Abstract

Fiber Bragg grating (FBG) based sensors have recently been introduced to the field of magnetic resonance imaging (MRI). Real-time MRI applications demand highly amplitude and phase sensitive MRI compatible sensors. Thus, a model and detailed analysis of FBG based ultrasound detection are required for designing better performing sensors. A hybrid FBG model incorporating numerical and FEA methods was developed and used for sensitivity and linearity analysis. The transfer matrix method was used for the modeling of optical modulation whereas FEA was used for pressure field calculations within the grating. The model was verified through reflection spectrum and acoustic pressure sensitivity testing of two π-phase shifted FBGs in a side slope read-out configuration. The sensitivity curves with respect to the operation point on the side slope was characterized in terms of amplitude and phase, and nonlinearity of the phase response has been quantified. Lastly, the impact of phase linearity of the FBG based acousto-optic sensor was tested under MRI when the sensor was used as a position marker and an analog phase shifter based solution was demonstrated.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2019 (1)

Y. S. Yaras, S. Satir, C. Ozsoy, R. Ramasawmy, A. E. Campbell-Washburn, R. J. Lederman, O. Kocaturk, and F. L. Degertekin, “Acousto-Optic Catheter Tracking Sensor for Interventional MRI Procedures,” IEEE Trans. Biomed. Eng. 66(4), 1148–1154 (2019).
[Crossref]

2016 (1)

C. J. Keulen, E. Akay, F. F. Melemez, E. S. Kocaman, A. Deniz, C. Yilmaz, T. Boz, M. Yildiz, H. S. Turkmen, and A. Suleman, “Prediction of fatigue response of composite structures by monitoring the strain energy release rate with embedded fiber Bragg gratings,” J. Intell. Mater. Syst. Struct. 27(1), 17–27 (2016).
[Crossref]

2013 (2)

D. Lau, Z. Chen, J. T. Teo, S. H. Ng, H. Rumpel, Y. Lian, H. Yang, and P. L. Kei, “Intensity-Modulated Microbend Fiber Optic Sensor for Respiratory Monitoring and Gating During MRI,” IEEE Trans. Biomed. Eng. 60(9), 2655–2662 (2013).
[Crossref]

Ł Dziuda, M. Krej, and F. W. Skibniewski, “Fiber Bragg Grating Strain Sensor Incorporated to Monitor Patient Vital Signs During MRI,” IEEE Sens. J. 13(12), 4986–4991 (2013).
[Crossref]

2012 (3)

H.-J. Bang, H.-I. Kim, and K.-S. Lee, “Measurement of strain and bending deflection of a wind turbine tower using arrayed FBG sensors,” Int. J. Precis. Eng. Manuf. 13(12), 2121–2126 (2012).
[Crossref]

Q. Wu and Y. Okabe, “High-sensitivity ultrasonic phase-shifted fiber Bragg grating balanced sensing system,” Opt. Express 20(27), 28353 (2012).
[Crossref]

T. Liu and M. Han, “Analysis of fiber bragg gratings for ultrasonic detection,” IEEE Sens. J. 12(7), 2368–2373 (2012).
[Crossref]

2011 (2)

A. Rosenthal, D. Razansky, and V. Ntziachristos, “High-sensitivity compact ultrasonic detector based on a pi-phase-shifted fiber Bragg grating,” Opt. Lett. 36(10), 1833–1835 (2011).
[Crossref]

C. J. Keulen, M. Yildiz, and A. Suleman, “Multiplexed FBG and etched fiber sensors for process and health monitoring of 2-&3-D RTM components,” J. Reinf. Plast. Compos. 30(12), 1055–1064 (2011).
[Crossref]

2010 (2)

Y. Park, S. Elayaperumal, B. Daniel, S. C. Ryu, M. Shin, J. Savall, R. J. Black, B. Moslehi, and M. R. Cutkosky, “Real-Time Estimation of 3-D Needle Shape and Deflection for MRI-Guided Interventions,” IEEE/ASME Trans. Mechatron. 15(6), 906–915 (2010).
[Crossref]

P. Polygerinos, D. Zbyszewski, T. Schaeffter, R. Razavi, L. D. Seneviratne, and K. Althoefer, “MRI-Compatible Fiber-Optic Force Sensors for Catheterization Procedures,” IEEE Sens. J. 10(10), 1598–1608 (2010).
[Crossref]

2008 (2)

G. Wild and S. Hinckley, “Acousto-ultrasonic optical fiber sensors: Overview and state-of-the-art,” IEEE Sens. J. 8(7), 1184–1193 (2008).
[Crossref]

M. Majumder, T. K. Gangopadhyay, A. K. Chakraborty, K. Dasgupta, and D. K. Bhattacharya, “Fibre Bragg gratings in structural health monitoring—Present status and applications,” Sens. Actuators, A 147(1), 150–164 (2008).
[Crossref]

2006 (1)

A. Cusano, P. Capoluongo, S. Campopiano, A. Cutolo, M. Giordano, F. Felli, A. Paolozzi, and M. Caponero, “Experimental modal analysis of an aircraft model wing by embedded fiber Bragg grating sensors,” IEEE Sens. J. 6(1), 67–77 (2006).
[Crossref]

2005 (1)

P. Moyo, J. M. W. Brownjohn, R. Suresh, and S. C. Tjin, “Development of fiber Bragg grating sensors for monitoring civil infrastructure,” Eng. Struct. 27(12), 1828–1834 (2005).
[Crossref]

2004 (1)

H.-N. Li, D.-S. Li, and G.-B. Song, “Recent applications of fiber optic sensors to health monitoring in civil engineering,” Eng. Struct. 26(11), 1647–1657 (2004).
[Crossref]

2003 (3)

P. Fomitchov and S. Krishnaswamy, “Response of a fiber Bragg grating ultrasonic sensor,” Opt. Eng. 42(4), 956–963 (2003).
[Crossref]

B. Yu, D. W. Kim, J. Deng, H. Xiao, and A. Wang, “Fiber Fabry-Perot sensors for detection of partial discharges in power transformers,” Appl. Opt. 42(16), 3241–3250 (2003).
[Crossref]

D. C. Betz, G. Thursby, B. Culshaw, and W. J. Staszewski, “Acousto-ultrasonic sensing using fiber Bragg gratings,” Smart Mater. Struct. 12(1), 122–128 (2003).
[Crossref]

1999 (2)

