Abstract

A distributed fiber optic vibration sensor is described, in which two Michelson interferometers are used as phase detectors and two 3×3 couplers are deployed to demodulate the time-varying phase change caused by vibration. The two interferometers are separated by four wavelength division multiplexers. The position of the vibration is obtained by signal correlation, which can be used as a perimeter security sensor to locate the intruder. The experimental results with a 4012m fiber sensor are discussed.

© 2011 Optical Society of America

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  1. F. Xiaojun, “A variable-loop Sagnac interferometer for distributed impact sensing,” J. Lightwave Technol. 14, 2250–2254(1996).
    [CrossRef]
  2. S. J. Spammer, P. L. Swart, and A. A. Chtcherbakov, “Distributed dual-wavelength Sagnac impact sensor,” Microw. Opt. Technol. Lett. 17, 170–173 (1998).
    [CrossRef]
  3. S. J. Russell, K. R. C. Brady, and J. P. Dakin, “Real-time location of multiple time-varying strain disturbances, acting over a 40 km fiber section, using a novel dual-Sagnac interferometer,” J. Lightwave Technol. 19, 205–213 (2001).
    [CrossRef]
  4. A. A. ChtcherbakovP. L. Swart, S. J. Spammer, and B. M. Lacquet, “A modified Sagnac/Mach–Zehnder interferometer for distributed disturbance sensing,” Proc. SPIE 3489, 60–64(1998).
    [CrossRef]
  5. S. J. Spammer, P. L. Swart, and A. A. Chtcherbakov, “Merged Sagnac–Michelson interferometer for distributed disturbance detection,” J. Lightwave Technol. 15, 972–976 (1997).
    [CrossRef]
  6. J. Tapanes and E. Edwardo, “Perimeter security system and perimeter monitoring method,” International patent WO 02/ 071356 (12 September 2002).
  7. C. Juarez, E. W. Maier, N. C. Kyoo, and H. F. Taylor, “Distributed fiber-optic intrusion sensor system,” J. Lightwave Technol. 23, 2081–2087 (2005).
    [CrossRef]
  8. Y. Lu, T. Zhu, L. Chen, and X. Bao, “Distributed vibration sensor based on coherent detection of phase-OTDR,” J. Lightwave Technol. 28, 3243–3249 (2010).
    [CrossRef]
  9. X. Hong, H. Guo, J. Wu, K. Xu, Y. Zuo, Y. Li, and J. Lin, “An intrusion detection sensor based on coherent optical time domain reflector,” Microw. Opt. Technol. Lett. 52, 2746–2748(2010).
    [CrossRef]
  10. J. C. Juan and T. F. Henry, “Field test of a distributed fiber-optic intrusion sensor system for long perimeters,” Appl. Opt. 46, 1968–1971 (2007).
    [CrossRef]
  11. Z. Zhang and X. Bao, “Distributed optical fiber vibration sensor based on spectrum analysis of polarization-OTDR system,” Opt. Express 16, 10240–10247 (2008).
    [CrossRef] [PubMed]
  12. H. Xiaobin, G. Hongxiang, and W. Jian, “A Brillouin optical time domain analysis based distributed fiber optic intrusion sensor system,” Chin. J. Lasers 37, 1037–1041(2010).
    [CrossRef]
  13. X. Hong, H. Guo, J. Wu, K. Xu, Y. Zuo, Y. Li, J. Lin, and C. Zuo, “Distributed fiber optics sensors for intrusion detection,” The 2nd Asia-Pacific Optical Sensors Conference 2010, (Asia-Pacific Optical Sensor Conference Committee, 2010), paper TU7.
  14. A. D. Kersey, M. J. Marrone, A. Dandridge, and A. B. Tveten, “Optimization and stabilization of visibility in interferometric fiber-optic sensors using input-polarization control,” J. Lightwave Technol. 6, 1599–1609 (1988).
    [CrossRef]
  15. A. D. Kersey, M. J. Marrone, and M. A. Davis, “Polarization insensitive fiber optic Michelson interferometer,” Electron. Lett. 27, 518–519 (1991).
    [CrossRef]
  16. L. A. Ferreira, J. L. Santos, and F. Farahi, “Polarization-induced noise in a fiber-optic Michelson interferometer with Faraday rotator mirror elements,” Appl. Opt. 34, 6399–6402(1995).
    [CrossRef] [PubMed]
  17. K. P. Koo, A. B. Tveten, and A. Dandridge, “Passive stabilization scheme for fiber interferometers using (3×3) fiber directional couplers,” Appl. Phys. Lett. 41, 616–618(1982).
    [CrossRef]
  18. V. Zetterberg, M. I. Pettersson, and I. Claesson, “Comparison between whitened generalized cross correlation and adaptive filter for time delay estimation with scattered arrays for passive positioning of moving targets in Baltic Sea shallow waters,” in Proceedings of IEEE Conference on OCEANS (IEEE, 2005), pp. 2356–2361.

