Abstract

The traditional methods of extracting sensing information of a Brillouin distributed sensor is by curve fitting the Brillouin gain profiles along the fiber, we propose two concise and time-saving methods to process signals from only the frequency shifted section(s) of the fiber by subtracting the original spectrum from the sensing Brillouin spectrum. Experimental results validate that our methods can provide up to over 9 times faster information acquisition for a 10 km sensing fiber with 800 m frequency shifted section comparing with the traditional way. The proposed methods could be potentially attractive in getting information for the Brillouin distributed sensors as well as the Raman and Rayleigh distributed sensors especially when the processing speed is concerned.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2017 (4)

2016 (6)

A. Lopez-Gil, M. A. Soto, X. Angulo-Vinuesa, A. Dominguez-Lopez, S. Martin-Lopez, L. Thévenaz, and M. Gonzalez-Herraez, “Evaluation of the accuracy of BOTDA systems based on the phase spectral response,” Opt. Express 24(15), 17200–17214 (2016).
[Crossref] [PubMed]

Y. London, Y. Antman, E. Preter, N. Levanon, and A. Zadok, “Brillouin Optical Correlation Domain Analysis Addressing 440 000 Resolution Points,” J. Lightwave Technol. 34(19), 4421–4429 (2016).
[Crossref]

M. A. Soto, J. A. Ramírez, and L. Thévenaz, “Intensifying the response of distributed optical fibre sensors using 2D and 3D image restoration,” Nat. Commun. 7(1), 10870 (2016).
[Crossref] [PubMed]

G. Soto, J. Fontbona, R. Cortez, and L. Mujica, “An online two-stage adaptive algorithm for strain profile estimation from noisy and abruptly changing BOTDR data and application to underground mines,” Measurement 92, 340–351 (2016).
[Crossref]

A. Motil, A. Bergman, and M. Tur, “[INVITED] State of the art of Brillouin fiber-optic distributed sensing,” Opt. Laser Technol. 78, 81–103 (2016).
[Crossref]

A. Denisov, M. A. Soto, and L. Thévenaz, “Going beyond 1000000 resolved points in a Brillouin distributed fiber sensor: theoretical analysis and experimental demonstration,” Light Sci. Appl. 5(5), e16074 (2016).
[Crossref] [PubMed]

2015 (1)

2014 (1)

2013 (5)

2012 (6)

2011 (3)

2009 (2)

2008 (1)

1997 (1)

M. Nikles, L. Thevenaz, and P. A. Robert, “Brillouin gain spectrum characterization in single-mode optical fibers,” J. Lightwave Technol. 15(10), 1842–1851 (1997).
[Crossref]

Alarashi, W. H.

Amiri Farahani, M.

Angulo-Vinuesa, X.

Antman, Y.

Azad, A. K.

Bao, X.

Belal, M.

Bergman, A.

A. Motil, A. Bergman, and M. Tur, “[INVITED] State of the art of Brillouin fiber-optic distributed sensing,” Opt. Laser Technol. 78, 81–103 (2016).
[Crossref]

Bernini, R.

Castillo-Guerra, E.

Chen, L.

Colpitts, B. G.

Cortez, R.

G. Soto, J. Fontbona, R. Cortez, and L. Mujica, “An online two-stage adaptive algorithm for strain profile estimation from noisy and abruptly changing BOTDR data and application to underground mines,” Measurement 92, 340–351 (2016).
[Crossref]

Dang, Y.

Denisov, A.

A. Denisov, M. A. Soto, and L. Thévenaz, “Going beyond 1000000 resolved points in a Brillouin distributed fiber sensor: theoretical analysis and experimental demonstration,” Light Sci. Appl. 5(5), e16074 (2016).
[Crossref] [PubMed]

Diao, D.

Dominguez-Lopez, A.

Dong, Y.

Farahani, M. A.

M. A. Farahani, E. Castillo-Guerra, and B. G. Colpitts, “A detailed evaluation of the correlation-based method used for estimation of the Brillouin frequency shift in BOTDA sensors,” IEEE Sens. J. 13(12), 4589–4598 (2013).
[Crossref]

M. A. Farahani, E. Castillo-Guerra, and B. G. Colpitts, “Accurate estimation of Brillouin frequency shift in Brillouin optical time domain analysis sensors using cross correlation,” Opt. Lett. 36(21), 4275–4277 (2011).
[Crossref] [PubMed]

Fontbona, J.

