1. Introduction

Since the publication of Refs. [1, 2] in 2005, the effect of optically controlled pulse delay (so-called slow light) via stimulated Brillouin scattering (SBS) in single-mode optical fibers has been widely considered as a hot topic of the modern optics. It is explained by the great expectations from this effect for development of important elements of optical communication systems, such as those enabling data synchronization, buffering, memory and processing. It was experimentally demonstrated that the delay of pulses of several tens of nanoseconds or longer can be effectively controlled simply by varying the pump power [1, 2, 3, 4]. These observations were theoretically confirmed by estimations based on steady-state small-signal approximation of SBS, resulting in the delay proportional to the Brillouin gain and inversely proportional to the Brillouin gain linewidth [2].

For practical applications it is more important to achieve slow light for short pulses of nanosecond and sub-nanosecond durations corresponding to Gb/s data streams. Obviously, as the Brillouin gain bandwidth is of the order of tens of MHz (defined by the inverse phonon lifetime in silica fiber which is in the 10-ns range) the slow light for short pulses is not as effective as for the pulses longer than the phonon lifetime. A narrow gain line causes the amplification of only a small portion of the pulse spectrum near the Brillouin resonance and inevitably causes pulse broadening (distortion) while propagating along the fiber. To overcome this obstacle of slow-light effect, broadening of Brillouin gain line by using multi-component [5, 6], frequency-modulated [7, 8], amplitude-modulated [9, 10] or phase-modulated pump [11, 12] has been proposed. At broadband pumping all pulse spectral components could potentially be amplified in accordance with their initial shape and acquire the same delay keeping the pulse shape unchanged. Though, it can be achieved at the cost of a considerable pump power growth. It was experimentally demonstrated that spectrally broadened pump by random variation of the pump frequency around the carrier has resulted in delay comparable to the pulse duration of sub-100-ps pulses [8]. However, fidelity of the delayed pulse was relatively low and the problem of optically controlled delay of short pulses at broadband pump provided that the pulse distortions are small remains unsolved both theoretically and experimentally.

We note that the simplified theoretical estimations [6, 7, 8, 9, 12] are applicable for the case of broadband pump and short pulses in rather limited extent. They totally ignore the nonlinear nature of coupling between pump and Stokes spectral components through the acoustic field excitation. More specifically, the approach developed to estimate the delay versus pump ignores pump depletion (gain saturation) which can be considered only by numerical solution of wave equations describing counter-propagating pump and Stokes waves along with equation for the acoustic field. As it was shown in Ref. [13], the dependence of the delay versus the gain substantially deviates from the proportionality even for pulses several times longer than phonon lifetime. In particular, ignoring this effect inherent to long fibers, strong pump and Stokes pulse powers obviously results in overestimation of the delay versus gain (see, e. g., [14] and [15, 16]). Therefore, to estimate prospective applications of the slow-light effect at Gb/s data transmission rates there is a need to consider it for short pulses and broadband pump on the basis of exact SBS model.

In the present paper we study slow-light effect via SBS in single-mode optical fibers for short nanosecond pulses. The analysis is based on numerical solution of three-wave coupled equations beyond steady-state small-signal approximation for both cw and broadband pumps and for long fibers. It is shown that at increasing power of cw pump the short pulse distortion originates from pulse peak shift to the maximum of the acoustic excitation extended over the characteristic phonon lifetime. A different slope of the gain-dependent delay for different pulse durations is shown at cw pumping. A way to obtain large delay and suppress pulse distortions at spectrally broadened pumping by multiple cw components, frequency-modulated pump and pulse train is studied for short pulses. Pump by the pulse train allows to obtain pump spectrum corresponding to pulse spectrum and this is shown to be advantageous over other types of spectrally broadened pump for large time delay as well as advancement with minimum pulse shape distortion.

2. System modeling

We consider SBS-based slow-light effect in single-mode fibers by solving the coupled wave equations for counter-propagation of the pump and Stokes waves with the slowly-varying field amplitudes E _p(t, z) and E _s(t, z), respectively, which nonlinearly interact by the excitation of the acoustic wave Q(t, z) [17]:

where t and z are the time and longitudinal coordinates, respectively, V is the phase velocity of the fiber fundamental mode, and α is the fiber linear loss coefficient.

