Abstract

In practical applications of fiber optic sensors based on Fabry-Perot interferometers (FPIs), the lead-in optical fiber often experiences dynamic or static bending due to environmental perturbations or limited installation space. Bending introduces wavelength-dependent losses to the sensors, which can cause erroneous readings for sensors based on wavelength demodulation interrogation. Here, we investigate the bending-induced wavelength shift (BIWS) to sensors based on FPIs. Partially explicit expressions of BIWSs for the reflection fringe peaks and valleys have been derived for sensors based on low-finesse FPI. The theoretical model predicts these findings: 1) provided that a fringe peak experiences the same modulation slope by bending losses with a fringe valley, BIWSs for the peak and valley have opposite signs and the BIWS for the valley has a smaller absolute value; 2) BIWS is a linear function of the length of the bending section; 3) a FPI with higher visibility and longer optical path length is more resistant to the influence of bending. Experiments have been carried out and the results agree well with the theoretical predictions.

© 2016 Optical Society of America

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References

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2015 (2)

X. Yang, Z. Chen, C. S. M. Elvin, L. H. Y. Janice, S. H. Ng, J. T. Teo, and R. Wu, “Textile fiber optic microbend sensor used for heartbeat and respiration monitoring,” IEEE Sens. J. 15(2), 757–761 (2015).
[Crossref]

G. Liu, M. Han, and W. Hou, “High-resolution and fast-response fiber-optic temperature sensor using silicon Fabry-Pérot cavity,” Opt. Express 23(6), 7237–7247 (2015).
[Crossref] [PubMed]

2013 (3)

D. Jauregui-Vazquez, J. M. Estudillo-Ayala, A. Castillo-Guzman, R. Rojas-Laguna, R. Selvas-Aguilar, E. Vargas-Rodriguez, J. M. Sierra-Hernandez, V. Guzman-Ramos, and A. Flores-Balderas, “Highly sensitive curvature and displacement sensing setup based on an all fiber micro Fabry–Perot interferometer,” Opt. Commun. 308, 289–292 (2013).
[Crossref]

G. Liu, K. Li, P. Hao, W. Zhou, Y. Wu, and M. Xuan, “Bent optical fiber taper for refractive index detection with a high sensitivity,” Sens. Actuators A Phys. 201, 352–356 (2013).
[Crossref]

J. B. Rosolem, D. C. Dini, R. S. Penze, C. Floridia, A. A. Leonardi, M. D. Loichate, and A. S. Durelli, “Fiber optic bending sensor for water level monitoring: development and field test: a review,” IEEE Sens. J. 13(11), 4113–4120 (2013).
[Crossref]

2012 (1)

2011 (3)

X. Dong, Y. Liu, L.-Y. Shao, J. Kang, and C.-L. Zhao, “Temperature-independent fiber bending sensor based on a superimposed grating,” IEEE Sens. J. 11(11), 3019–3022 (2011).
[Crossref]

A. Iadicicco, D. Paladino, S. Campopiano, W. J. Bock, A. Cutolo, and A. Cusano, “Evanescent wave sensor based on permanently bent single mode optical fiber,” Sens. Actuators B Chem. 155(2), 903–908 (2011).
[Crossref]

R. Yang, Y.-S. Yu, Y. Xue, C. Chen, Q.-D. Chen, and H.-B. Sun, “Single S-tapered fiber Mach-Zehnder interferometers,” Opt. Lett. 36(23), 4482–4484 (2011).
[Crossref] [PubMed]

2006 (1)

2005 (1)

2003 (1)

B. Lee, “Review of the present status of optical fiber sensors,” Opt. Fiber Technol. 9(2), 57–79 (2003).
[Crossref]

1997 (1)

L. Faustini and G. Martini, “Bend loss in single-mode fibers,” J. Lightwave Technol. 15(4), 671–679 (1997).
[Crossref]

1995 (1)

