Abstract

We analyze and predict the performance of a fiber-optic temperature sensor from the measured fluorescence spectrum to optimize its design. We apply this analysis to an erbium-doped silica fiber by employing the power-ratio technique. We develop expressions for the signal-to-noise ratio in a band to optimize sensor performance in each spectral channel. We improve the signal-to-noise ratio by a factor of 5 for each channel, compared with earlier results. We evaluate the analytical expression for the sensor sensitivity and predict it to be approximately 0.02 °C-1 for the temperature interval from room temperature to above 200 °C, increasing from 0.01 °C-1 at the edges of the interval to 0.03 °C-1 at the center, at 100–130 °C. The sensitivity again increases at temperatures higher than 300 °C, delineating its useful temperature intervals.

© 2003 Optical Society of America

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References

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  7. M. S. Scholl, J. R. Trimmier, “Luminescence of YAG:TM:Tb,” J. Electrochem. Soc. 133, 643–648 (1986).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  18. Z.-Y. Zhang, K. T. V. Grattan, A. W. Palmer, B. T. Meggitt, “Thulium-doped intrinsic fiber optic sensor for high temperature measurements (1100 °C),” Rev. Sci. Instrum. 69, 3210–3214 (1998).
    [CrossRef]
  19. T. Sun, Z. Y. Zhang, K. T. V. Grattan, A. W. Palmer, S. F. Collins, “Temperature dependence of the fluorescence lifetime in Pr3+: ZBLAN glass for fiber optic thermometry,” Rev. Sci. Instrum. 68, 3447–3451 (1997).
    [CrossRef]
  20. E. Maurice, S. A. Wade, S. F. Collins, G. Monnom, G. W. Baxter, “Self-referenced point temperature sensor based on a fluorescence intensity ratio in Yb3+-doped silica fiber,” Appl. Opt. 36, 8264–8269 (1997).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  24. E. Maurice, G. Monnom, B. Dussardier, A. Saissy, D. B. Ostrowsky, G. Baxter, “Thermalization effects between upper levels of green fluorescence in Er-doped silica fibers,” Opt. Lett. 19, 990–992 (1994).
    [CrossRef] [PubMed]
  25. C. G. Atkins, J. R. Armitage, R. Wyatt, B. J. Ainslie, S. P. Craig-Ryan, “Pump excited state absorption in Er3+ doped optical fibres,” Opt. Commun. 73, 217–222 (1989).
    [CrossRef]
  26. R. A. Betts, F. F. Kuhl, T. M. Kwok, G. F. Zheng, “Optical amplifiers based on phosphorus co-doped rare-earth-doped optical fibres,” Int. J. Optoelectron. 6, 47–64 (1991).
  27. G. Paez, M. S. Scholl, “Thermal contrast detected with a thermal detector,” Infrared Phys. Technol. 40, 109–118 (1999).
    [CrossRef]
  28. G. Paez, M. S. Scholl, “Thermal contrast detected with a quantum detector,” Infrared Phys. Technol. 40, 261–266 (1999).
    [CrossRef]

2002

J. Castrellon, G. Paez, M. Strojnik, “Remote temperature sensor employing erbium-doped silica fiber,” Infrared Phys. Technol. 43, 219–222 (2002).
[CrossRef]

1999

G. Paez, M. S. Scholl, “Thermal contrast detected with a thermal detector,” Infrared Phys. Technol. 40, 109–118 (1999).
[CrossRef]

G. Paez, M. S. Scholl, “Thermal contrast detected with a quantum detector,” Infrared Phys. Technol. 40, 261–266 (1999).
[CrossRef]

1998

P. V. dos Santos, M. T. de Araujo, A. S. Gouveia-Neto, J. A. Medeiros Neto, A. S. B. Sombra, “Optical temperature sensing using upconversion fluorescence emission in Er3+/Yb3+-codoped chalcogenide glass,” Appl. Phys. Lett. 73, 578–580 (1998).
[CrossRef]

Z.-Y. Zhang, K. T. V. Grattan, A. W. Palmer, B. T. Meggitt, “Thulium-doped intrinsic fiber optic sensor for high temperature measurements (1100 °C),” Rev. Sci. Instrum. 69, 3210–3214 (1998).
[CrossRef]

1997

1995

1994

1991

P. A. Krug, M. G. Sceats, G. R. Atkins, S. C. Guy, S. B. Poole, “Intermediate excited-state absorption in erbium-doped fiber strongly pumped at 980 nm,” Opt. Lett. 16, 1976–1978 (1991).
[CrossRef] [PubMed]

R. A. Betts, F. F. Kuhl, T. M. Kwok, G. F. Zheng, “Optical amplifiers based on phosphorus co-doped rare-earth-doped optical fibres,” Int. J. Optoelectron. 6, 47–64 (1991).

