Abstract

Slow-light effect via stimulated Brillouin scattering (SBS) in single-mode optical fibers was considered for short probe pulses of nanosecond duration relevant to Gb/s data streams. 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 achieves advancement.

© 2006 Optical Society of America

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  1. K. Y. Song, M. Herraez, and L. Th´evenaz, "Observation of pulse delaying and advancement in optical fibers using stimulated Brillouin scattering," Opt. Express 13, 82-88 (2005).
    [CrossRef] [PubMed]
  2. Y. Okawachi, M. S. Bigelow, J. E. Sharping, Z. Zhu, A. Schweinsberg, D. J. Gauthier, R. W. Boyd, and A. L. Gaeta, "Tunable all-optical delays via Brillouin slow light in an optical fiber," Phys. Rev. Lett. 94, 153902 (2005).
    [CrossRef] [PubMed]
  3. K. Y. Song, M. G. Herraez, and L. Thevenaz, "Long optically controlled delays in optical fibers," Opt. Lett. 30, 1782-1784 (2005).
    [CrossRef] [PubMed]
  4. M. G. Herraez, K. Y. Song, and L. Thevenaz, "Optically controlled slow and fast light in optical fibers using stimulated Brillouin scattering," Appl. Phys. Lett. 87, 081113 (2005).
    [CrossRef]
  5. K. Y. Song, M. G. Herraez, and L. Thevenaz, "Gain-assisted pulse advancement using single and double Brillouin gain peaks in optical fibers," Opt. Express 13, 9758-9765 (2005).
    [CrossRef] [PubMed]
  6. 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, 9995-10002 (2005).
    [CrossRef] [PubMed]
  7. M. G. Herraez, K. Y. Song, and L. Thevenaz, "Arbitrary-bandwidth Brillouin slow light in optical fibers," Opt. Express 14, 1395-1400 (2006).
    [CrossRef]
  8. <jrn>. Z. Zhu, A. M. C. Dawes, D. J. Gauthier, L. Zhang, and A. E. Willner, "12-GHz-Bandwidth SBS Slow Light in optical Fibers," Postdeadline paper PDP1, OFC 2006, Anaheim, CA, March 5-20, 2006.</jrn>
  9. A. Minardo, R. Bernini, and L. Zeni, "Low distortion Brillouin slow light in optical fibers using AMmodulation," Opt. Express 14, 5866-5876 (2006).
    [CrossRef] [PubMed]
  10. 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, 5877-5884 (2006).
    [CrossRef] [PubMed]
  11. L. Yi, L. Zhan, Y. Su, W. Hu, L. Leng, Y. Song, H. Shen, and Y. Xia, "Delay of RZ PRBS data based on wideband SBS by phase-modulating the Brollouin pump," in Slow and Fast Light 2006 Technical Digest (Optical Society of America, Washington, DC, 2006), presentation WB4.
  12. 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, 8498-8505 (2006).
    [CrossRef] [PubMed]
  13. Z. Zhu, D. J. Gauthier, Y. Okawachi, J. E. Sharping, A. L. Gaeta, R. W. Boyd, and A. E. Willner, "Numerical study of all-optical slow-light delays via stimulated Brillouin scattering in an optical fiber," J. Opt. Soc. Am. B 22, 2378-2384 (2005).
    [CrossRef]
  14. L. Thevenaz, K. Y. Song, and M. G. Herraez, "Time biasing due to the slow-light effect in distributed fiber-optic Brillouin sensors," Opt. Lett. 31, 715-717 (2006).
    [CrossRef] [PubMed]
  15. L. Zou, X. Bao, S. Yang, L. Chen, and F. Ravet, "Effect of Brillouin slow light on distributed Brillouin fiber sensors," Opt. Lett. 31, 2698-2700 (2006).
    [CrossRef] [PubMed]
  16. F. Ravet, L. Chen, X. Bao, L. Zou, and V. P. Kalosha, "Theoretical study of the effect of slow light on BOTDA spatial resolution," Opt. Express 14, 10351-10358 (2006).
    [CrossRef] [PubMed]
  17. R. W. Boyd, Nonlinear Optics (Academic Press, San Diego, 2003).
  18. V. P. Kalosha, E. Ponomarev, L. Chen, and X. Bao, "How to obtain high spectral resolution of SBS-based distributed sensing by using nanosecond pulses," Opt. Express 14, 2071-2078 (2006).
    [CrossRef] [PubMed]
  19. V. P. Kalosha and J. Herrmann, "Ultrawide spectral broadening and compression of single extremely short pulses in the visible, uv-vuv, and middle infrared by high-order stimulated Raman scattering," Phys. Rev. A 68, 023812 (2003).
    [CrossRef]
  20. C. Montes, A. Mikhailov, A. Picozzi, and F. Ginovart, "Dissipative three-wave structures in stimulated backscattering. I. A subluminous solitary attractor," Phys. Rev. E 55, 1086-1091 (1997).
    [CrossRef]
  21. C. Montes, A. Picozzi, and D. Bahloul,"Dissipative three-wave structures in stimulated backscattering. II. Superluminous and subluminous solitons," Phys. Rev. E 55, 1092-1105 (1997).
    [CrossRef]
  22. X. Bao, Q. Yu, V. P. Kalosha, and L. Chen, "Influence of transient phonon relaxation on the Brillouin loss spectrum of nanosecond pulses," Opt. Lett. 31, 888-890 (2006).
    [CrossRef] [PubMed]
  23. J. Coste and C. Montes, "Asymptotic evolution of stimulated Brillouin scattering: Implication for optical fibers," Phys. Rev. A 34, 3940-3949 (1986).
    [CrossRef] [PubMed]