P. C. Beard, F. Perennes, and T. N. Mills, “Transduction mechanisms of the Fabry-Perot polymer film sensing concept for wideband ultrasound detection,” IEEE Trans. Ultrason., Ferroelect., Freq. Contr. 46(6), 1575–1582 (1999).
[Crossref]

B. Sutapun, M. Tabib-Azar, and A. Kazemi, “Pd-coated elastooptic fiber optic Bragg grating sensors for multiplexed hydrogen sensing,” Sens. Actuators, B 60(1), 27–34 (1999).
[Crossref]

1997 (3)

Y.-J. Rao, “In-fibre Bragg grating sensors,” Meas. Sci. Technol. 8(4), 355–375 (1997).
[Crossref]

T. Erdogan, “Fiber grating spectra,” J. Lightwave Technol. 15(8), 1277–1294 (1997).
[Crossref]

S. M. Norton, T. Erdogan, and G. M. Morris, “Coupled-mode theory of resonant-grating filters,” J. Opt. Soc. Am. A 14(3), 629 (1997).
[Crossref]

1994 (2)

W.-P. Huang, “Coupled-mode theory for optical waveguides: an overview,” J. Opt. Soc. Am. A 11(3), 963 (1994).
[Crossref]

G. P. Agrawal and S. Radic, “Phase-Shifted Fiber Bragg Gratings and their Application for Wavelength Demultiplexing,” IEEE Photonics Technol. Lett. 6(8), 995–997 (1994).
[Crossref]

1993 (2)

1992 (1)

J. G. Och, G. D. Clarke, W. T. Sobol, C. W. Rosen, and S. K. Mun, “Acceptance testing of magnetic resonance imaging systems: report of AAPM Nuclear Magnetic Resonance Task Group No. 6,” Med. Phys. 19(1), 217–229 (1992).
[Crossref]

1991 (1)

W. W. Morey, J. R. Dunphy, and G. Meltz, “Multiplexing fiber bragg grating sensors,” Fiber Integr. Opt. 10(4), 351–360 (1991).
[Crossref]

Agrawal, G. P.

G. P. Agrawal and S. Radic, “Phase-Shifted Fiber Bragg Gratings and their Application for Wavelength Demultiplexing,” IEEE Photonics Technol. Lett. 6(8), 995–997 (1994).
[Crossref]

Akay, E.

C. J. Keulen, E. Akay, F. F. Melemez, E. S. Kocaman, A. Deniz, C. Yilmaz, T. Boz, M. Yildiz, H. S. Turkmen, and A. Suleman, “Prediction of fatigue response of composite structures by monitoring the strain energy release rate with embedded fiber Bragg gratings,” J. Intell. Mater. Syst. Struct. 27(1), 17–27 (2016).
[Crossref]

Althoefer, K.

P. Polygerinos, D. Zbyszewski, T. Schaeffter, R. Razavi, L. D. Seneviratne, and K. Althoefer, “MRI-Compatible Fiber-Optic Force Sensors for Catheterization Procedures,” IEEE Sens. J. 10(10), 1598–1608 (2010).
[Crossref]

Bang, H.-J.

H.-J. Bang, H.-I. Kim, and K.-S. Lee, “Measurement of strain and bending deflection of a wind turbine tower using arrayed FBG sensors,” Int. J. Precis. Eng. Manuf. 13(12), 2121–2126 (2012).
[Crossref]

Beard, P. C.

P. C. Beard, F. Perennes, and T. N. Mills, “Transduction mechanisms of the Fabry-Perot polymer film sensing concept for wideband ultrasound detection,” IEEE Trans. Ultrason., Ferroelect., Freq. Contr. 46(6), 1575–1582 (1999).
[Crossref]

Berkoff, T. A.

Betz, D. C.

D. C. Betz, G. Thursby, B. Culshaw, and W. J. Staszewski, “Acousto-ultrasonic sensing using fiber Bragg gratings,” Smart Mater. Struct. 12(1), 122–128 (2003).
[Crossref]

Bhattacharya, D. K.

M. Majumder, T. K. Gangopadhyay, A. K. Chakraborty, K. Dasgupta, and D. K. Bhattacharya, “Fibre Bragg gratings in structural health monitoring—Present status and applications,” Sens. Actuators, A 147(1), 150–164 (2008).
[Crossref]

Black, R. J.

Y. Park, S. Elayaperumal, B. Daniel, S. C. Ryu, M. Shin, J. Savall, R. J. Black, B. Moslehi, and M. R. Cutkosky, “Real-Time Estimation of 3-D Needle Shape and Deflection for MRI-Guided Interventions,” IEEE/ASME Trans. Mechatron. 15(6), 906–915 (2010).
[Crossref]

Boz, T.

C. J. Keulen, E. Akay, F. F. Melemez, E. S. Kocaman, A. Deniz, C. Yilmaz, T. Boz, M. Yildiz, H. S. Turkmen, and A. Suleman, “Prediction of fatigue response of composite structures by monitoring the strain energy release rate with embedded fiber Bragg gratings,” J. Intell. Mater. Syst. Struct. 27(1), 17–27 (2016).
[Crossref]

Brownjohn, J. M. W.

P. Moyo, J. M. W. Brownjohn, R. Suresh, and S. C. Tjin, “Development of fiber Bragg grating sensors for monitoring civil infrastructure,” Eng. Struct. 27(12), 1828–1834 (2005).
[Crossref]

Campbell-Washburn, A. E.

Y. S. Yaras, S. Satir, C. Ozsoy, R. Ramasawmy, A. E. Campbell-Washburn, R. J. Lederman, O. Kocaturk, and F. L. Degertekin, “Acousto-Optic Catheter Tracking Sensor for Interventional MRI Procedures,” IEEE Trans. Biomed. Eng. 66(4), 1148–1154 (2019).
[Crossref]

Y. S. Yaras, S. Satir, C. Ozsoy, R. Ramasawmy, A. E. Campbell-Washburn, A. Z. Faranesh, R. J. Lederman, O. Kocaturk, and F. L. Degertekin, “Acousto-optic Based Active MRI Marker for Interventional MRI Devices,” in Proc. 25th Annu. Meeting Exhib. ISMRM, 2017.

Campopiano, S.

A. Cusano, P. Capoluongo, S. Campopiano, A. Cutolo, M. Giordano, F. Felli, A. Paolozzi, and M. Caponero, “Experimental modal analysis of an aircraft model wing by embedded fiber Bragg grating sensors,” IEEE Sens. J. 6(1), 67–77 (2006).
[Crossref]

Capoluongo, P.

