## Abstract

It has been recently shown that in stimulated Brillouin amplification (pulsed pump & CW probe) the line-shape of the normalized logarithmic Brillouin Gain Spectrum (BGS) broadens with increasing gain. Most pronounced for short pump pulses, a linewidth increase of ~3 MHz (~1.5 MHz) per dB of additional gain was observed for a pump pulse width of 15 ns (30 ns), respectively. This gain-dependency of the shape of the BGS compromises the accuracy of the otherwise attractive, highly dynamic and distributed slope-assisted BOTDA techniques, where measurand-induced gain variations of a single probe, are converted to strain/temperature values through a calibration factor that depends on the line-shape of the BGS. A previously developed technique with built-in compensation for Brillouin gain variations, namely: the Ratio Double Slope-Assisted BOTDA (RDSA-BOTDA) method, where both slopes of the BGS are interrogated, fails to meet this new challenge of the gain-induced shape dependence of the BGS, resulting, for instance, in significant measurement errors of ~5% per dB of gain change for a 15 ns pump pulse. Here, we propose and demonstrate an extension of the RDSA-BOTDA method, which now offers immunity also to Brillouin gain-dependent line-shape variations. Requiring a prior characterization of the gain-induced line-shape dependency of the fiber and pump-pulse-width in use, this mitigation technique takes advantage of the fact that the sum of the measured logarithmic gains at judiciously chosen two fixed frequency points of the BGS can be used to determine the local peak gain, via a pre-established calibration curve. Based on the deduced correct peak gain, its associated BGS shape can now be used in the application of the previously introduced RDSA-BOTDA technique to obtain error-free results, independent of the gain dependence of the line-shape. The proposed technique has been successfully put to test in an experiment, involving a RDSA-BOTDA measurement of a fiber segment, vibrating at 50 Hz with a constant, peak-to-peak amplitude of 640 microstrain. As the Brillouin gain was manually varied from 1 to 3.5 dB, classical data processing, based on a single gain value, predicted amplitudes which varied by as much as 90 microstrain, while the proposed mitigation technique produced the correct constant amplitude, regardless of the gain changes. This restored accuracy of the RDSA-BOTDA technique is of importance, especially for monitoring real-world dynamic scenarios, where high Brillouin gains, which often locally vary due to dynamically introduced losses, can successfully achieve fast gain-independent double-slope-assisted Brillouin measurements (many kHz’s of sampling rates over hundreds of meters), with enhanced spatial resolution and signal to noise ratio.

© 2017 Optical Society of America

## 1. Introduction

The Brillouin amplification of a CW probe wave by a CW pump wave in optical fibers is characterized by a logarithmic gain spectrum (BGS) of Lorentzian-shape, centered around a specific value of the frequency difference between the two interacting waves, *ν _{B}* =

*ν*-

_{Pump}*ν*[1, 2].

_{Probe}*ν*, the so-called Brillouin Frequency Shift (BFS) (~11 GHz for standard single-mode optical fibers around 1550 nm) is a function of the local strain and temperature, and its accurate measurement is facilitated by the rather narrow linewidth (Full Width Half Maximum - FWHM) of the BGS: Δ

_{B}*ν*= 1/(2

_{B,CW}*πτ*)~30 MHz, where

_{A}*τ*is the acoustic phonon lifetime [3]. When the pump wave is pulsed, as in the well-known Brillouin Optical Time Domain Analysis (BOTDA) sensing scheme [1], the linewidth Δ

_{A}*ν*broadens from its CW value in proportion to the spectral contents of the pulse (

_{B}*e.g*. for a 15 ns wide pulse the FWHM is ~70 MHz). In the CW-CW case, theory and experiments clearly indicate that in the absence of depletion and non-linear effects both the shape and width of the logarithmic Brillouin gain spectrum are independent of the pump power, or equivalently, of the Brillouin gain [2]. Recently, however [4], it has been observed (and later independently corroborated [5]) that for the more common case of a

*pulsed*pump, both the linewidth and shape of the BGS are influenced not only by the width of the pump pulse, $T$, but also by the gain it provides, Fig. 1. According to [4], the gain-dependent BGS linewidth increase is approximately given by:

*T*. Thus, for example, Eq. (1) and [4], for a 15 ns (30 ns) pump pulse, a 1dB increase in the Brillouin logarithmic gain leads to a ~3 MHz (~1.5 MHz) increase of the BGS FWHM. This Gain-Dependent Line Shape (GDLS) effect is quite noticeable when high Brillouin gains of a few dB’s are used to avoid the need for averaging in dynamic short range scenarios (≤1 km), which are less prone to optical nonlinear effects [1].