G. Soto, J. Fontbona, R. Cortez, and L. Mujica, “An online two-stage adaptive algorithm for strain profile estimation from noisy and abruptly changing BOTDR data and application to underground mines,” Measurement 92, 340–351 (2016).
[Crossref]

Fu, S.

Gan, L.

Gonzalez-Herraez, M.

Guo, N.

A. K. Azad, F. N. Khan, W. H. Alarashi, N. Guo, A. P. T. Lau, and C. Lu, “Temperature extraction in Brillouin optical time-domain analysis sensors using principal component analysis based pattern recognition,” Opt. Express 25(14), 16534–16549 (2017).
[Crossref] [PubMed]

L. Wang, N. Guo, C. Jin, K. Zhong, X. Zhou, J. Yuan, Z. Kang, B. Zhou, C. Yu, H.-Y. Tam, and C. Lu, “Coherent BOTDA Using Phase- and Polarization-Diversity Heterodyne Detection and Embedded Digital Signal Processing,” IEEE Sens. J. 17(12), 3728–3734 (2017).
[Crossref]

Y. Mao, N. Guo, K. L. Yu, H. Y. Tam, and C. Lu, “1-cm-spatial-resolution Brillouin optical time-domain analysis based on bright pulse Brillouin gain and complementary code,” IEEE Photonics J. 4(6), 2243–2248 (2012).
[Crossref]

L. Wang, N. Guo, C. Jin, C. Yu, H. Y. Tam, and C. Lu, “BOTDA system using artificial neural network,” in Proceedings of 2017 Opto-Electronics and Communications Conference (OECC) and Photonics Global Conference (PGC), (IEEE, 2017).
[Crossref]

He, Q.

He, Z.

Hotate, K.

Jia, X.

Jin, C.

L. Wang, N. Guo, C. Jin, K. Zhong, X. Zhou, J. Yuan, Z. Kang, B. Zhou, C. Yu, H.-Y. Tam, and C. Lu, “Coherent BOTDA Using Phase- and Polarization-Diversity Heterodyne Detection and Embedded Digital Signal Processing,” IEEE Sens. J. 17(12), 3728–3734 (2017).
[Crossref]

L. Wang, N. Guo, C. Jin, C. Yu, H. Y. Tam, and C. Lu, “BOTDA system using artificial neural network,” in Proceedings of 2017 Opto-Electronics and Communications Conference (OECC) and Photonics Global Conference (PGC), (IEEE, 2017).
[Crossref]

Kang, Z.

L. Wang, N. Guo, C. Jin, K. Zhong, X. Zhou, J. Yuan, Z. Kang, B. Zhou, C. Yu, H.-Y. Tam, and C. Lu, “Coherent BOTDA Using Phase- and Polarization-Diversity Heterodyne Detection and Embedded Digital Signal Processing,” IEEE Sens. J. 17(12), 3728–3734 (2017).
[Crossref]

Khan, F. N.

Kim, Y. H.

Lau, A. P. T.

Lee, K.

Levanon, N.

Li, W.

Li, Y.

Liu, D.

Liu, T.

London, Y.

Lopez-Gil, A.

Lu, C.

A. K. Azad, F. N. Khan, W. H. Alarashi, N. Guo, A. P. T. Lau, and C. Lu, “Temperature extraction in Brillouin optical time-domain analysis sensors using principal component analysis based pattern recognition,” Opt. Express 25(14), 16534–16549 (2017).
[Crossref] [PubMed]

L. Wang, N. Guo, C. Jin, K. Zhong, X. Zhou, J. Yuan, Z. Kang, B. Zhou, C. Yu, H.-Y. Tam, and C. Lu, “Coherent BOTDA Using Phase- and Polarization-Diversity Heterodyne Detection and Embedded Digital Signal Processing,” IEEE Sens. J. 17(12), 3728–3734 (2017).
[Crossref]

Y. Mao, N. Guo, K. L. Yu, H. Y. Tam, and C. Lu, “1-cm-spatial-resolution Brillouin optical time-domain analysis based on bright pulse Brillouin gain and complementary code,” IEEE Photonics J. 4(6), 2243–2248 (2012).
[Crossref]

L. Wang, N. Guo, C. Jin, C. Yu, H. Y. Tam, and C. Lu, “BOTDA system using artificial neural network,” in Proceedings of 2017 Opto-Electronics and Communications Conference (OECC) and Photonics Global Conference (PGC), (IEEE, 2017).
[Crossref]

Lu, Y.

