High-resolution Brillouin optical correlation domain analysis with no spectral scanning

Distributed Brillouin fiber sensors typically rely on the reconstruction of the steady-state Brillouin gain spectrum (BGS), through spectral scanning of the frequency offset between the pump and signal waves. In this work, we propose and demonstrate an alternative approach, in which the local Brillouin frequency shift (BFS) is extracted from temporal transient analysis of the step response of the amplified signal wave. Measurements are taken at only two arbitrary frequency offsets between pump and signal. No spectral scanning and no prior knowledge of a reference BGS are necessary. The principle is supported by analytic and numeric solutions of the differential equations of stimulated Brillouin scattering. The BFS of a 2 meters-long fiber under test was measured with 1 MHz accuracy and a dynamic range of 200 MHz. Transient measurements were also performed in a Brillouin optical correlation domain analysis (B-OCDA) experiment with 4 cm resolution, standard deviation of 2.4 MHz and 100 MHz dynamic range. A 4 cm-wide hot-spot was properly identified in the measurements. Multiple correlation peaks could be addressed in a single flight of a pump pulse. The results represent the first B-OCDA that is free of spectral scanning. This new measurement concept may be applicable to random-access distributed and dynamic monitoring of sound and vibration. © 2016 Optical Society of America OCIS codes: (290.5900) Scattering, stimulated Brillouin; (060.2370) Fiber optics sensors; (190.4370) Nonlinear optics, fibers. References and Links 1. R. W. Boyd, Nonlinear Optics, 3rd ed. (Academic, 2008). 2. T. Horiguchi, T. Kurashima, and M. Tateda, “A technique to measure distributed strain in optical fibers,” IEEE Photonics Technol. Lett. 2(5), 352–354 (1990). 3. M. Niklès, L. Thévenaz, and P. A. 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Introduction
Stimulated Brillouin scattering (SBS) in optical fibers is a third-order nonlinear interaction between counter-propagating pump and signal waves that are coupled by an acoustic wave [1].Efficient coupling requires that the difference between the optical frequencies of pump and signal should closely match the Brillouin frequency shift (BFS) of the fiber.The BFS, in turn, varies with both temperature and mechanical strain.Hence the precise mapping of the local BFS is being used in the distributed monitoring of both quantities [2,3].Key performance metrics of distributed Brillouin sensors include the measurement range, spatial resolution, precision of the BFS estimate, and the acquisition duration.For a recent review of the state-of-the-art in Brillouin sensors see [4], and references therein.The measurement duration is particularly significant in the emerging field of distributed dynamic monitoring of sound and vibration [5][6][7].
The most widely-employed Brillouin sensing approach is that of Brillouin optical timedomain analysis (B-OTDA), in which a continuous signal wave is amplified by a pump pulse [2,3].The vast majority of B-OTDA setups recover the local BFS through repeating measurements at different values of the frequency offset between pump and signal, and reconstruction of the Brillouin gain spectrum (BGS).Some B-OTDA setups acquire data very quickly.In one recent example, the BGS was mapped along an entire 150 m-long fiber in only a few ms [5].
Another approach for high-rate B-OTDA involves slope-assisted measurements [6,7], which are also widely employed in the monitoring of fiber Bragg gratings [8].In this method the frequency difference between pump and signal is deliberately set to either side of the BFS, where the variations in signal gain with frequency offset are the steepest.With that choice, changes in the local BFS modify the intensity of the output signal.Spectral scanning of the entire BGS is not required.Slope-assisted B-OTDA is successfully employed in high-rate dynamic monitoring of vibrations [5,9].The technique is very effective in the identification of disturbances, however the quantitative measurement of the BFS is sometimes restricted to a comparatively narrow dynamic range.In addition, slope assisted measurements are sensitive to intensity noise and often require prior mapping of the 'nominal' BFS at each position.The principle was extended to the simultaneous monitoring of both spectral slopes [10].Another technique relies on the simultaneous analysis of multiple pump-signal pairs, each adjusted to a different frequency offset [11].All B-OTDA protocols, however, even the fastest-scanning ones and those that avoid spectral scanning altogether, rely on measurement of the output signal wave after the SBS interaction has reached its steady state in several acoustic lifetimes.
An alternative sensing paradigm is that of Brillouin optical correlation domain analysis (B-OCDA), which is based on the close relation between the magnitude of the stimulated acoustic field at a given location and the temporal cross-correlation between the complex envelopes of the pump and signal at that point [12,13].The joint phase or frequency modulation of the pump and signal waves may effectively confine their correlation to discrete and narrow peaks, and largely inhibits SBS interactions elsewhere.B-OCDA-based protocols have reached mm-scale resolution [14].Hundreds of thousands of high-resolution points were addressed [15], and the potential for covering over 2 million points was convincingly demonstrated [16].B-OCDA is fundamentally slower than B-OTDA.Nevertheless, several B-OCDA setups have reached dynamic measurement capabilities [17,18].Here too, however, all realizations of B-OCDA thus far involved spectral scanning of the BGS at the steady state.
In this work we propose, analyze and demonstrate a high-resolution Brillouin sensing protocol that is able to quantitatively recover the local BFS based on the transient analysis of the output signal wave, during the stimulation of Brillouin scattering.We show that the transient step response of the output signal is uniquely determined by the detuning between the BFS and the frequency of the stimulated acoustic field.Therefore, the comparison between the recorded traces and a library of modeled step-response functions may properly retrieve the BFS.Only a pair of traces, taken at two arbitrary values of frequency detuning between pump and signal, are required.No prior knowledge of the local BFS is necessary, and no spectral scanning is performed.The two traces are necessary to remove sign ambiguity.A single trace is sufficient with prior knowledge of the local BFS.
The protocol is supported by analytic solutions of the transient step-response of the acoustic and signal waves, and by direct numerical integration of the coupled differential equations of SBS subject to the appropriate boundary conditions.The principle is first demonstrated in measurements of the BFS of a 2 meter long fiber section, with 1 MHz precision and a dynamic range of 200 MHz.Transients-based B-OCDA is then performed with spatial resolution of 4 cm.The standard deviation in the transient BFS measurement is 2.4 MHz, and the dynamic range of the analysis is 100 MHz.A local hot-spot is properly identified.The results are validated by standard, phase-coded B-OCDA measurements of the local BGS at the steady state [19].
The remainder of this paper is organized as follows: analysis and numerical simulations are described in section 2 and 3, respectively.Experiments are reported in section 4, and a concluding discussion is given in section 5. Partial, preliminary results were reported in [20,21].

