“Slow Light” in stimulated Brillouin scattering: on the influence of the spectral width of pump radiation on the group index

V. I. Kovalev; N. E. Kotova; R. G. Harrison

doi:10.1364/OE.17.017317

Currently stimulated Brillouin scattering (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]. The underlying idea of this technology is that an externally injected Stokes signal is amplified through SBS [2]. The main obstacle to direct application of this SBS based technology is the narrow SBS spectral bandwidth, Γ_B, which is typically ~2π(20-30) MHz in silica fiber in the spectral range of telecom optical radiation (~1.3-1.6 μm). It was proposed in [3] that spectral broadening of the pump radiation may help overcome this problem. This approach has since been the focus of many subsequent publications [4–24].

In this paper we examine validity of this assertion through a rigorous analytical analysis of the basic coupled SBS equations in the Fourier domain. From this we determine the spectra of the medium’s response and the output Stokes signal when an input Stokes pulse is injected into an SBS medium pumped by non-monochromatic continuous wave (CW) radiation. We show that spectral broadening of the pump radiation by any reasonable amount results in only minute increase of the spectral width of the resonant material’s excitation at the acoustic frequency in SBS. Because this frequency is far away from the frequency of the Stokes optical signal to be delayed, it cannot be effective in modifying the natural group index of the medium for the Stokes signal as claimed in the literature [3–24].

The basic idea which underpins the concept of slow light is the creation of a resonantly enhanced normal dispersion of the refractive index in a medium for a propagating optical pulse [25]. The group index is then n_g(ω₀) = n(ω₀) + ω₀(dn/dω)_ω0 where ω₀ is the carrier frequency of the optical pulse and n(ω₀) is the refractive index of the medium. In the case of SBS the resonance in a medium’s response occurs at the Brillouin frequency, Ω_B, which is the central frequency of the variation of density in a medium, δρ(z,t) = 1/2{ρ(z,t)exp[-i(Ω_Bt + qz)] + c.c.}. This density variation is resonantly induced by an electrostrictive force resulting from interference of two plane counter-propagating waves, the forward-going (+z direction) Stokes and backward-going (-z direction) pump optical fields, E_S(z,t) = 1/2{E_S(z,t)exp[-i(ω_St-k_Sz)] + c.c.} and E_p(z,t) = 1/2{E_p(z,t)exp[-i(ω_pt + k_pz)] + c.c.}, respectively, where ρ(z,t), E_S(z,t) and E_p(z,t) are the amplitudes of the acoustic wave and of Stokes and pump fields, with Ω = ω_p - ω_S and q = k_p + k_S, ω_S and k_S, and ω_p and k_p being their radian frequencies and wavevectors, and c.c. is the abbreviation for complex conjugate. In an isotropic medium δρ(z,t) is [26],

\frac{\partial^{2} δ ρ}{\partial t^{2}} - v_{s}^{2} \nabla^{2} δ ρ - A \nabla^{2} \frac{\partial δ ρ}{\partial t} = - ρ_{0} \frac{\partial ε}{\partial ρ} \frac{1}{16 π} \nabla^{2} {| Ε (z, t) |}^{2},

where v_s is the speed of a free acoustic wave, A is its damping parameter, ∇² ≡ ∂²/∂z² in the chosen plane wave model, ε and ρ ₀ are the dielectric function and equilibrium density of the medium, and E(z,t) = E_p(z,t) + E_S(z,t). Since the amplitude ρ(z,t) is supposed to be slowly varying in space, then ∇²δρ(z,t) ≅ -q ²δρ(z,t) and Eq. (1) is usually reduced to

\frac{\partial^{2} δ ρ}{\partial t^{2}} + Ω_{B}^{2} δ ρ + \frac{Γ_{B}}{2} \frac{\partial δ ρ}{\partial t} = - ρ_{0} \frac{\partial ε}{\partial ρ} \frac{1}{8 π} \frac{\partial^{2}}{\partial z^{2}} [Ε_{p} (z, t) Ε_{S} (z, t)] .

This is then the equation for the induced acoustic wave. It is a typical equation for an externally driven damped resonant oscillator, in which the right-hand side is the driving force, Ω_B = qv_s = 2v_sω_p/(c/n+v_s) ≅ 2nv_sω_p/c is the resonant frequency of the oscillator, known as the Brillouin frequency, Γ_B is the FWHM spectral width of the resonant profile with 2/Γ_B being the decay time of the acoustic wave.