J. W. Berthold, “Historical review of microbend fiber-optic sensors,” J. Lightwave Technol. 13(7), 1193–1199 (1995).
[Crossref]

1992 (1)

H. Renner, “Bending losses of coated single-mode fibers: a simple approach,” J. Lightwave Technol. 10(5), 544–551 (1992).
[Crossref]

1990 (1)

1986 (1)

A. J. Harris and P. F. Castle, “Bend loss measurements on high numerical aperture single-mode fibers as a function of wavelength and bend radius,” J. Lightwave Technol. 4(1), 34–40 (1986).
[Crossref]

1976 (1)

Barton, J. S.

Berthold, J. W.

J. W. Berthold, “Historical review of microbend fiber-optic sensors,” J. Lightwave Technol. 13(7), 1193–1199 (1995).
[Crossref]

Bock, W. J.

A. Iadicicco, D. Paladino, S. Campopiano, W. J. Bock, A. Cutolo, and A. Cusano, “Evanescent wave sensor based on permanently bent single mode optical fiber,” Sens. Actuators B Chem. 155(2), 903–908 (2011).
[Crossref]

Campopiano, S.

A. Iadicicco, D. Paladino, S. Campopiano, W. J. Bock, A. Cutolo, and A. Cusano, “Evanescent wave sensor based on permanently bent single mode optical fiber,” Sens. Actuators B Chem. 155(2), 903–908 (2011).
[Crossref]

Castillo-Guzman, A.

D. Jauregui-Vazquez, J. M. Estudillo-Ayala, A. Castillo-Guzman, R. Rojas-Laguna, R. Selvas-Aguilar, E. Vargas-Rodriguez, J. M. Sierra-Hernandez, V. Guzman-Ramos, and A. Flores-Balderas, “Highly sensitive curvature and displacement sensing setup based on an all fiber micro Fabry–Perot interferometer,” Opt. Commun. 308, 289–292 (2013).
[Crossref]

Castle, P. F.

A. J. Harris and P. F. Castle, “Bend loss measurements on high numerical aperture single-mode fibers as a function of wavelength and bend radius,” J. Lightwave Technol. 4(1), 34–40 (1986).
[Crossref]

Chen, C.

Chen, Q.-D.

Chen, Z.

X. Yang, Z. Chen, C. S. M. Elvin, L. H. Y. Janice, S. H. Ng, J. T. Teo, and R. Wu, “Textile fiber optic microbend sensor used for heartbeat and respiration monitoring,” IEEE Sens. J. 15(2), 757–761 (2015).
[Crossref]

Cusano, A.

A. Iadicicco, D. Paladino, S. Campopiano, W. J. Bock, A. Cutolo, and A. Cusano, “Evanescent wave sensor based on permanently bent single mode optical fiber,” Sens. Actuators B Chem. 155(2), 903–908 (2011).
[Crossref]

Cutolo, A.

A. Iadicicco, D. Paladino, S. Campopiano, W. J. Bock, A. Cutolo, and A. Cusano, “Evanescent wave sensor based on permanently bent single mode optical fiber,” Sens. Actuators B Chem. 155(2), 903–908 (2011).
[Crossref]

Dini, D. C.

J. B. Rosolem, D. C. Dini, R. S. Penze, C. Floridia, A. A. Leonardi, M. D. Loichate, and A. S. Durelli, “Fiber optic bending sensor for water level monitoring: development and field test: a review,” IEEE Sens. J. 13(11), 4113–4120 (2013).
[Crossref]

Dong, X.

X. Dong, Y. Liu, L.-Y. Shao, J. Kang, and C.-L. Zhao, “Temperature-independent fiber bending sensor based on a superimposed grating,” IEEE Sens. J. 11(11), 3019–3022 (2011).
[Crossref]

Durelli, A. S.

