Abstract

A low-cost yet high-sensitivity temperature fiber sensor is proposed and demonstrated in this paper. A single-mode fiber with coating is simply bent in a droplet-like circle with a radius of several millimeters. The strong bending induces mode interferences between the silica core mode and the excited modes propagating in the polymer coating. Many resonant dips were observed in the transmission spectra and are found to shift to a shorter wavelength with the increase of environmental temperature. Our linear fitting result of the experimental data shows that the proposed sensor presents high temperature sensitivity up to 3.102nm/°C, which is even comparable with sensors based on selective liquid-filled photonic crystal fibers. Such high temperature sensitivity results from the large thermo-optical coefficient difference between the silica core and the polymer coating. The influence of a circle radius to the sensitivities is also discussed.

© 2014 Optical Society of America

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References

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  1. L. Yuan, T. Wei, Q. Han, H. Z. Wang, J. Huang, L. Jiang, and H. Xiao, “Fiber inline Michelson interferometer fabricated by a femtosecond laser,” Opt. Lett. 37, 3753–3755 (2012).
    [CrossRef]
  2. T. Y. Hu, Y. Wang, C. R. Liao, and D. N. Wang, “Miniaturized fiber in-line Mach–Zehnder interferometer based on inner air-cavity for high-temperature sensing,” Opt. Lett. 37, 5083–5085 (2012).
  3. S. H. Nam and S. Yin, “High-temperature sensing using whispering gallery mode resonance in bent optical fibers,” IEEE Photon. Technol. Lett. 17, 2391–2393 (2005).
    [CrossRef]
  4. S. W. James and R. P. Tatam, “Optical fibre long-period grating sensors: characteristics and application,” Meas. Sci. Technol. 14, R49–R61 (2003).
    [CrossRef]
  5. G. Brambilla and H. Rutt, “Fiber Bragg gratings with enhanced thermal stability,” Appl. Phys. Lett. 80, 3259–3261 (2002).
    [CrossRef]
  6. Y. Liu, B. Liu, X. Feng, W. Zhang, G. Zhou, S. Yuan, G. Kai, and X. Dong, “High-birefringence fiber loop mirrors and their applications as sensors,” Appl. Opt. 44, 2382–2390 (2005).
    [CrossRef]
  7. W. W. Qian, C. L. Zhao, S. L. He, X. Y. Dong, S. Q. Zhang, Z. X. Zhang, S. Z. Jin, J. T. Guo, and H. F. Wei, “High-sensitivity temperature sensor based on an alcohol-filled photonic crystal fiber loop mirror,” Opt. Lett. 36, 1549–1551 (2011).
  8. Y. Peng, J. Hou, Y. Zhang, Z. H. Huang, R. Xiao, and Q. S. Lu, “Temperature sensing using the bandgap-like effect in a selectively liquid-filled photonic crystal fiber,” Opt. Lett. 38, 263–265 (2013).
    [CrossRef]
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    [CrossRef]
  10. H. Liang, W. G. Zhang, P. C. Geng, Y. Liu, Z. Wang, J. Q. Guo, S. C. Gao, and S. Y. Yan, “Simultaneous measurement of temperature and force with high sensitivities based on filling different index liquids into photonic crystal fiber,” Opt. Lett. 38, 1071–1073 (2013).
    [CrossRef]
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    [CrossRef]
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2013 (2)

2012 (3)

2011 (1)

W. W. Qian, C. L. Zhao, S. L. He, X. Y. Dong, S. Q. Zhang, Z. X. Zhang, S. Z. Jin, J. T. Guo, and H. F. Wei, “High-sensitivity temperature sensor based on an alcohol-filled photonic crystal fiber loop mirror,” Opt. Lett. 36, 1549–1551 (2011).

2009 (1)

P. Wang, Y. Semenova, G. Rajan, T. Freir, and G. Farrell, “The temperature dependence of polarization-dependent loss for a macrobending single-mode-fiber-based edge filter,” IEEE Photon. Technol. Lett. 21, 516–518 (2009).
[CrossRef]

2008 (1)

P. Wang, G. Rajan, Y. Semenova, and G. Farrell, “Temperature dependence of a macrobending edge filter based on a high-bend loss fiber,” Opt. Lett. 33, 2471–2473 (2008).

