Abstract

A method for phase sensitive quasi-distributed vibration and acoustical sensing is presented. The method is based on double optical frequency domain reflectometry interrogation of a sensing fiber with an array of discrete weak reflectors. Two replicas of the interrogation signal are launched into the sensing fiber. The time delay between the replicas is equal to the roundtrip time between two consecutive reflectors. Each peak in the spectrum of the returning signal is made from a coherent addition of the reflections of two consecutive reflectors. Its magnitude is highly sensitive to the optical phase in the fiber segment between the reflectors. The system was used to detect and locate the fall of a paperclip from height of 40cm onto a sandbox where a 15cm segment of the fiber was buried. In a different experiment the system successfully detected and located minute vibrations at 440Hz that were induced by touching the fiber with a tuning fork.

© 2014 Optical Society of America

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  1. J. C. Juarez, E. W. Maier, K. N. Choi, H. F. Taylor, “Distributed fiber-optic intrusion sensor system,” J. Lightwave Technol. 23(6), 2081–2087 (2005).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
  5. Y. L. Lu, T. Zhu, L. A. Chen, X. Y. Bao, “Distributed Vibration Sensor Based on Coherent Detection of Phase-OTDR,” J. Lightwave Technol. 28, 3243–3249 (2010).
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  10. Z. G. Qin, L. Chen, X. Y. Bao, “Wavelet Denoising Method for Improving Detection Performance of Distributed Vibration Sensor,” IEEE Photon. Technol. Lett. 24(7), 542–544 (2012).
    [CrossRef]
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    [CrossRef]
  12. H. Wu, J. Wang, X. Wu, Y. Wu, “Real Intrusion Detection for Distributed Fiber Fence in Practical Strong Fluctuated Noisy Backgrounds,” Sens. Lett. 10(7), 1557–1563 (2012).
    [CrossRef]
  13. D. P. Zhou, Z. G. Qin, W. H. Li, L. Chen, X. Y. Bao, “Distributed vibration sensing with time-resolved optical frequency-domain reflectometry,” Opt. Express 20(12), 13138–13145 (2012).
    [CrossRef] [PubMed]
  14. T. Zhu, Q. He, X. H. Xiao, X. Y. Bao, “Modulated pulses based distributed vibration sensing with high frequency response and spatial resolution,” Opt. Express 21(3), 2953–2963 (2013).
    [CrossRef] [PubMed]
  15. N. Linze, P. Tihon, O. Verlinden, P. Mégret, M. Wuilpart, “Development of a multi-point polarization-based vibration sensor,” Opt. Express 21(5), 5606–5624 (2013).
    [CrossRef] [PubMed]
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    [CrossRef]
  17. M. Nazarathy, S. A. Newton, R. P. Giffard, D. S. Moberly, F. Sischka, W. R. Trutna, S. Foster, “Real-time long range complementary correlation optical time domain reflectometer,” J. Lightwave Technol. 7(1), 24–38 (1989).
    [CrossRef]
  18. P. Oberson, B. Huttner, O. Guinnard, L. Guinnard, G. Ribordy, N. Gisin, “Optical frequency domain reflectometry with a narrow linewidth fiber laser,” IEEE Photon. Technol. Lett. 12(7), 867–869 (2000).
    [CrossRef]
  19. B. Culshaw, A. Kersey, “Fiber-optic sensing: A historical perspective,” J. Lightwave Technol. 26(9), 1064–1078 (2008).
    [CrossRef]

2013 (2)

2012 (6)

D. P. Zhou, Z. G. Qin, W. H. Li, L. Chen, X. Y. Bao, “Distributed vibration sensing with time-resolved optical frequency-domain reflectometry,” Opt. Express 20(12), 13138–13145 (2012).
[CrossRef] [PubMed]

Z. Ding, X. S. Yao, T. Liu, Y. Du, K. Liu, Q. Han, Z. Meng, H. Chen, “Long-range vibration sensor based on correlation analysis of optical frequency-domain reflectometry signals,” Opt. Express 20(27), 28319–28329 (2012).
[CrossRef] [PubMed]

X. Y. Bao, L. Chen, “Recent Progress in Distributed Fiber Optic Sensors,” Sensors (Basel) 12(7), 8601–8639 (2012).
[CrossRef] [PubMed]

M. M. Chen, X. P. Zhang, F. Wang, X. C. Wang, “Development of Distributed Vibration Sensing System Based on Polarization Optical Time Domain Reflectometry,” Sens. Lett. 10, 1536–1540 (2012).

