Abstract

The large-range and high-sensitivity strain measurement in high-temperature ambiance is a great challenge in engineering applications. Because of the frangibility of the glass material, the traditional optical fiber strain sensors cannot endure a limit strain of 1%. To break through the limit, we propose a hybrid silica/polymer optical fiber sensor. It can endure extraordinarily large strain. The maximum strain of 35% is confirmed by experiments. To achieve high sensitivity and detect a small change in strain, a phase tracking method is used. The sensitivity of the sensor is 28 pm/με which is 28 times larger than that of the traditional FBG sensors. In addition, because of the excellent high-temperature endurance of polyimide (PI) and adhesive, the sensor can survive in the high temperature up to 220 °C. The proposed hybrid silica/polymer optical fiber sensor has potentials to monitor deformation in plastic products, structure health in composite materials, and even strain in biomaterials.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  1. Y. W. Huang, J. Tao, and X. G. Huang, “Research Progress on F-P Interference-Based Fiber-Optic Sensors,” Sensors (Basel) 16(9), 1424 (2016).
    [Crossref] [PubMed]
  2. Y. Liu and J. Zhang, “Model Study of the Influence of Ambient Temperature and Installation Types on Surface Temperature Measurement by Using a Fiber Bragg Grating Sensor,” Sensors (Basel) 16(7), 975 (2016).
    [Crossref] [PubMed]
  3. Y. Liu, Y. Li, P. Huang, H. Song, and G. Zhang, “Modeling of hydrogen atom diffusion and response behavior of hydrogen sensors in Pd-Y alloy nanofilm,” Sci. Rep. 6(1), 37043 (2016).
    [Crossref] [PubMed]
  4. M. Yucel and N. F. Ozturk, “Real-time monitoring of railroad track tension using a fiber Bragg grating-based strain sensor,” Instrum. Sci. Technol. 46(5), 519–533 (2018).
    [Crossref]
  5. X. W. Ye, Y. H. Su, and P. S. Xi, “Statistical Analysis of Stress Signals from Bridge Monitoring by FBG System,” Sensors (Basel) 18(2), 491 (2018).
    [Crossref] [PubMed]
  6. M. J. Nicolas, R. W. Sullivan, and W. L. Richards, “Large Scale Applications Using FBG Sensors: Determination of In-Flight Loads and Shape of a Composite Aircraft Wing,” Aerospace 3(3), 18 (2016).
    [Crossref]
  7. M. Liu, E. Zhang, Z. Zhou, Y. Tan, and Y. Liu, “Measurement of Temperature Field for the Spindle of Machine Tool Based on Optical Fiber Bragg Grating Sensors,” Adv. Mech. Eng. 5, 940626 (2013).
    [Crossref]
  8. Y. Huang, Z. Zhou, Y. Zhang, G. Chen, and H. Xiao, “A Temperature Self-Compensated LPFG Sensor for Large Strain Measurements at High Temperature,” IEEE Trans. Instrum. Meas. 59(11), 2997–3004 (2010).
    [Crossref]
  9. L. Zhang, X. Chang, Y. Zhang, and F. Yang, “Large Strain Detection of SRM Composite Shell Based on Fiber Bragg Grating Sensor,” Photonic Sens. 7(4), 350–356 (2017).
    [Crossref]
  10. H. Y. Liu, H. B. Liu, and G. D. Peng, “Tensile strain characterization of polymer optical fibre Bragg gratings,” Opt. Commun. 251(1-3), 37–43 (2005).
    [Crossref]
  11. J. Huang, X. Lan, H. Wang, L. Yuan, T. Wei, Z. Gao, and H. Xiao, “Polymer optical fiber for large strain measurement based on multimode interference,” Opt. Lett. 37(20), 4308–4310 (2012).
    [Crossref] [PubMed]
  12. Y. Huang, T. Wei, Z. Zhou, Y. Zhang, G. Chen, and H. Xiao, “An extrinsic Fabry–Perot interferometer-based large strain sensor with high resolution,” Meas. Sci. Technol. 21(10), 105308 (2010).
    [Crossref]
  13. H. Huang, A. Majumdar, and J. Cho, “Fabrication and evaluation of hybrid silica/polymer optical fiber sensors for large strain measurement,” T. I. Meas. Control 31(3-4), 247–257 (2009).
    [Crossref]

2018 (2)

