. 3. Schematic of the system. Black dotted box: OPLL-assisted linear frequency chirp generation; Purple dotted box: OFDR measurement construction. PMC: polarization-maintaining fiber coupler. BPD: balanced photodetector. PFD: phase frequency discriminator. AWG: arbitrary waveform generator. PC: polarization controller. DAQ: data acquisition card. RSA: real-time spectrum analyzer. PZS: piezo stack.
Meanwhile, a large amount of the output is guided to the sensing fiber connected through a circulator within the interferometer in the sensing part. The state-of-polarization of the RBS along the FUT was fine-adjusted using a polarization controller to maximize the interference efficiency. A frequency shift of 40 MHz is induced by an acousto-optic frequency shifter in the reference arm to constitute a heterodyne detection to avoid low frequency technical noises and the unwanted internal interference, particularly in the low-frequency region. Accordingly, the resulting beat note photocurrent detected by BPD2 is captured and converted for digital domain processing with an acquisition card and hardware platform.
In our demonstration, because of the tuning ability of the fiber laser, the sweep range of the probe was set to 8 GHz at a 160 GHz/s sweep rate with 100 ms sweep period. This corresponds to a theoretical transform-limited spatial resolution of 1.25 cm in the fiber. Accounting for the above considerations, the grating size of the fabricated FBG fiber was chosen ${d_{\textrm{FBG}}} = 0.1\textrm{mm}$ and ${D_{\textrm{FBG}}} = 2.5\textrm{cm}$, that is, with $\eta = 4 \times {10^{ - 3}}$. And the grating spacing is designed twice the theoretical transform-limited spatial resolution. The average center wavelength of the grating is ∼1549.8 nm, with a 3 dB optical bandwidth of ∼10.3 nm and reflectivity of ∼0.48%. Due to the limits in fabrication, for gratings with such set of parameters, it can be rather difficult for the further reduction of $\eta $. Nonetheless, current design is still sufficient for the demonstration of the proposed system. Concerning the fact that gratings possibly suffer from the above limits such as operation wavelength, optical bandwidth, and fluctuations due to temperature or strain changes. Alternative solutions such as weak reflection point which is in principle free from most of the above issues, should be quite interesting and will be thus considered in our future studies. Owing to the limited memory of the acquisition card, with a sampling rate of 20 MSa/s, the corresponding achievable recording length can only reach up to 2 s. Considering the sweep parameters, a total of 20 sequential interrogations can be recorded in continuous measurements.
It is worth mentioning that because of the restricted tuning characteristics of the intra-cavity piezo based fiber laser, the temporal sampling rate, namely the inverse of the sweep period is quite limited in the current implementation. This way, the verifications can only be carried out with relatively low frequency vibrations. Nonetheless, such verification is sufficient and valid because the causes that hinder the detection of multiple vibration events over consecutive sensing spatial resolutions are not relevant to the vibration frequencies. Instead, by using swept laser sources with a faster tuning speed, such as external modulations or semiconductor lasers, a higher sampling rate can be readily achieved, which will be pursued in our future verifications.
More importantly, sensing of slowly-varying vibrations is an important practical challenge due to the higher stability required for the probe laser. Benefiting from the OPLL based FMCW optical probe generation, the enhanced stability in the proposed system intrinsically addresses this issue and permit the sensing of low frequency vibrations as will be shown in the verified experiment results in the following.
3. φ-OFDR verification
3.1 Characterization of the OPLL effects
The beat note spectra from BPD1 were observed using a real-time spectrum analyzer under different locking conditions, as indicated in Fig. 4. In the free-running case, a severe noise bulge appears in the beat note spectrum because of the sweep nonlinearity and laser phase noise. In the closed-loop case, that is, when the OPLL is activated, a clear coherent beat note peak is obtained, indicating a rectified FMCW optical probe with enhanced coherence. Noise suppression within a loop bandwidth of up to ∼9 kHz can be achieved with a peak carrier-to-noise ratio of more than ∼40 dB. With the above results, a well-prepared φ-OFDR system with low phase noise, high sweep linearity, and excellent stability performance that allows achieving precise phase measurement with high spatial resolution over a long distance has been realized.
Fig. 4. Observed beat note spectrum at BPD1 for the yielded FMCW optical probe with 8 GHz sweep range in 50 ms in both free-running (Blue) and closed-loop (Red) cases.
