Abstract

We propose and experimentally demonstrate a stable homodyne phase demodulation technique in a ϕ-OTDR using a double-pulse probe and a simple direct detection receiver. The technique uses selective phase modulation of one of a pair of pulses to generate a carrier for dynamic phase changes and involves an enhanced phase demodulation scheme suitable for distributed sensing by being robust against light intensity fluctuations, independent of the modulation depth, and convenient for analogue signal processing. The capability of the technique to quantify distributed dynamic phase change due to a generic external impact is experimentally demonstrated by measuring the phase change induced by a nonlinear actuator generating a 2 kHz perturbation at a distance of 1.5 km on a standard singlemode fiber with an SNR of ~24 dB. The demodulated nonlinear response is shown to have a spectrum consistent with one obtained using an FBG sensor and a commercial reading unit.

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

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2017 (4)

2016 (6)

A. Zhang and S. Zhang, “High Stability Fiber-Optics Sensors With an Improved PGC Demodulation Algorithm,” IEEE Sens. J. 16(21), 7681–7684 (2016).

Y. Muanenda, C. J. Oton, S. Faralli, and F. Di Pasquale, “A Cost-Effective Distributed Acoustic Sensor Using a Commercial Off-the-Shelf DFB Laser and Direct Detection Phase-OTDR,” IEEE Photonics J. 8(1), 1–10 (2016).

Y. Muanenda, C. J. Oton, S. Faralli, T. Nannipieri, A. Signorini, and F. Di Pasquale, “Hybrid distributed acoustic and temperature sensor using a commercial off-the-shelf DFB laser and direct detection,” Opt. Lett. 41(3), 587–590 (2016).
[PubMed]

A. Garcia-Ruiz, J. Pastor-Graells, H. F. Martins, S. Martin-Lopez, and M. Gonzalez-Herraez, “Speckle analysis method for distributed detection of temperature gradients with phi-OTDR,” IEEE Photonics Technol. Lett. 28(18), 2000–2003 (2016).

J. Pastor-Graells, H. F. Martins, A. Garcia-Ruiz, S. Martin-Lopez, and M. Gonzalez-Herraez, “Single-shot distributed temperature and strain tracking using direct detection phase-sensitive OTDR with chirped pulses,” Opt. Express 24(12), 13121–13133 (2016).
[PubMed]

M. Ren, P. Lu, L. Chen, and X. Bao, “Study of Ф-OTDR stability for dynamic strain measurement in piezoelectric vibration,” Photonic Sensors 6(3), 199–208 (2016).

2015 (3)

2014 (1)

A. E. Alekseev, V. S. Vdovenko, B. G. Gorshkov, V. T. Potapov, and D. E. Simikin, “A phase-sensitive optical time-domain reflectometer with dual-pulse phase modulated probe signal,” Laser Phys. 24(11), 115106 (2014).

2013 (2)

Y. Li, Z. Liu, Y. Liu, L. Ma, Z. Tan, and S. Jian, “Interferometric vibration sensor using phase-generated carrier method,” Appl. Opt. 52(25), 6359–6363 (2013).
[PubMed]

A. Masoudi, M. Belal, and T. P. Newson, “A distributed optical fibre dynamic strain sensor based on phase OTDR,” Meas. Sci. Technol. 24(8), 085204 (2013).

2012 (2)

2011 (2)

C. Kopp, C. Kopp, S. Bernabé, B. B. Bakir, J. Fedeli, R. Orobtchouk, F. Schrank, H. Porte, L. Zimmermann, and T. Tekin, “Silicon Photonic Circuits: On-CMOS Integration, Fiber Optical Coupling, and Packaging,” IEEE J. Sel. Top. Quantum Electron. 17(3), 498–509 (2011).

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

2010 (4)

J. He, L. Wang, F. Li, and Y. Liu, “An Ameliorated Phase Generated Carrier Demodulation Algorithm With Low Harmonic Distortion and High Stability,” J. Lightwave Technol. 28(22), 3258–3265 (2010).

F. Bettarello, P. Fausti, V. Baccan, and M. Caniato, “Impact sound pressure level performances of basic beam floor structures,” J. Build. Acoust. 17(4), 305–316 (2010).

J. A. Langley, R. G. Brice, and Q. Zhao, “Recursive approach to the moment-based phase unwrapping method,” Appl. Opt. 49(16), 3096–3101 (2010).
[PubMed]

J. He, L. Wang, F. Li, and Y. Liu, “An Ameliorated Phase Generated Carrier Demodulation Algorithm With Low Harmonic Distortion and High Stability,” J. Lightwave Technol. 28(22), 258–3265 (2010).

2009 (1)

2008 (2)

2007 (1)

2005 (1)

2003 (1)

J. Park and H. F. Taylor, “Fiber Optic Intrusion Sensor using Coherent Optical Time Domain Reflectometer,” Jpn. J. Appl. Phys. 42(1), 3481–3482 (2003).

1998 (1)

L. S. Christensen and D. Manvell, “Sound level meters in building acoustic measurements,” J. Build. Acoust. 5(3), 217–222 (1998).

1996 (1)

1994 (1)

C. McGarrity and D. A. Jackson, “Improvement on phase generated carrier technique for passive demodulation of miniature interferometric sensors,” Opt. Commun. 109(3–4), 246–248 (1994).

1992 (1)

N. H. Ching, D. Rosenfeld, and M. Braun, “Two-dimensional phase unwrapping using a minimum spanning tree algorithm,” IEEE Trans. Image Process. 1(3), 355–365 (1992).
[PubMed]

1982 (1)

A. Dandridge, A. B. Tveten, and T. G. Giallorenzi, “Homodyne Demodulation Scheme for Fiber Optic Sensors Using Phase Generated Carrier,” IEEE Trans. Microw. Theory Tech. 30(10), 1635–1641 (1982).

Albason, A. C. B.

A. C. B. Albason, N. M. L. Axalan, M. T. A. Gusad, J. R. E. Hizon, and M. D. Rosales, “Design Methodologies for Low-Power CMOS Operational Amplifiers in a 0.25μm Digital CMOS Process,” in Proceedings of TENCON 2006 - 2006 IEEE Region 10 Conference (IEEE, 2006), pp. 1–4.

