Abstract

Interferometric noise in fiber-optic grating sensors is investigated. Interference between a signal wave and reflected waves causes signal fluctuation in the output that limits the wavelength detection accuracy of the sensing system. The measurement error limited by interferometric noise is calculated for both reflective-type and transmission-type sensors.

© 1998 Optical Society of America

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References

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  1. K. O. Hill, F. Fujii, D. C. Johnson, B. S. Kawasaki, “Photosensitivity on optical fiber waveguide: application to reflection filter fabrication,” Appl. Phys. Lett. 32, 647–649 (1978).
    [CrossRef]
  2. G. Meltz, W. W. Morey, W. H. Glenn, “Formation of Bragg gratings in optical fiber by a transverse holographic method,” Opt. Lett. 14, 823–825 (1989).
    [CrossRef] [PubMed]
  3. W. W. Morey, G. Meltz, W. H. Glenn, “Fiber optic Bragg grating sensors,” in Fiber Optic and Laser Sensors VII, R. P. Depaula, E. Udd, eds., Proc. SPIE1169, 98–107 (1989).
    [CrossRef]
  4. Y. J. Rao, “In-fiber Bragg grating sensor,” Meas. Sci. Technol. 8, 355–375 (1997).
    [CrossRef]
  5. A. D. Kersey, “Multiplexing techniques for fiber-optic sensors,” in Optical Fiber Sensors: Applications, Analysis, and Future Trends, J. Dakin, B. Culshaw, eds. (Artech House, Norwood, Mass., 1997), pp. 373–376.
  6. M. G. Xu, H. Geiger, J. P. Dakin, “Modeling and performance analysis of a fiber Bragg grating interrogation system using an acousto-optic tunable filter,” J. Lightwave Technol. 14, 391–396 (1996).
    [CrossRef]
  7. J. Dunphy, G. Ball, F. D’Amato, P. Ferraro, S. Inserra, A. Vannucci, M. Varasi, “Instrumentation development in support of fiber grating sensor arrays,” in Distributed and Multiplexed Fiber-Optic Sensors III, A. D. Kersey, J. P. Dakin, eds., Proc. SPIE2071, 2–11 (1993).
    [CrossRef]
  8. L. Bjerkan, D. R. Hjelme, K. Johannessen, “Bragg grating sensor demodulation scheme using semiconductor laser for measuring slamming forces of marine vehicle models,” in Proceedings of Eleventh International Conference on Optical Fiber Sensors (Japan Society of Applied Physics, Sapporo, Japan, 1996), pp. 236–239.
  9. D. R. Hjelme, L. Bjerkan, S. Neergard, J. S. Rambech, J. V. Aarsnes, “Application of Bragg grating sensor in the characterization of scaled marine vehicle models,” Appl. Opt. 36, 328–336 (1997).
    [CrossRef] [PubMed]
  10. P. St. J. Russel and J. L. Archambault, “Fiber gratings,” in Optical Fiber Sensors: Components and Subsystems, Vol. 3, J. Dakin, B. Culshaw, eds. (Artech House, Norwood, Mass., 1996), pp. 15–16.
  11. W. Jin, M. S. Demokan, G. Stewart, W. Philp, B. Culshaw, “Coherent backscatter noise in fiber optic gas sensors,” in Chemical, Biochemical, and Environmental Fiber Sensors IX, R. A. Lieberman, ed., Proc. SPIE3105, 266–274 (1997).
    [CrossRef]
  12. B. Moslehi, “Noise power spectra of optical two-beam interferometers induced by laser phase noise,” J. Lightwave Technol. LT-4, 1704–1710 (1986).
    [CrossRef]

1997 (2)

1996 (1)

M. G. Xu, H. Geiger, J. P. Dakin, “Modeling and performance analysis of a fiber Bragg grating interrogation system using an acousto-optic tunable filter,” J. Lightwave Technol. 14, 391–396 (1996).
[CrossRef]

1989 (1)

1986 (1)

B. Moslehi, “Noise power spectra of optical two-beam interferometers induced by laser phase noise,” J. Lightwave Technol. LT-4, 1704–1710 (1986).
[CrossRef]

1978 (1)

K. O. Hill, F. Fujii, D. C. Johnson, B. S. Kawasaki, “Photosensitivity on optical fiber waveguide: application to reflection filter fabrication,” Appl. Phys. Lett. 32, 647–649 (1978).
[CrossRef]

Aarsnes, J. V.

Ball, G.