The pump field reflected by the induced acoustic wave is a new Stokes field, which in turn interacts with the pump field to further electrostrictively enhance the acoustic wave and so the Stokes field and so forth. Increase of a Stokes field in SBS is therefore a direct consequence of increase of reflectivity of the acoustic wave for the pump field. As such, so called “SBS gain” characteristics are determined by the reflectivity, spectral characteristics and dynamics of the acoustic wave. In the approximation that the CW pump radiation is not depleted over the interaction length, L, the spatial/temporal evolution of the Stokes signal is described by the nonlinear wave equation,

\frac{\partial^{2} Ε_{S}}{\partial z^{2}} - \frac{ε}{c^{2}} \frac{\partial^{2} Ε_{S}}{\partial t^{2}} = \frac{1}{c^{2}} \frac{\partial ε}{\partial ρ} \frac{\partial^{2}}{\partial t^{2}} [δ ρ (z, t) Ε_{p} (z, t)] .

Equations (2) and (3) describe the SBS phenomenon in an optically lossless medium in the small signal plane wave approximation. Since the density and Stokes field amplitudes, ρ(z,t) and E_S(z,t), vary slowly in both space and time, and the acoustic wave in SBS attenuates strongly, their evolution is usually reduced to two well known first order equations: the relaxation equation for ρ(z,t),

\frac{\partial ρ}{\partial t} + (\frac{Γ_{B}}{2} + i δ Ω) ρ = - i ρ_{0} \frac{\partial ε}{\partial ρ} \frac{Ω_{B}}{8 π v_{s}^{2}} E_{p} (t) E_{S}^{*} (z, t),

which describes the amplitude of the driven damped resonant oscillator (see Eq. (2)), and the partial differential equation for E_S(z,t),

\frac{\partial E_{S}}{\partial z} + \frac{n}{c} \frac{\partial E_{S}}{\partial t} = - i \frac{ω_{S}}{2 c n} \frac{\partial ε}{\partial ρ} ρ * (z, t) E_{p} (t) .

Here δΩ = Ω - Ω_B is the difference between the acoustic drive frequency, Ω, and the resonant Brillouin frequency and asterisk, *, marks complex conjugate. The right-hand side of Eq. (5) is a source of the Stokes emission.

Let us examine the spectral features of the medium’s response and the Stokes emission, (we do not consider the absolute amplitudes of the fields in this work), which are revealed through Fourier transformation of these equations using the following properties of Fourier transforms, F(ω) ≡ S[f(t)] [27],

S [f (t)] = \int_{- \infty}^{\infty} f (t) e^{- i ω t} d t, S [\frac{d}{d t} f (t)] = i ω S [f (t)], a n d S [f_{1} (t) f_{2} (t)] = \frac{1}{2 π} \int_{- \infty}^{\infty} F_{1} (ν) F_{2} (ω - ν) d ν,

where f(t) is a function of time. We emphasize that ω and ν are Fourier transform frequencies; they are the difference frequencies of the acoustic, Stokes and pump signals from their respective line centers. One then gets from Eqs. (4) and (5)

(\frac{Γ_{B}}{2} - i (ω + δ Ω)) \tilde{ρ} * (z, ω) = i ρ_{0} \frac{\partial ε}{\partial ρ} \frac{Ω_{B}}{8 π v_{s}^{2}} \frac{1}{2 π} \int_{- \infty}^{\infty} {\tilde{E}}_{p}^{' *} (ν) {\tilde{E}}_{S}^{'} (z, ω - ν) d ν,

and

\frac{d {\tilde{E}}_{S} (z, ω)}{d z} + i ω \frac{n}{c} {\tilde{E}}_{S} (z, ω) = - i \frac{ω_{S}}{2 c n} \frac{\partial ε}{\partial ρ} \frac{1}{2 π} \int_{- \infty}^{\infty} {\tilde{E}}_{p} (ν) \tilde{ρ} * (z, ω - ν) d ν .

