Abstract

We present a novel fiber Fabry-Perot (FP) interference salinity sensor based on polyimide (PI) diaphragm. With an increase in water salinity, the PI diaphragm shrinks, and the PI diaphragm constriction causes the increase of the width of the air-gap, which causes the red shift of the interference fringes. We fabricated salinity sensor prototypes with different air-gap lengths and 20μm PI diaphragm. When salinity increases from 0mol/L to 5.47mol/L, the maximum sensitivity is 0.45nm/(mol/L). We verify that the sensitivity can be enhanced by reducing air-gap cavity length. We also choose appropriate air cavity length and PI diaphragm length to solve the cross-sensitivity between temperature and salinity. As a robust and ultra-compact salinity sensor, which is easy to be fabricated and need no alignment, this fiber interferometer can be applied for real-time salinity sensing applications.

© 2015 Optical Society of America

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References

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  1. X. D. Wang and O. S. Wolfbeis, “Fiber-optic chemical sensors and biosensors (2008-2012),” Anal. Chem. 85(2), 487–508 (2013).
    [Crossref] [PubMed]
  2. N. Díaz-Herrera, O. Esteban, M. C. Navarrete, M. L. Haitre, and A. González-Cano, “In situ salinity measurements in seawater with a fibre-optic probe,” Meas. Sci. Technol. 17(8), 2227–2232 (2006).
    [Crossref]
  3. D. J. Gentleman and K. S. Booksh, “Determining salinity using a multimode fiber optic surface plasmon resonance dip-probe,” Talanta 68(3), 504–515 (2006).
    [Crossref] [PubMed]
  4. R. Falate, O. Frazão, G. Rego, J. L. Fabris, and J. L. Santos, “Refractometric sensor based on a phase-shifted long-period fiber grating,” Appl. Opt. 45(21), 5066–5072 (2006).
    [Crossref] [PubMed]
  5. G. R. C. Possetti, R. C. Kamikawachi, C. L. Prevedello, M. Muller, and J. L. Fabris, “Salinity measurement in water environment with a long period grating based interferometer,” Meas. Sci. Technol. 20(3), 034003 (2009).
    [Crossref]
  6. D. A. Pereira, O. Frazão, and J. L. Santos, “Fiber Bragg grating sensing system for simultaneous measurement of salinity and temperature,” Opt. Eng. 43(2), 299–304 (2004).
    [Crossref]
  7. J. R. Guzman-Sepulveda, V. I. Ruiz-Perez, M. Torres-Cisneros, J. J. Sanchez-Mondragon, and D. A. May-Arrioja, “Fiber optic sensor for high-sensitivity salinity measurement,” IEEE Photon. Technol. Lett. 25(23), 2323–2326 (2013).
    [Crossref]
  8. H. A. Rahman, S. W. Harun, M. Yasin, S. W. Phang, S. S. Damanhuria, H. Arof, and H. Ahmad, “Tapered plastic multimode fiber sensor for salinity detection,” Sens. Actuators A Phys. 171(2), 219–222 (2011).
    [Crossref]
  9. L. V. Nguyen, M. Vasiliev, and K. Alameh, “Three-wave fiber Fabry-Perot interferometer for simultaneous measurement of temperature and water salinity of seawater,” IEEE Photon. Technol. Lett. 23(7), 450–452 (2011).
    [Crossref]
  10. J. Cong, X. M. Zhang, K. S. Chen, and J. Xu, “Fiber optic Bragg grating sensor based on hydrogels for measuring salinity,” Sens. Actuators B Chem. 87(3), 487–490 (2002).
    [Crossref]
  11. X. Liu, X. Zhang, J. Cong, J. Xu, and K. Chen, “Demonstration of etched cladding fiber Bragg grating-based sensors with hydrogel coating,” Sens. Actuators B Chem. 96(1–2), 468–472 (2003).
    [Crossref]
  12. L. Q. Men, P. Lu, and Q. Y. Chen, “A multiplexed fiber Bragg grating sensor for simultaneous salinity and temperature measurement,” J. Appl. Phys. 103(5), 053107 (2008).
    [Crossref]
  13. C. Wu, B. O. Guan, C. Lu, and H. Y. Tam, “Salinity sensor based on polyimide-coated photonic crystal fiber,” Opt. Express 19(21), 20003–20008 (2011).
    [Crossref] [PubMed]