2007 (1)

P. Wang, Q. Wang, G. Farrell, T. Freir, and J. Cassidy, “Investigation of macrobending losses of standard single mode fiber with small bend radii,” Microw. Opt. Technol. Lett. 49, 9 (2007).
[CrossRef]

2006 (1)

Z. Y. Zhang, P. Zhao, P. Lin, and F. G. Sun, “Thermo-optic coefficients of polymers for optical waveguide applications,” Polymer 47, 4893–4896 (2006).
[CrossRef]

2005 (2)

S. H. Nam and S. Yin, “High-temperature sensing using whispering gallery mode resonance in bent optical fibers,” IEEE Photon. Technol. Lett. 17, 2391–2393 (2005).
[CrossRef]

Y. Liu, B. Liu, X. Feng, W. Zhang, G. Zhou, S. Yuan, G. Kai, and X. Dong, “High-birefringence fiber loop mirrors and their applications as sensors,” Appl. Opt. 44, 2382–2390 (2005).
[CrossRef]

2003 (1)

S. W. James and R. P. Tatam, “Optical fibre long-period grating sensors: characteristics and application,” Meas. Sci. Technol. 14, R49–R61 (2003).
[CrossRef]

2002 (1)

G. Brambilla and H. Rutt, “Fiber Bragg gratings with enhanced thermal stability,” Appl. Phys. Lett. 80, 3259–3261 (2002).
[CrossRef]

1995 (1)

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

1981 (1)

1975 (1)

M. Heiblum and J. Harris, “Analysis of curved optical wave-guides by conformal transformation,” IEEE J. Quantum Electron. 11, 75–83 (1975).
[CrossRef]

Berthold, J. W.

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

Brambilla, G.

G. Brambilla and H. Rutt, “Fiber Bragg gratings with enhanced thermal stability,” Appl. Phys. Lett. 80, 3259–3261 (2002).
[CrossRef]

Cassidy, J.

P. Wang, Q. Wang, G. Farrell, T. Freir, and J. Cassidy, “Investigation of macrobending losses of standard single mode fiber with small bend radii,” Microw. Opt. Technol. Lett. 49, 9 (2007).
[CrossRef]

Dong, X.

Dong, X. Y.

W. W. Qian, C. L. Zhao, S. L. He, X. Y. Dong, S. Q. Zhang, Z. X. Zhang, S. Z. Jin, J. T. Guo, and H. F. Wei, “High-sensitivity temperature sensor based on an alcohol-filled photonic crystal fiber loop mirror,” Opt. Lett. 36, 1549–1551 (2011).

Farrell, G.

P. Wang, Y. Semenova, G. Rajan, T. Freir, and G. Farrell, “The temperature dependence of polarization-dependent loss for a macrobending single-mode-fiber-based edge filter,” IEEE Photon. Technol. Lett. 21, 516–518 (2009).
[CrossRef]

P. Wang, G. Rajan, Y. Semenova, and G. Farrell, “Temperature dependence of a macrobending edge filter based on a high-bend loss fiber,” Opt. Lett. 33, 2471–2473 (2008).

P. Wang, Q. Wang, G. Farrell, T. Freir, and J. Cassidy, “Investigation of macrobending losses of standard single mode fiber with small bend radii,” Microw. Opt. Technol. Lett. 49, 9 (2007).
[CrossRef]

Feng, X.

Freir, T.

P. Wang, Y. Semenova, G. Rajan, T. Freir, and G. Farrell, “The temperature dependence of polarization-dependent loss for a macrobending single-mode-fiber-based edge filter,” IEEE Photon. Technol. Lett. 21, 516–518 (2009).
[CrossRef]

P. Wang, Q. Wang, G. Farrell, T. Freir, and J. Cassidy, “Investigation of macrobending losses of standard single mode fiber with small bend radii,” Microw. Opt. Technol. Lett. 49, 9 (2007).
[CrossRef]

Gao, S. C.

Geng, P. C.

Guo, J. Q.

Guo, J. T.