Transient step response of stimulated Brillouin scattering
Consider a continuous, undepleted pump wave of magnitude 0 p A , propagating from 0 z = in the positive z direction along a short fiber under test (FUT) of length L .A counter- propagating signal wave is launched in the opposite direction, from z L = .The profile of the input signal wave as a function of time t , ( ) , is given by a step function: We suppose further that changes to the signal wave magnitude due to SBS along the short FUT are comparatively small.In this condition, we may express the complex magnitude of the stimulated acoustic wave in the following form [1,[20][21][22]: Here 1 g is an electro-strictive parameter, and A Γ denotes a complex linewidth: , 2 . The local acoustic field magnitude reaches a steady state value of ( ) ( ) following several acoustic lifetimes: ( ) . Unlike most Brillouin sensing protocols, in this work we are interested in the transient buildup of the acoustic field and its effect on the signal wave.
The differential equation describing the propagation of the signal wave complex envelope ( ) , s A z t is given by: Here 2 g is a second electro-strictive constant parameter that is determined by the fiber properties.The solution to Eq. ( 2) is of the form: The result is valid for . Note that the expression converges to the expected steady-state amplification of the signal wave for t → ∞ .For convenience, we introduce below an offset time variable: ' . The instantaneous power of the signal wave at the output of the FUT, 0 z = , is given by: Next, let us define the instantaneous gain coefficient of the output signal power, ( ) ', g t Ω , so that: . Using the definition of the complex linewidth A Γ above, we find [20,21,23]: In Eq. ( 5), B ΔΩ = Ω − Ω is the offset between the stimulated acoustic wave frequency and the BFS, and ( ) ( ) denotes the frequency-dependent, steady-state gain coefficient.Figure 1 shows several examples and a two-dimensional map of ( ) ', g t ΔΩ , calculated using Eq. ( 5).The results are normalized by ( ) Standard Brillouin sensors rely on measurements of ( ) ss g ΔΩ , taken at ' t τ >> .However, a single reading of ( ) ss g ΔΩ cannot retrieve the BFS, since it is susceptible to variations in the input power level of the pump 2 0 p A [4].Hence the BFS is usually recovered through spectral scanning over multiple values of frequency detuning between pump and signal, and the reconstruction of the entire BGS.Transient information is seldom considered.Nevertheless, Fig. 1 suggests that the temporal transient profile of ( ) ', g t ΔΩ may also be used to recover the BFS, through the following protocol.
The difference between the optical frequencies of pump and signal is set at an arbitrary known 0 B Ω ≈ Ω , and the instantaneous gain coefficient of the output signal is measured: The measured trace is cross-correlated with a library of previously calculated reference functions ( ) 0 ', i i g t Ω − Ω , of the form of Eq. ( 5), where { } i Ω is a set of guesses for the BFS.The maximum cross-correlation value is noted for each choice of i Ω : Here the symbol * denotes the temporal cross-correlation operation.The BFS B Ω is identified as the value i Ω for which the score ( ) C Ω is the highest.Since ( ) ', g t ΔΩ is an even function of ΔΩ for all ' t , a second measurement with a different choice of 0 Ω is required to remove sign ambiguity.Even so, a complete spectral scanning over many frequency offset values may be avoided.Figure 1 indicates that the output signal power goes through a local maximum, at an instance peak t that is determined by ΔΩ : Therefore, the measurement of the exact timing of the local maximum alone may already retrieve ΔΩ .However, more accurate analysis of noisy traces is obtained through fitting of the entire ( ) ' h t curve, as in Eq. (7).The analysis has been extended to the modulation of the signal wave by a step function with a finite rise time, which describes experimental conditions more realistically (for details see [20]).