2013 (2)

J. R. Guzman-Sepulveda, V. I. Ruiz-Perez, M. Torres-Cisneros, J. J. Sanchez-Mondragon, and D. A. May-Arrioja, “Fiber optic sensor for high-sensitivity salinity measurement,” IEEE Photon. Technol. Lett. 25(23), 2323–2326 (2013).
[Crossref]

X. D. Wang and O. S. Wolfbeis, “Fiber-optic chemical sensors and biosensors (2008-2012),” Anal. Chem. 85(2), 487–508 (2013).
[Crossref] [PubMed]

2011 (3)

C. Wu, B. O. Guan, C. Lu, and H. Y. Tam, “Salinity sensor based on polyimide-coated photonic crystal fiber,” Opt. Express 19(21), 20003–20008 (2011).
[Crossref] [PubMed]

H. A. Rahman, S. W. Harun, M. Yasin, S. W. Phang, S. S. Damanhuria, H. Arof, and H. Ahmad, “Tapered plastic multimode fiber sensor for salinity detection,” Sens. Actuators A Phys. 171(2), 219–222 (2011).
[Crossref]

L. V. Nguyen, M. Vasiliev, and K. Alameh, “Three-wave fiber Fabry-Perot interferometer for simultaneous measurement of temperature and water salinity of seawater,” IEEE Photon. Technol. Lett. 23(7), 450–452 (2011).
[Crossref]

2009 (1)

G. R. C. Possetti, R. C. Kamikawachi, C. L. Prevedello, M. Muller, and J. L. Fabris, “Salinity measurement in water environment with a long period grating based interferometer,” Meas. Sci. Technol. 20(3), 034003 (2009).
[Crossref]

2008 (1)

L. Q. Men, P. Lu, and Q. Y. Chen, “A multiplexed fiber Bragg grating sensor for simultaneous salinity and temperature measurement,” J. Appl. Phys. 103(5), 053107 (2008).
[Crossref]

2006 (3)

N. Díaz-Herrera, O. Esteban, M. C. Navarrete, M. L. Haitre, and A. González-Cano, “In situ salinity measurements in seawater with a fibre-optic probe,” Meas. Sci. Technol. 17(8), 2227–2232 (2006).
[Crossref]

D. J. Gentleman and K. S. Booksh, “Determining salinity using a multimode fiber optic surface plasmon resonance dip-probe,” Talanta 68(3), 504–515 (2006).
[Crossref] [PubMed]

R. Falate, O. Frazão, G. Rego, J. L. Fabris, and J. L. Santos, “Refractometric sensor based on a phase-shifted long-period fiber grating,” Appl. Opt. 45(21), 5066–5072 (2006).
[Crossref] [PubMed]

2004 (1)

D. A. Pereira, O. Frazão, and J. L. Santos, “Fiber Bragg grating sensing system for simultaneous measurement of salinity and temperature,” Opt. Eng. 43(2), 299–304 (2004).
[Crossref]

2003 (1)

X. Liu, X. Zhang, J. Cong, J. Xu, and K. Chen, “Demonstration of etched cladding fiber Bragg grating-based sensors with hydrogel coating,” Sens. Actuators B Chem. 96(1–2), 468–472 (2003).
[Crossref]

2002 (1)

J. Cong, X. M. Zhang, K. S. Chen, and J. Xu, “Fiber optic Bragg grating sensor based on hydrogels for measuring salinity,” Sens. Actuators B Chem. 87(3), 487–490 (2002).
[Crossref]

Ahmad, H.

H. A. Rahman, S. W. Harun, M. Yasin, S. W. Phang, S. S. Damanhuria, H. Arof, and H. Ahmad, “Tapered plastic multimode fiber sensor for salinity detection,” Sens. Actuators A Phys. 171(2), 219–222 (2011).
[Crossref]

Alameh, K.

L. V. Nguyen, M. Vasiliev, and K. Alameh, “Three-wave fiber Fabry-Perot interferometer for simultaneous measurement of temperature and water salinity of seawater,” IEEE Photon. Technol. Lett. 23(7), 450–452 (2011).
[Crossref]

Arof, H.

H. A. Rahman, S. W. Harun, M. Yasin, S. W. Phang, S. S. Damanhuria, H. Arof, and H. Ahmad, “Tapered plastic multimode fiber sensor for salinity detection,” Sens. Actuators A Phys. 171(2), 219–222 (2011).
[Crossref]

Booksh, K. S.