Brillouin sensing techniques, based on frequency scanning of the full BGS, such as in the classical BOTDA, as well as in faster Brillouin sensing methods [8–10], should not suffer from this broadening since they are looking for the peak of the BGS, whose location is independent of the GDLS. However, the accuracy of the much faster, vibration-measurement-oriented slope-assisted techniques [6, 7, 11–13], critically depends on a calibration factor derived from the slope of the BGS, making them sensitive to this GDLS effect, which changes the shape of the BGS [14].

Slope-Assisted BOTDA [6] (SA-BOTDA) is a modification of the classical BOTDA technique, where a fixed (rather than swept) frequency pump pulse propagates against a CW probe such that their frequency difference sits close to or at the middle (−3dB point) of either slope of the BGS. For sensing fibers, where the average BFS varies along their lengths, a variable optical frequency probe is used [6], whose time dependent optical frequency is tailored such that wherever the probe meets the pump pulse, their frequency difference again sits at the middle of the local BGS slope. Since practically no time is spent on frequency sweeping of either probe or pump, these slope-assisted methods offer extremely high sampling speeds, fundamentally limited only by the time-of-flight through the sensing fiber.

In SA-BOTDA, a measurand-induced shift of the BFS by δ*ν _{B}* induces a change in the probe logarithmic Brillouin gain, δ

*G*, which can then be used to measure δ

*ν*by dividing δ

_{B}*G*by the slope of the BGS at the probing point (a known δ

*ν*is then converted to strain and temperature by fiber-dependent conversion factors of ~20 με/MHz and ~1

_{B}^{0}/MHz, respectively. με stands for microstrain). Clearly, even in the absence of the GDLS effect, the slope at the probing frequency linearly depends on the pump pulse power,

*i.e.*, on the Brillouin gain [11]. This makes the accurate determination of the gain-to-frequency conversion factor quite difficult, since the value of the pump pulse power is not always known at all time and for every point along the fiber.

This linear dependence of the slope on pump power has been eliminated by the Double SA-BOTDA (DSA-BOTDA) methods of [7, 11], using readings taken on both slopes of the BGS, while still employing the same hardware as in the distributed SA-BOTDA technique. Two post processing approaches for the collected DSA-BOTDA data were demonstrated, one based on the sum-and-difference [7] of these two readings and the other, called the Ratio Double Slope Assisted BOTDA (RDSA-BOTDA) technique, on their ratio [11], offering a wider measurand dynamic range of the order of the BGS linewidth.

The RDSA-BOTDA method requires Ratio Curves [11], defined by the ratio of the actual gains at two reference frequencies, preferably sitting near the middle of the rising/falling slopes of the BGS, respectively denoted by ${\nu}_{-3dB}^{-}$and ${\nu}_{-3dB}^{+}$(for example, see Fig. 1 for ${\nu}_{-3dB}^{-}$ of the normalized BGS corresponding to a peak gain of 3.55 dB). If ${\nu}_{avg}(z)$ is the average BFS at distance $z$ and $\delta {\nu}_{B}(t,z)$ represents the measurand-induced variations around${\nu}_{avg}(z)$, the measured ratio is given by:

*shape*of the

*BGS*, see Fig. 1. Indeed, for a given choice of reference frequencies (say the −3dB frequencies of the

*BGS*of peak gain of 3.55 dB), BGSs of different peak gains are associated with ratio curves of

_{n}*different*slopes, Fig. 2.

Thus, for example, when ${R}_{B}$ is measured to be 5 dB, its conversion to *a frequency deviation* is not unique but rather depends on the actual peak gain! Lack of knowledge of the actual gain may, therefore, result in an erroneous determination of the measurand-induced frequency shift, and consequently in an erroneous estimation of the strain. Based on pump-pulse-power-induced linewidth broadening data for pulse widths of 45 ns, 30 ns and 15 ns [4] the maximum percentage errors in strain estimation due to using a ratio curve belonging to a wrong actual gain are ~3%, 4% and 5%, respectively.

Finally, eliminating the linear dependence of the slope on pump power can be alternatively achieved (to first order and at the expense of additional hardware) by working on the slope of the *phase* of the BGS [13]. Unfortunately, this method too suffers from the GDLS effect, although to somewhat lesser degree [5].