In the case of short pulses, the usually used slowly-varying amplitude approximation for acoustic field may no longer be valid. Therefore from the acoustic wave equation [17] we have obtained the equation for the field Q as follows [18]:

where Γ=1/τ _ph is the relaxation rate, τ _ph is the phonon lifetime, ω _p and ω _s are the pump and Stokes frequencies, respectively, Ω=ω _p-ω _s is the acoustic field frequency, Ω_B is the Brillouin frequency, g=ΓΩg _B, and g _B is the SBS gain factor. For typical silica fibers Γ≪Ω, Ω_B and Eq. (2) describes the stronger pump and Stokes interaction at the resonance Ω=Ω_B than at out-of-resonance owing to the contribution of the “oscillatory” term ∝${\mathrm{\Omega }}_{\mathrm{B}}^2$-Ω² on its left-hand side. This is the cause of a sharp Brillouin line for the output pump and Stokes pulse parameters as function of the frequency detuning.

By omitting the term containing the second-order time derivative in Eq. (2), using the approximation ${\mathrm{\Omega }}_{\mathrm{B}}^2$-Ω²≈2Ω(Ω_B-Ω) and for Γ≪Ω we obtain commonly used equation for the acoustic field [13]:

The results presented below were obtained by using both Eqs. (2) and (3) and a close coincidence was obtained even for pulses of duration 1 ns. However, we anticipate that the difference between Eqs. (2) and (3) could be substantial for still shorter pulses used in existing communication systems operating over 10 GB/s. The reason for that is that unlike Eq. (3) the actual gain lineshape corresponding to Eq. (2) includes two Lorentzian resonance lines, since by Fourier transformation we have

where (…)_ω denotes the spectral component of the corresponding quantity at frequency ω, and ω _1,2=Ω∓(${\mathrm{\Omega }}_{\mathrm{B}}^2$-Γ²)^1/2. Acoustic spectrum has one peak at the frequency ω ₁≈Ω-Ω_B corresponding to usually considered SBS gain line position, whereas the another peak is located as far away as at the frequency ω ₂≈Ω+Ω_B. However, the latter might be still inside the spectrum of sub-ns Stokes pulses or broadband pump like those used in Ref. [8] and contributes considerably into SBS.

To evaluate the slow-light effect, Fourier-transformed Eqs. (1b) and (3) are usually used under assumption of undepleted cw pump (small-signal approximation). Then amplitude and phase transformation of the Stokes pulse via SBS can be found. At the Brillouin resonance Ω=Ω_B the power and the delay of the spectral components of the Stokes pulse varying with distance along the fiber are obtained as follows [2, 13, 14]:

where G(z)=g _B|E _p|²(e^αz -1)e ^-αL/α≈g _B|E _p|² z is the gain parameter at the distance 0<z<L, and L is the fiber length. We show below in Section 3 that these results, usually applied to the carrier frequency component ω=0 (steady-state approximation), are strong overestimation of the slow-light effect for pulses shorter than the phonon lifetime even in small-signal regime.

To complete formulation of the exact model of SBS in the optical fibers, the boundary conditions of Eqs. (1–2) are established as an input Stokes pulse at one fiber end z=0 and input pump wave at another z=L. We describe the input Stokes pulse by Gaussian function as E _s(t, z=0)=E _s exp[-2ln2(t/τ _s)²] with duration τ _s (FWHM for the power). We have considered several different types of input pump, both cw and spectrally broadened. It was considered to be a cw wave of the power P _p, or consist of 2N+1 spectral lines of the same power as E _p(t, z=L)=E _p${\mathrm{\Sigma }}_{n=-N}^N$exp(inΔΩt), where ΔΩ is the line spectral spacing, or to be a frequency-modulated cw wave as E _p(t, z=L)=E _p exp(iη sinΔΩt), where η and ΔΩ are the amplitude and frequency of the modulation, respectively, or present pulse train E _p(t, z=L)=E _p${\mathrm{\Sigma }}_{n=-\infty }^{\infty }$ exp[-2ln2(t-nΔt)²/${\tau }_{\mathrm{p}}^2$], where τ _p is the pump pulse duration, Δt>τ _p is the train period. Here for the field normalization used in Eqs. (1)–(2) the input amplitudes E _p,s=(P _p,s/A _eff)^1/2, where P _p,s are the power of the spectral lines and/or peak power of the corresponding pulses, and A _eff is the fiber effective area.

 

Fig. 1. Normalized input spectra of the Stokes pulse for τ _s=1 ns (red curves), Lorentzian gain lineshape of steady-state SBS given in Eq. (5a) [green curve in (a)], and (a) cw, (b) multi-component, (c) frequency-modulated pump and (d) pump by pulse train (blue curves). (b) N=3, ΔΩ=50 MHz; (c) η=3, ΔΩ=50 MHz; (d) τ _p=1 ns, Δt=20 ns.

