Abstract

Brillouin correlation-domain techniques used for fiber optic distributed sensing have been widely studied owing to their unique feature in which the spatial resolution is not limited by the phonon lifetime, unlike with time-resolved methods. This approach can be divided into two main classes according to the scattering type, i.e., spontaneous or stimulated Brillouin scattering. In this study, we derived a formula for the measurement spectrum of the correlation-domain reflectometry using spontaneous Brillouin scattering by considering its stochastic properties. The derived formula is equivalent to the formula of the system using stimulated Brillouin scattering. Our results indicate that the methods developed thus far for improving the system’s performance can be commonly applied.

© 2020 Optical Society of America

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References

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  1. X. Bao and L. Chen, “Recent progress in Brillouin scattering based fiber sensors,” Sensors 11, 4152–4187 (2011).
    [Crossref]
  2. A. H. Hartog, An Introduction to Distributed Optical Fibre Sensors, 1st ed. (CRC Press, 2018).
  3. K. Hotate, “Brillouin optical correlation-domain technologies based on synthesis of optical coherence function as fiber optic nerve systems for structural health monitoring,” Appl. Sci. 9, 187 (2019).
    [Crossref]
  4. T. Horiguchi and M. Tateda, “BOTDA-nondestructive measurement of single-mode optical fiber attenuation characteristics using Brillouin interaction: theory,” J. Lightwave Technol. 7, 1170–1176 (1989).
    [Crossref]
  5. K. Hotate and T. Hasegawa, “Measurement of Brillouin gain spectrum distribution along an optical fiber using a correlation-based technique–proposal, experiment and simulation,” IEICE Trans. Electron. E83-C, 405–412 (2000).
  6. Y. Mizuno, W. Zou, Z. He, and K. Hotate, “Proposal of Brillouin optical correlation-domain reflectometry (BOCDR),” Opt. Express 16, 12148–12153 (2008).
    [Crossref]
  7. Y. Mizuno, H. Lee, and K. Nakamura, “Recent advances in Brillouin optical correlation-domain reflectometry,” Appl. Sci. 8, 1845 (2018).
    [Crossref]
  8. Y. Mizuno, Z. He, and K. Hotate, “One-end-access high-speed distributed strain measurement with 13-mm spatial resolution based on brillouin optical correlation-domain reflectometry,” IEEE Photon. Technol. Lett. 21, 474–476 (2009).
    [Crossref]
  9. Y. Mizuno, Z. He, and K. Hotate, “Measurement range enlargement in Brillouin optical correlation-domain reflectometry based on temporal gating scheme,” Opt. Express 17, 9040–9046 (2009).
    [Crossref]
  10. Y. Mizuno, Z. He, and K. Hotate, “Measurement range enlargement in Brillouin optical correlation-domain reflectometry based on double-modulation scheme,” Opt. Express 18, 5926–5933 (2010).
    [Crossref]
  11. Y. Mizuno, N. Hayashi, H. Fukuda, K. Y. Song, and K. Nakamura, “Ultrahigh-speed distributed Brillouin reflectometry,” Light Sci. Appl. 5, e16184 (2016).
    [Crossref]
  12. Y. Mizuno, W. Zou, Z. He, and K. Hotate, “Operation of Brillouin optical correlation-domain reflectometry: theoretical analysis and experimental validation,” J. Lightwave Technol. 28, 3300–3306 (2010).
    [Crossref]
  13. K.-Y. Song, Z. He, and K. Hotate, “Effects of intensity modulation of light source on Brillouin optical correlation domain analysis,” J. Lightwave Technol. 25, 1238–1246 (2007).
    [Crossref]
  14. J. H. Jeong, K. Lee, K. Y. Song, J.-M. Jeong, and S. B. Lee, “Differential measurement scheme for Brillouin optical correlation domain analysis,” Opt. Express 20, 27094–27101 (2012).
    [Crossref]
  15. R. W. Boyd, K. Rząewski, and P. Narum, “Noise initiation of stimulated Brillouin scattering,” Phys. Rev. A 42, 5514–5521 (1990).
    [Crossref]
  16. R. W. Boyd, Nonlinear Optics, 3rd ed. (Academic, 2008).
  17. R. B. Jenkins, R. M. Sova, and R. I. Joseph, “Steady-state noise analysis of spontaneous and stimulated Brillouin scattering in optical fibers,” J. Lightwave Technol. 25, 763–770 (2007).
    [Crossref]
  18. A. Papoulis, Probability, Random Variables and Stochastic Processes (McGraw-Hill, 1965).
  19. This can also be explained from the standpoint of quantum theory. That is, each frequency of the Stokes photon owing to a spontaneous emission is related to the phonon frequency to satisfy the energy conservation law ($\hbar {\omega _{\rm pump}} = \hbar {\omega _{\rm phonon}} + \hbar {\omega _{\rm stokes}}$ℏωpump=ℏωphonon+ℏωstokes), and the state of the composite system should be entangled within the time-frequency domain. Therefore, if we trace out the phonon system, each frequency state of the Stokes photon will be in a mixed state.