A. Cusano, P. Capoluongo, S. Campopiano, A. Cutolo, M. Giordano, F. Felli, A. Paolozzi, and M. Caponero, “Experimental modal analysis of an aircraft model wing by embedded fiber Bragg grating sensors,” IEEE Sens. J. 6(1), 67–77 (2006).
[Crossref]

Caponero, M.

A. Cusano, P. Capoluongo, S. Campopiano, A. Cutolo, M. Giordano, F. Felli, A. Paolozzi, and M. Caponero, “Experimental modal analysis of an aircraft model wing by embedded fiber Bragg grating sensors,” IEEE Sens. J. 6(1), 67–77 (2006).
[Crossref]

Chakraborty, A. K.

M. Majumder, T. K. Gangopadhyay, A. K. Chakraborty, K. Dasgupta, and D. K. Bhattacharya, “Fibre Bragg gratings in structural health monitoring—Present status and applications,” Sens. Actuators, A 147(1), 150–164 (2008).
[Crossref]

Chen, Z.

D. Lau, Z. Chen, J. T. Teo, S. H. Ng, H. Rumpel, Y. Lian, H. Yang, and P. L. Kei, “Intensity-Modulated Microbend Fiber Optic Sensor for Respiratory Monitoring and Gating During MRI,” IEEE Trans. Biomed. Eng. 60(9), 2655–2662 (2013).
[Crossref]

Clarke, G. D.

J. G. Och, G. D. Clarke, W. T. Sobol, C. W. Rosen, and S. K. Mun, “Acceptance testing of magnetic resonance imaging systems: report of AAPM Nuclear Magnetic Resonance Task Group No. 6,” Med. Phys. 19(1), 217–229 (1992).
[Crossref]

Culshaw, B.

D. C. Betz, G. Thursby, B. Culshaw, and W. J. Staszewski, “Acousto-ultrasonic sensing using fiber Bragg gratings,” Smart Mater. Struct. 12(1), 122–128 (2003).
[Crossref]

Cusano, A.

A. Cusano, P. Capoluongo, S. Campopiano, A. Cutolo, M. Giordano, F. Felli, A. Paolozzi, and M. Caponero, “Experimental modal analysis of an aircraft model wing by embedded fiber Bragg grating sensors,” IEEE Sens. J. 6(1), 67–77 (2006).
[Crossref]

Cutkosky, M. R.

Y. Park, S. Elayaperumal, B. Daniel, S. C. Ryu, M. Shin, J. Savall, R. J. Black, B. Moslehi, and M. R. Cutkosky, “Real-Time Estimation of 3-D Needle Shape and Deflection for MRI-Guided Interventions,” IEEE/ASME Trans. Mechatron. 15(6), 906–915 (2010).
[Crossref]

Cutolo, A.

A. Cusano, P. Capoluongo, S. Campopiano, A. Cutolo, M. Giordano, F. Felli, A. Paolozzi, and M. Caponero, “Experimental modal analysis of an aircraft model wing by embedded fiber Bragg grating sensors,” IEEE Sens. J. 6(1), 67–77 (2006).
[Crossref]

Daniel, B.

Y. Park, S. Elayaperumal, B. Daniel, S. C. Ryu, M. Shin, J. Savall, R. J. Black, B. Moslehi, and M. R. Cutkosky, “Real-Time Estimation of 3-D Needle Shape and Deflection for MRI-Guided Interventions,” IEEE/ASME Trans. Mechatron. 15(6), 906–915 (2010).
[Crossref]

Dasgupta, K.

M. Majumder, T. K. Gangopadhyay, A. K. Chakraborty, K. Dasgupta, and D. K. Bhattacharya, “Fibre Bragg gratings in structural health monitoring—Present status and applications,” Sens. Actuators, A 147(1), 150–164 (2008).
[Crossref]

Degertekin, F. L.

Y. S. Yaras, S. Satir, C. Ozsoy, R. Ramasawmy, A. E. Campbell-Washburn, R. J. Lederman, O. Kocaturk, and F. L. Degertekin, “Acousto-Optic Catheter Tracking Sensor for Interventional MRI Procedures,” IEEE Trans. Biomed. Eng. 66(4), 1148–1154 (2019).
[Crossref]

Y. S. Yaras, S. Satir, C. Ozsoy, R. Ramasawmy, A. E. Campbell-Washburn, A. Z. Faranesh, R. J. Lederman, O. Kocaturk, and F. L. Degertekin, “Acousto-optic Based Active MRI Marker for Interventional MRI Devices,” in Proc. 25th Annu. Meeting Exhib. ISMRM, 2017.

Deng, J.

Deniz, A.

C. J. Keulen, E. Akay, F. F. Melemez, E. S. Kocaman, A. Deniz, C. Yilmaz, T. Boz, M. Yildiz, H. S. Turkmen, and A. Suleman, “Prediction of fatigue response of composite structures by monitoring the strain energy release rate with embedded fiber Bragg gratings,” J. Intell. Mater. Syst. Struct. 27(1), 17–27 (2016).
[Crossref]

Dunphy, J. R.

W. W. Morey, J. R. Dunphy, and G. Meltz, “Multiplexing fiber bragg grating sensors,” Fiber Integr. Opt. 10(4), 351–360 (1991).
[Crossref]

Dziuda, L

Ł Dziuda, M. Krej, and F. W. Skibniewski, “Fiber Bragg Grating Strain Sensor Incorporated to Monitor Patient Vital Signs During MRI,” IEEE Sens. J. 13(12), 4986–4991 (2013).
[Crossref]

Elayaperumal, S.

Y. Park, S. Elayaperumal, B. Daniel, S. C. Ryu, M. Shin, J. Savall, R. J. Black, B. Moslehi, and M. R. Cutkosky, “Real-Time Estimation of 3-D Needle Shape and Deflection for MRI-Guided Interventions,” IEEE/ASME Trans. Mechatron. 15(6), 906–915 (2010).
[Crossref]

Erdogan, T.

Faranesh, A. Z.

Y. S. Yaras, S. Satir, C. Ozsoy, R. Ramasawmy, A. E. Campbell-Washburn, A. Z. Faranesh, R. J. Lederman, O. Kocaturk, and F. L. Degertekin, “Acousto-optic Based Active MRI Marker for Interventional MRI Devices,” in Proc. 25th Annu. Meeting Exhib. ISMRM, 2017.

