Phase-based , high spatial resolution and distributed , static and dynamic strain sensing using Brillouin dynamic gratings in optical fibers

A novel technique combining Brillouin phase-shift measurements with Brillouin dynamic gratings (BDGs) reflectometry in polarization-maintaining fibers is presented here for the first time. While a direct measurement of the optical phase in standard BDG setups is impractical due to non-local phase contributions, their detrimental effect is reduced by ~4 orders of magnitude through the coherent addition of Stokes and anti-Stokes reflections from two counter-propagating BDGs in the fiber. The technique advantageously combines the high-spatial-resolution of BDGs reflectometry with the increased tolerance to optical power fluctuations of phasorial measurements, to enhance the performance of fiber-optic strain sensors. We demonstrate a distributed measurement (20cm spatial-resolution) of both static and dynamic (5kHz of vibrations at a sampling rate of 1MHz) strain fields acting on the fiber, in good agreement with theory and (for the static case) with the results of commercial reflectometers. © 2017 Optical Society of America OCIS codes: (290.5900) Scattering, stimulated Brillouin; (190.2055) Dynamic gratings; (050.5080) Phase shift; (060.2840) Heterodyne; (060.2370) Fiber optics sensors; (120.7280) Vibration analysis. References and links 1. A. Motil, A. Bergman, and M. Tur, “State of the art of Brillouin fiber-optic distributed sensing,” Opt. Laser Technol. 78, 81–103 (2016). 2. I. Sovran, A. Motil, and M. Tur, “Frequency-scanning BOTDA with ultimately fast acquisition speed,” IEEE Photonics Technol. Lett. 27(13), 1426–1429 (2015). 3. C. Zhang, M. Kishi, and K. 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Introduction
Brillouin dynamic sensing is of importance in many applications [1].Recent implementations of the Brillouin Optical Time Domain Analysis (BOTDA) [2] and Brillouin Optical Correlation Domain Analysis (BOCDA) [3] techniques, have demonstrated sampling rates of the order of kilohertz's with a centimetric spatial resolution (10cm over a range of 145m for the fully distributed case of [2] and 3cm over 6m for the random access approach of [3]).Both techniques, however, require some form of time-consuming scanning of the probe frequency against that of the pump, which limits their acquisition speed.In contrast, slopeassisted (SA) techniques, using a single (or at most a few) pair(s) of pump and probe frequencies can be much faster.As such, they have played a key role in taking the Brillouin distributed fiber optic sensing to the fast dynamic regime [1,4], including demonstrations of its practical utilization for monitoring the propagation of mechanical waves [5,6] (for the use of slope-assisted interrogation of a fiber-Bragg grating see [7]).
Most commonly, the SA techniques employ a tunable laser source (TLS) adjusted to the linear region of the slope of either the reflection spectrum of a fiber Bragg grating (FBG) [8] or the intrinsic Brillouin gain spectrum (BGS) [9], such that changes induced by measurand variations (e.g., strain) are translated to changes in the measured quantity (usually optical power).However, SA techniques are inherently sensitive to source optical power fluctuations and frequency drifts, fiber bend losses and spectral shape longitudinal inhomogeneity, introducing errors to the strain measurement.Much ingenuity has been spent on finding sophisticated solutions for these problems, such as using the ratio between readings taken on both slopes of the BGS [10], locking the laser frequency via a feedback loop [11], and tailoring the probe frequency to the BGS profile of the fiber [12].However, problems still remain and new ones are frequently discovered, as evidenced by [13], where it was shown that the BGS linewidth broadens with increasing pump power (with obvious ramifications on its shape and slopes), which affects the performance of the slope-assisted Brillouin optical time-domain analysis (SA-BOTDA) techniques, indicating an additional drawback of techniques based on the direct detection of optical power.
An alternative method, which might avoid such problems, is to exploit the measurand information encoded in the optical phase, which is widely recognized as the workhorse of distributed acoustic sensors (DAS) based on Rayleigh backscattering in optical fibers.These methods employ a coherent interference between the backscattered components of the interrogating pulse, resulting in a speckle-like trace whose amplitude and phase can be detected by means of coherent detection [14,15].To obtain quantitative information of the measurand, rather than merely detect dynamic perturbations, the phase difference between two reflections can be measured using an imbalanced Mach Zehnder interferometer with predetermined path difference [16].
Recently, interesting SA-BOTDA techniques, harnessing Brillouin phase-shift, have emerged [17][18][19].It should be noted that the spatial resolution of both gain-and phase-based slope-assisted BOTDA techniques is practically limited by the phonon lifetime to ≥1m.Recently proposed combinations of the differential pulse-width pair (DPP) [20] with either the gain [21] or the phasorial [22] BOTDA techniques showed an improved spatial resolution of <1m, at the expense of a decreased signal to noise ratio, leading to an increased number of averages and slower dynamic capabilities.
A quite different distributed approach to enhance the spatial resolution, without sacrificing the sampling speed, is to take advantage of Brillouin dynamic gratings (BDGs) in polarization maintaining (PM) fibers [23].These moving Bragg gratings are generated by two strong counter-propagating pumps, whose polarizations are aligned with the slow axis of the fiber.While both the magnitude and phase of the gratings are affected by the measurand, all recent demonstrations of this high-spatial-resolution sensing technique, e.g., [24, static] and [25, dynamic, slope-assisted], have only used the gratings' magnitude, as measured by the reflectivity of an orthogonally polarized narrow probe pulse.While offering the advantage of probe-power-independent measurements, the correct estimation of the local Brillouin phaseshift (BPS) in BDG setups is quite challenging, mainly due to non-local contributions to phase of the reflected probe, from which the measurand-induced BPS is to be deduced.Indeed, the phase of the gratings at the location of interest is critically affected not only by the measurand but also by the phase of the interference pattern generated by the counterpropagating pumps.This latter phase is governed by the environmentally-dependent optical lengths of the down-lead fibers, feeding the two writing pumps.As for the probe itself, on its journey to the point of interest and back it also collects non-local phase contributions.Furthermore, it will be shown below that the probe phase is also affected by inherent longitudinal non-uniformity of the birefringence in PM fibers [4].Proper measurement of the phase is also an issue.While in BOTDA setups, operating in transmission, measurement of the BPS can be accomplished, with minimum phase drifts, by interference with a copropagating reference [26], BDG setups operate in reflection.By the same reasoning and due to the fact that in BDG setups the reflected probe is also shifted in frequency, the technique that employs the nonreciprocal phase shift between the two paths of Sagnac interferometer allowing for the measurement of BPS in BOTDA setup [18], cannot be efficiently harnessed in BDG setups.
In this paper, we present a novel technique, which practically combines the benefits of phasorial measurements and high spatial resolution BDG reflectometry.Using coherent addition of the Stokes and anti-Stokes reflections from two simultaneously counterpropagating BDGs in the fiber, the technique advantageously offers distributed Brillouininduced Phase-Shift (BiPS) measurement with high spatial resolution.The technique is largely immune to variations in laser optical power and frequency drifts, fiber bend losses, and similarly to phasorial SA-BOTDA techniques, offers an extended dynamic range.Detrimental non-local phases and birefringence-non-uniformity-induced contributions are shown to be significantly reduced, if not completely cancel out.Finally, a measurement of static and dynamic strain fields is demonstrated.

