Abstract

In this paper we propose a novel kind of multi-point vibration sensor based on the polarization properties of light. Its principle relies on the combination of mechanical transducers with fiber Bragg gratings. When subject to vibrations, the mechanical transducers induce birefringence variations within the fiber and in turn modify the state of polarization, which appears as a power variation after going through a polarizer. The FBGs reflect light from different positions of the sensing fiber and provide wavelength multiplexing. We show that this sensor can provide the vibration frequencies in a quasi-distributed manner.

© 2013 OSA

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2012

G. Rajan, M. Ramakrishnan, Y. Semenova, A. Domanski, A. Boczkowska, T. Wolinski, and G. Farrell, “Analysis of vibration measurements in a composite material using an embedded PM-PCF polarimetric sensor and an FBG sensor,” IEEE Sens. J.12, 1365–1371 (2012).
[CrossRef]

N. Linze, P. Tihon, O. Verlinden, P. Mégret, and M. Wuilpart, “Linearity considerations in polarization-based vibration sensors,” Appl. Opt.51, 6997–7004 (2012).
[CrossRef] [PubMed]

2011

2010

2009

R. Bernini, A. Minardo, and L. Zini, “Dynamic strain measurement in optical fibers by stimulated Brillouin scattering,” Opt. Lett.34, 2613–2615 (2009).
[CrossRef] [PubMed]

T. Guo, L. Shao, H-Y. Tam, P. A. Krug, and J. Albert, “Tilted fiber grating accelerometer incorporating an abrupt biconical taper for cladding to core recoupling,” Opt. Express17, 20651–20660 (2009).
[CrossRef] [PubMed]

F. Qin, H. Li, W. Fan, and Q. Sheng, “Experimental study on vibration frequency response of micro-bend optic-fiber sensor,” Chinese Opt. Lett.7, 556–559 (2009).
[CrossRef]

A. Galtarossa, D. Grosso, and L. Palmieri, “Accurate characterization of twist-induced optical activity in single-mode fibers by means of polarization-sensitive reflectometry,” IEEE Photon. Technol. Lett.21, 1713–1715 (2009).
[CrossRef]

2008

Z. Zhang and X. Bao, “Continuous and damped vibraton detection based on fiber diversity detection sensor by Rayleigh backscattering,” J. Lightwave Technol.26, 832–838 (2008).
[CrossRef]

Z. Zhang and X. Bao, “Distributed optical fiber vibration sensor based on spectrum analysis of polarization-OTDR system,” Opt. Express16, 10240–10247 (2008).
[CrossRef] [PubMed]

R. M. Manuel, M. G. Shlyagin, and S. V. Miridonov, “Location of a time-varying disturbance using an array of identical fiber-optic interferometers interrogated by CW DFB laser,” Opt. Express16, 20666–20675 (2008).
[CrossRef] [PubMed]

P. Chaube, B. G. Colpitts, D. Jagannathan, and A. W. Brown, “Distributed fiber-optic sensor for dynamic strain measurement,” IEEE Sens. J.8, 1067–1072 (2008).
[CrossRef]

A. Fender, W. N. Macpherson, R. R. J. Maier, J. S. Barton, D. S. George, R. I. Howden, G. W Smith, B. J. S. Jones, S. Mcculloch, X. Chen, R. Suo, L. Zhang, and I. Bennion, “Two-axis temperature-insensitive accelerometer based on multicore fiber Bragg gratings,” IEEE Sens. J.8, 1292–1298 (2008).
[CrossRef]

2007

2005

2004

X. Guo, Z. Yin, and N. Song, “Measuring vibration by using fiber Bragg grating and demodulating it by blazed grating,” Chinese Opt. Lett.2, 393–395 (2004).