2006 (9)

M. G. Herraez, K. Y. Song, and L. Thevenaz, "Arbitrary-bandwidth Brillouin slow light in optical fibers," Opt. Express 14, 1395-1400 (2006).
[CrossRef]

L. Thevenaz, K. Y. Song, and M. G. Herraez, "Time biasing due to the slow-light effect in distributed fiber-optic Brillouin sensors," Opt. Lett. 31, 715-717 (2006).
[CrossRef] [PubMed]

V. P. Kalosha, E. Ponomarev, L. Chen, and X. Bao, "How to obtain high spectral resolution of SBS-based distributed sensing by using nanosecond pulses," Opt. Express 14, 2071-2078 (2006).
[CrossRef] [PubMed]

X. Bao, Q. Yu, V. P. Kalosha, and L. Chen, "Influence of transient phonon relaxation on the Brillouin loss spectrum of nanosecond pulses," Opt. Lett. 31, 888-890 (2006).
[CrossRef] [PubMed]

A. Minardo, R. Bernini, and L. Zeni, "Low distortion Brillouin slow light in optical fibers using AMmodulation," Opt. Express 14, 5866-5876 (2006).
[CrossRef] [PubMed]

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, 5877-5884 (2006).
[CrossRef] [PubMed]

L. Zou, X. Bao, S. Yang, L. Chen, and F. Ravet, "Effect of Brillouin slow light on distributed Brillouin fiber sensors," Opt. Lett. 31, 2698-2700 (2006).
[CrossRef] [PubMed]

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, 8498-8505 (2006).
[CrossRef] [PubMed]

F. Ravet, L. Chen, X. Bao, L. Zou, and V. P. Kalosha, "Theoretical study of the effect of slow light on BOTDA spatial resolution," Opt. Express 14, 10351-10358 (2006).
[CrossRef] [PubMed]

2005 (7)

2003 (1)

V. P. Kalosha and J. Herrmann, "Ultrawide spectral broadening and compression of single extremely short pulses in the visible, uv-vuv, and middle infrared by high-order stimulated Raman scattering," Phys. Rev. A 68, 023812 (2003).
[CrossRef]

1997 (1)

C. Montes, A. Picozzi, and D. Bahloul,"Dissipative three-wave structures in stimulated backscattering. II. Superluminous and subluminous solitons," Phys. Rev. E 55, 1092-1105 (1997).
[CrossRef]

1986 (1)

J. Coste and C. Montes, "Asymptotic evolution of stimulated Brillouin scattering: Implication for optical fibers," Phys. Rev. A 34, 3940-3949 (1986).
[CrossRef] [PubMed]

Bahloul, D.

