1. Introduction

Distributed optical fiber sensors based on stimulated Brillouin scattering (SBS) have been used for structural deformation and strain measurement for civil structural health monitoring [1,2], as they have the unique capability of truly distributed sensing with high spatial resolution. An important figure of merit of a distributed sensor is its spatial resolution. A high spatial resolution makes the sensor capable to detect spatially-localized perturbations across the monitored structure. In applications such as structural health monitoring of aeronautic structures, a spatial resolution in the cm-range is essential to detect small defects such as cracks or delaminations in their early formation. State-of-the-art Brillouin-based commercial instruments provide a spatial resolution of about ten cm or more. The main obstacle to reaching a cm-scale spatial resolution is the broadening of the Brillouin gain spectrum (BGS) and reduction of its peak as the pulse-width is shorter than 10 ns (corresponding to a 1-m spatial resolution), due to convolution effect between the optical pulse spectrum and the Brillouin spectrum of the fiber [3]. On the other hand, it is known that this broadening effect can be partly or totally cancelled out by adding a baseline to the pulse [4,5]. Experimentally, one can control the baseline level by adjusting the bias voltage of the electro-optic modulator used for pulse forming. However, some detrimental effects may result from the use of an intense pulse baseline, i.e. of a pulse with a low extinction ratio (ER), such as distortion of the BGS and pump depletion due to distributed Stokes beam amplification [6,7].

Several attempts have been made to reach a cm-scale spatial resolution without incurring in BGS broadening [8–10]. Among them, one of the most valuable methods involves the use of a differential pulse-width pair (DPP-BOTDA). The latter can be easily implemented by using a convention BOTDA (Brillouin Optical Time-domain analysis) set-up. The idea is quite simple: two BOTDA measurements are carried out in sequence over the same fiber, by choosing two different pulse-widths for the pump. The acquired traces are successively subtracted at each scanned Brillouin frequency, in order to obtain a differential BGS having a spatial resolution much higher than those obtained in each single measurement, provided that the pulse-width difference is lower than the pulse-width of the two pump pulses. High spectral resolution, i.e. a BGS with narrow linewidth has been demonstrated by this approach, as a result of the use of long duration pump pulses [10,11].

In this paper, an extensive numerical analysis of single-pulse (SP) and differential pulse-width pair (DPP) BOTDA is presented, by taking advantage of the model reported in Ref [12]. The model permits to simulate accurately and efficiently the SBS interaction between a cw probe beam and a pulse beam with arbitrary waveform. The effects of the pulse extinction ratio, rise time and fall time are analyzed in detail. It is concluded that the fall time has a larger influence on spatial resolution than rise time, for both conventional and DPP-BOTDA systems. Moreover, the simulations lead to the introduction of a novel concept of spatial resolution, based on the analysis of the SBS signals in the pump-probe frequency shift domain. The results reported in this paper can be usefully employed to perform an optimization of BOTDA systems in terms of pulse characteristics.

2. Background

In Ref. [12]. it is shown that the Brillouin interaction between a pulsed pump wave with a finite cw component and a Stokes continuous wave counter-propagating along a single-mode optical fiber can be efficiently described by a simple integral equation, relating the transmitted Stokes amplitude (in the frequency domain), with the pump characteristics and the fiber conditions [12]:

3. Numerical results

The model described by Eqs. (1) and (2) has been used to simulate both conventional (single pulse) and DPP-BOTDA systems. In the former case, the equations are solved so as to retrieve the Stokes gain signals, while in the DPP case the model is solved twice, for two different values of the pulse-width. The calculated ac gain signals are then subtracted in order to retrieve the differential gain signal.