C. L. Zhao, Z. Q. Wang, S. Q. Zhang, L. Qi, C. Zhong, Z. X. Zhang, S. Z. Jin, J. T. Guo, and H. F. Wei, “Phenomenon in an alcohol not full-filled temperature sensor based on an optical fiber Sagnac interferometer,” Opt. Lett. 37, 4789–4791 (2012).
[CrossRef]

W. W. Qian, C. L. Zhao, S. L. He, X. Y. Dong, S. Q. Zhang, Z. X. Zhang, S. Z. Jin, J. T. Guo, and H. F. Wei, “High-sensitivity temperature sensor based on an alcohol-filled photonic crystal fiber loop mirror,” Opt. Lett. 36, 1549–1551 (2011).

Han, Q.

Harris, J.

M. Heiblum and J. Harris, “Analysis of curved optical wave-guides by conformal transformation,” IEEE J. Quantum Electron. 11, 75–83 (1975).
[CrossRef]

He, S. L.

W. W. Qian, C. L. Zhao, S. L. He, X. Y. Dong, S. Q. Zhang, Z. X. Zhang, S. Z. Jin, J. T. Guo, and H. F. Wei, “High-sensitivity temperature sensor based on an alcohol-filled photonic crystal fiber loop mirror,” Opt. Lett. 36, 1549–1551 (2011).

Heiblum, M.

M. Heiblum and J. Harris, “Analysis of curved optical wave-guides by conformal transformation,” IEEE J. Quantum Electron. 11, 75–83 (1975).
[CrossRef]

Hou, J.

Hu, T. Y.

T. Y. Hu, Y. Wang, C. R. Liao, and D. N. Wang, “Miniaturized fiber in-line Mach–Zehnder interferometer based on inner air-cavity for high-temperature sensing,” Opt. Lett. 37, 5083–5085 (2012).

Huang, J.

Huang, Z. H.

James, S. W.

S. W. James and R. P. Tatam, “Optical fibre long-period grating sensors: characteristics and application,” Meas. Sci. Technol. 14, R49–R61 (2003).
[CrossRef]

Jiang, L.

Jin, S. Z.

C. L. Zhao, Z. Q. Wang, S. Q. Zhang, L. Qi, C. Zhong, Z. X. Zhang, S. Z. Jin, J. T. Guo, and H. F. Wei, “Phenomenon in an alcohol not full-filled temperature sensor based on an optical fiber Sagnac interferometer,” Opt. Lett. 37, 4789–4791 (2012).
[CrossRef]

W. W. Qian, C. L. Zhao, S. L. He, X. Y. Dong, S. Q. Zhang, Z. X. Zhang, S. Z. Jin, J. T. Guo, and H. F. Wei, “High-sensitivity temperature sensor based on an alcohol-filled photonic crystal fiber loop mirror,” Opt. Lett. 36, 1549–1551 (2011).

Kai, G.

Lagakos, N.

Liang, H.

Liao, C. R.

T. Y. Hu, Y. Wang, C. R. Liao, and D. N. Wang, “Miniaturized fiber in-line Mach–Zehnder interferometer based on inner air-cavity for high-temperature sensing,” Opt. Lett. 37, 5083–5085 (2012).

Lin, P.

Z. Y. Zhang, P. Zhao, P. Lin, and F. G. Sun, “Thermo-optic coefficients of polymers for optical waveguide applications,” Polymer 47, 4893–4896 (2006).
[CrossRef]

Litovitz, T.

Liu, B.

Liu, Y.

Lu, Q. S.

Macedo, P.

Meister, R.

Mohr, R.

Nam, S. H.

S. H. Nam and S. Yin, “High-temperature sensing using whispering gallery mode resonance in bent optical fibers,” IEEE Photon. Technol. Lett. 17, 2391–2393 (2005).
[CrossRef]

Peng, Y.

Qi, L.

Qian, W. W.

W. W. Qian, C. L. Zhao, S. L. He, X. Y. Dong, S. Q. Zhang, Z. X. Zhang, S. Z. Jin, J. T. Guo, and H. F. Wei, “High-sensitivity temperature sensor based on an alcohol-filled photonic crystal fiber loop mirror,” Opt. Lett. 36, 1549–1551 (2011).

Rajan, G.