Z. G. Qin, L. Chen, X. Y. Bao, “Wavelet Denoising Method for Improving Detection Performance of Distributed Vibration Sensor,” IEEE Photon. Technol. Lett. 24(7), 542–544 (2012).
[CrossRef]

H. Wu, J. Wang, X. Wu, Y. Wu, “Real Intrusion Detection for Distributed Fiber Fence in Practical Strong Fluctuated Noisy Backgrounds,” Sens. Lett. 10(7), 1557–1563 (2012).
[CrossRef]

2011 (1)

Z. G. Qin, T. Zhu, L. Chen, X. Y. Bao, “High Sensitivity Distributed Vibration Sensor Based on Polarization-Maintaining Configurations of Phase-OTDR,” IEEE Photon. Technol. Lett. 23(15), 1091–1093 (2011).
[CrossRef]

2010 (2)

X. B. Hong, H. X. Guo, J. A. Wu, K. Xu, Y. Zuo, Y. Li, J. Lin, “An Intrusion Detection Sensor Based on Coherent Optical Time Domain Reflector,” Microw. Opt. Technol. Lett. 52(12), 2746–2748 (2010).
[CrossRef]

Y. L. Lu, T. Zhu, L. A. Chen, X. Y. Bao, “Distributed Vibration Sensor Based on Coherent Detection of Phase-OTDR,” J. Lightwave Technol. 28, 3243–3249 (2010).

2008 (2)

2007 (1)

2005 (1)

2000 (1)

P. Oberson, B. Huttner, O. Guinnard, L. Guinnard, G. Ribordy, N. Gisin, “Optical frequency domain reflectometry with a narrow linewidth fiber laser,” IEEE Photon. Technol. Lett. 12(7), 867–869 (2000).
[CrossRef]

1997 (1)

A. D. Kersey, M. A. Davis, H. J. Patrick, M. LeBlanc, K. P. Koo, C. G. Askins, M. A. Putnam, E. J. Friebele, “Fiber grating sensors,” J. Lightwave Technol. 15(8), 1442–1463 (1997).
[CrossRef]

1989 (1)

M. Nazarathy, S. A. Newton, R. P. Giffard, D. S. Moberly, F. Sischka, W. R. Trutna, S. Foster, “Real-time long range complementary correlation optical time domain reflectometer,” J. Lightwave Technol. 7(1), 24–38 (1989).
[CrossRef]

Askins, C. G.

A. D. Kersey, M. A. Davis, H. J. Patrick, M. LeBlanc, K. P. Koo, C. G. Askins, M. A. Putnam, E. J. Friebele, “Fiber grating sensors,” J. Lightwave Technol. 15(8), 1442–1463 (1997).
[CrossRef]

Bao, X. Y.

Chen, H.

Chen, L.

D. P. Zhou, Z. G. Qin, W. H. Li, L. Chen, X. Y. Bao, “Distributed vibration sensing with time-resolved optical frequency-domain reflectometry,” Opt. Express 20(12), 13138–13145 (2012).
[CrossRef] [PubMed]

Z. G. Qin, L. Chen, X. Y. Bao, “Wavelet Denoising Method for Improving Detection Performance of Distributed Vibration Sensor,” IEEE Photon. Technol. Lett. 24(7), 542–544 (2012).
[CrossRef]

X. Y. Bao, L. Chen, “Recent Progress in Distributed Fiber Optic Sensors,” Sensors (Basel) 12(7), 8601–8639 (2012).
[CrossRef] [PubMed]

Z. G. Qin, T. Zhu, L. Chen, X. Y. Bao, “High Sensitivity Distributed Vibration Sensor Based on Polarization-Maintaining Configurations of Phase-OTDR,” IEEE Photon. Technol. Lett. 23(15), 1091–1093 (2011).
[CrossRef]

Chen, L. A.

Chen, M. M.

M. M. Chen, X. P. Zhang, F. Wang, X. C. Wang, “Development of Distributed Vibration Sensing System Based on Polarization Optical Time Domain Reflectometry,” Sens. Lett. 10, 1536–1540 (2012).

Choi, K. N.

Culshaw, B.

Davis, M. A.

A. D. Kersey, M. A. Davis, H. J. Patrick, M. LeBlanc, K. P. Koo, C. G. Askins, M. A. Putnam, E. J. Friebele, “Fiber grating sensors,” J. Lightwave Technol. 15(8), 1442–1463 (1997).
[CrossRef]

Ding, Z.

