Abstract

We report on a systematic investigation of the dependence of both temperature and strain sensitivities on the fiber Bragg grating type, including the well-known Type I, Type IIA, and a new type that we have designated Type IA, using both hydrogen-free and hydrogenated B/Ge codoped fibres. We have identified distinct sensitivity characteristics for each grating type, and we have used them to implement a novel dual-grating, dual-parameter sensor device. Three dual-grating sensing schemes with different combinations of grating type have been constructed and compared, and that of a Type IA-Type IIA combination exhibits the best performance, which is also superior to that of previously reported grating-based structures. The characteristics of the measurement errors in such dual-grating sensor systems is also presented in detail.

© 2004 Optical Society of America

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  1. A. D. Kersey, M. A. Davis, H. J. Patrick, M. LeBlanc, K. P. Koo, C. G. Askins, M. A. Putnam, E. J. Friebele, “Fiber grating sensor,” J. Lightwave Technol. 15, 1442–1463 (1997).
    [CrossRef]
  2. M. G. Xu, J.-L. Archambault, L. Reekie, J. P. Dakin, “Discrimination between strain and temperature effects using dual-wavelength fiber grating sensor,” Electron. Lett. 30, 1085–1087 (1994).
    [CrossRef]
  3. G. P. Brady, K. Kalli, D. J. Webb, D. A. Jackson, L. Reekie, J. L. Archambault, “Simultaneous measurement of strain and temperature using the first- and second-order diffraction wavelength of Bragg gratings,” IEE Proc. Optoelectron. 44, 156–161 (1997).
    [CrossRef]
  4. H. J. Patrick, G. M. Williams, A. D. Kersey, J. P. Pedrazzani, A. M. Vengsarkar, “Hybrid fiber Bragg grating/long period fiber grating sensor for strain/temperature discrimination,” IEEE Photon. Technol. Lett. 8, 1223–1225 (1996).
    [CrossRef]
  5. S. E. Kanellopoulos, V. A. Handerek, A. J. Rogers, “Simultaneous strain and temperature sensing with photogenerated in-fiber gratings,” Opt. Lett. 20, 333–335 (1995).
    [CrossRef] [PubMed]
  6. S. W. James, M. L. Dockney, R. P. Tatam, “Simultaneous independent temperature and strain measurement using in-fiber Bragg grating sensor,” Electron. Lett. 32, 1133–1134 (1996).
    [CrossRef]
  7. B. O. Guan, H. Y. Tam, S. L. Ho, W. H. Chung, X. Y. Dong, “Simultaneous strain and temperature measurement using a single fiber Bragg grating,” Electron. Lett. 36, 1018–1019 (2000).
    [CrossRef]
  8. Y. Liu, J. A. R. Williams, L. Zhang, I. Bennion, “Abnormal spectral evolution of fiber Bragg gratings in hydrogenated fibers,” Opt. Lett. 27, 583–585 (2002).
    [CrossRef]
  9. X. Shu, Y. Liu, D. Zhao, B. Gwandu, F. Floreani, L. Zhang, I. Bennion, “Dependence of temperature and strain coefficients on fiber grating type and its application to simultaneous temperature and strain measurement,” Opt. Lett. 27, 701–703 (2002).
    [CrossRef]
  10. W. X. Xie, P. Niay, P. Bernage, M. Douay, J. F. Bayon, T. Georges, M. Monerie, B. Poumellec, “Experimental evidence of two types of photorefractive effects occurring during photoinscriptions of Bragg gratings within germanosilicate fibers,” Opt. Commun. 104, 185–195 (1993).
    [CrossRef]
  11. L. Dong, W. F. Liu, L. Reekie, “Negative-index gratings formed by a 193-nm excimer laser,” Opt. Lett. 21, 2032–2034 (1996).
    [CrossRef]
  12. W. Jin, W. C. Michie, G. Thursby, M. Konstantaki, B. Culshaw, “Simultaneous measurement of strain and temperature: error analysis,” Opt. Eng. 36, 598–609 (1997).
    [CrossRef]

2002

2000

B. O. Guan, H. Y. Tam, S. L. Ho, W. H. Chung, X. Y. Dong, “Simultaneous strain and temperature measurement using a single fiber Bragg grating,” Electron. Lett. 36, 1018–1019 (2000).
[CrossRef]

