Abstract

A novel fiber-optic refractometer is proposed and demonstrated to achieve temperature- and axial strain-compensated refractive index measurement using highly sensitive outer-cladding modes in a tapered bend-insensitive fiber based Mach-Zehnder interferometer. Peak wavelength shifts associated with different spatial frequency peaks are calibrated to obtain a wavelength-related character matrix λMRI,T,ε for simultaneous measurement of multiple environmental variables. A phase-related character matrix ΦMRI,T,ε is also acquired by direct determination of refractive index, temperature, and axial strain induced phase shifts of the corresponding sensing modes.

© 2013 OSA

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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]

2012 (6)

2011 (3)

2010 (4)

2009 (4)

2008 (3)

C. Caucheteur, M. Wuilpart, C. Chen, P. Mégret, and J. Albert, “Quasi-distributed refractometer using tilted Bragg gratings and time domain reflectometry,” Opt. Express 16(22), 17882–17890 (2008).
[Crossref] [PubMed]

L. Men, P. Lu, and Q. Chen, “Intelligent multiparameter sensing with fiber Bragg gratings,” Appl. Phys. Lett. 93(7), 071110 (2008).
[Crossref]

T. Mawatari and D. Nelson, “A multi-parameter Bragg grating fiber optic sensor and triaxial strain measurement,” Smart Mater. Struct. 17(3), 035033 (2008).
[Crossref]

2007 (2)

C. Zhan, Y. Zhu, S. Yin, and P. Ruffin, “Multi-parameter harsh environment sensing using asymmetric Bragg gratings inscribed by IR femtosecond irradiation,” Opt. Fiber Technol. 13(2), 98–107 (2007).
[Crossref]

M. C. Phan Huy, G. Laffont, V. Dewynter, P. Ferdinand, P. Roy, J.-L. Auguste, D. Pagnoux, W. Blanc, and B. Dussardier, “Three-hole microstructured optical fiber for efficient fiber Bragg grating refractometer,” Opt. Lett. 32(16), 2390–2392 (2007).
[Crossref] [PubMed]

2006 (2)

2005 (2)

P. Polynkin, A. Polynkin, N. Peyghambarian, and M. Mansuripur, “Evanescent field-based optical fiber sensing device for measuring the refractive index of liquids in microfluidic channels,” Opt. Lett. 30(11), 1273–1275 (2005).
[Crossref] [PubMed]

O. Frazão, L. A. Ferreira, and F. M. Araújo, “Applications of fiber optic grating technology to multi-parameter measurement,” Fiber Integr. Opt. 24(3–4), 227–244 (2005).

2003 (1)

2002 (2)

S. Singh, “Refractive index measurement and its applications,” Phys. Scr. 65(2), 167–180 (2002).
[Crossref]

H.-K. Kang, H.-J. Bang, C.-S. Hong, and C.-G. Kim, “Simultaneous measurement of strain, temperature and vibration frequency using a fiber-optic sensor,” Meas. Sci. Technol. 13(8), 1191–1196 (2002).
[Crossref]

1997 (2)

W. Jin, W. C. Michie, G. Thursby, M. Konstantaki, and B. Culshaw, “Simultaneous measurement of strain and temperature: error analysis,” Opt. Eng. 36(2), 598–609 (1997).
[Crossref]

Y. J. Rao, P. J. Henderson, D. A. Jackson, L. Zhang, and I. Bennion, “Simultaneous strain, temperature and vibration measurement using a multiplexed in-fibre-Bragg-grating/fibre-Fabry-Perot sensor system,” Electron. Lett. 33(24), 2063–2064 (1997).
[Crossref]

Albert, J.

Alberto, N. J.

Araújo, F. M.

O. Frazão, L. A. Ferreira, and F. M. Araújo, “Applications of fiber optic grating technology to multi-parameter measurement,” Fiber Integr. Opt. 24(3–4), 227–244 (2005).

Auguste, J.-L.

Baldini, F.

C. Trono, F. Baldini, M. Brenci, F. Chiavaioli, and M. Mugnaini, “Flow cell for strain- and temperature-compensated refractive index measurements by means of cascaded optical fibre long period and Bragg gratings,” Meas. Sci. Technol. 22(7), 075204 (2011).
[Crossref]

Bang, H.-J.

H.-K. Kang, H.-J. Bang, C.-S. Hong, and C.-G. Kim, “Simultaneous measurement of strain, temperature and vibration frequency using a fiber-optic sensor,” Meas. Sci. Technol. 13(8), 1191–1196 (2002).
[Crossref]

Bao, X.