P. Wang, Y. Semenova, G. Rajan, T. Freir, and G. Farrell, “The temperature dependence of polarization-dependent loss for a macrobending single-mode-fiber-based edge filter,” IEEE Photon. Technol. Lett. 21, 516–518 (2009).
[CrossRef]

P. Wang, G. Rajan, Y. Semenova, and G. Farrell, “Temperature dependence of a macrobending edge filter based on a high-bend loss fiber,” Opt. Lett. 33, 2471–2473 (2008).

Rutt, H.

G. Brambilla and H. Rutt, “Fiber Bragg gratings with enhanced thermal stability,” Appl. Phys. Lett. 80, 3259–3261 (2002).
[CrossRef]

Semenova, Y.

P. Wang, Y. Semenova, G. Rajan, T. Freir, and G. Farrell, “The temperature dependence of polarization-dependent loss for a macrobending single-mode-fiber-based edge filter,” IEEE Photon. Technol. Lett. 21, 516–518 (2009).
[CrossRef]

P. Wang, G. Rajan, Y. Semenova, and G. Farrell, “Temperature dependence of a macrobending edge filter based on a high-bend loss fiber,” Opt. Lett. 33, 2471–2473 (2008).

Sun, F. G.

Z. Y. Zhang, P. Zhao, P. Lin, and F. G. Sun, “Thermo-optic coefficients of polymers for optical waveguide applications,” Polymer 47, 4893–4896 (2006).
[CrossRef]

Tatam, R. P.

S. W. James and R. P. Tatam, “Optical fibre long-period grating sensors: characteristics and application,” Meas. Sci. Technol. 14, R49–R61 (2003).
[CrossRef]

Wang, D. N.

T. Y. Hu, Y. Wang, C. R. Liao, and D. N. Wang, “Miniaturized fiber in-line Mach–Zehnder interferometer based on inner air-cavity for high-temperature sensing,” Opt. Lett. 37, 5083–5085 (2012).

Wang, H. Z.

Wang, P.

P. Wang, Y. Semenova, G. Rajan, T. Freir, and G. Farrell, “The temperature dependence of polarization-dependent loss for a macrobending single-mode-fiber-based edge filter,” IEEE Photon. Technol. Lett. 21, 516–518 (2009).
[CrossRef]

P. Wang, G. Rajan, Y. Semenova, and G. Farrell, “Temperature dependence of a macrobending edge filter based on a high-bend loss fiber,” Opt. Lett. 33, 2471–2473 (2008).

P. Wang, Q. Wang, G. Farrell, T. Freir, and J. Cassidy, “Investigation of macrobending losses of standard single mode fiber with small bend radii,” Microw. Opt. Technol. Lett. 49, 9 (2007).
[CrossRef]

Wang, Q.

P. Wang, Q. Wang, G. Farrell, T. Freir, and J. Cassidy, “Investigation of macrobending losses of standard single mode fiber with small bend radii,” Microw. Opt. Technol. Lett. 49, 9 (2007).
[CrossRef]

Wang, Y.

T. Y. Hu, Y. Wang, C. R. Liao, and D. N. Wang, “Miniaturized fiber in-line Mach–Zehnder interferometer based on inner air-cavity for high-temperature sensing,” Opt. Lett. 37, 5083–5085 (2012).

Wang, Z.

Wang, Z. Q.

Wei, H. F.

C. L. Zhao, Z. Q. Wang, S. Q. Zhang, L. Qi, C. Zhong, Z. X. Zhang, S. Z. Jin, J. T. Guo, and H. F. Wei, “Phenomenon in an alcohol not full-filled temperature sensor based on an optical fiber Sagnac interferometer,” Opt. Lett. 37, 4789–4791 (2012).
[CrossRef]

W. W. Qian, C. L. Zhao, S. L. He, X. Y. Dong, S. Q. Zhang, Z. X. Zhang, S. Z. Jin, J. T. Guo, and H. F. Wei, “High-sensitivity temperature sensor based on an alcohol-filled photonic crystal fiber loop mirror,” Opt. Lett. 36, 1549–1551 (2011).

Wei, T.

Xiao, H.

Xiao, R.

Yan, S. Y.

Yin, S.

S. H. Nam and S. Yin, “High-temperature sensing using whispering gallery mode resonance in bent optical fibers,” IEEE Photon. Technol. Lett. 17, 2391–2393 (2005).
[CrossRef]

Yuan, L.

Yuan, S.

