Abstract

Polarization optical time domain reflectometers (POTDR) can detect vibration of fiber via the change of the state of polarization (SOP) of the Rayleigh backscattered light. For traditional POTDR systems, one key problem is the high misdiagnosis rate when multiple vibrations are simultaneously applied on the sensing fiber due to the random birefringence along the fiber. To solve this problem, we propose in this paper a novel implementation of the POTDR using probe pulses with ergodic SOPs. A series of vibration spectra along the fiber are obtained by sweeping the SOP of the probe pulse. The sum of these vibration spectra, which should be immune to the birefringence of the sensing fiber, is used to analyze the vibration information. Numerical simulation and experiments are carried out to analyze the performance of the proposed system when the input SOPs are traversed with uniform distribution and random distribution. Results show that the misdiagnosis rate of detecting multi-vibration with different frequencies is greatly reduced. In addition, detection of more-than-two vibrations with the same frequency based on POTDR is successfully performed for the first time to the best of our knowledge.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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2018 (1)

J. Hu, X. Wang, and S. Pan, “Application of the probe pulse with ergodic SOPs in detecting multi-vibrations using POTDR,” Proc. SPIE 10618, 1061805 (2018).

2017 (3)

F. Wang, Y. Pan, M. Zhang, C. Cao, and X. Zhang, “Detection of two identical frequency vibrations by phase discrimination in polarization-OTDR,” Opt. Commun. 389, 247–252 (2017).
[Crossref]

X. Wang, C. Wang, M. Tang, S. Fu, and P. Shum, “Multiplexed polarization OTDR system with high DOP and ability of multi-event detection,” Appl. Opt. 56(13), 3709–3713 (2017).
[Crossref] [PubMed]

Q. Sun, F. Ai, D. Liu, J. Cheng, H. Luo, K. Peng, Y. Luo, Z. Yan, and P. P. Shum, “M-OTDR sensing system based on 3D encoded microstructures,” Sci. Rep. 7(1), 41137 (2017).
[Crossref] [PubMed]

2016 (2)

2015 (3)

C. Jin, N. Guo, Y. Feng, L. Wang, H. Liang, J. Li, Z. Li, C. Yu, and C. Lu, “Scanning-free BOTDA based on ultra-fine digital optical frequency comb,” Opt. Express 23(4), 5277–5284 (2015).
[Crossref] [PubMed]

Y. Tong, H. Dong, Y. Wang, W. Sun, X. Wang, J. Bai, H. Yuan, N. Zhu, and J. Liu, “Distributed incomplete polarization-OTDR based on polarization maintaining fiber for multi-event detection,” Opt. Commun. 357, 41–44 (2015).
[Crossref]

Q. Zhao, L. Xia, C. Wan, J. Hu, T. Jia, M. Gu, L. Zhang, L. Kang, J. Chen, X. Zhang, and P. Wu, “Long-haul and high-resolution optical time domain reflectometry using superconducting nanowire single-photon detectors,” Sci. Rep. 5(1), 10441 (2015).
[Crossref] [PubMed]

2014 (2)

F. Peng, H. Wu, X. H. Jia, Y. J. Rao, Z. N. Wang, and Z. P. Peng, “Ultra-long high-sensitivity Φ-OTDR for high spatial resolution intrusion detection of pipelines,” Opt. Express 22(11), 13804–13810 (2014).
[Crossref] [PubMed]

F. Peng, N. Duan, Y. J. Rao, and J. Li, “Real-Time Position and Speed Monitoring of Trains Using Phase-Sensitive OTDR,” IEEE Photonics Technol. Lett. 26(20), 2055–2057 (2014).
[Crossref]

2013 (1)

2011 (1)

X. Wang, X. Zhang, F. Wang, M. Chen, and C. Li, “Application of frequency spectrum analysis in measuring multi-vibrations by using POTDR,” Proc. SPIE 8198, 819808 (2011).
[Crossref]

2010 (1)

2008 (1)

2005 (1)

2002 (1)

1999 (1)

1998 (1)

1981 (1)

1980 (1)

Ai, F.