2003

Y. Zhu, P. Shum, C. Lu, B. M. Lacquet, P. L. Swart, and S. J. Spammer, “Temperature-insensitive fiber Bragg grating accelerometer,” IEEE Photon. Technol. Lett.15, 1437–1439 (2003).
[CrossRef]

K. Hotate and S. S. L. Ong, “Distributed dynamic strain measurement using a correlation-based Brillouin sensing system,” IEEE Photon. Technol. Lett.15, 272–274 (2003).
[CrossRef]

2001

M. Wuilpart, P. Mégret, M. Blondel, A. J. Rogers, and Y. Defosse, “Measurement of the spatial distribution of birefringence in optical fibers,” IEEE Photon. Technol. Lett.13, 836–838 (2001).
[CrossRef]

1998

M. D. Todd, G. A. Johnson, B. A. Althouse, and S. T. Vohra, “Flexural beam-based fiber Bragg grating accelerometers,” IEEE Photon. Technol. Lett.10, 1605–1607 (1998).
[CrossRef]

1996

1988

A. Bertholds and R. Dandliker, “Determination of the individual strain-optic coefficients in single mode optical fibre,” J. Lightwave Technol.6, 17–20 (1988).
[CrossRef]

1983

S. C. Rashleigh, “Origins and control of polarization effects in single-mode fibers,” J. Lightwave Technol.1, 312–331 (1983).
[CrossRef]

Albert, J.

Althouse, B. A.

M. D. Todd, G. A. Johnson, B. A. Althouse, and S. T. Vohra, “Flexural beam-based fiber Bragg grating accelerometers,” IEEE Photon. Technol. Lett.10, 1605–1607 (1998).
[CrossRef]

Azzam, R. M. A.

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, 1977).

Bao, X.

Barton, J. S.

A. Fender, W. N. Macpherson, R. R. J. Maier, J. S. Barton, D. S. George, R. I. Howden, G. W Smith, B. J. S. Jones, S. Mcculloch, X. Chen, R. Suo, L. Zhang, and I. Bennion, “Two-axis temperature-insensitive accelerometer based on multicore fiber Bragg gratings,” IEEE Sens. J.8, 1292–1298 (2008).
[CrossRef]

Bashara, N. M.

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, 1977).

Bennion, I.

A. Fender, W. N. Macpherson, R. R. J. Maier, J. S. Barton, D. S. George, R. I. Howden, G. W Smith, B. J. S. Jones, S. Mcculloch, X. Chen, R. Suo, L. Zhang, and I. Bennion, “Two-axis temperature-insensitive accelerometer based on multicore fiber Bragg gratings,” IEEE Sens. J.8, 1292–1298 (2008).
[CrossRef]

Bernini, R.

Bertholds, A.

A. Bertholds and R. Dandliker, “Determination of the individual strain-optic coefficients in single mode optical fibre,” J. Lightwave Technol.6, 17–20 (1988).
[CrossRef]

Bette, S.

Blondel, M.

M. Wuilpart, P. Mégret, M. Blondel, A. J. Rogers, and Y. Defosse, “Measurement of the spatial distribution of birefringence in optical fibers,” IEEE Photon. Technol. Lett.13, 836–838 (2001).
[CrossRef]

Boczkowska, A.

G. Rajan, M. Ramakrishnan, Y. Semenova, A. Domanski, A. Boczkowska, T. Wolinski, and G. Farrell, “Analysis of vibration measurements in a composite material using an embedded PM-PCF polarimetric sensor and an FBG sensor,” IEEE Sens. J.12, 1365–1371 (2012).
[CrossRef]

Brown, A. W.

P. Chaube, B. G. Colpitts, D. Jagannathan, and A. W. Brown, “Distributed fiber-optic sensor for dynamic strain measurement,” IEEE Sens. J.8, 1067–1072 (2008).
[CrossRef]

Caucheteur, C.

Chaube, P.

P. Chaube, B. G. Colpitts, D. Jagannathan, and A. W. Brown, “Distributed fiber-optic sensor for dynamic strain measurement,” IEEE Sens. J.8, 1067–1072 (2008).
[CrossRef]

Chen, L.

Chen, X.