3.2 Measurement results of φ-OFDR
The intensity traces in the spatial domain for the ultra-weak FBG and standard SMF at the near (∼3 m) and far (∼2200 m) ends, respectively, are compared in Fig. 5(a) and 5(b). Compared with the poor SNR in SMF case which may leads to drastic crosstalk and large noises, it can be inferred that the SNR of the trace is appropriately enhanced by ∼38 dB for the ultra-weak FBG arrays compared with that of the SMF. It decreases to ∼20 dB when reaching the far end, as indicated in Fig. 5(b). This is mainly attributed to the deterioration in the phase noise, which is magnified owing to the associated relatively larger reflectivity of the gratings.
Fig. 5. Intensity traces of OFDR at the (a) near and (b) far ends, respectively; (c) phase results when the ultra-weak FBG at the far end in the absence of the disturbances.
To further verify the phase variations which constitute a noise floor for phase extraction, the phases at the far end in the absence of disturbances are shown in Fig. 5(c). Compared to the free-running case, in which severe phase variations within a range of $2\pi $ are observed, in the closed-loop case, a significantly smoothed phase that varies slightly over a fairly narrow range can be found, inferring an efficiently suppressed phase fluctuation. Owing to the high-quality FMCW optical probe enabled by the OPLL and the SNR enhancement caused by the grating fibers, this is prerequisite for highly sensitive DVS.
1. Experiment results for quantitative DVS
To verify the capability of the proposed system, three piezo stacks (PZSs) were adhered to the ultra-weak FBG fiber at different distances to emulate varied axial vibrations. The detailed configuration is shown in Fig. 6, where the PZSs are driven individually by arbitrary signal generators (ASGs).
Fig. 6. Configuration chart of the ultra-weak FBG fiber and axial vibration generating structures. ASG: arbitrary signal generator.
Two PZSs of the same length ${L_a} = {L_b} = 1.8\textrm{cm}$, are installed at different relative locations with respect to the grating section. PZS-a is set right between two gratings, namely completely within a whole sensing unit. The vibrations that act on PZS-a can be directly obtained by the phase differential between a1 and a2 and in particular, the results from PZS-a can be regarded as a reference for the following experiments.
In the contrary, PZS-b is placed across two sensing units, i.e. the center of which is located at the junction of two sensing units b1-b2 and b2-b3. The vibration in this area is spread equally on these two sensing units since the lengths of PZS-b adhered on the two sensing units are set almost the same. In other words, the vibration applied on PZS-b both acts on the sensing units b1-b2 and b2-b3, each of which is subject to half in amplitude of the vibration. This way, the actual vibration detected on the sensing unit b1-b3 should be twice in amplitude as that from b1-b2 or b2-b3, respectively. And the results from these two sensing unit should be equal.
Meanwhile, PZS-c, with a length ${L_c} = 3.6\textrm{cm}$ (larger than one sensing spatial resolution), twice as PZS-a and -b, is firmly glued at both ends across two sensing units with one grating lying right upon it. More precisely, approximately 35%, i.e. ∼1.26 cm of PZS-c, lies on the first sensing unit, while the rest is laid on a second sensing unit. As such configuration is similar to PZS-b only with different lengths, the detected vibration amplitude on the first sensor unit c1-c2 should be 35% of that from c1-c3, while that on the second sensor unit c2-c3 should be 65% of the detected vibration amplitude of c1-c3.
