Abstract

A highly accurate, fully analytic solution for the continuous wave and the probe wave in Brillouin amplification, in lossless optical fibers, is given. It is experimentally confirmed that the reported analytic solution can account for spectral distortion and pump depletion in the parameter space that is relevant to Brillouin fiber sensor applications, as well as applications in photonic logic. The analytic solutions are valid characterizations of Brillouin amplification in both the low and high nonlinearity regime, for short fiber lengths.

© 2013 Chinese Laser Press

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References

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  1. R. Boyd, Nonlinear Optics, 2nd ed. (Academic, 1992).
  2. P. Bayvel and P. M. Radmore, “Solutions of the SBS equations in single mode optical fibers and implications for fibre transmission systems,” Electron. Lett. 26, 434–436 (1990).
    [CrossRef]
  3. E. Geinitz, S. Jetschke, U. Röpke, S. Schröter, R. Willsch, and H. Bartelt, “The influence of pulse amplification on distributed fiber-optic Brillouin sensing and a method to compensate for systematic errors,” Meas. Sci. Technol. 10, 112–116 (1999).
    [CrossRef]
  4. X. Bao and L. Chen, “Recent progress in distributed fiber optic sensors,” Sensors 12, 8601–8639 (2012).
    [CrossRef]
  5. S. L. Zhang and J. J. O’Reilly, “Effect of stimulated Brillouin scattering on distributed erbium-doped fiber amplifier,” IEEE Photon. Technol. Lett. 5, 537–539 (1993).
    [CrossRef]
  6. B. Foley, M. L. Dakss, R. W. Davies, and P. Melman, “Gain saturation in fiber Raman amplifiers due to stimulated Brillouin scattering,” J. Lightwave Technol. 7, 2024–2032 (1989).
    [CrossRef]
  7. F. S. Gokhan, “Moderate-gain Brillouin amplification: an analytical solution below pump threshold,” Opt. Commun. 284, 4869–4873 (2011).
    [CrossRef]
  8. C. L. Tang, “Saturation and spectral characteristics of the Stokes emission in the stimulated Brillouin process,” J. Appl. Phys. 37, 2945–2955 (1966).
    [CrossRef]
  9. L. Chen and X. Bao, “Analytical and numerical solutions for steady state stimulated Brillouin scattering in a single-mode fiber,” Opt. Commun. 152, 65–70 (1998).
    [CrossRef]
  10. A. Kobyakov, S. Darmanyan, M. Sauer, and D. Chowdhury, “High-gain Brillouin amplification: an analytical approach,” Opt. Lett. 31, 1960–1962 (2006).
    [CrossRef]
  11. F. Ravet, Performance of the Distributed Brillouin Sensor: Benefits and Penalties Due to Pump Depletion, 1st ed. (University of Ottawa, 2007).
  12. T. Horiguchi, K. Shimizu, T. Kurashima, M. Tateda, and Y. Koyamada, “Development of a distributed sensing technique using Brillouin scattering,” J. Lightwave Technol. 13, 1296–1302 (1995).
    [CrossRef]
  13. L. Thévenaz and S. F. Mafang, “Depletion in a distributed Brillouin fiber sensor: practical limitation and strategy to avoid it,” Proc. SPIE 7753A5, 1–4 (2011).
  14. D. Williams, X. Bao, and L. Chen, “All-optical NAND/NOT/AND/OR logic gates based on combined Brillouin gain and loss in an optical fiber,” Appl. Opt. 52, 3404–3411 (2013).
    [CrossRef]
  15. A. Minardo, R. Bernini, L. Zeni, L. Thevenaz, and F. Briffod, “A reconstruction technique for long-range stimulated Brillouin scattering distributed fibre-optic sensors: experimental results,” Meas. Sci. Technol. 16, 900–908 (2005).
    [CrossRef]
  16. T. H. Russell and W. B. Roh, “Threshold of second-order stimulated Brillouin scattering in optical fiber,” J. Opt. Soc. Am. B 19, 2341–2345 (2002).
    [CrossRef]

2013 (1)

2012 (1)

X. Bao and L. Chen, “Recent progress in distributed fiber optic sensors,” Sensors 12, 8601–8639 (2012).
[CrossRef]

2011 (2)

F. S. Gokhan, “Moderate-gain Brillouin amplification: an analytical solution below pump threshold,” Opt. Commun. 284, 4869–4873 (2011).
[CrossRef]

L. Thévenaz and S. F. Mafang, “Depletion in a distributed Brillouin fiber sensor: practical limitation and strategy to avoid it,” Proc. SPIE 7753A5, 1–4 (2011).