J. B. Rosolem, D. C. Dini, R. S. Penze, C. Floridia, A. A. Leonardi, M. D. Loichate, and A. S. Durelli, “Fiber optic bending sensor for water level monitoring: development and field test: a review,” IEEE Sens. J. 13(11), 4113–4120 (2013).
[Crossref]

Elvin, C. S. M.

X. Yang, Z. Chen, C. S. M. Elvin, L. H. Y. Janice, S. H. Ng, J. T. Teo, and R. Wu, “Textile fiber optic microbend sensor used for heartbeat and respiration monitoring,” IEEE Sens. J. 15(2), 757–761 (2015).
[Crossref]

Estudillo-Ayala, J. M.

D. Jauregui-Vazquez, J. M. Estudillo-Ayala, A. Castillo-Guzman, R. Rojas-Laguna, R. Selvas-Aguilar, E. Vargas-Rodriguez, J. M. Sierra-Hernandez, V. Guzman-Ramos, and A. Flores-Balderas, “Highly sensitive curvature and displacement sensing setup based on an all fiber micro Fabry–Perot interferometer,” Opt. Commun. 308, 289–292 (2013).
[Crossref]

Fan, Y.

Farrell, G.

Faustini, L.

L. Faustini and G. Martini, “Bend loss in single-mode fibers,” J. Lightwave Technol. 15(4), 671–679 (1997).
[Crossref]

Flores-Balderas, A.

D. Jauregui-Vazquez, J. M. Estudillo-Ayala, A. Castillo-Guzman, R. Rojas-Laguna, R. Selvas-Aguilar, E. Vargas-Rodriguez, J. M. Sierra-Hernandez, V. Guzman-Ramos, and A. Flores-Balderas, “Highly sensitive curvature and displacement sensing setup based on an all fiber micro Fabry–Perot interferometer,” Opt. Commun. 308, 289–292 (2013).
[Crossref]

Floridia, C.

J. B. Rosolem, D. C. Dini, R. S. Penze, C. Floridia, A. A. Leonardi, M. D. Loichate, and A. S. Durelli, “Fiber optic bending sensor for water level monitoring: development and field test: a review,” IEEE Sens. J. 13(11), 4113–4120 (2013).
[Crossref]

Freir, T.

Guzman-Ramos, V.

D. Jauregui-Vazquez, J. M. Estudillo-Ayala, A. Castillo-Guzman, R. Rojas-Laguna, R. Selvas-Aguilar, E. Vargas-Rodriguez, J. M. Sierra-Hernandez, V. Guzman-Ramos, and A. Flores-Balderas, “Highly sensitive curvature and displacement sensing setup based on an all fiber micro Fabry–Perot interferometer,” Opt. Commun. 308, 289–292 (2013).
[Crossref]

Han, M.

Hao, P.

G. Liu, K. Li, P. Hao, W. Zhou, Y. Wu, and M. Xuan, “Bent optical fiber taper for refractive index detection with a high sensitivity,” Sens. Actuators A Phys. 201, 352–356 (2013).
[Crossref]

Harper, P. G.

Harris, A. J.

A. J. Harris and P. F. Castle, “Bend loss measurements on high numerical aperture single-mode fibers as a function of wavelength and bend radius,” J. Lightwave Technol. 4(1), 34–40 (1986).
[Crossref]

Hou, W.

Iadicicco, A.

A. Iadicicco, D. Paladino, S. Campopiano, W. J. Bock, A. Cutolo, and A. Cusano, “Evanescent wave sensor based on permanently bent single mode optical fiber,” Sens. Actuators B Chem. 155(2), 903–908 (2011).
[Crossref]

Janice, L. H. Y.

X. Yang, Z. Chen, C. S. M. Elvin, L. H. Y. Janice, S. H. Ng, J. T. Teo, and R. Wu, “Textile fiber optic microbend sensor used for heartbeat and respiration monitoring,” IEEE Sens. J. 15(2), 757–761 (2015).
[Crossref]

Jauregui-Vazquez, D.