Du, Y.

Foster, S.

M. Nazarathy, S. A. Newton, R. P. Giffard, D. S. Moberly, F. Sischka, W. R. Trutna, S. Foster, “Real-time long range complementary correlation optical time domain reflectometer,” J. Lightwave Technol. 7(1), 24–38 (1989).
[CrossRef]

Friebele, E. J.

A. D. Kersey, M. A. Davis, H. J. Patrick, M. LeBlanc, K. P. Koo, C. G. Askins, M. A. Putnam, E. J. Friebele, “Fiber grating sensors,” J. Lightwave Technol. 15(8), 1442–1463 (1997).
[CrossRef]

Giffard, R. P.

M. Nazarathy, S. A. Newton, R. P. Giffard, D. S. Moberly, F. Sischka, W. R. Trutna, S. Foster, “Real-time long range complementary correlation optical time domain reflectometer,” J. Lightwave Technol. 7(1), 24–38 (1989).
[CrossRef]

Gisin, N.

P. Oberson, B. Huttner, O. Guinnard, L. Guinnard, G. Ribordy, N. Gisin, “Optical frequency domain reflectometry with a narrow linewidth fiber laser,” IEEE Photon. Technol. Lett. 12(7), 867–869 (2000).
[CrossRef]

Guinnard, L.

P. Oberson, B. Huttner, O. Guinnard, L. Guinnard, G. Ribordy, N. Gisin, “Optical frequency domain reflectometry with a narrow linewidth fiber laser,” IEEE Photon. Technol. Lett. 12(7), 867–869 (2000).
[CrossRef]

Guinnard, O.

P. Oberson, B. Huttner, O. Guinnard, L. Guinnard, G. Ribordy, N. Gisin, “Optical frequency domain reflectometry with a narrow linewidth fiber laser,” IEEE Photon. Technol. Lett. 12(7), 867–869 (2000).
[CrossRef]

Guo, H. X.

X. B. Hong, H. X. Guo, J. A. Wu, K. Xu, Y. Zuo, Y. Li, J. Lin, “An Intrusion Detection Sensor Based on Coherent Optical Time Domain Reflector,” Microw. Opt. Technol. Lett. 52(12), 2746–2748 (2010).
[CrossRef]

Han, Q.

He, Q.

Hong, X. B.

X. B. Hong, H. X. Guo, J. A. Wu, K. Xu, Y. Zuo, Y. Li, J. Lin, “An Intrusion Detection Sensor Based on Coherent Optical Time Domain Reflector,” Microw. Opt. Technol. Lett. 52(12), 2746–2748 (2010).
[CrossRef]

Huttner, B.

P. Oberson, B. Huttner, O. Guinnard, L. Guinnard, G. Ribordy, N. Gisin, “Optical frequency domain reflectometry with a narrow linewidth fiber laser,” IEEE Photon. Technol. Lett. 12(7), 867–869 (2000).
[CrossRef]

Juarez, J. C.

Kersey, A.

Kersey, A. D.

A. D. Kersey, M. A. Davis, H. J. Patrick, M. LeBlanc, K. P. Koo, C. G. Askins, M. A. Putnam, E. J. Friebele, “Fiber grating sensors,” J. Lightwave Technol. 15(8), 1442–1463 (1997).
[CrossRef]

Koo, K. P.

A. D. Kersey, M. A. Davis, H. J. Patrick, M. LeBlanc, K. P. Koo, C. G. Askins, M. A. Putnam, E. J. Friebele, “Fiber grating sensors,” J. Lightwave Technol. 15(8), 1442–1463 (1997).
[CrossRef]

LeBlanc, M.

A. D. Kersey, M. A. Davis, H. J. Patrick, M. LeBlanc, K. P. Koo, C. G. Askins, M. A. Putnam, E. J. Friebele, “Fiber grating sensors,” J. Lightwave Technol. 15(8), 1442–1463 (1997).
[CrossRef]

Li, W. H.

Li, Y.

X. B. Hong, H. X. Guo, J. A. Wu, K. Xu, Y. Zuo, Y. Li, J. Lin, “An Intrusion Detection Sensor Based on Coherent Optical Time Domain Reflector,” Microw. Opt. Technol. Lett. 52(12), 2746–2748 (2010).
[CrossRef]

Lin, J.