Baptista, J. M.

Bennion, I.

K. Zhou, Z. Yan, L. Zhang, and I. Bennion, “Refractometer based on fiber Bragg grating Fabry-Pérot cavity embedded with a narrow microchannel,” Opt. Express 19(12), 11769–11779 (2011).
[Crossref] [PubMed]

Y. J. Rao, P. J. Henderson, D. A. Jackson, L. Zhang, and I. Bennion, “Simultaneous strain, temperature and vibration measurement using a multiplexed in-fibre-Bragg-grating/fibre-Fabry-Perot sensor system,” Electron. Lett. 33(24), 2063–2064 (1997).
[Crossref]

Bickham, S. R.

Blanc, W.

Bookbinder, D. C.

Brenci, M.

C. Trono, F. Baldini, M. Brenci, F. Chiavaioli, and M. Mugnaini, “Flow cell for strain- and temperature-compensated refractive index measurements by means of cascaded optical fibre long period and Bragg gratings,” Meas. Sci. Technol. 22(7), 075204 (2011).
[Crossref]

Caucheteur, C.

Chen, C.

Chen, L.

Chen, Q.

P. Lu, L. Men, K. Sooley, and Q. Chen, “Tapered fiber Mach-Zehnder interferometer for simultaneous measurement of refractive index and temperature,” Appl. Phys. Lett. 94(13), 131110 (2009).
[Crossref]

L. Men, P. Lu, and Q. Chen, “Intelligent multiparameter sensing with fiber Bragg gratings,” Appl. Phys. Lett. 93(7), 071110 (2008).
[Crossref]

Chen, Q.-D.

Chiavaioli, F.

C. Trono, F. Baldini, M. Brenci, F. Chiavaioli, and M. Mugnaini, “Flow cell for strain- and temperature-compensated refractive index measurements by means of cascaded optical fibre long period and Bragg gratings,” Meas. Sci. Technol. 22(7), 075204 (2011).
[Crossref]

Cui, Y.

Culshaw, B.

W. Jin, W. C. Michie, G. Thursby, M. Konstantaki, and B. Culshaw, “Simultaneous measurement of strain and temperature: error analysis,” Opt. Eng. 36(2), 598–609 (1997).
[Crossref]

da Cruz, A. R.

D. A. C. Enríquez, A. R. da Cruz, and M. T. M. R. Giraldi, “Hybrid FBG-LPG sensor for surrounding refractive index and temperature simultaneous discrimination,” Opt. Laser Technol. 44(4), 981–986 (2012).
[Crossref]

Desorcie, R. B.

Dewynter, V.

Dussardier, B.

Englebert, J. J.

Enríquez, D. A. C.

D. A. C. Enríquez, A. R. da Cruz, and M. T. M. R. Giraldi, “Hybrid FBG-LPG sensor for surrounding refractive index and temperature simultaneous discrimination,” Opt. Laser Technol. 44(4), 981–986 (2012).
[Crossref]

Farrell, G.

Ferdinand, P.

Ferreira, L. A.

O. Frazão, L. A. Ferreira, and F. M. Araújo, “Applications of fiber optic grating technology to multi-parameter measurement,” Fiber Integr. Opt. 24(3–4), 227–244 (2005).

Frazão, O.

Giraldi, M. T. M. R.

D. A. C. Enríquez, A. R. da Cruz, and M. T. M. R. Giraldi, “Hybrid FBG-LPG sensor for surrounding refractive index and temperature simultaneous discrimination,” Opt. Laser Technol. 44(4), 981–986 (2012).
[Crossref]

Guo, F. W.

Guo, J.-C.

Guo, T.

Han, M.

Han, Y.-G.

Harris, J.

Henderson, P. J.

Y. J. Rao, P. J. Henderson, D. A. Jackson, L. Zhang, and I. Bennion, “Simultaneous strain, temperature and vibration measurement using a multiplexed in-fibre-Bragg-grating/fibre-Fabry-Perot sensor system,” Electron. Lett. 33(24), 2063–2064 (1997).
[Crossref]

Hong, C.-S.

H.-K. Kang, H.-J. Bang, C.-S. Hong, and C.-G. Kim, “Simultaneous measurement of strain, temperature and vibration frequency using a fiber-optic sensor,” Meas. Sci. Technol. 13(8), 1191–1196 (2002).
[Crossref]

Hu, D. J. J.

Jackson, D. A.