1990

1989

C. G. Atkins, J. R. Armitage, R. Wyatt, B. J. Ainslie, S. P. Craig-Ryan, “Pump excited state absorption in Er3+ doped optical fibres,” Opt. Commun. 73, 217–222 (1989).
[CrossRef]

1986

M. C. Farries, M. E. Fernmann, R. I. Laming, S. B. Poole, D. N. Payne, A. P. Leach, “Distributed temperature sensor using Nd3+-doped optical fibre,” Electron. Lett. 22, 418–419 (1986).
[CrossRef]

M. S. Scholl, J. R. Trimmier, “Luminescence of YAG:TM:Tb,” J. Electrochem. Soc. 133, 643–648 (1986).
[CrossRef]

1982

1980

Ainslie, B. J.

C. G. Atkins, J. R. Armitage, R. Wyatt, B. J. Ainslie, S. P. Craig-Ryan, “Pump excited state absorption in Er3+ doped optical fibres,” Opt. Commun. 73, 217–222 (1989).
[CrossRef]

Allard, F. C.

F. C. Allard, Fiber Optics Handbook for Engineers and Scientists (McGraw-Hill, New York, 1990).

Armitage, J. R.

C. G. Atkins, J. R. Armitage, R. Wyatt, B. J. Ainslie, S. P. Craig-Ryan, “Pump excited state absorption in Er3+ doped optical fibres,” Opt. Commun. 73, 217–222 (1989).
[CrossRef]

Atkins, C. G.

C. G. Atkins, J. R. Armitage, R. Wyatt, B. J. Ainslie, S. P. Craig-Ryan, “Pump excited state absorption in Er3+ doped optical fibres,” Opt. Commun. 73, 217–222 (1989).
[CrossRef]

Atkins, G. R.

Baxter, G.

Baxter, G. W.

Berthou, H.

Betts, R. A.

R. A. Betts, F. F. Kuhl, T. M. Kwok, G. F. Zheng, “Optical amplifiers based on phosphorus co-doped rare-earth-doped optical fibres,” Int. J. Optoelectron. 6, 47–64 (1991).

Castrellon, J.

J. Castrellon, G. Paez, M. Strojnik, “Remote temperature sensor employing erbium-doped silica fiber,” Infrared Phys. Technol. 43, 219–222 (2002).
[CrossRef]

J. Castrellon, G. Paez, “Radiometric figures of merit of a fiber optic temperature sensor,” in Infrared Spaceborne Remote Sensing VII, M. Strojnik, B. F. Andresen, eds., Proc. SPIE3759, 410–421 (1999).
[CrossRef]

Choi, H.-S.

Collins, S. F.

T. Sun, Z. Y. Zhang, K. T. V. Grattan, A. W. Palmer, S. F. Collins, “Temperature dependence of the fluorescence lifetime in Pr3+: ZBLAN glass for fiber optic thermometry,” Rev. Sci. Instrum. 68, 3447–3451 (1997).
[CrossRef]

E. Maurice, S. A. Wade, S. F. Collins, G. Monnom, G. W. Baxter, “Self-referenced point temperature sensor based on a fluorescence intensity ratio in Yb3+-doped silica fiber,” Appl. Opt. 36, 8264–8269 (1997).
[CrossRef]

Craig-Ryan, S. P.

C. G. Atkins, J. R. Armitage, R. Wyatt, B. J. Ainslie, S. P. Craig-Ryan, “Pump excited state absorption in Er3+ doped optical fibres,” Opt. Commun. 73, 217–222 (1989).
[CrossRef]

de Araujo, M. T.

P. V. dos Santos, M. T. de Araujo, A. S. Gouveia-Neto, J. A. Medeiros Neto, A. S. B. Sombra, “Optical temperature sensing using upconversion fluorescence emission in Er3+/Yb3+-codoped chalcogenide glass,” Appl. Phys. Lett. 73, 578–580 (1998).
[CrossRef]

dos Santos, P. V.

P. V. dos Santos, M. T. de Araujo, A. S. Gouveia-Neto, J. A. Medeiros Neto, A. S. B. Sombra, “Optical temperature sensing using upconversion fluorescence emission in Er3+/Yb3+-codoped chalcogenide glass,” Appl. Phys. Lett. 73, 578–580 (1998).
[CrossRef]

Dussardier, B.