M. Yucel and N. F. Ozturk, “Real-time monitoring of railroad track tension using a fiber Bragg grating-based strain sensor,” Instrum. Sci. Technol. 46(5), 519–533 (2018).
[Crossref]

X. W. Ye, Y. H. Su, and P. S. Xi, “Statistical Analysis of Stress Signals from Bridge Monitoring by FBG System,” Sensors (Basel) 18(2), 491 (2018).
[Crossref] [PubMed]

2017 (1)

L. Zhang, X. Chang, Y. Zhang, and F. Yang, “Large Strain Detection of SRM Composite Shell Based on Fiber Bragg Grating Sensor,” Photonic Sens. 7(4), 350–356 (2017).
[Crossref]

2016 (4)

Y. W. Huang, J. Tao, and X. G. Huang, “Research Progress on F-P Interference-Based Fiber-Optic Sensors,” Sensors (Basel) 16(9), 1424 (2016).
[Crossref] [PubMed]

Y. Liu and J. Zhang, “Model Study of the Influence of Ambient Temperature and Installation Types on Surface Temperature Measurement by Using a Fiber Bragg Grating Sensor,” Sensors (Basel) 16(7), 975 (2016).
[Crossref] [PubMed]

Y. Liu, Y. Li, P. Huang, H. Song, and G. Zhang, “Modeling of hydrogen atom diffusion and response behavior of hydrogen sensors in Pd-Y alloy nanofilm,” Sci. Rep. 6(1), 37043 (2016).
[Crossref] [PubMed]

M. J. Nicolas, R. W. Sullivan, and W. L. Richards, “Large Scale Applications Using FBG Sensors: Determination of In-Flight Loads and Shape of a Composite Aircraft Wing,” Aerospace 3(3), 18 (2016).
[Crossref]

2013 (1)

M. Liu, E. Zhang, Z. Zhou, Y. Tan, and Y. Liu, “Measurement of Temperature Field for the Spindle of Machine Tool Based on Optical Fiber Bragg Grating Sensors,” Adv. Mech. Eng. 5, 940626 (2013).
[Crossref]

2012 (1)

2010 (2)

Y. Huang, T. Wei, Z. Zhou, Y. Zhang, G. Chen, and H. Xiao, “An extrinsic Fabry–Perot interferometer-based large strain sensor with high resolution,” Meas. Sci. Technol. 21(10), 105308 (2010).
[Crossref]

Y. Huang, Z. Zhou, Y. Zhang, G. Chen, and H. Xiao, “A Temperature Self-Compensated LPFG Sensor for Large Strain Measurements at High Temperature,” IEEE Trans. Instrum. Meas. 59(11), 2997–3004 (2010).
[Crossref]

2009 (1)

H. Huang, A. Majumdar, and J. Cho, “Fabrication and evaluation of hybrid silica/polymer optical fiber sensors for large strain measurement,” T. I. Meas. Control 31(3-4), 247–257 (2009).
[Crossref]

2005 (1)

H. Y. Liu, H. B. Liu, and G. D. Peng, “Tensile strain characterization of polymer optical fibre Bragg gratings,” Opt. Commun. 251(1-3), 37–43 (2005).
[Crossref]

Chang, X.

L. Zhang, X. Chang, Y. Zhang, and F. Yang, “Large Strain Detection of SRM Composite Shell Based on Fiber Bragg Grating Sensor,” Photonic Sens. 7(4), 350–356 (2017).
[Crossref]

Chen, G.

Y. Huang, Z. Zhou, Y. Zhang, G. Chen, and H. Xiao, “A Temperature Self-Compensated LPFG Sensor for Large Strain Measurements at High Temperature,” IEEE Trans. Instrum. Meas. 59(11), 2997–3004 (2010).
[Crossref]

Y. Huang, T. Wei, Z. Zhou, Y. Zhang, G. Chen, and H. Xiao, “An extrinsic Fabry–Perot interferometer-based large strain sensor with high resolution,” Meas. Sci. Technol. 21(10), 105308 (2010).
[Crossref]

Cho, J.

H. Huang, A. Majumdar, and J. Cho, “Fabrication and evaluation of hybrid silica/polymer optical fiber sensors for large strain measurement,” T. I. Meas. Control 31(3-4), 247–257 (2009).
[Crossref]

Gao, Z.

Huang, H.