Numerical tests were performed on a 5m-long fiber with a natural SBS gain width of 35 MHz (corresponding to a 4.5-ns decay time of the acoustic wave) and a Brillouin gain g_B/A_eff = 0.2 (W·m)⁻¹. The pulse peak power was set to 100 mW, while the probe power was set to 1 mW. Each pulse was modeled according to the following equation [4]:

Simulations for the SP-BOTDA have been carried out for a pulse-width ranging from 3 ns to 10 ns, while DPP-BOTDA simulations were done for a pulse-width pair consisting of a first pulse with width fixed to 20 ns, and a second pulse with width ranging from 23 to 30 ns. These pairs represent a good compromise between the requirement of a narrow BGS linewidth (calling for longer pulses) and the necessity, in practical cases, to avoid saturation at the photodetector [14]. Unless otherwise specified, the rise and fall times of the pulses are set to 0.5 ns. For each test case, the Stokes spectrum was calculated for a frequency interval ranging from 0 to 2 GHz, with a step of 2 MHz. Time-domain Stokes signals were then obtained by IFFT of the corresponding signals in the frequency domain.

The first numerical test was performed on a uniform fiber, i.e. by considering a constant Brillouin frequency shift along the fiber. The aim was analyzing the influence of pulse-width and pulse ER on the BGS linewidth. To this aim, the Stokes gain signals were calculated for a detuning ranging from −200 MHz to 200 MHz at a step of 2 MHz. Figure 1 reports the full-width-at-half-maximum (FWHM) linewidth of the calculated BGS, as a function of the pulse-width (or pulse-width difference) and for three different levels of ER (20 dB, 30 dB, 40 dB). The ER level is expressed as the ratio, in dB, between the pulse peak power and the pulse baseline. Note that the BGS at the middle section of the fiber (i.e., at t = 25 ns) was chosen to estimate the linewidth. The results for a conventional BOTDA system (Fig. 1(a)) show that reducing the ER leads to a narrowing of the BGS for a given pulse-width [5]. Actually, the steady-state interaction between the pulse baseline and the Stokes beam produces a stationary acoustic wave acting as pre-trigger for the SBS interaction between the pulse and the Stokes beam. Note that, although an ER lower than 20 dB would permit to narrow further the BGS linewidth, a high pulse baseline produces an amplification of the probe wave all over the fiber that may lead to significant pump depletion. Obviously, this is especially true for long fibers. A low ER has another drawback: in nonuniform fibers the interaction between the pulse baseline and the acoustic wave decaying after pulse passage produces a distorted BGS with a consequent loss of accuracy in Brillouin frequency shift retrieval [5,6]. This nonlocal effect will be treated in more detail in next simulations. Figure 1(b) reports the results relative to a DPP-BOTDA system. In this case, the BGS linewidth keeps nearly constant while varying the pulse-width pair difference or the ER (the same ER is supposed for the pulses forming the pair). In fact, the BGS linewidth is dominated in this case by the duration of the two pulses, which are at least 20 ns-long. For these durations, the influence of the pulse ER is much weaker (see Fig. 1(b)), and the BGS linewidth approaches its natural value (35 MHz).

 

Fig. 1 (a) BGS linewidth calculated for an SP-BOTDA system as a function of the pulse-width and pulse ER; (b) BGS linewidth calculated for a DPP-BOTDA system as a function of the pulse-width difference and pulses ER.

Download Full Size | PDF

It is also interesting to compare the Stokes peak gain in SP and DDP-BOTDA systems. Figure 2 reports the SBS peak gain normalized to the Stokes cw input power. It is seen that in SP systems the gain slightly increases with pulse baseline, while in DPP systems the ER has low influence, its contribution being canceled out by the subtraction underlying the differential BGS computation. Moreover, the relative gain enhancement associated to the differential approach is more pronounced at low spatial resolutions. This result agrees qualitatively with the experimental data reported in Ref. [10]. For example, with ER = 40 dB and a pulse-width (or pulse-width difference) of 3 ns, the calculated gain improvement is ≈7 dB.

 

Fig. 2 (a) BGS gain peak calculated for an SP-BOTDA system as a function of the pulse-width and pulse ER; (b) BGS gain peak calculated for a DPP-BOTDA system as a function of the pulse-width difference and pulses ER.

Download Full Size | PDF

Next numerical tests refer to a nonuniform fiber. The aim was to verify the spatial resolution offered by SP and DPP systems as a function of the pulse characteristics.