Felli, F.

A. Cusano, P. Capoluongo, S. Campopiano, A. Cutolo, M. Giordano, F. Felli, A. Paolozzi, and M. Caponero, “Experimental modal analysis of an aircraft model wing by embedded fiber Bragg grating sensors,” IEEE Sens. J. 6(1), 67–77 (2006).
[Crossref]

Fomitchov, P.

P. Fomitchov and S. Krishnaswamy, “Response of a fiber Bragg grating ultrasonic sensor,” Opt. Eng. 42(4), 956–963 (2003).
[Crossref]

Gangopadhyay, T. K.

M. Majumder, T. K. Gangopadhyay, A. K. Chakraborty, K. Dasgupta, and D. K. Bhattacharya, “Fibre Bragg gratings in structural health monitoring—Present status and applications,” Sens. Actuators, A 147(1), 150–164 (2008).
[Crossref]

Giordano, M.

A. Cusano, P. Capoluongo, S. Campopiano, A. Cutolo, M. Giordano, F. Felli, A. Paolozzi, and M. Caponero, “Experimental modal analysis of an aircraft model wing by embedded fiber Bragg grating sensors,” IEEE Sens. J. 6(1), 67–77 (2006).
[Crossref]

Han, M.

T. Liu and M. Han, “Analysis of fiber bragg gratings for ultrasonic detection,” IEEE Sens. J. 12(7), 2368–2373 (2012).
[Crossref]

Hinckley, S.

G. Wild and S. Hinckley, “Acousto-ultrasonic optical fiber sensors: Overview and state-of-the-art,” IEEE Sens. J. 8(7), 1184–1193 (2008).
[Crossref]

Huang, W.-P.

Kazemi, A.

B. Sutapun, M. Tabib-Azar, and A. Kazemi, “Pd-coated elastooptic fiber optic Bragg grating sensors for multiplexed hydrogen sensing,” Sens. Actuators, B 60(1), 27–34 (1999).
[Crossref]

Kei, P. L.

D. Lau, Z. Chen, J. T. Teo, S. H. Ng, H. Rumpel, Y. Lian, H. Yang, and P. L. Kei, “Intensity-Modulated Microbend Fiber Optic Sensor for Respiratory Monitoring and Gating During MRI,” IEEE Trans. Biomed. Eng. 60(9), 2655–2662 (2013).
[Crossref]

Kersey, A. D.

Keulen, C. J.

C. J. Keulen, E. Akay, F. F. Melemez, E. S. Kocaman, A. Deniz, C. Yilmaz, T. Boz, M. Yildiz, H. S. Turkmen, and A. Suleman, “Prediction of fatigue response of composite structures by monitoring the strain energy release rate with embedded fiber Bragg gratings,” J. Intell. Mater. Syst. Struct. 27(1), 17–27 (2016).
[Crossref]

C. J. Keulen, M. Yildiz, and A. Suleman, “Multiplexed FBG and etched fiber sensors for process and health monitoring of 2-&3-D RTM components,” J. Reinf. Plast. Compos. 30(12), 1055–1064 (2011).
[Crossref]

Kim, D. W.

Kim, H.-I.

H.-J. Bang, H.-I. Kim, and K.-S. Lee, “Measurement of strain and bending deflection of a wind turbine tower using arrayed FBG sensors,” Int. J. Precis. Eng. Manuf. 13(12), 2121–2126 (2012).
[Crossref]

Kocaman, E. S.

C. J. Keulen, E. Akay, F. F. Melemez, E. S. Kocaman, A. Deniz, C. Yilmaz, T. Boz, M. Yildiz, H. S. Turkmen, and A. Suleman, “Prediction of fatigue response of composite structures by monitoring the strain energy release rate with embedded fiber Bragg gratings,” J. Intell. Mater. Syst. Struct. 27(1), 17–27 (2016).
[Crossref]

Kocaturk, O.

Y. S. Yaras, S. Satir, C. Ozsoy, R. Ramasawmy, A. E. Campbell-Washburn, R. J. Lederman, O. Kocaturk, and F. L. Degertekin, “Acousto-Optic Catheter Tracking Sensor for Interventional MRI Procedures,” IEEE Trans. Biomed. Eng. 66(4), 1148–1154 (2019).
[Crossref]

Y. S. Yaras, S. Satir, C. Ozsoy, R. Ramasawmy, A. E. Campbell-Washburn, A. Z. Faranesh, R. J. Lederman, O. Kocaturk, and F. L. Degertekin, “Acousto-optic Based Active MRI Marker for Interventional MRI Devices,” in Proc. 25th Annu. Meeting Exhib. ISMRM, 2017.

Krej, M.

Ł Dziuda, M. Krej, and F. W. Skibniewski, “Fiber Bragg Grating Strain Sensor Incorporated to Monitor Patient Vital Signs During MRI,” IEEE Sens. J. 13(12), 4986–4991 (2013).
[Crossref]

Krishnaswamy, S.

P. Fomitchov and S. Krishnaswamy, “Response of a fiber Bragg grating ultrasonic sensor,” Opt. Eng. 42(4), 956–963 (2003).
[Crossref]

Lau, D.

D. Lau, Z. Chen, J. T. Teo, S. H. Ng, H. Rumpel, Y. Lian, H. Yang, and P. L. Kei, “Intensity-Modulated Microbend Fiber Optic Sensor for Respiratory Monitoring and Gating During MRI,” IEEE Trans. Biomed. Eng. 60(9), 2655–2662 (2013).
[Crossref]

Lederman, R. J.

Y. S. Yaras, S. Satir, C. Ozsoy, R. Ramasawmy, A. E. Campbell-Washburn, R. J. Lederman, O. Kocaturk, and F. L. Degertekin, “Acousto-Optic Catheter Tracking Sensor for Interventional MRI Procedures,” IEEE Trans. Biomed. Eng. 66(4), 1148–1154 (2019).
[Crossref]

Y. S. Yaras, S. Satir, C. Ozsoy, R. Ramasawmy, A. E. Campbell-Washburn, A. Z. Faranesh, R. J. Lederman, O. Kocaturk, and F. L. Degertekin, “Acousto-optic Based Active MRI Marker for Interventional MRI Devices,” in Proc. 25th Annu. Meeting Exhib. ISMRM, 2017.