Here we have primed the Stokes and pump fields within (7), which are responsible for inducing the acoustic wave, to distinguish them from the generated Stokes field, (LHS of (8)), and from the pump field, which generates the new Stokes field, (the field ${\tilde{E}}_{p} (ω)$ under the integral in (8)). We note that here δΩ is a detuning parameter, which can influence only the strength of the medium’s response to the drive force at frequency, Ω. (In the case of non-monochromatic pump and Stokes fields δΩ can be considered as the difference between the central frequencies of their spectral contours.)

Let us consider the case of a typical SBS slow light experiment in which the spectrum of the Stokes signal corresponds to that of a temporally smooth pulse and the spectrum of the pump radiation is the Fourier-transform of a CW field the amplitude of which is randomly fluctuating in time.

As seen, the right-hand sides of these equations are proportional to the convolution integrals of spectra ${\tilde{E}}_{p}^{'} (ω)$ and ${\tilde{E}}_{S}^{'} (z, ω)$ in Eq. (7), and ${\tilde{E}}_{p} (ω)$ and ${\tilde{ρ}}^{*} (z, ω)$ in Eq. (8), respectively.

Equation (7) is an algebraic equation, the solution of which is the spectrum of the medium’s response

{\tilde{ρ}}^{*} (z, ω) = i ρ_{0} \frac{\partial ε}{\partial ρ} \frac{Ω_{B}}{8 π v_{s}^{2}} \frac{F_{ρ} (z, ω)}{(Γ_{B} / 2 - i (ω + δ Ω))},

where

F_{ρ} (z, ω) = \frac{1}{2 π} \int_{- \infty}^{\infty} {\tilde{E}}_{p}^{*} (ν) {\tilde{E}}_{S} (z, ω - ν) d ν

is the function, which determines the spectrum of the driving force for the medium’s response. The spectrum of the medium’s response is then given by the modulus of

{\tilde{ρ}}^{*} (z, ω)

,

| \tilde{ρ} (z, ω) |

.

The spectrum of the Stokes field is described Eq. (8), the solution of which is

| {\tilde{E}}_{S} (z, ω) | = | e^{- i \frac{ω n}{c} z} [{\tilde{E}}_{S} (0, ω) + i \frac{ω_{S}}{2 n c} \frac{\partial ε}{\partial ρ} \int_{0}^{z} F_{E} (x, ω) e^{i \frac{ω n}{c} x} d x] |,

where

{\tilde{E}}_{S} (0, ω)

is the spectrum of an input Stokes signal at z = 0 and

F_{E} (z, ω) = \frac{1}{2 π} \int_{- \infty}^{\infty} {\tilde{E}}_{p} (ν) {\tilde{ρ}}^{*} (z, ω - ν) d ν

is the function which determines the spectrum of the source for the generated Stokes field.

Equations (9)-(12) describe the spectral features of the SBS-induced material response and Stokes field when both optical fields, pump and Stokes, are non-monochromatic. We note that solution Eq. (11) in the spectral domain is entirely consistent with the analytical solution of Eqs. (4) and (5), previously obtained in the temporal domain for stimulated scattering induced by non-monochromatic pump and monochromatic Stokes fields in [29–32], and for monochromatic pump and non-monochromatic Stokes fields in [33].

It is easily seen that the solution (11) for the Stokes field differs substantially from that usually deduced in textbooks from (4) and (5) in the steady state approximation (otherwise when both pump and Stokes fields are considered monochromatic),

E_{S} (z) = E_{S} (0) e^{\frac{g_{0} | E_{p} E_{p}^{*} | z}{2 (1 - i 2 δ Ω Γ_{B}^{- 1})}},

which results from the equation for the Stokes field of the form [26],

\frac{d E_{S} (z)}{d z} = [\frac{g_{0} | E_{p} E_{p}^{*} |}{2 (1 - i 2 δ Ω Γ_{B}^{- 1})}] E_{S} (z) .

