Abstract

Starting from the standard three-wave SBS coupled equations, we derive a novel expression describing Brillouin interaction between a pulsed pump wave with a finite cw component, and a Stokes continuous wave counter-propagating along a single-mode optical fiber. The derived integral equation relates the time-domain Stokes beam amplification to the Brillouin frequency distribution. The proposed model permits an accurate description of the Brillouin interaction even for arbitrarily-shaped pump pulses, and can be efficiently employed for improving the accuracy and the resolution of SBS-based distributed sensors. The validity and the limits of the proposed model are numerically analyzed and discussed.

© 2007 Optical Society of America

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References

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  1. T. Horiguchi, K. Shimizu, T. Kurashima, and Y. Koyamada, "Advances in distributed sensing techniques using Brillouin scattering," Proc. SPIE  2507, 126-135 (1995).
    [CrossRef]
  2. X. Bao, A. Brown, M. DeMerchant, and J. Smith, "Characterization of the Brillouin-loss spectrum of single-mode fibers by use of very short (< 10-ns) pulses," Opt. Lett. 24, 510-512 (1999),http://www.opticsinfobase.org/abstract.cfm?URI=ol-24-8-510>.
    [CrossRef]
  3. V. Lecoeuche, D. J. Webb, C. N. Pannell, and D. A. Jackson, "Transient response in high-resolution Brillouin-based distributed sensing using probe pulses shorter than the acoustic relaxation time, " Opt. Lett. 25, 156-158 (2000), http://www.opticsinfobase.org/abstract.cfm?URI=ol-25-3-156>.
    [CrossRef]
  4. 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), http://www.opticsinfobase.org/abstract.cfm?URI=ol-31-7-888>.
    [CrossRef] [PubMed]
  5. Y. Wan, S. Afshar V., L. Zou, L. Chen, and X. Bao, "Subpeaks in the Brillouin loss spectra of distributed fiber-optic sensors," Opt. Lett. 30, 1099-1101 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=ol-30-10-1099>.
    [CrossRef] [PubMed]
  6. R. Bernini, A. Minardo, and L. Zeni, "An accurate high-resolution technique for distributed sensing based on frequency-domain Brillouin scattering," IEEE Photon. Technol. Lett. 18, 280-282 (2006).
    [CrossRef]
  7. S. -B. Cho, J. -J. Lee, and I. -B. Kwon, "Strain event detection using a double-pulse technique of a Brillouin scattering-based distributed optical fiber sensor," Opt. Express 12, 4339-4346 (2004), http://www.opticsinfobase.org/abstract.cfm?URI=oe-12-18-4339>.
    [CrossRef] [PubMed]
  8. K. Kishida, C.-H Lee, and K. Nishiguchi, "Pulse pre-pump method for cm-order spatial resolution of BOTDA," Proc. SPIE 5855, 559-562 (2005).
    [CrossRef]
  9. A. W. Brown, B. G. Colpitts, and K. Brown, "Distributed sensor based on dark-pulse Brillouin scattering," IEEE Photon. Technol. Lett. 17, 1501-1503 (2005).
    [CrossRef]
  10. G. P. Agrawal, Nonlinear Fiber Optics (Academic, San Diego, Calif., 1995).
  11. R. J. LeVeque, "Wave propagation method algorithms for multi-dimensional hyperbolic systems," J. Comp. Physiol. 131, 327-353 (1997).
    [CrossRef]
  12. A. Minardo R. Bernini, L. Zeni, L. Thevenaz, F. Briffod, "A reconstruction technique for long-range stimulated Brillouin Scattering distributed fiber-optic sensors: experimental results," Meas. Sci. Technol. 16, 900-908 (2005).
    [CrossRef]
  13. 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, 1340-1348, (1995).
    [CrossRef]
  14. E. Licthman, R. G. Waarts, and A. A. Friesem, "Stimulated Brillouin scattering excited by a modulated pump wave in single-mode fibers," J. Lightwave Technol. 7, 171-174 (1989).
    [CrossRef]

2006

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), http://www.opticsinfobase.org/abstract.cfm?URI=ol-31-7-888>.
[CrossRef] [PubMed]

R. Bernini, A. Minardo, and L. Zeni, "An accurate high-resolution technique for distributed sensing based on frequency-domain Brillouin scattering," IEEE Photon. Technol. Lett. 18, 280-282 (2006).
[CrossRef]

2005

K. Kishida, C.-H Lee, and K. Nishiguchi, "Pulse pre-pump method for cm-order spatial resolution of BOTDA," Proc. SPIE 5855, 559-562 (2005).
[CrossRef]

A. W. Brown, B. G. Colpitts, and K. Brown, "Distributed sensor based on dark-pulse Brillouin scattering," IEEE Photon. Technol. Lett. 17, 1501-1503 (2005).
[CrossRef]