A. Fender, W. N. Macpherson, R. R. J. Maier, J. S. Barton, D. S. George, R. I. Howden, G. W Smith, B. J. S. Jones, S. Mcculloch, X. Chen, R. Suo, L. Zhang, and I. Bennion, “Two-axis temperature-insensitive accelerometer based on multicore fiber Bragg gratings,” IEEE Sens. J.8, 1292–1298 (2008).
[CrossRef]

Colpitts, B. G.

P. Chaube, B. G. Colpitts, D. Jagannathan, and A. W. Brown, “Distributed fiber-optic sensor for dynamic strain measurement,” IEEE Sens. J.8, 1067–1072 (2008).
[CrossRef]

Dandliker, R.

A. Bertholds and R. Dandliker, “Determination of the individual strain-optic coefficients in single mode optical fibre,” J. Lightwave Technol.6, 17–20 (1988).
[CrossRef]

Defosse, Y.

M. Wuilpart, P. Mégret, M. Blondel, A. J. Rogers, and Y. Defosse, “Measurement of the spatial distribution of birefringence in optical fibers,” IEEE Photon. Technol. Lett.13, 836–838 (2001).
[CrossRef]

Domanski, A.

G. Rajan, M. Ramakrishnan, Y. Semenova, A. Domanski, A. Boczkowska, T. Wolinski, and G. Farrell, “Analysis of vibration measurements in a composite material using an embedded PM-PCF polarimetric sensor and an FBG sensor,” IEEE Sens. J.12, 1365–1371 (2012).
[CrossRef]

Fan, W.

F. Qin, H. Li, W. Fan, and Q. Sheng, “Experimental study on vibration frequency response of micro-bend optic-fiber sensor,” Chinese Opt. Lett.7, 556–559 (2009).
[CrossRef]

Fang, X.

Farrell, G.

G. Rajan, M. Ramakrishnan, Y. Semenova, A. Domanski, A. Boczkowska, T. Wolinski, and G. Farrell, “Analysis of vibration measurements in a composite material using an embedded PM-PCF polarimetric sensor and an FBG sensor,” IEEE Sens. J.12, 1365–1371 (2012).
[CrossRef]

Fender, A.

A. Fender, W. N. Macpherson, R. R. J. Maier, J. S. Barton, D. S. George, R. I. Howden, G. W Smith, B. J. S. Jones, S. Mcculloch, X. Chen, R. Suo, L. Zhang, and I. Bennion, “Two-axis temperature-insensitive accelerometer based on multicore fiber Bragg gratings,” IEEE Sens. J.8, 1292–1298 (2008).
[CrossRef]

Feng, Z.

Y. Weng, X. Qiao, T. Guo, M. Hu, Z. Feng, R. Wang, and J. Zhang, “A robust and compact fiber Bragg grating vibration sensor for seismic measurement,” IEEE Sens. J.12, 800–804 (2011).
[CrossRef]

Galtarossa, A.

A. Galtarossa, D. Grosso, and L. Palmieri, “Accurate characterization of twist-induced optical activity in single-mode fibers by means of polarization-sensitive reflectometry,” IEEE Photon. Technol. Lett.21, 1713–1715 (2009).
[CrossRef]

George, D. S.

A. Fender, W. N. Macpherson, R. R. J. Maier, J. S. Barton, D. S. George, R. I. Howden, G. W Smith, B. J. S. Jones, S. Mcculloch, X. Chen, R. Suo, L. Zhang, and I. Bennion, “Two-axis temperature-insensitive accelerometer based on multicore fiber Bragg gratings,” IEEE Sens. J.8, 1292–1298 (2008).
[CrossRef]

Grosso, D.

A. Galtarossa, D. Grosso, and L. Palmieri, “Accurate characterization of twist-induced optical activity in single-mode fibers by means of polarization-sensitive reflectometry,” IEEE Photon. Technol. Lett.21, 1713–1715 (2009).
[CrossRef]

Guo, H.