4.1 Quantitative DVS on single-point vibration event
The sensing characteristics were experimentally verified with a sinusoidal voltage of $\textrm{8}\textrm{.0}\;{\textrm{V}_{\textrm{pp}}}$ exerted on PZS-a to imitate a single vibration event. As discussed before, without loss of generality, the vibration frequency is set 1 Hz throughout the verification. As we can observe, the spatio-temporal plots around 2.60 and 2213.70 m are exhibited in Fig. 7(a) and 7(e), respectively, where periodical vibrations can be identified with a clean background in both regions. The standard deviations (SD) of the phases at different distances are shown in Fig. 7(b) and 7(f), demonstrating their ability to unambiguously detect vibration events at both the near and far ends. By reconstructing the temporal vibration waveforms in both cases, as presented in Fig. 7(c) and 7(g), the corresponding phase amplitude can be derived from the power spectra as 1.3850 and 1.3600 rad, respectively, in near and far ends, as shown in Fig. 7(d) and 7(h), where the SNR of the vibration still reaches up to ∼32 dB at a distance of over ∼2200 m, even though the noise level has already increased by at least ∼10 dB owing to the deterioration of the phase noise. Although vibration is concerned here, the strain sensitivity can still be taken as the sensitivity metric for quantitative DVS, which can be directly deduced from the minimum measurable phase changes. The resulting strain sensitivity at the near end is ${\sim} \textrm{0}\mathrm{.086\;\ n\varepsilon }$, while it degrades to $\textrm{1}\mathrm{.941\;\ n\varepsilon }$ at the far end. Such differences in the different distances are mainly attributed to the influences of the residual laser phase noise and nonlinearity, while the gratings induced losses can also be a possible cause.
Fig. 7. Spatio-temporal domain perspectives, SD of the phases, temporal vibration waveforms, and the corresponding power spectra at the (a, b, c, d) near and (e, f, g, h) far ends of the sensing fiber, respectively, in case of single vibration event imitated by PZS-a.
Meanwhile, from the spatio-temporal views in Fig. 7(a) and 7(e) for single vibration event applied at a single sensing unit a1-a2, the observed clear background noise at the other sensing units infers the negligible crosstalk in this case. A cross-verification can also be confirmed by the clean noise floor in the SD of phase as shown in Fig. 7(b) and 7(f) with the presence of the single point vibration. In addition, the phase variances in the presence of vibrations are in accordance with the case when no vibration was applied as shown in Fig. 5(c). These results confirm the cross-verification of the effective crosstalk suppression in both cases.
To further explore the linearity response of the DVS, we performed experiments by varying the amplitude of the voltage for the sinusoidal driving signal on PZS-a from $\textrm{3}\textrm{.0}$ to $\textrm{8}\textrm{.0}\;{\textrm{V}_{\textrm{pp}}}$. The relationship between the demodulated phases with respect to the increase in amplitude at both distances is shown in Fig. 8. It can be concluded from their respective linear fitting that almost the same slope can be obtained in both cases, confirming the consistency of the linearity of the DVS response along the sensing fiber.
Fig. 8. The measured strain amplitude versus voltage amplitude for 1 Hz frequency sinusoidal wave, at the (a) near and (b) far ends of the sensing fiber, respectively.
Concerning the fact that the phase-difference between successive temporal samples should remain within the range of $\pi $, the dynamic range with which the linear response can be preserved should take into account both the vibration amplitude and frequency. This way, for the 1 Hz vibration in this demonstration, the linear response can be achieved with an amplitude of $\textrm{21}\mathrm{.536\;\ \mu \varepsilon }$ corresponding to a applied voltage of ∼29$\textrm{}{\textrm{V}_{\textrm{pp}}}$ by current PZS in our system.
4.2 Quantitative DVS on multi-vibration events
To assess the capability of the distributed sensing of multiple vibration events across spatially consecutive sensing units, all three installed PZSs, as indicated in Fig. 6, are simultaneously driven with the same but independent sinusoidal driving voltage. The overall spatio-temporal domain perspectives for the asynchronous vibration events at both ends are preliminarily shown in Fig. 9(a) and 9(b). Apart from the periodical vibrations identified at approximately 2.60, 2.90, 3.05 m, and 2213.70, 2214.00, 2214.20 m at the near and far ends, respectively, the clean and flat background noise in both cases provided a sounding confirmation for the suppression of the crosstalk at both distances. This can be further verified by comparing the SD of the phases along the distance in both cases as depicted in Fig. 10(a) and 10(b), respectively, where the corresponding vibrations are distinguishable from the non-vibrated areas with plainer phase variations. Slight degradations can be found at long distances as inferred by the merely increased fluctuations in the SD, because it tenderly experiences the influence of phase noise, nonlinearity, and SNR variations owing to the rectified FMCW optical probe generated by the OPLL and the enhancement in the SNR of the amplitude owing to the gratings.
Fig. 9. Spatio-temporal domain perspectives on multi-vibration events, when the ultra-weak FBG at the (a) near and (b) far ends of the sensing fiber, respectively.