The GDLS effect cannot be dealt with by simply keeping the launched pump power constant. Micro- and macro-bending in the sensing matrix and/or in the down-lead fibers are responsible for dynamically and spatially varying large optical power losses (up to a few dB’s), which are often encountered in real-world moving structures, where strain distribution is monitored by embedded/attached fiber-optic sensing networks, most often in composite materials involving carbon fibers and epoxies. In Structural Health Monitoring (SHM) of a fiber-optic-load-monitored airplane wing or tail [15], for example, the actual wing/tail motion has been observed to introduce optical losses, which in a Brillouin sensing scenario would dynamically modify the local Brillouin gain in a spatially and temporally non-uniform manner. This unknown (and often dynamically varying) spatial distribution of the optical losses makes it difficult to fight the GDLS effect by just monitoring the varying optical power at the fiber output end.

It is, therefore, the goal of this paper to propose and demonstrate a novel mitigation technique, which identifies the correct local gain, and consequently the correct shape of the normalized BGS, from which, methods, such as the RDSA-BOTDA method of [11], can overcome the GDLS effect, thereby providing gain-independent correct results. This mitigation technique is based on a prior characterization of the dependency of the BGS line-shape on the Brillouin gain for the fiber under test and for the chosen pump pulse width. It also takes advantage of the fact that for measurand vibrations lying within the dynamic range of the RDSA-BOTDA method, the *sum* of the measured logarithmic gains at two judiciously chosen fixed frequency points on the BGS (*e.g.*, the two −3dB frequency points of the BGS at its vibration-averaged position) can be used to determine the local, possibly varying peak gain, via a pre-established calibration curve. Once the peak gain is found, the previously obtained database of gain-dependent BGSs is searched for the correct BGS shape, together with the correct Ratio Curve of the RDSA-BOTDA method [11], which relates the ratio of the gains at the two chosen frequencies to the actual shift of the BFS. This gain-independent approach is tested on a 3 m segment of the fiber under test (FUT), which is vibrated at 50Hz with a constant peak-to-peak amplitude of 640 με, and interrogated by a 15 ns pump pulse at an effective sampling rate of 1.2 kHz, under Brillouin gain variations of 1-3.5 dB. It is shown that the proposed mitigation algorithm removes the otherwise observed errors of ~5% per dB (90 με for the 2.55 dB of gain change in our case), without compromising the measurement time.

The principle of the mitigating algorithm is presented in Sec. 2, while Sec. 3 describes the experimental setup used for this study. The same section also demonstrates the successful application of the algorithm. A discussion and concluding remarks appear in Sec. 4.

## 2. A method to determine the true local Brillouin gain, thereby mitigating the GDLS effect in the application of the dynamic RDSA-BOTDA method

The proposed mitigation method comprises three steps. In the first one, Sec. 2.1, BGSs of the fiber, corresponding to a few peak Brillouin gains, are measured for the pump pulse width of choice, producing a database of curves, as in Fig. 1. Next, as in the double slope-assisted BOTDA methods, two monitoring (reference) frequencies are chosen, according to criteria described in Sec. 2.2. The last step, Sec. 2.3, describes an algorithm, which determines the *actual peak gain* at the measurement point from the sum of the measured gains at the two reference frequencies. Then, the BGS associated with that estimated peak gain is pulled (either directly or through interpolation) from the previously acquired database of Brillouin gain spectra, for which the correct Ratio Curve is easily obtained. The use of the correct shape of the BGS at the time of measurement renders the RDSA-BOTDA measurement independent of the GDLS effect, as will be demonstrated below in Sec. 3.

#### 2.1 Mapping the BGS as function of peak gain

We assume a vibration measurement scenario, where the measurand induces oscillations of the BFS around an average value, ${\nu}_{avg}(z)$, at range $z$. As in both previously published DSA-BOTDA methods [7, 11], a preliminary Brillouin mapping of the fiber around ${\nu}_{avg}(z)$ is performed, using the pump-pulse-width of choice. For our proposed algorithm to succeed, this characterization is not limited to one value of the Brillouin gain at range $z$ but must be repeated for different gains (*i.e.*, for different pump powers), spanning the range of the expected gain variations. For each tested peak gain, ${G}_{Peak}(z)$, this procedure provides the spectral shape of the local logarithmic BGS as a function of the peak gain,$BGS\text{\hspace{0.17em}}\text{\hspace{0.17em}}({G}_{Peak}(z),\nu -{\nu}_{avg}(z))$, thereby forming a database of BGS curves to be later used in Sec. 2.3. Figure 1 shows a family of such experimentally obtained curves, presented in a normalized form:$BG{S}_{n}({G}_{Peak}(z),\nu -{\nu}_{avg}(z))=$$BGS\text{\hspace{0.17em}}\text{\hspace{0.17em}}({G}_{Peak}(z),\nu -{\nu}_{avg}(z))/{G}_{Peak}(z)$, for a pump pulse width of 15 ns.