Lee, K.-S.

H.-J. Bang, H.-I. Kim, and K.-S. Lee, “Measurement of strain and bending deflection of a wind turbine tower using arrayed FBG sensors,” Int. J. Precis. Eng. Manuf. 13(12), 2121–2126 (2012).
[Crossref]

Li, D.-S.

H.-N. Li, D.-S. Li, and G.-B. Song, “Recent applications of fiber optic sensors to health monitoring in civil engineering,” Eng. Struct. 26(11), 1647–1657 (2004).
[Crossref]

Li, H.-N.

H.-N. Li, D.-S. Li, and G.-B. Song, “Recent applications of fiber optic sensors to health monitoring in civil engineering,” Eng. Struct. 26(11), 1647–1657 (2004).
[Crossref]

Lian, Y.

D. Lau, Z. Chen, J. T. Teo, S. H. Ng, H. Rumpel, Y. Lian, H. Yang, and P. L. Kei, “Intensity-Modulated Microbend Fiber Optic Sensor for Respiratory Monitoring and Gating During MRI,” IEEE Trans. Biomed. Eng. 60(9), 2655–2662 (2013).
[Crossref]

Liu, T.

T. Liu and M. Han, “Analysis of fiber bragg gratings for ultrasonic detection,” IEEE Sens. J. 12(7), 2368–2373 (2012).
[Crossref]

Majumder, M.

M. Majumder, T. K. Gangopadhyay, A. K. Chakraborty, K. Dasgupta, and D. K. Bhattacharya, “Fibre Bragg gratings in structural health monitoring—Present status and applications,” Sens. Actuators, A 147(1), 150–164 (2008).
[Crossref]

Melemez, F. F.

C. J. Keulen, E. Akay, F. F. Melemez, E. S. Kocaman, A. Deniz, C. Yilmaz, T. Boz, M. Yildiz, H. S. Turkmen, and A. Suleman, “Prediction of fatigue response of composite structures by monitoring the strain energy release rate with embedded fiber Bragg gratings,” J. Intell. Mater. Syst. Struct. 27(1), 17–27 (2016).
[Crossref]

Meltz, G.

W. W. Morey, J. R. Dunphy, and G. Meltz, “Multiplexing fiber bragg grating sensors,” Fiber Integr. Opt. 10(4), 351–360 (1991).
[Crossref]

Mills, T. N.

P. C. Beard, F. Perennes, and T. N. Mills, “Transduction mechanisms of the Fabry-Perot polymer film sensing concept for wideband ultrasound detection,” IEEE Trans. Ultrason., Ferroelect., Freq. Contr. 46(6), 1575–1582 (1999).
[Crossref]

Morey, W. W.

Morris, G. M.

Moslehi, B.

Y. Park, S. Elayaperumal, B. Daniel, S. C. Ryu, M. Shin, J. Savall, R. J. Black, B. Moslehi, and M. R. Cutkosky, “Real-Time Estimation of 3-D Needle Shape and Deflection for MRI-Guided Interventions,” IEEE/ASME Trans. Mechatron. 15(6), 906–915 (2010).
[Crossref]

Moyo, P.

P. Moyo, J. M. W. Brownjohn, R. Suresh, and S. C. Tjin, “Development of fiber Bragg grating sensors for monitoring civil infrastructure,” Eng. Struct. 27(12), 1828–1834 (2005).
[Crossref]

Mun, S. K.

J. G. Och, G. D. Clarke, W. T. Sobol, C. W. Rosen, and S. K. Mun, “Acceptance testing of magnetic resonance imaging systems: report of AAPM Nuclear Magnetic Resonance Task Group No. 6,” Med. Phys. 19(1), 217–229 (1992).
[Crossref]

Ng, S. H.

D. Lau, Z. Chen, J. T. Teo, S. H. Ng, H. Rumpel, Y. Lian, H. Yang, and P. L. Kei, “Intensity-Modulated Microbend Fiber Optic Sensor for Respiratory Monitoring and Gating During MRI,” IEEE Trans. Biomed. Eng. 60(9), 2655–2662 (2013).
[Crossref]

Norton, S. M.

Ntziachristos, V.

Och, J. G.

J. G. Och, G. D. Clarke, W. T. Sobol, C. W. Rosen, and S. K. Mun, “Acceptance testing of magnetic resonance imaging systems: report of AAPM Nuclear Magnetic Resonance Task Group No. 6,” Med. Phys. 19(1), 217–229 (1992).
[Crossref]

Okabe, Y.

Ozsoy, C.

Y. S. Yaras, S. Satir, C. Ozsoy, R. Ramasawmy, A. E. Campbell-Washburn, R. J. Lederman, O. Kocaturk, and F. L. Degertekin, “Acousto-Optic Catheter Tracking Sensor for Interventional MRI Procedures,” IEEE Trans. Biomed. Eng. 66(4), 1148–1154 (2019).
[Crossref]

Y. S. Yaras, S. Satir, C. Ozsoy, R. Ramasawmy, A. E. Campbell-Washburn, A. Z. Faranesh, R. J. Lederman, O. Kocaturk, and F. L. Degertekin, “Acousto-optic Based Active MRI Marker for Interventional MRI Devices,” in Proc. 25th Annu. Meeting Exhib. ISMRM, 2017.

Paolozzi, A.

A. Cusano, P. Capoluongo, S. Campopiano, A. Cutolo, M. Giordano, F. Felli, A. Paolozzi, and M. Caponero, “Experimental modal analysis of an aircraft model wing by embedded fiber Bragg grating sensors,” IEEE Sens. J. 6(1), 67–77 (2006).
[Crossref]

Park, Y.

Y. Park, S. Elayaperumal, B. Daniel, S. C. Ryu, M. Shin, J. Savall, R. J. Black, B. Moslehi, and M. R. Cutkosky, “Real-Time Estimation of 3-D Needle Shape and Deflection for MRI-Guided Interventions,” IEEE/ASME Trans. Mechatron. 15(6), 906–915 (2010).
[Crossref]

Perennes, F.

P. C. Beard, F. Perennes, and T. N. Mills, “Transduction mechanisms of the Fabry-Perot polymer film sensing concept for wideband ultrasound detection,” IEEE Trans. Ultrason., Ferroelect., Freq. Contr. 46(6), 1575–1582 (1999).
[Crossref]

Polygerinos, P.