Principle of operation
BDGs are optically generated longitudinal density (acoustic) waves in optical fibers [23], whose magnitude and phase depend on the amplitudes, phases, and frequency difference of the optical pump waves that generate them, as well as on the electrostrictive properties of the interaction medium.Most commonly, BDG-based sensors employ PM fibers, where two counter-propagating optical pump waves (PumpH and PumpL, ν PumpH >ν PumpL ) are polarized along the slow axis of the fiber, and the Probe pulse is orthogonally polarized and propagates along the fast axis of the fiber.For a Stokes-BDG scenario, the Probe pulse is launched into the fiber from the same side as PumpH.It is then reflected from a co-propagating refractive index grating (the BDG), which was generated by PumpH and PumpL.The reflected signal is also Doppler-downshifted by the BDG frequency, ν ( = ν PumpH -ν PumpL ).The grating amplitude and phase depend on the frequency difference between the writing pumps, as well as on the local strain/temperature of the fiber.Therefore, in classical BDG sensing, to obtain the measurand information, the frequency difference between the writing pumps is scanned, looking for the frequency difference that maximizes the intensity of the reflected probe.Much like the case of the SA-BOTDA technique, a major speed advantage can be achieved if the frequencies of the signals involved in the interaction are tuned to the slope of the BDG spectrum [25], so that rapid strain variations are translated to changes in the intensity of the probe reflection.However, the intensity-based slope-assisted BDG (SA-BDG) and SA-BOTDA techniques share the same disadvantage of measurand dependence on the local optical power, which impairs their performance.Furthermore, in SA-BDG setups, the PM fibers' birefringence longitudinal variations introduce errors to the measurement through the modification of the conversion factor between the intensity and strain/temperature.While cannot be mitigated using the pre-compensation technique of [12], these manufacturingrelated and measurand-induced birefringence variations introduce additional impediments to dynamic strain measurements.To address these disadvantages, we hereby propose a phasorial SA-BDG technique which overcomes most if not all these disadvantages.
In our proposal, two counter-propagating BDGs are generated by the same PumpH and PumpL, both of which are now launched from both sides of the PM fiber (polarized along its slow axis), Fig. 1.To attain maximum gratings strength, the frequency difference between the pumps, ν, is tuned to the Brillouin frequency shift (BFS) of the slow axis of the fiber, ν B (~11GHz).An orthogonally polarized dual-tone Probe pulse can be launched from either side of the fiber and propagates along the fast axis of the fiber.The Probe pulse carrier frequency comprises two tones: a higher-frequency tone (ν Probe_HF ) which is reflected from the Stokes-BDG (a reflection from a receding grating), attaining maximum reflection for ν Probe_Stokes ≈ν PumpH + ν BDG (ν BDG primarily depends on the fiber birefringence [23], Δn = n slown fast , ~46GHz in PM Panda fibers), and a lower-frequency tone (ν Probe_LF ≡ν Probe_HF -ν), which is reflected from the anti-Stokes-BDG (a reflection from an oncoming grating), attaining its maximum value for ν Probe_anti-Stokes = ν Probe_Stokes -ν [27].Upon reflection, the Stokes component of the Probe pulse is Doppler-downshifted by the frequency of the receding grating, ν, while the anti-Stokes component of the Probe pulse is Doppler-upshifted by the frequency ν of the oncoming grating.The resultant electrical signal from direct photo-detection comprises a beat term oscillating at the RF frequency of ν.In the following section we show that the phase of this electrical RF signal depends almost exclusively on the local Brillouin interaction of the pumps, since all other contributions are dramatically reduced.