Guo, T.

Y. Weng, X. Qiao, T. Guo, M. Hu, Z. Feng, R. Wang, and J. Zhang, “A robust and compact fiber Bragg grating vibration sensor for seismic measurement,” IEEE Sens. J.12, 800–804 (2011).
[CrossRef]

T. Guo, L. Shao, H-Y. Tam, P. A. Krug, and J. Albert, “Tilted fiber grating accelerometer incorporating an abrupt biconical taper for cladding to core recoupling,” Opt. Express17, 20651–20660 (2009).
[CrossRef] [PubMed]

Guo, X.

X. Guo, Z. Yin, and N. Song, “Measuring vibration by using fiber Bragg grating and demodulating it by blazed grating,” Chinese Opt. Lett.2, 393–395 (2004).

Hong, X.

Horinaka, H.

Hotate, K.

K. Hotate and S. S. L. Ong, “Distributed dynamic strain measurement using a correlation-based Brillouin sensing system,” IEEE Photon. Technol. Lett.15, 272–274 (2003).
[CrossRef]

Howden, R. I.

A. Fender, W. N. Macpherson, R. R. J. Maier, J. S. Barton, D. S. George, R. I. Howden, G. W Smith, B. J. S. Jones, S. Mcculloch, X. Chen, R. Suo, L. Zhang, and I. Bennion, “Two-axis temperature-insensitive accelerometer based on multicore fiber Bragg gratings,” IEEE Sens. J.8, 1292–1298 (2008).
[CrossRef]

Hu, M.

Y. Weng, X. Qiao, T. Guo, M. Hu, Z. Feng, R. Wang, and J. Zhang, “A robust and compact fiber Bragg grating vibration sensor for seismic measurement,” IEEE Sens. J.12, 800–804 (2011).
[CrossRef]

Jagannathan, D.

P. Chaube, B. G. Colpitts, D. Jagannathan, and A. W. Brown, “Distributed fiber-optic sensor for dynamic strain measurement,” IEEE Sens. J.8, 1067–1072 (2008).
[CrossRef]

Johnson, G. A.

M. D. Todd, G. A. Johnson, B. A. Althouse, and S. T. Vohra, “Flexural beam-based fiber Bragg grating accelerometers,” IEEE Photon. Technol. Lett.10, 1605–1607 (1998).
[CrossRef]

Jones, B. J. S.

A. Fender, W. N. Macpherson, R. R. J. Maier, J. S. Barton, D. S. George, R. I. Howden, G. W Smith, B. J. S. Jones, S. Mcculloch, X. Chen, R. Suo, L. Zhang, and I. Bennion, “Two-axis temperature-insensitive accelerometer based on multicore fiber Bragg gratings,” IEEE Sens. J.8, 1292–1298 (2008).
[CrossRef]

Juarez, J. C.

Krug, P. A.

Lacquet, B. M.

Y. Zhu, P. Shum, C. Lu, B. M. Lacquet, P. L. Swart, and S. J. Spammer, “Temperature-insensitive fiber Bragg grating accelerometer,” IEEE Photon. Technol. Lett.15, 1437–1439 (2003).
[CrossRef]

Li, H.

F. Qin, H. Li, W. Fan, and Q. Sheng, “Experimental study on vibration frequency response of micro-bend optic-fiber sensor,” Chinese Opt. Lett.7, 556–559 (2009).
[CrossRef]

Linze, N.

Liu, F.

Lu, C.

Y. Zhu, P. Shum, C. Lu, B. M. Lacquet, P. L. Swart, and S. J. Spammer, “Temperature-insensitive fiber Bragg grating accelerometer,” IEEE Photon. Technol. Lett.15, 1437–1439 (2003).
[CrossRef]

Lu, Y.

Macpherson, W. N.