H. Huang, A. Majumdar, and J. Cho, “Fabrication and evaluation of hybrid silica/polymer optical fiber sensors for large strain measurement,” T. I. Meas. Control 31(3-4), 247–257 (2009).
[Crossref]

Huang, J.

Huang, P.

Y. Liu, Y. Li, P. Huang, H. Song, and G. Zhang, “Modeling of hydrogen atom diffusion and response behavior of hydrogen sensors in Pd-Y alloy nanofilm,” Sci. Rep. 6(1), 37043 (2016).
[Crossref] [PubMed]

Huang, X. G.

Y. W. Huang, J. Tao, and X. G. Huang, “Research Progress on F-P Interference-Based Fiber-Optic Sensors,” Sensors (Basel) 16(9), 1424 (2016).
[Crossref] [PubMed]

Huang, Y.

Y. Huang, Z. Zhou, Y. Zhang, G. Chen, and H. Xiao, “A Temperature Self-Compensated LPFG Sensor for Large Strain Measurements at High Temperature,” IEEE Trans. Instrum. Meas. 59(11), 2997–3004 (2010).
[Crossref]

Y. Huang, T. Wei, Z. Zhou, Y. Zhang, G. Chen, and H. Xiao, “An extrinsic Fabry–Perot interferometer-based large strain sensor with high resolution,” Meas. Sci. Technol. 21(10), 105308 (2010).
[Crossref]

Huang, Y. W.

Y. W. Huang, J. Tao, and X. G. Huang, “Research Progress on F-P Interference-Based Fiber-Optic Sensors,” Sensors (Basel) 16(9), 1424 (2016).
[Crossref] [PubMed]

Lan, X.

Li, Y.

Y. Liu, Y. Li, P. Huang, H. Song, and G. Zhang, “Modeling of hydrogen atom diffusion and response behavior of hydrogen sensors in Pd-Y alloy nanofilm,” Sci. Rep. 6(1), 37043 (2016).
[Crossref] [PubMed]

Liu, H. B.

H. Y. Liu, H. B. Liu, and G. D. Peng, “Tensile strain characterization of polymer optical fibre Bragg gratings,” Opt. Commun. 251(1-3), 37–43 (2005).
[Crossref]

Liu, H. Y.

H. Y. Liu, H. B. Liu, and G. D. Peng, “Tensile strain characterization of polymer optical fibre Bragg gratings,” Opt. Commun. 251(1-3), 37–43 (2005).
[Crossref]

Liu, M.

M. Liu, E. Zhang, Z. Zhou, Y. Tan, and Y. Liu, “Measurement of Temperature Field for the Spindle of Machine Tool Based on Optical Fiber Bragg Grating Sensors,” Adv. Mech. Eng. 5, 940626 (2013).
[Crossref]

Liu, Y.

Y. Liu, Y. Li, P. Huang, H. Song, and G. Zhang, “Modeling of hydrogen atom diffusion and response behavior of hydrogen sensors in Pd-Y alloy nanofilm,” Sci. Rep. 6(1), 37043 (2016).
[Crossref] [PubMed]

Y. Liu and J. Zhang, “Model Study of the Influence of Ambient Temperature and Installation Types on Surface Temperature Measurement by Using a Fiber Bragg Grating Sensor,” Sensors (Basel) 16(7), 975 (2016).
[Crossref] [PubMed]

M. Liu, E. Zhang, Z. Zhou, Y. Tan, and Y. Liu, “Measurement of Temperature Field for the Spindle of Machine Tool Based on Optical Fiber Bragg Grating Sensors,” Adv. Mech. Eng. 5, 940626 (2013).
[Crossref]

Majumdar, A.

H. Huang, A. Majumdar, and J. Cho, “Fabrication and evaluation of hybrid silica/polymer optical fiber sensors for large strain measurement,” T. I. Meas. Control 31(3-4), 247–257 (2009).
[Crossref]

Nicolas, M. J.

M. J. Nicolas, R. W. Sullivan, and W. L. Richards, “Large Scale Applications Using FBG Sensors: Determination of In-Flight Loads and Shape of a Composite Aircraft Wing,” Aerospace 3(3), 18 (2016).
[Crossref]

Ozturk, N. F.

M. Yucel and N. F. Ozturk, “Real-time monitoring of railroad track tension using a fiber Bragg grating-based strain sensor,” Instrum. Sci. Technol. 46(5), 519–533 (2018).
[Crossref]

Peng, G. D.