Y. J. Rao, P. J. Henderson, D. A. Jackson, L. Zhang, and I. Bennion, “Simultaneous strain, temperature and vibration measurement using a multiplexed in-fibre-Bragg-grating/fibre-Fabry-Perot sensor system,” Electron. Lett. 33(24), 2063–2064 (1997).
[Crossref]

Jeong, M.-Y.

S.-M. Lee, S. S. Saini, and M.-Y. Jeong, “Simultaneous Measurement of refractive index, temperature, and strain using etched-core fiber Bragg grating sensors,” IEEE Photon. Technol. Lett. 22(19), 1431–1433 (2010).
[Crossref]

Jiang, M.

Jin, W.

W. Jin, W. C. Michie, G. Thursby, M. Konstantaki, and B. Culshaw, “Simultaneous measurement of strain and temperature: error analysis,” Opt. Eng. 36(2), 598–609 (1997).
[Crossref]

Johnson, J. J.

Kang, H.-K.

H.-K. Kang, H.-J. Bang, C.-S. Hong, and C.-G. Kim, “Simultaneous measurement of strain, temperature and vibration frequency using a fiber-optic sensor,” Meas. Sci. Technol. 13(8), 1191–1196 (2002).
[Crossref]

Kim, C.-G.

H.-K. Kang, H.-J. Bang, C.-S. Hong, and C.-G. Kim, “Simultaneous measurement of strain, temperature and vibration frequency using a fiber-optic sensor,” Meas. Sci. Technol. 13(8), 1191–1196 (2002).
[Crossref]

Kim, H.-J.

Kobelke, J.

Konstantaki, M.

W. Jin, W. C. Michie, G. Thursby, M. Konstantaki, and B. Culshaw, “Simultaneous measurement of strain and temperature: error analysis,” Opt. Eng. 36(2), 598–609 (1997).
[Crossref]

Krug, P. A.

Kwon, O.-J.

Laffont, G.

Lee, J.

Lee, S. B.

Lee, S.-M.

S.-M. Lee, S. S. Saini, and M.-Y. Jeong, “Simultaneous Measurement of refractive index, temperature, and strain using etched-core fiber Bragg grating sensors,” IEEE Photon. Technol. Lett. 22(19), 1431–1433 (2010).
[Crossref]

Lewis, K. A.

Li, M.-J.

Liang, R.

Lim, J. L.

Liu, D.

Lu, P.

P. Lu, J. Harris, Y. Xu, Y. Lu, L. Chen, and X. Bao, “Simultaneous refractive index and temperature measurements using a tapered bend-resistant fiber interferometer,” Opt. Lett. 37(22), 4567–4569 (2012).
[Crossref] [PubMed]

P. Lu, L. Men, K. Sooley, and Q. Chen, “Tapered fiber Mach-Zehnder interferometer for simultaneous measurement of refractive index and temperature,” Appl. Phys. Lett. 94(13), 131110 (2009).
[Crossref]

L. Men, P. Lu, and Q. Chen, “Intelligent multiparameter sensing with fiber Bragg gratings,” Appl. Phys. Lett. 93(7), 071110 (2008).
[Crossref]

Lu, Y.

Lu, Y. F.

Luan, F.

Mansuripur, M.

Marques, C. A.

Martynkien, T.

Mawatari, T.

T. Mawatari and D. Nelson, “A multi-parameter Bragg grating fiber optic sensor and triaxial strain measurement,” Smart Mater. Struct. 17(3), 035033 (2008).
[Crossref]

McDermott, M. A.

Mégret, P.

Men, L.

P. Lu, L. Men, K. Sooley, and Q. Chen, “Tapered fiber Mach-Zehnder interferometer for simultaneous measurement of refractive index and temperature,” Appl. Phys. Lett. 94(13), 131110 (2009).
[Crossref]

L. Men, P. Lu, and Q. Chen, “Intelligent multiparameter sensing with fiber Bragg gratings,” Appl. Phys. Lett. 93(7), 071110 (2008).
[Crossref]

Michie, W. C.

W. Jin, W. C. Michie, G. Thursby, M. Konstantaki, and B. Culshaw, “Simultaneous measurement of strain and temperature: error analysis,” Opt. Eng. 36(2), 598–609 (1997).
[Crossref]

Mugnaini, M.

C. Trono, F. Baldini, M. Brenci, F. Chiavaioli, and M. Mugnaini, “Flow cell for strain- and temperature-compensated refractive index measurements by means of cascaded optical fibre long period and Bragg gratings,” Meas. Sci. Technol. 22(7), 075204 (2011).
[Crossref]

Nelson, D.