C. Montes, A. Picozzi, and D. Bahloul,"Dissipative three-wave structures in stimulated backscattering. II. Superluminous and subluminous solitons," Phys. Rev. E 55, 1092-1105 (1997).
[CrossRef]

Bao, X.

Bernini, R.

Bigelow, M. S.

Y. Okawachi, M. S. Bigelow, J. E. Sharping, Z. Zhu, A. Schweinsberg, D. J. Gauthier, R. W. Boyd, and A. L. Gaeta, "Tunable all-optical delays via Brillouin slow light in an optical fiber," Phys. Rev. Lett. 94, 153902 (2005).
[CrossRef] [PubMed]

Boyd, R. W.

Y. Okawachi, M. S. Bigelow, J. E. Sharping, Z. Zhu, A. Schweinsberg, D. J. Gauthier, R. W. Boyd, and A. L. Gaeta, "Tunable all-optical delays via Brillouin slow light in an optical fiber," Phys. Rev. Lett. 94, 153902 (2005).
[CrossRef] [PubMed]

Z. Zhu, D. J. Gauthier, Y. Okawachi, J. E. Sharping, A. L. Gaeta, R. W. Boyd, and A. E. Willner, "Numerical study of all-optical slow-light delays via stimulated Brillouin scattering in an optical fiber," J. Opt. Soc. Am. B 22, 2378-2384 (2005).
[CrossRef]

Chen, L.

Coste, J.

J. Coste and C. Montes, "Asymptotic evolution of stimulated Brillouin scattering: Implication for optical fibers," Phys. Rev. A 34, 3940-3949 (1986).
[CrossRef] [PubMed]

Dahan, D.

Dawes, A.M. C.

Eisenstein, G.

Eyal, A.

Gaeta, A. L.

Z. Zhu, D. J. Gauthier, Y. Okawachi, J. E. Sharping, A. L. Gaeta, R. W. Boyd, and A. E. Willner, "Numerical study of all-optical slow-light delays via stimulated Brillouin scattering in an optical fiber," J. Opt. Soc. Am. B 22, 2378-2384 (2005).
[CrossRef]

Y. Okawachi, M. S. Bigelow, J. E. Sharping, Z. Zhu, A. Schweinsberg, D. J. Gauthier, R. W. Boyd, and A. L. Gaeta, "Tunable all-optical delays via Brillouin slow light in an optical fiber," Phys. Rev. Lett. 94, 153902 (2005).
[CrossRef] [PubMed]

Gauthier, D. J.

Herr´aez, M. G.

M. G. Herraez, K. Y. Song, and L. Thevenaz, "Optically controlled slow and fast light in optical fibers using stimulated Brillouin scattering," Appl. Phys. Lett. 87, 081113 (2005).
[CrossRef]

K. Y. Song, M. G. Herraez, and L. Thevenaz, "Gain-assisted pulse advancement using single and double Brillouin gain peaks in optical fibers," Opt. Express 13, 9758-9765 (2005).
[CrossRef] [PubMed]

Herraez, M.

Herraez, M. G.

Herrmann, J.

V. P. Kalosha and J. Herrmann, "Ultrawide spectral broadening and compression of single extremely short pulses in the visible, uv-vuv, and middle infrared by high-order stimulated Raman scattering," Phys. Rev. A 68, 023812 (2003).
[CrossRef]

Kalosha, V. P.

Minardo, A.

Montes, C.

C. Montes, A. Picozzi, and D. Bahloul,"Dissipative three-wave structures in stimulated backscattering. II. Superluminous and subluminous solitons," Phys. Rev. E 55, 1092-1105 (1997).
[CrossRef]

J. Coste and C. Montes, "Asymptotic evolution of stimulated Brillouin scattering: Implication for optical fibers," Phys. Rev. A 34, 3940-3949 (1986).
[CrossRef] [PubMed]

Neifeld, M. A.