2010 (4)

X. Hong, H. Guo, J. Wu, K. Xu, Y. Zuo, Y. Li, and J. Lin, “An intrusion detection sensor based on coherent optical time domain reflector,” Microw. Opt. Technol. Lett. 52, 2746–2748(2010).
[CrossRef]

H. Xiaobin, G. Hongxiang, and W. Jian, “A Brillouin optical time domain analysis based distributed fiber optic intrusion sensor system,” Chin. J. Lasers 37, 1037–1041(2010).
[CrossRef]

X. Hong, H. Guo, J. Wu, K. Xu, Y. Zuo, Y. Li, J. Lin, and C. Zuo, “Distributed fiber optics sensors for intrusion detection,” The 2nd Asia-Pacific Optical Sensors Conference 2010, (Asia-Pacific Optical Sensor Conference Committee, 2010), paper TU7.

Y. Lu, T. Zhu, L. Chen, and X. Bao, “Distributed vibration sensor based on coherent detection of phase-OTDR,” J. Lightwave Technol. 28, 3243–3249 (2010).
[CrossRef]

2008 (1)

2007 (1)

2005 (2)

C. Juarez, E. W. Maier, N. C. Kyoo, and H. F. Taylor, “Distributed fiber-optic intrusion sensor system,” J. Lightwave Technol. 23, 2081–2087 (2005).
[CrossRef]

V. Zetterberg, M. I. Pettersson, and I. Claesson, “Comparison between whitened generalized cross correlation and adaptive filter for time delay estimation with scattered arrays for passive positioning of moving targets in Baltic Sea shallow waters,” in Proceedings of IEEE Conference on OCEANS (IEEE, 2005), pp. 2356–2361.

2001 (1)

1998 (2)

S. J. Spammer, P. L. Swart, and A. A. Chtcherbakov, “Distributed dual-wavelength Sagnac impact sensor,” Microw. Opt. Technol. Lett. 17, 170–173 (1998).
[CrossRef]

A. A. ChtcherbakovP. L. Swart, S. J. Spammer, and B. M. Lacquet, “A modified Sagnac/Mach–Zehnder interferometer for distributed disturbance sensing,” Proc. SPIE 3489, 60–64(1998).
[CrossRef]

1997 (1)

S. J. Spammer, P. L. Swart, and A. A. Chtcherbakov, “Merged Sagnac–Michelson interferometer for distributed disturbance detection,” J. Lightwave Technol. 15, 972–976 (1997).
[CrossRef]

1996 (1)

F. Xiaojun, “A variable-loop Sagnac interferometer for distributed impact sensing,” J. Lightwave Technol. 14, 2250–2254(1996).
[CrossRef]

1995 (1)

1991 (1)

A. D. Kersey, M. J. Marrone, and M. A. Davis, “Polarization insensitive fiber optic Michelson interferometer,” Electron. Lett. 27, 518–519 (1991).
[CrossRef]

1988 (1)

A. D. Kersey, M. J. Marrone, A. Dandridge, and A. B. Tveten, “Optimization and stabilization of visibility in interferometric fiber-optic sensors using input-polarization control,” J. Lightwave Technol. 6, 1599–1609 (1988).
[CrossRef]

1982 (1)

K. P. Koo, A. B. Tveten, and A. Dandridge, “Passive stabilization scheme for fiber interferometers using (3×3) fiber directional couplers,” Appl. Phys. Lett. 41, 616–618(1982).
[CrossRef]

Bao, X.

Brady, K. R. C.

Chen, L.

Chtcherbakov, A. A.