Transient step response of Brillouin optical correlation domain analysis
The analysis of the previous section requires that the SBS interaction is confined to a short FUT of uniform BFS.This condition is inherent to B-OCDA, in which phase or frequency coding effectively confines the stimulation of the acoustic wave to discrete and narrow correlation peaks [12,13].Therefore, B-OCDA is particularly suitable to the transient, spectral scanning-free measurement procedure proposed in this work.In the following section we examine the transient amplification of the signal wave in a phase-coded B-OCDA setup.The envelope of the signal wave is modulated be a repeating, high-rate binary phase code: Here ( ) and equals zero elsewhere, and { } n c are the elements of a pseudo-random binary sequence (PRBS) with symbol duration phase T which is repeated with a period of phase N bits [13,15,16,19].The magnitude of the signal wave is constant.The pump wave is phase modulated using the same sequence.In addition, the magnitude of the pump wave is modulated by a single pulse of duration pulse T : The complex envelope of the acoustic wave is given by [22]: In Eq. ( 11) we assumed again that the pump wave is undepleted, and that changes to the magnitude of the signal wave are small.The expression reduces to that of Eq. ( 1) within correlation peaks of width , and oscillates around zero mean value elsewhere [22].The spacing between adjacent peaks is phase N z ⋅ Δ .Suppose for the time being that L is sufficiently short so that only a single peak is established along the FUT.The magnitude of the signal wave ( ) , s A z t may be obtained by numerically integrating the following differential equation, with the complex magnitude of the acoustic wave taken from Eq. ( 11): Figure 2(a) shows several examples of the calculated ( ) ', g t ΔΩ , obtained by solving Eq. (12).The FUT was 4 cm long, the PRBS symbol duration was 400 ps, and its period was 1000 bits.The amplitude of the pump wave was modulated by a pulse of pulse T = 30 ns duration and a finite rise time of 5 ns.With this choice of parameters, the correlation peak spanned the entire simulated segment ( ) L z = Δ .Hence the contribution of residual off-peak interactions was not considered.was used in all calculations, however the sequence was offset by an arbitrary number of bits 0 0 phase n N ≤ ≤ in each repetition.An arbitrary starting index 0 n of the PRBS phase modulation at 0 t = is inherent to the experimental setup, since the waveform generators used in phase and amplitude modulation of pump and/or signal are free-running without synchronization.The random offset leads to the reduction of noise due to residual Brillouin interactions at off-peak fiber locations [15].Therefore, the simulation of SBS only within the correlation peak provides a clean reference for the subsequent analysis of experimental traces, following the procedure described in the previous section.
Lastly, the above analysis would also hold for time-multiplexed B-OCDA setups [15,16,19], in which SBS is built up in multiple correlation peaks, one after the other, during a single pass of the pump pulse along the FUT [15,16,19].When τ < < , individual gain events can be separated by direct time-domain analysis of the output signal wave.Up to several thousand locations were successfully addressed in a single trace [15,16,19].Here too, previous realizations of the concept involved measurements of ( ) ss g ΔΩ , taken towards the end of each gain event, and reconstruction of the BGS.Instead, the analysis of the transient profile of each event may retrieve the BFS in the same set of locations, using only a pair of frequencies 0 Ω as described above.A proof-of-concept experiment of transients-based, time-multiplexed B-OCDA is reported next.

