Abstract

A signal processing method based on group delay calculations is introduced for distributed measurements of long-length fiber Bragg gratings (FBGs) based on optical frequency domain reflectometry (OFDR). Bragg wavelength shifts in interfered signals of OFDR are regarded as group delay. By calculating group delay, the distribution of Bragg wavelength shifts is obtained with high computational efficiency. We introduce weighted averaging process for noise reduction. This method required only 3.5% of signal processing time which was necessary for conventional equivalent signal processing based on short-time Fourier transform. The method also showed high sensitivity to experimental signals where non-uniform strain distributions existed in a long-length FBG.

© 2014 Optical Society of America

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2014 (1)

D. Wada, H. Murayama, “Analytical investigation of response of birefringent fiber Bragg grating sensors in distributed monitoring system based on optical frequency domain reflectometry,” Opt. Lasers Eng. 52, 99–105 (2014).
[CrossRef]

2013 (2)

H. Murayama, D. Wada, H. Igawa, “Structural Health Monitoring by Using Fiber-Optic Distributed Strain Sensors With High Spatial Resolution,” Photonic Sensors 3(4), 355–376 (2013).
[CrossRef]

S. T. Kreger, A. K. Sang, N. Garg, J. Michel, “High resolution, high sensitivity, dynamic distributed structural monitoring using optical frequency domain reflectometry,” Proc. SPIE 8722, 87220D (2013).
[CrossRef]

2012 (2)

2011 (3)

D. K. Gifford, M. E. Froggatt, S. T. Kreger, “High precision, high sensitivity distributed displacement and temperature measurements using OFDR-based phase tracking,” Proc. SPIE 7753, 775331 (2011).
[CrossRef]

A. K. Sang, M. E. Froggatt, S. T. Kreger, D. K. Gifford, “Millimeter resolution distributed dynamic strain measurements using optical frequency domain reflectometry,” Proc. SPIE 7753, 77532S (2011).
[CrossRef]

D. Wada, H. Murayama, H. Igawa, K. Kageyama, K. Uzawa, K. Omichi, “Simultaneous distributed measurement of strain and temperature by polarization maintaining fiber Bragg grating based on optical frequency domain reflectometry,” Smart Mater. Struct. 20(8), 085028 (2011).
[CrossRef]

2010 (1)

Y. Mizuno, Z. He, K. Hotate, “Distributed strain measurement using a tellurite glass fiber with Brillouin optical correlation-domain reflectometry,” Opt. Commun. 283(11), 2438–2441 (2010).
[CrossRef]

2009 (3)

S. T. Kreger, A. K. Sang, D. K. Gifford, M. E. Froggatt, “Distributed strain and temperature sensing in plastic optical fiber using Rayleigh scatter,” Proc. SPIE 7316, 73160A (2009).
[CrossRef]

S. Shen, Z. Wu, C. Yang, Y. Tang, G. Wu, W. Hong, “A new optical fiber sensor with improved strain sensitivity based on distributed optical fiber sensing technique,” Proc. SPIE 7293, 729315 (2009).
[CrossRef]

Y. Dong, X. Bao, W. Li, “Differential Brillouin gain for improving the temperature accuracy and spatial resolution in a long-distance distributed fiber sensor,” Appl. Opt. 48(22), 4297–4301 (2009).
[CrossRef] [PubMed]

2008 (1)

H. Igawa, K. Ohta, T. Kasai, I. Yamaguchi, H. Murayama, K. Kageyama, “Distributed measurements with a long gauge FBG sensor using optical frequency domain reflectometry (1st report, system investigation using optical simulation model),” J. Solid Mech. Mater. Eng. 2(9), 1242–1252 (2008).
[CrossRef]

2007 (1)

D. K. Gifford, S. T. Kreger, A. K. Sang, M. E. Froggatt, R. G. Duncan, M. S. Wolfe, B. J. Soller, “Swept-wavelength interferometric interrogation of fiber Rayleigh scatter for distributed sensing applications,” Proc. SPIE 6770, 67700F (2007).
[CrossRef]

2006 (1)

1998 (1)

1997 (1)

M. Volanthen, H. Geiger, J. P. Dakin, “Distributed grating sensors using low-coherence reflectometry,” J. Lightwave Technol. 15(11), 2076–2082 (1997).
[CrossRef]

1995 (1)

