Abstract

Due to the resonant nature of Brillouin scattering, delay occurs while pulse is propagating in an optical fiber. This effect influences the location accuracy of distributed Brillouin sensors. The maximum delay in sensing fibers depends on length, position, pump and Stokes powers. Considering pump depletion, we have obtained integral solutions for the coupled amplitude equations under steady state conditions and then calculated the group delay. The results show that moderate pump depletion (which is the optimized sensor working range) mitigates significantly the delay, and the maximum delay induced at resonance is only a fraction of Brillouin Optical Time Domain (BOTDA) spatial resolution, which means that the use of pulse width to define the spatial resolution is valid when Brillouin slow light is considered. We have shown that uniform strain and temperature distribution in a fiber gives the maximum delay induced uncertainty.

© 2006 Optical Society of America

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References

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  1. K.-Y. Song, M. González-Herráez, and L. Thévenaz, "Observation of pulse delaying and advancement in optical fibers using stimulated Brillouin scattering," Opt. Express. 13, 82 (2005).
    [CrossRef] [PubMed]
  2. Y. Okawachi, M. S. Bigelow, J. E. Sharping, Z. Zhu, A. Schweinberg. 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. X. Bao, J. Dhliwayo, N. Heron, D. J. Webb, and, D. A. Jackson, "Experimental and theoretical studies on a distributed temperature sensor based on Brillouin scattering," J. Lightwave Technol. 13, 1340 (1995).
    [CrossRef]
  4. X. Bao, D. J. Webb, and D. A. Jackson, "32-km distributed temperature sensor based on Brillouin loss in an optical fiber," Opt. Lett. 18, 1561 (1993).
    [CrossRef] [PubMed]
  5. L. Zou, X. Bao, S. Afshar V. and L. Chen, "Dependence of the Brillouin frequency shift on strain and temperature in a photonic crystal fiber," Opt. Lett. 29, 1485 (2004).
    [CrossRef] [PubMed]
  6. L. Thévenaz, K.-Y. Song, and M. González Herráez, "Time biasing due to the slow-light effect in distributed fiber-optic Brillouin sensors," Opt. Lett. 31, 715 (2006).
    [CrossRef] [PubMed]
  7. R. W. Boyd, Nonlinear Optics, (San Diego,2003).
  8. X. Bao, Q. Yu, V. P. Kalosha, and L. Chen, "The influence of prolonged phonon relaxation on the Brillouin loss spectrum for the nanosecond pulses," Opt Lett. 31, 888-890 (2006).
    [CrossRef] [PubMed]
  9. E. Geinitz, S. Jetshke, U. Röpke, S. Schröter, R. Wilsch, and H. Bartelt, "The influence of pulse amplification on distributed fibre-optic Brillouin sensing and a method to compensate for systematic errors," Meas. Sci. Technol. 10, 112 (1999).
    [CrossRef]
  10. L. Chen, and X. Bao, "Analytical and numerical solutions for steady state stimulated Brillouin scattering in a single-mode fiber," Opt. Commun. 152, 65 (1998).
    [CrossRef]
  11. W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C: the Art of Scientific Computing, (Cambridge, New York, New Rochelle, Melbourne, Sydney, 1999).
  12. 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]

2006 (3)

2005 (2)

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

Y. Okawachi, M. S. Bigelow, J. E. Sharping, Z. Zhu, A. Schweinberg. 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]

2004 (1)

1999 (1)

E. Geinitz, S. Jetshke, U. Röpke, S. Schröter, R. Wilsch, and H. Bartelt, "The influence of pulse amplification on distributed fibre-optic Brillouin sensing and a method to compensate for systematic errors," Meas. Sci. Technol. 10, 112 (1999).
[CrossRef]

1998 (1)

L. Chen, and X. Bao, "Analytical and numerical solutions for steady state stimulated Brillouin scattering in a single-mode fiber," Opt. Commun. 152, 65 (1998).
[CrossRef]

1995 (1)

X. Bao, J. Dhliwayo, N. Heron, D. J. Webb, and, D. A. Jackson, "Experimental and theoretical studies on a distributed temperature sensor based on Brillouin scattering," J. Lightwave Technol. 13, 1340 (1995).
[CrossRef]

1993 (1)

Bao, X.