H. Y. Liu, H. B. Liu, and G. D. Peng, “Tensile strain characterization of polymer optical fibre Bragg gratings,” Opt. Commun. 251(1-3), 37–43 (2005).
[Crossref]

Richards, W. L.

M. J. Nicolas, R. W. Sullivan, and W. L. Richards, “Large Scale Applications Using FBG Sensors: Determination of In-Flight Loads and Shape of a Composite Aircraft Wing,” Aerospace 3(3), 18 (2016).
[Crossref]

Song, H.

Y. Liu, Y. Li, P. Huang, H. Song, and G. Zhang, “Modeling of hydrogen atom diffusion and response behavior of hydrogen sensors in Pd-Y alloy nanofilm,” Sci. Rep. 6(1), 37043 (2016).
[Crossref] [PubMed]

Su, Y. H.

X. W. Ye, Y. H. Su, and P. S. Xi, “Statistical Analysis of Stress Signals from Bridge Monitoring by FBG System,” Sensors (Basel) 18(2), 491 (2018).
[Crossref] [PubMed]

Sullivan, R. W.

M. J. Nicolas, R. W. Sullivan, and W. L. Richards, “Large Scale Applications Using FBG Sensors: Determination of In-Flight Loads and Shape of a Composite Aircraft Wing,” Aerospace 3(3), 18 (2016).
[Crossref]

Tan, Y.

M. Liu, E. Zhang, Z. Zhou, Y. Tan, and Y. Liu, “Measurement of Temperature Field for the Spindle of Machine Tool Based on Optical Fiber Bragg Grating Sensors,” Adv. Mech. Eng. 5, 940626 (2013).
[Crossref]

Tao, J.

Y. W. Huang, J. Tao, and X. G. Huang, “Research Progress on F-P Interference-Based Fiber-Optic Sensors,” Sensors (Basel) 16(9), 1424 (2016).
[Crossref] [PubMed]

Wang, H.

Wei, T.

J. Huang, X. Lan, H. Wang, L. Yuan, T. Wei, Z. Gao, and H. Xiao, “Polymer optical fiber for large strain measurement based on multimode interference,” Opt. Lett. 37(20), 4308–4310 (2012).
[Crossref] [PubMed]

Y. Huang, T. Wei, Z. Zhou, Y. Zhang, G. Chen, and H. Xiao, “An extrinsic Fabry–Perot interferometer-based large strain sensor with high resolution,” Meas. Sci. Technol. 21(10), 105308 (2010).
[Crossref]

Xi, P. S.

X. W. Ye, Y. H. Su, and P. S. Xi, “Statistical Analysis of Stress Signals from Bridge Monitoring by FBG System,” Sensors (Basel) 18(2), 491 (2018).
[Crossref] [PubMed]

Xiao, H.

J. Huang, X. Lan, H. Wang, L. Yuan, T. Wei, Z. Gao, and H. Xiao, “Polymer optical fiber for large strain measurement based on multimode interference,” Opt. Lett. 37(20), 4308–4310 (2012).
[Crossref] [PubMed]

Y. Huang, T. Wei, Z. Zhou, Y. Zhang, G. Chen, and H. Xiao, “An extrinsic Fabry–Perot interferometer-based large strain sensor with high resolution,” Meas. Sci. Technol. 21(10), 105308 (2010).
[Crossref]

Y. Huang, Z. Zhou, Y. Zhang, G. Chen, and H. Xiao, “A Temperature Self-Compensated LPFG Sensor for Large Strain Measurements at High Temperature,” IEEE Trans. Instrum. Meas. 59(11), 2997–3004 (2010).
[Crossref]

Yang, F.

L. Zhang, X. Chang, Y. Zhang, and F. Yang, “Large Strain Detection of SRM Composite Shell Based on Fiber Bragg Grating Sensor,” Photonic Sens. 7(4), 350–356 (2017).
[Crossref]

Ye, X. W.

X. W. Ye, Y. H. Su, and P. S. Xi, “Statistical Analysis of Stress Signals from Bridge Monitoring by FBG System,” Sensors (Basel) 18(2), 491 (2018).
[Crossref] [PubMed]

Yuan, L.

Yucel, M.

M. Yucel and N. F. Ozturk, “Real-time monitoring of railroad track tension using a fiber Bragg grating-based strain sensor,” Instrum. Sci. Technol. 46(5), 519–533 (2018).
[Crossref]

Zhang, E.