X. Bao, J. Dhliwayo, N. Heron, D. J. Webb, D. A. Jackson, “Experimental and theoretical studies on a distributed temperature sensor based on Brillouin scattering,” J. Lightwave Technol. 13(7), 1340–1348 (1995).
[CrossRef]

1990 (1)

1989 (1)

D. Culverhouse, F. Ferahi, C. N. Pannell, D. A. Jackson, “Exploitation of stimulated Brillouin scattering as a sensing mechanism for distributed temperature sensors and as a mean of realizing a tunable microwave generator,” Springer Proc. Phys. 44, 552–559 (1989).
[CrossRef]

1985 (1)

J. P. Dakin, D. J. Pratt, G. W. Bibby, J. N. Ross, “Distributed Optical Fibre Raman temperature sensor using a semiconductor light source and detector,” Electron. Lett. 21(13), 569–570 (1985).
[CrossRef]

Bao, X.

Bibby, G. W.

J. P. Dakin, D. J. Pratt, G. W. Bibby, J. N. Ross, “Distributed Optical Fibre Raman temperature sensor using a semiconductor light source and detector,” Electron. Lett. 21(13), 569–570 (1985).
[CrossRef]

Chen, L.

Culverhouse, D.

D. Culverhouse, F. Ferahi, C. N. Pannell, D. A. Jackson, “Exploitation of stimulated Brillouin scattering as a sensing mechanism for distributed temperature sensors and as a mean of realizing a tunable microwave generator,” Springer Proc. Phys. 44, 552–559 (1989).
[CrossRef]

Dakin, J. P.

M. Volanthen, H. Geiger, J. P. Dakin, “Distributed grating sensors using low-coherence reflectometry,” J. Lightwave Technol. 15(11), 2076–2082 (1997).
[CrossRef]

J. P. Dakin, D. J. Pratt, G. W. Bibby, J. N. Ross, “Distributed Optical Fibre Raman temperature sensor using a semiconductor light source and detector,” Electron. Lett. 21(13), 569–570 (1985).
[CrossRef]

Dhliwayo, J.

X. Bao, J. Dhliwayo, N. Heron, D. J. Webb, D. A. Jackson, “Experimental and theoretical studies on a distributed temperature sensor based on Brillouin scattering,” J. Lightwave Technol. 13(7), 1340–1348 (1995).
[CrossRef]

Dong, Y.

Duncan, R. G.

D. K. Gifford, S. T. Kreger, A. K. Sang, M. E. Froggatt, R. G. Duncan, M. S. Wolfe, B. J. Soller, “Swept-wavelength interferometric interrogation of fiber Rayleigh scatter for distributed sensing applications,” Proc. SPIE 6770, 67700F (2007).
[CrossRef]

Ferahi, F.

D. Culverhouse, F. Ferahi, C. N. Pannell, D. A. Jackson, “Exploitation of stimulated Brillouin scattering as a sensing mechanism for distributed temperature sensors and as a mean of realizing a tunable microwave generator,” Springer Proc. Phys. 44, 552–559 (1989).
[CrossRef]

Froggatt, M.

Froggatt, M. E.

D. K. Gifford, M. E. Froggatt, S. T. Kreger, “High precision, high sensitivity distributed displacement and temperature measurements using OFDR-based phase tracking,” Proc. SPIE 7753, 775331 (2011).
[CrossRef]

A. K. Sang, M. E. Froggatt, S. T. Kreger, D. K. Gifford, “Millimeter resolution distributed dynamic strain measurements using optical frequency domain reflectometry,” Proc. SPIE 7753, 77532S (2011).
[CrossRef]

S. T. Kreger, A. K. Sang, D. K. Gifford, M. E. Froggatt, “Distributed strain and temperature sensing in plastic optical fiber using Rayleigh scatter,” Proc. SPIE 7316, 73160A (2009).
[CrossRef]

D. K. Gifford, S. T. Kreger, A. K. Sang, M. E. Froggatt, R. G. Duncan, M. S. Wolfe, B. J. Soller, “Swept-wavelength interferometric interrogation of fiber Rayleigh scatter for distributed sensing applications,” Proc. SPIE 6770, 67700F (2007).
[CrossRef]

Garg, N.

S. T. Kreger, A. K. Sang, N. Garg, J. Michel, “High resolution, high sensitivity, dynamic distributed structural monitoring using optical frequency domain reflectometry,” Proc. SPIE 8722, 87220D (2013).
[CrossRef]

Geiger, H.