P. Polygerinos, D. Zbyszewski, T. Schaeffter, R. Razavi, L. D. Seneviratne, and K. Althoefer, “MRI-Compatible Fiber-Optic Force Sensors for Catheterization Procedures,” IEEE Sens. J. 10(10), 1598–1608 (2010).
[Crossref]

Radic, S.

G. P. Agrawal and S. Radic, “Phase-Shifted Fiber Bragg Gratings and their Application for Wavelength Demultiplexing,” IEEE Photonics Technol. Lett. 6(8), 995–997 (1994).
[Crossref]

Ramasawmy, R.

Y. S. Yaras, S. Satir, C. Ozsoy, R. Ramasawmy, A. E. Campbell-Washburn, R. J. Lederman, O. Kocaturk, and F. L. Degertekin, “Acousto-Optic Catheter Tracking Sensor for Interventional MRI Procedures,” IEEE Trans. Biomed. Eng. 66(4), 1148–1154 (2019).
[Crossref]

Y. S. Yaras, S. Satir, C. Ozsoy, R. Ramasawmy, A. E. Campbell-Washburn, A. Z. Faranesh, R. J. Lederman, O. Kocaturk, and F. L. Degertekin, “Acousto-optic Based Active MRI Marker for Interventional MRI Devices,” in Proc. 25th Annu. Meeting Exhib. ISMRM, 2017.

Rao, Y.-J.

Y.-J. Rao, “In-fibre Bragg grating sensors,” Meas. Sci. Technol. 8(4), 355–375 (1997).
[Crossref]

Razansky, D.

Razavi, R.

P. Polygerinos, D. Zbyszewski, T. Schaeffter, R. Razavi, L. D. Seneviratne, and K. Althoefer, “MRI-Compatible Fiber-Optic Force Sensors for Catheterization Procedures,” IEEE Sens. J. 10(10), 1598–1608 (2010).
[Crossref]

Rosen, C. W.

J. G. Och, G. D. Clarke, W. T. Sobol, C. W. Rosen, and S. K. Mun, “Acceptance testing of magnetic resonance imaging systems: report of AAPM Nuclear Magnetic Resonance Task Group No. 6,” Med. Phys. 19(1), 217–229 (1992).
[Crossref]

Rosenthal, A.

Rumpel, H.

D. Lau, Z. Chen, J. T. Teo, S. H. Ng, H. Rumpel, Y. Lian, H. Yang, and P. L. Kei, “Intensity-Modulated Microbend Fiber Optic Sensor for Respiratory Monitoring and Gating During MRI,” IEEE Trans. Biomed. Eng. 60(9), 2655–2662 (2013).
[Crossref]

Ryu, S. C.

Y. Park, S. Elayaperumal, B. Daniel, S. C. Ryu, M. Shin, J. Savall, R. J. Black, B. Moslehi, and M. R. Cutkosky, “Real-Time Estimation of 3-D Needle Shape and Deflection for MRI-Guided Interventions,” IEEE/ASME Trans. Mechatron. 15(6), 906–915 (2010).
[Crossref]

Satir, S.

Y. S. Yaras, S. Satir, C. Ozsoy, R. Ramasawmy, A. E. Campbell-Washburn, R. J. Lederman, O. Kocaturk, and F. L. Degertekin, “Acousto-Optic Catheter Tracking Sensor for Interventional MRI Procedures,” IEEE Trans. Biomed. Eng. 66(4), 1148–1154 (2019).
[Crossref]

Y. S. Yaras, S. Satir, C. Ozsoy, R. Ramasawmy, A. E. Campbell-Washburn, A. Z. Faranesh, R. J. Lederman, O. Kocaturk, and F. L. Degertekin, “Acousto-optic Based Active MRI Marker for Interventional MRI Devices,” in Proc. 25th Annu. Meeting Exhib. ISMRM, 2017.

Savall, J.

Y. Park, S. Elayaperumal, B. Daniel, S. C. Ryu, M. Shin, J. Savall, R. J. Black, B. Moslehi, and M. R. Cutkosky, “Real-Time Estimation of 3-D Needle Shape and Deflection for MRI-Guided Interventions,” IEEE/ASME Trans. Mechatron. 15(6), 906–915 (2010).
[Crossref]

Schaeffter, T.

P. Polygerinos, D. Zbyszewski, T. Schaeffter, R. Razavi, L. D. Seneviratne, and K. Althoefer, “MRI-Compatible Fiber-Optic Force Sensors for Catheterization Procedures,” IEEE Sens. J. 10(10), 1598–1608 (2010).
[Crossref]

Seneviratne, L. D.

P. Polygerinos, D. Zbyszewski, T. Schaeffter, R. Razavi, L. D. Seneviratne, and K. Althoefer, “MRI-Compatible Fiber-Optic Force Sensors for Catheterization Procedures,” IEEE Sens. J. 10(10), 1598–1608 (2010).
[Crossref]

Shin, M.

Y. Park, S. Elayaperumal, B. Daniel, S. C. Ryu, M. Shin, J. Savall, R. J. Black, B. Moslehi, and M. R. Cutkosky, “Real-Time Estimation of 3-D Needle Shape and Deflection for MRI-Guided Interventions,” IEEE/ASME Trans. Mechatron. 15(6), 906–915 (2010).
[Crossref]

Skibniewski, F. W.

Ł Dziuda, M. Krej, and F. W. Skibniewski, “Fiber Bragg Grating Strain Sensor Incorporated to Monitor Patient Vital Signs During MRI,” IEEE Sens. J. 13(12), 4986–4991 (2013).
[Crossref]

Sobol, W. T.

J. G. Och, G. D. Clarke, W. T. Sobol, C. W. Rosen, and S. K. Mun, “Acceptance testing of magnetic resonance imaging systems: report of AAPM Nuclear Magnetic Resonance Task Group No. 6,” Med. Phys. 19(1), 217–229 (1992).
[Crossref]

Song, G.-B.

H.-N. Li, D.-S. Li, and G.-B. Song, “Recent applications of fiber optic sensors to health monitoring in civil engineering,” Eng. Struct. 26(11), 1647–1657 (2004).
[Crossref]

Staszewski, W. J.

D. C. Betz, G. Thursby, B. Culshaw, and W. J. Staszewski, “Acousto-ultrasonic sensing using fiber Bragg gratings,” Smart Mater. Struct. 12(1), 122–128 (2003).
[Crossref]

Suleman, A.