A. Fender, W. N. Macpherson, R. R. J. Maier, J. S. Barton, D. S. George, R. I. Howden, G. W Smith, B. J. S. Jones, S. Mcculloch, X. Chen, R. Suo, L. Zhang, and I. Bennion, “Two-axis temperature-insensitive accelerometer based on multicore fiber Bragg gratings,” IEEE Sens. J.8, 1292–1298 (2008).
[CrossRef]

Maier, R. R. J.

A. Fender, W. N. Macpherson, R. R. J. Maier, J. S. Barton, D. S. George, R. I. Howden, G. W Smith, B. J. S. Jones, S. Mcculloch, X. Chen, R. Suo, L. Zhang, and I. Bennion, “Two-axis temperature-insensitive accelerometer based on multicore fiber Bragg gratings,” IEEE Sens. J.8, 1292–1298 (2008).
[CrossRef]

Manuel, R. M.

Matsuyama, T.

Mcculloch, S.

A. Fender, W. N. Macpherson, R. R. J. Maier, J. S. Barton, D. S. George, R. I. Howden, G. W Smith, B. J. S. Jones, S. Mcculloch, X. Chen, R. Suo, L. Zhang, and I. Bennion, “Two-axis temperature-insensitive accelerometer based on multicore fiber Bragg gratings,” IEEE Sens. J.8, 1292–1298 (2008).
[CrossRef]

Mégret, P.

Minardo, A.

Miridonov, S. V.

Narui, H.

Ong, S. S. L.

K. Hotate and S. S. L. Ong, “Distributed dynamic strain measurement using a correlation-based Brillouin sensing system,” IEEE Photon. Technol. Lett.15, 272–274 (2003).
[CrossRef]

Palmieri, L.

A. Galtarossa, D. Grosso, and L. Palmieri, “Accurate characterization of twist-induced optical activity in single-mode fibers by means of polarization-sensitive reflectometry,” IEEE Photon. Technol. Lett.21, 1713–1715 (2009).
[CrossRef]

Qiao, X.

Y. Weng, X. Qiao, T. Guo, M. Hu, Z. Feng, R. Wang, and J. Zhang, “A robust and compact fiber Bragg grating vibration sensor for seismic measurement,” IEEE Sens. J.12, 800–804 (2011).
[CrossRef]

Qin, F.

F. Qin, H. Li, W. Fan, and Q. Sheng, “Experimental study on vibration frequency response of micro-bend optic-fiber sensor,” Chinese Opt. Lett.7, 556–559 (2009).
[CrossRef]

Rajan, G.

G. Rajan, M. Ramakrishnan, Y. Semenova, A. Domanski, A. Boczkowska, T. Wolinski, and G. Farrell, “Analysis of vibration measurements in a composite material using an embedded PM-PCF polarimetric sensor and an FBG sensor,” IEEE Sens. J.12, 1365–1371 (2012).
[CrossRef]

Ramakrishnan, M.

G. Rajan, M. Ramakrishnan, Y. Semenova, A. Domanski, A. Boczkowska, T. Wolinski, and G. Farrell, “Analysis of vibration measurements in a composite material using an embedded PM-PCF polarimetric sensor and an FBG sensor,” IEEE Sens. J.12, 1365–1371 (2012).
[CrossRef]

Rashleigh, S. C.

S. C. Rashleigh, “Origins and control of polarization effects in single-mode fibers,” J. Lightwave Technol.1, 312–331 (1983).
[CrossRef]

Rogers, A. J.

M. Wuilpart, P. Mégret, M. Blondel, A. J. Rogers, and Y. Defosse, “Measurement of the spatial distribution of birefringence in optical fibers,” IEEE Photon. Technol. Lett.13, 836–838 (2001).
[CrossRef]

Semenova, Y.

G. Rajan, M. Ramakrishnan, Y. Semenova, A. Domanski, A. Boczkowska, T. Wolinski, and G. Farrell, “Analysis of vibration measurements in a composite material using an embedded PM-PCF polarimetric sensor and an FBG sensor,” IEEE Sens. J.12, 1365–1371 (2012).
[CrossRef]

Shao, L.