Experimental setup and results
A schematic illustration of the experimental setup used in temporal transient B-OCDA with no spectral scanning is shown in Fig. 3. Light from a laser diode at 1550 nm wavelength was used as the common source for both pump and signal waves.In some of the experiments, the laser output passed through an electro-optic phase modulator, driven by an arbitrary waveform generator (AWG) that was programmed to repeatedly generate a PRBS.In other experiments, the phase modulator was bypassed.The modulated waveform was amplified by an erbium-doped fiber amplifier (EDFA) to 250 mW power, and split into pump and probe branches.Light in the pump branch was offset in frequency by an adjustable shift ( ) using a single-sideband (SSB) electro-optic modulator, driven by the output voltage of a microwave signal generator.The pump wave was then further modulated into low duty cycle pulses, using a semiconductor optical amplifier (SOA) that was driven by a second AWG.Last, the pump pulses were amplified by a second EDFA to an average power of few hundreds of mW, and launched through a circulator into one end of a short FUT.
In the probe branch, a polarization scrambler was used to prevent polarization-related fading of the SBS interaction [24].A 2.3 km-long fiber delay imbalance was connected in the probe path, to allow for the precise scanning of the correlation peaks positions along the FUT (see [13] for details).In some experiments, the signal wave was further modulated in an electro-optic amplitude modulator, driven by pulses from a third AWG.In other experiments the amplitude modulator was removed.The probe wave was launched through an optical isolator into the opposite end of the FUT.At the FUT output, the probe wave was detected by a photo-receiver of 200 MHz bandwidth, and sampled by a real-time digitizing oscilloscope.Data was averaged over 1,024 repeating pulses.