Here it is again important to remember that δΩ is a detuning parameter, the value of which is the difference between the frequencies of the monochromatic pump and Stokes fields as chosen, Ω = ω_p - ω_S, and the resonant Brillouin frequency, Ω_B, |E_pE_p*| is the pump radiation intensity, and g ₀ is the standard steady state SBS gain coefficient. Equation (14) means that the Stokes field is amplified in the medium with a gain proportional to the pump radiation intensity. To appreciate the difference between this case and our case let us consider both pump and Stokes fields in Eqs. (7)-(12) to be monochromatic. The spectra of the amplitudes of driving forces and of the medium’s response then reduce to δ-functions at ω = 0 and these depend on z only. The equation for the Stokes field is then,

\frac{d E_{S} (z)}{d z} = [\frac{g_{0} (E_{p}^{' *} E_{S}^{'} (z))}{2 (1 - i 2 δ Ω Γ_{B}^{- 1})}] E_{p} .

The physical meaning Eq. (15) is substantially different from that of Eq. (14), though their mathematical forms may look similar. This equation describes the spatial evolution of the amplitude of the Stokes field, which results from reflection of the pump field by the induced acoustic wave. Since the Stokes field on the RHS of Eq. (15) is responsible for creating the acoustic wave, it is not the same as the reflected the Stokes field on the LHS and therefore it is distinguished by its prime. Though for this monochromatic case the Stokes fields have the same frequency, their roles still remain physically distinct as in our general treatment above. Such distinction is not made in the text-book treatment that leads to Eq. (14) and to its familiar exponential solution, Eq. (13), which displays “gain” and a “modified propagation constant” for the Stokes field (see [10]). Evidently the solution for Eq. (15) cannot be the same. As such, though the RHS of this equation has both real and imaginary parts (in the case of non-zero detuning, δΩ), this does not modify the propagation constant for the generated Stokes field and therefore it can have no bearing on changing the refractive and group indexes for this field.

Returning now to the solutions (9)-(12) of Eqs. (7) and (8) for the general case in which either one of the fields or both have nonzero bandwidth, they display three important features of the SBS interaction: i) the external input Stokes signal, as seen in Eq. (11), propagates through a non-absorbing medium without gain or measurable loss (its energy losses for creating the acoustic wave is usually negligible), ii) the SBS-generated Stokes signal is a result of reflection of the pump radiation by the acoustic wave, which is created by the pump and the original Stokes fields (see Eqs. (4) and (7)), and iii) each spectral component of the generated Stokes signal arises from a range of spectral components of the non-monochromatic pump and Stokes fields, see Eqs. (11) and (12).

To see the consequences of this let us consider the case when the growth of the Stokes field along z is small. As such, we can drop the z dependence of ${\tilde{E}}_{S} (z, ω)$ and ${\tilde{ρ}}^{*} (z, ω)$ . While this approximation does not allow us to describe gain narrowing of the SBS spectrum, typical for higher amplification [26,28], it still captures reasonably well the trends in the spectral features of the Stokes field, ${\tilde{E}}_{S} (z, ω)$ , the medium response, $| \tilde{ρ} (ω) |$ , and so the modified refractive and group indices. It then follows from Eq. (11) that the output spectrum of the Stokes signal, $| {\tilde{E}}_{S} (ω) |$ , is the sum of the Fourier spectra of the input Stokes signal, $| {\tilde{E}}_{S} (0, ω) |$ , and the convoluted spectrum of the pump field and medium’s excitation, characteristics which are described by the function F _E(ω) (see Eq. (12)). The latter is the SBS-induced contribution to the Stokes signal, and is determined by which of ${\tilde{E}}_{p} (ω)$ or ${\tilde{ρ}}^{*} (ω)$ is spectrally broadest. These spectral features of the output Stokes radiation are consistent with those predicted earlier in the low gain approximation using the temporal domain treatment of stimulated Raman scattering [30–32] and SBS [26] with non-monochromatic pump fields.

In contrast to the spectral features of the Stokes emission, those of the medium’s response, $| \tilde{ρ} (ω) |$ , have to date received little attention. According to Eq. (9) this case is not as straightforward as that of the Stokes spectrum since a Lorentzian shape multiplier of bandwidth Γ_B appears in addition to the convolution integral, F_ρ(ω). From Eq. (9), $| \tilde{ρ} (ω) |$ is given as

| \tilde{ρ} (ω) | = ρ_{0} \frac{\partial ε}{\partial ρ} \frac{Ω_{B}}{8 π v_{s}^{2}} \frac{| F_{ρ} (ω) |}{\sqrt{Γ_{B}^{2} / 4 + {(ω + δ Ω)}^{2}}} .