D. Jauregui-Vazquez, J. M. Estudillo-Ayala, A. Castillo-Guzman, R. Rojas-Laguna, R. Selvas-Aguilar, E. Vargas-Rodriguez, J. M. Sierra-Hernandez, V. Guzman-Ramos, and A. Flores-Balderas, “Highly sensitive curvature and displacement sensing setup based on an all fiber micro Fabry–Perot interferometer,” Opt. Commun. 308, 289–292 (2013).
[Crossref]

Jones, J. D. C.

Kang, J.

X. Dong, Y. Liu, L.-Y. Shao, J. Kang, and C.-L. Zhao, “Temperature-independent fiber bending sensor based on a superimposed grating,” IEEE Sens. J. 11(11), 3019–3022 (2011).
[Crossref]

Lee, B.

B. Lee, “Review of the present status of optical fiber sensors,” Opt. Fiber Technol. 9(2), 57–79 (2003).
[Crossref]

Leonardi, A. A.

J. B. Rosolem, D. C. Dini, R. S. Penze, C. Floridia, A. A. Leonardi, M. D. Loichate, and A. S. Durelli, “Fiber optic bending sensor for water level monitoring: development and field test: a review,” IEEE Sens. J. 13(11), 4113–4120 (2013).
[Crossref]

Li, K.

G. Liu, K. Li, P. Hao, W. Zhou, Y. Wu, and M. Xuan, “Bent optical fiber taper for refractive index detection with a high sensitivity,” Sens. Actuators A Phys. 201, 352–356 (2013).
[Crossref]

Lin, F.

Liu, G.

G. Liu, M. Han, and W. Hou, “High-resolution and fast-response fiber-optic temperature sensor using silicon Fabry-Pérot cavity,” Opt. Express 23(6), 7237–7247 (2015).
[Crossref] [PubMed]

G. Liu, K. Li, P. Hao, W. Zhou, Y. Wu, and M. Xuan, “Bent optical fiber taper for refractive index detection with a high sensitivity,” Sens. Actuators A Phys. 201, 352–356 (2013).
[Crossref]

Liu, Y.

X. Dong, Y. Liu, L.-Y. Shao, J. Kang, and C.-L. Zhao, “Temperature-independent fiber bending sensor based on a superimposed grating,” IEEE Sens. J. 11(11), 3019–3022 (2011).
[Crossref]

Loichate, M. D.

J. B. Rosolem, D. C. Dini, R. S. Penze, C. Floridia, A. A. Leonardi, M. D. Loichate, and A. S. Durelli, “Fiber optic bending sensor for water level monitoring: development and field test: a review,” IEEE Sens. J. 13(11), 4113–4120 (2013).
[Crossref]

Marcuse, D.

Martini, G.

L. Faustini and G. Martini, “Bend loss in single-mode fibers,” J. Lightwave Technol. 15(4), 671–679 (1997).
[Crossref]

Morgan, R.

Ng, S. H.

X. Yang, Z. Chen, C. S. M. Elvin, L. H. Y. Janice, S. H. Ng, J. T. Teo, and R. Wu, “Textile fiber optic microbend sensor used for heartbeat and respiration monitoring,” IEEE Sens. J. 15(2), 757–761 (2015).
[Crossref]

Paladino, D.

A. Iadicicco, D. Paladino, S. Campopiano, W. J. Bock, A. Cutolo, and A. Cusano, “Evanescent wave sensor based on permanently bent single mode optical fiber,” Sens. Actuators B Chem. 155(2), 903–908 (2011).
[Crossref]

Penze, R. S.

J. B. Rosolem, D. C. Dini, R. S. Penze, C. Floridia, A. A. Leonardi, M. D. Loichate, and A. S. Durelli, “Fiber optic bending sensor for water level monitoring: development and field test: a review,” IEEE Sens. J. 13(11), 4113–4120 (2013).
[Crossref]

Rajan, G.