2006 (1)

2005 (1)

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

2002 (1)

1999 (1)

E. Geinitz, S. Jetschke, U. Röpke, S. Schröter, R. Willsch, and H. Bartelt, “The influence of pulse amplification on distributed fiber-optic Brillouin sensing and a method to compensate for systematic errors,” Meas. Sci. Technol. 10, 112–116 (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, 65–70 (1998).
[CrossRef]

1995 (1)

T. Horiguchi, K. Shimizu, T. Kurashima, M. Tateda, and Y. Koyamada, “Development of a distributed sensing technique using Brillouin scattering,” J. Lightwave Technol. 13, 1296–1302 (1995).
[CrossRef]

1993 (1)

S. L. Zhang and J. J. O’Reilly, “Effect of stimulated Brillouin scattering on distributed erbium-doped fiber amplifier,” IEEE Photon. Technol. Lett. 5, 537–539 (1993).
[CrossRef]

1990 (1)

P. Bayvel and P. M. Radmore, “Solutions of the SBS equations in single mode optical fibers and implications for fibre transmission systems,” Electron. Lett. 26, 434–436 (1990).
[CrossRef]

1989 (1)

B. Foley, M. L. Dakss, R. W. Davies, and P. Melman, “Gain saturation in fiber Raman amplifiers due to stimulated Brillouin scattering,” J. Lightwave Technol. 7, 2024–2032 (1989).
[CrossRef]

1966 (1)

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

Bao, X.

D. Williams, X. Bao, and L. Chen, “All-optical NAND/NOT/AND/OR logic gates based on combined Brillouin gain and loss in an optical fiber,” Appl. Opt. 52, 3404–3411 (2013).
[CrossRef]

X. Bao and L. Chen, “Recent progress in distributed fiber optic sensors,” Sensors 12, 8601–8639 (2012).
[CrossRef]

L. Chen and X. Bao, “Analytical and numerical solutions for steady state stimulated Brillouin scattering in a single-mode fiber,” Opt. Commun. 152, 65–70 (1998).
[CrossRef]

Bartelt, H.

E. Geinitz, S. Jetschke, U. Röpke, S. Schröter, R. Willsch, and H. Bartelt, “The influence of pulse amplification on distributed fiber-optic Brillouin sensing and a method to compensate for systematic errors,” Meas. Sci. Technol. 10, 112–116 (1999).
[CrossRef]

Bayvel, P.

P. Bayvel and P. M. Radmore, “Solutions of the SBS equations in single mode optical fibers and implications for fibre transmission systems,” Electron. Lett. 26, 434–436 (1990).
[CrossRef]

Bernini, R.

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

Boyd, R.

R. Boyd, Nonlinear Optics, 2nd ed. (Academic, 1992).

Briffod, F.

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

Chen, L.

D. Williams, X. Bao, and L. Chen, “All-optical NAND/NOT/AND/OR logic gates based on combined Brillouin gain and loss in an optical fiber,” Appl. Opt. 52, 3404–3411 (2013).
[CrossRef]

X. Bao and L. Chen, “Recent progress in distributed fiber optic sensors,” Sensors 12, 8601–8639 (2012).
[CrossRef]

L. Chen and X. Bao, “Analytical and numerical solutions for steady state stimulated Brillouin scattering in a single-mode fiber,” Opt. Commun. 152, 65–70 (1998).
[CrossRef]

Chowdhury, D.

Dakss, M. L.

B. Foley, M. L. Dakss, R. W. Davies, and P. Melman, “Gain saturation in fiber Raman amplifiers due to stimulated Brillouin scattering,” J. Lightwave Technol. 7, 2024–2032 (1989).
[CrossRef]

Darmanyan, S.

Davies, R. W.

B. Foley, M. L. Dakss, R. W. Davies, and P. Melman, “Gain saturation in fiber Raman amplifiers due to stimulated Brillouin scattering,” J. Lightwave Technol. 7, 2024–2032 (1989).
[CrossRef]

Foley, B.

B. Foley, M. L. Dakss, R. W. Davies, and P. Melman, “Gain saturation in fiber Raman amplifiers due to stimulated Brillouin scattering,” J. Lightwave Technol. 7, 2024–2032 (1989).
[CrossRef]

Geinitz, E.

