Currently SBS generated in optical fiber is widely considered to be a promising and efficient room temperature approach towards realizing slow light fiber devices compatible with telecommunication needs. Foreseen applications include fibre based all-optical delay lines, optical buffers, optical equalizers and signal processors [1]. Many experimental papers on slow light (SL) via SBS in optical fiber have now been published, in which delayed pulses of durations, t_p, from ~40 ps to ~100 ns were observed [2–8]. The expression ΔT_d ≅ G/Γ₀, where ΔT_d is the time delay, G = g ₀I_p L is the SBS exponential gain and Γ₀ is the FWHM spectral width of the SBS gain profile, is usually used in interpreting the observed results. Here g ₀ is the SBS gain coefficient, I_p, is the pump intensity and L is the length of the medium. It is interesting to note that originally this expression was obtained for describing the time required for the output Stokes pulse to reach its maximum power, the so called build-up time, when a δ-function input pulse is sent into the medium pumped by monochromatic CW pump radiation [9]. It was proposed there and later shown in [10] that in the approximation that the input pulse duration is much longer than the decay time of the acoustic wave, t_p ≫ τ= 2/Γ₀, ΔT_d = Gτ/2 can then be treated as the group-delay time of the transmitted pulse. However, since the above condition does not hold in the majority of the SL experiments so far reported, it is pertinent to address this problem with more rigor. In this paper we investigate through analytic theory the effect of the SBS gain build-up on the characteristics of a Stokes pulse, its delay, duration, and amplification, as the pulse propagates through an extended SBS medium pumped by counter propagating CW monochromatic pump radiation. We show that there exist no conditions, which can give rise to group index induced delay of the Stokes pulse.

By its physical nature the gain for Stokes emission in SBS is a consequence of reflection of the pump radiation by a variation of density in a medium, ρ’(z,t), which is induced by an electrostrictive force resulting from interference of the pump and Stokes emissions. The evolution of the slowly varying amplitude of the Stokes field, E_S(z,t), in a medium is described by the equation,

where the nonlinear polarization, P_S(z,t), at the frequency of the Stokes field is

E_p is the pump field amplitude, Δε(z,t) is the variation of the dielectric function of the medium induced through electrostriction and ρ’(z,t) obeys the acoustic wave amplitude equation,

Here the pump and Stokes fields are considered to be plane counter-propagating waves of frequencies ω_p and ω_s= ω_p - Ω, n, ε and ρ ₀ are the refractive index, dielectric function and equilibrium density of a medium, c is the speed of light in vacuum, δΩ = Ω - Ω_B, where Ω and Ω_B are the acoustic and resonant Brillouin frequencies, ν is the speed of sound.

Since in typical SB S-based slow light experiments the CW pump power is kept below the value at which the SBS interaction begins to experience pump depletion, it remains constant throughout the interaction length (in lossless media). This is also the reason why spontaneous scattering is not taken into consideration (which is the usual practice in theoretical treatments of this problem [2,3,11]). The set of Eqs. (1) to (3) with appropriate boundary conditions is therefore sufficient for describing the evolution of a Stokes pulse in a medium.

It is convenient to introduce the new temporal coordinate, t’ = t - zn/c. We also suppose that the centre frequency of the Stokes pulse spectrum coincides with the resonant Brillouin Stokes frequency, that is δΩ = 0. In terms of the new variables Eqs (1)–(3) can be rewritten as

and

This set of equations has an analytic solution, which can be obtained using Reimann’s method [12]. So far such a solution was analyzed for both stimulated Raman and Brillouin scattering with pulsed pump radiation and a short interaction length [13–16]. We consider in this work the case addressed in the typical SL experiments, in which the duration of the Stokes pulse is much less than its transit time in the medium and the pump is CW monochromatic radiation.

We assume that there are no acoustic waves in the medium before a Stokes pulse enters, and E_s(t’ ≤ 0) = 0, E_s(z=0,t’) = E_s0(t’) and E_p(z,t’) = E_p = const. The solutions for the Stokes field and the density variation are then

and

Here I_p = ∣E_p∣² is the pump radiation intensity in [W/cm²], I _0,1(x) are the Bessel functions of imaginary argument x, and g is the SBS gain coefficient,

Let us suppose that the input Stokes signal is an optical pulse, the time dependent intensity of which is given by

where I_s0 is the intensity at the peak of the pulse, t_p is the FWHM pulse duration. The shape of the pulse is shown in Figs. 1 and 2 (curves for G = 0) and it is a good approximation for pulses actually used in experiments [17]. Such a shape is more appropriate to modelling compared to that Gaussian, which is traditionally used in the literature [2,3,8,11], because it has zero amplitude at t = 0. Since the SBS exponential gain, G, in the fiber is supposed to be below the SBS threshold, this, according to our recent work [18], limits G to ≤ 12 for standard silica fibers of > 0.1 km length.