Alekseev, A. E.

A. E. Alekseev, V. S. Vdovenko, B. G. Gorshkov, V. T. Potapov, and D. E. Simikin, “A phase-sensitive optical time-domain reflectometer with dual-pulse phase modulated probe signal,” Laser Phys. 24(11), 115106 (2014).

Asada, H. H.

J. Torres and H. H. Asada, “Harmonic analysis of a PZT poly-actuator,” in Proceedings of IEEE International Conference on Robotics and Automation (IEEE, 2015), pp. 842–849.

Axalan, N. M. L.

A. C. B. Albason, N. M. L. Axalan, M. T. A. Gusad, J. R. E. Hizon, and M. D. Rosales, “Design Methodologies for Low-Power CMOS Operational Amplifiers in a 0.25μm Digital CMOS Process,” in Proceedings of TENCON 2006 - 2006 IEEE Region 10 Conference (IEEE, 2006), pp. 1–4.

Baccan, V.

F. Bettarello, P. Fausti, V. Baccan, and M. Caniato, “Impact sound pressure level performances of basic beam floor structures,” J. Build. Acoust. 17(4), 305–316 (2010).

Bai, J.

Bakir, B. B.

C. Kopp, C. Kopp, S. Bernabé, B. B. Bakir, J. Fedeli, R. Orobtchouk, F. Schrank, H. Porte, L. Zimmermann, and T. Tekin, “Silicon Photonic Circuits: On-CMOS Integration, Fiber Optical Coupling, and Packaging,” IEEE J. Sel. Top. Quantum Electron. 17(3), 498–509 (2011).

Bao, X.

M. Ren, P. Lu, L. Chen, and X. Bao, “Study of Ф-OTDR stability for dynamic strain measurement in piezoelectric vibration,” Photonic Sensors 6(3), 199–208 (2016).

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).
[PubMed]

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

Bao, Z.

H. Liu, M. Xing, and Z. Bao, “A Cluster-Analysis-Based Noise-Robust Phase-Unwrapping Algorithm for Multibaseline Interferograms,” IEEE Trans. Geosci. Remote Sens. 53(1), 494–504 (2015).

Belal, M.

A. Masoudi, M. Belal, and T. P. Newson, “A distributed optical fibre dynamic strain sensor based on phase OTDR,” Meas. Sci. Technol. 24(8), 085204 (2013).

Bernabé, S.

C. Kopp, C. Kopp, S. Bernabé, B. B. Bakir, J. Fedeli, R. Orobtchouk, F. Schrank, H. Porte, L. Zimmermann, and T. Tekin, “Silicon Photonic Circuits: On-CMOS Integration, Fiber Optical Coupling, and Packaging,” IEEE J. Sel. Top. Quantum Electron. 17(3), 498–509 (2011).

Bettarello, F.

F. Bettarello, P. Fausti, V. Baccan, and M. Caniato, “Impact sound pressure level performances of basic beam floor structures,” J. Build. Acoust. 17(4), 305–316 (2010).

Braun, M.

N. H. Ching, D. Rosenfeld, and M. Braun, “Two-dimensional phase unwrapping using a minimum spanning tree algorithm,” IEEE Trans. Image Process. 1(3), 355–365 (1992).
[PubMed]

Brice, R. G.

Caniato, M.

F. Bettarello, P. Fausti, V. Baccan, and M. Caniato, “Impact sound pressure level performances of basic beam floor structures,” J. Build. Acoust. 17(4), 305–316 (2010).

Chen, L.

M. Ren, P. Lu, L. Chen, and X. Bao, “Study of Ф-OTDR stability for dynamic strain measurement in piezoelectric vibration,” Photonic Sensors 6(3), 199–208 (2016).

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).
[PubMed]

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

Chen, X.

Cheng, Z.

Ching, N. H.

N. H. Ching, D. Rosenfeld, and M. Braun, “Two-dimensional phase unwrapping using a minimum spanning tree algorithm,” IEEE Trans. Image Process. 1(3), 355–365 (1992).
[PubMed]

Choi, K. N.

Christensen, L. S.

L. S. Christensen and D. Manvell, “Sound level meters in building acoustic measurements,” J. Build. Acoust. 5(3), 217–222 (1998).

Crain, E.

M. G. Guvench, M. Miske, and E. Crain, “Design, fabrication and testing of CMOS operational amplifiers as training tool in analog integrated circuit design,” in Proceedings of the Fourteenth Biennial University/ Government/ Industry Microelectronics Symposium (IEEE, 2001), pp. 193–196.

Dandridge, A.

A. Dandridge, A. B. Tveten, and T. G. Giallorenzi, “Homodyne Demodulation Scheme for Fiber Optic Sensors Using Phase Generated Carrier,” IEEE Trans. Microw. Theory Tech. 30(10), 1635–1641 (1982).

Dardikman, G.

Di Pasquale, F.

Y. Muanenda, C. J. Oton, S. Faralli, T. Nannipieri, A. Signorini, and F. Di Pasquale, “Hybrid distributed acoustic and temperature sensor using a commercial off-the-shelf DFB laser and direct detection,” Opt. Lett. 41(3), 587–590 (2016).
[PubMed]

Y. Muanenda, C. J. Oton, S. Faralli, and F. Di Pasquale, “A Cost-Effective Distributed Acoustic Sensor Using a Commercial Off-the-Shelf DFB Laser and Direct Detection Phase-OTDR,” IEEE Photonics J. 8(1), 1–10 (2016).

Fang, X.

Faralli, S.

Y. Muanenda, C. J. Oton, S. Faralli, and F. Di Pasquale, “A Cost-Effective Distributed Acoustic Sensor Using a Commercial Off-the-Shelf DFB Laser and Direct Detection Phase-OTDR,” IEEE Photonics J. 8(1), 1–10 (2016).

Y. Muanenda, C. J. Oton, S. Faralli, T. Nannipieri, A. Signorini, and F. Di Pasquale, “Hybrid distributed acoustic and temperature sensor using a commercial off-the-shelf DFB laser and direct detection,” Opt. Lett. 41(3), 587–590 (2016).
[PubMed]

Fausti, P.