E. Geinitz, S. Jetschke, U. Röpke, S. Schröter, R. Willsch, and H. Bartelt, “The influence of pulse amplification on distributed fiber-optic Brillouin sensing and a method to compensate for systematic errors,” Meas. Sci. Technol. 10, 112–116 (1999).
[CrossRef]

Gokhan, F. S.

F. S. Gokhan, “Moderate-gain Brillouin amplification: an analytical solution below pump threshold,” Opt. Commun. 284, 4869–4873 (2011).
[CrossRef]

Horiguchi, T.

T. Horiguchi, K. Shimizu, T. Kurashima, M. Tateda, and Y. Koyamada, “Development of a distributed sensing technique using Brillouin scattering,” J. Lightwave Technol. 13, 1296–1302 (1995).
[CrossRef]

Jetschke, S.

E. Geinitz, S. Jetschke, U. Röpke, S. Schröter, R. Willsch, and H. Bartelt, “The influence of pulse amplification on distributed fiber-optic Brillouin sensing and a method to compensate for systematic errors,” Meas. Sci. Technol. 10, 112–116 (1999).
[CrossRef]

Kobyakov, A.

Koyamada, Y.

T. Horiguchi, K. Shimizu, T. Kurashima, M. Tateda, and Y. Koyamada, “Development of a distributed sensing technique using Brillouin scattering,” J. Lightwave Technol. 13, 1296–1302 (1995).
[CrossRef]

Kurashima, T.

T. Horiguchi, K. Shimizu, T. Kurashima, M. Tateda, and Y. Koyamada, “Development of a distributed sensing technique using Brillouin scattering,” J. Lightwave Technol. 13, 1296–1302 (1995).
[CrossRef]

Mafang, S. F.

L. Thévenaz and S. F. Mafang, “Depletion in a distributed Brillouin fiber sensor: practical limitation and strategy to avoid it,” Proc. SPIE 7753A5, 1–4 (2011).

Melman, P.

B. Foley, M. L. Dakss, R. W. Davies, and P. Melman, “Gain saturation in fiber Raman amplifiers due to stimulated Brillouin scattering,” J. Lightwave Technol. 7, 2024–2032 (1989).
[CrossRef]

Minardo, A.

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

O’Reilly, J. J.

S. L. Zhang and J. J. O’Reilly, “Effect of stimulated Brillouin scattering on distributed erbium-doped fiber amplifier,” IEEE Photon. Technol. Lett. 5, 537–539 (1993).
[CrossRef]

Radmore, P. M.

P. Bayvel and P. M. Radmore, “Solutions of the SBS equations in single mode optical fibers and implications for fibre transmission systems,” Electron. Lett. 26, 434–436 (1990).
[CrossRef]

Ravet, F.

F. Ravet, Performance of the Distributed Brillouin Sensor: Benefits and Penalties Due to Pump Depletion, 1st ed. (University of Ottawa, 2007).

Roh, W. B.

Röpke, U.

E. Geinitz, S. Jetschke, U. Röpke, S. Schröter, R. Willsch, and H. Bartelt, “The influence of pulse amplification on distributed fiber-optic Brillouin sensing and a method to compensate for systematic errors,” Meas. Sci. Technol. 10, 112–116 (1999).
[CrossRef]

Russell, T. H.

Sauer, M.

Schröter, S.

E. Geinitz, S. Jetschke, U. Röpke, S. Schröter, R. Willsch, and H. Bartelt, “The influence of pulse amplification on distributed fiber-optic Brillouin sensing and a method to compensate for systematic errors,” Meas. Sci. Technol. 10, 112–116 (1999).
[CrossRef]

Shimizu, K.

T. Horiguchi, K. Shimizu, T. Kurashima, M. Tateda, and Y. Koyamada, “Development of a distributed sensing technique using Brillouin scattering,” J. Lightwave Technol. 13, 1296–1302 (1995).
[CrossRef]

Tang, C. L.

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

Tateda, M.

T. Horiguchi, K. Shimizu, T. Kurashima, M. Tateda, and Y. Koyamada, “Development of a distributed sensing technique using Brillouin scattering,” J. Lightwave Technol. 13, 1296–1302 (1995).
[CrossRef]

Thevenaz, L.

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

Thévenaz, L.

L. Thévenaz and S. F. Mafang, “Depletion in a distributed Brillouin fiber sensor: practical limitation and strategy to avoid it,” Proc. SPIE 7753A5, 1–4 (2011).

Williams, D.

Willsch, R.