Nevet, A.

Okawachi, Y.

Y. Okawachi, M. S. Bigelow, J. E. Sharping, Z. Zhu, A. Schweinsberg, D. J. Gauthier, R. W. Boyd, and A. L. Gaeta, "Tunable all-optical delays via Brillouin slow light in an optical fiber," Phys. Rev. Lett. 94, 153902 (2005).
[CrossRef] [PubMed]

Z. Zhu, D. J. Gauthier, Y. Okawachi, J. E. Sharping, A. L. Gaeta, R. W. Boyd, and A. E. Willner, "Numerical study of all-optical slow-light delays via stimulated Brillouin scattering in an optical fiber," J. Opt. Soc. Am. B 22, 2378-2384 (2005).
[CrossRef]

Orbach, N.

Picozzi, A.

C. Montes, A. Picozzi, and D. Bahloul,"Dissipative three-wave structures in stimulated backscattering. II. Superluminous and subluminous solitons," Phys. Rev. E 55, 1092-1105 (1997).
[CrossRef]

Ponomarev, E.

Ravet, F.

Schweinsberg, A.

Y. Okawachi, M. S. Bigelow, J. E. Sharping, Z. Zhu, A. Schweinsberg, D. J. Gauthier, R. W. Boyd, and A. L. Gaeta, "Tunable all-optical delays via Brillouin slow light in an optical fiber," Phys. Rev. Lett. 94, 153902 (2005).
[CrossRef] [PubMed]

Sharping, J. E.

Y. Okawachi, M. S. Bigelow, J. E. Sharping, Z. Zhu, A. Schweinsberg, D. J. Gauthier, R. W. Boyd, and A. L. Gaeta, "Tunable all-optical delays via Brillouin slow light in an optical fiber," Phys. Rev. Lett. 94, 153902 (2005).
[CrossRef] [PubMed]

Z. Zhu, D. J. Gauthier, Y. Okawachi, J. E. Sharping, A. L. Gaeta, R. W. Boyd, and A. E. Willner, "Numerical study of all-optical slow-light delays via stimulated Brillouin scattering in an optical fiber," J. Opt. Soc. Am. B 22, 2378-2384 (2005).
[CrossRef]

Shumakher, E.

Song, K. Y.

Stenner, M. D.

Th´evenaz, L.

Thevenaz, L.

Tur, M.

Willner, A. E.

Yang, S.

Yu, Q.

Zadok, A.

Zeni, L.

Zhu, Z.

Zou, L.

Appl. Phys. Lett. (1)

M. G. Herraez, K. Y. Song, and L. Thevenaz, "Optically controlled slow and fast light in optical fibers using stimulated Brillouin scattering," Appl. Phys. Lett. 87, 081113 (2005).
[CrossRef]

J. Opt. Soc. Am. B (1)

Opt. Express (9)

K. Y. Song, M. G. Herraez, and L. Thevenaz, "Gain-assisted pulse advancement using single and double Brillouin gain peaks in optical fibers," Opt. Express 13, 9758-9765 (2005).
[CrossRef] [PubMed]

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, 9995-10002 (2005).
[CrossRef] [PubMed]

M. G. Herraez, K. Y. Song, and L. Thevenaz, "Arbitrary-bandwidth Brillouin slow light in optical fibers," Opt. Express 14, 1395-1400 (2006).
[CrossRef]

K. Y. Song, M. Herraez, and L. Th´evenaz, "Observation of pulse delaying and advancement in optical fibers using stimulated Brillouin scattering," Opt. Express 13, 82-88 (2005).
[CrossRef] [PubMed]

V. P. Kalosha, E. Ponomarev, L. Chen, and X. Bao, "How to obtain high spectral resolution of SBS-based distributed sensing by using nanosecond pulses," Opt. Express 14, 2071-2078 (2006).
[CrossRef] [PubMed]