S. J. Spammer, P. L. Swart, and A. A. Chtcherbakov, “Distributed dual-wavelength Sagnac impact sensor,” Microw. Opt. Technol. Lett. 17, 170–173 (1998).
[CrossRef]

A. A. ChtcherbakovP. L. Swart, S. J. Spammer, and B. M. Lacquet, “A modified Sagnac/Mach–Zehnder interferometer for distributed disturbance sensing,” Proc. SPIE 3489, 60–64(1998).
[CrossRef]

S. J. Spammer, P. L. Swart, and A. A. Chtcherbakov, “Merged Sagnac–Michelson interferometer for distributed disturbance detection,” J. Lightwave Technol. 15, 972–976 (1997).
[CrossRef]

Claesson, I.

V. Zetterberg, M. I. Pettersson, and I. Claesson, “Comparison between whitened generalized cross correlation and adaptive filter for time delay estimation with scattered arrays for passive positioning of moving targets in Baltic Sea shallow waters,” in Proceedings of IEEE Conference on OCEANS (IEEE, 2005), pp. 2356–2361.

Dakin, J. P.

Dandridge, A.

A. D. Kersey, M. J. Marrone, A. Dandridge, and A. B. Tveten, “Optimization and stabilization of visibility in interferometric fiber-optic sensors using input-polarization control,” J. Lightwave Technol. 6, 1599–1609 (1988).
[CrossRef]

K. P. Koo, A. B. Tveten, and A. Dandridge, “Passive stabilization scheme for fiber interferometers using (3×3) fiber directional couplers,” Appl. Phys. Lett. 41, 616–618(1982).
[CrossRef]

Davis, M. A.

A. D. Kersey, M. J. Marrone, and M. A. Davis, “Polarization insensitive fiber optic Michelson interferometer,” Electron. Lett. 27, 518–519 (1991).
[CrossRef]

Edwardo, E.

J. Tapanes and E. Edwardo, “Perimeter security system and perimeter monitoring method,” International patent WO 02/ 071356 (12 September 2002).

Farahi, F.

Ferreira, L. A.

Guo, H.

X. Hong, H. Guo, J. Wu, K. Xu, Y. Zuo, Y. Li, and J. Lin, “An intrusion detection sensor based on coherent optical time domain reflector,” Microw. Opt. Technol. Lett. 52, 2746–2748(2010).
[CrossRef]

X. Hong, H. Guo, J. Wu, K. Xu, Y. Zuo, Y. Li, J. Lin, and C. Zuo, “Distributed fiber optics sensors for intrusion detection,” The 2nd Asia-Pacific Optical Sensors Conference 2010, (Asia-Pacific Optical Sensor Conference Committee, 2010), paper TU7.

Henry, T. F.

Hong, X.

X. Hong, H. Guo, J. Wu, K. Xu, Y. Zuo, Y. Li, J. Lin, and C. Zuo, “Distributed fiber optics sensors for intrusion detection,” The 2nd Asia-Pacific Optical Sensors Conference 2010, (Asia-Pacific Optical Sensor Conference Committee, 2010), paper TU7.

X. Hong, H. Guo, J. Wu, K. Xu, Y. Zuo, Y. Li, and J. Lin, “An intrusion detection sensor based on coherent optical time domain reflector,” Microw. Opt. Technol. Lett. 52, 2746–2748(2010).
[CrossRef]

Hongxiang, G.

H. Xiaobin, G. Hongxiang, and W. Jian, “A Brillouin optical time domain analysis based distributed fiber optic intrusion sensor system,” Chin. J. Lasers 37, 1037–1041(2010).
[CrossRef]

Jian, W.

H. Xiaobin, G. Hongxiang, and W. Jian, “A Brillouin optical time domain analysis based distributed fiber optic intrusion sensor system,” Chin. J. Lasers 37, 1037–1041(2010).
[CrossRef]

Juan, J. C.

Juarez, C.

Kersey, A. D.

A. D. Kersey, M. J. Marrone, and M. A. Davis, “Polarization insensitive fiber optic Michelson interferometer,” Electron. Lett. 27, 518–519 (1991).
[CrossRef]

A. D. Kersey, M. J. Marrone, A. Dandridge, and A. B. Tveten, “Optimization and stabilization of visibility in interferometric fiber-optic sensors using input-polarization control,” J. Lightwave Technol. 6, 1599–1609 (1988).
[CrossRef]

Koo, K. P.

K. P. Koo, A. B. Tveten, and A. Dandridge, “Passive stabilization scheme for fiber interferometers using (3×3) fiber directional couplers,” Appl. Phys. Lett. 41, 616–618(1982).
[CrossRef]

Kyoo, N. C.