M. Volanthen, H. Geiger, J. P. Dakin, “Distributed grating sensors using low-coherence reflectometry,” J. Lightwave Technol. 15(11), 2076–2082 (1997).
[CrossRef]

Gifford, D. K.

A. K. Sang, M. E. Froggatt, S. T. Kreger, D. K. Gifford, “Millimeter resolution distributed dynamic strain measurements using optical frequency domain reflectometry,” Proc. SPIE 7753, 77532S (2011).
[CrossRef]

D. K. Gifford, M. E. Froggatt, S. T. Kreger, “High precision, high sensitivity distributed displacement and temperature measurements using OFDR-based phase tracking,” Proc. SPIE 7753, 775331 (2011).
[CrossRef]

S. T. Kreger, A. K. Sang, D. K. Gifford, M. E. Froggatt, “Distributed strain and temperature sensing in plastic optical fiber using Rayleigh scatter,” Proc. SPIE 7316, 73160A (2009).
[CrossRef]

D. K. Gifford, S. T. Kreger, A. K. Sang, M. E. Froggatt, R. G. Duncan, M. S. Wolfe, B. J. Soller, “Swept-wavelength interferometric interrogation of fiber Rayleigh scatter for distributed sensing applications,” Proc. SPIE 6770, 67700F (2007).
[CrossRef]

He, Z.

Y. Mizuno, Z. He, K. Hotate, “Distributed strain measurement using a tellurite glass fiber with Brillouin optical correlation-domain reflectometry,” Opt. Commun. 283(11), 2438–2441 (2010).
[CrossRef]

K. Y. Song, Z. He, K. Hotate, “Distributed strain measurement with millimeter-order spatial resolution based on Brillouin optical correlation domain analysis,” Opt. Lett. 31(17), 2526–2528 (2006).
[CrossRef] [PubMed]

Heron, N.

X. Bao, J. Dhliwayo, N. Heron, D. J. Webb, D. A. Jackson, “Experimental and theoretical studies on a distributed temperature sensor based on Brillouin scattering,” J. Lightwave Technol. 13(7), 1340–1348 (1995).
[CrossRef]

Hong, W.

S. Shen, Z. Wu, C. Yang, Y. Tang, G. Wu, W. Hong, “A new optical fiber sensor with improved strain sensitivity based on distributed optical fiber sensing technique,” Proc. SPIE 7293, 729315 (2009).
[CrossRef]

Horiguchi, T.

Hotate, K.

Y. Mizuno, Z. He, K. Hotate, “Distributed strain measurement using a tellurite glass fiber with Brillouin optical correlation-domain reflectometry,” Opt. Commun. 283(11), 2438–2441 (2010).
[CrossRef]

K. Y. Song, Z. He, K. Hotate, “Distributed strain measurement with millimeter-order spatial resolution based on Brillouin optical correlation domain analysis,” Opt. Lett. 31(17), 2526–2528 (2006).
[CrossRef] [PubMed]

Igawa, H.

H. Murayama, D. Wada, H. Igawa, “Structural Health Monitoring by Using Fiber-Optic Distributed Strain Sensors With High Spatial Resolution,” Photonic Sensors 3(4), 355–376 (2013).
[CrossRef]

D. Wada, H. Murayama, H. Igawa, “Lateral load measurements based on a distributed sensing system of optical frequency domain reflectometry using long-length fiber Bragg gratings,” J. Lightwave Technol. 30(14), 2337–2344 (2012).
[CrossRef]

D. Wada, H. Murayama, H. Igawa, K. Kageyama, K. Uzawa, K. Omichi, “Simultaneous distributed measurement of strain and temperature by polarization maintaining fiber Bragg grating based on optical frequency domain reflectometry,” Smart Mater. Struct. 20(8), 085028 (2011).
[CrossRef]

H. Igawa, K. Ohta, T. Kasai, I. Yamaguchi, H. Murayama, K. Kageyama, “Distributed measurements with a long gauge FBG sensor using optical frequency domain reflectometry (1st report, system investigation using optical simulation model),” J. Solid Mech. Mater. Eng. 2(9), 1242–1252 (2008).
[CrossRef]

Jackson, D. A.