In Ref. [9]. the spatial resolution is defined by the average of the rise and fall time equivalent fiber length of the Stokes gain signal for a small stress section, calculated from 10% to 90% (rise time) or 90% to 10% (fall time). We can name it as time-domain spatial resolution, as it is evaluated by analyzing the temporal variations of the transmitted Stokes intensity. In the following, the time-domain spatial resolution is calculated by considering a 300-MHz shift of the Brillouin frequency compared with the rest of the fiber, extending from z = 2 m to z = 4 m. The choice of a large shift (300 MHz) has been done in order to have a negligible Brillouin gain in the perturbed section, when the pump-probe frequency offset is tuned to the resonance of the unperturbed fiber. The simulated traces of the transmitted Stokes intensity, calculated for a resonant interaction in the perturbed section were used to calculate the time-domain spatial resolution. We underline that Stokes signals were calculated over a bandwidth of 2 GHz. In practical cases, a limited bandwidth of the detector may affect the final spatial resolution. Note that our model calculates the signals in the frequency domain, therefore this effect can be easily taken into account by restricting the computation of the Stokes spectrum over the detector bandwidth.

Results are plotted in Fig. 3 . It is seen that, while ER has a weak effect on spatial resolution in DPP systems, the opposite is true in SP systems: a low ER reduces drastically the time-domain spatial resolution, especially for shorter pulses. As the spatial resolution in Fig. 3 is calculated on the basis of the rise and fall times of the time-domain waveforms, it is useful to show the computed temporal variations of the transmitted Stokes intensity, for a given pulse-width or pulse-width difference. Figure 4 compares the numerical results for an SP-BOTDA system with those relative to a DPP-BOTDA system. It is seen that, while in DPP systems the subtraction of waveforms performed to calculate the differential BGS makes the latter almost independent of ER, in conventional BOTDA systems a low ER results in a waveform with long rising and falling trails. This effect can be explained by considering that the acoustic wave produced by pulse passage over the perturbed region continues to modulate the Stokes field when a cw pulse baseline is present [5]. While for high extinction ratio this interaction is negligible, for modest ER it produces an additional Stokes signal whose duration is dominated by the phonon lifetime. Therefore, a low ER has a detrimental effect on time-domain spatial resolution. Observe that the 20-ns time offset in the waveforms shown in Fig. 4(b) is a direct consequence of the use of the differential approach [10].

 

Fig. 3 (a) Time-domain spatial resolution calculated for an SP-BOTDA system as a function of the pulse-width and pulse ER; (b) Time-domain spatial resolution calculated for a DPP-BOTDA system as a function of the pulse-width difference and pulses ER.

Download Full Size | PDF

 

Fig. 4 (a) Time-domain waveforms calculated for an SP-BOTDA system in a nonuniform fiber (pulse-width set to 3 ns); (b) Time-domain waveforms calculated for a DPP-BOTDA system in a nonuniform fiber (pulse-width difference set to 3 ns).

Download Full Size | PDF

While defining spatial resolution on the basis of the time-domain traces appears as a reasonable criterion, it makes sense to wonder if it is really appropriate for a distributed sensor. Actually, a high spatial resolution should be related to the sensor capability to detect spatially localized perturbations.

In BOTDA systems, the time-domain traces acquired for various pump-probe frequency shifts are processed simultaneously, in order to infer the Brillouin frequency shift from each BGS. When the perturbation width is smaller than spatial resolution, a multi-peaked BGS will be acquired at the perturbation, indicating that the various Brillouin frequency shifts are not spatially resolved [15]. In this case, it becomes hard or even impossible to retrieve the actual Brillouin frequency shift. Therefore, the capability of the sensor to detect a narrow perturbation is mostly related to the BGS shape, rather to the temporal characteristics of a single Stokes gain trace. These considerations suggest that a more adequate criterion to define spatial resolution may be adopted. In particular, we introduce a frequency-domain spatial resolution, based on the ratio between the correct BGS peak and the spurious BGS peak due to nonlocal effect, in the simple case in which only two Brillouin frequency shifts are present along the fiber. To give a quantitative value, we define the frequency-domain spatial resolution as the perturbation width such that the spurious peak is 30% of the correct peak in the BGS calculated at the middle of the perturbation. The choice of this percentage is made in order to match time-domain and frequency-domain spatial resolutions at high ER.