2019 (1)

K. Hotate, “Brillouin optical correlation-domain technologies based on synthesis of optical coherence function as fiber optic nerve systems for structural health monitoring,” Appl. Sci. 9, 187 (2019).
[Crossref]

2018 (1)

Y. Mizuno, H. Lee, and K. Nakamura, “Recent advances in Brillouin optical correlation-domain reflectometry,” Appl. Sci. 8, 1845 (2018).
[Crossref]

2016 (1)

Y. Mizuno, N. Hayashi, H. Fukuda, K. Y. Song, and K. Nakamura, “Ultrahigh-speed distributed Brillouin reflectometry,” Light Sci. Appl. 5, e16184 (2016).
[Crossref]

2012 (1)

2011 (1)

X. Bao and L. Chen, “Recent progress in Brillouin scattering based fiber sensors,” Sensors 11, 4152–4187 (2011).
[Crossref]

2010 (2)

2009 (2)

Y. Mizuno, Z. He, and K. Hotate, “One-end-access high-speed distributed strain measurement with 13-mm spatial resolution based on brillouin optical correlation-domain reflectometry,” IEEE Photon. Technol. Lett. 21, 474–476 (2009).
[Crossref]

Y. Mizuno, Z. He, and K. Hotate, “Measurement range enlargement in Brillouin optical correlation-domain reflectometry based on temporal gating scheme,” Opt. Express 17, 9040–9046 (2009).
[Crossref]

2008 (1)

2007 (2)

2000 (1)

K. Hotate and T. Hasegawa, “Measurement of Brillouin gain spectrum distribution along an optical fiber using a correlation-based technique–proposal, experiment and simulation,” IEICE Trans. Electron. E83-C, 405–412 (2000).

1990 (1)

R. W. Boyd, K. Rząewski, and P. Narum, “Noise initiation of stimulated Brillouin scattering,” Phys. Rev. A 42, 5514–5521 (1990).
[Crossref]

1989 (1)

T. Horiguchi and M. Tateda, “BOTDA-nondestructive measurement of single-mode optical fiber attenuation characteristics using Brillouin interaction: theory,” J. Lightwave Technol. 7, 1170–1176 (1989).
[Crossref]

Bao, X.

X. Bao and L. Chen, “Recent progress in Brillouin scattering based fiber sensors,” Sensors 11, 4152–4187 (2011).
[Crossref]

Boyd, R. W.

R. W. Boyd, K. Rząewski, and P. Narum, “Noise initiation of stimulated Brillouin scattering,” Phys. Rev. A 42, 5514–5521 (1990).
[Crossref]

R. W. Boyd, Nonlinear Optics, 3rd ed. (Academic, 2008).

Chen, L.

X. Bao and L. Chen, “Recent progress in Brillouin scattering based fiber sensors,” Sensors 11, 4152–4187 (2011).
[Crossref]

Fukuda, H.