Renner, H.

H. Renner, “Bending losses of coated single-mode fibers: a simple approach,” J. Lightwave Technol. 10(5), 544–551 (1992).
[Crossref]

Rojas-Laguna, R.

D. Jauregui-Vazquez, J. M. Estudillo-Ayala, A. Castillo-Guzman, R. Rojas-Laguna, R. Selvas-Aguilar, E. Vargas-Rodriguez, J. M. Sierra-Hernandez, V. Guzman-Ramos, and A. Flores-Balderas, “Highly sensitive curvature and displacement sensing setup based on an all fiber micro Fabry–Perot interferometer,” Opt. Commun. 308, 289–292 (2013).
[Crossref]

Rosolem, J. B.

J. B. Rosolem, D. C. Dini, R. S. Penze, C. Floridia, A. A. Leonardi, M. D. Loichate, and A. S. Durelli, “Fiber optic bending sensor for water level monitoring: development and field test: a review,” IEEE Sens. J. 13(11), 4113–4120 (2013).
[Crossref]

Selvas-Aguilar, R.

D. Jauregui-Vazquez, J. M. Estudillo-Ayala, A. Castillo-Guzman, R. Rojas-Laguna, R. Selvas-Aguilar, E. Vargas-Rodriguez, J. M. Sierra-Hernandez, V. Guzman-Ramos, and A. Flores-Balderas, “Highly sensitive curvature and displacement sensing setup based on an all fiber micro Fabry–Perot interferometer,” Opt. Commun. 308, 289–292 (2013).
[Crossref]

Shao, L.-Y.

X. Dong, Y. Liu, L.-Y. Shao, J. Kang, and C.-L. Zhao, “Temperature-independent fiber bending sensor based on a superimposed grating,” IEEE Sens. J. 11(11), 3019–3022 (2011).
[Crossref]

Sierra-Hernandez, J. M.

D. Jauregui-Vazquez, J. M. Estudillo-Ayala, A. Castillo-Guzman, R. Rojas-Laguna, R. Selvas-Aguilar, E. Vargas-Rodriguez, J. M. Sierra-Hernandez, V. Guzman-Ramos, and A. Flores-Balderas, “Highly sensitive curvature and displacement sensing setup based on an all fiber micro Fabry–Perot interferometer,” Opt. Commun. 308, 289–292 (2013).
[Crossref]

Sun, H.-B.

Teo, J. T.

X. Yang, Z. Chen, C. S. M. Elvin, L. H. Y. Janice, S. H. Ng, J. T. Teo, and R. Wu, “Textile fiber optic microbend sensor used for heartbeat and respiration monitoring,” IEEE Sens. J. 15(2), 757–761 (2015).
[Crossref]

Vargas-Rodriguez, E.

D. Jauregui-Vazquez, J. M. Estudillo-Ayala, A. Castillo-Guzman, R. Rojas-Laguna, R. Selvas-Aguilar, E. Vargas-Rodriguez, J. M. Sierra-Hernandez, V. Guzman-Ramos, and A. Flores-Balderas, “Highly sensitive curvature and displacement sensing setup based on an all fiber micro Fabry–Perot interferometer,” Opt. Commun. 308, 289–292 (2013).
[Crossref]

Wang, P.

Wang, Q.

Wei, W.

Wu, G.

Wu, R.

X. Yang, Z. Chen, C. S. M. Elvin, L. H. Y. Janice, S. H. Ng, J. T. Teo, and R. Wu, “Textile fiber optic microbend sensor used for heartbeat and respiration monitoring,” IEEE Sens. J. 15(2), 757–761 (2015).
[Crossref]

Wu, X.

Wu, Y.

G. Liu, K. Li, P. Hao, W. Zhou, Y. Wu, and M. Xuan, “Bent optical fiber taper for refractive index detection with a high sensitivity,” Sens. Actuators A Phys. 201, 352–356 (2013).
[Crossref]

Xuan, M.