Simulations were done by considering a single, 300-MHz perturbation located in the middle of the 5m-long sensor. As for time-domain spatial resolution calculations, a perturbation of 300 MHz results in a negligible Brillouin gain along the perturbation, when the pump-probe frequency offset is tuned to the unperturbed fiber. In other words, even at low ER (and therefore for large BGS bandwidths) the correct and spurious BGS do not overlap in the spectral domain. The perturbation width was varied at each simulation, in the range between 0.18 m and 1 m. At the same time, the pulse-width or pulse-width difference was scanned from 3 ns to 10 ns at a 1-ns step. The ratio of the correct BGS peak (at the perturbed Brillouin frequency shift) to the spurious BGS peak (at the unperturbed Brillouin frequency shift) at the midsection was calculated for each test-case. Results for SP-BOTDA and DPP-BOTDA are plotted in Figs. 5 and 6 , respectively.

 

Fig. 5 Ratio of the correct BGS peak to the spurious BGS peak in an SP-BOTDA system, as a function of the perturbation width and pulse-width. The different curves refer to a pulse-width ranging from 3 ns to 10 ns at 1-ns step. The black dashed line indicates the level chosen for defining the frequency-domain spatial resolution. (a) ER = 40 dB; (b) ER = 30 dB; (c) ER = 20 dB.

Download Full Size | PDF

 

Fig. 6 Ratio of the correct BGS peak to the spurious BGS peak in a DPP-BOTDA system, as a function of the perturbation width and pulse-width. The different curves refer to a pulse-width difference ranging from 3 ns to 10 ns at 1-ns step. The black dashed line indicates the level chosen for defining the frequency-domain spatial resolution. (a) ER = 40 dB; (b) ER = 30 dB; (c) ER = 20 dB.

Download Full Size | PDF

These figures suggest that spatial resolution can be defined only in an approximate way, as the capability to detect a given perturbation degrades gradually, and not abruptly, when reducing the pulse-width. For example, the blue (leftmost) curve in Fig. 5(a), calculated for an SP-BOTDA system with a pulse-width of 3 ns and an ER of 40 dB, indicates that even a perturbation as large as 1 m (hence well above the time-domain spatial resolution of 23 cm) gives rise to a dual-peaked BGS at the middle of the perturbation, with the spurious peak being about 5% of the correct peak. As discussed earlier, such a spurious peak arises from the interaction between the pump leakage and the acoustic wave produced by pulse passage in the region of the sensor extending about one meter at the left of the perturbation. Nonetheless, for any given perturbation width, reducing the pulse-width (or the pulse-width difference) gradually enhances the spectral purity (i.e. reduces the energy of the spurious peak). Another consideration is that the lower the ER is, the lower is the improvement in terms of spectral purity associated to the use of shorter pulses, for a given perturbation width (i.e. for low ER the curves are more closely spaced). From the above data we can extract the frequency-domain spatial resolution for SP and DPP-BOTDA systems (see Fig. 7 ). It is interesting to observe that, although the results for a DPP system shown in Fig. 7(b) suggest that the ER has low influence on spatial resolution, the data presented in Fig. 6 indicate that much higher spectral purity can be obtained with the use of pulses with high ER. Comparing the results in Fig. 7 with the time-domain spatial resolutions shown in Fig. 3, we note that, while for DPP-BOTDA sensors time-domain and frequency-domain resolutions are very close, major differences can be observed in SP-BOTDA simulations, especially for ER = 20 dB. While a large degradation of the time-domain spatial resolution is predicted for a modest ER, the newly defined spatial resolution shows a less severe deterioration.

 

Fig. 7 (a) Frequency-domain spatial resolution calculated for an SP-BOTDA system as a function of pulse-width and ER; (b) Frequency-domain spatial resolution calculated for a DPP-BOTDA system as a function of pulse-width difference and ER.

