Abstract

Radio-frequency (rf) beat frequencies between two longitudinal modes and two polarization modes of a birefringent dual-longitudinal-mode moiré distributed-feedback fiber laser are employed to measure strain and temperature simultaneously. Operating entirely in the rf domain, this approach potentially allows one to employ low-cost and precise rf measuring techniques. A strain–temperature cross sensitivity of the strain- and the thermo-optic coefficients, which can be neglected in wavelength-based grating sensors, has been observed. The achieved sensor accuracy was ±15 µ∊ and ±0.2 °C.

© 2001 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 sensors,” J. Lightwave Technol. 15, 1442–1463 (1997).
    [CrossRef]
  2. P. J. Henderson, D. J. Webb, D. A. Jackson, L. Zhang, I. Bennion, “Highly-multiplexed grating-sensors for temperature-referenced quasi-static measurements of strain in concrete bridges,” in 13th International Conference on Optical Fiber Sensors, B. Kim, K. Hotate, eds., Proc. SPIE3746, 320–323 (1999).
  3. M. G. Xu, J.-L. Archambault, L. Reekie, J. P. Dakin, “Discrimination between strain and temperature effects using dual-wavelength fibre grating sensors,” Electron. Lett. 30, 1085–1087 (1994).
    [CrossRef]
  4. M. Sudo, M. Nakai, K. Himeno, S. Suzaki, A. Wada, R. Yamauchi, “Simultaneous measurement of temperature and strain using PANDA fiber grating,” in 12th International Conference on Optical Fiber Sensors, Vol. 16 of 1997 OSA Technical Digest Series (Optical Society of America, Washington D.C., 1997), pp. 170–173.
  5. O. Hadeler, E. Rønnekleiv, M. Ibsen, R. I. Laming, “Polarimetric distributed feedback fiber laser sensor for simultaneous strain and temperature measurements,” Appl. Opt. 38, 1953–1958 (1999).
  6. M. Ibsen, E. Rønnekleiv, O. Hadeler, G. J. Cowle, M. N. Zervas, R. I. Laming, “Stable multiple wavelength generation in all-fibre DFB lasers,” in Bragg Gratings, Photosensitivity, and Poling in Glass Waveguides, E. J. Friebele, R. Kashyap, T. Erdogan, eds., Vol. 33 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1999), Paper FA4.
  7. S.-Y. Huang, J. N. Blake, B. Y. Kim, “Perturbation effects on mode propagation in highly elliptical core two-mode fibers,” J. Lightwave Technol. 8, 23–33 (1990).
    [CrossRef]
  8. L. Dong, W. H. Loh, J. E. Caplen, J. D. Minelly, “Efficient single-frequency fiber lasers with novel photosensitive Er/Yb optical fibers,” Opt. Lett. 22, 694–696 (1997).
    [CrossRef] [PubMed]
  9. O. Hadeler, M. N. Zervas, “Application of a DFB fibre laser temperature sensor for characterizing pump induced temperature distributions along another DFB fibre laser,” in 14th International Conference on Optical Fiber Sensors, A. G. Mignani, ed., Proc. SPIE4185, 142–145 (2000).
  10. G. Meltz, W. W. Morrey, W. H. Glenn, J. D. Farina, “In-fiber Bragg grating sensors,” in 5th International Conference on Optical Fiber Sensors, in Vol. 2 of 1988 OSA Technical Digest Series (Optical Society of America, Washington D.C., 1988), pp. 163–166.
  11. R. M. Waxler, G. W. Cleek, “The effect of temperature and pressure on the refractive index of some oxide glasses,” J. Res. Nat. Bur. Stand. Sect. A 77, 755–763 (1973).
    [CrossRef]

1999 (1)

1997 (2)

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 sensors,” J. Lightwave Technol. 15, 1442–1463 (1997).
[CrossRef]

L. Dong, W. H. Loh, J. E. Caplen, J. D. Minelly, “Efficient single-frequency fiber lasers with novel photosensitive Er/Yb optical fibers,” Opt. Lett. 22, 694–696 (1997).
[CrossRef] [PubMed]

1994 (1)

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

1990 (1)

S.-Y. Huang, J. N. Blake, B. Y. Kim, “Perturbation effects on mode propagation in highly elliptical core two-mode fibers,” J. Lightwave Technol. 8, 23–33 (1990).
[CrossRef]

1973 (1)

R. M. Waxler, G. W. Cleek, “The effect of temperature and pressure on the refractive index of some oxide glasses,” J. Res. Nat. Bur. Stand. Sect. A 77, 755–763 (1973).
[CrossRef]

Archambault, J.-L.