G. Liu, K. Li, P. Hao, W. Zhou, Y. Wu, and M. Xuan, “Bent optical fiber taper for refractive index detection with a high sensitivity,” Sens. Actuators A Phys. 201, 352–356 (2013).
[Crossref]

Xue, Y.

Yang, R.

Yang, X.

X. Yang, Z. Chen, C. S. M. Elvin, L. H. Y. Janice, S. H. Ng, J. T. Teo, and R. Wu, “Textile fiber optic microbend sensor used for heartbeat and respiration monitoring,” IEEE Sens. J. 15(2), 757–761 (2015).
[Crossref]

Yu, Y.-S.

Yuan, Y.

Zhao, C.-L.

X. Dong, Y. Liu, L.-Y. Shao, J. Kang, and C.-L. Zhao, “Temperature-independent fiber bending sensor based on a superimposed grating,” IEEE Sens. J. 11(11), 3019–3022 (2011).
[Crossref]

Zhou, W.

G. Liu, K. Li, P. Hao, W. Zhou, Y. Wu, and M. Xuan, “Bent optical fiber taper for refractive index detection with a high sensitivity,” Sens. Actuators A Phys. 201, 352–356 (2013).
[Crossref]

IEEE Sens. J. (3)

J. B. Rosolem, D. C. Dini, R. S. Penze, C. Floridia, A. A. Leonardi, M. D. Loichate, and A. S. Durelli, “Fiber optic bending sensor for water level monitoring: development and field test: a review,” IEEE Sens. J. 13(11), 4113–4120 (2013).
[Crossref]

X. Dong, Y. Liu, L.-Y. Shao, J. Kang, and C.-L. Zhao, “Temperature-independent fiber bending sensor based on a superimposed grating,” IEEE Sens. J. 11(11), 3019–3022 (2011).
[Crossref]

X. Yang, Z. Chen, C. S. M. Elvin, L. H. Y. Janice, S. H. Ng, J. T. Teo, and R. Wu, “Textile fiber optic microbend sensor used for heartbeat and respiration monitoring,” IEEE Sens. J. 15(2), 757–761 (2015).
[Crossref]

J. Lightwave Technol. (4)

A. J. Harris and P. F. Castle, “Bend loss measurements on high numerical aperture single-mode fibers as a function of wavelength and bend radius,” J. Lightwave Technol. 4(1), 34–40 (1986).
[Crossref]

H. Renner, “Bending losses of coated single-mode fibers: a simple approach,” J. Lightwave Technol. 10(5), 544–551 (1992).
[Crossref]

L. Faustini and G. Martini, “Bend loss in single-mode fibers,” J. Lightwave Technol. 15(4), 671–679 (1997).
[Crossref]

J. W. Berthold, “Historical review of microbend fiber-optic sensors,” J. Lightwave Technol. 13(7), 1193–1199 (1995).
[Crossref]

J. Opt. Soc. Am. (1)

Opt. Commun. (1)

D. Jauregui-Vazquez, J. M. Estudillo-Ayala, A. Castillo-Guzman, R. Rojas-Laguna, R. Selvas-Aguilar, E. Vargas-Rodriguez, J. M. Sierra-Hernandez, V. Guzman-Ramos, and A. Flores-Balderas, “Highly sensitive curvature and displacement sensing setup based on an all fiber micro Fabry–Perot interferometer,” Opt. Commun. 308, 289–292 (2013).
[Crossref]

Opt. Express (3)

Opt. Fiber Technol. (1)

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[Crossref]

Opt. Lett. (3)

Sens. Actuators A Phys. (1)

G. Liu, K. Li, P. Hao, W. Zhou, Y. Wu, and M. Xuan, “Bent optical fiber taper for refractive index detection with a high sensitivity,” Sens. Actuators A Phys. 201, 352–356 (2013).
[Crossref]