F. Bettarello, P. Fausti, V. Baccan, and M. Caniato, “Impact sound pressure level performances of basic beam floor structures,” J. Build. Acoust. 17(4), 305–316 (2010).

Fedeli, J.

C. Kopp, C. Kopp, S. Bernabé, B. B. Bakir, J. Fedeli, R. Orobtchouk, F. Schrank, H. Porte, L. Zimmermann, and T. Tekin, “Silicon Photonic Circuits: On-CMOS Integration, Fiber Optical Coupling, and Packaging,” IEEE J. Sel. Top. Quantum Electron. 17(3), 498–509 (2011).

Fujikata, J.

D. Okamoto, Y. Suzuki, K. Yashiki, Y. Hagihara, M. Tokushima, J. Fujikata, M. Kurihara, J. Tsuchida, T. Nedachi, J. Inasaka, and K. Kurata, “25-Gbps 5×5 mm chip-scale silicon-photonic receiver integrated with 28-nm CMOS transimpedance amplifier,” in Proceedings of 2015 IEEE Optical Interconnects Conference (IEEE, 2015), pp. 56–57.

Gao, W.

Garcia-Ruiz, A.

Giallorenzi, T. G.

A. Dandridge, A. B. Tveten, and T. G. Giallorenzi, “Homodyne Demodulation Scheme for Fiber Optic Sensors Using Phase Generated Carrier,” IEEE Trans. Microw. Theory Tech. 30(10), 1635–1641 (1982).

Gonzalez-Herraez, M.

Gorshkov, B. G.

A. E. Alekseev, V. S. Vdovenko, B. G. Gorshkov, V. T. Potapov, and D. E. Simikin, “A phase-sensitive optical time-domain reflectometer with dual-pulse phase modulated probe signal,” Laser Phys. 24(11), 115106 (2014).

Gusad, M. T. A.

A. C. B. Albason, N. M. L. Axalan, M. T. A. Gusad, J. R. E. Hizon, and M. D. Rosales, “Design Methodologies for Low-Power CMOS Operational Amplifiers in a 0.25μm Digital CMOS Process,” in Proceedings of TENCON 2006 - 2006 IEEE Region 10 Conference (IEEE, 2006), pp. 1–4.

Guvench, M. G.

M. G. Guvench, M. Miske, and E. Crain, “Design, fabrication and testing of CMOS operational amplifiers as training tool in analog integrated circuit design,” in Proceedings of the Fourteenth Biennial University/ Government/ Industry Microelectronics Symposium (IEEE, 2001), pp. 193–196.

Habaza, M.

Hagihara, Y.

D. Okamoto, Y. Suzuki, K. Yashiki, Y. Hagihara, M. Tokushima, J. Fujikata, M. Kurihara, J. Tsuchida, T. Nedachi, J. Inasaka, and K. Kurata, “25-Gbps 5×5 mm chip-scale silicon-photonic receiver integrated with 28-nm CMOS transimpedance amplifier,” in Proceedings of 2015 IEEE Optical Interconnects Conference (IEEE, 2015), pp. 56–57.

He, J.

J. He, L. Wang, F. Li, and Y. Liu, “An Ameliorated Phase Generated Carrier Demodulation Algorithm With Low Harmonic Distortion and High Stability,” J. Lightwave Technol. 28(22), 258–3265 (2010).

J. He, L. Wang, F. Li, and Y. Liu, “An Ameliorated Phase Generated Carrier Demodulation Algorithm With Low Harmonic Distortion and High Stability,” J. Lightwave Technol. 28(22), 3258–3265 (2010).

Hey Tow, K.

Hizon, J. R. E.

A. C. B. Albason, N. M. L. Axalan, M. T. A. Gusad, J. R. E. Hizon, and M. D. Rosales, “Design Methodologies for Low-Power CMOS Operational Amplifiers in a 0.25μm Digital CMOS Process,” in Proceedings of TENCON 2006 - 2006 IEEE Region 10 Conference (IEEE, 2006), pp. 1–4.

Hogari, K.

Hu, Y.

Hu, Z.

Huang, S. C.

Huang, W.

Imahama, M.

Inasaka, J.

D. Okamoto, Y. Suzuki, K. Yashiki, Y. Hagihara, M. Tokushima, J. Fujikata, M. Kurihara, J. Tsuchida, T. Nedachi, J. Inasaka, and K. Kurata, “25-Gbps 5×5 mm chip-scale silicon-photonic receiver integrated with 28-nm CMOS transimpedance amplifier,” in Proceedings of 2015 IEEE Optical Interconnects Conference (IEEE, 2015), pp. 56–57.

Jackson, D. A.

C. McGarrity and D. A. Jackson, “Improvement on phase generated carrier technique for passive demodulation of miniature interferometric sensors,” Opt. Commun. 109(3–4), 246–248 (1994).

Jia, J.

Y. Liu, J. Jia, P. Tao, G. Yin, Z. Tan, W. Ren, and S. Jian, “Signal processing of Sagnac fiber interferometer used as distributed sensor with wavelets,” in Proceedings of Asia Communications and Photonics Conference and Exhibition (IEEE, 2010), pp. 290–291.

Jian, S.

Y. Li, Z. Liu, Y. Liu, L. Ma, Z. Tan, and S. Jian, “Interferometric vibration sensor using phase-generated carrier method,” Appl. Opt. 52(25), 6359–6363 (2013).
[PubMed]

Y. Liu, J. Jia, P. Tao, G. Yin, Z. Tan, W. Ren, and S. Jian, “Signal processing of Sagnac fiber interferometer used as distributed sensor with wavelets,” in Proceedings of Asia Communications and Photonics Conference and Exhibition (IEEE, 2010), pp. 290–291.

Juarez, J. C.

Kemao, Q.

Kitahara, D.

D. Kitahara, M. Yamagishi, and I. Yamada, “A virtual resampling technique for algebraic two-dimensional phase unwrapping,” in Proceedings of 2015 IEEE International Conference on Acoustics, Speech and Signal Processing (IEEE, 2015), pp. 3871–3875.

Kopp, C.

