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

Aldo Minardo; Romeo Bernini; Luigi Zeni

doi:10.1364/OE.18.001788

Equations of the paper [1] are referenced below as (C#). In Ref [1]. the authors claim that spectral broadening of pump radiation cannot be effective in modifying the natural group index of the medium for the Stokes pulse. Furthermore, it is argued that, as the resonant material’s excitation in SBS is spectrally far away from the Stokes optical signal, the latter cannot be delayed by any significant amount from this excitation. The above claims are based on an analytic solution of the coupled SBS equations in the Fourier domain. The solution for the spectrum of the medium’s response [Eq. (C9)], suggests that the acoustic wave bandwidth cannot be increased indefinitely, rather it saturates at ~1.7Γ_B (Γ_B is the SBS spectral bandwidth), even though the convolution of pump and Stokes signals can result in a spectrally broader driving force. The solution is correct, and it is formally identical to the one previously reported in Ref [2]. (Eq. (12)). The solution for the Stokes pulse [Eq. (C11)] was obtained by separating the Stokes field generated by SBS reflection of the pump, from the Stokes field inducing the acoustic wave. In a similar manner, the authors distinguish the pump field involved in acoustic wave generation, from the pump field generating the new Stokes field [Eqs. (C7)-(C8)]. The distinctions permit to decouple the two sides of Eq. (C8), so that the newly generated Stokes field appears only in the LHS. However, this has a meaning only when the fields have different frequencies and/or orthogonal polarization. In standard fibers such a distinction is purely formal, as the roles played by the fields cannot be separated: the external input Stokes and pump fields generate an acoustic wave via electrostriction; the acoustic wave reflects the pump field, generating a new Stokes field. In turns, this newly generated Stokes field does interact with the pump field, reinforcing the acoustic wave, and so on. This positive feedback is responsible for creating a slow light response in the medium, with respect to a propagating Stokes pulse. Let us start our analysis by noting that, when the fields cannot be separated, Eq. (C11) is not a real solution (Stokes field is implicitly present on the RHS through the medium’s response), rather is only a reformulation, in terms of an integral equation, of Eq. (C8). The latter can be alternatively written as [2]:

\frac{d {\tilde{E}}_{S} (z, ω)}{d z} = - \frac{g_{0} Γ_{B}}{2} \int_{ω''} {\tilde{E}}_{p} (z, ω^{''}) \exp (- 2 j \frac{n}{c} ω^{''} z) \times [\int_{ω^{'}} \frac{{\tilde{E}}_{p}^{*} (z, ω^{'}) {\tilde{E}}_{S} (z, ω^{'} - ω^{''} + ω)}{Γ_{B} + j (ω - ω'')} \cdot \exp (+ 2 j \frac{n}{c} ω^{'} z) d ω^{'}] d ω^{''} .

The integral in the square brackets represents the acoustic wave spectral component at ω”-ω, which interacts with the pump spectral component at ω”, contributing to the Stokes signal at ω. In Eq. (1) we have supposed that the pump-Stokes frequency shift is tuned exactly to the SBS resonant frequency, and that the medium is lossless. Integration of Eq. (1) is difficult due to the intercoupling between the different Stokes spectral components. The problem can be greatly simplified if supposing that pump radiation is monochromatic and that it is not depleted over the interaction length, L. Equation (1) then reduces to:

\frac{d {\tilde{E}}_{S} (z, ω)}{d z} = - \frac{g_{0} {| E_{p} |}^{2}}{2} \frac{Γ_{B}}{Γ_{B} + j ω} {\tilde{E}}_{S} (z, ω),

whose solution is

{\tilde{E}}_{S} (0, ω) = {\tilde{E}}_{S} (L, ω) \exp ((g_{0} {| E_{p} |}^{2} L Γ_{B}) / [2 (Γ_{B} + j ω)])

, in which

{\tilde{E}}_{S} (L, ω)

is the spectrum of the input Stokes pulse. The fiber then acts as a linear medium, whose transfer function is a single Lorentzian gain line, according to which the input Stokes pulse will be amplified and delayed as it propagates trough the extended SBS medium. Contrarily to what affirmed in Ref [3], the SBS-induced delay bandwidth can be larger than the Stokes pulse spectrum, provided that the pulse duration is sufficiently longer than the acoustic decay time τ = 2/Γ_B. Hence, SBS is able to efficiently delay an external Stokes pulse with minimal distortion, without the necessity to independently prepare suitable group index features in the medium. In case of pump radiation with a bandwidth larger than Γ_B, a broadband slow light medium will be produced according to the leftmost convolution integral in Eq. (1), even though the acoustic bandwidth is restricted to ~1.7Γ_B. This is fully consistent with the several experimental observations of broadband slow-light by use of a broadband pump (e.g [4].).

Let us suppose now that a distinction can be done between the Stokes field responsible for acoustic wave generation ( ${\hat{E}}_{S} (z, ω)$ ), and the Stokes field produced by pump SBS reflection ( ${\tilde{E}}_{S} (z, ω)$ ). This happens e.g. in highly birefringent fibers, in which the acoustic wave may be generated by pumping along one polarization axis, while “reading” is performed along the orthogonal axis, with a readout pump distant several GHz from the acoustic wave-inducing fields [5]. Even in standard fibers, the two Stokes fields can be separated when a strong input cw Stokes field interacts with a counterpropagating pulsed pump field. Assuming a small SBS exponential gain, the generated Stokes ac component is so weak that its contribution to the generation of the acoustic wave will be negligible. Note that such a situation is very common in SBS-based distributed sensing schemes. Equation (2) now becomes:

\frac{d {\tilde{E}}_{S} (z, ω)}{d z} = - \frac{g_{0} {| E_{p} |}^{2}}{2} \frac{Γ_{B}}{Γ_{B} + j ω} {\hat{E}}_{S} (z, ω) .

A more general expression for the Stokes generated field, valid for a non-monochromatic pump radiation, is reported in Ref [1]. [Eq. (C11)]. As discussed in Ref [1], in this situation the RHS on Eq. (3) does not modify the propagation constant of the generated Stokes field. Please note that in case the Stokes field responsible for acoustic wave generation is purely stationary, the RHS on Eq. (3) is identically null for any ω≠0, therefore it does not modify the propagation constant for any spectral component of Stokes signal at ω≠0. Of course, this is true as long as the ω-component of the Stokes field is so small that its contribution to electrostriction is negligible.

References and links

1. V. I. Kovalev, N. E. Kotova, and R. G. Harrison, ““Slow Light” in stimulated Brillouin scattering: on the influence of the spectral width of pump radiation on the group index,” Opt. Express 17(20), 17317–17323 (2009). [CrossRef] [PubMed]

2. A. Minardo, R. Bernini, and L. Zeni, “Stimulated Brillouin scattering modeling for high-resolution, time-domain distributed sensing,” Opt. Express 15(16), 10397–10407 (2007). [CrossRef] [PubMed]

3. 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). [CrossRef] [PubMed]

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

5. K. Y. Song, W. Zou, Z. He, and K. Hotate, “Optical time-domain measurement of Brillouin dynamic grating spectrum in a polarization-maintaining fiber,” Opt. Lett. 34(9), 1381–1383 (2009). [CrossRef] [PubMed]

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

Abstract

References and links

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Equations (3)

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