Transient analysis of step response in a 2 meters-long fiber under test
In a first set of experiments, the transient step response of SBS was used to measure the BFS in a uniform 2 meters-long FUT.The BFS and Brillouin gain linewidth of the FUT were determined in a B-OTDA control experiment as 2π⋅10.151GHz and 2π⋅39 MHz, respectively.The BFS in the FUT was lower than that of standard single-mode fibers, which make up the rest of the setup, by about 700 MHz.The SBS interactions therefore could be confined to the FUT.The maximum steady-state SBS gain coefficient of the FUT ( ) No phase coding was used in this experiment.The amplitude of the pump wave was modulated by 300 ns pulses with period of 1.3 µs, and the probe wave was modulated by 50 ns pulses with period of 100 ns.Transient traces were acquired for multiple values of ν , from 10.05 GHz to 10.25 GHz in 1 MHz increments [21].The BFS of the FUT was independently evaluated using each measured ( ) exp ' h t trace, following the procedure described in section 2. Figure 5(a) shows the recovered BFS and the measurement error ε with respect to the true BFS as a function of ν , and Fig. 5(b) shows the histogram and the cumulative probability of ε .The standard deviation of the BFS estimate error is 0.87 MHz.80% of the experiments yielded ε that was better than 1 MHz.The results demonstrate the quantitative recovery of the BFS using transient analysis, with a dynamic range of 200 MHz [21].

Transient analysis in time-multiplexed B-OCDA
In a second set of experiments, B-OCDA was performed over a 42 m-long section of standard single-mode fiber.Light at the output of the laser diode source was phase modulated by a repeating PRBS of symbol duration phase T = 400 ps and period phase N = 500 bits.The symbol duration corresponds to B-OCDA spatial resolution z Δ of 4 cm.The amplitude of the probe wave was continuous, and the pump wave was modulated by pulses of 30 ns duration and a period of 2 µs.The Brillouin gain linewidth in the FUT was measured separately as 2π⋅31 MHz.
Figure 6 (solid lines) shows several examples of the output probe wave as a function of time.Two gain events are observed, corresponding to two correlation peaks along the FUT [15,16,19].The maximum relative SBS power gain at steady state was 10%.The exponential gain coefficient ', g t ΔΩ (see Fig. 2 in section 3).The black, dashed lines in Fig. 6 show the simulated traces of the output signal wave, calculated using the best-fitting ν Δ .The results illustrate the ability of transient analysis to address multiple correlation peaks in a single flight of a pump pulse, in a manner that is analogous to B-OCDA measurements of the BGS that involve spectral scanning [15,16,19].

Monitoring of a local hot-spot
In a final experiment, a 4 cm-long section of the FUT of the previous experiment was heated to 57 °C above room temperature.Nine contiguous, 4 cm-wide resolution cells were addressed using phase-coded B-OCDA.The fourth resolution cell was in overlap with the hot-spot.The period of the PRBS used in phase modulation was increased to 1000 bits, so that only a single correlation peak was generated along the FUT.The BFS in each resolution cell was measured using transient temporal analysis of the output signal wave.The same resolution cells were also monitored by standard, spectral scanning B-OCDA measurements of the BGS at steady state.The BGS analysis served as a control experiment.The measured steady state Brillouin gain as a function of z and ν is shown in Fig. 7(a).The BFS of the non-heated fiber segments is 10.755 GHz ± 1 MHz.    Figure 10 shows the histogram and the cumulative probability of the BFS measurement error ε , over all 25 pairs of traces in all 9 resolution cells.The mean value of ε is 2.9 MHz.75% of the BFS estimates had an error smaller than 4 MHz.The standard deviation in the BFS measurements was 2.4 MHz.The estimates are considerably less accurate than that of the analysis of the 2 m-long FUT reported in section 4.1 (see Fig. 5).Possible reasons for the lower performance are the weak SBS gain over 4 cm of fiber, residual noise due to offpeak SBS interactions, possible small deviations of the output voltage of the AWG used in phase modulation from V π of the modulator, and an imperfect temporal shape of the pump pulses.Figure 11 shows the breakup of the average measurement error ε according to resolution cell (panel (a)), and according to offset ν (panel (b)).