X. Bao, Q. Yu, V. P. Kalosha, and L. Chen, "The influence of prolonged phonon relaxation on the Brillouin loss spectrum for the nanosecond pulses," Opt Lett. 31, 888-890 (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]

L. Zou, X. Bao, S. Afshar V. and L. Chen, "Dependence of the Brillouin frequency shift on strain and temperature in a photonic crystal fiber," Opt. Lett. 29, 1485 (2004).
[CrossRef] [PubMed]

L. Chen, and X. Bao, "Analytical and numerical solutions for steady state stimulated Brillouin scattering in a single-mode fiber," Opt. Commun. 152, 65 (1998).
[CrossRef]

X. Bao, J. Dhliwayo, N. Heron, D. J. Webb, and, D. A. Jackson, "Experimental and theoretical studies on a distributed temperature sensor based on Brillouin scattering," J. Lightwave Technol. 13, 1340 (1995).
[CrossRef]

X. Bao, D. J. Webb, and D. A. Jackson, "32-km distributed temperature sensor based on Brillouin loss in an optical fiber," Opt. Lett. 18, 1561 (1993).
[CrossRef] [PubMed]

Bartelt, H.

E. Geinitz, S. Jetshke, U. Röpke, S. Schröter, R. Wilsch, and H. Bartelt, "The influence of pulse amplification on distributed fibre-optic Brillouin sensing and a method to compensate for systematic errors," Meas. Sci. Technol. 10, 112 (1999).
[CrossRef]

Bigelow, M. S.

Y. Okawachi, M. S. Bigelow, J. E. Sharping, Z. Zhu, A. Schweinberg. 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]

Chen, L.

X. Bao, Q. Yu, V. P. Kalosha, and L. Chen, "The influence of prolonged phonon relaxation on the Brillouin loss spectrum for the nanosecond pulses," Opt Lett. 31, 888-890 (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]

L. Chen, and X. Bao, "Analytical and numerical solutions for steady state stimulated Brillouin scattering in a single-mode fiber," Opt. Commun. 152, 65 (1998).
[CrossRef]

Dhliwayo, J.

X. Bao, J. Dhliwayo, N. Heron, D. J. Webb, and, D. A. Jackson, "Experimental and theoretical studies on a distributed temperature sensor based on Brillouin scattering," J. Lightwave Technol. 13, 1340 (1995).
[CrossRef]

Geinitz, E.

E. Geinitz, S. Jetshke, U. Röpke, S. Schröter, R. Wilsch, and H. Bartelt, "The influence of pulse amplification on distributed fibre-optic Brillouin sensing and a method to compensate for systematic errors," Meas. Sci. Technol. 10, 112 (1999).
[CrossRef]

González Herráez, M.

González-Herráez, M.

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

Heron, N.

X. Bao, J. Dhliwayo, N. Heron, D. J. Webb, and, D. A. Jackson, "Experimental and theoretical studies on a distributed temperature sensor based on Brillouin scattering," J. Lightwave Technol. 13, 1340 (1995).
[CrossRef]

Jackson, D. A.

X. Bao, J. Dhliwayo, N. Heron, D. J. Webb, and, D. A. Jackson, "Experimental and theoretical studies on a distributed temperature sensor based on Brillouin scattering," J. Lightwave Technol. 13, 1340 (1995).
[CrossRef]

X. Bao, D. J. Webb, and D. A. Jackson, "32-km distributed temperature sensor based on Brillouin loss in an optical fiber," Opt. Lett. 18, 1561 (1993).
[CrossRef] [PubMed]

Jetshke, S.

E. Geinitz, S. Jetshke, U. Röpke, S. Schröter, R. Wilsch, and H. Bartelt, "The influence of pulse amplification on distributed fibre-optic Brillouin sensing and a method to compensate for systematic errors," Meas. Sci. Technol. 10, 112 (1999).
[CrossRef]

Kalosha, V. P.