X. B. Hong, H. X. Guo, J. A. Wu, K. Xu, Y. Zuo, Y. Li, J. Lin, “An Intrusion Detection Sensor Based on Coherent Optical Time Domain Reflector,” Microw. Opt. Technol. Lett. 52(12), 2746–2748 (2010).
[CrossRef]

Linze, N.

Liu, K.

Liu, T.

Lu, Y. L.

Maier, E. W.

Mégret, P.

Meng, Z.

Moberly, D. S.

M. Nazarathy, S. A. Newton, R. P. Giffard, D. S. Moberly, F. Sischka, W. R. Trutna, S. Foster, “Real-time long range complementary correlation optical time domain reflectometer,” J. Lightwave Technol. 7(1), 24–38 (1989).
[CrossRef]

Nazarathy, M.

M. Nazarathy, S. A. Newton, R. P. Giffard, D. S. Moberly, F. Sischka, W. R. Trutna, S. Foster, “Real-time long range complementary correlation optical time domain reflectometer,” J. Lightwave Technol. 7(1), 24–38 (1989).
[CrossRef]

Newton, S. A.

M. Nazarathy, S. A. Newton, R. P. Giffard, D. S. Moberly, F. Sischka, W. R. Trutna, S. Foster, “Real-time long range complementary correlation optical time domain reflectometer,” J. Lightwave Technol. 7(1), 24–38 (1989).
[CrossRef]

Oberson, P.

P. Oberson, B. Huttner, O. Guinnard, L. Guinnard, G. Ribordy, N. Gisin, “Optical frequency domain reflectometry with a narrow linewidth fiber laser,” IEEE Photon. Technol. Lett. 12(7), 867–869 (2000).
[CrossRef]

Patrick, H. J.

A. D. Kersey, M. A. Davis, H. J. Patrick, M. LeBlanc, K. P. Koo, C. G. Askins, M. A. Putnam, E. J. Friebele, “Fiber grating sensors,” J. Lightwave Technol. 15(8), 1442–1463 (1997).
[CrossRef]

Putnam, M. A.

A. D. Kersey, M. A. Davis, H. J. Patrick, M. LeBlanc, K. P. Koo, C. G. Askins, M. A. Putnam, E. J. Friebele, “Fiber grating sensors,” J. Lightwave Technol. 15(8), 1442–1463 (1997).
[CrossRef]

Qin, Z. G.

Z. G. Qin, L. Chen, X. Y. Bao, “Wavelet Denoising Method for Improving Detection Performance of Distributed Vibration Sensor,” IEEE Photon. Technol. Lett. 24(7), 542–544 (2012).
[CrossRef]

D. P. Zhou, Z. G. Qin, W. H. Li, L. Chen, X. Y. Bao, “Distributed vibration sensing with time-resolved optical frequency-domain reflectometry,” Opt. Express 20(12), 13138–13145 (2012).
[CrossRef] [PubMed]

Z. G. Qin, T. Zhu, L. Chen, X. Y. Bao, “High Sensitivity Distributed Vibration Sensor Based on Polarization-Maintaining Configurations of Phase-OTDR,” IEEE Photon. Technol. Lett. 23(15), 1091–1093 (2011).
[CrossRef]

Ribordy, G.

P. Oberson, B. Huttner, O. Guinnard, L. Guinnard, G. Ribordy, N. Gisin, “Optical frequency domain reflectometry with a narrow linewidth fiber laser,” IEEE Photon. Technol. Lett. 12(7), 867–869 (2000).
[CrossRef]

Sischka, F.

M. Nazarathy, S. A. Newton, R. P. Giffard, D. S. Moberly, F. Sischka, W. R. Trutna, S. Foster, “Real-time long range complementary correlation optical time domain reflectometer,” J. Lightwave Technol. 7(1), 24–38 (1989).
[CrossRef]

Taylor, H. F.

Tihon, P.

Trutna, W. R.

M. Nazarathy, S. A. Newton, R. P. Giffard, D. S. Moberly, F. Sischka, W. R. Trutna, S. Foster, “Real-time long range complementary correlation optical time domain reflectometer,” J. Lightwave Technol. 7(1), 24–38 (1989).
[CrossRef]

Verlinden, O.

Wang, F.

M. M. Chen, X. P. Zhang, F. Wang, X. C. Wang, “Development of Distributed Vibration Sensing System Based on Polarization Optical Time Domain Reflectometry,” Sens. Lett. 10, 1536–1540 (2012).