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Spectra of 1-ns Stokes pulse and considered pumps as compared to the Lorentzian gain lineshape of steady-state SBS are presented in Fig. 1. Pumping by multiple components is in some extent equivalent to pumping by pulse train, since the interference of spectral lines results in periodic beating of the pump power with the period 2π/ΔΩ and sharp pulse-like power increase up to N ² P _p with duration ~1/NΔΩ. In contrast, the case of pulse train allows to assume independently the train repetition rate and duration of the individual pulse. In particular, by choosing equal duration of the Stokes pulse and pump pulse in the train we match the intensity distribution of the pump lines to the spectrum of the Stokes pulse [Fig. 1(d)], whereas the spectral spacing of the lines 2π/Δt is varied by the train period. On the other hand, frequency-modulated cw pump has a spectrum consisted of multiple spectral lines with spectral spacing ΔΩ and is similar to multi-component pump [Figs. 1(b) and 1(c)]. Its line intensities, however, are distributed non-uniformly around the carrier frequency, accordingly to squared Bessel functions $J_n^2$(η), n=0,±1, … (see, e.g., [19]). It is of the great interest to study the impact of that close-related and widely used spectrally-broadened pumps on slow-light delay.

Obviously, there is no scalability of SBS process in respect to fiber length, power and pulse duration for strong pump and under the condition of pump depletion. Therefore we need to solve nonlinear Eqs. (1)–(2) numerically for the parameter ranges of the pump and Stokes pulses and fiber lengths which are as close to the realistic conditions as possible. A numerical solution of corresponding two-point boundary value problem is a very time-consuming and memory-demanding problem when short pulses and fiber lengths of the order of 1 km are considered. Besides the nonlinear type of the problem, the reason is a tremendous space-time domain (t, z) of the order T ₀×L, where T ₀=2L/V, needed to be discretesized with a dense grid to resolve 1-ns pulse with spatial width of ~20 cm.

However, we note that for short pulses the pump-probe interaction occurs only in a very narrow time-space domain with some temporal “width” T ₁≪T ₀ along “light” line z=Vt. Here we assume that the pulse is launched at the moment 0<t<T ₁. In the time-space domain z>Vt the pump is propagated freely without changing its boundary (at z=L) and initial values (at t=0) parallel to the line z=L-Vt since the Stokes pulse is absent there. The same occurs after interaction with the Stokes pulse and after complete acoustic field relaxation at z<V(t-T ₁). These two domains occupy the most area of the whole computational domain T ₀×L and require no numerical calculation but identical transfer of the initial pump to z=Vt and pump after interaction from z=V(t-T ₁) to the output z=0, respectively. Hence, one can consider a domain T ₁×L, where T ₁ can be taken of several phonon lifetimes depending of the Stokes pulse duration. For 1-km long fiber the size of this time-space domain is at least two orders of magnitude smaller than the starting domain and the same reduction is achieved in computation time and memory.

Realization of this advantage requires transformation of Eqs. (1)–(2) to the moving frame coordinates for the Stokes pulse with retarded time coordinate τ=t-z/V [20, 21]. In this frame the Stokes pulse is shifted in time just via slow-light effect within the interval 0<t<T ₁. Transformed Eqs. (1)–(2) can be solved by the time update of the initial pump, Stokes and acoustic waves to obtain output pump and Stokes pulse.

Finally, all the simulations below were performed for the Stokes and pump waves at the Brillouin resonance Ω=Ω_B, the peak power of the Stokes pulse of 1 mW, the fiber length L=1 km and for typical Brillouin parameters of standard 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, α=2.1 dB/km, A _eff=50 µm², V=0.2 m/ns.

3. CW pump

Figures 2, 3, 4, 5, and 6 report the results of the simulations for the case of cw pumping. To illustrate a delay of the Stokes pulse, in Figs. 2 and 3 we show the evolution of the normalized pulse with input duration 10 (a) and 3 ns (b) while it propagates along 1-km-long fiber for pump powers 10 and 20 mW, respectively. Low and high powers correspond to SBS under the conditions without and with depletion. Shift of the Stokes pulse peak in the retarded time frame which gradually increase with distance is clearly visible. It is related to the acoustic field excitation induced by pump-pulse interaction through SBS behind the Stokes pulse (Fig. 4). Acoustic field has a characteristic temporal prolongation defined initially by the phonon relaxation time τ _ph and increased later due to its induced character [22].