D. J. Gentleman and K. S. Booksh, “Determining salinity using a multimode fiber optic surface plasmon resonance dip-probe,” Talanta 68(3), 504–515 (2006).
[Crossref] [PubMed]

Chen, K.

X. Liu, X. Zhang, J. Cong, J. Xu, and K. Chen, “Demonstration of etched cladding fiber Bragg grating-based sensors with hydrogel coating,” Sens. Actuators B Chem. 96(1–2), 468–472 (2003).
[Crossref]

Chen, K. S.

J. Cong, X. M. Zhang, K. S. Chen, and J. Xu, “Fiber optic Bragg grating sensor based on hydrogels for measuring salinity,” Sens. Actuators B Chem. 87(3), 487–490 (2002).
[Crossref]

Chen, Q. Y.

L. Q. Men, P. Lu, and Q. Y. Chen, “A multiplexed fiber Bragg grating sensor for simultaneous salinity and temperature measurement,” J. Appl. Phys. 103(5), 053107 (2008).
[Crossref]

Cong, J.

X. Liu, X. Zhang, J. Cong, J. Xu, and K. Chen, “Demonstration of etched cladding fiber Bragg grating-based sensors with hydrogel coating,” Sens. Actuators B Chem. 96(1–2), 468–472 (2003).
[Crossref]

J. Cong, X. M. Zhang, K. S. Chen, and J. Xu, “Fiber optic Bragg grating sensor based on hydrogels for measuring salinity,” Sens. Actuators B Chem. 87(3), 487–490 (2002).
[Crossref]

Damanhuria, S. S.

H. A. Rahman, S. W. Harun, M. Yasin, S. W. Phang, S. S. Damanhuria, H. Arof, and H. Ahmad, “Tapered plastic multimode fiber sensor for salinity detection,” Sens. Actuators A Phys. 171(2), 219–222 (2011).
[Crossref]

Díaz-Herrera, N.

N. Díaz-Herrera, O. Esteban, M. C. Navarrete, M. L. Haitre, and A. González-Cano, “In situ salinity measurements in seawater with a fibre-optic probe,” Meas. Sci. Technol. 17(8), 2227–2232 (2006).
[Crossref]

Esteban, O.

N. Díaz-Herrera, O. Esteban, M. C. Navarrete, M. L. Haitre, and A. González-Cano, “In situ salinity measurements in seawater with a fibre-optic probe,” Meas. Sci. Technol. 17(8), 2227–2232 (2006).
[Crossref]

Fabris, J. L.

G. R. C. Possetti, R. C. Kamikawachi, C. L. Prevedello, M. Muller, and J. L. Fabris, “Salinity measurement in water environment with a long period grating based interferometer,” Meas. Sci. Technol. 20(3), 034003 (2009).
[Crossref]

R. Falate, O. Frazão, G. Rego, J. L. Fabris, and J. L. Santos, “Refractometric sensor based on a phase-shifted long-period fiber grating,” Appl. Opt. 45(21), 5066–5072 (2006).
[Crossref] [PubMed]

Falate, R.

Frazão, O.

R. Falate, O. Frazão, G. Rego, J. L. Fabris, and J. L. Santos, “Refractometric sensor based on a phase-shifted long-period fiber grating,” Appl. Opt. 45(21), 5066–5072 (2006).
[Crossref] [PubMed]

D. A. Pereira, O. Frazão, and J. L. Santos, “Fiber Bragg grating sensing system for simultaneous measurement of salinity and temperature,” Opt. Eng. 43(2), 299–304 (2004).
[Crossref]

Gentleman, D. J.

D. J. Gentleman and K. S. Booksh, “Determining salinity using a multimode fiber optic surface plasmon resonance dip-probe,” Talanta 68(3), 504–515 (2006).
[Crossref] [PubMed]

González-Cano, A.

N. Díaz-Herrera, O. Esteban, M. C. Navarrete, M. L. Haitre, and A. González-Cano, “In situ salinity measurements in seawater with a fibre-optic probe,” Meas. Sci. Technol. 17(8), 2227–2232 (2006).
[Crossref]

Guan, B. O.

Guzman-Sepulveda, J. R.