M. G. Xu, J.-L. Archambault, L. Reekie, J. P. Dakin, “Discrimination between strain and temperature effects using dual-wavelength fibre grating sensors,” 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 sensors,” J. Lightwave Technol. 15, 1442–1463 (1997).
[CrossRef]

Bennion, I.

P. J. Henderson, D. J. Webb, D. A. Jackson, L. Zhang, I. Bennion, “Highly-multiplexed grating-sensors for temperature-referenced quasi-static measurements of strain in concrete bridges,” in 13th International Conference on Optical Fiber Sensors, B. Kim, K. Hotate, eds., Proc. SPIE3746, 320–323 (1999).

Blake, J. N.

S.-Y. Huang, J. N. Blake, B. Y. Kim, “Perturbation effects on mode propagation in highly elliptical core two-mode fibers,” J. Lightwave Technol. 8, 23–33 (1990).
[CrossRef]

Caplen, J. E.

Cleek, G. W.

R. M. Waxler, G. W. Cleek, “The effect of temperature and pressure on the refractive index of some oxide glasses,” J. Res. Nat. Bur. Stand. Sect. A 77, 755–763 (1973).
[CrossRef]

Cowle, G. J.

M. Ibsen, E. Rønnekleiv, O. Hadeler, G. J. Cowle, M. N. Zervas, R. I. Laming, “Stable multiple wavelength generation in all-fibre DFB lasers,” in Bragg Gratings, Photosensitivity, and Poling in Glass Waveguides, E. J. Friebele, R. Kashyap, T. Erdogan, eds., Vol. 33 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1999), Paper FA4.

Dakin, J. P.

M. G. Xu, J.-L. Archambault, L. Reekie, J. P. Dakin, “Discrimination between strain and temperature effects using dual-wavelength fibre grating sensors,” 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 sensors,” J. Lightwave Technol. 15, 1442–1463 (1997).
[CrossRef]

Dong, L.

Farina, J. D.

G. Meltz, W. W. Morrey, W. H. Glenn, J. D. Farina, “In-fiber Bragg grating sensors,” in 5th International Conference on Optical Fiber Sensors, in Vol. 2 of 1988 OSA Technical Digest Series (Optical Society of America, Washington D.C., 1988), pp. 163–166.

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 sensors,” J. Lightwave Technol. 15, 1442–1463 (1997).
[CrossRef]

Glenn, W. H.

G. Meltz, W. W. Morrey, W. H. Glenn, J. D. Farina, “In-fiber Bragg grating sensors,” in 5th International Conference on Optical Fiber Sensors, in Vol. 2 of 1988 OSA Technical Digest Series (Optical Society of America, Washington D.C., 1988), pp. 163–166.

Hadeler, O.

O. Hadeler, E. Rønnekleiv, M. Ibsen, R. I. Laming, “Polarimetric distributed feedback fiber laser sensor for simultaneous strain and temperature measurements,” Appl. Opt. 38, 1953–1958 (1999).

M. Ibsen, E. Rønnekleiv, O. Hadeler, G. J. Cowle, M. N. Zervas, R. I. Laming, “Stable multiple wavelength generation in all-fibre DFB lasers,” in Bragg Gratings, Photosensitivity, and Poling in Glass Waveguides, E. J. Friebele, R. Kashyap, T. Erdogan, eds., Vol. 33 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1999), Paper FA4.

O. Hadeler, M. N. Zervas, “Application of a DFB fibre laser temperature sensor for characterizing pump induced temperature distributions along another DFB fibre laser,” in 14th International Conference on Optical Fiber Sensors, A. G. Mignani, ed., Proc. SPIE4185, 142–145 (2000).

Henderson, P. J.

P. J. Henderson, D. J. Webb, D. A. Jackson, L. Zhang, I. Bennion, “Highly-multiplexed grating-sensors for temperature-referenced quasi-static measurements of strain in concrete bridges,” in 13th International Conference on Optical Fiber Sensors, B. Kim, K. Hotate, eds., Proc. SPIE3746, 320–323 (1999).

Himeno, K.

M. Sudo, M. Nakai, K. Himeno, S. Suzaki, A. Wada, R. Yamauchi, “Simultaneous measurement of temperature and strain using PANDA fiber grating,” in 12th International Conference on Optical Fiber Sensors, Vol. 16 of 1997 OSA Technical Digest Series (Optical Society of America, Washington D.C., 1997), pp. 170–173.

Huang, S.-Y.