#### 2.2 Choosing the two monitoring frequencies

First, an initial gain value, ${G}_{Peak}^{0}(z)$, is chosen, preferably in the middle of the expected range of gain changes. Then, in order to provide an optimal dynamic range for the RDSA-BOTDA method, the two monitoring frequencies are chosen at the mid-slopes of the BGS, which corresponds to ${G}_{Peak}^{0}(z)$. These frequencies, denoted by ${\nu}_{-3dB}^{\pm}(z,{G}_{Peak}^{0}(z))$, obey: $BG{S}_{n}({G}_{Peak}^{0}(z),{\nu}_{-3dB}^{\pm}({G}_{Peak}^{0}(z)))=0.5$(the ± superscripts refer, respectively, to points on the rising (-) and falling ( + ) slopes of the BGS *vs*.$\nu $ curve). Note that in view of the GDLS effect, gain changes at $z$ will change the shape of the local BGS and the pre-assigned ${\nu}_{-3dB}^{\pm}(z,{G}_{Peak}^{0}(z))$values will sample the BGS *away* from the −3 dB points of the BGS, which now corresponds to the new local gain, see Fig. 1. Consequently, the wrong Ratio Curves will be used, giving rise to errors in the estimation of the measurand [14].

In general, the BFS, *i.e.*,${\nu}_{avg}$, may vary along the fiber, making the chosen reference frequencies range-dependent. Once previously characterized, this dependence can be easily handled by the tailored probe approach of [6,11]. Here, a tailored, *variable* optical frequency probe wave is used, where the time evolution of its frequency is designed so that when the probe wave meets the counter-propagating fixed-frequency pump pulse at location $z$along the fiber, the optical frequency difference between these two waves is equal to the mid-slope frequency at location $z$, namely: at ${\nu}_{-3dB}^{-}(z,{G}_{Peak}^{0}(z))$when the rising slope is interrogated, and at ${\nu}_{-3dB}^{+}(z,{G}_{Peak}^{0}(z))$for the falling slope. The application of this technique to the RDSA-BOTDA method has been demonstrated in [11]. While this tailored probe approach can easily handle the *known*, *previously characterized*, $z$-dependence of ${\nu}_{-3dB}^{\pm}(z,{G}_{Peak}^{0}(z))$, it fails to follow variations due to the GDLS effect.

#### 2.3 Establishing the relation between the sum of probes readings and the local peak gain

We now move on to use the available gain measurements at ${\nu}_{-3dB}^{\pm}(z,{G}_{Peak}^{0}(z))$ to deduce the real gain, ${G}_{Peak}(z)$, at $z$. Once ${G}_{Peak}(z)$is found, hopefully lying within the range of gains used in the original mapping of Sec. 2.1, one can use the previously established database of BGS curves to find the BGS associated with the obtained ${G}_{Peak}(z)$, from which the correct shape at the chosen reference frequencies can be determined.

Our key claim is that for any actual gain value ${G}_{Peak}(z)$, the sum of the probe gains at the two reference frequencies, ${\nu}_{-3dB}^{\pm}(z,{G}_{Peak}^{0}(z))$:

Returning to Eq. (5), we note that it establishes a one-to-one mathematical connection between the measured ${G}_{SUM}(z)$ and the unknown actual peak gain, ${G}_{Peak}(z)$. This mathematical connection can be numerically solved using the previously obtained database of Brillouin BGSs,$\{{G}_{Peak}(z)\cdot BG{S}_{n}({G}_{Peak}(z),\nu -{\nu}_{avg})\}$. This procedure establishes a set of calibration curves that relate the measurable ${G}_{SUM}(z)$ with the sought-for ${G}_{Peak}(z)$ for different choices of the reference frequencies, ${\nu}_{-3dB}^{\pm}(z)$, whose values were dictated by the choice of ${G}_{Peak}^{0}(z)$, Sec. 2.1. Calculated from the BGSs of Fig. 1, such calibration curves are presented in Fig. 3, each drawn for a different choice of the reference gain, ${G}_{Peak}^{0}(z)$, from which the reference frequencies, ${\nu}_{-3dB}^{\pm}(z,{G}_{Peak}^{0}(z))$were derived.