A. Minardo, R. Bernini, and L. Zeni, "Low distortion Brillouin slow light in optical fibers using AMmodulation," Opt. Express 14, 5866-5876 (2006).
[CrossRef] [PubMed]

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, 5877-5884 (2006).
[CrossRef] [PubMed]

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, 8498-8505 (2006).
[CrossRef] [PubMed]

F. Ravet, L. Chen, X. Bao, L. Zou, and V. P. Kalosha, "Theoretical study of the effect of slow light on BOTDA spatial resolution," Opt. Express 14, 10351-10358 (2006).
[CrossRef] [PubMed]

Opt. Lett. (4)

Phys. Rev. A (2)

V. P. Kalosha and J. Herrmann, "Ultrawide spectral broadening and compression of single extremely short pulses in the visible, uv-vuv, and middle infrared by high-order stimulated Raman scattering," Phys. Rev. A 68, 023812 (2003).
[CrossRef]

J. Coste and C. Montes, "Asymptotic evolution of stimulated Brillouin scattering: Implication for optical fibers," Phys. Rev. A 34, 3940-3949 (1986).
[CrossRef] [PubMed]

Phys. Rev. E (1)

C. Montes, A. Picozzi, and D. Bahloul,"Dissipative three-wave structures in stimulated backscattering. II. Superluminous and subluminous solitons," Phys. Rev. E 55, 1092-1105 (1997).
[CrossRef]

Phys. Rev. Lett. (1)

Y. Okawachi, M. S. Bigelow, J. E. Sharping, Z. Zhu, A. Schweinsberg, D. J. Gauthier, R. W. Boyd, and A. L. Gaeta, "Tunable all-optical delays via Brillouin slow light in an optical fiber," Phys. Rev. Lett. 94, 153902 (2005).
[CrossRef] [PubMed]

Other (4)

<jrn>. Z. Zhu, A. M. C. Dawes, D. J. Gauthier, L. Zhang, and A. E. Willner, "12-GHz-Bandwidth SBS Slow Light in optical Fibers," Postdeadline paper PDP1, OFC 2006, Anaheim, CA, March 5-20, 2006.</jrn>

L. Yi, L. Zhan, Y. Su, W. Hu, L. Leng, Y. Song, H. Shen, and Y. Xia, "Delay of RZ PRBS data based on wideband SBS by phase-modulating the Brollouin pump," in Slow and Fast Light 2006 Technical Digest (Optical Society of America, Washington, DC, 2006), presentation WB4.

R. W. Boyd, Nonlinear Optics (Academic Press, San Diego, 2003).

C. Montes, A. Mikhailov, A. Picozzi, and F. Ginovart, "Dissipative three-wave structures in stimulated backscattering. I. A subluminous solitary attractor," Phys. Rev. E 55, 1086-1091 (1997).
[CrossRef]

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Figures (10)

Fig. 1.
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.

Fig. 2.
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.

Fig. 3.
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).

Fig. 4.
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.

Fig. 5.
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.

Fig. 6.
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.

Fig. 7.
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.

Fig. 8.
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.

Fig. 9.
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).

Fig. 10.
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).

Equations (7)

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( 1 V t z ) E p = Q E s α 2 E p ,
( 1 V t + z ) E s = Q * E p α 2 E s ,
[ 2 t 2 + 2 ( Γ i Ω ) t + ( Ω B 2 Ω 2 2 i Γ Ω ) ] Q = i g E p E s * ,
[ t + Γ + i ( Ω B Ω ) ] Q = 1 2 Γ g B E p E s * .
Q ω = i g ( E p E s * ) ω ( ω ω 1 + i Γ ) ( ω ω 2 + i Γ ) ,
P s ( ω , z ) = P s ( ω , z = 0 ) exp [ G ( z ) 1 + ω 2 Γ 2 α z ] ,
Δ T ( ω , z ) = G ( z ) 2 Γ 1 ω 2 Γ 2 ( 1 + ω 2 Γ 2 ) 2 ,

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