Y. Mizuno, N. Hayashi, H. Fukuda, K. Y. Song, and K. Nakamura, “Ultrahigh-speed distributed Brillouin reflectometry,” Light Sci. Appl. 5, e16184 (2016).
[Crossref]

Hartog, A. H.

A. H. Hartog, An Introduction to Distributed Optical Fibre Sensors, 1st ed. (CRC Press, 2018).

Hasegawa, T.

K. Hotate and T. Hasegawa, “Measurement of Brillouin gain spectrum distribution along an optical fiber using a correlation-based technique–proposal, experiment and simulation,” IEICE Trans. Electron. E83-C, 405–412 (2000).

Hayashi, N.

Y. Mizuno, N. Hayashi, H. Fukuda, K. Y. Song, and K. Nakamura, “Ultrahigh-speed distributed Brillouin reflectometry,” Light Sci. Appl. 5, e16184 (2016).
[Crossref]

He, Z.

Horiguchi, T.

T. Horiguchi and M. Tateda, “BOTDA-nondestructive measurement of single-mode optical fiber attenuation characteristics using Brillouin interaction: theory,” J. Lightwave Technol. 7, 1170–1176 (1989).
[Crossref]

Hotate, K.

K. Hotate, “Brillouin optical correlation-domain technologies based on synthesis of optical coherence function as fiber optic nerve systems for structural health monitoring,” Appl. Sci. 9, 187 (2019).
[Crossref]

Y. Mizuno, W. Zou, Z. He, and K. Hotate, “Operation of Brillouin optical correlation-domain reflectometry: theoretical analysis and experimental validation,” J. Lightwave Technol. 28, 3300–3306 (2010).
[Crossref]

Y. Mizuno, Z. He, and K. Hotate, “Measurement range enlargement in Brillouin optical correlation-domain reflectometry based on double-modulation scheme,” Opt. Express 18, 5926–5933 (2010).
[Crossref]

Y. Mizuno, Z. He, and K. Hotate, “Measurement range enlargement in Brillouin optical correlation-domain reflectometry based on temporal gating scheme,” Opt. Express 17, 9040–9046 (2009).
[Crossref]

Y. Mizuno, Z. He, and K. Hotate, “One-end-access high-speed distributed strain measurement with 13-mm spatial resolution based on brillouin optical correlation-domain reflectometry,” IEEE Photon. Technol. Lett. 21, 474–476 (2009).
[Crossref]

Y. Mizuno, W. Zou, Z. He, and K. Hotate, “Proposal of Brillouin optical correlation-domain reflectometry (BOCDR),” Opt. Express 16, 12148–12153 (2008).
[Crossref]

K.-Y. Song, Z. He, and K. Hotate, “Effects of intensity modulation of light source on Brillouin optical correlation domain analysis,” J. Lightwave Technol. 25, 1238–1246 (2007).
[Crossref]

K. Hotate and T. Hasegawa, “Measurement of Brillouin gain spectrum distribution along an optical fiber using a correlation-based technique–proposal, experiment and simulation,” IEICE Trans. Electron. E83-C, 405–412 (2000).

Jenkins, R. B.

Jeong, J. H.

Jeong, J.-M.

Joseph, R. I.

Lee, H.

Y. Mizuno, H. Lee, and K. Nakamura, “Recent advances in Brillouin optical correlation-domain reflectometry,” Appl. Sci. 8, 1845 (2018).
[Crossref]

Lee, K.

Lee, S. B.

Mizuno, Y.

Nakamura, K.

Y. Mizuno, H. Lee, and K. Nakamura, “Recent advances in Brillouin optical correlation-domain reflectometry,” Appl. Sci. 8, 1845 (2018).
[Crossref]

Y. Mizuno, N. Hayashi, H. Fukuda, K. Y. Song, and K. Nakamura, “Ultrahigh-speed distributed Brillouin reflectometry,” Light Sci. Appl. 5, e16184 (2016).
[Crossref]

Narum, P.