Download Full Size | PDF

For example, with a pulse-width of 3 ns, the conventional definition produces a spatial resolution of ≈1 m, while the new definition gives a 0.4 m-spatial resolution. It is opportune to analyze this discrepancy in more detail. Let us consider the same simulation used to calculate the data reported in Fig. 4(a), i.e. a 2m-long, 300-MHz perturbation is located along the 5m-long sensor. We plot in Fig. 8 the computed waveforms for an SP-BOTDA system at ER = 20 dB and τ = 3 ns. The blue curve refers to a frequency shift resonant along the perturbation, while the red curve refers to a frequency shift resonant outside the perturbation. The latter extends from t = 20 to t = 40 ns. It is seen that, although both traces are characterized by relatively long rising and falling edges, more than 75% of the overall swing of the signals occur in the first 3 ns, i.e. in a time interval set by the pulse-width. As a consequence, at a temporal distance of 3 ns from the instant at which the pulse enters the perturbation (i.e. at t = 23 ns) the SBS gain at the correct Brillouin frequency shift is more than three times larger than the spurious gain peak. Following this first, faster transient, a slower transient with a time-constant imposed by the phonon lifetime starts up, with the signals evolving towards steady state [13].

 

Fig. 8 Time-domain waveforms calculated for an SP-BOTDA system in a nonuniform fiber. The pulse-width is 3 ns, while the pulse ER is 20 dB. The blue curve refers to a pump-probe offset resonant at perturbation, while red curve refers to a pump-probe frequency offset tuned to the Brillouin resonance outside the perturbation. The inset shows a zoom of the transition indicated by the arrow.

Download Full Size | PDF

Actually, the partitioning of the transient into two intervals is somewhat artificial, as both intervals overlap. However, if the two time-constants (i.e. the pulse-width and the phonon lifetime) are sufficiently distant, this assumption is reasonable. Note that the peak ratios curves reported in Figs. 5 and 6 have been computed under the hypothesis of a large ν_B-shift. As discussed earlier, in case of a smaller shift, the Brillouin gain at the perturbed section may be not negligible even when the pump-probe frequency offset is tuned to the unperturbed fiber, so it will contribute to the spurious peak appearing in the associated BGS. Therefore, in these conditions we may expect a smaller ratio of correct peak to spurious peak, and thus a smaller frequency-domain spatial resolution. As our analysis is focused on the spatial resolution degradation due to interaction between acoustic wave transient produced by pulse passage and pump leakage, the choice of a large shift assures that the contribution to spurious peak due to a finite Brillouin gain is negligible.

As regards the data reported in Fig. 7(b) and related to a DPP-BOTDA system, let us remind that in this case the signals are obtained by subtraction of two traces, each one associated to a pulse-width longer than phonon lifetime. As a consequence, each BOTDA signal is dominated by a time-constant imposed by the pulse-width. Furthermore, the contribution associated to the interaction between the decaying acoustic wave and the pulse leakage, with time-constant equal to the phonon lifetime, is canceled out by subtraction of the BOTDA waveforms. As the response is dominated in this case by a single time-constant, no significant discrepancy is observed between the time-domain and the frequency-domain definitions of spatial resolution.

It is interesting to show the combination of spatial resolution and BGS linewidth simultaneously achievable in an SP or DPP system. Figure 9 reports the two parameters, calculated using the above specified pulse lengths. We see that lowering the ER the performances of SP and DPP systems become comparable. Let us remind, however, that in this picture the gain improvement allowed by DPP, reported in Fig. 2, is not taken into account.

 

Fig. 9 Combination of BGS linewidth and frequency-domain spatial resolution calculated for an SP-BOTDA system (solid lines) or a DPP-BPTDA system (dashed line).

Download Full Size | PDF

We further study the influence of the rise and fall time of the pulse(s) light on spatial resolution. To this aim, we carried out numerical analyses aimed at determining both time-domain and frequency-domain spatial resolutions in SP and DPP systems, while varying either rise or fall time of the pulse.