Stokes-BDG and anti-Stokes-BDG field-reflection
We start with the full differential equations governing the Stokes-BDG [28] and anti-Stokes-BDG [29] interactions (Fig. 2).For not too long BDGs, PumpH depletion and PumpL amplification, as well as their linear losses, are neglected, and the equations reduce to: ( ) is the mean refractive index).While Δν B ≡ν-ν B ( = ΔΩ B /2π, the detuning parameter) measures the deviation from phase-matching conditions for the creation of the acoustic fields (mainly depending on the acoustic velocity in the fiber), Δν BDG ( = ΔΩ BDG /2π) is a measure of the phase matching between the induced acoustic fields (i.e., the BDGs) and the Probe waves (solely depending on birefringence).Equation sets (1) and ( 2) are readily identified as coupled-mode equations which govern the wave reflection in Bragg gratings under the "synchronous approximation" [31].For an undepleted Probe, a condition which BDG interactions certainly satisfy, the impulse responses of the Stokes-and anti-Stokes-BDGs can be found using the technique of [32]: Here L is the length of the fiber where BDG interactions take place.The leading ratio in Eqs.
(3)-( 4 where the longitudinally-dependent RF phase-shift is given by: We already note that the birefringence-nonuniformity-induced contribution (the last term in the first line of Eqs. ( 3) and ( 4 ν ε = which was found to be ~0.05MHz/με[33].These BFS changes are translated to Brillouin phase-shift changes and demodulated from the RF phase-shift in the electrical domain.Figure 3 shows the theoretical effect of periodic variations in BFS (black curves) on the detected RF phase-shift.It is important to note, that since the RF beat term oscillates at ~11GHz, the spatial resolution of this technique is practically limited to ~1cm, due to the increasing phase uncertainty in the electrical signal demodulation process.
We now show how the proposed technique is independent of non-local phase contributions rapidly accumulating due to changes in the optical path induced by temperature and/or strain changes along the leading fiber, and to which degree it is immune to variations in laser optical power or/and fiber bend losses.