Zhang, S. Q.

C. L. Zhao, Z. Q. Wang, S. Q. Zhang, L. Qi, C. Zhong, Z. X. Zhang, S. Z. Jin, J. T. Guo, and H. F. Wei, “Phenomenon in an alcohol not full-filled temperature sensor based on an optical fiber Sagnac interferometer,” Opt. Lett. 37, 4789–4791 (2012).
[CrossRef]

W. W. Qian, C. L. Zhao, S. L. He, X. Y. Dong, S. Q. Zhang, Z. X. Zhang, S. Z. Jin, J. T. Guo, and H. F. Wei, “High-sensitivity temperature sensor based on an alcohol-filled photonic crystal fiber loop mirror,” Opt. Lett. 36, 1549–1551 (2011).

Zhang, W.

Zhang, W. G.

Zhang, Y.

Zhang, Z. X.

C. L. Zhao, Z. Q. Wang, S. Q. Zhang, L. Qi, C. Zhong, Z. X. Zhang, S. Z. Jin, J. T. Guo, and H. F. Wei, “Phenomenon in an alcohol not full-filled temperature sensor based on an optical fiber Sagnac interferometer,” Opt. Lett. 37, 4789–4791 (2012).
[CrossRef]

W. W. Qian, C. L. Zhao, S. L. He, X. Y. Dong, S. Q. Zhang, Z. X. Zhang, S. Z. Jin, J. T. Guo, and H. F. Wei, “High-sensitivity temperature sensor based on an alcohol-filled photonic crystal fiber loop mirror,” Opt. Lett. 36, 1549–1551 (2011).

Zhang, Z. Y.

Z. Y. Zhang, P. Zhao, P. Lin, and F. G. Sun, “Thermo-optic coefficients of polymers for optical waveguide applications,” Polymer 47, 4893–4896 (2006).
[CrossRef]

Zhao, C. L.

C. L. Zhao, Z. Q. Wang, S. Q. Zhang, L. Qi, C. Zhong, Z. X. Zhang, S. Z. Jin, J. T. Guo, and H. F. Wei, “Phenomenon in an alcohol not full-filled temperature sensor based on an optical fiber Sagnac interferometer,” Opt. Lett. 37, 4789–4791 (2012).
[CrossRef]

W. W. Qian, C. L. Zhao, S. L. He, X. Y. Dong, S. Q. Zhang, Z. X. Zhang, S. Z. Jin, J. T. Guo, and H. F. Wei, “High-sensitivity temperature sensor based on an alcohol-filled photonic crystal fiber loop mirror,” Opt. Lett. 36, 1549–1551 (2011).

Zhao, P.

Z. Y. Zhang, P. Zhao, P. Lin, and F. G. Sun, “Thermo-optic coefficients of polymers for optical waveguide applications,” Polymer 47, 4893–4896 (2006).
[CrossRef]

Zhong, C.

Zhou, G.

Appl. Opt. (2)

Appl. Phys. Lett. (1)

G. Brambilla and H. Rutt, “Fiber Bragg gratings with enhanced thermal stability,” Appl. Phys. Lett. 80, 3259–3261 (2002).
[CrossRef]

IEEE J. Quantum Electron. (1)

M. Heiblum and J. Harris, “Analysis of curved optical wave-guides by conformal transformation,” IEEE J. Quantum Electron. 11, 75–83 (1975).
[CrossRef]

IEEE Photon. Technol. Lett. (2)

S. H. Nam and S. Yin, “High-temperature sensing using whispering gallery mode resonance in bent optical fibers,” IEEE Photon. Technol. Lett. 17, 2391–2393 (2005).
[CrossRef]

P. Wang, Y. Semenova, G. Rajan, T. Freir, and G. Farrell, “The temperature dependence of polarization-dependent loss for a macrobending single-mode-fiber-based edge filter,” IEEE Photon. Technol. Lett. 21, 516–518 (2009).
[CrossRef]

J. Lightwave Technol. (1)

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

Meas. Sci. Technol. (1)

S. W. James and R. P. Tatam, “Optical fibre long-period grating sensors: characteristics and application,” Meas. Sci. Technol. 14, R49–R61 (2003).
[CrossRef]

Microw. Opt. Technol. Lett. (1)

P. Wang, Q. Wang, G. Farrell, T. Freir, and J. Cassidy, “Investigation of macrobending losses of standard single mode fiber with small bend radii,” Microw. Opt. Technol. Lett. 49, 9 (2007).
[CrossRef]

Opt. Lett. (7)

T. Y. Hu, Y. Wang, C. R. Liao, and D. N. Wang, “Miniaturized fiber in-line Mach–Zehnder interferometer based on inner air-cavity for high-temperature sensing,” Opt. Lett. 37, 5083–5085 (2012).