Lacquet, B. M.

A. A. ChtcherbakovP. L. Swart, S. J. Spammer, and B. M. Lacquet, “A modified Sagnac/Mach–Zehnder interferometer for distributed disturbance sensing,” Proc. SPIE 3489, 60–64(1998).
[CrossRef]

Li, Y.

X. Hong, H. Guo, J. Wu, K. Xu, Y. Zuo, Y. Li, J. Lin, and C. Zuo, “Distributed fiber optics sensors for intrusion detection,” The 2nd Asia-Pacific Optical Sensors Conference 2010, (Asia-Pacific Optical Sensor Conference Committee, 2010), paper TU7.

X. Hong, H. Guo, J. Wu, K. Xu, Y. Zuo, Y. Li, and J. Lin, “An intrusion detection sensor based on coherent optical time domain reflector,” Microw. Opt. Technol. Lett. 52, 2746–2748(2010).
[CrossRef]

Lin, J.

X. Hong, H. Guo, J. Wu, K. Xu, Y. Zuo, Y. Li, and J. Lin, “An intrusion detection sensor based on coherent optical time domain reflector,” Microw. Opt. Technol. Lett. 52, 2746–2748(2010).
[CrossRef]

X. Hong, H. Guo, J. Wu, K. Xu, Y. Zuo, Y. Li, J. Lin, and C. Zuo, “Distributed fiber optics sensors for intrusion detection,” The 2nd Asia-Pacific Optical Sensors Conference 2010, (Asia-Pacific Optical Sensor Conference Committee, 2010), paper TU7.

Lu, Y.

Maier, E. W.

Marrone, M. J.

A. D. Kersey, M. J. Marrone, and M. A. Davis, “Polarization insensitive fiber optic Michelson interferometer,” Electron. Lett. 27, 518–519 (1991).
[CrossRef]

A. D. Kersey, M. J. Marrone, A. Dandridge, and A. B. Tveten, “Optimization and stabilization of visibility in interferometric fiber-optic sensors using input-polarization control,” J. Lightwave Technol. 6, 1599–1609 (1988).
[CrossRef]

Pettersson, M. I.

V. Zetterberg, M. I. Pettersson, and I. Claesson, “Comparison between whitened generalized cross correlation and adaptive filter for time delay estimation with scattered arrays for passive positioning of moving targets in Baltic Sea shallow waters,” in Proceedings of IEEE Conference on OCEANS (IEEE, 2005), pp. 2356–2361.

Russell, S. J.

Santos, J. L.

Spammer, S. J.

A. A. ChtcherbakovP. L. Swart, S. J. Spammer, and B. M. Lacquet, “A modified Sagnac/Mach–Zehnder interferometer for distributed disturbance sensing,” Proc. SPIE 3489, 60–64(1998).
[CrossRef]

S. J. Spammer, P. L. Swart, and A. A. Chtcherbakov, “Distributed dual-wavelength Sagnac impact sensor,” Microw. Opt. Technol. Lett. 17, 170–173 (1998).
[CrossRef]

S. J. Spammer, P. L. Swart, and A. A. Chtcherbakov, “Merged Sagnac–Michelson interferometer for distributed disturbance detection,” J. Lightwave Technol. 15, 972–976 (1997).
[CrossRef]

Swart, P. L.

S. J. Spammer, P. L. Swart, and A. A. Chtcherbakov, “Distributed dual-wavelength Sagnac impact sensor,” Microw. Opt. Technol. Lett. 17, 170–173 (1998).
[CrossRef]

A. A. ChtcherbakovP. L. Swart, S. J. Spammer, and B. M. Lacquet, “A modified Sagnac/Mach–Zehnder interferometer for distributed disturbance sensing,” Proc. SPIE 3489, 60–64(1998).
[CrossRef]

S. J. Spammer, P. L. Swart, and A. A. Chtcherbakov, “Merged Sagnac–Michelson interferometer for distributed disturbance detection,” J. Lightwave Technol. 15, 972–976 (1997).
[CrossRef]

Tapanes, J.

J. Tapanes and E. Edwardo, “Perimeter security system and perimeter monitoring method,” International patent WO 02/ 071356 (12 September 2002).

Taylor, H. F.

Tveten, A. B.