exp( )
Δ , to appear in the cosine of Eq. ( 5), whose impact is ~4 orders of magnitude smaller.Similar conclusion arises from the analysis of strain-induced non-local effects (here, however, the change in the physical length is the main contributor to the accumulated phase).
As for the probing signal, the two tones of both the forward propagating Probe and the reflected ProbeR are separated by the frequency ν, and therefore the total phase difference will be zero.This excludes the optical path difference between the 'source→FUT' path and the 'FUT→detector' path, which can in principle be balanced or thermally controlled.
Our technique is also quite insensitive to variations in the pumps optical power and/or fiber bend losses.Based on the assumption of a constant PumpL, Eq. ( 6) shows complete independence of the RF phase-shift on the local powers of PumpL and PumpH.In practice, however, PumpL may experience some gain, altering both its amplitude and phase.PumpH will be somewhat affected as well.To investigate the implications of these practically encountered power variations on the RF phase shift we have numerically [13] solved Eqs.(1) for the phase shift of the induced BDG, as PumpL assumes different optical powers.For the scenarios of interest in this paper, with pumps power below 1W and a few meters long sensing fibers, gain is low (≤0.5dB).In this regime of operation, our simulation shows that many dB's of variation in the power of the pumps merely affect the phase of the BDG by a few tens of a milliradian.Consequently, the resulting strain inaccuracy is of the order of a few microstrains (see below Section 4), practically making this technique quite immune to power variations.

Experimental setup
A complete phasorial PM-BDG system, Fig. 4, was built to experimentally demonstrate the proposed technique.A single narrow-band laser diode was used as a source for all the optical signals in the system.Half the laser power is routed to the 'pumps' branch, where the PumpH and PumpL waves are, respectively, the higher-and the lower-frequency sidebands generated by a low-V π electro-optic Mach-Zehnder modulator (MZM1), biased at its minimum transmission to maximally suppress the carrier.The modulation frequency of the feeding RF signal generator (SG1) lies in the vicinity of ν B /2 (5.425GHz).The modulator output is split by a -3dB coupler, whose outputs are both amplified by Erbium-doped fiber amplifiers (EDFA1 and EDFA2) to 20dBm, and after passing through high power PM fiber isolators (ISO1 and ISO2) are launched into the slow axis of a 5m PM FUT from both its sides (entering the fiber from the side from which the probe also enters requires the use of a polarization beam splitter (PBS)).In the 'probes' branch, first, the laser frequency is upshifted using MZM2.The modulation frequency of SG2 is equal to ν BDG (45.5GHz).A tunable optical filter (TOF1) removes the lower frequency sideband, as well as most of the amplified spontaneous emission (ASE) of EDFA3, which immediately follows MZM2.The ν Probe_HF and ν Probe_LF tones are, respectively, the higher-and the lower-frequency sidebands generated by MZM3, also fed by SG1 (modulation frequency of ~νB /2).Finally, the Probe 2ns pulses are generated by a pulse generator (PG), which feeds a semiconductor optical amplifier (SOA) with a high extinction ratio of >40dB.Subject to the constraint that no two Probe pulses are allowed to be simultaneously present inside the 5m FUT (plus the leading fibers for a total of ≤10m), the pulse repetition rate of PG is set at 10MHz.The SOA output is amplified by EDFA4, providing a peak pulse power of 15dBm.Half the Probe power is launched into the fiber through a PBS to propagate along the fast axis of the PM FUT.The other half, serving as a reference signal, is routed to the fast photodiode (PD1), whose output is acquired by a wideband digital oscilloscope with a sampling rate of 80 Gsamples/sec.The probe reflection, ProbeR, is guided through the coupler into an acquisition channel comprised of EDFA5 and TOF2, which removes the pumps leakage into the fast axis as well as the ASE of EDFA5.The amplified and spectrally filtered ProbeR signal is then detected by a second fast photodiode (PD2), and acquired by the oscilloscope.The RF phase-shifts are then demodulated in the electrical domain.A low-V π electro-optic Mach-Zehnder modulators, biased at their minimum transmission to maximally suppress the carrier; SG1/2: RF signal generators whose modulation frequencies lie in the vicinity of ν B /2 and ν BDG , respectively; EDFA1-5: Erbium-doped fiber amplifiers; TOF1: A tunable optical filter which removes the higher frequency sideband of MZM2 as well as the amplified spontaneous emission (ASE) of EDFA3; SOA: A high extinction ratio semiconductor optical amplifier; PG: Pulse generator; ISO1-3: Isolators; PBS: Polarization beam splitter; FUT: Fiber under test; TOF2: A second tunable optical filter which removes the pumps leakage into the fast axis as well as the ASE of EDFA5; PD1/2; Fast photodiodes.