Farries, M. C.

M. C. Farries, M. E. Fernmann, R. I. Laming, S. B. Poole, D. N. Payne, A. P. Leach, “Distributed temperature sensor using Nd3+-doped optical fibre,” Electron. Lett. 22, 418–419 (1986).
[CrossRef]

Fernmann, M. E.

M. C. Farries, M. E. Fernmann, R. I. Laming, S. B. Poole, D. N. Payne, A. P. Leach, “Distributed temperature sensor using Nd3+-doped optical fibre,” Electron. Lett. 22, 418–419 (1986).
[CrossRef]

Glenn, W. H.

E. Snitzer, W. W. Morey, W. H. Glenn, “Fiber optic rare earth temperature sensors,” in First International Conference on Optical Fibre Sensors, Vol. CDP01 (SPIE Press, Bellingham, Wash., 1983), pp. 79–82.

Gouveia-Neto, A. S.

P. V. dos Santos, M. T. de Araujo, A. S. Gouveia-Neto, J. A. Medeiros Neto, A. S. B. Sombra, “Optical temperature sensing using upconversion fluorescence emission in Er3+/Yb3+-codoped chalcogenide glass,” Appl. Phys. Lett. 73, 578–580 (1998).
[CrossRef]

Grattan, K. T. V.

Z.-Y. Zhang, K. T. V. Grattan, A. W. Palmer, B. T. Meggitt, “Thulium-doped intrinsic fiber optic sensor for high temperature measurements (1100 °C),” Rev. Sci. Instrum. 69, 3210–3214 (1998).
[CrossRef]

T. Sun, Z. Y. Zhang, K. T. V. Grattan, A. W. Palmer, S. F. Collins, “Temperature dependence of the fluorescence lifetime in Pr3+: ZBLAN glass for fiber optic thermometry,” Rev. Sci. Instrum. 68, 3447–3451 (1997).
[CrossRef]

Guy, S. C.

Hyatt, W. D.

K. A. Wickersheim, W. D. Hyatt, “Commercial applications of fiber optic temperature measurement,” in Fiber Optic Sensors IV, R. T. Kersten, ed., Proc. SPIE1267, 84–96 (1990).
[CrossRef]

Jörgensen, C. K.

Krug, P. A.

Kuhl, F. F.

R. A. Betts, F. F. Kuhl, T. M. Kwok, G. F. Zheng, “Optical amplifiers based on phosphorus co-doped rare-earth-doped optical fibres,” Int. J. Optoelectron. 6, 47–64 (1991).

Kwok, T. M.

R. A. Betts, F. F. Kuhl, T. M. Kwok, G. F. Zheng, “Optical amplifiers based on phosphorus co-doped rare-earth-doped optical fibres,” Int. J. Optoelectron. 6, 47–64 (1991).

Laming, R. I.

M. C. Farries, M. E. Fernmann, R. I. Laming, S. B. Poole, D. N. Payne, A. P. Leach, “Distributed temperature sensor using Nd3+-doped optical fibre,” Electron. Lett. 22, 418–419 (1986).
[CrossRef]

Leach, A. P.

M. C. Farries, M. E. Fernmann, R. I. Laming, S. B. Poole, D. N. Payne, A. P. Leach, “Distributed temperature sensor using Nd3+-doped optical fibre,” Electron. Lett. 22, 418–419 (1986).
[CrossRef]

Lee, C. E.

Maurice, E.

Medeiros Neto, J. A.

P. V. dos Santos, M. T. de Araujo, A. S. Gouveia-Neto, J. A. Medeiros Neto, A. S. B. Sombra, “Optical temperature sensing using upconversion fluorescence emission in Er3+/Yb3+-codoped chalcogenide glass,” Appl. Phys. Lett. 73, 578–580 (1998).
[CrossRef]

Meggitt, B. T.

Z.-Y. Zhang, K. T. V. Grattan, A. W. Palmer, B. T. Meggitt, “Thulium-doped intrinsic fiber optic sensor for high temperature measurements (1100 °C),” Rev. Sci. Instrum. 69, 3210–3214 (1998).
[CrossRef]

Monnom, G.

Morey, W. W.

E. Snitzer, W. W. Morey, W. H. Glenn, “Fiber optic rare earth temperature sensors,” in First International Conference on Optical Fibre Sensors, Vol. CDP01 (SPIE Press, Bellingham, Wash., 1983), pp. 79–82.