R. W. Boyd, K. Rząewski, and P. Narum, “Noise initiation of stimulated Brillouin scattering,” Phys. Rev. A 42, 5514–5521 (1990).
[Crossref]

Papoulis, A.

A. Papoulis, Probability, Random Variables and Stochastic Processes (McGraw-Hill, 1965).

Rzaewski, K.

R. W. Boyd, K. Rząewski, and P. Narum, “Noise initiation of stimulated Brillouin scattering,” Phys. Rev. A 42, 5514–5521 (1990).
[Crossref]

Song, K. Y.

Y. Mizuno, N. Hayashi, H. Fukuda, K. Y. Song, and K. Nakamura, “Ultrahigh-speed distributed Brillouin reflectometry,” Light Sci. Appl. 5, e16184 (2016).
[Crossref]

J. H. Jeong, K. Lee, K. Y. Song, J.-M. Jeong, and S. B. Lee, “Differential measurement scheme for Brillouin optical correlation domain analysis,” Opt. Express 20, 27094–27101 (2012).
[Crossref]

Song, K.-Y.

Sova, R. M.

Tateda, M.

T. Horiguchi and M. Tateda, “BOTDA-nondestructive measurement of single-mode optical fiber attenuation characteristics using Brillouin interaction: theory,” J. Lightwave Technol. 7, 1170–1176 (1989).
[Crossref]

Zou, W.

Appl. Sci. (2)

K. Hotate, “Brillouin optical correlation-domain technologies based on synthesis of optical coherence function as fiber optic nerve systems for structural health monitoring,” Appl. Sci. 9, 187 (2019).
[Crossref]

Y. Mizuno, H. Lee, and K. Nakamura, “Recent advances in Brillouin optical correlation-domain reflectometry,” Appl. Sci. 8, 1845 (2018).
[Crossref]

IEEE Photon. Technol. Lett. (1)

Y. Mizuno, Z. He, and K. Hotate, “One-end-access high-speed distributed strain measurement with 13-mm spatial resolution based on brillouin optical correlation-domain reflectometry,” IEEE Photon. Technol. Lett. 21, 474–476 (2009).
[Crossref]

IEICE Trans. Electron. (1)

K. Hotate and T. Hasegawa, “Measurement of Brillouin gain spectrum distribution along an optical fiber using a correlation-based technique–proposal, experiment and simulation,” IEICE Trans. Electron. E83-C, 405–412 (2000).

J. Lightwave Technol. (4)

Light Sci. Appl. (1)

Y. Mizuno, N. Hayashi, H. Fukuda, K. Y. Song, and K. Nakamura, “Ultrahigh-speed distributed Brillouin reflectometry,” Light Sci. Appl. 5, e16184 (2016).
[Crossref]

Opt. Express (4)

Phys. Rev. A (1)

R. W. Boyd, K. Rząewski, and P. Narum, “Noise initiation of stimulated Brillouin scattering,” Phys. Rev. A 42, 5514–5521 (1990).
[Crossref]

Sensors (1)

X. Bao and L. Chen, “Recent progress in Brillouin scattering based fiber sensors,” Sensors 11, 4152–4187 (2011).
[Crossref]

Other (4)

A. H. Hartog, An Introduction to Distributed Optical Fibre Sensors, 1st ed. (CRC Press, 2018).

R. W. Boyd, Nonlinear Optics, 3rd ed. (Academic, 2008).

A. Papoulis, Probability, Random Variables and Stochastic Processes (McGraw-Hill, 1965).

This can also be explained from the standpoint of quantum theory. That is, each frequency of the Stokes photon owing to a spontaneous emission is related to the phonon frequency to satisfy the energy conservation law ($\hbar {\omega _{\rm pump}} = \hbar {\omega _{\rm phonon}} + \hbar {\omega _{\rm stokes}}$ℏωpump=ℏωphonon+ℏωstokes), and the state of the composite system should be entangled within the time-frequency domain. Therefore, if we trace out the phonon system, each frequency state of the Stokes photon will be in a mixed state.