During the propagation along the fiber and the amplification, the pulse gradually shifts to-wards the maximum of retarded acoustic field. For high input pump power almost a complete pump depletion occurs for both long and short input Stokes pulse (Fig. 5). Induced acoustic field is substantially enhanced (Fig. 4) and results in a substantial amplification of the Stokes pulse via SBS at the ending fiber sections before output. For short pulse this is accompanied by an abrupt shift towards acoustic field maximum at some location after a gradual shift of the peak related to the input pulse maximum [Fig. 3(b)]. At this stage the acoustic field prolongation is almost independent on input pulse duration. Hence, under the conditions of strong depletion and for input pulses shorter than the phonon lifetime, the pulse at the output has a duration related mostly to the prolonged excited phonon field and is strongly distorted.

Figure 6 summerizes characteristics of output Stokes pulse versus pump power for different input pulse durations. Pulse peak, duration (FWHM) as a measure of the pulse distortion, and peak temporal position in respect to the location without SBS as a measure of the delay via slow-light effect are shown. As one can see, the pulse peak amplification substantially depends on the input duration. The same refers to the output pulse duration and the delay. These parameters are especially large and weakly dependent on the input duration under the conditions of the pump depletion at large pump power, when they are defined by the prolonged induced acoustic field. At small pump, the shorter input pulse, the smaller part of pulse spectrum is amplified and delayed via SBS. Therefore, in contrast to the pulses longer than phonon lifetime with a constant slope of gain-dependant delay [1, 2, 3, 4] (see also Eq. (5b), for short pulses the slope of the delay versus gain decreases with pulse duration (Fig. 6(c)). Abrupt increase of the duration and delay at large pump power in Figs. 6(b) and 6(c) is related to the peak shift to the maximum of strong acoustic field induced behind the short Stokes pulse.

 

Fig. 2. Evolution of the Stokes pulse in 1-km-long fiber with input power 1 mW and duration 10 (a) and 3 ns (b) for cw pump of power 10 mW. Input pulse peak position is at τ=0, pulse power is normalized to the current peak power along the fiber.

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Fig. 3. Evolution of the Stokes pulse in 1-km-long fiber with input power 1 mW and duration 10 (a) and 3 ns (b) for cw pump of power 20 mW. Note different scale for the retarded time τ in (a) and (b).

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Fig. 4. Acoustic field ℜ(Q) at different positions along 1-km-long fiber as indicated for 1-mW Stokes pulse with duration 10 (a) and 3 ns (b) and for 20-mW cw pump power.

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Both pulse peak and delay as functions of the pump power substantially deviates from those predicted by Eqs. (5) [dashed curves in Figs. 6(a) and 6(c)] demonstrating a failure of the small-signal approximation to predict slow-light effect in the case of short pulses. At further increase of the pump power larger than those used in Fig. 6 we have observed even more complicated wave behavior when the Stokes wave envelope exhibits a set of peaks of decreasing amplitude similar to asymptotical evolution described in Ref. [21, 23].

In terms of the pulse frequency components and Brillouin gain line (Fig. 1), the amplification of a small spectrum portion results in pulse broadening and distortion which do not permit even to obtain a delay for short pulses comparable with duration by increasing pump. Eventually, together with spontaneous Brillouin scattering, this diminishes the practical importance of the slow-light effect via SBS in the typical configuration with cw pump. The possible way to obtain a large delay equally for all spectral components while the pulse shape remains undistorted is to use broadband pump reported in Refs. [5, 6, 7, 8, 9, 10, 11].

 

Fig. 5. Pump power at different positions along 1-km-long fiber as indicated for 1-mW Stokes pulse with duration 10 (a) and 3 ns (b) and for 20-mW cw pump power.

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Fig. 6. Stokes pulse peak power at the fiber output (a), output duration (b) and delay (c) versus cw pump power for different input pulse durations as indicated and power 1 mW. The dashed curves in (a) and (c) are obtained from Eqs. (5a) and (5b) at ω=0, respectively.

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4. Broadband pump

As it is described in section 2 we have considered pump by a frequency comb, frequency-modulated pump and pump by short pulse train as typical and easy to implement broadband pumps of SBS in optical fibers to obtain a large delay of short Stokes pulses comparable to the initial duration and with a small shape distortion. Figures 7, 8, and 9 present output duration and delay of the pulses with input duration shorter than the phonon lifetime for these types of the pump, respectively. In all cases with increasing pump power we observe larger slope of delay increase with pump power, smaller duration of the pulse and larger maximum possible delay value (before the maximum of the pulse is shifted to the maximum of the acoustic field behind the pulse due to strong pump depletion and strong acoustic field excitation). These results clearly show the advantages of the broadband pump against the cw pump.