M. Liu, E. Zhang, Z. Zhou, Y. Tan, and Y. Liu, “Measurement of Temperature Field for the Spindle of Machine Tool Based on Optical Fiber Bragg Grating Sensors,” Adv. Mech. Eng. 5, 940626 (2013).
[Crossref]

Zhang, G.

Y. Liu, Y. Li, P. Huang, H. Song, and G. Zhang, “Modeling of hydrogen atom diffusion and response behavior of hydrogen sensors in Pd-Y alloy nanofilm,” Sci. Rep. 6(1), 37043 (2016).
[Crossref] [PubMed]

Zhang, J.

Y. Liu and J. Zhang, “Model Study of the Influence of Ambient Temperature and Installation Types on Surface Temperature Measurement by Using a Fiber Bragg Grating Sensor,” Sensors (Basel) 16(7), 975 (2016).
[Crossref] [PubMed]

Zhang, L.

L. Zhang, X. Chang, Y. Zhang, and F. Yang, “Large Strain Detection of SRM Composite Shell Based on Fiber Bragg Grating Sensor,” Photonic Sens. 7(4), 350–356 (2017).
[Crossref]

Zhang, Y.

L. Zhang, X. Chang, Y. Zhang, and F. Yang, “Large Strain Detection of SRM Composite Shell Based on Fiber Bragg Grating Sensor,” Photonic Sens. 7(4), 350–356 (2017).
[Crossref]

Y. Huang, Z. Zhou, Y. Zhang, G. Chen, and H. Xiao, “A Temperature Self-Compensated LPFG Sensor for Large Strain Measurements at High Temperature,” IEEE Trans. Instrum. Meas. 59(11), 2997–3004 (2010).
[Crossref]

Y. Huang, T. Wei, Z. Zhou, Y. Zhang, G. Chen, and H. Xiao, “An extrinsic Fabry–Perot interferometer-based large strain sensor with high resolution,” Meas. Sci. Technol. 21(10), 105308 (2010).
[Crossref]

Zhou, Z.

M. Liu, E. Zhang, Z. Zhou, Y. Tan, and Y. Liu, “Measurement of Temperature Field for the Spindle of Machine Tool Based on Optical Fiber Bragg Grating Sensors,” Adv. Mech. Eng. 5, 940626 (2013).
[Crossref]

Y. Huang, Z. Zhou, Y. Zhang, G. Chen, and H. Xiao, “A Temperature Self-Compensated LPFG Sensor for Large Strain Measurements at High Temperature,” IEEE Trans. Instrum. Meas. 59(11), 2997–3004 (2010).
[Crossref]

Y. Huang, T. Wei, Z. Zhou, Y. Zhang, G. Chen, and H. Xiao, “An extrinsic Fabry–Perot interferometer-based large strain sensor with high resolution,” Meas. Sci. Technol. 21(10), 105308 (2010).
[Crossref]

Adv. Mech. Eng. (1)

M. Liu, E. Zhang, Z. Zhou, Y. Tan, and Y. Liu, “Measurement of Temperature Field for the Spindle of Machine Tool Based on Optical Fiber Bragg Grating Sensors,” Adv. Mech. Eng. 5, 940626 (2013).
[Crossref]

Aerospace (1)

M. J. Nicolas, R. W. Sullivan, and W. L. Richards, “Large Scale Applications Using FBG Sensors: Determination of In-Flight Loads and Shape of a Composite Aircraft Wing,” Aerospace 3(3), 18 (2016).
[Crossref]

IEEE Trans. Instrum. Meas. (1)

Y. Huang, Z. Zhou, Y. Zhang, G. Chen, and H. Xiao, “A Temperature Self-Compensated LPFG Sensor for Large Strain Measurements at High Temperature,” IEEE Trans. Instrum. Meas. 59(11), 2997–3004 (2010).
[Crossref]

Instrum. Sci. Technol. (1)

M. Yucel and N. F. Ozturk, “Real-time monitoring of railroad track tension using a fiber Bragg grating-based strain sensor,” Instrum. Sci. Technol. 46(5), 519–533 (2018).
[Crossref]

Meas. Sci. Technol. (1)

Y. Huang, T. Wei, Z. Zhou, Y. Zhang, G. Chen, and H. Xiao, “An extrinsic Fabry–Perot interferometer-based large strain sensor with high resolution,” Meas. Sci. Technol. 21(10), 105308 (2010).
[Crossref]