T. Mawatari and D. Nelson, “A multi-parameter Bragg grating fiber optic sensor and triaxial strain measurement,” Smart Mater. Struct. 17(3), 035033 (2008).
[Crossref]

Nogueira, R. N.

Nolan, D. A.

Pagnoux, D.

Peyghambarian, N.

Phan Huy, M. C.

Pinto, J. L.

Polynkin, A.

Polynkin, P.

Ran, Z.-L.

Rao, Y. J.

Y. J. Rao, P. J. Henderson, D. A. Jackson, L. Zhang, and I. Bennion, “Simultaneous strain, temperature and vibration measurement using a multiplexed in-fibre-Bragg-grating/fibre-Fabry-Perot sensor system,” Electron. Lett. 33(24), 2063–2064 (1997).
[Crossref]

Rao, Y.-J.

Roy, P.

Ruffin, P.

C. Zhan, Y. Zhu, S. Yin, and P. Ruffin, “Multi-parameter harsh environment sensing using asymmetric Bragg gratings inscribed by IR femtosecond irradiation,” Opt. Fiber Technol. 13(2), 98–107 (2007).
[Crossref]

Z. Yong, C. Zhan, J. Lee, S. Yin, and P. Ruffin, “Multiple parameter vector bending and high-temperature sensors based on asymmetric multimode fiber Bragg gratings inscribed by an infrared femtosecond laser,” Opt. Lett. 31(12), 1794–1796 (2006).
[Crossref] [PubMed]

Saini, S. S.

S.-M. Lee, S. S. Saini, and M.-Y. Jeong, “Simultaneous Measurement of refractive index, temperature, and strain using etched-core fiber Bragg grating sensors,” IEEE Photon. Technol. Lett. 22(19), 1431–1433 (2010).
[Crossref]

Santos, J. L.

Schuster, K.

Semenova, Y.

Shum, P. P.

Silva, R. M.

Singh, S.

S. Singh, “Refractive index measurement and its applications,” Phys. Scr. 65(2), 167–180 (2002).
[Crossref]

Sooley, K.

P. Lu, L. Men, K. Sooley, and Q. Chen, “Tapered fiber Mach-Zehnder interferometer for simultaneous measurement of refractive index and temperature,” Appl. Phys. Lett. 94(13), 131110 (2009).
[Crossref]

Sun, H.-B.

Sun, Q.

Tam, H.-Y.

Tandon, P.

Thursby, G.

W. Jin, W. C. Michie, G. Thursby, M. Konstantaki, and B. Culshaw, “Simultaneous measurement of strain and temperature: error analysis,” Opt. Eng. 36(2), 598–609 (1997).
[Crossref]

Tong, W.

Trono, C.

C. Trono, F. Baldini, M. Brenci, F. Chiavaioli, and M. Mugnaini, “Flow cell for strain- and temperature-compensated refractive index measurements by means of cascaded optical fibre long period and Bragg gratings,” Meas. Sci. Technol. 22(7), 075204 (2011).
[Crossref]

Urbanczyk, W.

Wang, C.

Wang, G.

Wang, P.

Wang, Q.

Wang, Y.

Wang, Y.-P.

Wei, H.

Wo, J.

Wojcik, J.

Wu, Q.

Wuilpart, M.

Xu, Y.

Xue, Y.

Yan, Z.

Yang, R.

Yin, S.

C. Zhan, Y. Zhu, S. Yin, and P. Ruffin, “Multi-parameter harsh environment sensing using asymmetric Bragg gratings inscribed by IR femtosecond irradiation,” Opt. Fiber Technol. 13(2), 98–107 (2007).
[Crossref]

Z. Yong, C. Zhan, J. Lee, S. Yin, and P. Ruffin, “Multiple parameter vector bending and high-temperature sensors based on asymmetric multimode fiber Bragg gratings inscribed by an infrared femtosecond laser,” Opt. Lett. 31(12), 1794–1796 (2006).
[Crossref] [PubMed]

Yong, Z.

Yu, Y.-S.

Zhan, C.