Ostrowsky, D. B.

Paez, G.

J. Castrellon, G. Paez, M. Strojnik, “Remote temperature sensor employing erbium-doped silica fiber,” Infrared Phys. Technol. 43, 219–222 (2002).
[CrossRef]

G. Paez, M. S. Scholl, “Thermal contrast detected with a thermal detector,” Infrared Phys. Technol. 40, 109–118 (1999).
[CrossRef]

G. Paez, M. S. Scholl, “Thermal contrast detected with a quantum detector,” Infrared Phys. Technol. 40, 261–266 (1999).
[CrossRef]

J. Castrellon, G. Paez, “Radiometric figures of merit of a fiber optic temperature sensor,” in Infrared Spaceborne Remote Sensing VII, M. Strojnik, B. F. Andresen, eds., Proc. SPIE3759, 410–421 (1999).
[CrossRef]

M. Strojnik, G. Paez, “Radiometry,” in Handbook of Optical Engineering, D. Malacara, B. Thompson, eds. (Marcel Dekker, New York, 2001), pp. 649–699.

Palmer, A. W.

Z.-Y. Zhang, K. T. V. Grattan, A. W. Palmer, B. T. Meggitt, “Thulium-doped intrinsic fiber optic sensor for high temperature measurements (1100 °C),” Rev. Sci. Instrum. 69, 3210–3214 (1998).
[CrossRef]

T. Sun, Z. Y. Zhang, K. T. V. Grattan, A. W. Palmer, S. F. Collins, “Temperature dependence of the fluorescence lifetime in Pr3+: ZBLAN glass for fiber optic thermometry,” Rev. Sci. Instrum. 68, 3447–3451 (1997).
[CrossRef]

Payne, D. N.

M. C. Farries, M. E. Fernmann, R. I. Laming, S. B. Poole, D. N. Payne, A. P. Leach, “Distributed temperature sensor using Nd3+-doped optical fibre,” Electron. Lett. 22, 418–419 (1986).
[CrossRef]

Poole, S. B.

P. A. Krug, M. G. Sceats, G. R. Atkins, S. C. Guy, S. B. Poole, “Intermediate excited-state absorption in erbium-doped fiber strongly pumped at 980 nm,” Opt. Lett. 16, 1976–1978 (1991).
[CrossRef] [PubMed]

M. C. Farries, M. E. Fernmann, R. I. Laming, S. B. Poole, D. N. Payne, A. P. Leach, “Distributed temperature sensor using Nd3+-doped optical fibre,” Electron. Lett. 22, 418–419 (1986).
[CrossRef]

Saissy, A.

Sceats, M. G.

Scholl, M. S.

Snitzer, E.

E. Snitzer, W. W. Morey, W. H. Glenn, “Fiber optic rare earth temperature sensors,” in First International Conference on Optical Fibre Sensors, Vol. CDP01 (SPIE Press, Bellingham, Wash., 1983), pp. 79–82.

Sombra, A. S. B.

P. V. dos Santos, M. T. de Araujo, A. S. Gouveia-Neto, J. A. Medeiros Neto, A. S. B. Sombra, “Optical temperature sensing using upconversion fluorescence emission in Er3+/Yb3+-codoped chalcogenide glass,” Appl. Phys. Lett. 73, 578–580 (1998).
[CrossRef]

Strojnik, M.

J. Castrellon, G. Paez, M. Strojnik, “Remote temperature sensor employing erbium-doped silica fiber,” Infrared Phys. Technol. 43, 219–222 (2002).
[CrossRef]

M. Strojnik, G. Paez, “Radiometry,” in Handbook of Optical Engineering, D. Malacara, B. Thompson, eds. (Marcel Dekker, New York, 2001), pp. 649–699.

Sun, T.

T. Sun, Z. Y. Zhang, K. T. V. Grattan, A. W. Palmer, S. F. Collins, “Temperature dependence of the fluorescence lifetime in Pr3+: ZBLAN glass for fiber optic thermometry,” Rev. Sci. Instrum. 68, 3447–3451 (1997).
[CrossRef]

Taylor, H. F.

Trimmier, J. R.

M. S. Scholl, J. R. Trimmier, “Luminescence of YAG:TM:Tb,” J. Electrochem. Soc. 133, 643–648 (1986).
[CrossRef]

Wade, S. A.

Wickersheim, K. A.