Q. Sun, F. Ai, D. Liu, J. Cheng, H. Luo, K. Peng, Y. Luo, Z. Yan, and P. P. Shum, “M-OTDR sensing system based on 3D encoded microstructures,” Sci. Rep. 7(1), 41137 (2017).
[Crossref] [PubMed]

Atubga, D.

Bai, J.

Y. Tong, H. Dong, Y. Wang, W. Sun, X. Wang, J. Bai, H. Yuan, N. Zhu, and J. Liu, “Distributed incomplete polarization-OTDR based on polarization maintaining fiber for multi-event detection,” Opt. Commun. 357, 41–44 (2015).
[Crossref]

Bao, X.

Cao, C.

F. Wang, Y. Pan, M. Zhang, C. Cao, and X. Zhang, “Detection of two identical frequency vibrations by phase discrimination in polarization-OTDR,” Opt. Commun. 389, 247–252 (2017).
[Crossref]

Chen, J.

Q. Zhao, L. Xia, C. Wan, J. Hu, T. Jia, M. Gu, L. Zhang, L. Kang, J. Chen, X. Zhang, and P. Wu, “Long-haul and high-resolution optical time domain reflectometry using superconducting nanowire single-photon detectors,” Sci. Rep. 5(1), 10441 (2015).
[Crossref] [PubMed]

Chen, L.

Chen, M.

X. Wang, X. Zhang, F. Wang, M. Chen, and C. Li, “Application of frequency spectrum analysis in measuring multi-vibrations by using POTDR,” Proc. SPIE 8198, 819808 (2011).
[Crossref]

Cheng, J.

Q. Sun, F. Ai, D. Liu, J. Cheng, H. Luo, K. Peng, Y. Luo, Z. Yan, and P. P. Shum, “M-OTDR sensing system based on 3D encoded microstructures,” Sci. Rep. 7(1), 41137 (2017).
[Crossref] [PubMed]

Corsi, F.

Dong, H.

Y. Tong, H. Dong, Y. Wang, W. Sun, X. Wang, J. Bai, H. Yuan, N. Zhu, and J. Liu, “Distributed incomplete polarization-OTDR based on polarization maintaining fiber for multi-event detection,” Opt. Commun. 357, 41–44 (2015).
[Crossref]

Duan, N.

F. Peng, N. Duan, Y. J. Rao, and J. Li, “Real-Time Position and Speed Monitoring of Trains Using Phase-Sensitive OTDR,” IEEE Photonics Technol. Lett. 26(20), 2055–2057 (2014).
[Crossref]

Eickhoff, W.

Feng, Y.

Fu, S.

Fukuda, H.

Y. Mizuno, N. Hayashi, H. Fukuda, K. Y. Song, and K. Nakamura, “Ultrahigh-speed distributed Brillouin reflectometry,” Light Sci. Appl. 5(12), e16184 (2016).
[Crossref] [PubMed]

Galtarossa, A.

Gu, M.

Q. Zhao, L. Xia, C. Wan, J. Hu, T. Jia, M. Gu, L. Zhang, L. Kang, J. Chen, X. Zhang, and P. Wu, “Long-haul and high-resolution optical time domain reflectometry using superconducting nanowire single-photon detectors,” Sci. Rep. 5(1), 10441 (2015).
[Crossref] [PubMed]

Guo, N.

Hayashi, N.

Y. Mizuno, N. Hayashi, H. Fukuda, K. Y. Song, and K. Nakamura, “Ultrahigh-speed distributed Brillouin reflectometry,” Light Sci. Appl. 5(12), e16184 (2016).
[Crossref] [PubMed]

Hu, J.

J. Hu, X. Wang, and S. Pan, “Application of the probe pulse with ergodic SOPs in detecting multi-vibrations using POTDR,” Proc. SPIE 10618, 1061805 (2018).

Q. Zhao, L. Xia, C. Wan, J. Hu, T. Jia, M. Gu, L. Zhang, L. Kang, J. Chen, X. Zhang, and P. Wu, “Long-haul and high-resolution optical time domain reflectometry using superconducting nanowire single-photon detectors,” Sci. Rep. 5(1), 10441 (2015).
[Crossref] [PubMed]

Jia, T.