Results
In Eqs. ( 1)-( 2), we have tacitly assumed that the two counter-propagating BDGs do not interact.To establish whether this assumption was justified, we have performed an experiment in which we have measured the reflection from an oncoming grating (standard anti-Stokes-BDG configuration) while altering the intensity of the receding grating.To that aim, we have disconnected the MZM3 feeding RF signal and changed its bias to maximum transmission point.We have changed SG2 modulation frequency to ν BDG -ν B /2 (40.075GHz), such that the Probe pulse carrier frequency comprised only one tone, matched to the anti-Stokes-BDG.In the lower 'pumps' branch of Fig. 4, after ISO2, we have placed a tunable optical filter whose roll-off was adjusted to ν PumpL , while ν PumpH remained in the pass-band.By tuning the filter, we have manipulated the transmitted intensity of PumpL without affecting PumpH, Fig. 5(a).Figure 5(b) shows the ProbeR signal of PD2 for different intensities of the receding grating and a constant intensity of the oncoming grating, to which the Probe signal was matched.It can be seen that the anti-Stokes-BDG reflection remains unchanged, justifying our assumption that counter-propagating BDGs do not interact.Next, we have experimentally addressed the concern of Rayleigh backscattering of the Probe.Returning to the full phasorial PM-BDG setup of Fig. 4, we have placed an optical spectrum analyzer (OSA) before EDFA5 and measured the spectrum of ProbeR for two cases: one when the pumps where turned off (representing only the Rayleigh backscattering contribution) and other when pumps where turned on, Fig. 6.The lower/higher wavelength sideband for the case when pumps where turned off, is the Rayleigh backscattering of the higher/lower frequency Probe, and for the case when then pumps where turned on, it's the anti-Stokes-/ Stokes-BDG reflection of the lower/higher frequency Probe.It can be seen that the Rayleigh backscattering is ~15dB weaker than the BDG reflection.Though in principle sufficient, better dynamic range can be achieved by increasing the intensity of the generated acoustic wave with higher intensity pumps or/and through the use of chalcogenide glass fibers with a larger nonlinear coefficient.The expected linear dependence of the BDG peak reflectivity on the power of the pumps and the Probe was also measured.To measure the power of PumpH and PumpL, we have placed a "tap" coupler and an OSA1 in the lower 'pumps' branch of Fig. 4, after ISO2. Figure 7(a) shows the average measured power of the two tones comprising ProbeR (using OSA2 placed before EDFA5) as a function of the average power of PumpH and PumpL that was tuned by altering the output power of EDFA2.Next, Fig. 7(b), we have replaced PD1 with OSA1 and measured again the average power of the two tones comprising ProbeR, but this time as a function of the average power of the two tones comprising the Probe that was tuned by altering the output power of EDFA4.Both graphs show a distinct linear behaviour.A static strain experiment was then conducted.First, we have performed a calibration procedure in which a static strain was applied to a 20cm section of the PM FUT, bonded between two linear translation stages, Fig. 8(a).The resulting strain was monitored using another single-mode fiber, bonded in parallel to the PM FUT and interrogated by a commercial Rayleigh-backscattering-based optical frequency-domain reflectometer (OFDR).ν ε = ~0.05MHz/με, in excellent agreement with [33].The dynamic range for the proposed slope-assisted method may extend over the full slope, Fig. 8(b), while its linear part occupies a few hundred microstrains, making the method quite appropriate for the measurement of vibrations.Following the calibration procedure, a second static experiment was performed.This time, an additional 1m section of the FUT was bonded to a second pair of linear translation stages, Fig. 9(a).Two types of reference traces of the loose fiber were recorded: one of the Rayleighbackscattered signal using a commercial OFDR interrogator, and other of the Brillouininduced phase-shift.The Brillouin-induced phase-shift (BiPS) was obtained by scanning a range of modulation frequencies of SG1 around ν B , and analyzing the phase-shift of the beat signal of PD2, using our phasorial BDG system of Fig. 4. The BiPS was acquired for two modulation frequencies of SG2, ν BDG and ν BDG + 100MHz, emulating fiber birefringence nonuniformity.A strain of ~110με and ~90με was then applied to the 20cm and the 1m sections of the fiber, respectively, and a second set of OFDR and BiPS traces was acquired.Finally, a dynamic strain experiment was conducted.A 20cm section of the FUT was bonded to a linear translation stage at one end, and to a mechanical shaker at the other end, Fig. 10(a).The FUT was periodically stretched by the shaker, driven by an electrical function generator at 1kHz / 5kHz.Averaging over 10 repetitions was applied to the raw data, representing an effective sampling rate of 1MHz.RF phase-shifts as a function of time, measured at the periodically stretched section of the fiber, are depicted in Fig. 10(b) / 10(c), clearly showing the 1kHz / 5kHz periodic variations, as predicted by our model shown in Fig. 3.In Fig. 10(c), 10kHz low pass filter was applied.The RMS noise level of the measured RF phase-shift, normalized to 1Hz, was found to be 0.1mrad/√Hz (equivalent to 10nε/√Hz).