1997

A. D. Kersey, M. A. Davis, H. J. Patrick, M. LeBlanc, K. P. Koo, C. G. Askins, M. A. Putnam, E. J. Friebele, “Fiber grating sensor,” J. Lightwave Technol. 15, 1442–1463 (1997).
[CrossRef]

G. P. Brady, K. Kalli, D. J. Webb, D. A. Jackson, L. Reekie, J. L. Archambault, “Simultaneous measurement of strain and temperature using the first- and second-order diffraction wavelength of Bragg gratings,” IEE Proc. Optoelectron. 44, 156–161 (1997).
[CrossRef]

W. Jin, W. C. Michie, G. Thursby, M. Konstantaki, B. Culshaw, “Simultaneous measurement of strain and temperature: error analysis,” Opt. Eng. 36, 598–609 (1997).
[CrossRef]

1996

L. Dong, W. F. Liu, L. Reekie, “Negative-index gratings formed by a 193-nm excimer laser,” Opt. Lett. 21, 2032–2034 (1996).
[CrossRef]

H. J. Patrick, G. M. Williams, A. D. Kersey, J. P. Pedrazzani, A. M. Vengsarkar, “Hybrid fiber Bragg grating/long period fiber grating sensor for strain/temperature discrimination,” IEEE Photon. Technol. Lett. 8, 1223–1225 (1996).
[CrossRef]

S. W. James, M. L. Dockney, R. P. Tatam, “Simultaneous independent temperature and strain measurement using in-fiber Bragg grating sensor,” Electron. Lett. 32, 1133–1134 (1996).
[CrossRef]

1995

1994

M. G. Xu, J.-L. Archambault, L. Reekie, J. P. Dakin, “Discrimination between strain and temperature effects using dual-wavelength fiber grating sensor,” Electron. Lett. 30, 1085–1087 (1994).
[CrossRef]

1993

W. X. Xie, P. Niay, P. Bernage, M. Douay, J. F. Bayon, T. Georges, M. Monerie, B. Poumellec, “Experimental evidence of two types of photorefractive effects occurring during photoinscriptions of Bragg gratings within germanosilicate fibers,” Opt. Commun. 104, 185–195 (1993).
[CrossRef]

Archambault, J. L.

G. P. Brady, K. Kalli, D. J. Webb, D. A. Jackson, L. Reekie, J. L. Archambault, “Simultaneous measurement of strain and temperature using the first- and second-order diffraction wavelength of Bragg gratings,” IEE Proc. Optoelectron. 44, 156–161 (1997).
[CrossRef]

Archambault, J.-L.

M. G. Xu, J.-L. Archambault, L. Reekie, J. P. Dakin, “Discrimination between strain and temperature effects using dual-wavelength fiber grating sensor,” Electron. Lett. 30, 1085–1087 (1994).
[CrossRef]

Askins, C. G.

A. D. Kersey, M. A. Davis, H. J. Patrick, M. LeBlanc, K. P. Koo, C. G. Askins, M. A. Putnam, E. J. Friebele, “Fiber grating sensor,” J. Lightwave Technol. 15, 1442–1463 (1997).
[CrossRef]

Bayon, J. F.

W. X. Xie, P. Niay, P. Bernage, M. Douay, J. F. Bayon, T. Georges, M. Monerie, B. Poumellec, “Experimental evidence of two types of photorefractive effects occurring during photoinscriptions of Bragg gratings within germanosilicate fibers,” Opt. Commun. 104, 185–195 (1993).
[CrossRef]

Bennion, I.

Bernage, P.

W. X. Xie, P. Niay, P. Bernage, M. Douay, J. F. Bayon, T. Georges, M. Monerie, B. Poumellec, “Experimental evidence of two types of photorefractive effects occurring during photoinscriptions of Bragg gratings within germanosilicate fibers,” Opt. Commun. 104, 185–195 (1993).
[CrossRef]

Brady, G. P.

G. P. Brady, K. Kalli, D. J. Webb, D. A. Jackson, L. Reekie, J. L. Archambault, “Simultaneous measurement of strain and temperature using the first- and second-order diffraction wavelength of Bragg gratings,” IEE Proc. Optoelectron. 44, 156–161 (1997).
[CrossRef]

Chung, W. H.