C. Kopp, C. Kopp, S. Bernabé, B. B. Bakir, J. Fedeli, R. Orobtchouk, F. Schrank, H. Porte, L. Zimmermann, and T. Tekin, “Silicon Photonic Circuits: On-CMOS Integration, Fiber Optical Coupling, and Packaging,” IEEE J. Sel. Top. Quantum Electron. 17(3), 498–509 (2011).

C. Kopp, C. Kopp, S. Bernabé, B. B. Bakir, J. Fedeli, R. Orobtchouk, F. Schrank, H. Porte, L. Zimmermann, and T. Tekin, “Silicon Photonic Circuits: On-CMOS Integration, Fiber Optical Coupling, and Packaging,” IEEE J. Sel. Top. Quantum Electron. 17(3), 498–509 (2011).

Koyamada, Y.

Krebber, K.

Kubota, K.

Kurata, K.

D. Okamoto, Y. Suzuki, K. Yashiki, Y. Hagihara, M. Tokushima, J. Fujikata, M. Kurihara, J. Tsuchida, T. Nedachi, J. Inasaka, and K. Kurata, “25-Gbps 5×5 mm chip-scale silicon-photonic receiver integrated with 28-nm CMOS transimpedance amplifier,” in Proceedings of 2015 IEEE Optical Interconnects Conference (IEEE, 2015), pp. 56–57.

Kurihara, M.

D. Okamoto, Y. Suzuki, K. Yashiki, Y. Hagihara, M. Tokushima, J. Fujikata, M. Kurihara, J. Tsuchida, T. Nedachi, J. Inasaka, and K. Kurata, “25-Gbps 5×5 mm chip-scale silicon-photonic receiver integrated with 28-nm CMOS transimpedance amplifier,” in Proceedings of 2015 IEEE Optical Interconnects Conference (IEEE, 2015), pp. 56–57.

Langley, J. A.

Li, F.

J. He, L. Wang, F. Li, and Y. Liu, “An Ameliorated Phase Generated Carrier Demodulation Algorithm With Low Harmonic Distortion and High Stability,” J. Lightwave Technol. 28(22), 3258–3265 (2010).

J. He, L. Wang, F. Li, and Y. Liu, “An Ameliorated Phase Generated Carrier Demodulation Algorithm With Low Harmonic Distortion and High Stability,” J. Lightwave Technol. 28(22), 258–3265 (2010).

Li, Y.

Liao, Y. B.

Liehr, S.

Lin, H.

Ling, T.

Liu, D.

Liu, H.

H. Liu, M. Xing, and Z. Bao, “A Cluster-Analysis-Based Noise-Robust Phase-Unwrapping Algorithm for Multibaseline Interferograms,” IEEE Trans. Geosci. Remote Sens. 53(1), 494–504 (2015).

Liu, Y.

Y. Li, Z. Liu, Y. Liu, L. Ma, Z. Tan, and S. Jian, “Interferometric vibration sensor using phase-generated carrier method,” Appl. Opt. 52(25), 6359–6363 (2013).
[PubMed]

J. He, L. Wang, F. Li, and Y. Liu, “An Ameliorated Phase Generated Carrier Demodulation Algorithm With Low Harmonic Distortion and High Stability,” J. Lightwave Technol. 28(22), 3258–3265 (2010).

J. He, L. Wang, F. Li, and Y. Liu, “An Ameliorated Phase Generated Carrier Demodulation Algorithm With Low Harmonic Distortion and High Stability,” J. Lightwave Technol. 28(22), 258–3265 (2010).

Y. Liu, L. W. Wang, C. D. Tian, M. Zhang, and Y. B. Liao, “Analysis and optimization of the PGC method in all digital demodulation systems,” J. Lightwave Technol. 26(18), 3225–3233 (2008).

Y. Liu, J. Jia, P. Tao, G. Yin, Z. Tan, W. Ren, and S. Jian, “Signal processing of Sagnac fiber interferometer used as distributed sensor with wavelets,” in Proceedings of Asia Communications and Photonics Conference and Exhibition (IEEE, 2010), pp. 290–291.

Liu, Z.

Lo, Y.

Lu, P.

M. Ren, P. Lu, L. Chen, and X. Bao, “Study of Ф-OTDR stability for dynamic strain measurement in piezoelectric vibration,” Photonic Sensors 6(3), 199–208 (2016).

Luo, H.

Ma, L.

Maier, E. W.

Manvell, D.

L. S. Christensen and D. Manvell, “Sound level meters in building acoustic measurements,” J. Build. Acoust. 5(3), 217–222 (1998).

Martin-Lopez, S.

Martins, H. F.

Masoudi, A.

A. Masoudi and T. P. Newson, “High spatial resolution distributed optical fiber dynamic strain sensor with enhanced frequency and strain resolution,” Opt. Lett. 42(2), 290–293 (2017).
[PubMed]

A. Masoudi, M. Belal, and T. P. Newson, “A distributed optical fibre dynamic strain sensor based on phase OTDR,” Meas. Sci. Technol. 24(8), 085204 (2013).

McGarrity, C.

C. McGarrity and D. A. Jackson, “Improvement on phase generated carrier technique for passive demodulation of miniature interferometric sensors,” Opt. Commun. 109(3–4), 246–248 (1994).

Miao, L.

Mirsky, S.

Miske, M.

M. G. Guvench, M. Miske, and E. Crain, “Design, fabrication and testing of CMOS operational amplifiers as training tool in analog integrated circuit design,” in Proceedings of the Fourteenth Biennial University/ Government/ Industry Microelectronics Symposium (IEEE, 2001), pp. 193–196.

Muanenda, Y.

Y. Muanenda, C. J. Oton, S. Faralli, and F. Di Pasquale, “A Cost-Effective Distributed Acoustic Sensor Using a Commercial Off-the-Shelf DFB Laser and Direct Detection Phase-OTDR,” IEEE Photonics J. 8(1), 1–10 (2016).

Y. Muanenda, C. J. Oton, S. Faralli, T. Nannipieri, A. Signorini, and F. Di Pasquale, “Hybrid distributed acoustic and temperature sensor using a commercial off-the-shelf DFB laser and direct detection,” Opt. Lett. 41(3), 587–590 (2016).
[PubMed]

Muanenda, Y. S.

Münzenberger, S.

Nannipieri, T.