Examples of the spectra, $| \tilde{ρ} (ω) |$ , are shown in Fig. 1 by solid curves for δΩ = 0 (Fig. 1(a)) and δΩ = 0.1 GHz (Fig. 1(b)). It follows from Eq. (16) that when the spectral width, δω_ρ, of the driving force, |F_ρ(ω)| (dashed curves in Fig. 1(a)), is narrower than Γ_B, the width, Γ, and centre frequency of the medium’s spectrum is determined by |F_ρ(ω)|, (the convolution of ${\tilde{E}}_{p}^{'} (ω)$ and ${\tilde{E}}_{S}^{'} (z, ω)$ as given in Eq. (10)), the central frequency of which is detuned by δΩ as shown by curve 1 in Fig. 1(b). Nonzero detuning δΩ results also in decreased amplitude of the medium’s response (compare solid curves 1 of Figs. 1(a) and 1(b)). As δω_ρ increases, Γ grows and for δω_ρ > Γ_B it saturates at ~1.7Γ_B (solid curves 3 and 4 in Fig. 1(a)), and the effect of the detuning δΩ on the features of the medium’s spectrum becomes negligible (compare solid curves 3 and 4 in Figs. 1(a) and 1(b)). In essence, this means that irrespective of how broad the bandwidth of the broadband pump and/or Stokes emission is/are, the bandwidth of the material response spectrum is predominantly determined by the features of the material and can never be much greater than that of the medium’s resonant response (~Γ_B). This is exactly the features expected from an externally driven damped resonant oscillator (see Eq. (2)).

Fig. 1 (a) LHS and RHS halves of the (symmetrical) spectra of |F_ρ(ω)| (dashed) and of $| \tilde{ρ} (ω) |$ (solid) respectively for the widths of F_ρ(ω) 0.02(1), 0.2(2), 2(3) and 20(4) GHz, when Γ_B = 0.2 GHz and δΩ = 0. The dotted line 5 is the Lorentzian with Γ_B = 0.2 GHz; (b) spectra of $| \tilde{ρ} (ω) |$ for the same spectral widths of F_ρ(ω) when δΩ = 0.1 GHz.

Download Full Size | PDF

From Eq. (16) the SBS induced dispersion of refractive index, Δn(ω) ≅ (∂ε/∂ρ)| $\tilde{ρ} (ω)$ |/2n ₀, and its corresponding induced group index, n_g(ω ₀) = ω ₀(dΔn/dω)|_ω ₀ may be directly determined. To do so let us replace the relative frequency ω in the equations above with the absolute frequency, ω’, and consider the spectrum of |F_ρ(ω’)| to be Gaussian in shape centred at frequency Ω_B, F_ρ(ω’) = F_ρ _aexp[-(ω’-Ω_B)²/δω_ρ ²], where $F_{ρ a} = \sqrt{I_{p} I_{S}}$ is the spectral amplitude and I_S is the Stokes signal intensity. The induced group index at the absolute frequency ω’ is then

n_{g} (ω') = - ρ_{0} {(\frac{\partial ε}{\partial ρ})}^{2} \frac{Ω_{B} \sqrt{I_{p} I_{S}}}{16 π n_{0} v_{s}^{2}} \frac{ω' (ω' - Ω_{B})}{δ ω_{ρ}^{2}} \frac{[Γ_{B}^{2} / 2 + 2 {(ω' - Ω_{B})}^{2} + δ ω_{ρ}^{2}]}{{[Γ_{B}^{2} / 2 + {(ω' - Ω_{B})}^{2}]}^{- \frac{3}{2}}} e^{- \frac{{(ω' - Ω_{B})}^{2}}{δ ω_{ρ}^{2}}} .

Its dispersion has a maximum at ω’ = Ω_B - Δ and a minimum at ω’ = Ω_B + Δ, where Δ is functionally dependent on δω_ρ and is < Γ_B/2. These maximum and minimum, in principle, may be of very high amplitude, however, they are located at ω’ ≈Ω_B. At the Stokes frequency, ω’ = ω _S, since ω _S >> Ω_B > Γ_B (ω _S ≈1.7×10⁴Ω_B in silica), Eq. (17) reduces to

n_{g} (ω_{S}) = - \frac{2 B ω_{S}}{δ ω_{ρ}^{2}} e^{- ω_{S}^{2} / δ ω_{ρ}^{2}},

which shows that the SBS-induced n_g(ω _S) is independent of both Γ_B and Ω_B, and is negligibly small for any reasonable bandwidth, δω_ρ, of

| \tilde{ρ} (ω) |

. As such the acoustic resonance can have next to no effect in enhancing or modifying the natural group index in the medium for the Stokes signal.