S.-Y. Huang, J. N. Blake, B. Y. Kim, “Perturbation effects on mode propagation in highly elliptical core two-mode fibers,” J. Lightwave Technol. 8, 23–33 (1990).
[CrossRef]

Ibsen, M.

O. Hadeler, E. Rønnekleiv, M. Ibsen, R. I. Laming, “Polarimetric distributed feedback fiber laser sensor for simultaneous strain and temperature measurements,” Appl. Opt. 38, 1953–1958 (1999).

M. Ibsen, E. Rønnekleiv, O. Hadeler, G. J. Cowle, M. N. Zervas, R. I. Laming, “Stable multiple wavelength generation in all-fibre DFB lasers,” in Bragg Gratings, Photosensitivity, and Poling in Glass Waveguides, E. J. Friebele, R. Kashyap, T. Erdogan, eds., Vol. 33 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1999), Paper FA4.

Jackson, D. A.

P. J. Henderson, D. J. Webb, D. A. Jackson, L. Zhang, I. Bennion, “Highly-multiplexed grating-sensors for temperature-referenced quasi-static measurements of strain in concrete bridges,” in 13th International Conference on Optical Fiber Sensors, B. Kim, K. Hotate, eds., Proc. SPIE3746, 320–323 (1999).

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 sensors,” J. Lightwave Technol. 15, 1442–1463 (1997).
[CrossRef]

Kim, B. Y.

S.-Y. Huang, J. N. Blake, B. Y. Kim, “Perturbation effects on mode propagation in highly elliptical core two-mode fibers,” J. Lightwave Technol. 8, 23–33 (1990).
[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 sensors,” J. Lightwave Technol. 15, 1442–1463 (1997).
[CrossRef]

Laming, R. I.

O. Hadeler, E. Rønnekleiv, M. Ibsen, R. I. Laming, “Polarimetric distributed feedback fiber laser sensor for simultaneous strain and temperature measurements,” Appl. Opt. 38, 1953–1958 (1999).

M. Ibsen, E. Rønnekleiv, O. Hadeler, G. J. Cowle, M. N. Zervas, R. I. Laming, “Stable multiple wavelength generation in all-fibre DFB lasers,” in Bragg Gratings, Photosensitivity, and Poling in Glass Waveguides, E. J. Friebele, R. Kashyap, T. Erdogan, eds., Vol. 33 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1999), Paper FA4.

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 sensors,” J. Lightwave Technol. 15, 1442–1463 (1997).
[CrossRef]

Loh, W. H.

Meltz, G.

G. Meltz, W. W. Morrey, W. H. Glenn, J. D. Farina, “In-fiber Bragg grating sensors,” in 5th International Conference on Optical Fiber Sensors, in Vol. 2 of 1988 OSA Technical Digest Series (Optical Society of America, Washington D.C., 1988), pp. 163–166.

Minelly, J. D.

Morrey, W. W.

G. Meltz, W. W. Morrey, W. H. Glenn, J. D. Farina, “In-fiber Bragg grating sensors,” in 5th International Conference on Optical Fiber Sensors, in Vol. 2 of 1988 OSA Technical Digest Series (Optical Society of America, Washington D.C., 1988), pp. 163–166.

Nakai, M.

M. Sudo, M. Nakai, K. Himeno, S. Suzaki, A. Wada, R. Yamauchi, “Simultaneous measurement of temperature and strain using PANDA fiber grating,” in 12th International Conference on Optical Fiber Sensors, Vol. 16 of 1997 OSA Technical Digest Series (Optical Society of America, Washington D.C., 1997), pp. 170–173.

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 sensors,” J. Lightwave Technol. 15, 1442–1463 (1997).
[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 sensors,” J. Lightwave Technol. 15, 1442–1463 (1997).
[CrossRef]

Reekie, L.

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

Rønnekleiv, E.

O. Hadeler, E. Rønnekleiv, M. Ibsen, R. I. Laming, “Polarimetric distributed feedback fiber laser sensor for simultaneous strain and temperature measurements,” Appl. Opt. 38, 1953–1958 (1999).

M. Ibsen, E. Rønnekleiv, O. Hadeler, G. J. Cowle, M. N. Zervas, R. I. Laming, “Stable multiple wavelength generation in all-fibre DFB lasers,” in Bragg Gratings, Photosensitivity, and Poling in Glass Waveguides, E. J. Friebele, R. Kashyap, T. Erdogan, eds., Vol. 33 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1999), Paper FA4.

Sudo, M.