Q. Zhao, L. Xia, C. Wan, J. Hu, T. Jia, M. Gu, L. Zhang, L. Kang, J. Chen, X. Zhang, and P. Wu, “Long-haul and high-resolution optical time domain reflectometry using superconducting nanowire single-photon detectors,” Sci. Rep. 5(1), 10441 (2015).
[Crossref] [PubMed]

Jia, X. H.

Jin, C.

Juarez, J. C.

Kang, L.

Q. Zhao, L. Xia, C. Wan, J. Hu, T. Jia, M. Gu, L. Zhang, L. Kang, J. Chen, X. Zhang, and P. Wu, “Long-haul and high-resolution optical time domain reflectometry using superconducting nanowire single-photon detectors,” Sci. Rep. 5(1), 10441 (2015).
[Crossref] [PubMed]

Li, C.

X. Wang, X. Zhang, F. Wang, M. Chen, and C. Li, “Application of frequency spectrum analysis in measuring multi-vibrations by using POTDR,” Proc. SPIE 8198, 819808 (2011).
[Crossref]

Li, J.

C. Jin, N. Guo, Y. Feng, L. Wang, H. Liang, J. Li, Z. Li, C. Yu, and C. Lu, “Scanning-free BOTDA based on ultra-fine digital optical frequency comb,” Opt. Express 23(4), 5277–5284 (2015).
[Crossref] [PubMed]

F. Peng, N. Duan, Y. J. Rao, and J. Li, “Real-Time Position and Speed Monitoring of Trains Using Phase-Sensitive OTDR,” IEEE Photonics Technol. Lett. 26(20), 2055–2057 (2014).
[Crossref]

Li, Z.

Liang, H.

Linze, N.

Liu, D.

Q. Sun, F. Ai, D. Liu, J. Cheng, H. Luo, K. Peng, Y. Luo, Z. Yan, and P. P. Shum, “M-OTDR sensing system based on 3D encoded microstructures,” Sci. Rep. 7(1), 41137 (2017).
[Crossref] [PubMed]

Liu, J.

H. Wu, J. Liu, L. Lu, X. Sun, D. Atubga, and Y. Rao, “Multi-Point Disturbance Detection and High-Precision Positioning of Polarization-Sensitive Optical Time-Domain Reflectometry,” J. Lightwave Technol. 34(23), 5371–5377 (2016).
[Crossref]

Y. Tong, H. Dong, Y. Wang, W. Sun, X. Wang, J. Bai, H. Yuan, N. Zhu, and J. Liu, “Distributed incomplete polarization-OTDR based on polarization maintaining fiber for multi-event detection,” Opt. Commun. 357, 41–44 (2015).
[Crossref]

Lu, C.

Lu, L.

Lu, Y.

Luo, H.

Q. Sun, F. Ai, D. Liu, J. Cheng, H. Luo, K. Peng, Y. Luo, Z. Yan, and P. P. Shum, “M-OTDR sensing system based on 3D encoded microstructures,” Sci. Rep. 7(1), 41137 (2017).
[Crossref] [PubMed]

Luo, Y.

Q. Sun, F. Ai, D. Liu, J. Cheng, H. Luo, K. Peng, Y. Luo, Z. Yan, and P. P. Shum, “M-OTDR sensing system based on 3D encoded microstructures,” Sci. Rep. 7(1), 41137 (2017).
[Crossref] [PubMed]

Mégret, P.

Mizuno, Y.

Y. Mizuno, N. Hayashi, H. Fukuda, K. Y. Song, and K. Nakamura, “Ultrahigh-speed distributed Brillouin reflectometry,” Light Sci. Appl. 5(12), e16184 (2016).
[Crossref] [PubMed]

Nakamura, K.

Y. Mizuno, N. Hayashi, H. Fukuda, K. Y. Song, and K. Nakamura, “Ultrahigh-speed distributed Brillouin reflectometry,” Light Sci. Appl. 5(12), e16184 (2016).
[Crossref] [PubMed]

Palmieri, L.

Pan, S.

J. Hu, X. Wang, and S. Pan, “Application of the probe pulse with ergodic SOPs in detecting multi-vibrations using POTDR,” Proc. SPIE 10618, 1061805 (2018).

Pan, Y.

F. Wang, Y. Pan, M. Zhang, C. Cao, and X. Zhang, “Detection of two identical frequency vibrations by phase discrimination in polarization-OTDR,” Opt. Commun. 389, 247–252 (2017).
[Crossref]

Peng, F.