Fig. 10. SD of the phases along the distance on multi-vibration events, when the ultra-weak FBG at the (a) near and (b) far ends of the sensing fiber, respectively.
Meanwhile, the crosstalk suppression not only from non-grating but also among the FBGs can also be verified. It can be observed that at the FBGs located after the vibrations (red-dotted box), hardly any vibrations is detected at these non-vibrated FBGs. The corresponding SDs of the phases at those FBGs are almost at the same level of the noise floor as presented in Fig. 10 (red-dotted box), confirming the negligible crosstalk among the FBGs.
Insightful views in terms of the power spectra for the results obtained at both distances are displayed in Fig. 11 and Fig. 12, respectively, along with the vibration waveforms. At the near end, the phase amplitudes at each sensing unit (as denoted in Fig. 6) were extracted as 1.3725 (a1–a2), 0.6243 (b1–b2), 0.6342 (b2–b3), 0.8912 (c1–c2) and 1.6541 rad (c2–c3), respectively, following the calculation in Eq. (5). The SNR for the phase extractions of the corresponding events degenerates relatively mildly at the far end, as verified by the power spectra, compared to those at the near end. This is mainly attributed to the residual laser phase noise and nonlinearity even with the OPLL, as well as unwanted perturbations in addition to the loss induced SNR decline for the RBS amplitude. Similar degradation, as described in the above sections, leads to a deterioration in the sensitivity from 0.091 (a1–a2), 0.147 (b1–b2), 0.151 (b2–b3), 0.129 (c1–c2) and $\textrm{0}\mathrm{.078\;\ n\varepsilon }$ (c2–c3), respectively, in the near end, to 2.588 (a1–a2), 3.235 (b1–b2), 3.278 (b2–b3), 2.933 (c1–c2) and $\textrm{1}\mathrm{.941\;\ n\varepsilon }$ (c2–c3) at the far end, respectively.
Fig. 11. Measurement DVS results in terms of the temporal waveforms (a, b, and c) and the corresponding power spectra (d, e, and f) at the near end when multiple vibration events are exerted at PZS-a, -b and -c, respectively.
Fig. 12. Measurement DVS results in terms of the temporal waveforms (a, b, and c) and the corresponding power spectra (d, e, and f) at the far end when multiple vibration events are exerted at PZS-a, -b and -c, respectively.
The results associated with a1-a2 constitute the verification for the sensing spatial resolution. Since the distance between the two FBGs (a1 and a2) involved in this sensing unit is 2.5 cm, when vibration occurs in between the two FBGs, the phase differential between them allows a direct retrieval of this vibration, testifying the sensing spatial resolution.
Moreover, the sum of the vibrations sensed by sensing units b1-b2 and b2-b3 almost coincident with that by the direct differential for b1-b3. That is to say, the vibration occurred across b1-b3 can be steadily identified as two vibration events at two consecutive sensing units b1-b2 and b2-b3. The actual vibration detected on the sensing unit b1-b3 is twice in amplitude as that from b1-b2 and b2-b3, respectively. And the results from these two are also equal in amplitude. Similar results but with different ratios can be found at PZS-c which are in accordance with the configuration shown in Fig. 6. All these strongly testify the actual 2.5 cm sensing spatial resolution achieved by the proposed system.
At this point, it is essential to assess the linearity performance of the DVS response at each sensing unit in consideration of consistency in the measurement of multiple vibration events. Therefore, we have conducted experiments by applying independent driving voltages with a varied amplitude from $\textrm{3}\textrm{.0}$ to $\textrm{8}\textrm{.0}\;{\textrm{V}_{\textrm{pp}}}$ to the PZSs, simultaneously. The results with their linear fittings for the vibrations exerted at the near and far ends are shown in Fig. 13. The corresponding sensing slopes obtained at different consecutive sensing units in both cases calculated to be ∼0.76 (a1–a2), ∼0.35 (b1–b2), ∼0.35 (b2–b3), ∼0.49 (c1–c2), and ${\sim} \textrm{0}\mathrm{.90\;\ \mu \varepsilon }\textrm{ / }\textrm{V}$ (c2–c3) are well in accordance with the configuration shown in Fig. 6. Note that even with the sensitivity penalty at the far end, the sensing slopes can be almost preserved with only slight discrepancies, verifying the linearity between the phase and the vibration amplitude within the effective measurement range.