Wang, J.

H. Wu, J. Wang, X. Wu, Y. Wu, “Real Intrusion Detection for Distributed Fiber Fence in Practical Strong Fluctuated Noisy Backgrounds,” Sens. Lett. 10(7), 1557–1563 (2012).
[CrossRef]

Wang, X. C.

M. M. Chen, X. P. Zhang, F. Wang, X. C. Wang, “Development of Distributed Vibration Sensing System Based on Polarization Optical Time Domain Reflectometry,” Sens. Lett. 10, 1536–1540 (2012).

Wu, H.

H. Wu, J. Wang, X. Wu, Y. Wu, “Real Intrusion Detection for Distributed Fiber Fence in Practical Strong Fluctuated Noisy Backgrounds,” Sens. Lett. 10(7), 1557–1563 (2012).
[CrossRef]

Wu, J. A.

X. B. Hong, H. X. Guo, J. A. Wu, K. Xu, Y. Zuo, Y. Li, J. Lin, “An Intrusion Detection Sensor Based on Coherent Optical Time Domain Reflector,” Microw. Opt. Technol. Lett. 52(12), 2746–2748 (2010).
[CrossRef]

Wu, X.

H. Wu, J. Wang, X. Wu, Y. Wu, “Real Intrusion Detection for Distributed Fiber Fence in Practical Strong Fluctuated Noisy Backgrounds,” Sens. Lett. 10(7), 1557–1563 (2012).
[CrossRef]

Wu, Y.

H. Wu, J. Wang, X. Wu, Y. Wu, “Real Intrusion Detection for Distributed Fiber Fence in Practical Strong Fluctuated Noisy Backgrounds,” Sens. Lett. 10(7), 1557–1563 (2012).
[CrossRef]

Wuilpart, M.

Xiao, X. H.

Xu, K.

X. B. Hong, H. X. Guo, J. A. Wu, K. Xu, Y. Zuo, Y. Li, J. Lin, “An Intrusion Detection Sensor Based on Coherent Optical Time Domain Reflector,” Microw. Opt. Technol. Lett. 52(12), 2746–2748 (2010).
[CrossRef]

Yao, X. S.

Zhang, X. P.

M. M. Chen, X. P. Zhang, F. Wang, X. C. Wang, “Development of Distributed Vibration Sensing System Based on Polarization Optical Time Domain Reflectometry,” Sens. Lett. 10, 1536–1540 (2012).

Zhang, Z. Y.

Zhou, D. P.

Zhu, T.

Zuo, Y.

X. B. Hong, H. X. Guo, J. A. Wu, K. Xu, Y. Zuo, Y. Li, J. Lin, “An Intrusion Detection Sensor Based on Coherent Optical Time Domain Reflector,” Microw. Opt. Technol. Lett. 52(12), 2746–2748 (2010).
[CrossRef]

Appl. Opt. (1)

IEEE Photon. Technol. Lett. (3)

P. Oberson, B. Huttner, O. Guinnard, L. Guinnard, G. Ribordy, N. Gisin, “Optical frequency domain reflectometry with a narrow linewidth fiber laser,” IEEE Photon. Technol. Lett. 12(7), 867–869 (2000).
[CrossRef]

Z. G. Qin, T. Zhu, L. Chen, X. Y. Bao, “High Sensitivity Distributed Vibration Sensor Based on Polarization-Maintaining Configurations of Phase-OTDR,” IEEE Photon. Technol. Lett. 23(15), 1091–1093 (2011).
[CrossRef]

Z. G. Qin, L. Chen, X. Y. Bao, “Wavelet Denoising Method for Improving Detection Performance of Distributed Vibration Sensor,” IEEE Photon. Technol. Lett. 24(7), 542–544 (2012).
[CrossRef]

J. Lightwave Technol. (5)

A. D. Kersey, M. A. Davis, H. J. Patrick, M. LeBlanc, K. P. Koo, C. G. Askins, M. A. Putnam, E. J. Friebele, “Fiber grating sensors,” J. Lightwave Technol. 15(8), 1442–1463 (1997).
[CrossRef]

M. Nazarathy, S. A. Newton, R. P. Giffard, D. S. Moberly, F. Sischka, W. R. Trutna, S. Foster, “Real-time long range complementary correlation optical time domain reflectometer,” J. Lightwave Technol. 7(1), 24–38 (1989).
[CrossRef]

J. C. Juarez, E. W. Maier, K. N. Choi, H. F. Taylor, “Distributed fiber-optic intrusion sensor system,” J. Lightwave Technol. 23(6), 2081–2087 (2005).
[CrossRef]

B. Culshaw, A. Kersey, “Fiber-optic sensing: A historical perspective,” J. Lightwave Technol. 26(9), 1064–1078 (2008).
[CrossRef]

Y. L. Lu, T. Zhu, L. A. Chen, X. Y. Bao, “Distributed Vibration Sensor Based on Coherent Detection of Phase-OTDR,” J. Lightwave Technol. 28, 3243–3249 (2010).