F. Peng, H. Wu, X. H. Jia, Y. J. Rao, Z. N. Wang, and Z. P. Peng, “Ultra-long high-sensitivity Φ-OTDR for high spatial resolution intrusion detection of pipelines,” Opt. Express 22(11), 13804–13810 (2014).
[Crossref] [PubMed]

F. Peng, N. Duan, Y. J. Rao, and J. Li, “Real-Time Position and Speed Monitoring of Trains Using Phase-Sensitive OTDR,” IEEE Photonics Technol. Lett. 26(20), 2055–2057 (2014).
[Crossref]

Peng, K.

Q. Sun, F. Ai, D. Liu, J. Cheng, H. Luo, K. Peng, Y. Luo, Z. Yan, and P. P. Shum, “M-OTDR sensing system based on 3D encoded microstructures,” Sci. Rep. 7(1), 41137 (2017).
[Crossref] [PubMed]

Peng, Z. P.

Rao, Y.

Rao, Y. J.

F. Peng, N. Duan, Y. J. Rao, and J. Li, “Real-Time Position and Speed Monitoring of Trains Using Phase-Sensitive OTDR,” IEEE Photonics Technol. Lett. 26(20), 2055–2057 (2014).
[Crossref]

F. Peng, H. Wu, X. H. Jia, Y. J. Rao, Z. N. Wang, and Z. P. Peng, “Ultra-long high-sensitivity Φ-OTDR for high spatial resolution intrusion detection of pipelines,” Opt. Express 22(11), 13804–13810 (2014).
[Crossref] [PubMed]

Rashleigh, S. C.

Rogers, A. J.

Shum, P.

Shum, P. P.

Q. Sun, F. Ai, D. Liu, J. Cheng, H. Luo, K. Peng, Y. Luo, Z. Yan, and P. P. Shum, “M-OTDR sensing system based on 3D encoded microstructures,” Sci. Rep. 7(1), 41137 (2017).
[Crossref] [PubMed]

Song, K. Y.

Y. Mizuno, N. Hayashi, H. Fukuda, K. Y. Song, and K. Nakamura, “Ultrahigh-speed distributed Brillouin reflectometry,” Light Sci. Appl. 5(12), e16184 (2016).
[Crossref] [PubMed]

Sun, Q.

Q. Sun, F. Ai, D. Liu, J. Cheng, H. Luo, K. Peng, Y. Luo, Z. Yan, and P. P. Shum, “M-OTDR sensing system based on 3D encoded microstructures,” Sci. Rep. 7(1), 41137 (2017).
[Crossref] [PubMed]

Sun, W.

Y. Tong, H. Dong, Y. Wang, W. Sun, X. Wang, J. Bai, H. Yuan, N. Zhu, and J. Liu, “Distributed incomplete polarization-OTDR based on polarization maintaining fiber for multi-event detection,” Opt. Commun. 357, 41–44 (2015).
[Crossref]

Sun, X.

Tang, M.

Taylor, H. F.

Tihon, P.

Tong, Y.

Y. Tong, H. Dong, Y. Wang, W. Sun, X. Wang, J. Bai, H. Yuan, N. Zhu, and J. Liu, “Distributed incomplete polarization-OTDR based on polarization maintaining fiber for multi-event detection,” Opt. Commun. 357, 41–44 (2015).
[Crossref]

Ulrich, R.

Verlinden, O.

Wan, C.

Q. Zhao, L. Xia, C. Wan, J. Hu, T. Jia, M. Gu, L. Zhang, L. Kang, J. Chen, X. Zhang, and P. Wu, “Long-haul and high-resolution optical time domain reflectometry using superconducting nanowire single-photon detectors,” Sci. Rep. 5(1), 10441 (2015).
[Crossref] [PubMed]

Wang, C.

Wang, F.

F. Wang, Y. Pan, M. Zhang, C. Cao, and X. Zhang, “Detection of two identical frequency vibrations by phase discrimination in polarization-OTDR,” Opt. Commun. 389, 247–252 (2017).
[Crossref]

X. Wang, X. Zhang, F. Wang, M. Chen, and C. Li, “Application of frequency spectrum analysis in measuring multi-vibrations by using POTDR,” Proc. SPIE 8198, 819808 (2011).
[Crossref]

Wang, L.