Nedachi, T.

D. Okamoto, Y. Suzuki, K. Yashiki, Y. Hagihara, M. Tokushima, J. Fujikata, M. Kurihara, J. Tsuchida, T. Nedachi, J. Inasaka, and K. Kurata, “25-Gbps 5×5 mm chip-scale silicon-photonic receiver integrated with 28-nm CMOS transimpedance amplifier,” in Proceedings of 2015 IEEE Optical Interconnects Conference (IEEE, 2015), pp. 56–57.

Newson, T. P.

A. Masoudi and T. P. Newson, “High spatial resolution distributed optical fiber dynamic strain sensor with enhanced frequency and strain resolution,” Opt. Lett. 42(2), 290–293 (2017).
[PubMed]

A. Masoudi, M. Belal, and T. P. Newson, “A distributed optical fibre dynamic strain sensor based on phase OTDR,” Meas. Sci. Technol. 24(8), 085204 (2013).

Okamoto, D.

D. Okamoto, Y. Suzuki, K. Yashiki, Y. Hagihara, M. Tokushima, J. Fujikata, M. Kurihara, J. Tsuchida, T. Nedachi, J. Inasaka, and K. Kurata, “25-Gbps 5×5 mm chip-scale silicon-photonic receiver integrated with 28-nm CMOS transimpedance amplifier,” in Proceedings of 2015 IEEE Optical Interconnects Conference (IEEE, 2015), pp. 56–57.

Orobtchouk, R.

C. Kopp, C. Kopp, S. Bernabé, B. B. Bakir, J. Fedeli, R. Orobtchouk, F. Schrank, H. Porte, L. Zimmermann, and T. Tekin, “Silicon Photonic Circuits: On-CMOS Integration, Fiber Optical Coupling, and Packaging,” IEEE J. Sel. Top. Quantum Electron. 17(3), 498–509 (2011).

Oton, C. J.

Y. Muanenda, C. J. Oton, S. Faralli, and F. Di Pasquale, “A Cost-Effective Distributed Acoustic Sensor Using a Commercial Off-the-Shelf DFB Laser and Direct Detection Phase-OTDR,” IEEE Photonics J. 8(1), 1–10 (2016).

Y. Muanenda, C. J. Oton, S. Faralli, T. Nannipieri, A. Signorini, and F. Di Pasquale, “Hybrid distributed acoustic and temperature sensor using a commercial off-the-shelf DFB laser and direct detection,” Opt. Lett. 41(3), 587–590 (2016).
[PubMed]

Park, J.

J. Park and H. F. Taylor, “Fiber Optic Intrusion Sensor using Coherent Optical Time Domain Reflectometer,” Jpn. J. Appl. Phys. 42(1), 3481–3482 (2003).

Pastor-Graells, J.

Porte, H.

C. Kopp, C. Kopp, S. Bernabé, B. B. Bakir, J. Fedeli, R. Orobtchouk, F. Schrank, H. Porte, L. Zimmermann, and T. Tekin, “Silicon Photonic Circuits: On-CMOS Integration, Fiber Optical Coupling, and Packaging,” IEEE J. Sel. Top. Quantum Electron. 17(3), 498–509 (2011).

Potapov, V. T.

A. E. Alekseev, V. S. Vdovenko, B. G. Gorshkov, V. T. Potapov, and D. E. Simikin, “A phase-sensitive optical time-domain reflectometer with dual-pulse phase modulated probe signal,” Laser Phys. 24(11), 115106 (2014).

Qin, Z.

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).
[PubMed]

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

Ren, M.

M. Ren, P. Lu, L. Chen, and X. Bao, “Study of Ф-OTDR stability for dynamic strain measurement in piezoelectric vibration,” Photonic Sensors 6(3), 199–208 (2016).

Ren, W.

Y. Liu, J. Jia, P. Tao, G. Yin, Z. Tan, W. Ren, and S. Jian, “Signal processing of Sagnac fiber interferometer used as distributed sensor with wavelets,” in Proceedings of Asia Communications and Photonics Conference and Exhibition (IEEE, 2010), pp. 290–291.

Roichman, Y.

Rosales, M. D.

A. C. B. Albason, N. M. L. Axalan, M. T. A. Gusad, J. R. E. Hizon, and M. D. Rosales, “Design Methodologies for Low-Power CMOS Operational Amplifiers in a 0.25μm Digital CMOS Process,” in Proceedings of TENCON 2006 - 2006 IEEE Region 10 Conference (IEEE, 2006), pp. 1–4.

Rosenfeld, D.

N. H. Ching, D. Rosenfeld, and M. Braun, “Two-dimensional phase unwrapping using a minimum spanning tree algorithm,” IEEE Trans. Image Process. 1(3), 355–365 (1992).
[PubMed]

Schrank, F.

C. Kopp, C. Kopp, S. Bernabé, B. B. Bakir, J. Fedeli, R. Orobtchouk, F. Schrank, H. Porte, L. Zimmermann, and T. Tekin, “Silicon Photonic Circuits: On-CMOS Integration, Fiber Optical Coupling, and Packaging,” IEEE J. Sel. Top. Quantum Electron. 17(3), 498–509 (2011).

Shaked, N. T.

Shen, Y.

Signorini, A.

Simikin, D. E.

A. E. Alekseev, V. S. Vdovenko, B. G. Gorshkov, V. T. Potapov, and D. E. Simikin, “A phase-sensitive optical time-domain reflectometer with dual-pulse phase modulated probe signal,” Laser Phys. 24(11), 115106 (2014).

Suzuki, Y.

D. Okamoto, Y. Suzuki, K. Yashiki, Y. Hagihara, M. Tokushima, J. Fujikata, M. Kurihara, J. Tsuchida, T. Nedachi, J. Inasaka, and K. Kurata, “25-Gbps 5×5 mm chip-scale silicon-photonic receiver integrated with 28-nm CMOS transimpedance amplifier,” in Proceedings of 2015 IEEE Optical Interconnects Conference (IEEE, 2015), pp. 56–57.

Tan, Z.