Mao, Y.

Y. Mao, N. Guo, K. L. Yu, H. Y. Tam, and C. Lu, “1-cm-spatial-resolution Brillouin optical time-domain analysis based on bright pulse Brillouin gain and complementary code,” IEEE Photonics J. 4(6), 2243–2248 (2012).
[Crossref]

Martin-Lopez, S.

Masoudi, A.

Minardo, A.

Motil, A.

A. Motil, A. Bergman, and M. Tur, “[INVITED] State of the art of Brillouin fiber-optic distributed sensing,” Opt. Laser Technol. 78, 81–103 (2016).
[Crossref]

Y. Peled, A. Motil, L. Yaron, and M. Tur, “Slope-assisted fast distributed sensing in optical fibers with arbitrary Brillouin profile,” Opt. Express 19(21), 19845–19854 (2011).
[Crossref] [PubMed]

Muanenda, Y.

Mujica, L.

G. Soto, J. Fontbona, R. Cortez, and L. Mujica, “An online two-stage adaptive algorithm for strain profile estimation from noisy and abruptly changing BOTDR data and application to underground mines,” Measurement 92, 340–351 (2016).
[Crossref]

Newson, T. P.

Nikles, M.

M. Nikles, L. Thevenaz, and P. A. Robert, “Brillouin gain spectrum characterization in single-mode optical fibers,” J. Lightwave Technol. 15(10), 1842–1851 (1997).
[Crossref]

Pasquale, F. D.

Peled, Y.

Preter, E.

Qian, X.

Qin, Z.

Ramírez, J. A.

M. A. Soto, J. A. Ramírez, and L. Thévenaz, “Intensifying the response of distributed optical fibre sensors using 2D and 3D image restoration,” Nat. Commun. 7(1), 10870 (2016).
[Crossref] [PubMed]

Robert, P. A.

M. Nikles, L. Thevenaz, and P. A. Robert, “Brillouin gain spectrum characterization in single-mode optical fibers,” J. Lightwave Technol. 15(10), 1842–1851 (1997).
[Crossref]

Santagiustina, M.

Shum, P. P.

Song, K. Y.

Soto, G.

G. Soto, J. Fontbona, R. Cortez, and L. Mujica, “An online two-stage adaptive algorithm for strain profile estimation from noisy and abruptly changing BOTDR data and application to underground mines,” Measurement 92, 340–351 (2016).
[Crossref]

Soto, M. A.

M. A. Soto, J. A. Ramírez, and L. Thévenaz, “Intensifying the response of distributed optical fibre sensors using 2D and 3D image restoration,” Nat. Commun. 7(1), 10870 (2016).
[Crossref] [PubMed]

A. Denisov, M. A. Soto, and L. Thévenaz, “Going beyond 1000000 resolved points in a Brillouin distributed fiber sensor: theoretical analysis and experimental demonstration,” Light Sci. Appl. 5(5), e16074 (2016).
[Crossref] [PubMed]

A. Lopez-Gil, M. A. Soto, X. Angulo-Vinuesa, A. Dominguez-Lopez, S. Martin-Lopez, L. Thévenaz, and M. Gonzalez-Herraez, “Evaluation of the accuracy of BOTDA systems based on the phase spectral response,” Opt. Express 24(15), 17200–17214 (2016).
[Crossref] [PubMed]

M. A. Soto and L. Thévenaz, “Modeling and evaluating the performance of Brillouin distributed optical fiber sensors,” Opt. Express 21(25), 31347–31366 (2013).
[Crossref] [PubMed]

Sun, W.

Taki, M.

Tam, H. Y.

Y. Mao, N. Guo, K. L. Yu, H. Y. Tam, and C. Lu, “1-cm-spatial-resolution Brillouin optical time-domain analysis based on bright pulse Brillouin gain and complementary code,” IEEE Photonics J. 4(6), 2243–2248 (2012).
[Crossref]

L. Wang, N. Guo, C. Jin, C. Yu, H. Y. Tam, and C. Lu, “BOTDA system using artificial neural network,” in Proceedings of 2017 Opto-Electronics and Communications Conference (OECC) and Photonics Global Conference (PGC), (IEEE, 2017).
[Crossref]

Tam, H.-Y.