Wang, X.

J. Hu, X. Wang, and S. Pan, “Application of the probe pulse with ergodic SOPs in detecting multi-vibrations using POTDR,” Proc. SPIE 10618, 1061805 (2018).

X. Wang, C. Wang, M. Tang, S. Fu, and P. Shum, “Multiplexed polarization OTDR system with high DOP and ability of multi-event detection,” Appl. Opt. 56(13), 3709–3713 (2017).
[Crossref] [PubMed]

Y. Tong, H. Dong, Y. Wang, W. Sun, X. Wang, J. Bai, H. Yuan, N. Zhu, and J. Liu, “Distributed incomplete polarization-OTDR based on polarization maintaining fiber for multi-event detection,” Opt. Commun. 357, 41–44 (2015).
[Crossref]

X. Wang, X. Zhang, F. Wang, M. Chen, and C. Li, “Application of frequency spectrum analysis in measuring multi-vibrations by using POTDR,” Proc. SPIE 8198, 819808 (2011).
[Crossref]

Wang, Y.

Y. Tong, H. Dong, Y. Wang, W. Sun, X. Wang, J. Bai, H. Yuan, N. Zhu, and J. Liu, “Distributed incomplete polarization-OTDR based on polarization maintaining fiber for multi-event detection,” Opt. Commun. 357, 41–44 (2015).
[Crossref]

Wang, Z. N.

Wu, H.

Wu, P.

Q. Zhao, L. Xia, C. Wan, J. Hu, T. Jia, M. Gu, L. Zhang, L. Kang, J. Chen, X. Zhang, and P. Wu, “Long-haul and high-resolution optical time domain reflectometry using superconducting nanowire single-photon detectors,” Sci. Rep. 5(1), 10441 (2015).
[Crossref] [PubMed]

Wuilpart, M.

Xia, L.

Q. Zhao, L. Xia, C. Wan, J. Hu, T. Jia, M. Gu, L. Zhang, L. Kang, J. Chen, X. Zhang, and P. Wu, “Long-haul and high-resolution optical time domain reflectometry using superconducting nanowire single-photon detectors,” Sci. Rep. 5(1), 10441 (2015).
[Crossref] [PubMed]

Yan, Z.

Q. Sun, F. Ai, D. Liu, J. Cheng, H. Luo, K. Peng, Y. Luo, Z. Yan, and P. P. Shum, “M-OTDR sensing system based on 3D encoded microstructures,” Sci. Rep. 7(1), 41137 (2017).
[Crossref] [PubMed]

Yu, C.

Yuan, H.

Y. Tong, H. Dong, Y. Wang, W. Sun, X. Wang, J. Bai, H. Yuan, N. Zhu, and J. Liu, “Distributed incomplete polarization-OTDR based on polarization maintaining fiber for multi-event detection,” Opt. Commun. 357, 41–44 (2015).
[Crossref]

Zhang, L.

Q. Zhao, L. Xia, C. Wan, J. Hu, T. Jia, M. Gu, L. Zhang, L. Kang, J. Chen, X. Zhang, and P. Wu, “Long-haul and high-resolution optical time domain reflectometry using superconducting nanowire single-photon detectors,” Sci. Rep. 5(1), 10441 (2015).
[Crossref] [PubMed]

Zhang, M.

F. Wang, Y. Pan, M. Zhang, C. Cao, and X. Zhang, “Detection of two identical frequency vibrations by phase discrimination in polarization-OTDR,” Opt. Commun. 389, 247–252 (2017).
[Crossref]

Zhang, X.

F. Wang, Y. Pan, M. Zhang, C. Cao, and X. Zhang, “Detection of two identical frequency vibrations by phase discrimination in polarization-OTDR,” Opt. Commun. 389, 247–252 (2017).
[Crossref]

Q. Zhao, L. Xia, C. Wan, J. Hu, T. Jia, M. Gu, L. Zhang, L. Kang, J. Chen, X. Zhang, and P. Wu, “Long-haul and high-resolution optical time domain reflectometry using superconducting nanowire single-photon detectors,” Sci. Rep. 5(1), 10441 (2015).
[Crossref] [PubMed]

X. Wang, X. Zhang, F. Wang, M. Chen, and C. Li, “Application of frequency spectrum analysis in measuring multi-vibrations by using POTDR,” Proc. SPIE 8198, 819808 (2011).
[Crossref]

Zhang, Z.