J. R. Guzman-Sepulveda, V. I. Ruiz-Perez, M. Torres-Cisneros, J. J. Sanchez-Mondragon, and D. A. May-Arrioja, “Fiber optic sensor for high-sensitivity salinity measurement,” IEEE Photon. Technol. Lett. 25(23), 2323–2326 (2013).
[Crossref]

Haitre, M. L.

N. Díaz-Herrera, O. Esteban, M. C. Navarrete, M. L. Haitre, and A. González-Cano, “In situ salinity measurements in seawater with a fibre-optic probe,” Meas. Sci. Technol. 17(8), 2227–2232 (2006).
[Crossref]

Harun, S. W.

H. A. Rahman, S. W. Harun, M. Yasin, S. W. Phang, S. S. Damanhuria, H. Arof, and H. Ahmad, “Tapered plastic multimode fiber sensor for salinity detection,” Sens. Actuators A Phys. 171(2), 219–222 (2011).
[Crossref]

Kamikawachi, R. C.

G. R. C. Possetti, R. C. Kamikawachi, C. L. Prevedello, M. Muller, and J. L. Fabris, “Salinity measurement in water environment with a long period grating based interferometer,” Meas. Sci. Technol. 20(3), 034003 (2009).
[Crossref]

Liu, X.

X. Liu, X. Zhang, J. Cong, J. Xu, and K. Chen, “Demonstration of etched cladding fiber Bragg grating-based sensors with hydrogel coating,” Sens. Actuators B Chem. 96(1–2), 468–472 (2003).
[Crossref]

Lu, C.

Lu, P.

L. Q. Men, P. Lu, and Q. Y. Chen, “A multiplexed fiber Bragg grating sensor for simultaneous salinity and temperature measurement,” J. Appl. Phys. 103(5), 053107 (2008).
[Crossref]

May-Arrioja, D. A.

J. R. Guzman-Sepulveda, V. I. Ruiz-Perez, M. Torres-Cisneros, J. J. Sanchez-Mondragon, and D. A. May-Arrioja, “Fiber optic sensor for high-sensitivity salinity measurement,” IEEE Photon. Technol. Lett. 25(23), 2323–2326 (2013).
[Crossref]

Men, L. Q.

L. Q. Men, P. Lu, and Q. Y. Chen, “A multiplexed fiber Bragg grating sensor for simultaneous salinity and temperature measurement,” J. Appl. Phys. 103(5), 053107 (2008).
[Crossref]

Muller, M.

G. R. C. Possetti, R. C. Kamikawachi, C. L. Prevedello, M. Muller, and J. L. Fabris, “Salinity measurement in water environment with a long period grating based interferometer,” Meas. Sci. Technol. 20(3), 034003 (2009).
[Crossref]

Navarrete, M. C.

N. Díaz-Herrera, O. Esteban, M. C. Navarrete, M. L. Haitre, and A. González-Cano, “In situ salinity measurements in seawater with a fibre-optic probe,” Meas. Sci. Technol. 17(8), 2227–2232 (2006).
[Crossref]

Nguyen, L. V.

L. V. Nguyen, M. Vasiliev, and K. Alameh, “Three-wave fiber Fabry-Perot interferometer for simultaneous measurement of temperature and water salinity of seawater,” IEEE Photon. Technol. Lett. 23(7), 450–452 (2011).
[Crossref]

Pereira, D. A.

D. A. Pereira, O. Frazão, and J. L. Santos, “Fiber Bragg grating sensing system for simultaneous measurement of salinity and temperature,” Opt. Eng. 43(2), 299–304 (2004).
[Crossref]

Phang, S. W.

H. A. Rahman, S. W. Harun, M. Yasin, S. W. Phang, S. S. Damanhuria, H. Arof, and H. Ahmad, “Tapered plastic multimode fiber sensor for salinity detection,” Sens. Actuators A Phys. 171(2), 219–222 (2011).
[Crossref]

Possetti, G. R. C.

G. R. C. Possetti, R. C. Kamikawachi, C. L. Prevedello, M. Muller, and J. L. Fabris, “Salinity measurement in water environment with a long period grating based interferometer,” Meas. Sci. Technol. 20(3), 034003 (2009).
[Crossref]

Prevedello, C. L.