Y. Li, Z. Liu, Y. Liu, L. Ma, Z. Tan, and S. Jian, “Interferometric vibration sensor using phase-generated carrier method,” Appl. Opt. 52(25), 6359–6363 (2013).
[PubMed]

Y. Liu, J. Jia, P. Tao, G. Yin, Z. Tan, W. Ren, and S. Jian, “Signal processing of Sagnac fiber interferometer used as distributed sensor with wavelets,” in Proceedings of Asia Communications and Photonics Conference and Exhibition (IEEE, 2010), pp. 290–291.

Tao, P.

Y. Liu, J. Jia, P. Tao, G. Yin, Z. Tan, W. Ren, and S. Jian, “Signal processing of Sagnac fiber interferometer used as distributed sensor with wavelets,” in Proceedings of Asia Communications and Photonics Conference and Exhibition (IEEE, 2010), pp. 290–291.

Taylor, H. F.

J. C. Juarez, E. W. Maier, K. N. Choi, and H. F. Taylor, “Distributed fiberoptic intrusion sensor system,” J. Lightwave Technol. 23(6), 2081–2087 (2005).

J. Park and H. F. Taylor, “Fiber Optic Intrusion Sensor using Coherent Optical Time Domain Reflectometer,” Jpn. J. Appl. Phys. 42(1), 3481–3482 (2003).

Tekin, T.

C. Kopp, C. Kopp, S. Bernabé, B. B. Bakir, J. Fedeli, R. Orobtchouk, F. Schrank, H. Porte, L. Zimmermann, and T. Tekin, “Silicon Photonic Circuits: On-CMOS Integration, Fiber Optical Coupling, and Packaging,” IEEE J. Sel. Top. Quantum Electron. 17(3), 498–509 (2011).

Thévenaz, L.

Tian, C. D.

Tokushima, M.

D. Okamoto, Y. Suzuki, K. Yashiki, Y. Hagihara, M. Tokushima, J. Fujikata, M. Kurihara, J. Tsuchida, T. Nedachi, J. Inasaka, and K. Kurata, “25-Gbps 5×5 mm chip-scale silicon-photonic receiver integrated with 28-nm CMOS transimpedance amplifier,” in Proceedings of 2015 IEEE Optical Interconnects Conference (IEEE, 2015), pp. 56–57.

Torres, J.

J. Torres and H. H. Asada, “Harmonic analysis of a PZT poly-actuator,” in Proceedings of IEEE International Conference on Robotics and Automation (IEEE, 2015), pp. 842–849.

Tsuchida, J.

D. Okamoto, Y. Suzuki, K. Yashiki, Y. Hagihara, M. Tokushima, J. Fujikata, M. Kurihara, J. Tsuchida, T. Nedachi, J. Inasaka, and K. Kurata, “25-Gbps 5×5 mm chip-scale silicon-photonic receiver integrated with 28-nm CMOS transimpedance amplifier,” in Proceedings of 2015 IEEE Optical Interconnects Conference (IEEE, 2015), pp. 56–57.

Tveten, A. B.

A. Dandridge, A. B. Tveten, and T. G. Giallorenzi, “Homodyne Demodulation Scheme for Fiber Optic Sensors Using Phase Generated Carrier,” IEEE Trans. Microw. Theory Tech. 30(10), 1635–1641 (1982).

Vdovenko, V. S.

A. E. Alekseev, V. S. Vdovenko, B. G. Gorshkov, V. T. Potapov, and D. E. Simikin, “A phase-sensitive optical time-domain reflectometer with dual-pulse phase modulated probe signal,” Laser Phys. 24(11), 115106 (2014).

Wang, F.

Wang, H.

Wang, L.

J. He, L. Wang, F. Li, and Y. Liu, “An Ameliorated Phase Generated Carrier Demodulation Algorithm With Low Harmonic Distortion and High Stability,” J. Lightwave Technol. 28(22), 3258–3265 (2010).

J. He, L. Wang, F. Li, and Y. Liu, “An Ameliorated Phase Generated Carrier Demodulation Algorithm With Low Harmonic Distortion and High Stability,” J. Lightwave Technol. 28(22), 258–3265 (2010).

Wang, L. W.

Weng, J.

Xie, J.

Xing, M.

H. Liu, M. Xing, and Z. Bao, “A Cluster-Analysis-Based Noise-Robust Phase-Unwrapping Algorithm for Multibaseline Interferograms,” IEEE Trans. Geosci. Remote Sens. 53(1), 494–504 (2015).

Xiong, S.

Yamada, I.

D. Kitahara, M. Yamagishi, and I. Yamada, “A virtual resampling technique for algebraic two-dimensional phase unwrapping,” in Proceedings of 2015 IEEE International Conference on Acoustics, Speech and Signal Processing (IEEE, 2015), pp. 3871–3875.

Yamagishi, M.

D. Kitahara, M. Yamagishi, and I. Yamada, “A virtual resampling technique for algebraic two-dimensional phase unwrapping,” in Proceedings of 2015 IEEE International Conference on Acoustics, Speech and Signal Processing (IEEE, 2015), pp. 3871–3875.

Yang, Y.

Yashiki, K.

D. Okamoto, Y. Suzuki, K. Yashiki, Y. Hagihara, M. Tokushima, J. Fujikata, M. Kurihara, J. Tsuchida, T. Nedachi, J. Inasaka, and K. Kurata, “25-Gbps 5×5 mm chip-scale silicon-photonic receiver integrated with 28-nm CMOS transimpedance amplifier,” in Proceedings of 2015 IEEE Optical Interconnects Conference (IEEE, 2015), pp. 56–57.

Yin, G.

Y. Liu, J. Jia, P. Tao, G. Yin, Z. Tan, W. Ren, and S. Jian, “Signal processing of Sagnac fiber interferometer used as distributed sensor with wavelets,” in Proceedings of Asia Communications and Photonics Conference and Exhibition (IEEE, 2010), pp. 290–291.

Zhang, A.

A. Zhang and S. Zhang, “High Stability Fiber-Optics Sensors With an Improved PGC Demodulation Algorithm,” IEEE Sens. J. 16(21), 7681–7684 (2016).

Zhang, L.

Zhang, M.

Zhang, S.

A. Zhang and S. Zhang, “High Stability Fiber-Optics Sensors With an Improved PGC Demodulation Algorithm,” IEEE Sens. J. 16(21), 7681–7684 (2016).