As an example, Fig. 8(c) illustrates evolution of the normalized pulse with the duration 1 ns along 1-km-long fiber for the frequency-modulated pump at the power 17 mW, modulation amplitude η=3 and frequency Ω=40MHz. One can see a gradual shift of the pulse position in the retarded time frame with the delay ~1 ns at the output and relatively low level of prolonged tail behind the pulse (cf. Fig. 3(b)).

There are optimal parameters of the broadband pump which provide better performance of slow light from the viewpoint of largest possible delay with no substantial shape distortion. Frequency separation of the pump lines (Fig. 7), modulation frequency (Fig. 8, thick curves) should exceed SBS linewidth to prevent acoustic field enhancement by neighboring lines and hence pulse broadening. Too small temporal separation of the pulses in pump train in comparison with phonon lifetime also results in acoustic field enhancement and pulse distortion due to high everage pump power (cf. thick and thin curves in Fig. 9 obtained for Δt=40 and 20 ns corresponding to the everage pump power of 27 and 42 mW, respectively). From the other hand, larger temporal spacing provides weaker slow-light effect, since during propagation along the fiber of fixed length the pulse could interact with too small amount of the pump pulses: for Δt=40 and 20 ns the Stokes pulse interacts at 1-km-long fiber with the pump pulses 250 and 500 times, respectively. Under real experiment conditions with longer fibers, the Stokes pulse could “collide” with the pump pulses much more times increasing substantially the efficiency of slow-light effect. Therefore, broadband pump allows a variety of possibilities to control delay and at proper choice of its parameters demonstrates superiority in slow-light effect over cw pump.

Figure 10 shows evolution of the normalized pulse along the fiber for different input Stokes pulse durations at pumping by pulse train. At pulse and pump pulse duration 1 ns one can see a gradual pulse delay due to slow light with output delay comparable to the pulse duration. At increasing duration to 3 and 5 ns and at τ _p=τ _s the pump energy is substantially increased (for Δt=40 ns the everage pump power is 80 and 133 mW, repectively) and results in a strong pump depletion and pulse amplification already at starting fiber sections close to the input. At further propagation the pulse shortens with decreasing of the delay which can be transformed at the output even into the advancement. This demonstates a possibility of fast light for short pulses at broadband pump (see also Fig. 9). Similar effect in gain-saturation regime was noticed for long pulses with τ _s≫τ _ph at cw pump [13].

 

Fig. 7. Output pulse duration (a) and delay (b) versus pump power for input pulse power 1 mW and different durations as indicated at multi-component pumping by 5 lines with frequency separation 40 MHz.

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Fig. 8. Output pulse duration (a) and delay (b) versus pump power for input pulse power 1 mW and different durations as indicated at frequency-modulated pumping with modulation amplitude η=3 and frequency Ω=40 (thick curves) and 20 MHz (thin curves). (c) Evolution of 1-ns, 1-mW pulse along 1-km fiber for the pump power 17 mW and η=3, Ω=40 MHz.

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5. Conclusion

In conclusion, we have studied theoretically SBS for short pulses of nanosecond duration aiming possibility to control the pulse delay via slow-light effect by pump, provided that the pulse envelope distortion is insignificant. Unlike recent estimations of delay versus pump based on steady-state small-signal approximation we have used numerical solution of three-wave equations describing SBS for a realistic fiber length. Both regimes of small signal and pump depletion (gain saturation) were considered. The physical origin of Stokes pulse distortion is revealed which is related to excitation of long-living acoustic field behind the pulse and prevents effective delay control by pump power increase at cw pumping. We have shown different slope of the gain-dependent delay for different pulse durations. Spectrally broadened pumping by multiple cw components, frequency-modulated pump and pulse train were studied for short pulses which allow to obtain large delay and suppress pulse distortion. In the pump-depletion regime of pumping by pulse train, both pulse delay and distortion decrease with increasing pump, and the pulse could achieve the advancement.

 

Fig. 9. Pulse duration (a) and delay (b) versus peak pump pulse power for input pulse power 1 mWand different durations as indicated at pumping by pulse train with τ _p=τ _s and pulse repetition period Δt=40 (thick curves) and 20 ns (thin curves).

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Fig. 10. Normalized pulse evolution for input pulse power 1 mW and duration τ _s=1 (a), 3 (b) and 5 ns (c) at pumping by pulse train with pulse peak power 1 W, duration τ _p=τ _s and repetition period 40 ns. Note different retarded time scale in (a), (b) and (c).

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