Summary
In this work, we have proposed and demonstrated a first Brillouin sensor protocol that relies on the transient analysis of the output signal wave, rather than its measurement at the steady state.The protocol is also a first B-OCDA setup that is free of spectral scanning of the frequency offset between pump and signal.The principle was first employed in measurements of the BFS in a uniform, 2 meters long fiber segment with a dynamic range of 200 MHz and 1 MHz accuracy.B-OCDA measurements were then performed with 4 cm resolution, 100 MHz dynamic range and a standard deviation of 2.4 MHz.A local hot-spot was properly identified in the measurements.The addressing of multiple correlation peaks in a single flight of a pump pulse was demonstrated as well.
A main drawback of the proposed protocol is the complexity of the post-processing of data: multiple digital calculations of cross-correlation functions are required, as opposed to simple sampling of the photo-detector output in steady-state analysis.The necessary detection bandwidth of 200 MHz is also four times broader than the typical bandwidths of steady-state setups.The signal-to-noise ratio (SNR) of transient measurements is inherently poor.The steady state SBS gain over few cm of fiber, at the BFS, is 10% at most.The details of the transient signal traces at arbitrary frequency offsets are smaller still, at least by another order of magnitude.The measurement of sub-percent Brillouin gain is challenging.The poor SNR is compensated, at least in part, by the processing gain of correlation over an entire trace.Still, the weak amplification remains a challenge to the proposed analysis.
Despite the low SNR, transient analysis of the output signal trace may provide two potential benefits in B-OCDA: a) Measurement of the BFS with a smaller number of frequency offsets, and/or b) better precision than that of steady-state analysis in measurement of the BFS, for a given number of frequency scans.Precision enhancement might be anticipated since the transient analysis of each trace provides more information than a single reading at the steady state.For example, the precision of a steady-state B-OCDA control experiment carried out as part of this work was ± 1 MHz.The analysis consisted of 50 frequency scans.Soto and Thevenaz showed that the measurement accuracy scales with the square root of the number of frequencies used [25].Therefore, the 2.4 MHz precision obtained using transient analysis of a pair of traces is similar to that of 'conventional' B-OCDA in similar conditions, involving 8 frequency scans.A comprehensive quantitative comparison between the accuracies of steady-state and transient analysis of the BFS using a large number of frequencies is outside the scope of this work.
The measurement accuracy scales with the SNR [25].It may improve with the square root of the number of repetitions in the acquisition of each trace, when longer averaging is permitted [25].The precision of the transient B-OCDA may be improved further with calibration of the exact temporal profile of pump pulses, and of the temporal shape of phase code symbols.Measurement errors are also reduced if thermal drifts in the group delay of the long fiber path imbalance are stabilized.We recently demonstrated the closed-loop stabilization of group delays in B-OCDA [26].The measurement resolution depends on the phase-coding symbol duration.Sub-cm resolution was previously achieved in steady-state, spectral-scanning B-OCDA [13].Similar resolution may be obtained in transient analysis, given sufficient averaging.
The comparison between different Brillouin sensor architectures is difficult, since measurement ranges, resolutions and precisions vary [25].That being said, the acquisition rate of the proposed transient B-OCDA protocol is slower than that of state-of-the-art dynamic B-OTDA demonstrations.The measurement duration of the proposed method is limited by the time-of-flight along the FUT, the number of position scans, and the number of averages required.The number of position scans in few-cm-resolution, phase-coded B-OCDA is on the order of 50-100 [19].The number of averages used in our experiment was 1024, and two traces were taken at each set of correlation peak positions.Altogether, the net experimental duration necessary to address a 150 m-long fiber, for example, with 4 cm resolution would be 0.2-0.4seconds.This estimate does not take into consideration the latencies involved in the switching of laboratory equipment, and off-line post-processing.Note that post-processing is not expected to restrict the acquisition rate in a practical system, as it may be carried out in real-time using commercially-available digital signal processing units.
In contrast, slope-assisted B-OTDA achieved kHz acquisition rates over 160 m of fiber, with a dynamic range of 120 MHz [9,27].The spatial resolution, however, was 1 m [9].Fast double-pulse pair (DPP) B-OTDA measured the steady-state BGS along 150 m of fiber in about 2 ms [4,5].The two main advantages of the fast DPP B-OTDA setup were the small number of repetitions in the averaging of each trace, only six [5], and the addressing of the entire FUT with a single pair of traces.The spatial resolution of the fast DPP B-OTDA experiment was 10 cm.It is anticipated that a larger number of averages might be necessary in fast DPP B-OTDA if resolution were to be further improved towards few cm.In addition, the number of averages used in our experiment might be reduced if polarization switching and diversity techniques are used instead of polarization scrambling [4,5,28,29].Even so, B-OCDA setups are inherently slower than their fast B-OTDA counterparts, due to the spatial scanning of correlation peaks positions.B-OCDA, on the other hand, provides the advantage of random access monitoring [4].
A closer, and perhaps more relevant, comparison may be drawn between the method proposed in this work and reported dynamic B-OCDA measurements.The highest-rate B-OCDA experiment, to the best of our knowledge, addressed 5,000 locations per second with a spatial resolution of 3 cm [17].The fiber under test was only few-meters long.Given the number of scans necessary in our experiment, transient B-OCDA could be carried out over such short FUTs at a comparable rate.Dynamic B-OCDA over 100 m of fiber achieved 20 Hz rates with a spatial resolution of 80 cm [18].Similar and perhaps better trade-off between acquisition rate and high resolution may be achieved in the transient analysis of fibers of such lengths.
In conclusion, a new concept of transients-based Brillouin sensing was proposed and demonstrated.With proper optimization, the approach may increase the acquisition rate of B-OCDA and/or improve its precision.Further work would look to explore the full potential of transient Brillouin analysis, in terms of measurement range, acquisition rate, the addressing of the entire fiber under test, and improving the precision of BFS measurements.In addition, future research would employ perfect Golomb phase codes for the further suppression of residual off-peak interactions, and examine the transient analysis of frequency-modulated B-OCDA.
difference between the optical frequencies of the pump and signal waves is denoted by 2πν group velocity of light in the FUT is g v .

