1. Introduction

Stimulated Brillouin scattering (SBS) in optical fibers has attracted considerable attention recently because of its very important application for the sensing of distributed strain and temperature profiles along extended objects [1, 2, 3, 4]. SBS occurs due to coupling of a counter-propagating cw pump wave and a Stokes-shifted probe pulse via an induced acoustic wave [5]. Sensing is obtained by temporally resolved Brillouin spectra, i.e. the depleted output pump as a function of the detuning from the Brillouin frequency. The Brillouin frequency gives the strain and/or temperature along the fiber. To obtain precise values of the strain/temperature and their locations the following contradictory conditions should be compromised: to get accurate frequency measurements, the spectra should be as narrow as possible implying the application of long Stokes pulses; however, to get an accurate spatial resolution, the short pulses should be used.

In tens km-long fibers the Stokes pulses of several-nanosecond duration provide the high spatial resolution, which is defined by the pulse length and is a few tens of centimeters, with a significant signal-to-noise ratio [1]. At the same time, the Brillouin spectrum linewidth defined by the spectral width of nanosecond Stokes pulses is expected to be larger than the natural Brillouin linewidth. The latter is defined by the phonon lifetime, τ _ph ≈ 10 ns for silica fibers, and corresponds to the stationary interaction of cw pump and cw Stokes waves. Surprisingly, it was experimentally shown that the Brillouin spectrum linewidth could be close to the natural for nanosecond Stokes pulses [6]. This was explained by the fact that in a real experiment the Stokes pulses have a small cw component (base) due to unavoidable leakage of any optical amplitude modulator used to generate nanosecond pulses. In this case the linewidth could still be defined by the stationary interaction of the cw pump and cw base and could therefore be close to the natural linewidth, provided the relative contribution of the transient regime on the output pump depletion is weak [7].

Based on this fact, in the present paper we have explored the transient SBS regime for nanosecond Stokes pulses with a base and have shown how to achieve the narrowest linewidth and, simultaneously, the smallest spatial resolution of SBS-based sensing. An important issue regarding the sensing, which was not addressed so far to the best of our knowledge, is the influence of AC or DC detection of the output pump power on the spectral measurements. We report that both high spectral and high spatial resolution can be obtained by time-domain analysis of AC-detected Brillouin spectra in the case of nanosecond Stokes pulses with a finite base. It is demonstrated by simulation of SBS-based sensing of longitudinally inhomogeneous distributions of the Brillouin frequency along the fiber.

2. Theoretical model

The theoretical model used to simulate the sensing in single-mode fiber is based on the wave equations for the pump and Stokes waves with the slowly-varying field amplitudes E _p(z,t) and E _s(z,t), respectively, which interact nonlinearly by the excitation of the acoustic wave Q(z,t) [5]:

where dot and prime stand for the derivatives over time and longitudinal coordinates t and z, respectively, and v is the phase velocity of the fiber’s fundamental mode. In the case of several-nanosecond optical pulses participating in SBS, the slowly-varying amplitude approximation normally used may no longer be valid for the acoustic field [8]. Therefore, from the acoustic wave equation [5] we have obtained the equation for the field Q(z,t) with second-order time derivative as follows:

where Γ = 1/τ _ph is the relaxation rate, Ω = ω _p - ω _s, ω _p, ω _s are the acoustic field, pump, and Stokes frequencies, respectively, Ω_B is the Brillouin frequency, g = vΓΩgB, and gB is the SBS gain factor.

Equations (1–2) are solved for the boundary conditions of input pump power P _p at one fiber boundary z = L and input Stokes pulse at another boundary z = 0. We describe the input Stokes pulse by the two-kink profile function A(t) = [(tanh t ₁ -tanh t ₂)/2]^1/2, where t _1,2 = (t ± τ _s/2)/a and τ _s is the pulse duration. This function has a flat peak and a short rise time at the pulse edges as compared to the pulse duration and presents the typical shape of the nanosecond optical pulses. When the rise time is defined as a time interval between pulse power levels 0.1 and 0.9 of the peak power at leading and tailing edges, then it is related to the parameter a as t _rise = a/0.45. Note that it is independent of the pulse duration. We have assumed t _rise = 0.1 ns for all results presented below.

Taking into account the existence of the base, we model the boundary input Stokes pulse as

where E _s,b = (P _s,b/A _eff)^1/2 for the field normalization used in Eqs. (1)-(2), P _s and P _b are the peak and cw base powers of the Stokes pulse, respectively, and A _eff is the fiber effective area. We characterize the base level of the Stokes pulse by the extinction ratio ER = 10log(P _s/Pb).

Equations (1)-(2) were solved by the time update [9] of the initial distribution of the pump, Stokes and acoustic waves obtained from the corresponding stationary equations of the SBS for cw pump and cw base. Our results below do not depend on the time moment, when the Stokes pulse enters the fiber (cf. [7]). This time moment was assumed to be equal to t ₀ = 5 ns in respect to the pulse leading edge for any pulse duration τ _s.