Zhao, Q.

Q. Zhao, L. Xia, C. Wan, J. Hu, T. Jia, M. Gu, L. Zhang, L. Kang, J. Chen, X. Zhang, and P. Wu, “Long-haul and high-resolution optical time domain reflectometry using superconducting nanowire single-photon detectors,” Sci. Rep. 5(1), 10441 (2015).
[Crossref] [PubMed]

Zhu, N.

Y. Tong, H. Dong, Y. Wang, W. Sun, X. Wang, J. Bai, H. Yuan, N. Zhu, and J. Liu, “Distributed incomplete polarization-OTDR based on polarization maintaining fiber for multi-event detection,” Opt. Commun. 357, 41–44 (2015).
[Crossref]

Zhu, T.

Appl. Opt. (2)

IEEE Photonics Technol. Lett. (1)

F. Peng, N. Duan, Y. J. Rao, and J. Li, “Real-Time Position and Speed Monitoring of Trains Using Phase-Sensitive OTDR,” IEEE Photonics Technol. Lett. 26(20), 2055–2057 (2014).
[Crossref]

J. Lightwave Technol. (5)

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(12), e16184 (2016).
[Crossref] [PubMed]

Opt. Commun. (2)

Y. Tong, H. Dong, Y. Wang, W. Sun, X. Wang, J. Bai, H. Yuan, N. Zhu, and J. Liu, “Distributed incomplete polarization-OTDR based on polarization maintaining fiber for multi-event detection,” Opt. Commun. 357, 41–44 (2015).
[Crossref]

F. Wang, Y. Pan, M. Zhang, C. Cao, and X. Zhang, “Detection of two identical frequency vibrations by phase discrimination in polarization-OTDR,” Opt. Commun. 389, 247–252 (2017).
[Crossref]

Opt. Express (4)

Opt. Lett. (2)

Proc. SPIE (2)

J. Hu, X. Wang, and S. Pan, “Application of the probe pulse with ergodic SOPs in detecting multi-vibrations using POTDR,” Proc. SPIE 10618, 1061805 (2018).

X. Wang, X. Zhang, F. Wang, M. Chen, and C. Li, “Application of frequency spectrum analysis in measuring multi-vibrations by using POTDR,” Proc. SPIE 8198, 819808 (2011).
[Crossref]

Sci. Rep. (2)

Q. Sun, F. Ai, D. Liu, J. Cheng, H. Luo, K. Peng, Y. Luo, Z. Yan, and P. P. Shum, “M-OTDR sensing system based on 3D encoded microstructures,” Sci. Rep. 7(1), 41137 (2017).
[Crossref] [PubMed]

Q. Zhao, L. Xia, C. Wan, J. Hu, T. Jia, M. Gu, L. Zhang, L. Kang, J. Chen, X. Zhang, and P. Wu, “Long-haul and high-resolution optical time domain reflectometry using superconducting nanowire single-photon detectors,” Sci. Rep. 5(1), 10441 (2015).
[Crossref] [PubMed]

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Figures (18)