Discussion and conclusions
In summary, we have proposed and successfully demonstrated a high-spatial-resolution and ultrafast fiber reflectometry technique based on the distributed measurement of Brillouininduced phase-shifts in Brillouin dynamic gratings.The main obstacles associated with localized phase measurements in BDG setups have been overcome by employing coherent addition of the Stokes and anti-Stokes reflections from two counter-propagating BDGs in the fiber, followed by heterodyne detection.As predicted by the analysis of the phasorial properties of BDG operation, most measurand-unrelated non-local common phases were cancelled-out.Two sources of measurement errors still remain, Sec.2.2: strain/temperatureinduced non-local differential phase contributions due to the difference in optical frequency between PumpL and PumpH; and large variations in the power of the pumps.Under a few degrees of temperature change along ~10m of fiber and up to several dB of pumps power change, the total measurement error is estimated to be of the order of 10 microstrains.While we have only demonstrated a spatial resolution of 20cm (limited by the 0.5ns switching time of our SOA), the fundamental limitation of the method probably lies in the vicinity of 1cm, where the bandwidth of the probing pulse approaches the ~11GHz BFS of silica fibers.It is likely that higher spatial resolutions may be achieved with materials having larger BFS, e.g., Sapphire-derived all-glass optical fibers [35] (the use of narrow probe pulses may be challenged by signal to noise limitations, which may be overcome by coding [32]).
Owing to its high-spatial-resolution and speed, this technique may be extremely attractive for applications such as monitoring the propagation of mechanical waves.Here, we have demonstrated a distributed measurement of Brillouin-induced phase-shift of a 5m-long fiber with a spatial resolution of 20cm.Measurement of both static and dynamic (5kHz) strain fields acting on the fiber were also demonstrated, in excellent agreement with the theory and reference measurements by a commercial strain interrogator.This technique is expected to manifest increased tolerance to laser optical power fluctuations, fiber bend losses and optical pumps depletion.This first demonstration of high-spatial-resolution Brillouin phase-shift measurement may have implications which go beyond the realm of fiber-optic sensors.For instance, it has potential important application in the characterization of BDG-based reconfigurable optical filters [36].

Fig. 1 .
Fig. 1.A schematic diagram of the proposed PM-BDG setup for distributed measurement of Brillouin-induced phase-shift using two simultaneously counter-propagating Brillouin dynamic gratings and a dual-tone probe.PBS: Polarization beam splitter; PD: Fast photodiode.