W. W. Qian, C. L. Zhao, S. L. He, X. Y. Dong, S. Q. Zhang, Z. X. Zhang, S. Z. Jin, J. T. Guo, and H. F. Wei, “High-sensitivity temperature sensor based on an alcohol-filled photonic crystal fiber loop mirror,” Opt. Lett. 36, 1549–1551 (2011).

P. Wang, G. Rajan, Y. Semenova, and G. Farrell, “Temperature dependence of a macrobending edge filter based on a high-bend loss fiber,” Opt. Lett. 33, 2471–2473 (2008).

L. Yuan, T. Wei, Q. Han, H. Z. Wang, J. Huang, L. Jiang, and H. Xiao, “Fiber inline Michelson interferometer fabricated by a femtosecond laser,” Opt. Lett. 37, 3753–3755 (2012).
[CrossRef]

C. L. Zhao, Z. Q. Wang, S. Q. Zhang, L. Qi, C. Zhong, Z. X. Zhang, S. Z. Jin, J. T. Guo, and H. F. Wei, “Phenomenon in an alcohol not full-filled temperature sensor based on an optical fiber Sagnac interferometer,” Opt. Lett. 37, 4789–4791 (2012).
[CrossRef]

Y. Peng, J. Hou, Y. Zhang, Z. H. Huang, R. Xiao, and Q. S. Lu, “Temperature sensing using the bandgap-like effect in a selectively liquid-filled photonic crystal fiber,” Opt. Lett. 38, 263–265 (2013).
[CrossRef]

H. Liang, W. G. Zhang, P. C. Geng, Y. Liu, Z. Wang, J. Q. Guo, S. C. Gao, and S. Y. Yan, “Simultaneous measurement of temperature and force with high sensitivities based on filling different index liquids into photonic crystal fiber,” Opt. Lett. 38, 1071–1073 (2013).
[CrossRef]

Polymer (1)

Z. Y. Zhang, P. Zhao, P. Lin, and F. G. Sun, “Thermo-optic coefficients of polymers for optical waveguide applications,” Polymer 47, 4893–4896 (2006).
[CrossRef]

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Figures (6)

Fig. 1.
Fig. 1.

(a) Schematic distribution of refractive index in normal SMF with coating. (b) The amplitude distribution of the light propagating at the most curved segment. The red, green, and white solid lines represent the profiles of core, cladding, and coating, respectively.

Fig. 2.
Fig. 2.

Experimental setup of a temperature sensor based on bending a SMF. Insert: photograph of the sensor with a curvature radius of around 3.6 mm.

Fig. 3.
Fig. 3.

Transmission spectra comparison of the sensor with a radius of 3.6 mm in air and water.

Fig. 4.
Fig. 4.

Transmission spectra of the sensor based on a droplet-like fiber circle with a curvature radius of 3.6 mm when the temperature increases from 18°C to 30°C.

Fig. 5.
Fig. 5.

Linear fitting of the central wavelength of the resonant dip in different temperatures with the sensor having a curvature radius of 3.6 mm.

Fig. 6.
Fig. 6.

Relationship between the temperature sensitivity and circle curvature radius of the sensors.

Equations (3)

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

I=Icore+Icoating+2IcoreIcoatingcos(2πLΔneffλ+ϕ0),
dλdT=λLdLdT+λneffcore(T)neffcoating(T)(dneffcore(T)dTdneffcoating(T)dT),
Δ(dλdT)=dLdTΔλ.LΔL.λL2+(dneffcore(T)dTdneffcoating(T)dT)Δλ(neffcore(T)neffcoating(T))(dneffcore(T)dTdneffcoating(T)dT).ΔT.λ(neffcore(T)neffcoating(T))2,

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