X. Bao, J. Dhliwayo, N. Heron, D. J. Webb, D. A. Jackson, “Experimental and theoretical studies on a distributed temperature sensor based on Brillouin scattering,” J. Lightwave Technol. 13(7), 1340–1348 (1995).
[CrossRef]

D. Culverhouse, F. Ferahi, C. N. Pannell, D. A. Jackson, “Exploitation of stimulated Brillouin scattering as a sensing mechanism for distributed temperature sensors and as a mean of realizing a tunable microwave generator,” Springer Proc. Phys. 44, 552–559 (1989).
[CrossRef]

Kageyama, K.

D. Wada, H. Murayama, H. Igawa, K. Kageyama, K. Uzawa, K. Omichi, “Simultaneous distributed measurement of strain and temperature by polarization maintaining fiber Bragg grating based on optical frequency domain reflectometry,” Smart Mater. Struct. 20(8), 085028 (2011).
[CrossRef]

H. Igawa, K. Ohta, T. Kasai, I. Yamaguchi, H. Murayama, K. Kageyama, “Distributed measurements with a long gauge FBG sensor using optical frequency domain reflectometry (1st report, system investigation using optical simulation model),” J. Solid Mech. Mater. Eng. 2(9), 1242–1252 (2008).
[CrossRef]

Kasai, T.

H. Igawa, K. Ohta, T. Kasai, I. Yamaguchi, H. Murayama, K. Kageyama, “Distributed measurements with a long gauge FBG sensor using optical frequency domain reflectometry (1st report, system investigation using optical simulation model),” J. Solid Mech. Mater. Eng. 2(9), 1242–1252 (2008).
[CrossRef]

Kreger, S. T.

S. T. Kreger, A. K. Sang, N. Garg, J. Michel, “High resolution, high sensitivity, dynamic distributed structural monitoring using optical frequency domain reflectometry,” Proc. SPIE 8722, 87220D (2013).
[CrossRef]

A. K. Sang, M. E. Froggatt, S. T. Kreger, D. K. Gifford, “Millimeter resolution distributed dynamic strain measurements using optical frequency domain reflectometry,” Proc. SPIE 7753, 77532S (2011).
[CrossRef]

D. K. Gifford, M. E. Froggatt, S. T. Kreger, “High precision, high sensitivity distributed displacement and temperature measurements using OFDR-based phase tracking,” Proc. SPIE 7753, 775331 (2011).
[CrossRef]

S. T. Kreger, A. K. Sang, D. K. Gifford, M. E. Froggatt, “Distributed strain and temperature sensing in plastic optical fiber using Rayleigh scatter,” Proc. SPIE 7316, 73160A (2009).
[CrossRef]

D. K. Gifford, S. T. Kreger, A. K. Sang, M. E. Froggatt, R. G. Duncan, M. S. Wolfe, B. J. Soller, “Swept-wavelength interferometric interrogation of fiber Rayleigh scatter for distributed sensing applications,” Proc. SPIE 6770, 67700F (2007).
[CrossRef]

Kurashima, T.

Li, W.

Michel, J.

S. T. Kreger, A. K. Sang, N. Garg, J. Michel, “High resolution, high sensitivity, dynamic distributed structural monitoring using optical frequency domain reflectometry,” Proc. SPIE 8722, 87220D (2013).
[CrossRef]

Mizuno, Y.

Y. Mizuno, Z. He, K. Hotate, “Distributed strain measurement using a tellurite glass fiber with Brillouin optical correlation-domain reflectometry,” Opt. Commun. 283(11), 2438–2441 (2010).
[CrossRef]

Moore, J.

Murayama, H.

D. Wada, H. Murayama, “Analytical investigation of response of birefringent fiber Bragg grating sensors in distributed monitoring system based on optical frequency domain reflectometry,” Opt. Lasers Eng. 52, 99–105 (2014).
[CrossRef]

H. Murayama, D. Wada, H. Igawa, “Structural Health Monitoring by Using Fiber-Optic Distributed Strain Sensors With High Spatial Resolution,” Photonic Sensors 3(4), 355–376 (2013).
[CrossRef]

D. Wada, H. Murayama, H. Igawa, “Lateral load measurements based on a distributed sensing system of optical frequency domain reflectometry using long-length fiber Bragg gratings,” J. Lightwave Technol. 30(14), 2337–2344 (2012).
[CrossRef]

D. Wada, H. Murayama, H. Igawa, K. Kageyama, K. Uzawa, K. Omichi, “Simultaneous distributed measurement of strain and temperature by polarization maintaining fiber Bragg grating based on optical frequency domain reflectometry,” Smart Mater. Struct. 20(8), 085028 (2011).
[CrossRef]

H. Igawa, K. Ohta, T. Kasai, I. Yamaguchi, H. Murayama, K. Kageyama, “Distributed measurements with a long gauge FBG sensor using optical frequency domain reflectometry (1st report, system investigation using optical simulation model),” J. Solid Mech. Mater. Eng. 2(9), 1242–1252 (2008).
[CrossRef]

Ohta, K.