Sheng, Q.

F. Qin, H. Li, W. Fan, and Q. Sheng, “Experimental study on vibration frequency response of micro-bend optic-fiber sensor,” Chinese Opt. Lett.7, 556–559 (2009).
[CrossRef]

Shlyagin, M. G.

Shum, P.

Y. Zhu, P. Shum, C. Lu, B. M. Lacquet, P. L. Swart, and S. J. Spammer, “Temperature-insensitive fiber Bragg grating accelerometer,” IEEE Photon. Technol. Lett.15, 1437–1439 (2003).
[CrossRef]

Smith, G. W

A. Fender, W. N. Macpherson, R. R. J. Maier, J. S. Barton, D. S. George, R. I. Howden, G. W Smith, B. J. S. Jones, S. Mcculloch, X. Chen, R. Suo, L. Zhang, and I. Bennion, “Two-axis temperature-insensitive accelerometer based on multicore fiber Bragg gratings,” IEEE Sens. J.8, 1292–1298 (2008).
[CrossRef]

Song, N.

X. Guo, Z. Yin, and N. Song, “Measuring vibration by using fiber Bragg grating and demodulating it by blazed grating,” Chinese Opt. Lett.2, 393–395 (2004).

Spammer, S. J.

Y. Zhu, P. Shum, C. Lu, B. M. Lacquet, P. L. Swart, and S. J. Spammer, “Temperature-insensitive fiber Bragg grating accelerometer,” IEEE Photon. Technol. Lett.15, 1437–1439 (2003).
[CrossRef]

Suo, R.

A. Fender, W. N. Macpherson, R. R. J. Maier, J. S. Barton, D. S. George, R. I. Howden, G. W Smith, B. J. S. Jones, S. Mcculloch, X. Chen, R. Suo, L. Zhang, and I. Bennion, “Two-axis temperature-insensitive accelerometer based on multicore fiber Bragg gratings,” IEEE Sens. J.8, 1292–1298 (2008).
[CrossRef]

Swart, P. L.

Y. Zhu, P. Shum, C. Lu, B. M. Lacquet, P. L. Swart, and S. J. Spammer, “Temperature-insensitive fiber Bragg grating accelerometer,” IEEE Photon. Technol. Lett.15, 1437–1439 (2003).
[CrossRef]

Tam, H-Y.

Taylor, H. F.

Tihon, P.

Todd, M. D.

M. D. Todd, G. A. Johnson, B. A. Althouse, and S. T. Vohra, “Flexural beam-based fiber Bragg grating accelerometers,” IEEE Photon. Technol. Lett.10, 1605–1607 (1998).
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Figures (15)

Fig. 1
Fig. 1

Experimental setup (plain lines: optical links; dashed lines: electrical links, FUT: fiber under test; MT: mechanical transducer; Si: sensor i; FBG: fiber Bragg grating.

Fig. 2
Fig. 2

Mechanical transducer used to induce birefringence into the fiber.

Fig. 3
Fig. 3

FFTs of the three reflected signals (FBG1 (a); FBG2 (b) and FBG3 (c)).

Fig. 4
Fig. 4

Experimental setup (plain lines: optical links ; dashed lines: electrical links, FUT: fiber under test ; MT: mechanical transducer ; Si: sensor i ; FBG: fiber Bragg grating ; OSC: oscilloscope).

Fig. 5
Fig. 5

Evolution of the signals reflected by FBG21 and FBG22 when: (a) a 120 Hz vibration is applied on S1; (b) 120 Hz vibrations are applied on MT1 and MT2.

Fig. 6
Fig. 6

Simulation of the proposed vibration sensor.

Fig. 7
Fig. 7

Multi-point sensing example: 300 Hz on MT1, 175 Hz and 240 Hz on MT2, 175 Hz and 300 Hz on MT3. The acceleration amplitudes are 30 m/s2.