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, "The influence of prolonged phonon relaxation on the Brillouin loss spectrum for the nanosecond pulses," Opt Lett. 31, 888-890 (2006).
[CrossRef] [PubMed]

Okawachi, Y.

Y. Okawachi, M. S. Bigelow, J. E. Sharping, Z. Zhu, A. Schweinberg. 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]

Ponomarev, E.

Röpke, U.

E. Geinitz, S. Jetshke, U. Röpke, S. Schröter, R. Wilsch, and H. Bartelt, "The influence of pulse amplification on distributed fibre-optic Brillouin sensing and a method to compensate for systematic errors," Meas. Sci. Technol. 10, 112 (1999).
[CrossRef]

Schröter, S.

E. Geinitz, S. Jetshke, U. Röpke, S. Schröter, R. Wilsch, and H. Bartelt, "The influence of pulse amplification on distributed fibre-optic Brillouin sensing and a method to compensate for systematic errors," Meas. Sci. Technol. 10, 112 (1999).
[CrossRef]

Schweinberg, A.

Y. Okawachi, M. S. Bigelow, J. E. Sharping, Z. Zhu, A. Schweinberg. 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. Schweinberg. 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]

Song, K.-Y.

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

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

Thévenaz, L.

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

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

Webb, D. J.

X. Bao, J. Dhliwayo, N. Heron, D. J. Webb, and, D. A. Jackson, "Experimental and theoretical studies on a distributed temperature sensor based on Brillouin scattering," J. Lightwave Technol. 13, 1340 (1995).
[CrossRef]

X. Bao, D. J. Webb, and D. A. Jackson, "32-km distributed temperature sensor based on Brillouin loss in an optical fiber," Opt. Lett. 18, 1561 (1993).
[CrossRef] [PubMed]

Wilsch, R.

E. Geinitz, S. Jetshke, U. Röpke, S. Schröter, R. Wilsch, and H. Bartelt, "The influence of pulse amplification on distributed fibre-optic Brillouin sensing and a method to compensate for systematic errors," Meas. Sci. Technol. 10, 112 (1999).
[CrossRef]

Yu, Q.

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

Zhu, Z.

Y. Okawachi, M. S. Bigelow, J. E. Sharping, Z. Zhu, A. Schweinberg. 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]

Zou, L.

J. Lightwave Technol. (1)

X. Bao, J. Dhliwayo, N. Heron, D. J. Webb, and, D. A. Jackson, "Experimental and theoretical studies on a distributed temperature sensor based on Brillouin scattering," J. Lightwave Technol. 13, 1340 (1995).
[CrossRef]

Meas. Sci. Technol. (1)

E. Geinitz, S. Jetshke, U. Röpke, S. Schröter, R. Wilsch, and H. Bartelt, "The influence of pulse amplification on distributed fibre-optic Brillouin sensing and a method to compensate for systematic errors," Meas. Sci. Technol. 10, 112 (1999).
[CrossRef]

Opt Lett. (1)

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

Opt. Commun. (1)

L. Chen, and X. Bao, "Analytical and numerical solutions for steady state stimulated Brillouin scattering in a single-mode fiber," Opt. Commun. 152, 65 (1998).
[CrossRef]

Opt. Express (1)

Opt. Express. (1)

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

Opt. Lett. (3)

Phys. Rev. Lett. (1)

Y. Okawachi, M. S. Bigelow, J. E. Sharping, Z. Zhu, A. Schweinberg. 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 (2)

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C: the Art of Scientific Computing, (Cambridge, New York, New Rochelle, Melbourne, Sydney, 1999).

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

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Figures (6)

Fig. 1.
Fig. 1.

(a). Group delay of Stokes beam with PsL = 0.1mW (blue), PsL = 1 mW (red), PsL = 10 mW (green) and undepleted (violet) for L = 10 km and Pp0 = 6 mW; (b) Group delay of three Stokes beam with PsL = 0.1 mW (blue), PsL = 1 mW (red), PsL = 10 mW (green) and undepleted (violet) for L = 100 m and Pp0 = 6 mW.