A. D. Kersey, M. J. Marrone, A. Dandridge, and A. B. Tveten, “Optimization and stabilization of visibility in interferometric fiber-optic sensors using input-polarization control,” J. Lightwave Technol. 6, 1599–1609 (1988).
[CrossRef]

K. P. Koo, A. B. Tveten, and A. Dandridge, “Passive stabilization scheme for fiber interferometers using (3×3) fiber directional couplers,” Appl. Phys. Lett. 41, 616–618(1982).
[CrossRef]

Wu, J.

X. Hong, H. Guo, J. Wu, K. Xu, Y. Zuo, Y. Li, J. Lin, and C. Zuo, “Distributed fiber optics sensors for intrusion detection,” The 2nd Asia-Pacific Optical Sensors Conference 2010, (Asia-Pacific Optical Sensor Conference Committee, 2010), paper TU7.

X. Hong, H. Guo, J. Wu, K. Xu, Y. Zuo, Y. Li, and J. Lin, “An intrusion detection sensor based on coherent optical time domain reflector,” Microw. Opt. Technol. Lett. 52, 2746–2748(2010).
[CrossRef]

Xiaobin, H.

H. Xiaobin, G. Hongxiang, and W. Jian, “A Brillouin optical time domain analysis based distributed fiber optic intrusion sensor system,” Chin. J. Lasers 37, 1037–1041(2010).
[CrossRef]

Xiaojun, F.

F. Xiaojun, “A variable-loop Sagnac interferometer for distributed impact sensing,” J. Lightwave Technol. 14, 2250–2254(1996).
[CrossRef]

Xu, K.

X. Hong, H. Guo, J. Wu, K. Xu, Y. Zuo, Y. Li, and J. Lin, “An intrusion detection sensor based on coherent optical time domain reflector,” Microw. Opt. Technol. Lett. 52, 2746–2748(2010).
[CrossRef]

X. Hong, H. Guo, J. Wu, K. Xu, Y. Zuo, Y. Li, J. Lin, and C. Zuo, “Distributed fiber optics sensors for intrusion detection,” The 2nd Asia-Pacific Optical Sensors Conference 2010, (Asia-Pacific Optical Sensor Conference Committee, 2010), paper TU7.

Zetterberg, V.

V. Zetterberg, M. I. Pettersson, and I. Claesson, “Comparison between whitened generalized cross correlation and adaptive filter for time delay estimation with scattered arrays for passive positioning of moving targets in Baltic Sea shallow waters,” in Proceedings of IEEE Conference on OCEANS (IEEE, 2005), pp. 2356–2361.

Zhang, Z.

Zhu, T.

Zuo, C.

X. Hong, H. Guo, J. Wu, K. Xu, Y. Zuo, Y. Li, J. Lin, and C. Zuo, “Distributed fiber optics sensors for intrusion detection,” The 2nd Asia-Pacific Optical Sensors Conference 2010, (Asia-Pacific Optical Sensor Conference Committee, 2010), paper TU7.

Zuo, Y.

X. Hong, H. Guo, J. Wu, K. Xu, Y. Zuo, Y. Li, J. Lin, and C. Zuo, “Distributed fiber optics sensors for intrusion detection,” The 2nd Asia-Pacific Optical Sensors Conference 2010, (Asia-Pacific Optical Sensor Conference Committee, 2010), paper TU7.

X. Hong, H. Guo, J. Wu, K. Xu, Y. Zuo, Y. Li, and J. Lin, “An intrusion detection sensor based on coherent optical time domain reflector,” Microw. Opt. Technol. Lett. 52, 2746–2748(2010).
[CrossRef]

Appl. Opt. (2)

Appl. Phys. Lett. (1)

K. P. Koo, A. B. Tveten, and A. Dandridge, “Passive stabilization scheme for fiber interferometers using (3×3) fiber directional couplers,” Appl. Phys. Lett. 41, 616–618(1982).
[CrossRef]

Chin. J. Lasers (1)

H. Xiaobin, G. Hongxiang, and W. Jian, “A Brillouin optical time domain analysis based distributed fiber optic intrusion sensor system,” Chin. J. Lasers 37, 1037–1041(2010).
[CrossRef]

Electron. Lett. (1)

A. D. Kersey, M. J. Marrone, and M. A. Davis, “Polarization insensitive fiber optic Michelson interferometer,” Electron. Lett. 27, 518–519 (1991).
[CrossRef]