Figures 1 and 2 show the calculated relative output Stokes pulse powers, P_S(t) = ∣E_S(t)∣² S (S is the effective area of fiber–mode cross section), shapes, amplitudes and delays for four input pulse durations, t_p, and different G. Here we take the decay time, τ, of the hyper-sound wave in silica to be τ = 18 ns at a pump radiation wavelength ~1.55 μm. We emphasize that the decay time of the acoustic wave, which is determined by the viscosity of fused silica [19], is used in our calculations rather than the SBS spectral width. This is because the latter depends on intrinsic characteristics of the fiber (core/cladding design, doping type and concentration, numerical aperture, etc. [20–22]) and the ambient environment (mechanical, thermal and electromagnetic fields [20]), all of which can result in substantial variation of the resonant Brillouin frequency, Ω_B. However they do not appreciably effect the viscosity of the medium.

It follows from Figs. 1 and 2 that the induced delay of the output Stokes pulse, its duration and peak power, which is estimated to be P_S0e^Gef, where G_ef is the effective SBS exponential gain, in all cases increase with increase of G. Rates of these growths depend substantially on the ratio of pulse duration to acoustic wave decay time, t_p/τ, as shown in Fig. 3.

For a long input Stokes pulse, t_p = 200 ns that is t_p/τ = 200/18 ≫ 1, the output pulse, on increase of G, remains similar in form (Fig. 1(a)), and is increasingly delayed following ΔT_d /G ≅ 11 ns as shown by the thick dashed line in Fig. 3(a). Additional calculations have shown that it may coincide with ΔT_d ≅ τG/2 ≅ 9G ns (shown by the thin solid line in Fig. 3(a)), but this happens only when t_p/τ ≅ 2.5±0.5 and >100. For t_p/τ < 2 the growth of ΔT_d falls below the value ΔT_d ≅ 9G ns. The duration of the output pulse compared to that of the input is slightly broadened with increase of G (see dashed line in Fig. 3(b)), and G_ef decreases very slightly compared to G (see dashed line in Fig. 3(c)).

 

Fig. 1. Output Stokes pulse shapes for a) t_p = 200 ns (≫τ ) and G = 0 (1), 1(0.42), 4(0.026), 8(0.0006), and 12(0.000012) and b) for t_p = 18 ns (=τ ) and G = 0(1), 1(0.9), 2(0.6), 4(0.2), 8(0.09), and 12(0.00026), where numbers in brackets are the amplitude magnification factors. The numbers on curves are Gs.

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Fig. 2. Output Stokes pulse shapes for a) t_p = 4 ns (<τ) and G = 0(1), 2(1), 4(1), 6(1), 8(0.2), and 12(0.007), and b) for t_p = 0.5 ns (≪τ) and G = 0(1), 4(1), 8(1), and 12(1 at the first peak and 0.3 at the tail). LHS of b) is temporally stretched to show profiles.

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Fig. 3. Output Stokes pulse a) delay, b) broadening factor, and c) effective exponential gain at peak of output pulse vs gain G for pulse durations t_p = 200 (dashed), 18 (dotted), 4 (thick solid in the main graph and in the insert), and 0.5 ns (dashed in the insert). Thin solid lines in a) and c) are ΔT_d = 9G ns and G_ef = G dependencies respectively.

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For t_p = 18 ns (Fig. 1(b)), that is t_p/τ = 1, ΔT_d for the output pulse again increases with G though not linearly; at lower G (0 to ~2) with slope ΔT_d /G ≅ 3 ns, and at higher G (> 2) with slope ΔT_d /G ≅ 9 ns (dotted line in Fig. 3(a)). The pulse broadening in this case increases substantially with G, by a factor ≥ 5 at G > 10 (dotted line in Fig. 3(b)) and G_ef decreases notably, by a factor of ~3 at lower G to ~1.5 at G ≥ 10 (dotted line in Fig. 3(c)).