K. A. Wickersheim, W. D. Hyatt, “Commercial applications of fiber optic temperature measurement,” in Fiber Optic Sensors IV, R. T. Kersten, ed., Proc. SPIE1267, 84–96 (1990).
[CrossRef]

Wyatt, R.

C. G. Atkins, J. R. Armitage, R. Wyatt, B. J. Ainslie, S. P. Craig-Ryan, “Pump excited state absorption in Er3+ doped optical fibres,” Opt. Commun. 73, 217–222 (1989).
[CrossRef]

Zhang, Z. Y.

T. Sun, Z. Y. Zhang, K. T. V. Grattan, A. W. Palmer, S. F. Collins, “Temperature dependence of the fluorescence lifetime in Pr3+: ZBLAN glass for fiber optic thermometry,” Rev. Sci. Instrum. 68, 3447–3451 (1997).
[CrossRef]

Zhang, Z.-Y.

Z.-Y. Zhang, K. T. V. Grattan, A. W. Palmer, B. T. Meggitt, “Thulium-doped intrinsic fiber optic sensor for high temperature measurements (1100 °C),” Rev. Sci. Instrum. 69, 3210–3214 (1998).
[CrossRef]

Zheng, G. F.

R. A. Betts, F. F. Kuhl, T. M. Kwok, G. F. Zheng, “Optical amplifiers based on phosphorus co-doped rare-earth-doped optical fibres,” Int. J. Optoelectron. 6, 47–64 (1991).

Appl. Opt.

Appl. Phys. Lett.

P. V. dos Santos, M. T. de Araujo, A. S. Gouveia-Neto, J. A. Medeiros Neto, A. S. B. Sombra, “Optical temperature sensing using upconversion fluorescence emission in Er3+/Yb3+-codoped chalcogenide glass,” Appl. Phys. Lett. 73, 578–580 (1998).
[CrossRef]

Electron. Lett.

M. C. Farries, M. E. Fernmann, R. I. Laming, S. B. Poole, D. N. Payne, A. P. Leach, “Distributed temperature sensor using Nd3+-doped optical fibre,” Electron. Lett. 22, 418–419 (1986).
[CrossRef]

Infrared Phys. Technol.

J. Castrellon, G. Paez, M. Strojnik, “Remote temperature sensor employing erbium-doped silica fiber,” Infrared Phys. Technol. 43, 219–222 (2002).
[CrossRef]

G. Paez, M. S. Scholl, “Thermal contrast detected with a thermal detector,” Infrared Phys. Technol. 40, 109–118 (1999).
[CrossRef]

G. Paez, M. S. Scholl, “Thermal contrast detected with a quantum detector,” Infrared Phys. Technol. 40, 261–266 (1999).
[CrossRef]

Int. J. Optoelectron.

R. A. Betts, F. F. Kuhl, T. M. Kwok, G. F. Zheng, “Optical amplifiers based on phosphorus co-doped rare-earth-doped optical fibres,” Int. J. Optoelectron. 6, 47–64 (1991).

J. Electrochem. Soc.

M. S. Scholl, J. R. Trimmier, “Luminescence of YAG:TM:Tb,” J. Electrochem. Soc. 133, 643–648 (1986).
[CrossRef]

J. Lightwave Technol.

E. Maurice, G. Monnom, D. B. Ostrowsky, G. W. Baxter, “High dynamic range temperature point sensor using green fluorescence intensity ratio in erbium-doped silica fiber,” J. Lightwave Technol. 13, 1349–1353 (1995).
[CrossRef]

Opt. Commun.

C. G. Atkins, J. R. Armitage, R. Wyatt, B. J. Ainslie, S. P. Craig-Ryan, “Pump excited state absorption in Er3+ doped optical fibres,” Opt. Commun. 73, 217–222 (1989).
[CrossRef]

Opt. Lett.

Rev. Sci. Instrum.

Z.-Y. Zhang, K. T. V. Grattan, A. W. Palmer, B. T. Meggitt, “Thulium-doped intrinsic fiber optic sensor for high temperature measurements (1100 °C),” Rev. Sci. Instrum. 69, 3210–3214 (1998).
[CrossRef]

T. Sun, Z. Y. Zhang, K. T. V. Grattan, A. W. Palmer, S. F. Collins, “Temperature dependence of the fluorescence lifetime in Pr3+: ZBLAN glass for fiber optic thermometry,” Rev. Sci. Instrum. 68, 3447–3451 (1997).
[CrossRef]

Other

E. Snitzer, W. W. Morey, W. H. Glenn, “Fiber optic rare earth temperature sensors,” in First International Conference on Optical Fibre Sensors, Vol. CDP01 (SPIE Press, Bellingham, Wash., 1983), pp. 79–82.