J. Lightwave Technol. (6)

Y. Lu, T. Zhu, L. Chen, and X. Bao, “Distributed vibration sensor based on coherent detection of phase-OTDR,” J. Lightwave Technol. 28, 3243–3249 (2010).
[CrossRef]

S. J. Russell, K. R. C. Brady, and J. P. Dakin, “Real-time location of multiple time-varying strain disturbances, acting over a 40 km fiber section, using a novel dual-Sagnac interferometer,” J. Lightwave Technol. 19, 205–213 (2001).
[CrossRef]

C. Juarez, E. W. Maier, N. C. Kyoo, and H. F. Taylor, “Distributed fiber-optic intrusion sensor system,” J. Lightwave Technol. 23, 2081–2087 (2005).
[CrossRef]

A. D. Kersey, M. J. Marrone, A. Dandridge, and A. B. Tveten, “Optimization and stabilization of visibility in interferometric fiber-optic sensors using input-polarization control,” J. Lightwave Technol. 6, 1599–1609 (1988).
[CrossRef]

S. J. Spammer, P. L. Swart, and A. A. Chtcherbakov, “Merged Sagnac–Michelson interferometer for distributed disturbance detection,” J. Lightwave Technol. 15, 972–976 (1997).
[CrossRef]

F. Xiaojun, “A variable-loop Sagnac interferometer for distributed impact sensing,” J. Lightwave Technol. 14, 2250–2254(1996).
[CrossRef]

Microw. Opt. Technol. Lett. (2)

S. J. Spammer, P. L. Swart, and A. A. Chtcherbakov, “Distributed dual-wavelength Sagnac impact sensor,” Microw. Opt. Technol. Lett. 17, 170–173 (1998).
[CrossRef]

X. Hong, H. Guo, J. Wu, K. Xu, Y. Zuo, Y. Li, and J. Lin, “An intrusion detection sensor based on coherent optical time domain reflector,” Microw. Opt. Technol. Lett. 52, 2746–2748(2010).
[CrossRef]

Opt. Express (1)

Proc. SPIE (1)

A. A. ChtcherbakovP. L. Swart, S. J. Spammer, and B. M. Lacquet, “A modified Sagnac/Mach–Zehnder interferometer for distributed disturbance sensing,” Proc. SPIE 3489, 60–64(1998).
[CrossRef]

Other (3)

X. Hong, H. Guo, J. Wu, K. Xu, Y. Zuo, Y. Li, J. Lin, and C. Zuo, “Distributed fiber optics sensors for intrusion detection,” The 2nd Asia-Pacific Optical Sensors Conference 2010, (Asia-Pacific Optical Sensor Conference Committee, 2010), paper TU7.

J. Tapanes and E. Edwardo, “Perimeter security system and perimeter monitoring method,” International patent WO 02/ 071356 (12 September 2002).

V. Zetterberg, M. I. Pettersson, and I. Claesson, “Comparison between whitened generalized cross correlation and adaptive filter for time delay estimation with scattered arrays for passive positioning of moving targets in Baltic Sea shallow waters,” in Proceedings of IEEE Conference on OCEANS (IEEE, 2005), pp. 2356–2361.

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Figures (9)

Fig. 1
Fig. 1

Schematic of distributed fiber optic intrusion sensor system.

Fig. 2
Fig. 2

The time sequence when a vibration happens in the system.

Fig. 3
Fig. 3

Acquired signals when a vibration happens at L = 4012 m . (a) Received signals at PD 1 and PD 2 varied with time, (b) signals at PD 1 varied with PD 2, (c) received signals at PD 3 and PD 4 varied with time, (d) signals at PD 3 varied with PD 4.

Fig. 4
Fig. 4

(a) Normalized sinusoidal and cosinusoidal signals at coupler C1, (b) normalized sinusoidal and cosinusoidal signals at coupler C2.

Fig. 5
Fig. 5

Recovered signals with delay information as described in Eqs. (15, 16) when Δ ϕ = π / 2 .

Fig. 6
Fig. 6

The variation of peak correlation value and its corresponding position via Δ ϕ .

Fig. 7
Fig. 7

Acquired signals when a vibration happens at L = 4012 m with the same conditions as Fig. 3 but different DC offset.

Fig. 8
Fig. 8

The variation of peak correlation value and its corresponding position via Δ ϕ .