Assume, for example, that the strain distribution along an oscillating structure is to be measured using an attached or embedded optical fiber using RDSA-BOTDA. Assume further, that in order to facilitate fast operation (no averaging), the Brillouin sensor was designed to have a high gain of 3.55 dB at range $z$for a pump pulse width of 15 ns. In view of possible lower achievable gains at $z$, due, for example, to bend-induced losses in the FUT, gain-dependent BGSs were mapped not only for the peak gain of 3.55 dB but also for a few other gains (1-3.55 dB), as in Fig. 1. Naturally, we chose to perform the RDSA-BOTDA measurements with ${\nu}_{-3dB}^{\pm}(z,{G}_{Peak}^{0}(z)=3.55\text{dB})$, which are the −3 dB frequencies associated with the peak gain of ${G}_{Peak}^{0}(z)=3.55\text{dB}$. Note, though, that this choice of ${G}_{Peak}^{0}(z)$ is not necessarily the actual pump-power-dependent gain,${G}_{Peak}(z)$ to be met in the operation of the sensor, which might be lower or potentially higher. Suppose now that the experimentally obtained value of ${G}_{Measured\_SUM}(z,{\nu}_{-3dB}^{\pm}(z,{G}_{Peak}^{0}(z),{G}_{Peak}(z))$ at point $z$ was 2dB. Using the top (purple) curve of Fig. 3, drawn for${G}_{Peak}^{0}(z)=3.55\text{dB}$, we find that the actual peak gain, ${G}_{Peak}(z)$, was not 3.55 dB but rather 2.25dB. Searching our previously obtained database for the BGS corresponding to peak gain of ${G}_{Peak}(z)$ = 2.25 dB, we either find it directly or can get its shape by interpolating over close-by available curves.

Once the true BGS is identified, its corresponding Ratio Curve for the already-assigned values of the two reference frequencies ${\nu}_{-3dB}^{\pm}$, can be calculated from its known shape, using the following more detailed version of Eq. (2), namely:

## 3. The experimental setup and results

The application of the proposed mitigation strategy of Sec. 2 to the RDSA-BOTDA dynamic method of [11] was experimentally demonstrated on a vibrating 3m of a stretched fiber segment (FUT), located at the end of a 50 m SMF-28 single-mode fiber, using the setup of Fig. 4 (see [4] for a more detailed description of the setup). The pump power, and consequently, the Brillouin gain, were controlled by the setting of polarization controller, PC2, which was followed by a polarizer.

Following the recipe of Sec. 2.1, the execution of the RDSA-BOTDA dynamic sensing method, was preceded by the measurement of BGSs of the FUT for 28 gain values, spanning the range of 1-3.55 dB, using a pump pulse width of 15 ns (the few Watts of peak pump power required for these high measured gain values together with the 10 milliwatts of probe power ensured non-depletion operation).

The Brillouin interaction was maximized by adjusting the state of polarization of the probe, using a polarization controller, PC1, (for a non-switching, polarization-independent measurement method see [9]. The frequency granularity was 1 MHz, 1024 averages were used, and the BFS of the 3 meters fiber section was found to be uniform with ${\nu}_{B}$ = 10.909 GHz.

For the sake of visual clarity, only four (normalized) BGS traces are shown in Fig. 1, along with their corresponding peak gain (${G}_{Peak}^{0}$) values, while the gain calibration curves of Fig. 3, relating the actual peak gain to the measured, were derived using all 28 traces. Using the data of Fig. 1 and Eq. (6), Ratio Curves were calculated for all measured BGSs, but Fig. 2 displays only those corresponding to the BGSs of Fig. 1.

Following these preliminary steps, measurements at ${\nu}_{-3dB}^{\pm}(z,{G}_{Peak}^{0}(z))$ can now be performed, leading to a proper estimation of ${G}_{Peak}$ (from Eq. (5) and Fig. 3) and the choice of the right ratio curve (from Fig. 2) for the conversion of measured ratio values to strain ones.