For the short pulses, t_p ≤ 4 ns (Fig. 2), that is for t_p/τ< 1, significant new features appear. For G < 4 the output pulses approximately retain their shape with only a slight increase of ΔT_d with G (ΔT_d /G ≅ 0.1–0.15 ns for t_p = 4 ns and ΔT_d /G ≅ 0.03 ns for t_p = 0.5 ns, as is shown in inset of Fig. 3(a), solid and dashed lines respectively). In both cases G_ef for the leading peak decreases substantially, by a factor ~10. For G between 4 and 6 there is substantial growth of the power in the tail of the pulses and for G > 8 the maximum of the pulse shifts to the tail (see Figs. 2(a) and 2(b)). This is because of the long decay of the acoustic wave excited by the short Stokes pulse interacting with the CW pump. The dependence of ΔT_don G for G > 8 then follows the linear relation ΔT_d ≅ (9G − 25) ns (thick solid line in Fig. 3(a)). The pulse broadening factor is then ~20 for t_p = 4 ns (solid line in Fig. 3(b) at G > 6) and ~200 for t_p = 0.5 ns.

It is interesting to note that our analytical results presented in Figs. 3(a) and 3(b), which are obtained in the small signal limit, give dependencies of pulse delay and broadening on G quite similar to these obtained numerically both for long pulses, t_p/τ ≅ 15 and 5 in [11], and for short pulses, 0.1 < t_p/τ < 2 in [23]. There are however some quantitative differences, which, we think, are important to highlight. The numerical modelling in [11] gives ΔT_d = τG/2 dependence for t_p/τ ≅ 15, while our calculations predict this only for t_p/τ in the region ~2.5±0.5, while for t_p/τ > 5 the slope ΔT_d /G is a factor 1.2–1.3 steeper. Also the shape of the ΔT_d upon G dependence for t_p/τ ≅ 5 in [11] is similar to that in our calculations but for t_p/τ ≅ 1. As to the numerical results presented in [23], the difference with our findings is most probably because all their results were obtained for an input Stokes power well above that for onset of pump depletion (input Stokes power is of 1 mW compared with pump power of ≤ 20 mW).

To understand the nature of the behaviour we find let us next consider through Eq. (7) the temporal and spectral characteristics of the complex dielectric function variation in the medium, Δε = (∂ε/∂ρ)ρ’ = Δε’ + iΔε”, induced by the interaction of the pump and Stokes signals; Δε(ω) results in the SBS gain and Δε’(ω) is responsible for modification of the refractive index, Δn(ω) ≅ Δε’(ω)/2n₀, of the medium, where n0 is the refractive index of a medium without SBS. In the limit of small gain, G < 1, the Bessel function I ₀(x) in Eq. (7) may be set to unity, and analysis is greatly simplified. While this approximation does not allow us to describe gain narrowing of the SBS spectrum typical for higher G, it still captures reasonably well the trends in the temporal and spectral features of ρ’, which determine those of the SBS gain and modified refractive index.

When I ₀(x) = 1, the integral in Eq. (7) can be taken for E_S0(t) given by Eq. (10). It results in the following analytic expression for ρ’(t),

where b = t_p/3.5, and $A=-i{\rho }_0\frac{\partial \epsilon }{\partial \rho }\frac{{\Omega }_B}{8\pi v^2}{E}_p{E}_S^{*}\left(0\right)$ The spectral characteristics of the acoustic wave amplitude follow from the Fourier transform of Eq. (11),

The temporal dynamics of the induced acoustic wave amplitudes and their spectra for values of t_p/τ considered above are shown in Fig. 4 (the dynamics for t_p = 0.5 ns is not shown on the timescale of Fig. 4(a)). The curves in Fig. 4(a) represent different characteristic types of SBS interaction: curves 1 give an example of quasi-steady state interaction when t_p/τ ≫ 1, curves 3 demonstrate transient type interaction when t_p/τ ≪ 1, and curves 2 are for an intermediate case when t_p/τ≅ 1.

In the first case of a long Stokes pulse, that is t_p/τ ≫ 1, the shape of the acoustic wave pulse shown in Fig. 4(a) as the solid curve 1 almost reproduces the shape of the input Stokes pulse (dotted curve 1). The spectrum of the excited acoustic wave in this case, Fig. 4(b), reproduces the spectrum of the input Stokes pulse, shown by the solid and dashed curves 1 respectively. These are both narrower than the Lorentzian-shaped spectrum corresponding to τ = 18 ns (dotted curve 5). This is to be expected since Eq. (5) is in essence the equation for the amplitude of a driven damped oscillator; for such a system the spectrum of the induced oscillations is fully determined by the spectrum of the driving force when it is narrower than reciprocal decay time of the oscillator.