Fig. 9
Fig. 9

The positioning error in 5 m , 1030 m , 1910 m , and 4012 m .

Equations (18)

Equations on this page are rendered with MathJax. Learn more.

S 1 ( t ) = I 11 + I 12 cos { ϕ [ t ( T 1 + τ ) ] + ϕ [ t ( T 1 + 2 T 2 τ ) ] + ϕ 1 } ,
S 2 ( t ) = I 21 + I 22 cos { ϕ [ t ( T 1 + τ ) ] + ϕ [ t ( T 1 + 2 T 2 τ ) ] + ϕ 1 + φ 1 } .
cos { ϕ [ t ( T 1 + τ ) ] + ϕ [ t ( T 1 + 2 T 2 τ ) ] + ϕ 1 } = S 1 ( t ) I 11 I 12 ,
sin { ϕ [ t ( T 1 + τ ) ] + ϕ [ t ( T 1 + 2 T 2 τ ) ] + ϕ 1 } = S 1 ( t ) I 11 I 12 cos φ 1 sin φ 1 S 2 ( t ) I 21 I 22 sin φ 1 .
S 3 ( t ) = I 31 + I 32 cos { ϕ [ t ( T 2 + T 3 τ ) ] + ϕ [ t ( T 2 + T 3 + τ ) ] + ϕ 2 } ,
S 4 ( t ) = I 41 + I 42 cos { ϕ [ t ( T 2 + T 3 τ ) ] + ϕ [ t ( T 2 + T 3 + τ ) ] + ϕ 2 + φ 2 } ,
cos { ϕ [ t ( T 2 + T 3 τ ) ] + ϕ [ t ( T 2 + T 3 + τ ) ] + ϕ 2 } = S 3 ( t ) I 31 I 32 ,
sin { ϕ [ t ( T 2 + T 3 τ ) ] + ϕ [ t ( T 2 + T 3 + τ ) ] + ϕ 2 } = S 3 ( t ) I 31 I 32 cos φ 2 sin φ 2 S 4 ( t ) I 41 I 42 sin φ 2 .
cos { ϕ [ t ( T 1 + τ ) ] ϕ [ t ( T 1 + 2 T 2 + τ ) ] + ϕ 1 ϕ 2 } = f 1 ( t ) ,
f 1 ( t ) = S 1 ( t ) I 11 I 12 S 3 [ t ( T 1 + T 2 T 3 ) ] I 31 I 32 + [ S 1 ( t ) I 11 I 12 cos φ 1 sin φ 1 S 2 ( t ) I 21 I 22 sin φ 1 ] × [ S 3 [ t ( T 1 + T 2 T 3 ) ] I 31 I 32 cos φ 2 sin φ 2 S 4 [ t ( T 1 + T 2 T 3 ) ] I 41 I 42 sin φ 2 ] .
cos { ϕ [ t ( T 2 + T 3 τ ) ] ϕ [ t ( 3 T 2 + T 3 τ ) ] + ϕ 2 ϕ 1 } = f 2 ( t ) ,
f 2 ( t ) = S 1 [ t ( T 2 + T 3 T 1 ) ] I 11 I 12 S 3 ( t ) I 31 I 32 + [ S 3 ( t ) I 31 I 32 cos φ 2 sin φ 2 S 4 ( t ) I 41 I 42 sin φ 2 ] × [ S 1 [ t ( T 2 + T 3 T 1 ) ] I 11 I 12 cos φ 1 sin φ 1 S 2 [ t ( T 2 + T 3 T 1 ) ] I 21 I 22 sin φ 1 ] .
cos [ ψ ( t ) + Δ ϕ ] = f 1 ( t ) ,
cos { ψ [ t ( T 2 + T 3 T 1 2 τ ) ] Δ ϕ } = f 2 ( t ) .
sin [ ψ ( t ) + Δ ϕ ] = f 3 ( t ) ,
sin { ψ [ t ( T 2 + T 3 T 1 2 τ ) ] Δ ϕ } = f 4 ( t ) .
cos [ ψ ( t ) ] = f 1 ( t ) cos Δ ϕ + f 3 ( t ) sin Δ ϕ ,
cos { ψ [ t ( T 2 + T 3 T 1 2 τ ) ] } = f 2 ( t ) cos Δ ϕ f 4 ( t ) sin Δ ϕ .

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