B. O. Guan, H. Y. Tam, S. L. Ho, W. H. Chung, X. Y. Dong, “Simultaneous strain and temperature measurement using a single fiber Bragg grating,” Electron. Lett. 36, 1018–1019 (2000).
[CrossRef]

Culshaw, B.

W. Jin, W. C. Michie, G. Thursby, M. Konstantaki, B. Culshaw, “Simultaneous measurement of strain and temperature: error analysis,” Opt. Eng. 36, 598–609 (1997).
[CrossRef]

Dakin, J. P.

M. G. Xu, J.-L. Archambault, L. Reekie, J. P. Dakin, “Discrimination between strain and temperature effects using dual-wavelength fiber grating sensor,” Electron. Lett. 30, 1085–1087 (1994).
[CrossRef]

Davis, M. A.

A. D. Kersey, M. A. Davis, H. J. Patrick, M. LeBlanc, K. P. Koo, C. G. Askins, M. A. Putnam, E. J. Friebele, “Fiber grating sensor,” J. Lightwave Technol. 15, 1442–1463 (1997).
[CrossRef]

Dockney, M. L.

S. W. James, M. L. Dockney, R. P. Tatam, “Simultaneous independent temperature and strain measurement using in-fiber Bragg grating sensor,” Electron. Lett. 32, 1133–1134 (1996).
[CrossRef]

Dong, L.

Dong, X. Y.

B. O. Guan, H. Y. Tam, S. L. Ho, W. H. Chung, X. Y. Dong, “Simultaneous strain and temperature measurement using a single fiber Bragg grating,” Electron. Lett. 36, 1018–1019 (2000).
[CrossRef]

Douay, M.

W. X. Xie, P. Niay, P. Bernage, M. Douay, J. F. Bayon, T. Georges, M. Monerie, B. Poumellec, “Experimental evidence of two types of photorefractive effects occurring during photoinscriptions of Bragg gratings within germanosilicate fibers,” Opt. Commun. 104, 185–195 (1993).
[CrossRef]

Floreani, F.

Friebele, E. J.

A. D. Kersey, M. A. Davis, H. J. Patrick, M. LeBlanc, K. P. Koo, C. G. Askins, M. A. Putnam, E. J. Friebele, “Fiber grating sensor,” J. Lightwave Technol. 15, 1442–1463 (1997).
[CrossRef]

Georges, T.

W. X. Xie, P. Niay, P. Bernage, M. Douay, J. F. Bayon, T. Georges, M. Monerie, B. Poumellec, “Experimental evidence of two types of photorefractive effects occurring during photoinscriptions of Bragg gratings within germanosilicate fibers,” Opt. Commun. 104, 185–195 (1993).
[CrossRef]

Guan, B. O.

B. O. Guan, H. Y. Tam, S. L. Ho, W. H. Chung, X. Y. Dong, “Simultaneous strain and temperature measurement using a single fiber Bragg grating,” Electron. Lett. 36, 1018–1019 (2000).
[CrossRef]

Gwandu, B.

Handerek, V. A.

Ho, S. L.

B. O. Guan, H. Y. Tam, S. L. Ho, W. H. Chung, X. Y. Dong, “Simultaneous strain and temperature measurement using a single fiber Bragg grating,” Electron. Lett. 36, 1018–1019 (2000).
[CrossRef]

Jackson, D. A.

G. P. Brady, K. Kalli, D. J. Webb, D. A. Jackson, L. Reekie, J. L. Archambault, “Simultaneous measurement of strain and temperature using the first- and second-order diffraction wavelength of Bragg gratings,” IEE Proc. Optoelectron. 44, 156–161 (1997).
[CrossRef]

James, S. W.

S. W. James, M. L. Dockney, R. P. Tatam, “Simultaneous independent temperature and strain measurement using in-fiber Bragg grating sensor,” Electron. Lett. 32, 1133–1134 (1996).
[CrossRef]

Jin, W.

W. Jin, W. C. Michie, G. Thursby, M. Konstantaki, B. Culshaw, “Simultaneous measurement of strain and temperature: error analysis,” Opt. Eng. 36, 598–609 (1997).
[CrossRef]

Kalli, K.

G. P. Brady, K. Kalli, D. J. Webb, D. A. Jackson, L. Reekie, J. L. Archambault, “Simultaneous measurement of strain and temperature using the first- and second-order diffraction wavelength of Bragg gratings,” IEE Proc. Optoelectron. 44, 156–161 (1997).
[CrossRef]

Kanellopoulos, S. E.