G. R. C. Possetti, R. C. Kamikawachi, C. L. Prevedello, M. Muller, and J. L. Fabris, “Salinity measurement in water environment with a long period grating based interferometer,” Meas. Sci. Technol. 20(3), 034003 (2009).
[Crossref]

Rahman, H. A.

H. A. Rahman, S. W. Harun, M. Yasin, S. W. Phang, S. S. Damanhuria, H. Arof, and H. Ahmad, “Tapered plastic multimode fiber sensor for salinity detection,” Sens. Actuators A Phys. 171(2), 219–222 (2011).
[Crossref]

Rego, G.

Ruiz-Perez, V. I.

J. R. Guzman-Sepulveda, V. I. Ruiz-Perez, M. Torres-Cisneros, J. J. Sanchez-Mondragon, and D. A. May-Arrioja, “Fiber optic sensor for high-sensitivity salinity measurement,” IEEE Photon. Technol. Lett. 25(23), 2323–2326 (2013).
[Crossref]

Sanchez-Mondragon, J. J.

J. R. Guzman-Sepulveda, V. I. Ruiz-Perez, M. Torres-Cisneros, J. J. Sanchez-Mondragon, and D. A. May-Arrioja, “Fiber optic sensor for high-sensitivity salinity measurement,” IEEE Photon. Technol. Lett. 25(23), 2323–2326 (2013).
[Crossref]

Santos, J. L.

R. Falate, O. Frazão, G. Rego, J. L. Fabris, and J. L. Santos, “Refractometric sensor based on a phase-shifted long-period fiber grating,” Appl. Opt. 45(21), 5066–5072 (2006).
[Crossref] [PubMed]

D. A. Pereira, O. Frazão, and J. L. Santos, “Fiber Bragg grating sensing system for simultaneous measurement of salinity and temperature,” Opt. Eng. 43(2), 299–304 (2004).
[Crossref]

Tam, H. Y.

Torres-Cisneros, M.

J. R. Guzman-Sepulveda, V. I. Ruiz-Perez, M. Torres-Cisneros, J. J. Sanchez-Mondragon, and D. A. May-Arrioja, “Fiber optic sensor for high-sensitivity salinity measurement,” IEEE Photon. Technol. Lett. 25(23), 2323–2326 (2013).
[Crossref]

Vasiliev, M.

L. V. Nguyen, M. Vasiliev, and K. Alameh, “Three-wave fiber Fabry-Perot interferometer for simultaneous measurement of temperature and water salinity of seawater,” IEEE Photon. Technol. Lett. 23(7), 450–452 (2011).
[Crossref]

Wang, X. D.

X. D. Wang and O. S. Wolfbeis, “Fiber-optic chemical sensors and biosensors (2008-2012),” Anal. Chem. 85(2), 487–508 (2013).
[Crossref] [PubMed]

Wolfbeis, O. S.

X. D. Wang and O. S. Wolfbeis, “Fiber-optic chemical sensors and biosensors (2008-2012),” Anal. Chem. 85(2), 487–508 (2013).
[Crossref] [PubMed]

Wu, C.

Xu, J.

X. Liu, X. Zhang, J. Cong, J. Xu, and K. Chen, “Demonstration of etched cladding fiber Bragg grating-based sensors with hydrogel coating,” Sens. Actuators B Chem. 96(1–2), 468–472 (2003).
[Crossref]

J. Cong, X. M. Zhang, K. S. Chen, and J. Xu, “Fiber optic Bragg grating sensor based on hydrogels for measuring salinity,” Sens. Actuators B Chem. 87(3), 487–490 (2002).
[Crossref]

Yasin, M.

H. A. Rahman, S. W. Harun, M. Yasin, S. W. Phang, S. S. Damanhuria, H. Arof, and H. Ahmad, “Tapered plastic multimode fiber sensor for salinity detection,” Sens. Actuators A Phys. 171(2), 219–222 (2011).
[Crossref]

Zhang, X.

X. Liu, X. Zhang, J. Cong, J. Xu, and K. Chen, “Demonstration of etched cladding fiber Bragg grating-based sensors with hydrogel coating,” Sens. Actuators B Chem. 96(1–2), 468–472 (2003).
[Crossref]

Zhang, X. M.