M. Sudo, M. Nakai, K. Himeno, S. Suzaki, A. Wada, R. Yamauchi, “Simultaneous measurement of temperature and strain using PANDA fiber grating,” in 12th International Conference on Optical Fiber Sensors, Vol. 16 of 1997 OSA Technical Digest Series (Optical Society of America, Washington D.C., 1997), pp. 170–173.

Suzaki, S.

M. Sudo, M. Nakai, K. Himeno, S. Suzaki, A. Wada, R. Yamauchi, “Simultaneous measurement of temperature and strain using PANDA fiber grating,” in 12th International Conference on Optical Fiber Sensors, Vol. 16 of 1997 OSA Technical Digest Series (Optical Society of America, Washington D.C., 1997), pp. 170–173.

Wada, A.

M. Sudo, M. Nakai, K. Himeno, S. Suzaki, A. Wada, R. Yamauchi, “Simultaneous measurement of temperature and strain using PANDA fiber grating,” in 12th International Conference on Optical Fiber Sensors, Vol. 16 of 1997 OSA Technical Digest Series (Optical Society of America, Washington D.C., 1997), pp. 170–173.

Waxler, R. M.

R. M. Waxler, G. W. Cleek, “The effect of temperature and pressure on the refractive index of some oxide glasses,” J. Res. Nat. Bur. Stand. Sect. A 77, 755–763 (1973).
[CrossRef]

Webb, D. J.

P. J. Henderson, D. J. Webb, D. A. Jackson, L. Zhang, I. Bennion, “Highly-multiplexed grating-sensors for temperature-referenced quasi-static measurements of strain in concrete bridges,” in 13th International Conference on Optical Fiber Sensors, B. Kim, K. Hotate, eds., Proc. SPIE3746, 320–323 (1999).

Xu, M. G.

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

Yamauchi, R.

M. Sudo, M. Nakai, K. Himeno, S. Suzaki, A. Wada, R. Yamauchi, “Simultaneous measurement of temperature and strain using PANDA fiber grating,” in 12th International Conference on Optical Fiber Sensors, Vol. 16 of 1997 OSA Technical Digest Series (Optical Society of America, Washington D.C., 1997), pp. 170–173.

Zervas, M. N.

M. Ibsen, E. Rønnekleiv, O. Hadeler, G. J. Cowle, M. N. Zervas, R. I. Laming, “Stable multiple wavelength generation in all-fibre DFB lasers,” in Bragg Gratings, Photosensitivity, and Poling in Glass Waveguides, E. J. Friebele, R. Kashyap, T. Erdogan, eds., Vol. 33 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1999), Paper FA4.

O. Hadeler, M. N. Zervas, “Application of a DFB fibre laser temperature sensor for characterizing pump induced temperature distributions along another DFB fibre laser,” in 14th International Conference on Optical Fiber Sensors, A. G. Mignani, ed., Proc. SPIE4185, 142–145 (2000).

Zhang, L.

P. J. Henderson, D. J. Webb, D. A. Jackson, L. Zhang, I. Bennion, “Highly-multiplexed grating-sensors for temperature-referenced quasi-static measurements of strain in concrete bridges,” in 13th International Conference on Optical Fiber Sensors, B. Kim, K. Hotate, eds., Proc. SPIE3746, 320–323 (1999).

Appl. Opt. (1)

Electron. Lett. (1)

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

J. Lightwave Technol. (2)

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 sensors,” J. Lightwave Technol. 15, 1442–1463 (1997).
[CrossRef]

S.-Y. Huang, J. N. Blake, B. Y. Kim, “Perturbation effects on mode propagation in highly elliptical core two-mode fibers,” J. Lightwave Technol. 8, 23–33 (1990).
[CrossRef]

J. Res. Nat. Bur. Stand. Sect. A (1)

R. M. Waxler, G. W. Cleek, “The effect of temperature and pressure on the refractive index of some oxide glasses,” J. Res. Nat. Bur. Stand. Sect. A 77, 755–763 (1973).
[CrossRef]

Opt. Lett. (1)

Other (5)

O. Hadeler, M. N. Zervas, “Application of a DFB fibre laser temperature sensor for characterizing pump induced temperature distributions along another DFB fibre laser,” in 14th International Conference on Optical Fiber Sensors, A. G. Mignani, ed., Proc. SPIE4185, 142–145 (2000).

G. Meltz, W. W. Morrey, W. H. Glenn, J. D. Farina, “In-fiber Bragg grating sensors,” in 5th International Conference on Optical Fiber Sensors, in Vol. 2 of 1988 OSA Technical Digest Series (Optical Society of America, Washington D.C., 1988), pp. 163–166.