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Figures (2)

Fig. 1.
Fig. 1. Basic experimental setup of (a) BOCDA and (b) BOCDR. LD, laser diode; FM, frequency modulation; EDFA, erbium-doped fiber amplifier; FUT, fiber under test; B. PD, balanced photodetector, ESA, electrical spectrum analyzer; SSBM, single side-band modulator; PD, photodetector; OS, oscilloscope.
Fig. 2.
Fig. 2. Configuration of pump and Stokes light in FUT.

Equations (45)

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E P z + 1 v g E P t = α E P + i κ ρ E S ,
E S z 1 v g E S t = α E S i κ ρ E P ,
ρ t + ( Γ a 2 + i δ ω B ) ρ = i Λ E P E S + f .
E P = j = 0 E P ( j ) , E S = j = 0 E S ( j ) , ρ = j = 0 ρ ( j ) E P ( j ) , E S ( j ) , ρ ( j ) γ e j .
ρ ( 0 ) t + Γ ρ ( 0 ) = f ,
f ( z , t ) f ( z , t ) = Q δ ( z z ) δ ( t t ) ,
ρ ( 0 ) ( z , t ) = e Γ t t e Γ τ f ( z , τ ) d τ .
ρ ^ ( 0 ) ( z , ω ) = D ^ ( z , ω ) f ^ ( z , ω )
D ^ ( z , ω ) = 1 Γ i ω
f ^ ( z , ω ) f ^ ( z , ω ) = d t d t f ( z , t ) f ( z , t ) e i ω t i ω t = d t d t Q δ ( z z ) δ ( t t ) e i ω t i ω t = Q δ ( z z ) d t e i ( ω ω ) t = Q δ ( z z ) δ ( ω ω ) .
ρ ^ ( 0 ) ( z , ω ) ρ ^ ( 0 ) ( z , ω ) = Q g B ( z , ω ) δ ( z z ) δ ( ω ω ) .
E P ( 0 ) z + 1 v g E P ( 0 ) t = 0.
E ´ P ( 0 ) ( z 1 , t 1 ) z 1 = 0
E ´ P ( 0 ) ( z 1 , t 1 ) = E ´ P ( 0 ) ( 0 , t 1 ) ,
E P ( 0 ) ( z , t ) = E P ( 0 ) ( 0 , t ζ z ) ( = e P ( t ζ z ) ) .
E S ( 0 ) z 1 v g E S ( 0 ) t = 0.
E S ( 0 ) ( z , t ) = E S ( 0 ) ( L , t ζ L z ) = 0.
E S ( 1 ) z 1 v g E S ( 1 ) t = i κ ρ ( 0 ) E P ( 0 ) .
E ´ S ( 1 ) ( z 2 , t 2 ) z 2 = i κ ρ ´ ( 0 ) ( z 2 , t 2 ) E ´ P ( 0 ) ( z 2 , t 2 ) .
E ´ S ( 1 ) ( 0 , t 2 ) = i κ 0 L d z 2 E ´ P ( 0 ) ( z 2 , t 2 ) ρ ´ ( 0 ) ( z 2 , t 2 ) ,
E S ( 1 ) ( 0 , t ) = i κ 0 L d z E P ( 0 ) ( z , t ζ z ) ρ ( 0 ) ( z , t ζ z ) .
| E P ( 0 , t ) e i ω P t + E S ( 0 , t ) e i ω S t | 2 | E P ( 0 , t ) e i ω P t E S ( 0 , t ) e i ω S t | 2 = 2 E P ( 0 , t ) E S ( 0 , t ) e i ( ω P ω S ) t + c . c .