Kersey, A. D.

A. D. Kersey, M. A. Davis, H. J. Patrick, M. LeBlanc, K. P. Koo, C. G. Askins, M. A. Putnam, E. J. Friebele, “Fiber grating sensor,” J. Lightwave Technol. 15, 1442–1463 (1997).
[CrossRef]

H. J. Patrick, G. M. Williams, A. D. Kersey, J. P. Pedrazzani, A. M. Vengsarkar, “Hybrid fiber Bragg grating/long period fiber grating sensor for strain/temperature discrimination,” IEEE Photon. Technol. Lett. 8, 1223–1225 (1996).
[CrossRef]

Konstantaki, M.

W. Jin, W. C. Michie, G. Thursby, M. Konstantaki, B. Culshaw, “Simultaneous measurement of strain and temperature: error analysis,” Opt. Eng. 36, 598–609 (1997).
[CrossRef]

Koo, K. P.

A. D. Kersey, M. A. Davis, H. J. Patrick, M. LeBlanc, K. P. Koo, C. G. Askins, M. A. Putnam, E. J. Friebele, “Fiber grating sensor,” J. Lightwave Technol. 15, 1442–1463 (1997).
[CrossRef]

LeBlanc, M.

A. D. Kersey, M. A. Davis, H. J. Patrick, M. LeBlanc, K. P. Koo, C. G. Askins, M. A. Putnam, E. J. Friebele, “Fiber grating sensor,” J. Lightwave Technol. 15, 1442–1463 (1997).
[CrossRef]

Liu, W. F.

Liu, Y.

Michie, W. C.

W. Jin, W. C. Michie, G. Thursby, M. Konstantaki, B. Culshaw, “Simultaneous measurement of strain and temperature: error analysis,” Opt. Eng. 36, 598–609 (1997).
[CrossRef]

Monerie, M.

W. X. Xie, P. Niay, P. Bernage, M. Douay, J. F. Bayon, T. Georges, M. Monerie, B. Poumellec, “Experimental evidence of two types of photorefractive effects occurring during photoinscriptions of Bragg gratings within germanosilicate fibers,” Opt. Commun. 104, 185–195 (1993).
[CrossRef]

Niay, P.

W. X. Xie, P. Niay, P. Bernage, M. Douay, J. F. Bayon, T. Georges, M. Monerie, B. Poumellec, “Experimental evidence of two types of photorefractive effects occurring during photoinscriptions of Bragg gratings within germanosilicate fibers,” Opt. Commun. 104, 185–195 (1993).
[CrossRef]

Patrick, H. J.

A. D. Kersey, M. A. Davis, H. J. Patrick, M. LeBlanc, K. P. Koo, C. G. Askins, M. A. Putnam, E. J. Friebele, “Fiber grating sensor,” J. Lightwave Technol. 15, 1442–1463 (1997).
[CrossRef]

H. J. Patrick, G. M. Williams, A. D. Kersey, J. P. Pedrazzani, A. M. Vengsarkar, “Hybrid fiber Bragg grating/long period fiber grating sensor for strain/temperature discrimination,” IEEE Photon. Technol. Lett. 8, 1223–1225 (1996).
[CrossRef]

Pedrazzani, J. P.

H. J. Patrick, G. M. Williams, A. D. Kersey, J. P. Pedrazzani, A. M. Vengsarkar, “Hybrid fiber Bragg grating/long period fiber grating sensor for strain/temperature discrimination,” IEEE Photon. Technol. Lett. 8, 1223–1225 (1996).
[CrossRef]

Poumellec, B.

W. X. Xie, P. Niay, P. Bernage, M. Douay, J. F. Bayon, T. Georges, M. Monerie, B. Poumellec, “Experimental evidence of two types of photorefractive effects occurring during photoinscriptions of Bragg gratings within germanosilicate fibers,” Opt. Commun. 104, 185–195 (1993).
[CrossRef]

Putnam, M. A.

A. D. Kersey, M. A. Davis, H. J. Patrick, M. LeBlanc, K. P. Koo, C. G. Askins, M. A. Putnam, E. J. Friebele, “Fiber grating sensor,” J. Lightwave Technol. 15, 1442–1463 (1997).
[CrossRef]

Reekie, L.