Zhao, Q.

Zhu, T.

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

Zimmermann, L.

C. Kopp, C. Kopp, S. Bernabé, B. B. Bakir, J. Fedeli, R. Orobtchouk, F. Schrank, H. Porte, L. Zimmermann, and T. Tekin, “Silicon Photonic Circuits: On-CMOS Integration, Fiber Optical Coupling, and Packaging,” IEEE J. Sel. Top. Quantum Electron. 17(3), 498–509 (2011).

Appl. Opt. (4)

IEEE J. Sel. Top. Quantum Electron. (1)

C. Kopp, C. Kopp, S. Bernabé, B. B. Bakir, J. Fedeli, R. Orobtchouk, F. Schrank, H. Porte, L. Zimmermann, and T. Tekin, “Silicon Photonic Circuits: On-CMOS Integration, Fiber Optical Coupling, and Packaging,” IEEE J. Sel. Top. Quantum Electron. 17(3), 498–509 (2011).

IEEE Photonics J. (1)

Y. Muanenda, C. J. Oton, S. Faralli, and F. Di Pasquale, “A Cost-Effective Distributed Acoustic Sensor Using a Commercial Off-the-Shelf DFB Laser and Direct Detection Phase-OTDR,” IEEE Photonics J. 8(1), 1–10 (2016).

IEEE Photonics Technol. Lett. (2)

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

A. Garcia-Ruiz, J. Pastor-Graells, H. F. Martins, S. Martin-Lopez, and M. Gonzalez-Herraez, “Speckle analysis method for distributed detection of temperature gradients with phi-OTDR,” IEEE Photonics Technol. Lett. 28(18), 2000–2003 (2016).

IEEE Sens. J. (1)

A. Zhang and S. Zhang, “High Stability Fiber-Optics Sensors With an Improved PGC Demodulation Algorithm,” IEEE Sens. J. 16(21), 7681–7684 (2016).

IEEE Trans. Geosci. Remote Sens. (1)

H. Liu, M. Xing, and Z. Bao, “A Cluster-Analysis-Based Noise-Robust Phase-Unwrapping Algorithm for Multibaseline Interferograms,” IEEE Trans. Geosci. Remote Sens. 53(1), 494–504 (2015).

IEEE Trans. Image Process. (1)

N. H. Ching, D. Rosenfeld, and M. Braun, “Two-dimensional phase unwrapping using a minimum spanning tree algorithm,” IEEE Trans. Image Process. 1(3), 355–365 (1992).
[PubMed]

IEEE Trans. Microw. Theory Tech. (1)

A. Dandridge, A. B. Tveten, and T. G. Giallorenzi, “Homodyne Demodulation Scheme for Fiber Optic Sensors Using Phase Generated Carrier,” IEEE Trans. Microw. Theory Tech. 30(10), 1635–1641 (1982).

J. Build. Acoust. (2)

L. S. Christensen and D. Manvell, “Sound level meters in building acoustic measurements,” J. Build. Acoust. 5(3), 217–222 (1998).

F. Bettarello, P. Fausti, V. Baccan, and M. Caniato, “Impact sound pressure level performances of basic beam floor structures,” J. Build. Acoust. 17(4), 305–316 (2010).

J. Lightwave Technol. (6)

Jpn. J. Appl. Phys. (1)

J. Park and H. F. Taylor, “Fiber Optic Intrusion Sensor using Coherent Optical Time Domain Reflectometer,” Jpn. J. Appl. Phys. 42(1), 3481–3482 (2003).

Laser Phys. (1)

A. E. Alekseev, V. S. Vdovenko, B. G. Gorshkov, V. T. Potapov, and D. E. Simikin, “A phase-sensitive optical time-domain reflectometer with dual-pulse phase modulated probe signal,” Laser Phys. 24(11), 115106 (2014).

Meas. Sci. Technol. (1)

A. Masoudi, M. Belal, and T. P. Newson, “A distributed optical fibre dynamic strain sensor based on phase OTDR,” Meas. Sci. Technol. 24(8), 085204 (2013).

Opt. Commun. (1)

C. McGarrity and D. A. Jackson, “Improvement on phase generated carrier technique for passive demodulation of miniature interferometric sensors,” Opt. Commun. 109(3–4), 246–248 (1994).

Opt. Express (7)

J. Weng and Y. Lo, “Novel rotation algorithm for phase unwrapping applications,” Opt. Express 20(16), 16838–16860 (2012).

Z. Cheng, D. Liu, Y. Yang, T. Ling, X. Chen, L. Zhang, J. Bai, Y. Shen, L. Miao, and W. Huang, “Practical phase unwrapping of interferometric fringes based on unscented Kalman filter technique,” Opt. Express 23(25), 32337–32349 (2015).
[PubMed]

G. Dardikman, S. Mirsky, M. Habaza, Y. Roichman, and N. T. Shaked, “Angular phase unwrapping of optically thick objects with a thin dimension,” Opt. Express 25(4), 3347–3357 (2017).
[PubMed]

J. Pastor-Graells, H. F. Martins, A. Garcia-Ruiz, S. Martin-Lopez, and M. Gonzalez-Herraez, “Single-shot distributed temperature and strain tracking using direct detection phase-sensitive OTDR with chirped pulses,” Opt. Express 24(12), 13121–13133 (2016).
[PubMed]

A. Garcia-Ruiz, J. Pastor-Graells, H. F. Martins, K. Hey Tow, L. Thévenaz, S. Martin-Lopez, and M. Gonzalez-Herraez, “Distributed photothermal spectroscopy in microstructured optical fibers: towards high-resolution mapping of gas presence over long distances,” Opt. Express 25(3), 1789–1805 (2017).

S. Liehr, Y. S. Muanenda, S. Münzenberger, and K. Krebber, “Relative change measurement of physical quantities using dual-wavelength coherent OTDR,” Opt. Express 25(2), 720–729 (2017).
[PubMed]

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).
[PubMed]

Opt. Lett. (3)

Photonic Sensors (1)

M. Ren, P. Lu, L. Chen, and X. Bao, “Study of Ф-OTDR stability for dynamic strain measurement in piezoelectric vibration,” Photonic Sensors 6(3), 199–208 (2016).