All the simulations presented here were performed with Brillouin parameters typical for the single-mode silica fibers at the wavelength 1.3 μm: τ _ph = 10 ns, ν _B = 12.8 GHz, g _B = 5 × 10^-11 m/W, A _eff = 50 μm², v = 0.2 m/ns.

3. Brillouin spectrum detection for the several-nanosecond pulses with base

Due to existence of the cw component of the Stokes waves, the pump depletion takes place at any time moment before the pulse arrival. The spectral properties of the output pump are different in the stationary regime, when the pump and the Stokes base interact, and in the transient regime, when the Stokes pulse comes into play. Therefore, it is important to clarify which value is represented in the time-domain analysis of the Brillouin spectra. The actual time-dependent output pump power or the time-dependent deviation from the stationary output power, induced by the Stokes pulse, can be measured in the experiment. Then the pump loss defined as

or

each as a function of the time and frequency detuning ΔΩ = Ω - Ω_B, can represent the Brillouin spectrum. Here P _p is the boundary input pump power at z = L, P _p (z = 0, t) is the time-dependent output pump power at z = 0, P _p(z = 0,t = 0) is the output pump power at the initial time moment, defined by the stationary pump-base SBS before the Stokes pulse arrives in the fiber, and P _p (z,t) = |E _p (z, t)|² A _eff. Hereafter we refer to these two possibilities of pump loss definition by Eq. (4a) and Eq. (4b) as DC or AC detection, respectively. We note that AC detection is most likely to be used in the experiment to measure the small relative variations of the output pump signal in the case of several-nanosecond Stokes pulses.

For the case of a 3-ns Stokes pulse with ER =15 dB, Fig. 1 demonstrates that DC and AC detections result in rather different time-resolved Brillouin spectra. At the time t < t ₀, before the probe pulse arrives in the fiber, and at t > 2L/v + t ₀ + τ _s, after it has left the fiber and the related pump depletion has propagated from z = 0 to the output z = L, the DC-detected Brillouin spectrum shows a Lorentzian line shape with the Brillouin natural width Δν _B = Γ/π ≈ 32 MHz resulting from the pump-base SBS [5]. For AC detection this corresponds to no signal before pulse arrival and transient relaxation again to zero signal through complicated Brillouin loss signal oscillations with even negative values at the out-of-resonance.

 

Fig. 1. Brillouin spectra resolved in the time domain by DC (a) and AC (b) detection for the pump power P _p = 5 mW and the Stokes pulse with duration τ _s = 3 ns, peak power P _s = 10 mW, extinction ratio ER = 15 dB, and for the fiber length L = 10 m.

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Besides the obvious difference in the Brillouin loss, at the times when the pulse is inside the fiber, DC and AC detections result in the different spectral linewidth Δν _B (FWHM). It is shown in Fig. 2 for the Brillouin spectra detected at the time moment t = 100 ns, when the output pump depletion at z = 0 corresponds to the Stokes pulse position just in front of the fiber end z = L. Both types of detection reveal increasing linewidths with Stokes pulse shortening up to the moderate durations. This is related to the fact that the Brillouin spectrum is a spectral convolution of the narrow natural line and the broad pulse spectrum, which takes the features of the broader spectrum, i.e. of the Stokes pulse. However, for several-nanosecond duration and finite ER, the Stokes pulse essentially probes the acoustic field induced by the stationary pump-base SBS, with negligable acoustic field change. Then the contribution of the stationary regime into the spectral characteristics prevails over that of the transient regime and the linewidth becomes close to the natural again. This effect was experimentally observed and theoretically analyzed in Refs. [6] and [7], respectively. It provides a possibility of the SBS-based sensing with both high spatial and spectral resolutions corresponding to the length of several-nanosecond pulses and the natural linewidth, respectively. Although, for extremely short Stokes pulses the problem of low signal-to-noise ratio could potentially arise.

 

Fig. 2. Brillouin spectrum linewidth vs Stokes pulse duration for DC (a) and AC (b) detection at t = 100 ns, different extinction ratio and the same pump and Stokes pulse parameters as in Fig. 1. Dashed line shows the natural Brillouin linewidth.

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4. SBS-based sensing of longitudinally inhomogeneous fibers

Now we address the SBS-based sensing for longitudinally inhomogeneous distribution of the Brillouin frequency along the fiber and give two examples showing that several-nanosecond Stokes pulses with finite base level provide a possibility to precisely detect different Brillouin frequencies along fibers. First we consider DC- and AC-detected Brillouin spectra of the fiber consisting of two 10-m long sections with Brillouin frequencies 12.800 and 12.875 GHz for the case of a 1-ns Stokes pulse (Fig. 3). We verify how the sections and especially the boundary between them are identified by both types of detection for two different base levels of 15 and 50 dB. The figure is limited by the time interval of the output pump corresponding to the time moments, when the Stokes pulse propagates in the vicinity of the boundary. The thick dashed curve is drawn at the time t = 105.5 ns, when the pulse peak center is passing through the middle of the fiber, and this point should be identified by sensing as a boundary between the sections.