Microw. Opt. Technol. Lett. (1)

X. B. Hong, H. X. Guo, J. A. Wu, K. Xu, Y. Zuo, Y. Li, J. Lin, “An Intrusion Detection Sensor Based on Coherent Optical Time Domain Reflector,” Microw. Opt. Technol. Lett. 52(12), 2746–2748 (2010).
[CrossRef]

Opt. Express (5)

Sens. Lett. (2)

M. M. Chen, X. P. Zhang, F. Wang, X. C. Wang, “Development of Distributed Vibration Sensing System Based on Polarization Optical Time Domain Reflectometry,” Sens. Lett. 10, 1536–1540 (2012).

H. Wu, J. Wang, X. Wu, Y. Wu, “Real Intrusion Detection for Distributed Fiber Fence in Practical Strong Fluctuated Noisy Backgrounds,” Sens. Lett. 10(7), 1557–1563 (2012).
[CrossRef]

Sensors (Basel) (1)

X. Y. Bao, L. Chen, “Recent Progress in Distributed Fiber Optic Sensors,” Sensors (Basel) 12(7), 8601–8639 (2012).
[CrossRef] [PubMed]

Other (1)

O. Y. Sagiv, D. Arbel, Y. Katz, Y. Grotas, and A. Eyal, “Dynamical strain sensing via discrete reflectors interrogated by optical frequency domain reflectometry,” Proc. SPIE 8421, OFS2012 22nd International Conference on Optical Fiber Sensors, 84218L (2012).
[CrossRef]

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Figures (9)

Fig. 1
Fig. 1

the experimental setup.

Fig. 2
Fig. 2

impulse response with a free fall of a paper clip.

Fig. 3
Fig. 3

Vibration excitation with a A440Hz tuning fork.

Fig. 4
Fig. 4

Backscatter response with single interrogation (top) and double interrogation (bottom).

Fig. 5
Fig. 5

PSDs of the sensors outputs with single interrogation (blue) and double interrogation (red).

Fig. 6
Fig. 6

The impulse response of the system with single interrogation (a) and double interrogation (b).

Fig. 7
Fig. 7

The impulse response of the system with double interrogation after HPF.

Fig. 8
Fig. 8

The system response to vibration excitation with single interrogation (a) and with double interrogation (b).

Fig. 9
Fig. 9

Number of backscattered photons from a 10m resolution cell.

Equations (7)

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

E( t )= E 0 exp[ j( ω 0 t+πγ t 2 ) ]
E S ( t )=r( z )[ E( t 2z v )+E( t τ d 2z v ) ]
V( t )=ar( z ){ exp{ j[ 2πγ 2z v t+ ϕ 1 ( z ) ] }+exp{ j[ 2πγ( 2z v + τ d )t+ ϕ 2 ( z ) ] } }
V( t )=a 0 L r( z' ){ exp{ j[ 2πγ 2z' v t+ ϕ 1 ( z' ) ] }+exp{ j[ 2πγ( 2z' v + τ d )t+ ϕ 2 ( z' ) ] } }dz'
V ˜ ( ω )=a[ r ˜ 1 ( ω )+ r ˜ 2 ( ω ) ]
r ˜ 1 ( ω )= r( z )exp[ j ϕ 1 ( z ) ] | z= ωv / 4πγ r ˜ 2 ( ω )= r( z )exp[ j ϕ 2 ( z ) ] | z= ωv / 4πγ τ d v /2
| V ˜ ( ω ) | 2 = a 2 { | r ˜ 1 ( ω ) | 2 + | r ˜ 2 ( ω ) | 2 +2| r ˜ 1 ( ω ) || r ˜ 2 ( ω ) |cos[ ϕ 1 ( ωv 4πγ ) ϕ 2 ( ωv 4πγ v τ d 2 ) ] }

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