G. P. Brady, K. Kalli, D. J. Webb, D. A. Jackson, L. Reekie, J. L. Archambault, “Simultaneous measurement of strain and temperature using the first- and second-order diffraction wavelength of Bragg gratings,” IEE Proc. Optoelectron. 44, 156–161 (1997).
[CrossRef]

L. Dong, W. F. Liu, L. Reekie, “Negative-index gratings formed by a 193-nm excimer laser,” Opt. Lett. 21, 2032–2034 (1996).
[CrossRef]

M. G. Xu, J.-L. Archambault, L. Reekie, J. P. Dakin, “Discrimination between strain and temperature effects using dual-wavelength fiber grating sensor,” Electron. Lett. 30, 1085–1087 (1994).
[CrossRef]

Rogers, A. J.

Shu, X.

Tam, H. Y.

B. O. Guan, H. Y. Tam, S. L. Ho, W. H. Chung, X. Y. Dong, “Simultaneous strain and temperature measurement using a single fiber Bragg grating,” Electron. Lett. 36, 1018–1019 (2000).
[CrossRef]

Tatam, R. P.

S. W. James, M. L. Dockney, R. P. Tatam, “Simultaneous independent temperature and strain measurement using in-fiber Bragg grating sensor,” Electron. Lett. 32, 1133–1134 (1996).
[CrossRef]

Thursby, G.

W. Jin, W. C. Michie, G. Thursby, M. Konstantaki, B. Culshaw, “Simultaneous measurement of strain and temperature: error analysis,” Opt. Eng. 36, 598–609 (1997).
[CrossRef]

Vengsarkar, A. M.

H. J. Patrick, G. M. Williams, A. D. Kersey, J. P. Pedrazzani, A. M. Vengsarkar, “Hybrid fiber Bragg grating/long period fiber grating sensor for strain/temperature discrimination,” IEEE Photon. Technol. Lett. 8, 1223–1225 (1996).
[CrossRef]

Webb, D. J.

G. P. Brady, K. Kalli, D. J. Webb, D. A. Jackson, L. Reekie, J. L. Archambault, “Simultaneous measurement of strain and temperature using the first- and second-order diffraction wavelength of Bragg gratings,” IEE Proc. Optoelectron. 44, 156–161 (1997).
[CrossRef]

Williams, G. M.

H. J. Patrick, G. M. Williams, A. D. Kersey, J. P. Pedrazzani, A. M. Vengsarkar, “Hybrid fiber Bragg grating/long period fiber grating sensor for strain/temperature discrimination,” IEEE Photon. Technol. Lett. 8, 1223–1225 (1996).
[CrossRef]

Williams, J. A. R.

Xie, W. X.

W. X. Xie, P. Niay, P. Bernage, M. Douay, J. F. Bayon, T. Georges, M. Monerie, B. Poumellec, “Experimental evidence of two types of photorefractive effects occurring during photoinscriptions of Bragg gratings within germanosilicate fibers,” Opt. Commun. 104, 185–195 (1993).
[CrossRef]

Xu, M. G.

M. G. Xu, J.-L. Archambault, L. Reekie, J. P. Dakin, “Discrimination between strain and temperature effects using dual-wavelength fiber grating sensor,” Electron. Lett. 30, 1085–1087 (1994).
[CrossRef]

Zhang, L.

Zhao, D.

Electron. Lett.

M. G. Xu, J.-L. Archambault, L. Reekie, J. P. Dakin, “Discrimination between strain and temperature effects using dual-wavelength fiber grating sensor,” Electron. Lett. 30, 1085–1087 (1994).
[CrossRef]

S. W. James, M. L. Dockney, R. P. Tatam, “Simultaneous independent temperature and strain measurement using in-fiber Bragg grating sensor,” Electron. Lett. 32, 1133–1134 (1996).
[CrossRef]

B. O. Guan, H. Y. Tam, S. L. Ho, W. H. Chung, X. Y. Dong, “Simultaneous strain and temperature measurement using a single fiber Bragg grating,” Electron. Lett. 36, 1018–1019 (2000).
[CrossRef]

IEE Proc. Optoelectron.

G. P. Brady, K. Kalli, D. J. Webb, D. A. Jackson, L. Reekie, J. L. Archambault, “Simultaneous measurement of strain and temperature using the first- and second-order diffraction wavelength of Bragg gratings,” IEE Proc. Optoelectron. 44, 156–161 (1997).
[CrossRef]

IEEE Photon. Technol. Lett.