E. Geinitz, S. Jetschke, U. Röpke, S. Schröter, R. Willsch, and H. Bartelt, “The influence of pulse amplification on distributed fiber-optic Brillouin sensing and a method to compensate for systematic errors,” Meas. Sci. Technol. 10, 112–116 (1999).
[CrossRef]

Zeni, L.

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

Zhang, S. L.

S. L. Zhang and J. J. O’Reilly, “Effect of stimulated Brillouin scattering on distributed erbium-doped fiber amplifier,” IEEE Photon. Technol. Lett. 5, 537–539 (1993).
[CrossRef]

Appl. Opt. (1)

Electron. Lett. (1)

P. Bayvel and P. M. Radmore, “Solutions of the SBS equations in single mode optical fibers and implications for fibre transmission systems,” Electron. Lett. 26, 434–436 (1990).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

S. L. Zhang and J. J. O’Reilly, “Effect of stimulated Brillouin scattering on distributed erbium-doped fiber amplifier,” IEEE Photon. Technol. Lett. 5, 537–539 (1993).
[CrossRef]

J. Appl. Phys. (1)

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

J. Lightwave Technol. (2)

T. Horiguchi, K. Shimizu, T. Kurashima, M. Tateda, and Y. Koyamada, “Development of a distributed sensing technique using Brillouin scattering,” J. Lightwave Technol. 13, 1296–1302 (1995).
[CrossRef]

B. Foley, M. L. Dakss, R. W. Davies, and P. Melman, “Gain saturation in fiber Raman amplifiers due to stimulated Brillouin scattering,” J. Lightwave Technol. 7, 2024–2032 (1989).
[CrossRef]

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

Meas. Sci. Technol. (2)

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

E. Geinitz, S. Jetschke, U. Röpke, S. Schröter, R. Willsch, and H. Bartelt, “The influence of pulse amplification on distributed fiber-optic Brillouin sensing and a method to compensate for systematic errors,” Meas. Sci. Technol. 10, 112–116 (1999).
[CrossRef]

Opt. Commun. (2)

F. S. Gokhan, “Moderate-gain Brillouin amplification: an analytical solution below pump threshold,” Opt. Commun. 284, 4869–4873 (2011).
[CrossRef]

L. Chen and X. Bao, “Analytical and numerical solutions for steady state stimulated Brillouin scattering in a single-mode fiber,” Opt. Commun. 152, 65–70 (1998).
[CrossRef]

Opt. Lett. (1)

Proc. SPIE (1)

L. Thévenaz and S. F. Mafang, “Depletion in a distributed Brillouin fiber sensor: practical limitation and strategy to avoid it,” Proc. SPIE 7753A5, 1–4 (2011).

Sensors (1)

X. Bao and L. Chen, “Recent progress in distributed fiber optic sensors,” Sensors 12, 8601–8639 (2012).
[CrossRef]

Other (2)

F. Ravet, Performance of the Distributed Brillouin Sensor: Benefits and Penalties Due to Pump Depletion, 1st ed. (University of Ottawa, 2007).

R. Boyd, Nonlinear Optics, 2nd ed. (Academic, 1992).

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Figures (10)

Fig. 1.
Fig. 1.

(a) Schematic arrangement of SBS in a fiber of length L. Pump and probe configuration: A1(z)—continuous wave, A2(z)—probe wave. (b) Schematic distribution of the pump and probe intensities during SBS.

Fig. 2.
Fig. 2.

Relative error of linear approximation of 3D parametric model of output CW. L=1000m, 0<Ppump<10mW, 0<Pprobe<40mW.

Fig. 3.
Fig. 3.

Relative error of quadratic approximation of 3D parametric model of output CW. L=1000m, 0<Ppump<10mW, 0<Pprobe<40mW.

Fig. 4.
Fig. 4.

Linear approximation of 3D parametric model of output CW. Dimensionless output intensity of the CW versus dimensionless parameters β1 and β3. γe=0.902, v=5616m/s, n=1.48, λ=1.319μm, ρ0=2.21g/cm3, ΓB=0.1GHz L=1000m, 0<Ppump<10mW, 0<Pprobe<40mW.

Fig. 5.
Fig. 5.

Linear approximation of 3D parametric model of output PW. Dimensionless output intensity of the PW versus dimensionless parameters β1 and β3. γe=0.902, v=5616m/s, n=1.48, λ=1.319μm, ρ0=2.21g/cm3, ΓB=0.1GHz L=1000m, 0<Ppump<10mW, 0<Pprobe<40mW.

Fig. 6.
Fig. 6.