Fig. 1 .
Fig. 1.(a) -calculated logarithmic gain coefficient of the output signal wave as a function of time, during the transient buildup of the stimulated Brillouin interaction, for several values of the frequency detuning

Fig. 2 .
Fig. 2. (a) -calculated logarithmic gain coefficient of the output signal wave as a function of time, during the transient buildup of the stimulated Brillouin interaction within a single correlation peak in phase-coded B-OCDA.The detuning ν Δ between the stimulated acoustic wave frequency and the BFS in traces 1 through 5 was 1 MHz, 15 MHz, 25 MHz, 50 MHz and 100 MHz.The gain coefficients are normalized to the steady state value at 0 ν Δ = .(b)-map of the calculated gain functions, evaluated as a function of time and frequency detuning.The calculated magnitude of the output signal wave was averaged over 83 realizations of the B-OCDA boundary conditions.The same PRBS { } n cwas used in all calculations, however the sequence was offset by an arbitrary number of bits

Fig. 4 .
Fig. 4. (a) -Measured output pulses of the probe wave, following SBS amplification by pump pulses over 2 meters of FUT, at different values of detuning ν Δ between the frequency of the stimulated acoustic field and the BFS (solid lines, see legend for colors).The black, solid line denotes an unamplified, reference pulse.(b) -Measured logarithmic gain factor of the probe wave at different values of ν Δ (dotted lines), alongside the corresponding calculated curves (solid lines, see legend on left panel for colors) [21].