Fig. 2.
Fig. 2.

(a). Gain of Stokes beam with PsL = 0.1 mW (blue), PsL = 1 mW (red), PsL = 10 mW (green) for L = 10 km and Pp0 = 6 mW; (b) Stokes and pump power distribution along a 10km long fiber for PsL = 0.1 mW (blue), PsL = 1 mW (red), PsL = 10 mW (green) and Pp0 = 6 mW; Solid lines refer to pump power and dashed lines refer to Stokes power.

Fig. 3.
Fig. 3.

(a). Brillouin loss spectrum of the stressed section for PsL = 10 mW (blue), PsL = 1 mW (red), PsL = 20 μW (green); (b) Peak frequency detected on the Brillouin loss spectrum of the stressed section as a function of input Stokes power. Simulation parameters are L = 10 km, Pp0 = 6 mW, Δl = 1m and vBs-vB = 35 MHz.

Fig. 4.
Fig. 4.

(a) Gain of three Stokes beam with PsL = 0.1 mW (blue), PsL = 1 mW (red), PsL = 10 mW (green) for L = 100 m and Pp0 = 6 mW; (b) Stokes and pump power distribution along a 100 m long fiber for PsL = 0.1 mW (blue), PsL = 1 mW (red), PsL = 10 mW (green) and Pp0 = 6 mW; Solid lines refer to pump power and dashed lines refer to Stokes power.

Fig. 5.
Fig. 5.

Group delay of output Stokes beam (z=0 m,) as a function of input Stokes power for three fiber lengths (L = 100 m, blue, L = 1 km, green, L= 10 km, red) and Pp0 = 6mW.

Fig. 6.
Fig. 6.

(a). Group delay of a Stokes beam at the Brillouin frequency of a fiber as a function of position for uniform Brillouin frequency profile (blue), Δl = 10 m (red), Δl = 100 m (green); (b) Gain of a Stokes beam at the Brillouin frequency of a fiber as a function of position for uniform Brillouin frequency profile (blue), Δl = 10 m (red), Δl = 100 m (green); parameters are Pp0 = 6 mW, Ps0 = 1 mW. The non-uniform cases are up-shifted by 50 MHz.

Equations (12)

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d A s d z = κ * 2 A p 2 A s + 1 2 α A s ,
d A p d z = κ 2 A s 2 A p 1 2 α A p ,
κ ( Δ v ) = g ( Δ v ) + j φ ( Δ v ) = g B 1 + ( 2 Δ v / Δ v B ) 2 + j ( 2 Δ v / Δ v B ) g B 1 + ( 2 Δ v / Δ v B ) 2 .
A p ( z , Δ v ) = A p ( 0 , Δ v ) exp G p ( z , Δ v ) + j Φ p ( z , Δ v ) ,
A s ( z , Δ v ) = A s ( L , Δ v ) exp G s ( z , Δ v ) + j Φ s ( z , Δ v ) ,
Φ s = L z φ 2 I p ( z′ , Δ v ) d z′ ,
Φ p = 0 z φ 2 I s ( z′ , Δ v ) d z′ .
= { ( 0 2 Δ 0 2 ) exp [ ( g / α ) ( Δ Δ 0 ) ] + Δ 2 } 1 / 2 ,
Δ 0 Δ { ( 0 2 Δ 0 2 ) exp [ ( g / α ) ( u Δ 0 ) ] + u 2 } 1 / 2 d u = α z ,
τ s = 1 2 π d Φ s d Δ v = g B 2 π Δ v B 1 ( 2 Δ v / Δ v B ) 2 [ 1 + ( 2 Δ v / Δ v B ) 2 ] 2 L z I p d z φ 4 π d d Δ v L z I p d z ,
τ p = 1 2 π d Φ p d Δ v = g B 2 π Δ v B 1 ( 2 Δ v / Δ v B ) 2 [ 1 + ( 2 Δ v / Δ v B ) 2 ] 2 0 z I s d z φ 4 π d d Δ v 0 z I s d z .
τ s = g B I p L eff 2 π Δ v B .

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