Moving on to dynamic measurements using the RDSA-BOTDA method, the probe frequency was periodically switched between the two reference frequencies of the highest gain, namely: ${\nu}_{-3dB}^{\pm}({G}_{Peak}^{0}=3.55\text{\hspace{0.17em}}\text{dB})$. The FUT was then vibrated at 50 Hz by a mechanical shaker, with a *constant-amplitude* of 320 με, independently measured by the Fast-BOTDA method of [8] (The resulting peak-to-peak of 640 με occupy 50% of the available dynamic range for the 15 ns pulse width [11]. Higher values could not be tested due to equipment limitations). Measurements were performed at a pump repetition rate of 614.4 kHz, and when accounting for the switching between the two reference frequencies and averaging, the effective sampling rate was 1.2 kHz. Since, all computational steps necessary for the correct application of the described mitigation technique are performed during data post-processing, the maximum achievable sampling rate is not compromised.

During the experiment, the pump-controlled Brillouin peak gain, ${G}_{Peak}$, was gradually raised from 1 to 3.55dB, and Fig. 5 shows the two raw SA-BOTDA data series, measured at the two probing frequencies, ${\nu}_{-3dB}^{\pm}({G}_{Peak}^{0}=3.55\text{\hspace{0.17em}}\text{dB})$.

As expected, the observed oscillating gains significantly depend on the pump power and the readings at the two opposite-values slopes are 180° out of phase. Their average values are slightly different, indicating that the BFS has shifted a bit so that two reference frequencies,$\{{\nu}_{-3dB}^{\pm}({G}_{Peak}^{0})\}$, no longer sit on precisely equal gain values. ${G}_{SUM}$, of Eq. (4), shown in Fig. 5, appears quite thin, *i.e.*, largely vibration-independent, as the contributions of the oscillating gains from the two slopes have almost completely cancelled. While the experimentally obtained ${G}_{SUM}$ is not completely free of measurand-induced temporal variations, its *rms* value for all gain settings is much smaller than its mean, leading to negligible errors in the determination of ${G}_{Peak}$(especially in view of the fairly shallow slopes of the calibration curves of Fig. 2).

Figure 6(a) is the result of the removal of the linear pump-power dependency of the measured data using the original algorithm of [11], which is based on the Ratio Curve corresponding to the BGS shape associated with the gain value of 3.55dB, for which the reference frequencies, ${\nu}_{-3dB}^{\pm}({G}_{Peak}^{0})$, were chosen. Indeed, when compared to Fig. 5, significant compensation is achieved, but only for the linear dependence of the measured values on the pump power.

However, a non-negligible residual dependence, amounting to an error of 90 με (13%), still remains: The peak-to-peak strain readings at the lowest gain are 730 με instead of the setting value of 640 με. This error is due to the GDLS effect, namely: to the wrong use of the Ratio Curve derived from the shape of the BGS of the 3.55 dB peak gain for the *other* peak gain values encountered in the experiment, whose associated Ratio Curves are different, Fig. 4. Looking back at Fig. 6(a) we note that there is a consistent trend of incorrectly higher predicted estimates for lower gains.

This trend nicely correlates with the observation, Fig. 4, that lower Brillouin peak gains are associated with steeper Ratio Curves. Thus, a given frequency shift will generate a higher ratio value for an *actual* peak gain of 1 dB than for an *actual* gain of 3.55 dB. However, if this higher measured ratio is erroneously converted to a frequency shift by a Ratio Curve belonging to ${G}_{Peak}^{0}=3.55\text{\hspace{0.17em}}\text{dB}$, rather than by the one belonging to ${G}_{Peak}^{0}=1\text{\hspace{0.17em}}\text{dB}$, an estimate higher than the correct one will result.

We now apply our mitigation algorithm of Sec. 2.3. This algorithm allows us at every measurement instant to use the 3.55 dB calibration curve of Fig. 2 to estimate the actual${G}_{Peak}$obtained from the measured ${G}_{SUM}$. For the ratio method, knowing ${G}_{Peak}$is sufficient to pick the right shape of the BGS, from which, the correct Ratio Curve is used, resulting in constant amplitude readings, as shown in Fig. 6(b), where the high strain error of 90 με is fully compensated.

## 4. Discussion and conclusions

An improvement of the Ratio-based Double-Slope Assisted BOTDA method of [11] was described that can mitigate the significant error-inflicting effect (~5% per 1 dB gain variation for 15 ns pump pulse) of the gain-dependence of the Brillouin line-shape. This proposed and demonstrated approach does not require additional hardware. It is based on the determination of the true value of the Brillouin gain using the sum of the logarithmic gains, measured at the two mid-slopes of a BGS, having an arbitrary but known peak gain. Preferably, a peak gain is to be selected that lies in the middle of the expected range of gain variations. Choosing the interrogating probe frequencies to reside close to the mid-slope (−3 dB) frequency points of the BGS corresponding to *ν _{avg}*, although not mandatory, should provide the widest dynamic range.