M. Strojnik, G. Paez, “Radiometry,” in Handbook of Optical Engineering, D. Malacara, B. Thompson, eds. (Marcel Dekker, New York, 2001), pp. 649–699.

J. Castrellon, G. Paez, “Radiometric figures of merit of a fiber optic temperature sensor,” in Infrared Spaceborne Remote Sensing VII, M. Strojnik, B. F. Andresen, eds., Proc. SPIE3759, 410–421 (1999).
[CrossRef]

F. C. Allard, Fiber Optics Handbook for Engineers and Scientists (McGraw-Hill, New York, 1990).

K. A. Wickersheim, W. D. Hyatt, “Commercial applications of fiber optic temperature measurement,” in Fiber Optic Sensors IV, R. T. Kersten, ed., Proc. SPIE1267, 84–96 (1990).
[CrossRef]

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Figures (7)

Fig. 1
Fig. 1

Fluorescence emission spectrum of erbium-doped silica as a function of wavelength for different temperature values (after Ref. 21). The emission of one peak increases while the other decreases when the temperature increases.

Fig. 2
Fig. 2

Predicted ratio of power at two peaks increases approximately as an exponential function of temperature. When shown on a logarithmic scale, it decreases nearly linearly with inverse temperature.

Fig. 3
Fig. 3

Predicted ratio of power of two transitions that integrates over spectral bands is approximately an exponential function of temperature. However, on a logarithmic scale, it decreases nearly linearly with inverse temperature.

Fig. 4
Fig. 4

Experimental setup to evaluate the performance of the erbium-doped fiber-optic sensor for remote temperature measurements. The setup uses fluorescence emission ratio in the 510–570-nm wavelength interval.

Fig. 5
Fig. 5

(a) Spectral SNR at the detector output for erbium-doped silica fiber as a function of wavelength with temperature as a parameter, including the transmission losses. (b) SNR integrated over the spectral band δλ and transmitted by the wavelength-selecting filters at the detector output for erbium-doped silica fiber as a function of wavelength. For a filter centered at a specific wavelength the SNR depends only on temperature.

Fig. 6
Fig. 6

Sensitivity in a band ΔI p T shown as a function of temperature for different spectral bands that are available in the ratio. The best band with a 10-nm spectral width for the 2 H 11/2 transition is 520–530 nm and for the 4 S 3/2 transition it is 550–560 nm.

Fig. 7
Fig. 7

Sensitivity of a sensor that uses the ratio technique can be defined as the change in power ratio ΔR(P 1/P 2) to the change in enclosure temperature ΔT. The sensitivity is shown as a function of temperature for a number of spectral bands in the ratio. The sensitivity between 300 and 500 K is shown in more detail at right.

Equations (14)

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R= P2H11/2P4S3/2= ν2H11/2ν4S3/2σe2H11/2σe4S3/2exp- ΔEkT,
P0Δλ, T=τ1τ2τ3PfΔλ, T W.
IpΔλ, T= qλP0Δλ, TηλhcA.
IJ=4kTdΔfRF1/2 A.
IsnΔλ, T=2qIpΔλ, T+IdTdΔf1/2 A.
ITn=IJ2+IdTd2+Isn21/2A.
ITnΔλ, T; Td=IdTd2+4kTdRF+2qIpΔλ, T+IdTdΔf1/2A.
SNRΔλ, T= IpΔλ, TITn2+Ia21/2 dB.
SNRΔλ, T= IpΔλ, TIdTd2+Ia2+4kTdRF+2qIpΔλ, T+IdTdΔf1/2 dB.
SΔλ1, Δλ2, T= ΔRΔλ1, Δλ2, TΔT = ΔIp1Δλ1, TIp2Δλ2, TΔT K-1.
Ip1Δλ1, T= qλ1P1Δλ1, Tηλ1hc A.
Ip2Δλ2, T= qλ2P2Δλ2, Tηλ2hc A.
SΔλ1, Δλ2, T= ΔP1Δλ1, TP2Δλ2, TΔT K-1.
SΔλ1, Δλ2, T= dP1Δλ1, TdTP2Δλ2, T-P1Δλ1, TdP2Δλ2, TdTP2Δλ2, T2K-1.

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