In summary we have shown that spectral broadening of the pump radiation by any reasonable amount results in only minute increase of the spectral width of the resonant material’s excitation at the acoustic frequency in SBS. Because this frequency is far away from the frequency of the Stokes optical signal to be delayed it cannot be effective in modifying the natural group index of the medium for the Stokes signal.

Acknowledgments

The work is supported by the Engineering and Physics Research Council, UK, Grant No GR/R56105/01, by the Scottish University Physics Alliance, and by the Educational Scientific Complex of the Lebedev Physical Institute of the Russian Academy of Sciences.

References and Links

1. D. J. Gauthier, “Slow light brings faster communications,” Phys. World 18, 30–32 (2005).

2. K. Y. Song, M. G. Herráez, and L. Thévenaz, “Gain-assisted pulse advancement using single and double Brillouin gain peaks in optical fibers,” Opt. Express 13(24), 9758–9765 (2005). [PubMed]

3. M. D. Stenner, M. A. Neifeld, Z. Zhu, A. M. C. Dawes, and D. J. Gauthier, “Distortion management in slow-light pulse delay,” Opt. Express 13(25), 9995–10002 (2005). [PubMed]

4. M. G. Herráez, K. Y. Song, and L. Thévenaz, “Arbitrary-bandwidth Brillouin slow light in optical fibers,” Opt. Express 14(4), 1395–1400 (2006).

5. A. Minardo, R. Bernini, and L. Zeni, “Low distortion Brillouin slow light in optical fibers using AM modulation,” Opt. Express 14(13), 5866–5876 (2006). [PubMed]

6. E. Shumakher, N. Orbach, A. Nevet, D. Dahan, and G. Eisenstein, “On the balance between delay, bandwidth and signal distortion in slow light systems based on stimulated Brillouin scattering in optical fibers,” Opt. Express 14(13), 5877–5884 (2006). [PubMed]

7. Z. Zhu and D. J. Gauthier, “Nearly transparent SBS slow light in an optical fiber,” Opt. Express 14(16), 7238–7245 (2006). [PubMed]

8. A. Zadok, A. Eyal, and M. Tur, “Extended delay of broadband signals in stimulated Brillouin scattering slow light using synthesized pump chirp,” Opt. Express 14(19), 8498–8505 (2006). [PubMed]

9. S. Chin, M. G. Herraez, and L. Thévenaz, “Zero-gain slow & fast light propagation in an optical fiber,” Opt. Express 14(22), 10684–10692 (2006). [PubMed]

10. T. Schneider, M. Junker, and K.-U. Lauterbach, “Potential ultra wide slow-light bandwidth enhancement,” Opt. Express 14(23), 11082–11087 (2006). [PubMed]

11. V. P. Kalosha, L. Chen, and X. Bao, “Slow and fast light via SBS in optical fibers for short pulses and broadband pump,” Opt. Express 14(26), 12693–12703 (2006). [PubMed]

12. Z. Zhu, A. M. C. Dawes, D. J. Gauthier, L. Zhang, and A. E. Willner, “Broadband SBS slow light in an optical fiber,” J. Lightwave Technol. 25(1), 201–206 (2007).

13. K. Y. Song and K. Hotate, “25 GHz bandwidth Brillouin slow light in optical fibers,” Opt. Lett. 32(3), 217–219 (2007). [PubMed]

14. Z. Lu, Y. Dong, and Q. Li, “Slow light in multi-line Brillouin gain spectrum,” Opt. Express 15(4), 1871–1877 (2007). [PubMed]

15. B. Zhang, L. Yan, I. Fazal, L. Zhang, A. E. Willner, Z. Zhu, and D. J. Gauthier, “Slow light on Gbit/s differential-phase-shift-keying signals,” Opt. Express 15(4), 1878–1883 (2007). [PubMed]

16. L. Yi, L. Zhan, W. Hu, and Y. Xia, “Delay of broadband signals using slow light in stimulated Brillouin scattering with phase-modulated pump,” IEEE Photon. Technol. Lett. 19(8), 619–621 (2007).