Figure 4 (Figure 4
Figure 4(a) shows several examples of the detected output probe pulses as a function of time curves), alongside the analytic predictions of Eq. (5), with the frequency detuning

Fig. 5 .
Fig. 5. (a) -Experimentally estimated BFS values (left axis), and residual difference error from the correct value (right axis), as a function of the frequency difference ν between pump and probe waves.(b) -Histogram (bars, left axis) and cumulative probability (solid line, right axis) of the absolute value of the BFS measurement error ε [21].
calculated for each gain event, and the frequency detuning ν Δ between ν and the BFS

Fig. 6 .
Fig. 6.Solid lines: measured output probe waves as a function of time, in phase-coded B-OCDA experiments.Two amplification events are observed, corresponding to SBS interactions in two correlation peaks.The detuning ν Δ between the frequency of the stimulated acoustic field and the BFS was 0, 20 MHz and 50 MHz (see legend).Dashed, solid lines show the simulated output probe waves, calculated using best-fitted values of ν Δ .

Figure 7 (
b) shows an example of the Brillouin gain as a function of z and time ' t , with ν set to 10.733 GHz.The temporal gain measured at the location of the hot spot goes through a local maximum, as anticipated (see section 2).

Fig. 7 .
Fig. 7. (a) -measured normalized steady-state Brillouin gain as a function of correlation peak position z and frequency offset ν between pump and probe, in a phase-coded B-OCDA experiment over 360 mm of fiber.A 4 cm-wide segment at z = 160 mm was locally heated.(b) -measured normalized transient Brillouin gain of the same fiber, as a function of correlation peak position z and time.The frequency offset was set to ν = 10.733GHz.The ability to estimate the local BFS using the temporal transient B-OCDA protocol proposed in this work was tested using 50 values of ν between 10.7 GHz and 10.8 GHz in 2 MHz increments.The B-OCDA phase modulation was set to sequentially address each of the nine resolution cells.A pair of traces was used in each BFS estimation attempt, with a separation of 50 MHz between ν values, in order to remove sign ambiguity as discussed in section 2. Altogether, 450 traces of the signal wave were analyzed.Figure 8 (solid blue curve) shows an example of the measured BFS as a function of position, obtained using the transient traces recorded with ν = 10.719GHz and 10.769 GHz.The BFS estimates obtained using steady state B-OCDA measurements of the local BGS are shown in the solid red curve, for comparison.The dashed solid curve shows the average value of the BFS estimates obtained using the transient analysis of all 25 different pairs of transient traces.

Fig. 8 .
Fig. 8. Measured BFS as a function of position.A local hot-spot was introduced between 12 and 16 cm.Blue, solid curve: estimates based on temporal transient B-OCDA with a single pair of arbitrary frequency offset values ν between pump and probe: 10.719 GHz and 10.769 GHz.Blue error bars show the uncertainty in the transient B-OCDA BFS measurement in each resolution cell (see below).Black, dashed curve: average of the estimates obtained using temporal transient B-OCDA with 25 different pairs of frequencies.Red, solid curve: measurements of a control experiment based on B-OCDA of the local Brillouin gain spectra at steady state.

Fig. 9 .
Fig. 9. Estimates of the BFS in resolution cells 2, 4, 6 and 8, obtained using temporal transient B-OCDA, as a function of the frequency offset ν between pump and probe waves.Resolution cell 4 was in overlap with the hot-spot.Horizontal black lines represent the BFS as measured by spectral scanning B-OCDA at steady state (control experiment).

Fig. 10 .
Fig. 10.Histogram (bars, left axis) and cumulative probability (solid line, right axis) of the BFS measurement error (absolute value), calculated using 225 pairs of traces.

Fig. 11 .
Fig. 11.Experimental errors in the measurements the local BFS using temporal-transient B-OCDA.(a) -mean absolute value of the measurement error as a function of resolution cell number, averaged over all 25 frequency pairs.(b) -mean absolute value of the measurement error as a function of the frequency offset ν between pump and signal, averaged over all nine resolution cells.The overall mean of the absolute value of the BFS measurement error was 2.9 MHz.The standard deviation of the measurement error was 2.4 MHz.