*Dynamic range considerations:* The actual value of the dynamic range is determined by the requirements underlying both the standard RDSA-BOTDA method and the proposed mitigation technique. As the BFS varies with respect to *ν _{avg}*, the accurate determination of${R}_{B}$ critically depends on the signal-to-noise ratio of the lowest gain value involved in its calculation, Eq. (6), as well as on the slope of the${R}_{B}.vs.\delta {\nu}_{B}$curve. Averaging can improve the signal-to-noise ratio, albeit at the expense of measurement speed. It appears, therefore, that subject to proper averaging, these considerations lead the RDSA-BOTDA to have a dynamic range of the order of the Brillouin linewidth [11]. Key to the mitigation technique is the assumption that the sum of the logarithmic gains at the two reference frequencies provides a reliable estimate of the correct peak gain. Analysis, based on our experimental data, indicates that while this assumption slowly starts to lose its validity as $\left|\delta {\nu}_{B}\right|$ increases, the resulting inaccuracy in the determination of the actual gain is below 0.2 dB, as long as $\delta {\nu}_{B}$ lies within a dynamic range comprising 85% of the relevant Brillouin linewidth. This 0.2 dB deviation translates into a loss of 1% in measurand accuracy.

While the dynamic range of slope-assisted Brillouin methods is significantly narrower than that of classical BOTDA implementations, these dynamic techniques may nevertheless find applications in many vibrating scenarios where they are the only available way to provide very fast distributed and quantitative measurements of practically encountered vibrations/fluctuations over many tens of meters, see [16]. Besides, this dynamic range can be readily extended by employing narrower pump pulse widths, which are associated with broader Brillouin gain spectra [17], obviously, at the expense of lower signal-to-noise ratios.

*Non-local effects*: Like in other BOTDA schemes, non-local effects may compromise the accuracy of all Slope-Assisted Brillouin sensing techniques. Since RDSA-BOTDA primarily addresses dynamic scenarios over short measurement ranges, it uses high power pump pulses and much weaker probe waves. Thus, pump depletion [1], is not of concern. BFS variations faster than the available spatial resolution introduce errors to all distributed sensing techniques, ours included. Here, the only viable solution, wherever possible, is to increase the spatial resolution until no change in fiber's BFS signature is observed. The finite extinction ratio of the pump pulse [1], while also an issue for classical peak-searching, frequency-scanning Brillouin BOTDA implementations, appears, however, to be more of a problem for Slope-Assisted Brillouin techniques. An extinction ratio of the order of the ratio between the total fiber length and the spatial resolution [18], may not be sufficiently high enough. Our analysis indicates that for a pump pulse width of 15 ns ( = 1.5 m of spatial resolution), and a fiber length of 100 m, where the 98.5 meters have a BFS different from the *ν _{avg}* of the 1.5 m long resolution cell, worst case scenario requires an extinction ratio of 50 dB to ensure errors below 1% for $\delta {\nu}_{B}$ deviations within the dynamic range of the previous paragraph. Indeed, our setup of Fig. 4 employs a semiconductor-based optical shutter/switch, which meets the required specifications. The effect of the finite extinction ratio of the pump pulse former on the performance of slope-assisted Brillouin sensing methods, as a function of their different relevant parameters, is under current investigation.

The proposed technique has been successfully demonstrated on a fiber segment, vibrating at 50 Hz with a constant, peak-to-peak amplitude of 640 με, well within the dynamic range of a 15 ns pump pulse (~1200 με in the demonstrating setup). Without the mitigation technique and under gain variations of 1-3.55 dB, gain-dependent strain errors as high 90 με were observed. On the other, with the application of the proposed procedure, constant, *i.e.*, gain-independent vibrations are obtained.

In summary, the restored accuracy of the RDAS-BOTDA method is of importance especially for the measurement of fast (kHz's) strain/temperature variations of limited dynamic range over relatively short ranges (hundreds of meters, weak nonlinear effects), where slope-assisted-based Brillouin methods can utilize high Brillouin gains in order to reduce or even eliminate the need for averaging. Finally, this dynamic gain estimation technique may find applications in other Brillouin sensing scenarios.

## Funding

This research was supported by the Israel Science Foundation (grant No. 1380/12).

## References and links

**1. **A. Motil, A. Bergman, and M. Tur, “[INVITED] State of the art of Brillouin fiber-optic distributed sensing,” Opt. Laser Technol. **78**, 1–23 (2015).