 

Fig. 4. (a) Dynamics and (b) spectra of the normalised acoustic wave amplitude (solid lines) for Stokes pulses of the shape given by Eq. (10) (dashed lines) with t_p = 200(1), 18(2), 4(3) and 0.5 ns(4), τ= 18 ns. The dotted line 5 in (b) is the Lorentzian-shaped spectrum with τ= 18 ns. Note, curves in (b) show half spectra of the pulses at LHS and of corresponding to them medium’s response at RHS.

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In the case of shorter Stokes pulses, t_p/τ ≤ 1, (dashed curves 2 and 3 in Fig. 4(a)) the spectrum of the driving force, that is that of the input Stokes pulse, is broad band, (dashed curves 2 and 3 and also curve 4 for t_p = 0.5 ns pulse in Fig. 4(b)). As seen the dynamics of the medium’s response (solid curves 2 and 3 in Fig. 4(a)) and its spectra (solid curves 2, 3 and 4 in Fig. 4(b)) differ substantially from those of the Stokes pulses and their spectra. In the temporal domain the maximum amplitude of the induced ρ’(t) decreases with decrease of pulse duration and a long tail appears after a Stokes pulse, which decays exponentially with a decay time of τ. The spectra of ρ’(t) in these cases are narrower, increasingly so for shorter pulses, than the spectra of the driving force (input Stokes pulse). Their width is then determined predominantly by the reciprocal decay time of the oscillator (dotted curve 5 in Fig. 4(b)) as is to be expected for a damped oscillator, which is driven by a broad-band force. Reduction of amplitude of the induced ρ’(t) with decrease of pulse duration results in its reduced amplification as shown in Fig. 3(c) for G_ef.

The imaginary part of ρ’(ω) described by Eq. (12) allows us to calculate the refractive index of the medium modified by the SBS interaction, n(ω) = n₀ + Δn(ω) ≅ n₀ + (∂ε/∂ρ)ρ’(ω)/2n₀ and its corresponding group index, n_g(ω) = n₀ + ω[dΔn(ω)/dω]. Spectra of the Stokes pulses and the group indices induced by these pulses are shown in Fig. 5 for the four different t_p/τ.

 

Fig. 5. Relation between the spectra of a Stokes pulse and of the group index induced by this pulse for t_p = 200 a), 18 b), 4 c), and 0.5 ns d). Horizontal scales are in GHz.

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It is clear from Fig. 5 that the spectral width of a Stokes pulse is bigger then that of the SBS induced group index regardless of t_p/τ. As such, while some, central, part of the input spectrum may experience a group delay, other parts of the spectrum will experience group advancement or neither delay or advancement (see Figs. 5(c) and 5(d)). We therefore conclude that regardless of the pulse length of the input Stokes pulse the pulse delays associated with SBS amplification of a Stokes pulse, as described above, cannot be attributed to SBS induced group delay. They are predominantly a consequence of the phenomenon of SBS build-up.

Our results therefore raise a question of whether slow light, as first discussed by Zeldovich [9,10], can be realised. To answer the question let us remind ourselves of the physical nature of the effect. It is a linear phenomenon exhibited by a pulse propagating through a medium with normal dispersion of refractive index [24]. The effect is greatly enhanced in the vicinity of a medium’s resonance, which for a gain medium is normally dispersive (and for absorptive medium is abnormally dispersive). As we have seen above the nonlinear effect of SBS amplification of an external Stokes pulse cannot give rise to group index induced delays of this pulse. However this is possible if suitable group index features, for the pulse to be delayed, are independently prepared in the medium. Actually this happens naturally in SBS through spontaneous Brillouin scattering, which is generic to SBS and so unavoidable. The amplification of the steady state spontaneously scattered signal induces a group index profile of bandwidth associated with the broadest possible SBS gain spectral width, Γ, the latter of which may be either homogeneously (Γ ≅ 2/τ) or inhomogeneously (Γ > 2/τ) broadened. We have shown in particular that in optical fibers the SBS gain spectral bandwidth, [22], and that of its corresponding group index profile [25] can be broadened up to ~10 GHz width due to the waveguiding effect [21]. In the ideal limit, that an input Stokes pulse acts as a probe with spectral width narrower than Γ, it will then experience classical group delay. Interestingly we see from Fig. 2(b) for the case of short pulses (sub-nanosecond) that the pulse can act as a probe for reasonable G < 10, since there is minimal pulse distortion by the SBS build-up processes. Classical group delay of this pulse can then be expected when it propagates through optical fibre with the waveguide induced broadening [21,25]. However, more generally, there is always a modification of the Stokes pulse by the build-up process, as described above, which must therefore be properly addressed in modelling SBS induced SL phenomenon.