C. Zhan, Y. Zhu, S. Yin, and P. Ruffin, “Multi-parameter harsh environment sensing using asymmetric Bragg gratings inscribed by IR femtosecond irradiation,” Opt. Fiber Technol. 13(2), 98–107 (2007).
[Crossref]

Z. Yong, C. Zhan, J. Lee, S. Yin, and P. Ruffin, “Multiple parameter vector bending and high-temperature sensors based on asymmetric multimode fiber Bragg gratings inscribed by an infrared femtosecond laser,” Opt. Lett. 31(12), 1794–1796 (2006).
[Crossref] [PubMed]

Zhang, B.-L.

Zhang, L.

K. Zhou, Z. Yan, L. Zhang, and I. Bennion, “Refractometer based on fiber Bragg grating Fabry-Pérot cavity embedded with a narrow microchannel,” Opt. Express 19(12), 11769–11779 (2011).
[Crossref] [PubMed]

Y. J. Rao, P. J. Henderson, D. A. Jackson, L. Zhang, and I. Bennion, “Simultaneous strain, temperature and vibration measurement using a multiplexed in-fibre-Bragg-grating/fibre-Fabry-Perot sensor system,” Electron. Lett. 33(24), 2063–2064 (1997).
[Crossref]

Zhang, X.-L.

Zhou, K.

Zhu, F.

Zhu, T.

Zhu, Y.

C. Zhan, Y. Zhu, S. Yin, and P. Ruffin, “Multi-parameter harsh environment sensing using asymmetric Bragg gratings inscribed by IR femtosecond irradiation,” Opt. Fiber Technol. 13(2), 98–107 (2007).
[Crossref]

Appl. Opt. (1)

Appl. Phys. Lett. (2)

P. Lu, L. Men, K. Sooley, and Q. Chen, “Tapered fiber Mach-Zehnder interferometer for simultaneous measurement of refractive index and temperature,” Appl. Phys. Lett. 94(13), 131110 (2009).
[Crossref]

L. Men, P. Lu, and Q. Chen, “Intelligent multiparameter sensing with fiber Bragg gratings,” Appl. Phys. Lett. 93(7), 071110 (2008).
[Crossref]

Electron. Lett. (1)

Y. J. Rao, P. J. Henderson, D. A. Jackson, L. Zhang, and I. Bennion, “Simultaneous strain, temperature and vibration measurement using a multiplexed in-fibre-Bragg-grating/fibre-Fabry-Perot sensor system,” Electron. Lett. 33(24), 2063–2064 (1997).
[Crossref]

Fiber Integr. Opt. (1)

O. Frazão, L. A. Ferreira, and F. M. Araújo, “Applications of fiber optic grating technology to multi-parameter measurement,” Fiber Integr. Opt. 24(3–4), 227–244 (2005).

IEEE Photon. Technol. Lett. (1)

S.-M. Lee, S. S. Saini, and M.-Y. Jeong, “Simultaneous Measurement of refractive index, temperature, and strain using etched-core fiber Bragg grating sensors,” IEEE Photon. Technol. Lett. 22(19), 1431–1433 (2010).
[Crossref]

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Meas. Sci. Technol. (2)

C. Trono, F. Baldini, M. Brenci, F. Chiavaioli, and M. Mugnaini, “Flow cell for strain- and temperature-compensated refractive index measurements by means of cascaded optical fibre long period and Bragg gratings,” Meas. Sci. Technol. 22(7), 075204 (2011).
[Crossref]

H.-K. Kang, H.-J. Bang, C.-S. Hong, and C.-G. Kim, “Simultaneous measurement of strain, temperature and vibration frequency using a fiber-optic sensor,” Meas. Sci. Technol. 13(8), 1191–1196 (2002).
[Crossref]

Opt. Eng. (1)

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[Crossref]

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Opt. Fiber Technol. (1)

C. Zhan, Y. Zhu, S. Yin, and P. Ruffin, “Multi-parameter harsh environment sensing using asymmetric Bragg gratings inscribed by IR femtosecond irradiation,” Opt. Fiber Technol. 13(2), 98–107 (2007).
[Crossref]

Opt. Laser Technol. (1)

D. A. C. Enríquez, A. R. da Cruz, and M. T. M. R. Giraldi, “Hybrid FBG-LPG sensor for surrounding refractive index and temperature simultaneous discrimination,” Opt. Laser Technol. 44(4), 981–986 (2012).
[Crossref]

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P. Polynkin, A. Polynkin, N. Peyghambarian, and M. Mansuripur, “Evanescent field-based optical fiber sensing device for measuring the refractive index of liquids in microfluidic channels,” Opt. Lett. 30(11), 1273–1275 (2005).
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[Crossref] [PubMed]

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T. Mawatari and D. Nelson, “A multi-parameter Bragg grating fiber optic sensor and triaxial strain measurement,” Smart Mater. Struct. 17(3), 035033 (2008).
[Crossref]

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Figures (9)

Fig. 1
Fig. 1

(a) A SEM cross-sectional image of the bend-insensitive fiber etched with 5% hydrofluoric acid solution for 2 minutes; (b) A schematic illustration of the tapered bend-insensitive fiber based Mach-Zehnder interferometer under various external disturbances with an inset of the abrupt taper picture.