Fig. 13. The measured strain amplitude versus voltage amplitude for 1 Hz frequency sinusoidal wave corresponding to PZS-a, -b and -c, at the near (a) and far (b) ends of the sensing fiber, respectively.
More importantly, though with slight discrepancies in the sensing slopes between the short and long distances which is mainly attributed to the residual laser phase noise, nonlinearity, and the gratings losses, the agreement in the linearity between these consecutive, in particular, adjacent sensing units, has strongly demonstrated the capability for efficient crosstalk suppression and mitigation of the requisite phase stability during the differential process in the proposed φ-OFDR system. Therefore, this has allowed to the successful realization of the quantitative quasi-DVS over consecutive sensing spatial resolutions with a high resolution at long distances.
For an insight of the experiment, the results obtained at the sensing unit a1-a2 when with or without the vibrations exerted on PZS-b as shown in Fig. 8 and Fig. 13 exhibit almost identical behaviors. Almost the same linear response can be detected in both cases. It strongly testified the fact that even with the vibrations acting on the nearby sensing unit, negligible impact has been found compared with the case when no vibration occurs on any of the nearby sensing units. Similar cases can be found when the vibrations are simultaneously acting on all the PZSs.
4.3 Quantitative DVS on multiple vibration events with arbitrary-waveforms
Further characterizations in support of crosstalk suppression for consecutive sensing have also been conducted in terms of arbitrary-waveform identification for multiple vibration events. Under different cases, the three PZSs were actuated with asynchronous triangular, sinusoidal, and sinusoidal signals of the same amplitude and frequency of $\textrm{8}\textrm{.0}\;{\textrm{V}_{\textrm{pp}}}$ at 1 Hz.
The spatio-temporal views at different distances are summarily shown in Fig. 14(a) and 14(e), where a similar performance in terms of the background noise and the periodical waveforms can be readily exhibited at both distances. In a closer view of the demodulated temporal signals together with their respective driving signals (plotted in dotted lines) at each sensing unit, as presented in Fig. 14(b), 14(c) and 14(d), distinct vibration events with their respective waveforms at each position can be identified from the non-vibrated regions with an expected reasonable SNR degradation at the far end, which is consistent with the exerted vibrations. In addition, it appears that the non-trivial distortions, particularly at the far end shown in Fig. 14(f), 14(g) and 14(h), can be found, mainly attributed to the deterioration in both the SNR of the RBS amplitude and the phase variations. Meanwhile, the similarity of each waveform with respect to the original driving signal are given. Those derived for the testified waveforms are all quite close to one in both near and far ends. The results indicated that the proposed φ-OFDR system can detect arbitrary-waveforms in addition to quantitative DVS for multiple vibration events across consecutive spatial resolutions.
Fig. 14. Spatio-temporal domain perspectives, temporal vibration waveforms corresponding to PZS-a, -b and -c, at the (a, b, c, d) near and (e, f, g, h) far ends of the sensing fiber, respectively, in case of multiple vibration events with arbitrary-waveforms.
2. Conclusions
In this study, an φ-OFDR based quantitative quasi-DVS that permits the measurement of multiple vibration events with arbitrary waveforms over spatially consecutive sensing spatial resolutions was proposed and demonstrated. To realize effective crosstalk suppression and mitigation of instability during the phase differential process, specially customized fibers with embedded ultra-weak grating arrays as the sensing fiber were adopted. It exhibits a low spatial duty-cycle, namely, a large length ratio between the grating size and spacing, while the spacing is additionally designed to match an optimized integer multiple of the theoretical spatial resolution. In conjunction with a high-quality FMCW optical probe supported by the OPLL, a quantitative quasi-DVS on multi-vibration events with arbitrary vibration waveforms over a spatially consecutive sensing spatial resolution of ∼2.5 cm along a distance of ∼2200 m was realized. The obtained linearity responses with sensitivity up to $\textrm{1}\mathrm{.941\;\ n\varepsilon }$ at the far end with high consistency over spatially consecutive sensing units testify to the high resolution distributed sensing capability at long distances.
Funding
National Natural Science Foundation of China (61827807, 61805014).
Disclosures
The authors declare no conflicts of interest.
Data availability
The 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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