L. Wang, N. Guo, C. Jin, K. Zhong, X. Zhou, J. Yuan, Z. Kang, B. Zhou, C. Yu, H.-Y. Tam, and C. Lu, “Coherent BOTDA Using Phase- and Polarization-Diversity Heterodyne Detection and Embedded Digital Signal Processing,” IEEE Sens. J. 17(12), 3728–3734 (2017).
[Crossref]

Tang, M.

Thevenaz, L.

M. Nikles, L. Thevenaz, and P. A. Robert, “Brillouin gain spectrum characterization in single-mode optical fibers,” J. Lightwave Technol. 15(10), 1842–1851 (1997).
[Crossref]

Thévenaz, L.

A. Denisov, M. A. Soto, and L. Thévenaz, “Going beyond 1000000 resolved points in a Brillouin distributed fiber sensor: theoretical analysis and experimental demonstration,” Light Sci. Appl. 5(5), e16074 (2016).
[Crossref] [PubMed]

M. A. Soto, J. A. Ramírez, and L. Thévenaz, “Intensifying the response of distributed optical fibre sensors using 2D and 3D image restoration,” Nat. Commun. 7(1), 10870 (2016).
[Crossref] [PubMed]

A. Lopez-Gil, M. A. Soto, X. Angulo-Vinuesa, A. Dominguez-Lopez, S. Martin-Lopez, L. Thévenaz, and M. Gonzalez-Herraez, “Evaluation of the accuracy of BOTDA systems based on the phase spectral response,” Opt. Express 24(15), 17200–17214 (2016).
[Crossref] [PubMed]

M. A. Soto and L. Thévenaz, “Modeling and evaluating the performance of Brillouin distributed optical fiber sensors,” Opt. Express 21(25), 31347–31366 (2013).
[Crossref] [PubMed]

Tong, W.

Tur, M.

Ursini, L.

Voskoboinik, A.

Wang, F.

Wang, L.

L. Wang, N. Guo, C. Jin, K. Zhong, X. Zhou, J. Yuan, Z. Kang, B. Zhou, C. Yu, H.-Y. Tam, and C. Lu, “Coherent BOTDA Using Phase- and Polarization-Diversity Heterodyne Detection and Embedded Digital Signal Processing,” IEEE Sens. J. 17(12), 3728–3734 (2017).
[Crossref]

L. Wang, N. Guo, C. Jin, C. Yu, H. Y. Tam, and C. Lu, “BOTDA system using artificial neural network,” in Proceedings of 2017 Opto-Electronics and Communications Conference (OECC) and Photonics Global Conference (PGC), (IEEE, 2017).
[Crossref]

Wang, Z.

Willner, A. W.

Wu, H.

Wylie, M. T. V.

Xiao, X.

Xue, N.

Yaron, L.

Yilmaz, O. F.

Yu, C.

L. Wang, N. Guo, C. Jin, K. Zhong, X. Zhou, J. Yuan, Z. Kang, B. Zhou, C. Yu, H.-Y. Tam, and C. Lu, “Coherent BOTDA Using Phase- and Polarization-Diversity Heterodyne Detection and Embedded Digital Signal Processing,” IEEE Sens. J. 17(12), 3728–3734 (2017).
[Crossref]

L. Wang, N. Guo, C. Jin, C. Yu, H. Y. Tam, and C. Lu, “BOTDA system using artificial neural network,” in Proceedings of 2017 Opto-Electronics and Communications Conference (OECC) and Photonics Global Conference (PGC), (IEEE, 2017).
[Crossref]

Yu, K. L.

Y. Mao, N. Guo, K. L. Yu, H. Y. Tam, and C. Lu, “1-cm-spatial-resolution Brillouin optical time-domain analysis based on bright pulse Brillouin gain and complementary code,” IEEE Photonics J. 4(6), 2243–2248 (2012).
[Crossref]

Yuan, J.

L. Wang, N. Guo, C. Jin, K. Zhong, X. Zhou, J. Yuan, Z. Kang, B. Zhou, C. Yu, H.-Y. Tam, and C. Lu, “Coherent BOTDA Using Phase- and Polarization-Diversity Heterodyne Detection and Embedded Digital Signal Processing,” IEEE Sens. J. 17(12), 3728–3734 (2017).
[Crossref]

Zadok, A.

Zeni, L.

Zhan, W.

Zhang, B.

Zhang, H.

Zhang, X.

Zhao, C.

Zhao, Z.

Zhong, K.