The pulse-width for SP and the pulse-width difference for DPP simulations were fixed to 5 ns. We report in Fig. 10 the impact of the rise time, evaluated by fixing the fall time to 0.5 ns. It can be noted that while rise time has some impact on SP-BOTDA spatial resolution, it has a negligible influence in DPP-BOTDA systems, at least for rise times not longer than the pulse-width difference. This result may be expected, owing to the differentiation carried out in the latter technique. An interesting result is that, in SP systems the frequency-domain spatial resolution always degrades for longer rise times, whereas the time-domain spatial resolution, calculated for ER = 20 dB, improves by increasing the rise time of the pump pulse. This apparently contradictory result can be explained by considering that a longer rise time has the effect of increasing the overall pulse energy. As in this regime the time-domain spatial resolution comes from a balance between the gain signal produced by the pulse and the gain signal produced by the leakage (the latter contribution being characterized by a longer time-constant), increasing the pulse energy results in a shorter equivalent time-constant.

 

Fig. 10 (a) Time-domain (solid lines) and frequency-domain (dashed lines) spatial resolution calculated for an SP-BOTDA system as a function of the pulse rise time and ER. The pulse-width is set to 5 ns; (b) Time-domain (solid lines) and frequency-domain (dashed lines) spatial resolution calculated for a DPP-BOTDA system as a function of the pulses rise time and ER. The pulse-width difference is set to 5 ns.

Download Full Size | PDF

Finally, Fig. 11 shows the impact of the pump light fall time. Still, we fixed the pulse-width (or the pulse-width difference) to 5 ns, while the rise time was set to 0.5 ns. Interestingly enough, in both SP and DPP configurations the pulse fall time has a higher influence than rise time on spatial resolution. For example, for an SP system with τ = 5 ns and ER = 40 dB, the frequency-domain spatial resolution increases from ≈33 cm to ≈42 cm when increasing the rise time from 0.5 ns to 5 ns, while it increases up to ≈61 cm when varying the fall time in the same range. Analogously, in DPP-BOTDA systems with Δτ = 5 ns and ER = 40 dB, the frequency-domain spatial resolution is practically constant at ≈41 cm when varying the rise time from 0.5 ns to 5 ns, whereas it increases from ≈41 cm to ≈70 cm when the fall time is 5 ns. The asymmetrical behavior of SP and DPP systems with respect to the pulse shape can be explained considering that the acoustic wave is mostly produced by the flat part of the pulse. A long trailing edge can interact with the acoustic wave produced by the preceding flat part, whereas the leading edge only interacts with the weaker acoustic wave produced by the leading edge itself. Hence, in the latter case the prolonging of the Stokes intensity temporal variation is smaller, resulting in a slighter degradation of spatial resolution. We note that, although in Ref [10]. a couple of experiments focused on the impact of the pulse light rise/fall time to the spatial resolution are reported, those experiments did not clarify the different roles played by the leading and trailing edges.

 

Fig. 11 (a) Time-domain (solid lines) and frequency-domain (dashed lines) spatial resolution calculated for an SP-BOTDA system as a function of the pulse fall time and pulse ER. The pulse-width is set to 5 ns; (b) Time-domain (solid lines) and frequency-domain (dashed lines) spatial resolution calculated for a DPP-BOTDA system as a function of the pulses fall time and pulses ER. The pulse-width difference is set to 5 ns.

Download Full Size | PDF

Although the curves in Fig. 10(b) suggest that spatial resolution may be further improved by reducing the fall time below 0.5 ns, it must be observed that a shorter fall time corresponds to higher bandwidth and more noise in the signal. Also, the spatial resolution improvement becomes smaller when the fall time drops below ≈10% of the pulse-width difference.

4. Conclusions

An extensive numerical analysis of conventional (single pulse) and differential pulse-pair BOTDA systems has been reported. The results shows that, different from the commonly accepted definition of spatial resolution based on the rise and fall times of the temporal variations of the transmitted Stokes intensity, a different definition can be given analyzing the signals in the frequency shift domain. This new definition appears to be more appropriate to describe the resolution of conventional BOTDA system under modest extinction ratio condition. Moreover, it has been shown that the pulse fall time has a higher influence on spatial resolution than pulse rise time. This is especially true in DPP-BOTDA systems, whose spatial resolution is virtually independent of pulse rise time, while being significantly sensitive to pulse fall time.