H. J. Patrick, G. M. Williams, A. D. Kersey, J. P. Pedrazzani, A. M. Vengsarkar, “Hybrid fiber Bragg grating/long period fiber grating sensor for strain/temperature discrimination,” IEEE Photon. Technol. Lett. 8, 1223–1225 (1996).
[CrossRef]

J. Lightwave Technol.

A. D. Kersey, M. A. Davis, H. J. Patrick, M. LeBlanc, K. P. Koo, C. G. Askins, M. A. Putnam, E. J. Friebele, “Fiber grating sensor,” J. Lightwave Technol. 15, 1442–1463 (1997).
[CrossRef]

Opt. Commun.

W. X. Xie, P. Niay, P. Bernage, M. Douay, J. F. Bayon, T. Georges, M. Monerie, B. Poumellec, “Experimental evidence of two types of photorefractive effects occurring during photoinscriptions of Bragg gratings within germanosilicate fibers,” Opt. Commun. 104, 185–195 (1993).
[CrossRef]

Opt. Eng.

W. Jin, W. C. Michie, G. Thursby, M. Konstantaki, B. Culshaw, “Simultaneous measurement of strain and temperature: error analysis,” Opt. Eng. 36, 598–609 (1997).
[CrossRef]

Opt. Lett.

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Figures (10)

Fig. 1
Fig. 1

Evolution of the transmission spectrum of a grating inscribed in (a) a hydrogen-free and (b) a hydrogenated B/Ge codoped fiber with long UV exposure. The arrows indicate the direction of increasing exposure time.

Fig. 2
Fig. 2

Plots of the FBG central wavelength and the transmission peak depth as a function of accumulated exposure time. (a) and (b) correspond to (a) and (b) in Fig. 1, respectively.

Fig. 3
Fig. 3

Distribution of (a) the measured temperature coefficient and (b) the strain coefficient for different types of FBG inscribed in B/Ge codoped fiber; A, Type IIA gratings; B, Type I gratings in hydrogen-free fiber; C, Type I gratings in hydrogenated fiber; D, Type IA gratings.

Fig. 4
Fig. 4

Schematic of the proposed dual-grating sensor employing concatenated different-type gratings.

Fig. 5
Fig. 5

Spectral response changes in the first dual-grating sensor induced by (a) temperature and (b) strain.

Fig. 6
Fig. 6

Plots of wavelength shift for the first dual-grating sensor as a function of (a) temperature and (b) strain. FBG1, Type IA grating; FBG2, Type IIA grating.

Fig. 7
Fig. 7

Plots of wavelength shift for the second dual-grating sensor as a function of (a) temperature and (b) strain. FBG1, Type IIA grating; FBG2, Type 1 grating.

Fig. 8
Fig. 8

Plots of wavelength shift for the third dual-grating sensor as a function of (a) temperature and (b) strain. FBG1: Type IA grating; FBG2, type I grating.

Fig. 9
Fig. 9

Maximum temperature and strain errors as a function of κ T1 in the dual-grating sensor system, assuming κ T2 = 10.02pm/°C, κε1 = 1.074pm/με, and κε2 = 1.075pm/με.

Fig. 10
Fig. 10

(a) Maximum temperature errors and (b) strain errors as a function of the ratio of κ T2 and κ T1 in the dual-grating sensor system, assuming κ T1 = 8, 10, and 12 pm/°C, κε1 = 1.074pm/με, and κε2 = 1.075pm/με.

Tables (2)

Tables Icon

Table 1 Combination of Two Different Types of Grating in a Dual-Grating Sensing Scheme

Tables Icon

Table 2 Maximum Temperature and Strain Errors for Three Different Combinations in a Dual-Grating Sensing Scheme

Equations (6)

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

Δλ1Δλ2=κT1κε1κT2κε2ΔTΔε
ΔTΔε=1κT2-κT1Δλ2-Δλ1κT2Δλ1-κT1Δλ2/κε1.
ΔTmaxΔεmax=±MT-MTε2/Mε- 12±(Mε-MTε2/MT- 12,
MT=κT12a2+κT22b2,
Mε=κε12a2+κε22b2,
MTε=κT1κε1a2+κT2κε2b2,

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