Sens. Actuators B Chem. (1)

A. Iadicicco, D. Paladino, S. Campopiano, W. J. Bock, A. Cutolo, and A. Cusano, “Evanescent wave sensor based on permanently bent single mode optical fiber,” Sens. Actuators B Chem. 155(2), 903–908 (2011).
[Crossref]

Other (3)

W. Hou, G. Liu, and M. Han, “A novel, high-resolution, high-speed fiber-optic temperature sensor for oceanographic applications,” In Proceedings of the 2015 IEEE/OES 11th Current, Waves and Turbulence Measurement Conference (CWTM 2015) (2015).

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Figures (6)

Fig. 1
Fig. 1 (a) Schematic representation of the radiation losses introduced by bending to the lead-in single-mode fiber of the sensor. (b) Reflection spectrum of a 200-µm-thick silicon FPI without disturbance from bending. (c) Schematic showing some parameters to model the sensor head.
Fig. 2
Fig. 2 Theoretical (a) transition loss coefficient and (b) pure bend loss coefficient as a function of wavelength with different bending radii. (c) The according reflection spectra with different bending radii. In (c), both transition loss and pure bend loss are included, the pink solid and void dots denote, respectively, the peak and valley wavelengths that are calculated in Fig. 3(a).
Fig. 3
Fig. 3 BIWS as functions of (a) bending radius, (b) length of bending section, (c) visibility of the FPI, and (d) bending radius for different cavity lengths of the silicon FPI.
Fig. 4
Fig. 4 Experimental setup. Inset shows a 200-µm-long, 100-µm-diameter silicon cylinder mounted on the end face of a SMF, and the box on the right shows top view of the bending angle.
Fig. 5
Fig. 5 Normalized reflection spectra (a) and relative wavelength and peak intensity versus time (b) during the bending-and-releasing experiment on a mandrel diameter of 13.5 mm. The spectrum at time t1, t2 and t3 in (a) were captured at the initial, intermediate, and ultimate bending moments, respectively, which are located by the green arrows in (b). The red, black and pink circles in (a) highlight the peak-valley pair 1, 2 and 3, respectively, which were tracked during the bending process and are shown in (b).
Fig. 6
Fig. 6 (a) BIWS versus bending angle. Angle of 180 degrees means a half-turn bend. Relative peak wavelength during the bending-and-releasing experiments for the silicon FPIs with different (b) visibilities of 0.45, 0.66, 0.79 and (c) cavity lengths of 10 and 200 µm. In these experiments, diameter of the mandrel was 21 mm.

Equations (23)