17. B. Zhang, L. Zhang, L.-S. Yan, I. Fazal, J.-Y. Yang, and A. E. Willner, “Continuously-tunable, bit-rate variable OTDM using broadband SBS slow-light delay line,” Opt. Express 15(13), 8317–8322 (2007). [PubMed]

18. Z. Shi, R. Pant, Z. Zhu, M. D. Stenner, M. A. Neifeld, D. J. Gauthier, and R. W. Boyd, “Design of a tunable time-delay element using multiple gain lines for increased fractional delay with high data fidelity,” Opt. Lett. 32(14), 1986–1988 (2007). [PubMed]

19. R. Pant, M. D. Stenner, M. A. Neifeld, and D. J. Gauthier, “Optimal pump profile designs for broadband SBS slow-light systems,” Opt. Express 16(4), 2764–2777 (2008). [PubMed]

20. L. Ren and Y. Tomita, “Reducing group-velocity-dispersion-dependent broadening of stimulated Brillouin scattering slow light in an optical fiber by use of a single pump laser,” J. Opt. Soc. Am. B 25(5), 741–746 (2008).

21. T. Sakamoto, T. Yamamoto, K. Shiraki, and T. Kurashima, “Low distortion slow light in flat Brillouin gain spectrum by using optical frequency comb,” Opt. Express 16(11), 8026–8032 (2008). [PubMed]

22. S. Wang, L. Ren, Y. Liu, and Y. Tomita, “Zero-broadening SBS slow light propagation in an optical fiber using two broadband pump beams,” Opt. Express 16(11), 8067–8076 (2008). [PubMed]

23. T. Schneider, R. Henker, K.-U. Lauterbach, and M. Junker, “Distortion reduction in Slow Light systems based on stimulated Brillouin scattering,” Opt. Express 16(11), 8280–8285 (2008). [PubMed]

24. A. Cheng, M. P. Fok, and C. Shu, “Wavelength-transparent, stimulated-Brillouin-scattering slow light using cross-gain-modulation-based wavelength converter and Brillouin fiber laser,” Opt. Lett. 33(22), 2596–2598 (2008). [PubMed]

25. R. W. Boyd, and D. J. Gauthier, “Slow and fast light,” in Progress in Optics, 43, 497 (2002).

26. B. Y. Zeldovich, N. F. Pilipetskii, and V. V. Shkunov, Principles of phase conjugation, Springer Verlag, Berlin, 1985.

27. G. A. Korn, and T. M. Korn, Manual of mathematics, McGraw-Hill, New York, 1967.

28. C. L. Tang, “Saturation and spectral characteristics of the Stokes emission in the stimulated Brillouin process,” J. Appl. Phys. 37(8), 2945–2955 (1966).

29. N. Kroll, “Excitation of hypersonic vibrations by means of photoelastic coupling of high-intensity light waves to elastic waves,” J. Appl. Phys. 36(1), 34–43 (1965).

30. R. L. Carman, F. Shimizu, C. S. Wang, and N. Bloembergen, “Theory of Stokes pulse shapes in transient Raman scattering,” Phys. Rev. A 2(1), 60–72 (1970).

31. S. A. Akhmanov, K. N. Drabovich, A. P. Sukhorukov, and A. S. Chirkin, “Stimulated Raman scattering in a field of ultrashort light pulses,” Sov. Phys. JETP 32, 266–273 (1971).

32. S. A. Akhmanov, Yu. E. Dyakov, and A. S. Chirkin, Introduction to Statistical Radiophysics and Optics, Berlin, Springer, 1988.

33. V. I. Kovalev, N. E. Kotova, and R. G. Harrison, “Effect of acoustic wave inertia and its implication to slow light via stimulated Brillouin scattering in an extended medium,” Opt. Express 17(4), 2826–2833 (2009). [PubMed]

“Slow Light” in stimulated Brillouin scattering: on the influence of the spectral width of pump radiation on the group index

Abstract

Acknowledgments

References and Links

Cited By

Figures (1)

Equations (18)

Optics Express