H. Igawa, K. Ohta, T. Kasai, I. Yamaguchi, H. Murayama, K. Kageyama, “Distributed measurements with a long gauge FBG sensor using optical frequency domain reflectometry (1st report, system investigation using optical simulation model),” J. Solid Mech. Mater. Eng. 2(9), 1242–1252 (2008).
[CrossRef]

Omichi, K.

D. Wada, H. Murayama, H. Igawa, K. Kageyama, K. Uzawa, K. Omichi, “Simultaneous distributed measurement of strain and temperature by polarization maintaining fiber Bragg grating based on optical frequency domain reflectometry,” Smart Mater. Struct. 20(8), 085028 (2011).
[CrossRef]

Pannell, C. N.

D. Culverhouse, F. Ferahi, C. N. Pannell, D. A. Jackson, “Exploitation of stimulated Brillouin scattering as a sensing mechanism for distributed temperature sensors and as a mean of realizing a tunable microwave generator,” Springer Proc. Phys. 44, 552–559 (1989).
[CrossRef]

Pratt, D. J.

J. P. Dakin, D. J. Pratt, G. W. Bibby, J. N. Ross, “Distributed Optical Fibre Raman temperature sensor using a semiconductor light source and detector,” Electron. Lett. 21(13), 569–570 (1985).
[CrossRef]

Qin, Z.

Ross, J. N.

J. P. Dakin, D. J. Pratt, G. W. Bibby, J. N. Ross, “Distributed Optical Fibre Raman temperature sensor using a semiconductor light source and detector,” Electron. Lett. 21(13), 569–570 (1985).
[CrossRef]

Sang, A. K.

S. T. Kreger, A. K. Sang, N. Garg, J. Michel, “High resolution, high sensitivity, dynamic distributed structural monitoring using optical frequency domain reflectometry,” Proc. SPIE 8722, 87220D (2013).
[CrossRef]

A. K. Sang, M. E. Froggatt, S. T. Kreger, D. K. Gifford, “Millimeter resolution distributed dynamic strain measurements using optical frequency domain reflectometry,” Proc. SPIE 7753, 77532S (2011).
[CrossRef]

S. T. Kreger, A. K. Sang, D. K. Gifford, M. E. Froggatt, “Distributed strain and temperature sensing in plastic optical fiber using Rayleigh scatter,” Proc. SPIE 7316, 73160A (2009).
[CrossRef]

D. K. Gifford, S. T. Kreger, A. K. Sang, M. E. Froggatt, R. G. Duncan, M. S. Wolfe, B. J. Soller, “Swept-wavelength interferometric interrogation of fiber Rayleigh scatter for distributed sensing applications,” Proc. SPIE 6770, 67700F (2007).
[CrossRef]

Shen, S.

S. Shen, Z. Wu, C. Yang, Y. Tang, G. Wu, W. Hong, “A new optical fiber sensor with improved strain sensitivity based on distributed optical fiber sensing technique,” Proc. SPIE 7293, 729315 (2009).
[CrossRef]

Soller, B. J.

D. K. Gifford, S. T. Kreger, A. K. Sang, M. E. Froggatt, R. G. Duncan, M. S. Wolfe, B. J. Soller, “Swept-wavelength interferometric interrogation of fiber Rayleigh scatter for distributed sensing applications,” Proc. SPIE 6770, 67700F (2007).
[CrossRef]

Song, K. Y.

Tang, Y.

S. Shen, Z. Wu, C. Yang, Y. Tang, G. Wu, W. Hong, “A new optical fiber sensor with improved strain sensitivity based on distributed optical fiber sensing technique,” Proc. SPIE 7293, 729315 (2009).
[CrossRef]

Tateda, M.

Uzawa, K.

D. Wada, H. Murayama, H. Igawa, K. Kageyama, K. Uzawa, K. Omichi, “Simultaneous distributed measurement of strain and temperature by polarization maintaining fiber Bragg grating based on optical frequency domain reflectometry,” Smart Mater. Struct. 20(8), 085028 (2011).
[CrossRef]

Volanthen, M.