Other (9)

J. Torres and H. H. Asada, “Harmonic analysis of a PZT poly-actuator,” in Proceedings of IEEE International Conference on Robotics and Automation (IEEE, 2015), pp. 842–849.

D. Okamoto, Y. Suzuki, K. Yashiki, Y. Hagihara, M. Tokushima, J. Fujikata, M. Kurihara, J. Tsuchida, T. Nedachi, J. Inasaka, and K. Kurata, “25-Gbps 5×5 mm chip-scale silicon-photonic receiver integrated with 28-nm CMOS transimpedance amplifier,” in Proceedings of 2015 IEEE Optical Interconnects Conference (IEEE, 2015), pp. 56–57.

Y. Liu, J. Jia, P. Tao, G. Yin, Z. Tan, W. Ren, and S. Jian, “Signal processing of Sagnac fiber interferometer used as distributed sensor with wavelets,” in Proceedings of Asia Communications and Photonics Conference and Exhibition (IEEE, 2010), pp. 290–291.

D. Kitahara, M. Yamagishi, and I. Yamada, “A virtual resampling technique for algebraic two-dimensional phase unwrapping,” in Proceedings of 2015 IEEE International Conference on Acoustics, Speech and Signal Processing (IEEE, 2015), pp. 3871–3875.

D. M. Scott, Industrial Process Sensors (CRC Press, 2008).

M. G. Guvench, M. Miske, and E. Crain, “Design, fabrication and testing of CMOS operational amplifiers as training tool in analog integrated circuit design,” in Proceedings of the Fourteenth Biennial University/ Government/ Industry Microelectronics Symposium (IEEE, 2001), pp. 193–196.

A. C. B. Albason, N. M. L. Axalan, M. T. A. Gusad, J. R. E. Hizon, and M. D. Rosales, “Design Methodologies for Low-Power CMOS Operational Amplifiers in a 0.25μm Digital CMOS Process,” in Proceedings of TENCON 2006 - 2006 IEEE Region 10 Conference (IEEE, 2006), pp. 1–4.

K. Johannessen and B. K. Drakeley, “Distributed Acoustic Sensing: a new way of listening to your well/reservoir,” SPE Intelligent Energy International, SPE-149602-MS (2012).

Persistence Market Research forecast, “Global Distributed Acoustic Sensing Market to Surpass US$ 2 Billion in Revenues by 2025,” (Persistence Market Research, 2017) http://www.persistencemarketresearch.com/mediarelease/distributed-acoustic-sensing-market.asp .

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Figures (10)

Fig. 1
Fig. 1 Schematic of a double pulse ϕ-OTDR showing the backscattering from two adjacent points.
Fig. 2
Fig. 2 Initial mixing in PGC demodulation to obtain intermediate signals to the I and Q components (LPF: Low-pass Filter).
Fig. 3
Fig. 3 Phase demodulation in the ϕ-OTDR sensor using PGC-DMS.
Fig. 4
Fig. 4 Experimental setup of the proposed ϕ-OTDR sensor.
Fig. 5
Fig. 5 Individual ϕ-OTDR traces for single and double pulse both with and without selective phase demodulation of one pulse.
Fig. 6
Fig. 6 Spectra of acquired signal with no perturbation at an arbitrary locations showing corresponding carriers resulting from selective phase modulation of one of the pulse pairs with 5, 8 and 10 kHz signals.
Fig. 7
Fig. 7 Modulated signals showing raw perturbation signals when 1 and 2 kHz vibrations applied to the PZT, before performing the mixing, filtering and PGC-DMS demodulation operations.
Fig. 8
Fig. 8 (a) Demodulated phase change for 2 kHz vibration applied to the PZT and (b) The spectrum of the demodulated phase change with PGC-DMS showing higher order harmonics of the response.
Fig. 9
Fig. 9 Spectra of demodulated phase using dual pulse probe with PGC-DMS demodulation.
Fig. 10
Fig. 10 Comparison of the response of a nonlinear PZT actuator using proposed PGC-DMS scheme for DAS with that of a point sensor and a commercial reading unit.

Equations (15)

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E m (t)= E m exp[j δ m (t)+j φ m ], E k (t)= E k exp[jϕ(t)+j δ k (t)+j φ k ],
I= E m 2 + E k 2 +2 E m E k cos(Δδ(t)+ϕ(t)),
I=A+ηBcos( Ccos ω o t+ϕ(t) ).
I=A+ηB{ cos( Ccos ω 0 t )cosϕ(t) sin( Ccos ω 0 t )sinϕ(t) }.
cos( Ccos(x) )= J 0 (C)+2 n=1 (1) n J 2n (C)cos(2nx ),
sin( Ccos(x) )=2 n=0 (1) n J 2n+1 (C)cos((2n+1)x ),
I=A+ηB[ Φ 1 + Φ 2 ].
Φ 1 (t)=[ J 0 (C)+2 n=1 (1) n J 2n (C)cos(2n ω o t )]cosϕ(t).
Φ 2 (t)=[2 n=0 (1) n J 2n+1 (C)cos((2n+1) ω o t )]sinϕ(t).
s 1 (t)=ηBG J 1 (C)sinϕ(t). s 2 (t)=ηBH J 2 (C)cosϕ(t).
s DCM (t) . =[ B 2 η 2 GH J 2 (C) J 1 (C)]× [ sin 2 ϕ(t)+ cos 2 ϕ(t)] ϕ(t) . = B 2 η 2 GH J 2 (C) J 1 (C) ϕ(t). .
s DCM (t)= B 2 η 2 GH J 2 (C) J 1 (C)ϕ(t).
s D1 (t) s 2 (t)= ( ηB ) 2 GH J 1 (C) J 2 (C) cos 2 ϕ(t) d dt ϕ(t), s SQ2 (t)= ( ηBH ) 2 J 2 2 (C) cos 2 ϕ(t).
s D1 (t) s 2 (t) s SQ2 (t) = J 1 (C) J 2 (C) d dt ϕ(t).
S DMS (t)= J 1 (C) J 2 (C) d dt ϕ(t) dt = J 1 (C) J 2 (C) ϕ(t)

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