Fig. 2 .
Fig. 2. Schematic description of the (a) Stokes-BDG and (b) anti-Stokes-BDG interactions, described by the equation sets (1) and (2), respectively.Here, S ρ and AS ρ (having the acoustic frequency of ν), , Probe S E , , Probe AS E , , ProbeR S E and ) is the amplitude of the reflection.It is a function of the longitudinally-distancedependent B ΔΩ .The next phase factors represent the dependence of the phases of the probe reflections from the counter-propagating gratings on the mismatch between the pumps frequency difference ν and ν B .These two mismatch-induced phases share their dependence on distance through that of B ΔΩ , but they are of opposite signs.This sign difference can be easily understood by way of example.Let's assume the frequency difference between the pumps to be larger than the BFS of the slow axis: ν>ν B .As a result, in the Stokes-BDG scenario, the density disturbance moves faster than the speed of sound away from the Stokes probe, -BDG scenario, the density disturbance moves faster than the speed of sound towards the anti-Stokes probe, identical phase factors in Eqs.(3)-(4) originate from a mismatch between the incoming probe frequency and the resonant frequencies of the moving Bragg gratings.These phases too are distance dependent due to the longitudinal variations of the fiber birefringence, Δn(z), having similar effects on reflections from the two BDGs.Equations (3)-(4) end with the phasors of the Stokes and anti-Stokes reflections, having corresponding frequencies of ν Probe_Stokes -ν and ν Probe_anti-Stokes + ν ( = ν Probe_Stokes ).The Stokes-reflected and anti-Stokes-reflected components back-propagate to the detector and interfere to produce the following AC photocurrent (oscillating at the difference between the frequencies of , )) vanishes.The phase spectra of the Stokes-BDG ( Γ ), as well as that characterizing their beat term, RF φ Δ , are plotted in Fig.3.Strain in the fiber will change the fiber sound velocity, and consequently, the BFS, ν B , according to
First, let's consider a standard Stokes-BDG scenario where PumpH signal is launched into the fiber under test (FUT) and propagates towards the location of interest at distance z from the FUT entrance point.The accumulated phase is / change of ΔT in the leading fiber, the corresponding change in the accumulated phase can be written as ( -optic coefficient, which is approximately equal to 8.6•10 −6 °C−1 for a germanium-doped, silica-core fiber[34] (the change in the physical length due to thermal expansion is negligible).This detrimental non-local phase rapidly accumulates over few meters even for small temperature changes of 1°C, practically precluding localized Brillouin-induced phase-shift measurements in standard BDG setups.In our setup, however, owing to the generation of two counter-propagating BDGs in the FUT, the phase change in the PumpH signal (PumpH ψ Δ) is imparted on ProbeR of the Stokes-BDG,

Fig. 4 .
Fig. 4. Experimental setup.LD: A narrow-band tunable laser diode set at 1550nm; MZM1-3:A low-V π electro-optic Mach-Zehnder modulators, biased at their minimum transmission to maximally suppress the carrier; SG1/2: RF signal generators whose modulation frequencies lie in the vicinity of ν B /2 and ν BDG , respectively; EDFA1-5: Erbium-doped fiber amplifiers; TOF1: A tunable optical filter which removes the higher frequency sideband of MZM2 as well as the amplified spontaneous emission (ASE) of EDFA3; SOA: A high extinction ratio semiconductor optical amplifier; PG: Pulse generator; ISO1-3: Isolators; PBS: Polarization beam splitter; FUT: Fiber under test; TOF2: A second tunable optical filter which removes the pumps leakage into the fast axis as well as the ASE of EDFA5; PD1/2; Fast photodiodes.

Fig. 5 .
Fig. 5. (a) The transmitted spectrum of the lower 'pumps' branch of Fig. 4, after ISO2.(b) The ProbeR signal of PD2 for different intensities of the receding grating and a constant intensity of the oncoming grating, to which the Probe signal was matched.

Fig. 6 .
Fig. 6.The OSA measured spectrum of ProbeR for two cases: one when pumps where turned off (representing only the Rayleigh backscattering contribution) and other when pumps where turned on.

Fig. 7 .
Fig. 7. Measured BDG reflectivity as a function of (a) pumps power and (b) the Probe power.

Figure 8 (
b) shows the RF phase-shift of the detected signal as a function of the measured strain (blue circles).Superimposed on the experimental points is the solid red curve which shows the theoretical RF phase-shift, RF φ Δ , from strain-induced shift in the BFS, obtained with a linear regression coefficient of

Figure 9 (
Figure 9(b) shows the strain analysis employing both methods.In our technique, the strain field was obtained by subtracting the current BiPS trace from the reference, and recording the maximum phase-shift, occurring in the vicinity of ν B .As evident from Fig. 9(b), the conversion coefficient for small strains is ~1rad/100με, in excellent agreement with the calibration procedure of Fig. 8(b).

Fig. 9 .
Fig. 9. (a) The setup of the second static strain experiment.(b) Longitudinal strain field along the fiber as measured by a commercial OFDR interrogator, compared vs. the maximum of the difference in the Brillouin-induced phase-shift.

Fig. 10 .
Fig. 10.(a) A 20cm section of the FUT was bonded to a linear translation stage at one end, and to a mechanical shaker on the other end.(b) RF phase-shift as a function of time, measured at the periodically stretched section of the FUT driven by an electrical function generator at 1kHz and (c) 5kHz (in Fig. 10(c), 10kHz low pass filter was applied).