Opt. Commun. (1)

H. Y. Liu, H. B. Liu, and G. D. Peng, “Tensile strain characterization of polymer optical fibre Bragg gratings,” Opt. Commun. 251(1-3), 37–43 (2005).
[Crossref]

Opt. Lett. (1)

Photonic Sens. (1)

L. Zhang, X. Chang, Y. Zhang, and F. Yang, “Large Strain Detection of SRM Composite Shell Based on Fiber Bragg Grating Sensor,” Photonic Sens. 7(4), 350–356 (2017).
[Crossref]

Sci. Rep. (1)

Y. Liu, Y. Li, P. Huang, H. Song, and G. Zhang, “Modeling of hydrogen atom diffusion and response behavior of hydrogen sensors in Pd-Y alloy nanofilm,” Sci. Rep. 6(1), 37043 (2016).
[Crossref] [PubMed]

Sensors (Basel) (3)

X. W. Ye, Y. H. Su, and P. S. Xi, “Statistical Analysis of Stress Signals from Bridge Monitoring by FBG System,” Sensors (Basel) 18(2), 491 (2018).
[Crossref] [PubMed]

Y. W. Huang, J. Tao, and X. G. Huang, “Research Progress on F-P Interference-Based Fiber-Optic Sensors,” Sensors (Basel) 16(9), 1424 (2016).
[Crossref] [PubMed]

Y. Liu and J. Zhang, “Model Study of the Influence of Ambient Temperature and Installation Types on Surface Temperature Measurement by Using a Fiber Bragg Grating Sensor,” Sensors (Basel) 16(7), 975 (2016).
[Crossref] [PubMed]

T. I. Meas. Control (1)

H. Huang, A. Majumdar, and J. Cho, “Fabrication and evaluation of hybrid silica/polymer optical fiber sensors for large strain measurement,” T. I. Meas. Control 31(3-4), 247–257 (2009).
[Crossref]

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Figures (8)

Fig. 1
Fig. 1 (a) Schematic of a hybrid silica/polymer F-P optical fiber sensor structure and (b) the test system.
Fig. 2
Fig. 2 Microscope image of the up-taper fiber and the PI tube
Fig. 3
Fig. 3 (a) Experimental photos of the sensor fixed on the precision stages and (b) stretch process of the sensor
Fig. 4
Fig. 4 (a) Reflected spectrum of the F-P cavity with different length and (b) FFT of the reflected spectrum
Fig. 5
Fig. 5 (a) Period of the reflected spectrum of the F-P cavity and (b) strain in the PI tube with different cavity length
Fig. 6
Fig. 6 (a) Reflected spectrum of the sensor; (b) relationship between the wavelength and the strain; (c) response of the sensor to the dynamic strain; (d) repeatability of the sensor in eight loading/unloading cycles
Fig. 7
Fig. 7 Reflected spectrum of the F-P cavity with different temperature
Fig. 8
Fig. 8 (a) FFT of the reflected spectrum in Fig. (7); (b) Calibration curve between the temperature and the period.

Equations (11)

Equations on this page are rendered with MathJax. Learn more.

I r = I 1 + I 2 +2 I 1 I 2 cos( 4πnl λ + φ 0 ),
4πnl λ + φ 0 =( 2m+1 )π (m= 1, 2,),
λ m = 4πnl ( 2m+1 )π- φ 0 .
S p = Δ λ m ε = 4πnΔl ( 2m+1 )π- φ 0 1 ε = λ m0 L l 0
S p =1500nm× 10mm 60μm =0.25nm/με
I r I 1 + I 2 +2 I 1 I 2 cos(4πnl( 1 λ 0 - 1 λ 0 2 ( λ- λ 0 ) )+ φ 0 ) = I 1 + I 2 +2 I 1 I 2 cos( 4πnl λ 0 2 λ+ Φ 0 ).
P= 2π 4πnl λ 0 2 = λ 0 2 2nl
l= λ 0 2 2nP
ε= Δl l 0 = λ 0 2 2n ( 1 P 1 P 0 )/ l 0 = λ 0 2 2n l 0 P 1= P 0 P 1
ΔT= 1 α Δl l 0 = 1 α ( P 0 P 1 )
S= dP dε = P 0 ( ε+1 ) 2 ,