C. J. Keulen, E. Akay, F. F. Melemez, E. S. Kocaman, A. Deniz, C. Yilmaz, T. Boz, M. Yildiz, H. S. Turkmen, and A. Suleman, “Prediction of fatigue response of composite structures by monitoring the strain energy release rate with embedded fiber Bragg gratings,” J. Intell. Mater. Syst. Struct. 27(1), 17–27 (2016).
[Crossref]

C. J. Keulen, M. Yildiz, and A. Suleman, “Multiplexed FBG and etched fiber sensors for process and health monitoring of 2-&3-D RTM components,” J. Reinf. Plast. Compos. 30(12), 1055–1064 (2011).
[Crossref]

Suresh, R.

P. Moyo, J. M. W. Brownjohn, R. Suresh, and S. C. Tjin, “Development of fiber Bragg grating sensors for monitoring civil infrastructure,” Eng. Struct. 27(12), 1828–1834 (2005).
[Crossref]

Sutapun, B.

B. Sutapun, M. Tabib-Azar, and A. Kazemi, “Pd-coated elastooptic fiber optic Bragg grating sensors for multiplexed hydrogen sensing,” Sens. Actuators, B 60(1), 27–34 (1999).
[Crossref]

Tabib-Azar, M.

B. Sutapun, M. Tabib-Azar, and A. Kazemi, “Pd-coated elastooptic fiber optic Bragg grating sensors for multiplexed hydrogen sensing,” Sens. Actuators, B 60(1), 27–34 (1999).
[Crossref]

Teo, J. T.

D. Lau, Z. Chen, J. T. Teo, S. H. Ng, H. Rumpel, Y. Lian, H. Yang, and P. L. Kei, “Intensity-Modulated Microbend Fiber Optic Sensor for Respiratory Monitoring and Gating During MRI,” IEEE Trans. Biomed. Eng. 60(9), 2655–2662 (2013).
[Crossref]

Thursby, G.

D. C. Betz, G. Thursby, B. Culshaw, and W. J. Staszewski, “Acousto-ultrasonic sensing using fiber Bragg gratings,” Smart Mater. Struct. 12(1), 122–128 (2003).
[Crossref]

Tjin, S. C.

P. Moyo, J. M. W. Brownjohn, R. Suresh, and S. C. Tjin, “Development of fiber Bragg grating sensors for monitoring civil infrastructure,” Eng. Struct. 27(12), 1828–1834 (2005).
[Crossref]

Turkmen, H. S.

C. J. Keulen, E. Akay, F. F. Melemez, E. S. Kocaman, A. Deniz, C. Yilmaz, T. Boz, M. Yildiz, H. S. Turkmen, and A. Suleman, “Prediction of fatigue response of composite structures by monitoring the strain energy release rate with embedded fiber Bragg gratings,” J. Intell. Mater. Syst. Struct. 27(1), 17–27 (2016).
[Crossref]

Wang, A.

Wild, G.

G. Wild and S. Hinckley, “Acousto-ultrasonic optical fiber sensors: Overview and state-of-the-art,” IEEE Sens. J. 8(7), 1184–1193 (2008).
[Crossref]

Wu, Q.

Xiao, H.

Yang, H.

D. Lau, Z. Chen, J. T. Teo, S. H. Ng, H. Rumpel, Y. Lian, H. Yang, and P. L. Kei, “Intensity-Modulated Microbend Fiber Optic Sensor for Respiratory Monitoring and Gating During MRI,” IEEE Trans. Biomed. Eng. 60(9), 2655–2662 (2013).
[Crossref]

Yaras, Y. S.

Y. S. Yaras, S. Satir, C. Ozsoy, R. Ramasawmy, A. E. Campbell-Washburn, R. J. Lederman, O. Kocaturk, and F. L. Degertekin, “Acousto-Optic Catheter Tracking Sensor for Interventional MRI Procedures,” IEEE Trans. Biomed. Eng. 66(4), 1148–1154 (2019).
[Crossref]

Y. S. Yaras, S. Satir, C. Ozsoy, R. Ramasawmy, A. E. Campbell-Washburn, A. Z. Faranesh, R. J. Lederman, O. Kocaturk, and F. L. Degertekin, “Acousto-optic Based Active MRI Marker for Interventional MRI Devices,” in Proc. 25th Annu. Meeting Exhib. ISMRM, 2017.

Yildiz, M.

C. J. Keulen, E. Akay, F. F. Melemez, E. S. Kocaman, A. Deniz, C. Yilmaz, T. Boz, M. Yildiz, H. S. Turkmen, and A. Suleman, “Prediction of fatigue response of composite structures by monitoring the strain energy release rate with embedded fiber Bragg gratings,” J. Intell. Mater. Syst. Struct. 27(1), 17–27 (2016).
[Crossref]

C. J. Keulen, M. Yildiz, and A. Suleman, “Multiplexed FBG and etched fiber sensors for process and health monitoring of 2-&3-D RTM components,” J. Reinf. Plast. Compos. 30(12), 1055–1064 (2011).
[Crossref]

Yilmaz, C.

C. J. Keulen, E. Akay, F. F. Melemez, E. S. Kocaman, A. Deniz, C. Yilmaz, T. Boz, M. Yildiz, H. S. Turkmen, and A. Suleman, “Prediction of fatigue response of composite structures by monitoring the strain energy release rate with embedded fiber Bragg gratings,” J. Intell. Mater. Syst. Struct. 27(1), 17–27 (2016).
[Crossref]

Yu, B.

Zbyszewski, D.