P. J. Henderson, D. J. Webb, D. A. Jackson, L. Zhang, I. Bennion, “Highly-multiplexed grating-sensors for temperature-referenced quasi-static measurements of strain in concrete bridges,” in 13th International Conference on Optical Fiber Sensors, B. Kim, K. Hotate, eds., Proc. SPIE3746, 320–323 (1999).

M. Sudo, M. Nakai, K. Himeno, S. Suzaki, A. Wada, R. Yamauchi, “Simultaneous measurement of temperature and strain using PANDA fiber grating,” in 12th International Conference on Optical Fiber Sensors, Vol. 16 of 1997 OSA Technical Digest Series (Optical Society of America, Washington D.C., 1997), pp. 170–173.

M. Ibsen, E. Rønnekleiv, O. Hadeler, G. J. Cowle, M. N. Zervas, R. I. Laming, “Stable multiple wavelength generation in all-fibre DFB lasers,” in Bragg Gratings, Photosensitivity, and Poling in Glass Waveguides, E. J. Friebele, R. Kashyap, T. Erdogan, eds., Vol. 33 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1999), Paper FA4.

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Figures (9)

Fig. 1
Fig. 1

Mode structure of the birefringent moiré DFB fiber laser and rf beat frequencies between the modes. ν x,y , longitudinal-mode beat frequencies at ∼14.9 GHz; ν 1,2, polarization-mode beat frequencies at ∼650 MHz.

Fig. 2
Fig. 2

Experimental arrangement: WDM, wavelength-division multiplexer; ISO, isolator; PIN, photodiode; RFSA, rf spectrum analyzer; OSA, optical spectrum analyzer.

Fig. 3
Fig. 3

Optical spectrum of the dual-longitudinal-mode DFB fiber laser taken with 1-pm resolution at different strain levels.

Fig. 4
Fig. 4

Polarization and longitudinal beat frequencies as a function of strain. Straight lines represent best fits, including the strain–temperature cross sensitivity of the longitudinal beat frequency response.

Fig. 5
Fig. 5

Polarization and longitudinal beat frequencies as a function of temperature. Data of only four strain levels are shown for clarity in the top graph. Straight lines, again, represent best fits, including the strain–temperature cross sensitivity of the longitudinal beat frequency response.

Fig. 6
Fig. 6

Contour plot of polarization beat frequency change Δν 1 and longitudinal beat frequency change Δν x in the strain–temperature plane. Error bars show the accuracy of the sensor calculated from the uncertainties of the fit parameters and the frequency measurements at the applied strains and temperatures.

Fig. 7
Fig. 7

Averaged strain and temperature calculated by use of all four pairs of longitudinal and polarization beat frequencies. The error bars represent the standard deviation of the averaged value.

Fig. 8
Fig. 8

Relative difference between the strain responses at the two longitudinal modes as a function of temperature, calculated from the planes fitted to the experimental data in Figs. 4 and 5.

Fig. 9
Fig. 9

Relative difference between the temperature responses at the two longitudinal modes as a function of strain, calculated from the planes fitted to the experimental data in Figs. 4 and 5.

Tables (1)

Tables Icon

Table 1 Beat Frequency Responses kij and Beat Frequencies at ∊ = 0 µ∊, T = 0 °C Found by Fitting of Eqs. (8) and (9) to the Experimental Data

Equations (14)

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

Δλm,e/λm,e=1+pm,e+α+ξm,eΔT, m=1, 2,  e=x, y,
νe=c λ2,e-λ1,eλ1,eλ2,e,  e=x, y,
Δνe, ΔTνe=1LdLd-2+p1,e+p2,e|T=const+1LdLdT-2α+ξ1,e+ξ2,eΔT|=const,
νm=cB/2nm2Λ,  m=1, 2,
Δνm, ΔTνm=1BdBd-1+2pm|T=const+1BdBdT-α+2ξmΔT|=const,
νx+ν2=νy+ν1.
ΔναΔνβ=KsensorΔT,
Δν1Δν2=k11 k12k21 k22ΔT=KeΔT.
ΔνxΔνy=k31 k32 k33k41 k42 k43ΔTΔT=KmΔTΔT.
Δν1Δνx=k11 k12 0k31 k32 k33ΔTΔT=K1,xΔTΔT,
p2-p1=1Δνν+3Lλ-L=fT,
ξ2-ξ1=1TΔνν+3αLλ-L=f,
p=-0.5 n2p12-μp11+p12,
ξ=1nnT+nρρT,

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