M ( t ) = E P ( 0 , t ) E S ( 0 , t ) ,
E P ( 0 ) ( 0 , t ) E S ( 1 ) ( 0 , t ) .
B ( z , t ) = E P ( 0 ) ( 0 , t ) E P ( 0 ) ( z , t ζ z ) ,
( = e P ( t ) e P ( t 2 ζ z ) ) .
M ( t ) = i κ E P ( 0 ) ( 0 , t ) 0 L d z E P ( 0 ) ( z , t ζ z ) ρ ( 0 ) ( z , t ζ z ) ,
= i κ 0 L d z B ( z , t ) ρ ( 0 ) ( z , t ζ z ) ,
M ^ ( ω ) = i κ 0 L d z [ B ^ ( z , ω ) ω ρ ^ ( 0 ) ( z , ω ) ] e i ω ζ z ,
| M ^ ( ω ) | 2 = κ 2 | 0 L d z B ^ ( z , ω ) ω ρ ^ ( 0 ) ( z , ω ) e i ω ζ z | 2 = κ 2 0 L d z 1 d z 2 d ω 1 d ω 2 B ^ ( z 1 , ω 1 ) B ^ ( z 2 , ω 2 ) × ρ ^ ( 0 ) ( z 1 , ω ω 1 ) ρ ^ ( 0 ) ( z 2 , ω ω 2 ) e i ω 1 ζ z 1 + i ω 2 ζ z 2 = κ 2 Q 0 L d z 1 d z 2 d ω 1 d ω 2 B ^ ( z 1 , ω 1 ) B ^ ( z 2 , ω 2 ) × g B ( z 1 , ω ω 1 ) δ ( z 1 z 2 ) δ ( ω 1 ω 2 ) e i ω 1 ζ z 1 + i ω 2 ζ z 2 = κ 2 Q 0 L d z g B ( z , ω ) ω S b ( z , ω ) ,
E P z + 1 v g E P t = i κ ρ E S
E S z 1 v g E S t = i κ ρ E P
ρ t + Γ ρ = i Λ E P E S
E P ( 0 ) ( z , t ) = E P ( 0 ) ( 0 , t ζ z )
E S ( 0 ) ( z , t ) = E S ( 0 ) ( 0 , t + ζ z )
ρ ( 0 ) ( z , t ) = 0
ρ ( 1 ) t + Γ ρ ( 1 ) = i Λ E P ( 0 ) E S ( 0 ) .
ρ ( 1 ) ( z , ω ) = i Λ D ^ ( z , ω ) B ^ ( z , ω ) ,
B ( z , t ) = E P ( 0 ) ( z , t ) E S ( 0 ) ( z , t ) ( = E P ( 0 ) ( 0 , t ζ z ) E S ( 0 ) ( 0 , t + ζ z ) ) .
E P ( 2 ) z + 1 v g E P ( 2 ) t = i κ ρ ( 1 ) E S ( 0 )
E S ( 2 ) z 1 v g E S ( 2 ) t = i κ ρ ( 1 ) E P ( 0 ) .
E S ( 2 ) ( 0 , t ) = i κ 0 L d z E P ( 0 ) ( z , t ζ z ) ρ ( 1 ) ( z , t ζ z ) .
Δ E S = d t ( | E S ( 0 ) ( 0 , t ) + E S ( 2 ) ( 0 , t ) | 2 | E S ( 0 ) ( 0 , t ) | 2 ) d t E S ( 0 ) ( 0 , t ) E S ( 2 ) ( 0 , t ) + c . c . = κ Λ 0 L d z d t E S ( 0 ) ( 0 , t ) E P ( 0 ) ( z , t ζ z ) × d ω D ^ ( z , ω ) B ^ ( z , ω ) e i ω ( t ζ z ) + c . c . = κ Λ 0 L d z d ω S b ( z , ω ) g B ( z , ω ) .
Δ E S ( ω ) = κ Λ 0 L d z d ω g B ( z , ω ) ω S b ( z , ω ) .
Δ E S ( z d , ω d ) = κ Λ 0 L d z d ω g B ( z , ω ) S b ( z d z , ω d ω ) .