Y. Wan, S. Afshar V., L. Zou, L. Chen, and X. Bao, "Subpeaks in the Brillouin loss spectra of distributed fiber-optic sensors," Opt. Lett. 30, 1099-1101 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=ol-30-10-1099>.
[CrossRef] [PubMed]

A. Minardo R. Bernini, L. Zeni, L. Thevenaz, F. Briffod, "A reconstruction technique for long-range stimulated Brillouin Scattering distributed fiber-optic sensors: experimental results," Meas. Sci. Technol. 16, 900-908 (2005).
[CrossRef]

2004

2000

1999

1997

R. J. LeVeque, "Wave propagation method algorithms for multi-dimensional hyperbolic systems," J. Comp. Physiol. 131, 327-353 (1997).
[CrossRef]

1995

T. Horiguchi, K. Shimizu, T. Kurashima, and Y. Koyamada, "Advances in distributed sensing techniques using Brillouin scattering," Proc. SPIE  2507, 126-135 (1995).
[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, 1340-1348, (1995).
[CrossRef]

1989

E. Licthman, R. G. Waarts, and A. A. Friesem, "Stimulated Brillouin scattering excited by a modulated pump wave in single-mode fibers," J. Lightwave Technol. 7, 171-174 (1989).
[CrossRef]

J. Comp. Physiol.

R. J. LeVeque, "Wave propagation method algorithms for multi-dimensional hyperbolic systems," J. Comp. Physiol. 131, 327-353 (1997).
[CrossRef]

J. Lightwave Technol.

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, 1340-1348, (1995).
[CrossRef]

E. Licthman, R. G. Waarts, and A. A. Friesem, "Stimulated Brillouin scattering excited by a modulated pump wave in single-mode fibers," J. Lightwave Technol. 7, 171-174 (1989).
[CrossRef]

Meas. Sci. Technol.

A. Minardo R. Bernini, L. Zeni, L. Thevenaz, F. Briffod, "A reconstruction technique for long-range stimulated Brillouin Scattering distributed fiber-optic sensors: experimental results," Meas. Sci. Technol. 16, 900-908 (2005).
[CrossRef]

Opt. Express

Opt. Lett.

Photon. Technol. Lett.

R. Bernini, A. Minardo, and L. Zeni, "An accurate high-resolution technique for distributed sensing based on frequency-domain Brillouin scattering," IEEE Photon. Technol. Lett. 18, 280-282 (2006).
[CrossRef]

A. W. Brown, B. G. Colpitts, and K. Brown, "Distributed sensor based on dark-pulse Brillouin scattering," IEEE Photon. Technol. Lett. 17, 1501-1503 (2005).
[CrossRef]

Proc. SPIE

T. Horiguchi, K. Shimizu, T. Kurashima, and Y. Koyamada, "Advances in distributed sensing techniques using Brillouin scattering," Proc. SPIE  2507, 126-135 (1995).
[CrossRef]

K. Kishida, C.-H Lee, and K. Nishiguchi, "Pulse pre-pump method for cm-order spatial resolution of BOTDA," Proc. SPIE 5855, 559-562 (2005).
[CrossRef]

Other

G. P. Agrawal, Nonlinear Fiber Optics (Academic, San Diego, Calif., 1995).

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Figures (5)

Fig. 1.
Fig. 1.

Stokes amplification time-domain waveforms, as calculated by solving the full model of Eqs. (1) (dashed blue lines) and the model of Eq. (7) (solid red lines). Solutions are calculated for L=10 m, τP=10 ns, and ER=10 dB (A) ER=20 dB (B).

Fig. 2.
Fig. 2.

Stokes amplification time-domain waveforms, as calculated by solving the full model of Eq.s (1) (dashed blue lines) and the model of Eq. (7) (solid red lines). The inset shows a zoom-in view of the perturbed region, at Δ=30 MHz. Solutions are calculated for L=50 m, τP=10 ns, and ER=20 dB.

Fig. 3.
Fig. 3.

Brillouin gain spectrum calculated at t=18 ns, for a uniform 1-meter-long fiber, τP=1.5 ns, ER=20 dB, by solving the full model of Eqs. (1) (dashed blue line) and the model of Eq. (7) (solid red line).

Fig. 4.
Fig. 4.

Brillouin gain spectrum calculated at t=12.1 ns (blue line), t=13.6 ns (red line) and t=15 ns (green line), for a perturbed 1-meter-long fiber, τP=1.5 ns, ER=20 dB, by solving the full model of Eqs. (1) (dashed lines) and the model of Eq. (7) (solid lines).

Fig. 5.
Fig. 5.

Stokes amplification time-domain waveforms, as calculated by solving the full model of Eq.s (1) (dashed blue lines) and the model of Eq. (6), in which pump depletion is taken into account (solid red lines). Solutions are calculated for L=50 m, τP=10 ns, and ER=20 dB.

Tables (1)

Tables Icon

Table 1. Normalized norm difference between the Brillouin signals calculated by solving Eq. (7) and the Brillouin signals calculated by solving Eqs. (1). Solutions are calculated for L=10 m, τP=10 ns, ER=20 dB, and input Stokes power=1 mW.