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R( λ )=B( λ )F( λ ),
dR( λ ) dλ = dB( λ ) dλ F( λ )+ dF( λ ) dλ B( λ )=0.
dF( λ ) dλ = dB( λ ) dλ F( λ ) B( λ ) .
T Tran =exp( α 1 )=exp[ 1 16 ( a R ) 2 V 4 ( lnV ) 3 n 1 4 ( n 1 2 n 2 2 ) 2 ],
T Bend =exp( α 2 L ),
α 2 = 2( k 2 n 1 2 β 0 2 ) β 0 V 2 K 1 2 ( aγ ) 0 exp[ a ( γ 2 + ζ 2 ) 1/2 ] ( γ 2 + ζ 2 ) 1/2 Ai[ X 2 ( 0,ζ ) ] Bi[ X 2 ( a,ζ ) ] x 2 1/2 x 3 1/3 x 2 cos 2 θ( ζ )+ x 3 sin 2 θ( ζ ) dζ,
{ X q ( y,ζ )= ( R 2 k 2 n q 2 ) 2/3 [ β 0 2 + ζ 2 k 2 n q 2 ( 1+ 2y R ) ], q=2,3 x q = X q ( b,ζ ) ( 2 k 2 n q 2 R ) 2/3 , q=2,3 θ( ζ )= 2 3 [ X 2 ( b,ζ ) ] 3/2 + π 4 ,
B( λ )= ( T Tran ) 4 ( T Bend ) 2 =exp( 4 α 1 2 α 2 L ).
F( λ )= R 1 + R 2 2 R 1 R 2 cos( 2π 2 n si d λ ) 1+ R 1 R 2 2 R 1 R 2 cos( 2π 2 n si d λ ) ,
F( λ )= R 1 + R 2 2 R 1 R 2 cos( 2π 2 n si d λ ).
dF( λ ) dλ | λ PN, λ VN = 8π n si d R 1 R 2 λ 2 sin( 4π n si d λ ) | λ PN, λ VN =0,
Δ λ PN =( 4 α 1 λ | λ ¯ PN +2L α 2 λ | λ ¯ PN )( 1 V b +1 ) λ PN 4 ( 4π n si d ) 2 .
Δ λ VN =( 4 α 1 λ | λ ¯ VN +2L α 2 λ | λ ¯ VN )( 1 V b 1 ) λ VN 4 ( 4π n si d ) 2 .
dB( λ ) dλ =( 4 d α 1 dλ +2L d α 2 dλ )exp( 4 α 1 2 α 2 L )=( 4 d α 1 dλ +2L d α 2 dλ )B( λ ).
8π n si d R 1 R 2 λ 2 sin( 4π n si d λ ) | λ ¯ PN , λ ¯ VN = ( 4 d α 1 dλ +2L d α 2 dλ )[ R 1 + R 2 2 R 1 R 2 cos( 2π 2 n si d λ ) ] | λ ¯ PN , λ ¯ VN
8π n si d R 1 R 2 ( λ PN +Δ λ PN ) 2 sin( 4π n si d λ PN +Δ λ PN )=( 4 d α 1 dλ | λ ¯ PN + 2L d α 2 dλ | λ ¯ PN )[ R 1 + R 2 2 R 1 R 2 cos( 2π 2 n si d λ PN +Δ λ PN ) ]
8π n si d R 1 R 2 ( λ VN +Δ λ VN ) 2 sin( 4π n si d λ VN +Δ λ VN )=( 4 d α 1 dλ | λ ¯ VN + 2L d α 2 dλ | λ ¯ VN )[ R 1 + R 2 2 R 1 R 2 cos( 2π 2 n si d λ VN +Δ λ VN ) ]
8π n si d R 1 R 2 λ PN 2 sin[ 4π n si d( 1 λ PN Δ λ PN λ PN 2 ) ]= ( 4 d α 1 dλ | λ ¯ PN + 2L d α 2 dλ | λ ¯ PN ){ R 1 + R 2 2 R 1 R 2 cos[ 4π n si d( 1 λ PN Δ λ PN λ PN 2 ) ] },
8π n si d R 1 R 2 λ VN 2 sin[ 4π n si d( 1 λ VN Δ λ VN λ VN 2 ) ]= ( 4 d α 1 dλ | λ ¯ VN + 2L d α 2 dλ | λ ¯ VN ){ R 1 + R 2 2 R 1 R 2 cos[ 4π n si d( 1 λ VN Δ λ VN λ VN 2 ) ] }.
{ sin( 4π n si d λ PN )=sin( 4π n si d λ VN )=0 cos( 4π n si d λ PN )=1, cos( 4π n si d λ VN )=1 .
Δ λ PN =( 4 α 1 λ | λ ¯ PN +2L α 2 λ | λ ¯ PN )( R 1 + R 2 2 R 1 R 2 +1 ) λ PN 4 ( 4π n si d ) 2 ,
Δ λ VN =( 4 α 1 λ | λ ¯ VN +2L α 2 λ | λ ¯ VN )( R 1 + R 2 2 R 1 R 2 1 ) λ VN 4 ( 4π n si d ) 2 .
V b = F max F min F max + F min = 2 R 1 R 2 R 1 + R 2 ,

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