Fig. 1
Fig. 1 The schematic diagram of the proposed POTDR system using probe pulses with ergodic SOPs. LD: laser diode; MZM: Mach-Zehner modulator; PC: polarization controller; PBS: polarization beam splitter; PD: photo-detector; AC: acquisition card.
Fig. 2
Fig. 2 The frequency spectra along the fiber generated from one SOP.
Fig. 3
Fig. 3 The frequency spectra along the fiber generated from one SOP within one beat length: (a) at 6000 m; (b) at 6010 m; (c) at 6020 m; (d) at 6030 m respectively.
Fig. 4
Fig. 4 The frequency spectra along the fiber generated from ergodic SOPs.
Fig. 5
Fig. 5 The frequency spectra along the fiber generated from ergodic SOPs within one beat length: (a) at 6000 m; (b) at 6010 m; (c) at 6020 m; (d) at 6030 m respectively.
Fig. 6
Fig. 6 The amplitude of 11 Hz component along the fiber with single SOP or with ergodic SOPs.
Fig. 7
Fig. 7 The standard deviation of the amplitude of 11 Hz component with different value of n.
Fig. 8
Fig. 8 The frequency spectra along the fiber generated from one SOP.
Fig. 9
Fig. 9 The frequency spectra along the fiber generated from ergodic SOPs with uniform distribution.
Fig. 10
Fig. 10 The frequency spectra along the fiber generated from ergodic SOPs with random distribution.
Fig. 11
Fig. 11 The frequency spectra generated from one fixed SOP when there are five vibrations along the sensing fiber.
Fig. 12
Fig. 12 The frequency spectra generated from ergodic SOPs with random distribution when there are five vibrations along the sensing fiber.
Fig. 13
Fig. 13 The spectra along the fiber with one fixed SOP when three vibrations with different frequencies are induced.
Fig. 14
Fig. 14 The spectra along the fiber with ergodic SOPs when three vibrations with different frequencies are induced.
Fig. 15
Fig. 15 The frequency spectra along the fiber generated from single fixed SOP and ergodic SOPs: (a) at 1600 m; (b) at 1610 m; (c) at 1620 m; (d) at 1630 m, respectively.
Fig. 16
Fig. 16 The amplitude of 12 Hz component along the fiber with single SOP or with ergodic SOPs.
Fig. 17
Fig. 17 The spectra obtained with one single SOP along the sensing fiber when the fiber is disturbed by three vibrations with the same frequency of 12 Hz.
Fig. 18
Fig. 18 The spectra obtained with ergodic SOPs along the sensing fiber when the fiber is disturbed by three vibrations with the same frequency of 12 Hz.

Equations (12)

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S ^ out = R S ^ in = R z ( τ ) R y ( σ ) R x ( γ ) R y ( σ ) R z ( τ ) S ^ in
R B = M R T M R
S ^ out ( N L Z ) = H R 1 ... R i ( ω i ) ... R j ( ω j ) ... R q ( ω q ) ... R N R N ... R q ( ω q ) ... R j ( ω j ) ... R i ( ω i ) ... R 1 S ^ in
Δ ϕ i = k i cos ( ω i t + θ i )
P i = 1 2 E x 2 + 1 2 E y 2 + R e ( E x E y e j ( ϕ + Δ ϕ ) )
E x = E cos ( δ ) E y = E sin ( δ )
Δ ϕ = 2 j = 1 i Δ ϕ j
P i = 1 2 E x 2 + 1 2 E y 2 + E x E y cos ( ϕ + Δ ϕ ) = 1 2 E x 2 + 1 2 E y 2 + E x E y [ cos ( ϕ ) cos ( Δ ϕ ) sin ( ϕ ) sin ( Δ ϕ ) ] = 1 2 E x 2 + 1 2 E y 2 + E x E y [ cos ( ϕ ) cos ( 2 j = 1 i k j cos ( ω j t + θ j ) ) sin ( ϕ ) sin ( 2 j = 1 i k j cos ( ω j t + θ j ) ) ]
P i | E x E y cos ( ϕ ) j = 1 i J 2 ( 2 k j ) cos ( 2 ω j t + 2 θ j ) E x E y sin ( ϕ ) j = 1 i J 1 ( 2 k j ) cos ( ω j t + θ j ) |
P i | E 2 cos ( δ ) sin ( δ ) sin ( ϕ ) j = 1 i J 1 ( 2 k j ) | cos ( ω j t + θ j )
A sum ( j ) = p = 1 l q = 1 m | E 2 cos ( δ p ) sin ( δ p ) sin ( ϕ q ) J 1 ( 2 k j ) |
A sum ( j ) = 0 π sin ( ϕ q ) d ϕ q * p = 1 l | E 2 cos ( δ p ) sin ( δ p ) J 1 ( 2 k j ) | =2 E 2 { 0 π / 2 cos ( δ p ) sin ( δ p ) J 1 ( 2 k j ) d δ p + π / 2 π cos ( δ p ) sin ( δ p ) J 1 ( 2 k j ) d δ p } = 2 E 2 J 1 ( 2 k j )

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