M. Volanthen, H. Geiger, J. P. Dakin, “Distributed grating sensors using low-coherence reflectometry,” J. Lightwave Technol. 15(11), 2076–2082 (1997).
[CrossRef]

Wada, D.

D. Wada, H. Murayama, “Analytical investigation of response of birefringent fiber Bragg grating sensors in distributed monitoring system based on optical frequency domain reflectometry,” Opt. Lasers Eng. 52, 99–105 (2014).
[CrossRef]

H. Murayama, D. Wada, H. Igawa, “Structural Health Monitoring by Using Fiber-Optic Distributed Strain Sensors With High Spatial Resolution,” Photonic Sensors 3(4), 355–376 (2013).
[CrossRef]

D. Wada, H. Murayama, H. Igawa, “Lateral load measurements based on a distributed sensing system of optical frequency domain reflectometry using long-length fiber Bragg gratings,” J. Lightwave Technol. 30(14), 2337–2344 (2012).
[CrossRef]

D. Wada, H. Murayama, H. Igawa, K. Kageyama, K. Uzawa, K. Omichi, “Simultaneous distributed measurement of strain and temperature by polarization maintaining fiber Bragg grating based on optical frequency domain reflectometry,” Smart Mater. Struct. 20(8), 085028 (2011).
[CrossRef]

Webb, D. J.

X. Bao, J. Dhliwayo, N. Heron, D. J. Webb, D. A. Jackson, “Experimental and theoretical studies on a distributed temperature sensor based on Brillouin scattering,” J. Lightwave Technol. 13(7), 1340–1348 (1995).
[CrossRef]

Wolfe, M. S.

D. K. Gifford, S. T. Kreger, A. K. Sang, M. E. Froggatt, R. G. Duncan, M. S. Wolfe, B. J. Soller, “Swept-wavelength interferometric interrogation of fiber Rayleigh scatter for distributed sensing applications,” Proc. SPIE 6770, 67700F (2007).
[CrossRef]

Wu, G.

S. Shen, Z. Wu, C. Yang, Y. Tang, G. Wu, W. Hong, “A new optical fiber sensor with improved strain sensitivity based on distributed optical fiber sensing technique,” Proc. SPIE 7293, 729315 (2009).
[CrossRef]

Wu, Z.

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L. Li, J. Yang, L. Liu, Z. Zhang, X. Chen, and M. Zhang, “Kilometers-range dark-pulse Brillouin optical time domain analyzer with centimeters spatial resolution,” presented at the 2010 Symposium on Photonics and Optoelectronics, Chengdu, China, 19–21 June 2010.
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Figures (6)

Fig. 1
Fig. 1

OFDR system configuration. TLS: tunable laser source; DAQ: data acquisition; C: optical 3 dB coupler; M: polarization-maintaining mirror; D: photodetector.

Fig. 2
Fig. 2

Spectrograms calculated by STFT. (a) Initial FBG. (b) Shifted FBG.

Fig. 3
Fig. 3

Distribution of Bragg wavelength shifts calculated by STFT.

Fig. 4
Fig. 4

Distribution of Bragg wavelength shifts calculated using group delay. (a) Without noise reduction process. (b) With noise reduction process.

Fig. 5
Fig. 5

Schematic of the aluminum plate with holes.

Fig. 6
Fig. 6

Results of strain distribution measurements using STFT and group delay calculation.

Tables (1)

Tables Icon

Table 1 Optical and geometric parameters for numerical model of OFDR

Equations (11)

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Δk= π n eff L R ,
h(k)= x 2 (k) x 1 (k),
H( e jωT )=A(ω) e jb(ω) ,
T g (ω)= db(ω) dω =Im dH( e jωT ) / dω H( e jωT ) ,
T g =TRe z dH(z) / dz H(z) | z= e jωT ,
kh(k) H k (z),
H k (z)=z dH(z) dz .
T g =TRe H k (z) H(z) | z= e jωT .
T g =TRe fft{ kh(k) } fft{ h(k) } .
T ^ g =TRe fft( h ^ k ) fft( h ^ ) ¯ fft( h ^ ) fft( h ^ ) ¯ .
T g =TRe i fft( h ^ k ) fft( h ^ ) ¯ i fft( h ^ ) fft( h ^ ) ¯ .

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