As it can be clearly seen, DC detection gives two spectral lines, which are centered at Δν= 0 and 75 MHz and are almost identical at all time moments [Fig. 3(a)]. This type of detection makes boundary identification impossible. At the same time, AC detection does reveal that the frequency difference of the Stokes pulse and the pump is getting into the resonance with the local Brillouin frequencies at different time moments, while the pulse propagates in the corresponding fiber sections. In spite of the fact that the pulse is as short as 1 ns, the Brillouin spectrum linewidth at the corresponding fiber sections, i.e. the spectral resolution provided by the pulse, is as fine as the natural (Fig. 2). Extremely short Stokes pulses behave like a probe of the acoustic field preliminarily induced by pump-base SBS and provide the best possible resolution both in the frequency and space domains. Additionally, at the time moment corresponding to the boundary, the spectrum transition from one line to another is clearly visible. Though, due to broad pulse spectrum this transition is getting complicated by the prolonged relaxation of the acoustic field at the frequency related to the previous section, when the pulse is already in the next section [Fig. 3(b)].

 

Fig. 3. SBS-based sensing of the boundary between two 10-m fibers with 75 MHz-shifted Brillouin frequencies by a 1-ns Stokes pulse with 15-dB (a,b) and 50-dB (c) base for DC (a) and AC (b,c) detection, and the same other parameters as in Fig. 1. The thick dashed curve is at time moment t = 105.5 ns and corresponds to the boundary between the fiber sections.

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Thus, the Stokes pulse with the finite cw base provides a precise identification of the local information along the fiber by the narrowest spectral linewidth. In contrast, when the base is small, the time-resolved Brillouin spectrum is defined by the broad spectrum of the Stokes pulse. The pulse does not probe the weak pump-base induced acoustic field but equally excites the acoustic field in the transient regime. In this case the local frequency resonances are hardly identified against the broad Brillouin spectrum related to the broad spectrum of the nanosecond Stokes pulse. The boundary “location” spreads over the long time period with the complicated oscillations comparable to the main peaks and is also hardly visible [Fig. 3(c)].

Figure 4 illustrates the SBS-based sensing of a 10-cm section in the middle of a 10-m fiber, where the Brillouin frequency of the section is shifted by 75 MHz relative to that of the rest of the fiber. We have used a 1-ns Stokes pulse with ER = 15 dB and the other parameters are taken from Fig. 1. Thick dashed curves are drawn at the time moments for the output pump when the pulse peak center passes through the section’s middle, and at the frequency shift corresponding to the Brillouin frequency of the section. They denote the time and the frequency, which should be revealed from the detection as a section’s spatial and spectral “location”. The case of AC detection is illustrated, whereas for the case of DC detection the contribution of the section to the time-resolved Brillouin spectra is hardly visible. Outside the time interval used in Fig. 4, the Brillouin spectra are similar to those shown in Fig. 1(b). As one can see, the existence of the Brillouin frequency-shifted section is clearly resolved by AC detection at both the frequency peak related to the main part of the fiber (cf. Fig. 1), and at the frequency peak related to the Brillouin frequency of the section. Both peaks mostly originate from the stationary pump-base SBS and have linewidths close to the natural. Such a resolution becomes impossible if the base is small. Then the local information from the small section spreads over both spectral and spatial domains and makes impossible to determine it from the Brillouin spectra precisely [Fig. 4(b)].

 

Fig. 4. SBS-based sensing of a 10-cm section with 75 MHz-shifted Brillouin frequency in the middle of a 10-m fiber by a 1-ns Stokes pulse with 15-dB (a) and 50-dB (b) base for AC detection and the same other parameters as in Fig. 1. Thick dashed curves are at the time moment t = 55.5 ns and the frequency detuning Δν= 75 MHz.

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5. Conclusion

In summary, we have shown the possibility to obtain the high spectral resolution along with the high spatial resolution in SBS-based sensors for 1-ns Stokes pulses with finite cw component (base). The precise identification of the local Brillouin frequency is provided by AC detection of the output pump, i.e. detection of the time-dependent deviation of the output pump induced by the pulse from the output pump induced by the stationary pump-base SBS. In this case the spectral resolution is close to the natural Brillouin linewidth and, simultaneously, the spatial resolution is defined by the Stokes pulse length. This was illustrated by the examples of the SBS-based detection of the two-section fiber and the fiber with the Brillouin frequency-shifted small section. Neither DC detection for a substantial base, nor both DC and AC detection for a low base makes possible precise sensing of longitudinally inhomogeneous fibers.