J. Dunphy, G. Ball, F. D’Amato, P. Ferraro, S. Inserra, A. Vannucci, M. Varasi, “Instrumentation development in support of fiber grating sensor arrays,” in Distributed and Multiplexed Fiber-Optic Sensors III, A. D. Kersey, J. P. Dakin, eds., Proc. SPIE2071, 2–11 (1993).
[CrossRef]

Bjerkan, L.

D. R. Hjelme, L. Bjerkan, S. Neergard, J. S. Rambech, J. V. Aarsnes, “Application of Bragg grating sensor in the characterization of scaled marine vehicle models,” Appl. Opt. 36, 328–336 (1997).
[CrossRef] [PubMed]

L. Bjerkan, D. R. Hjelme, K. Johannessen, “Bragg grating sensor demodulation scheme using semiconductor laser for measuring slamming forces of marine vehicle models,” in Proceedings of Eleventh International Conference on Optical Fiber Sensors (Japan Society of Applied Physics, Sapporo, Japan, 1996), pp. 236–239.

Culshaw, B.

W. Jin, M. S. Demokan, G. Stewart, W. Philp, B. Culshaw, “Coherent backscatter noise in fiber optic gas sensors,” in Chemical, Biochemical, and Environmental Fiber Sensors IX, R. A. Lieberman, ed., Proc. SPIE3105, 266–274 (1997).
[CrossRef]

D’Amato, F.

J. Dunphy, G. Ball, F. D’Amato, P. Ferraro, S. Inserra, A. Vannucci, M. Varasi, “Instrumentation development in support of fiber grating sensor arrays,” in Distributed and Multiplexed Fiber-Optic Sensors III, A. D. Kersey, J. P. Dakin, eds., Proc. SPIE2071, 2–11 (1993).
[CrossRef]

Dakin, J. P.

M. G. Xu, H. Geiger, J. P. Dakin, “Modeling and performance analysis of a fiber Bragg grating interrogation system using an acousto-optic tunable filter,” J. Lightwave Technol. 14, 391–396 (1996).
[CrossRef]

Demokan, M. S.

W. Jin, M. S. Demokan, G. Stewart, W. Philp, B. Culshaw, “Coherent backscatter noise in fiber optic gas sensors,” in Chemical, Biochemical, and Environmental Fiber Sensors IX, R. A. Lieberman, ed., Proc. SPIE3105, 266–274 (1997).
[CrossRef]

Dunphy, J.

J. Dunphy, G. Ball, F. D’Amato, P. Ferraro, S. Inserra, A. Vannucci, M. Varasi, “Instrumentation development in support of fiber grating sensor arrays,” in Distributed and Multiplexed Fiber-Optic Sensors III, A. D. Kersey, J. P. Dakin, eds., Proc. SPIE2071, 2–11 (1993).
[CrossRef]

Ferraro, P.

J. Dunphy, G. Ball, F. D’Amato, P. Ferraro, S. Inserra, A. Vannucci, M. Varasi, “Instrumentation development in support of fiber grating sensor arrays,” in Distributed and Multiplexed Fiber-Optic Sensors III, A. D. Kersey, J. P. Dakin, eds., Proc. SPIE2071, 2–11 (1993).
[CrossRef]

Fujii, F.

K. O. Hill, F. Fujii, D. C. Johnson, B. S. Kawasaki, “Photosensitivity on optical fiber waveguide: application to reflection filter fabrication,” Appl. Phys. Lett. 32, 647–649 (1978).
[CrossRef]

Geiger, H.

M. G. Xu, H. Geiger, J. P. Dakin, “Modeling and performance analysis of a fiber Bragg grating interrogation system using an acousto-optic tunable filter,” J. Lightwave Technol. 14, 391–396 (1996).
[CrossRef]

Glenn, W. H.

G. Meltz, W. W. Morey, W. H. Glenn, “Formation of Bragg gratings in optical fiber by a transverse holographic method,” Opt. Lett. 14, 823–825 (1989).
[CrossRef] [PubMed]

W. W. Morey, G. Meltz, W. H. Glenn, “Fiber optic Bragg grating sensors,” in Fiber Optic and Laser Sensors VII, R. P. Depaula, E. Udd, eds., Proc. SPIE1169, 98–107 (1989).
[CrossRef]

Hill, K. O.

K. O. Hill, F. Fujii, D. C. Johnson, B. S. Kawasaki, “Photosensitivity on optical fiber waveguide: application to reflection filter fabrication,” Appl. Phys. Lett. 32, 647–649 (1978).
[CrossRef]

Hjelme, D. R.