J. Cong, X. M. Zhang, K. S. Chen, and J. Xu, “Fiber optic Bragg grating sensor based on hydrogels for measuring salinity,” Sens. Actuators B Chem. 87(3), 487–490 (2002).
[Crossref]

Anal. Chem. (1)

X. D. Wang and O. S. Wolfbeis, “Fiber-optic chemical sensors and biosensors (2008-2012),” Anal. Chem. 85(2), 487–508 (2013).
[Crossref] [PubMed]

Appl. Opt. (1)

IEEE Photon. Technol. Lett. (2)

J. R. Guzman-Sepulveda, V. I. Ruiz-Perez, M. Torres-Cisneros, J. J. Sanchez-Mondragon, and D. A. May-Arrioja, “Fiber optic sensor for high-sensitivity salinity measurement,” IEEE Photon. Technol. Lett. 25(23), 2323–2326 (2013).
[Crossref]

L. V. Nguyen, M. Vasiliev, and K. Alameh, “Three-wave fiber Fabry-Perot interferometer for simultaneous measurement of temperature and water salinity of seawater,” IEEE Photon. Technol. Lett. 23(7), 450–452 (2011).
[Crossref]

J. Appl. Phys. (1)

L. Q. Men, P. Lu, and Q. Y. Chen, “A multiplexed fiber Bragg grating sensor for simultaneous salinity and temperature measurement,” J. Appl. Phys. 103(5), 053107 (2008).
[Crossref]

Meas. Sci. Technol. (2)

G. R. C. Possetti, R. C. Kamikawachi, C. L. Prevedello, M. Muller, and J. L. Fabris, “Salinity measurement in water environment with a long period grating based interferometer,” Meas. Sci. Technol. 20(3), 034003 (2009).
[Crossref]

N. Díaz-Herrera, O. Esteban, M. C. Navarrete, M. L. Haitre, and A. González-Cano, “In situ salinity measurements in seawater with a fibre-optic probe,” Meas. Sci. Technol. 17(8), 2227–2232 (2006).
[Crossref]

Opt. Eng. (1)

D. A. Pereira, O. Frazão, and J. L. Santos, “Fiber Bragg grating sensing system for simultaneous measurement of salinity and temperature,” Opt. Eng. 43(2), 299–304 (2004).
[Crossref]

Opt. Express (1)

Sens. Actuators A Phys. (1)

H. A. Rahman, S. W. Harun, M. Yasin, S. W. Phang, S. S. Damanhuria, H. Arof, and H. Ahmad, “Tapered plastic multimode fiber sensor for salinity detection,” Sens. Actuators A Phys. 171(2), 219–222 (2011).
[Crossref]

Sens. Actuators B Chem. (2)

J. Cong, X. M. Zhang, K. S. Chen, and J. Xu, “Fiber optic Bragg grating sensor based on hydrogels for measuring salinity,” Sens. Actuators B Chem. 87(3), 487–490 (2002).
[Crossref]

X. Liu, X. Zhang, J. Cong, J. Xu, and K. Chen, “Demonstration of etched cladding fiber Bragg grating-based sensors with hydrogel coating,” Sens. Actuators B Chem. 96(1–2), 468–472 (2003).
[Crossref]

Talanta (1)

D. J. Gentleman and K. S. Booksh, “Determining salinity using a multimode fiber optic surface plasmon resonance dip-probe,” Talanta 68(3), 504–515 (2006).
[Crossref] [PubMed]

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Figures (6)

Fig. 1
Fig. 1 Proposed salinity sensor’s production sequence and operating mechanism. Inset: Micrograph of the sensor head with 200μm air-gap and 20μm PI diaphragm.
Fig. 2
Fig. 2 (a) Typical interference pattern with 200μm air-gap and 20μm PI diaphragm. (b) Relationship between the free space range and the air-gap interferometer length.
Fig. 3
Fig. 3 Red shift of the fringes as the salinity increases.
Fig. 4
Fig. 4 Wavelength shift as a function of water salinity (a) and relationship between sensitivity and air-gap cavity length (b).
Fig. 5
Fig. 5 Reflection spectra with decreasing temperature.
Fig. 6
Fig. 6 Relationship between temperature and wavelength shift when the sensors immersed in air in air (a) and water (b). (c) The salinity sensitivity fluctuation versus temperature.

Equations (4)

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

I= I 1 + I 2 +2 I 1 I 2 cos( 4π n eff L λ + ϕ 0 )
λ m = 2L m
Δλ= 2ΔL m
δλ= λ 2 2L

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