L. Wang, N. Guo, C. Jin, K. Zhong, X. Zhou, J. Yuan, Z. Kang, B. Zhou, C. Yu, H.-Y. Tam, and C. Lu, “Coherent BOTDA Using Phase- and Polarization-Diversity Heterodyne Detection and Embedded Digital Signal Processing,” IEEE Sens. J. 17(12), 3728–3734 (2017).
[Crossref]

Zhou, B.

L. Wang, N. Guo, C. Jin, K. Zhong, X. Zhou, J. Yuan, Z. Kang, B. Zhou, C. Yu, H.-Y. Tam, and C. Lu, “Coherent BOTDA Using Phase- and Polarization-Diversity Heterodyne Detection and Embedded Digital Signal Processing,” IEEE Sens. J. 17(12), 3728–3734 (2017).
[Crossref]

Zhou, X.

L. Wang, N. Guo, C. Jin, K. Zhong, X. Zhou, J. Yuan, Z. Kang, B. Zhou, C. Yu, H.-Y. Tam, and C. Lu, “Coherent BOTDA Using Phase- and Polarization-Diversity Heterodyne Detection and Embedded Digital Signal Processing,” IEEE Sens. J. 17(12), 3728–3734 (2017).
[Crossref]

Zhu, T.

Zou, W.

Appl. Opt. (2)

IEEE Photonics J. (1)

Y. Mao, N. Guo, K. L. Yu, H. Y. Tam, and C. Lu, “1-cm-spatial-resolution Brillouin optical time-domain analysis based on bright pulse Brillouin gain and complementary code,” IEEE Photonics J. 4(6), 2243–2248 (2012).
[Crossref]

IEEE Sens. J. (2)

M. A. Farahani, E. Castillo-Guerra, and B. G. Colpitts, “A detailed evaluation of the correlation-based method used for estimation of the Brillouin frequency shift in BOTDA sensors,” IEEE Sens. J. 13(12), 4589–4598 (2013).
[Crossref]

L. Wang, N. Guo, C. Jin, K. Zhong, X. Zhou, J. Yuan, Z. Kang, B. Zhou, C. Yu, H.-Y. Tam, and C. Lu, “Coherent BOTDA Using Phase- and Polarization-Diversity Heterodyne Detection and Embedded Digital Signal Processing,” IEEE Sens. J. 17(12), 3728–3734 (2017).
[Crossref]

J. Lightwave Technol. (5)

Light Sci. Appl. (1)

A. Denisov, M. A. Soto, and L. Thévenaz, “Going beyond 1000000 resolved points in a Brillouin distributed fiber sensor: theoretical analysis and experimental demonstration,” Light Sci. Appl. 5(5), e16074 (2016).
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Measurement (1)

G. Soto, J. Fontbona, R. Cortez, and L. Mujica, “An online two-stage adaptive algorithm for strain profile estimation from noisy and abruptly changing BOTDR data and application to underground mines,” Measurement 92, 340–351 (2016).
[Crossref]

Nat. Commun. (1)

M. A. Soto, J. A. Ramírez, and L. Thévenaz, “Intensifying the response of distributed optical fibre sensors using 2D and 3D image restoration,” Nat. Commun. 7(1), 10870 (2016).
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Opt. Express (10)

Z. Qin, L. Chen, and X. Bao, “Continuous wavelet transform for non-stationary vibration detection with phase-OTDR,” Opt. Express 20(18), 20459–20465 (2012).
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A. Voskoboinik, O. F. Yilmaz, A. W. Willner, and M. Tur, “Sweep-free distributed Brillouin time-domain analyzer (SF-BOTDA),” Opt. Express 19(26), B842–B847 (2011).
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W. Li, X. Bao, Y. Li, and L. Chen, “Differential pulse-width pair BOTDA for high spatial resolution sensing,” Opt. Express 16(26), 21616–21625 (2008).
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A. K. Azad, F. N. Khan, W. H. Alarashi, N. Guo, A. P. T. Lau, and C. Lu, “Temperature extraction in Brillouin optical time-domain analysis sensors using principal component analysis based pattern recognition,” Opt. Express 25(14), 16534–16549 (2017).
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A. Lopez-Gil, M. A. Soto, X. Angulo-Vinuesa, A. Dominguez-Lopez, S. Martin-Lopez, L. Thévenaz, and M. Gonzalez-Herraez, “Evaluation of the accuracy of BOTDA systems based on the phase spectral response,” Opt. Express 24(15), 17200–17214 (2016).
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Y. H. Kim, K. Lee, and K. Y. Song, “Brillouin optical correlation domain analysis with more than 1 million effective sensing points based on differential measurement,” Opt. Express 23(26), 33241–33248 (2015).
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Y. Peled, A. Motil, L. Yaron, and M. Tur, “Slope-assisted fast distributed sensing in optical fibers with arbitrary Brillouin profile,” Opt. Express 19(21), 19845–19854 (2011).
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Y. Dang, Z. Zhao, M. Tang, C. Zhao, L. Gan, S. Fu, T. Liu, W. Tong, P. P. Shum, and D. Liu, “Towards large dynamic range and ultrahigh measurement resolution in distributed fiber sensing based on multicore fiber,” Opt. Express 25(17), 20183–20193 (2017).
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W. Zou, Z. He, and K. Hotate, “Complete discrimination of strain and temperature using Brillouin frequency shift and birefringence in a polarization-maintaining fiber,” Opt. Express 17(3), 1248–1255 (2009).
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M. A. Soto and L. Thévenaz, “Modeling and evaluating the performance of Brillouin distributed optical fiber sensors,” Opt. Express 21(25), 31347–31366 (2013).
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Opt. Laser Technol. (1)