**2. **L. Thévenaz, *Advanced Fiber Optics - Concepts and Technology* (EPFL University, 2011).

**3. **R. W. Boyd, *Nonlinear Optics*, 3rd ed. (Academic, 2008).

**4. **A. Motil, R. Hadar, I. Sovran, and M. Tur, “Gain dependence of the linewidth of Brillouin amplification in optical fibers,” Opt. Express **22**(22), 27535–27541 (2014). [CrossRef] [PubMed]

**5. **J. Marinelarena, J. Urricelqui, and A. Loayssa, “Gain dependence of the phase-shift spectra measured in coherent Brillouin optical time-domain analysis sensors,” J. Lightwave Technol. **34**(17), 3972–3980 (2016). [CrossRef]

**6. **Y. Peled, A. Motil, L. Yaron, and M. Tur, “Slope-assisted fast distributed sensing in optical fibers with arbitrary Brillouin profile,” Opt. Express **19**(21), 19845–19854 (2011). [CrossRef] [PubMed]

**7. **A. Motil, O. Danon, Y. Peled, and M. Tur, “Fast pump-power-independent brillouin fiber optic sensor,” in Optical Fiber Communication Conference (OFC, 2014), pp. 1–3.

**8. **Y. Peled, A. Motil, and M. Tur, “Fast Brillouin optical time domain analysis for dynamic sensing,” Opt. Express **20**(8), 8584–8591 (2012). [CrossRef] [PubMed]

**9. **I. Sovran, A. Motil, and M. Tur, “Frequency-scanning BOTDA with ultimately fast acquisition speed,” IEEE Photonics Technol. Lett. **27**(13), 1426–1429 (2015). [CrossRef]

**10. **A. Voskoboinik, O. F. Yilmaz, A. W. Willner, and M. Tur, “Sweep-free distributed Brillouin time-domain analyzer (SF-BOTDA),” Opt. Express **19**(26), B842–B847 (2011). [CrossRef] [PubMed]

**11. **A. Motil, O. Danon, Y. Peled, and M. Tur, “Pump-power-independent double slope-assisted distributed and fast brillouin fiber-optic sensor,” IEEE Photonics Technol. Lett. **26**(8), 797–800 (2014). [CrossRef]

**12. **R. Bernini, A. Minardo, and L. Zeni, “Dynamic strain measurement in optical fibers by stimulated Brillouin scattering,” Opt. Lett. **34**(17), 2613–2615 (2009). [CrossRef] [PubMed]

**13. **J. Urricelqui, A. Zornoza, M. Sagues, and A. Loayssa, “Dynamic BOTDA measurements based on Brillouin phase-shift and RF demodulation,” Opt. Express **20**(24), 26942–26949 (2012). [CrossRef] [PubMed]

**14. **A. Motil, I. Sovran, R. Hadar, and M. Tur, “Ramifications of the gain dependence of the Brillouin linewidth on the shape and slopes of the Brillouin gain spectrum,” in Fifth Asia-Pacific Optical Sensors Conference, (2015).

**15. **I. Kressel, B. Dorfman, Y. Botsev, A. Handelman, J. Balter, A. C. R. Pillai, M. H. Prasad, N. Gupta, A. M. Joseph, R. Sundaram, and M. Tur, “Flight validation of an embedded structural health monitoring system for an unmanned aerial vehicle,” Smart Mater. Struct. **24**(7), 075022 (2015). [CrossRef]

**16. **Y. Peled, A. Motil, I. Kressel, and M. Tur, “Monitoring the propagation of mechanical waves using an optical fiber distributed and dynamic strain sensor based on BOTDA,” Opt. Express **21**(9), 10697–10705 (2013). [CrossRef] [PubMed]

**17. **M. Alem, M. A. Soto, M. Tur, and J. Thévenaz, “Analytical expression and experimental validation of the Brillouin gain spectral broadening at any sensing spatial resolution,” Proc. SPIE **10323**, 103239J (2017). [CrossRef]

**18. **X. Angulo-Vinuesa, S. Martin-Lopez, J. Nuño, P. Corredera, J. D. Ania-Castañon, L. Thévenaz, and M. González-Herráez, “Raman-assisted Brillouin distributed temperature sensor over 100 km featuring 2 m resolution and 1.2 C uncertainty,” J. Lightwave Technol. **30**(8), 1060–1065 (2012). [CrossRef]