Equations (32)

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( z + n c t ) E p = Q E s ,
( z + n c t ) E s = Q * E p ,
( t + Γ ) Q = 1 2 Γ 1 g B E p E s *
E P ( z , t ) = + E P ( z , ω ) exp [ j ω ( t n c z ) ] d ω
E S ( z , t ) = + E S ( z , ω ) exp [ j ω ( t + n c z ) ] d ω
Q ( z , t ) = + Q ( z , Ω ) exp [ ( t z ν a ) ] d Ω
z E s ( z , ω ) = g B Γ 1 2 ω E p ( z , ω ) exp ( 2 j n c ω z )
× [ ω E p * ( z , ω ) E s ( z , ω ω + ω ) Γ 1 j ( Δ ( z ) + ω ω ) · exp ( 2 j n c ω z ) d ω ] d ω
z E s ( z , ω ) = g B Γ 1 2 ω E p ( z , ω ) E p * ( z , ω ω ) E s CW ( z ) Γ 1 j ( Δ ( z ) + ω ω ) exp ( 2 j n c ω z ) d ω
E s ( 0 , ω ) = E s ( 0 , ω ) E s ( L , ω ) = g B Γ 1 2 0 L G ( z , ω ) exp ( 2 j n c ω z ) d z
G ( z , ω ) = E s CW ( z ) ω E p ( 0 , ω ) E p * ( 0 , ω ω ) Γ 1 j ( Δ + ω ω ) d ω
G ( z , ω ) = E s CW ( z ) · ( E p ( 0 , ω ) E p ( 0 , ω ) Γ 1 j ( Δ ( z ) + ω ) )
( t + Γ ) Q = + Q ( z , Ω ) ( Γ + j Ω ) exp [ j Ω ( t z ν a ) ] d Ω
= 1 2 Γ 1 g B + + E P ( z , ω ) E S * ( z , ω ) exp [ j ( ω ω ) t ] exp [ j ( ω + ω ) n c z ] d ω d ω
+ Q ( z , Ω ) ( Γ + j Ω ) exp [ j Ω ( t z ν a ) ] d Ω
= 1 2 Γ 1 g B + + E P ( z , ω ) E S * ( z , ω Ω ) exp [ j Ω ( t z ν a ) ] d Ω d ω
Q ( z , Ω ) = 1 2 Γ 1 g B + E P ( z , ω ) E S * ( z , ω Ω ) ( Γ + j Ω ) d ω
Q ( z , t ) = 1 2 Γ 1 g B + + E P ( z , ω ) E S * ( z , ω ) Γ + j ( ω ω ) exp [ j ( ω ω ) t ] exp [ j ( ω + ω ) n c z ] d ω d ω
Q * E P = 1 2 Γ 1 g B + + + E P * ( z , ω ) E S ( z , ω ) E P ( z , ω ) Γ * j ( ω ω )
× exp [ j ( ω ω ω ) t ] exp [ j ( ω + ω ω ) n c z ] d ω d ω d ω
( z n c t ) E S
= ( z n c t ) + E S ( z , ω ¯ ) exp [ j ω ¯ ( t + n c z ) ] d ω ¯ = + d E S ( z , ω ¯ ) d z exp [ j ω ¯ ( t + n c z ) ] d ω ¯
−∞ +∞ E S ( z , ω ¯ ) z exp [ j ω ¯ ( t + n c z ) ] d ω ¯
= 1 2 Γ 1 g B + + + E P * ( z , ω ) E S ( z , ω ) E P ( z , ω ) Γ * j ( ω ω ) exp [ j ( ω ω ω ) t ]
× exp [ j ( ω + ω ω ) n c z ] d ω d ω d ω
−∞ +∞ E S ( z , ω ¯ ) z exp [ j ω ¯ ( t + n c z ) ] d ω ¯
= 1 2 Γ 1 g B + + + E P * ( z , ω ) E S ( z , ω ) E P ( z , ω ω + ω ¯ ) Γ * j ( ω ω ) exp [ j ω ¯ t ]
× exp [ j ( 2 ω + ω ¯ ) n c z ] d ω d ω d ω ¯
E S ( z , ω ¯ ) z = g B Γ 1 2 + + E P * ( z , ω ) E S ( z , ω ) E P ( z , ω ω " + ω ¯ ) Γ * j ( ω ω " ) exp [ j ( 2 ω " 2 ω ¯ ) n c z ] d ω d ω "
z E s ( z , ω ) = g B Γ 1 2 ω " E p ( z , ω ) E p * ( z , ω ω ) E s CW ( z ) Γ 1 j ( Δ ( z ) + ω ω ) exp ( 2 j n c ω z ) d ω
+ α 2 E s ( z , ω )
E s ( 0 , ω ) = g B Γ 1 2 0 L G ( z , ω ) exp ( 2 j n c ω z α ) d z

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