Fig. 2
Fig. 2

Flow chart of the demodulation algorithm for an interference spectrum between the fundamental mode and multiple cladding modes.

Fig. 3
Fig. 3

(a) Transmission spectrum of the BIF-MZI suspended in air at 24.6 °C with zero-strain; (b) FFT spectrum of the BIF-MZI with an inset of the simulated mode field patterns of the fundamental mode, IC mode, OC-1 mode, the OC-2 mode, the OC-3 mode, and OC-4 mode respectively.

Fig. 4
Fig. 4

(a) Filtered transmission spectra of the BIF-MZI corresponding to isolated spatial frequency components; (b) Spatial frequency dependent phase spectrum of the BIF-MZI.

Fig. 5
Fig. 5

Transmission spectra of the BIF-MZI submerged in solutions of 10% and 25% glycerol at 20.0 °C with zero-strain.

Fig. 6
Fig. 6

Filtered transmission spectra of the BIF-MZI in solutions of 10% and 25% glycerol corresponding to different groups of outer-cladding modes of (a) the OC-1 mode, (b) the OC-2 mode, and (c) the OC-3 mode.

Fig. 7
Fig. 7

Spatial frequency dependent phase spectra of the BIF-MZI in solutions of 10% and 25% glycerol.

Fig. 8
Fig. 8

Peak wavelength shift calibration curves for (a) refractive index, (b) temperature, and (c) axial strain, for the OC-1, OC-2, and OC-3 modes.

Fig. 9
Fig. 9

Phase shift calibration curves for (a) refractive index, (b) temperature, and (c) axial strain, for the OC-1, OC-2, and OC-3 modes.

Equations (18)

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Δ Φ ij = Φ 01 Φ ij = 2πL λ ( n eff 01 n eff ij )= 2πΔ n eff ij L λ ,
λ m ij = Δ n eff ij L m .
I= I 01 + ij I ij +2 ij I 01 I ij cos( 2πΔ n eff ij L λ ) .
Δ λ ij = λ m1 ij λ m ij λ 0 2 Δ n eff ij L ,
ξ ij = 1 Δ λ ij Δ n eff ij L λ 0 2 .
δλ=δ( Δ n eff L m )=( δ( Δ n eff ) Δ n eff )λ,
δΦ=δ( 2π λ 0 2 ξ λ )=2πξδλ.
δλ=( δ( Δ n eff,RI ) Δ n eff )λ,
δΦ= 2πLδ( Δ n eff,RI ) λ .
δλ=( δ( Δ n eff,T ) Δ n eff )λ,
δΦ= 2πLδ( Δ n eff,T ) λ .
δλ=( δL L + δ( Δ n eff,ε ) Δ n eff,ε )λ,
δΦ= 2π( Δ n eff,ε δL+Lδ( Δ n eff,ε ) ) λ .
( Δ λ 1 Δ λ 2 Δ λ 3 )= M λ RI,T,ε ( ΔRI ΔT Δε )=( C λ RI 1 C λ T 1 C λ ε 1 C λ RI 2 C λ T 2 C λ ε 2 C λ RI 3 C λ T 3 C λ ε 3 )( ΔRI ΔT Δε ),
( ΔRI ΔT Δε )= M λ RI,T,ε 1 ( Δ λ 1 Δ λ 2 Δ λ 3 ),
( Δ Φ 1 Δ Φ 2 Δ Φ 3 )= M Φ RI,T,ε ( ΔRI ΔT Δε )=( C Φ RI 1 C Φ T 1 C Φ ε 1 C Φ RI 2 C Φ T 2 C Φ ε 2 C Φ RI 3 C Φ T 3 C Φ ε 3 )( ΔRI ΔT Δε ),
( ΔRI ΔT Δε )= M Φ RI,T,ε 1 ( Δ Φ 1 Δ Φ 2 Δ Φ 3 ),
( δRI δT δε )= M Φ ' RI,T,ε Δ ( δ Φ 1 δ Φ 2 δ Φ 3 ),

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