Analytical results, normalized intensity units. PPW (mW) = ○ 0.01; ▵ 1.8; × 6.6; ◻ 12.1; ▿ 17.1; + 22.4; * 27.2; --- 31.8; ─ 36.3. n=1.48, γe=0.902, λ=1319nm, ρ0=2.21g/cm3, v=5616m/s, L=1000m, ΓB=0.1GHz, PCW=1.0mW.

Fig. 7.
Fig. 7.

Pump depletion as a function of probe spectral distortion. PPW (mW) = ○ 0.01; ▵ 1.8; × 6.6; ◻ 12.1; ▿ 17.1; + 22.4; * 27.2; ▪ 31.8; ♦ 36.3. n=1.48, γe=0.902, λ=1319nm, ρ0=2.21g/cm3, v=5616m/s, L=1000m, ΓB=0.1GHz, PCW=1.0mW.

Fig. 8.
Fig. 8.

Shaded area depicts range of β1 and β3 values that yield curvatures within 20% of the Lorentz curvature for both CW and PW spectra.

Fig. 9.
Fig. 9.

Experimental setup.

Fig. 10.
Fig. 10.

Experimental results, normalized intensity units. PPW (mW) = ○ 0.01; ▵ 1.8; × 6.6; ◻ 12.1; ▿ 17.1; + 22.4; * 27.2; --- 31.8; ─ 36.3. n=1.48, γe=0.902, λ=1319nm, ρ0=2.21g/cm3, v=5616m/s, L=1000m, ΓB=0.1GHz, PCW=1.0mW.

Equations (37)

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A1z=iω1γe2ncρ0ρ1A2,
A2z=iω2γe2ncρ0ρ1*A1,
(ΩB2Ω12iΩ1ΓB)ρ1=γeω12n2πc2A1A2*,
|A1(L)|2=A102;|A2(0)|2=A202,
dY1dl=β1·Y1Y2,
dY2dl=β3·Y1Y2,
|ρ1ρ0|2=β5·Y1Y2,
Y1(1)=1;Y2(0)=1.
β1=2γe2k2ω1I20Ln2c2ρ0Ω1ΓB·11+ξ2,
β3=2γe2k2ω2I10Ln2c2ρ0Ω1ΓB·11+ξ2,
β5=(2γe2k2ncρ0Ω1ΓB)2·11+ξ2·I10I20,
ξ=ΩB2Ω12Ω1ΓB,
Y1(l)=Y1(0)β1β31β1β3·1Y1(0)·e(Y1(0)β1β3)·β3·l;Y1(0)β1/β3,
Y2(l)=1+GPW;Y1(0)β1/β3,
GPW=e(Y1(0)β1β3)·β3·l11β1β3·1Y1(0)·e(Y1(0)β1β3)·β3·l.
1Y1(0)β1β3·ln{β3β1·Y1(0)·[1+β1β3Y1(0)]}β3=0;Y1(0)β1/β3.
1Y1(0)β1·eβ3·Y1(0)β11β3·Y1(0)β1=0.
Y2(1)=1+β3β1[1Y1(0)].
F(β1,β3,x)=1xβ1·eβ3·xβ11β3·xβ1,
F(β1,β3,x)=eβ1xeβ1n=1Cn·xn·β3n.
eβ1xeβ1n=1Cn·xn·β3n=0.
x=β1(β1+β3)·eβ1β3·(1+β1).
x=xlinear·112+14+xlinear2·β32·eβ11β112β12β12,
ΩBΩ11;ω2ω21;ΩB+Ω12ΩB.
x(ξ)=b(1+b)·eβ1(ξ)1β1(ξ),
μ(ξ)=1+1b1(1+b)·eβ1(ξ)1β1(ξ),
β1(ξ)=β101+ξ2;β3(ξ)=β301+ξ2;β10=2γe2kair2I20Lcρ0ΓBΩB;β30=2γe2kair2I10Lcρ0ΓBΩB;x0=x(0)=b(1+b)·eβ101β10.
FWHM=2β101+x01x0(12+121+21+bb1x01+x0)1(units ofΓB).
P(ξ)=x(ξ)+1=β301+ξ212β30(β30β10)(1+ξ2)2,
S(ξ)=μ(ξ)1=β101+ξ2+12β30(β30β10)(1+ξ2)2.
β101,β301.
CR=|CCWCLorentzCLorentz|.
CCW=2β10x0(β30x0+1),
CPW=2β30x0(β30x0+1),
CLorentz=2β10for theCW,
CLorentz=2β30for thePW.
CR<δ.

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