A. Motil, A. Bergman, and M. Tur, “[INVITED] State of the art of Brillouin fiber-optic distributed sensing,” Opt. Laser Technol. 78, 81–103 (2016).
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Opt. Lett. (5)

Sensors (Basel) (1)

X. Bao and L. Chen, “Recent progress in distributed fiber optic sensors,” Sensors (Basel) 12(7), 8601–8639 (2012).
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L. Wang, N. Guo, C. Jin, C. Yu, H. Y. Tam, and C. Lu, “BOTDA system using artificial neural network,” in Proceedings of 2017 Opto-Electronics and Communications Conference (OECC) and Photonics Global Conference (PGC), (IEEE, 2017).
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[Crossref]

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Figures (7)

Fig. 1
Fig. 1 (a) Original spectrum of 10 km fiber, (b) 9-10 km section at the far end with + 20 MHz shift, (c) subtracted spectrum, and (d), frequency shift from the traditional Lorentzian fitting (blue), the SFM (red) and the SCM (black). Insets: Details of the curves.
Fig. 2
Fig. 2 Theoretical curves. (a) Demonstration of subtracting two Brillouin gain curves, (b) Frequency difference between peak-dim distance and sensing induced frequency shift, ∆ν. Insert: Peak dim distance relationship with the induced frequency shift.
Fig. 3
Fig. 3 Time consumption for the SCM, SFM, and traditional methods for (a), different spatial points number N, (b), different length of fiber with sensing induced frequency shift, ∆ν.
Fig. 4
Fig. 4 B-OTDA experimental configuration. PC: polarization controller, EOM: electro-optic modulator, RF: radio frequency, PS: polarization scrambler, EDFA: Erbium doped fiber amplifier, FUT: fiber under test, PD: photodetector.
Fig. 5
Fig. 5 Brillouin spectra obtain with our system with 1000 time averages with the tail parts in (a) 30 degree Celsius and (b) 50 Celsius, (c) is the subtraction of the former two, (d) is the xz view of (a), (e) and (f) are details of (b) and (c) respectively.
Fig. 6
Fig. 6 Frequency shift with (a) the traditional Lorentzian fitting, (b) the SFM, and (c) the SCM for 20 ̊C temperature change, (d) is variance of the subtracted spectrum. The insets are details.
Fig. 7
Fig. 7 Time costs for the Lorentz fitting, the SFM and the SCM.

Tables (1)

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Table 1 Time and Accuracy Compare for the Three Methods

Equations (4)

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g B (ν)= g 0 (Δ ν B /2) 2 (ν ν B ) 2 + (Δ ν B /2) 2 ,
Δ g B (ν)= g 0 (Δ ν B /2) 2 [2Δν(ν+ ν B )Δ ν 2 ] [ (ν ν B Δν) 2 + (Δ ν B /2) 2 ][ (ν ν B ) 2 + (Δ ν B /2) 2 ] ,
0=3 (ν ν B ) 4 6Δν (ν ν B ) 3 +(4Δ ν 2 2 γ 2 ) (ν ν B ) 2 (2Δν γ 2 +Δ ν 3 )(ν ν B ) γ 4 .
Δν=± Δ ν PD 2 4 ( ν B /2) 2 +2 Δ ν PD 4 +4Δ ν PD 2 (Δ ν B /2) 2 .

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