P. Polygerinos, D. Zbyszewski, T. Schaeffter, R. Razavi, L. D. Seneviratne, and K. Althoefer, “MRI-Compatible Fiber-Optic Force Sensors for Catheterization Procedures,” IEEE Sens. J. 10(10), 1598–1608 (2010).
[Crossref]

Appl. Opt. (1)

Eng. Struct. (2)

H.-N. Li, D.-S. Li, and G.-B. Song, “Recent applications of fiber optic sensors to health monitoring in civil engineering,” Eng. Struct. 26(11), 1647–1657 (2004).
[Crossref]

P. Moyo, J. M. W. Brownjohn, R. Suresh, and S. C. Tjin, “Development of fiber Bragg grating sensors for monitoring civil infrastructure,” Eng. Struct. 27(12), 1828–1834 (2005).
[Crossref]

Fiber Integr. Opt. (1)

W. W. Morey, J. R. Dunphy, and G. Meltz, “Multiplexing fiber bragg grating sensors,” Fiber Integr. Opt. 10(4), 351–360 (1991).
[Crossref]

IEEE Photonics Technol. Lett. (1)

G. P. Agrawal and S. Radic, “Phase-Shifted Fiber Bragg Gratings and their Application for Wavelength Demultiplexing,” IEEE Photonics Technol. Lett. 6(8), 995–997 (1994).
[Crossref]

IEEE Sens. J. (5)

P. Polygerinos, D. Zbyszewski, T. Schaeffter, R. Razavi, L. D. Seneviratne, and K. Althoefer, “MRI-Compatible Fiber-Optic Force Sensors for Catheterization Procedures,” IEEE Sens. J. 10(10), 1598–1608 (2010).
[Crossref]

Ł Dziuda, M. Krej, and F. W. Skibniewski, “Fiber Bragg Grating Strain Sensor Incorporated to Monitor Patient Vital Signs During MRI,” IEEE Sens. J. 13(12), 4986–4991 (2013).
[Crossref]

A. Cusano, P. Capoluongo, S. Campopiano, A. Cutolo, M. Giordano, F. Felli, A. Paolozzi, and M. Caponero, “Experimental modal analysis of an aircraft model wing by embedded fiber Bragg grating sensors,” IEEE Sens. J. 6(1), 67–77 (2006).
[Crossref]

T. Liu and M. Han, “Analysis of fiber bragg gratings for ultrasonic detection,” IEEE Sens. J. 12(7), 2368–2373 (2012).
[Crossref]

G. Wild and S. Hinckley, “Acousto-ultrasonic optical fiber sensors: Overview and state-of-the-art,” IEEE Sens. J. 8(7), 1184–1193 (2008).
[Crossref]

IEEE Trans. Biomed. Eng. (2)

D. Lau, Z. Chen, J. T. Teo, S. H. Ng, H. Rumpel, Y. Lian, H. Yang, and P. L. Kei, “Intensity-Modulated Microbend Fiber Optic Sensor for Respiratory Monitoring and Gating During MRI,” IEEE Trans. Biomed. Eng. 60(9), 2655–2662 (2013).
[Crossref]

Y. S. Yaras, S. Satir, C. Ozsoy, R. Ramasawmy, A. E. Campbell-Washburn, R. J. Lederman, O. Kocaturk, and F. L. Degertekin, “Acousto-Optic Catheter Tracking Sensor for Interventional MRI Procedures,” IEEE Trans. Biomed. Eng. 66(4), 1148–1154 (2019).
[Crossref]

IEEE Trans. Ultrason., Ferroelect., Freq. Contr. (1)

P. C. Beard, F. Perennes, and T. N. Mills, “Transduction mechanisms of the Fabry-Perot polymer film sensing concept for wideband ultrasound detection,” IEEE Trans. Ultrason., Ferroelect., Freq. Contr. 46(6), 1575–1582 (1999).
[Crossref]

IEEE/ASME Trans. Mechatron. (1)

Y. Park, S. Elayaperumal, B. Daniel, S. C. Ryu, M. Shin, J. Savall, R. J. Black, B. Moslehi, and M. R. Cutkosky, “Real-Time Estimation of 3-D Needle Shape and Deflection for MRI-Guided Interventions,” IEEE/ASME Trans. Mechatron. 15(6), 906–915 (2010).
[Crossref]

Int. J. Precis. Eng. Manuf. (1)

H.-J. Bang, H.-I. Kim, and K.-S. Lee, “Measurement of strain and bending deflection of a wind turbine tower using arrayed FBG sensors,” Int. J. Precis. Eng. Manuf. 13(12), 2121–2126 (2012).
[Crossref]

J. Intell. Mater. Syst. Struct. (1)

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Figures (10)

Fig. 1.
Fig. 1. a) Schematic of FBG depicting transfer matrix method b) Side spe read out scheme utilizing a single wavelength laser source.
Fig. 2.
Fig. 2. Composite model of the FBG sensor.
Fig. 3.
Fig. 3. Schematic of side slope optical read out.
Fig. 4.
Fig. 4. a) Reflection spectrum of π-FBG-1 around the center notch. b) Reflection spectrum of π-FBG-2 around the center notch
Fig. 5.
Fig. 5. a) Experimental set-up for pressure sensitivity. The hydrophone is placed at the FBG location for pressure calibration b) FEA simulation showing pressure distribution inside the FBG and the surrounding water in the axial direction. Note that the pressure field inside the fiber is 85% of the surrounding pressure field.
Fig. 6.
Fig. 6. Pressure field captured by π-FBG-1 (a) and π-FBG-2 (b) with the hydrophone measurement. The time delay between the FBG and hydrophone signals is due to the ∼4 mm distance between them as shown in Fig. 5(a).
Fig. 7.
Fig. 7. Pressure sensitivity of π-FBG-1 (a) and π-FBG-2(b) with respect to reflectivity on the side slope.
Fig. 8.
Fig. 8. Phase change with respect to reflectivity on the side slope for π-FBG-1 (a) and π-FBG-2 (b).
Fig. 9.
Fig. 9. Schematic of the sensor with phase shifter inside MRI.
Fig. 10.
Fig. 10. Image of acousto-optic sensor without phase correction (a) and with phase correction (b) under MRI.

Tables (1)

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Table 1. Simulation parameters for π-FBG-1 and π-FBG-2.

Equations (12)

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λ B r a g g = 2 n e f f Λ
[ A i B i ] = T i [ A i 1 B i 1 ] = [ T 11 T 12 T 21 T 22 ] [ A i 1 B i 1 ]
T 11 = cosh ( γ B Δ z ) i σ ^ γ B sinh ( γ B Δ z )
T 12 = i κ γ B sinh ( γ B Δ z )
T 21 = T 12
T 22 = T 11
σ = δ + 2 π n ( z ) / λ
κ = π n ( z ) / λ
T π = [ i 0 0 i ] λ
[ A ( z = L ) B ( z = L ) ] = T M T M 1 T π T 2 T 1 [ A ( z = 0 ) B ( z = 0 ) ] = T [ A ( z = 0 ) B ( z = 0 ) ]
Δ n ( z , t ) n e f f = n 2 P ( z , t ) 2 E ( 1 2 ν ) ( 2 P 12 + P 11 )
Δ Λ ( z , t ) Λ = P ( z , t ) E ( 1 2 ν )

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