Fig. 8
Fig. 8

Magnitude spectra of the signals coming from FBG12 (a), FBG21 (b), FBG22 (c), FBG31 (d) and FBG32 (e) (simulation results).

Fig. 9
Fig. 9

Measured electrical signals corresponding to FBG12 (a), FBG21 (b), FBG22 (c), FBG31 (d) and FBG32 (e).

Fig. 10
Fig. 10

Magnitude spectra of the signals coming from FBG12 (a), FBG21 (b), FBG22 (c), FBG31 (d) and FBG32 (e).

Fig. 11
Fig. 11

(a): System configuration (vibration on MT2, no vibrations on MT1 and MT3); (b): Evolution of the measured phase shift vs. phase of the vibration applied on MT2.

Fig. 12
Fig. 12

(a): System configuration (vibrations on MT2 and MT3, no vibration on MT1) ; (b) : Evolution of the measured phase shift vs. phase of the vibration on MT3 for four vibration frequencies.

Fig. 13
Fig. 13

(a): System configuration (vibration on MT2 and MT3, no vibration on MT1) ; (b) : Evolution of the measured phase shift vs. amplitude of the vibration on MT3 (dashed line : 5° phase shift).

Fig. 14
Fig. 14

(a): System configuration (vibration on MT2 and MT3, no vibration on MT1) ; (b): Evolution of the measured phase shift vs. phase of the vibration on MT3 for two different input SOPs separated by 45°.

Fig. 15
Fig. 15

(a): System configuration (vibration on MT2 and MT3, no vibration on MT1) ; (b): Evolution of the measured phase shift vs. phase of the vibration on MT3 for two different input SOPs.

Tables (2)

Tables Icon

Table 1 Frequencies at the different MTs (Not relevant : the phase shift has not to be calculated; Δϕ: the phase shift has to be calculated in order to deduce if this frequency is really applied) (simulation results)

Tables Icon

Table 2 Frequencies at the different MTs (Not relevant : the phase shift has not to be calculated; Phase shift: the phase shift has to be calculated in order to deduce if this frequency is really applied)

Equations (13)

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δ β = 2 k n 0 3 ( p 11 p 12 ) ( 1 + ν P ) f π b E
δ = 2 k n 0 3 ( p 11 p 12 ) ( 1 + ν P ) f π b E L = K f
S in , λ i j = ( 1 cos 2 φ sin 2 φ 0 ) P in
M Pol , θ = 1 2 ( 1 cos 2 θ sin 2 θ 0 cos 2 θ cos 2 2 θ sin 2 θ cos 2 θ 0 sin 2 θ sin 2 θ cos θ sin 2 2 θ 0 0 0 0 0 )
M Fiber = ( 1 0 0 0 0 cos 2 δ int 2 + sin 2 δ int 2 cos 4 q sin 2 δ int 2 sin 4 q sin δ int sin 2 q 0 sin 2 δ int 2 sin 4 q cos 2 δ int 2 sin 2 δ int 2 cos 4 q sin δ int cos 2 q 0 sin δ int sin 2 q sin δ int cos 2 q cos δ int )
M Sensor = ( 1 0 0 0 0 cos 2 δ i 2 + sin 2 δ i 2 cos 4 q sin 2 δ i 2 sin 4 q sin δ i sin 2 q 0 sin 2 δ i 2 sin 4 q cos 2 δ i 2 sin 2 δ i 2 cos 4 q sin δ i cos 2 q 0 sin δ i sin 2 q sin δ i cos 2 q cos δ i )
f i = m L a i
M λ i j = M S M λ i j T M S
M S = ( 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 )
S r t , λ i j = M λ i j M λ i j S in , λ i j = M S M λ i j T M S M λ i j S in , λ i j
S out , λ i j = M Pol , θ S r t , λ i j
S out , λ 12 = M Pol , θ S r t , λ 12 = M Pol , θ M S M λ 12 T M S M λ 12 T S in , λ 12
M λ 12 = M F B G 12 , r M fiber 2 M Sensor 1 M fiber 1

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