D. R. Hjelme, L. Bjerkan, S. Neergard, J. S. Rambech, J. V. Aarsnes, “Application of Bragg grating sensor in the characterization of scaled marine vehicle models,” Appl. Opt. 36, 328–336 (1997).
[CrossRef] [PubMed]

L. Bjerkan, D. R. Hjelme, K. Johannessen, “Bragg grating sensor demodulation scheme using semiconductor laser for measuring slamming forces of marine vehicle models,” in Proceedings of Eleventh International Conference on Optical Fiber Sensors (Japan Society of Applied Physics, Sapporo, Japan, 1996), pp. 236–239.

Inserra, S.

J. Dunphy, G. Ball, F. D’Amato, P. Ferraro, S. Inserra, A. Vannucci, M. Varasi, “Instrumentation development in support of fiber grating sensor arrays,” in Distributed and Multiplexed Fiber-Optic Sensors III, A. D. Kersey, J. P. Dakin, eds., Proc. SPIE2071, 2–11 (1993).
[CrossRef]

Jin, W.

W. Jin, M. S. Demokan, G. Stewart, W. Philp, B. Culshaw, “Coherent backscatter noise in fiber optic gas sensors,” in Chemical, Biochemical, and Environmental Fiber Sensors IX, R. A. Lieberman, ed., Proc. SPIE3105, 266–274 (1997).
[CrossRef]

Johannessen, K.

L. Bjerkan, D. R. Hjelme, K. Johannessen, “Bragg grating sensor demodulation scheme using semiconductor laser for measuring slamming forces of marine vehicle models,” in Proceedings of Eleventh International Conference on Optical Fiber Sensors (Japan Society of Applied Physics, Sapporo, Japan, 1996), pp. 236–239.

Johnson, D. C.

K. O. Hill, F. Fujii, D. C. Johnson, B. S. Kawasaki, “Photosensitivity on optical fiber waveguide: application to reflection filter fabrication,” Appl. Phys. Lett. 32, 647–649 (1978).
[CrossRef]

Kawasaki, B. S.

K. O. Hill, F. Fujii, D. C. Johnson, B. S. Kawasaki, “Photosensitivity on optical fiber waveguide: application to reflection filter fabrication,” Appl. Phys. Lett. 32, 647–649 (1978).
[CrossRef]

Kersey, A. D.

A. D. Kersey, “Multiplexing techniques for fiber-optic sensors,” in Optical Fiber Sensors: Applications, Analysis, and Future Trends, J. Dakin, B. Culshaw, eds. (Artech House, Norwood, Mass., 1997), pp. 373–376.

Meltz, G.

G. Meltz, W. W. Morey, W. H. Glenn, “Formation of Bragg gratings in optical fiber by a transverse holographic method,” Opt. Lett. 14, 823–825 (1989).
[CrossRef] [PubMed]

W. W. Morey, G. Meltz, W. H. Glenn, “Fiber optic Bragg grating sensors,” in Fiber Optic and Laser Sensors VII, R. P. Depaula, E. Udd, eds., Proc. SPIE1169, 98–107 (1989).
[CrossRef]

Morey, W. W.

G. Meltz, W. W. Morey, W. H. Glenn, “Formation of Bragg gratings in optical fiber by a transverse holographic method,” Opt. Lett. 14, 823–825 (1989).
[CrossRef] [PubMed]

W. W. Morey, G. Meltz, W. H. Glenn, “Fiber optic Bragg grating sensors,” in Fiber Optic and Laser Sensors VII, R. P. Depaula, E. Udd, eds., Proc. SPIE1169, 98–107 (1989).
[CrossRef]

Moslehi, B.

B. Moslehi, “Noise power spectra of optical two-beam interferometers induced by laser phase noise,” J. Lightwave Technol. LT-4, 1704–1710 (1986).
[CrossRef]

Neergard, S.

Philp, W.

W. Jin, M. S. Demokan, G. Stewart, W. Philp, B. Culshaw, “Coherent backscatter noise in fiber optic gas sensors,” in Chemical, Biochemical, and Environmental Fiber Sensors IX, R. A. Lieberman, ed., Proc. SPIE3105, 266–274 (1997).
[CrossRef]

Rambech, J. S.

Rao, Y. J.

Y. J. Rao, “In-fiber Bragg grating sensor,” Meas. Sci. Technol. 8, 355–375 (1997).
[CrossRef]

St. J. Russel and J. L. Archambault, P.

P. St. J. Russel and J. L. Archambault, “Fiber gratings,” in Optical Fiber Sensors: Components and Subsystems, Vol. 3, J. Dakin, B. Culshaw, eds. (Artech House, Norwood, Mass., 1996), pp. 15–16.

Stewart, G.

W. Jin, M. S. Demokan, G. Stewart, W. Philp, B. Culshaw, “Coherent backscatter noise in fiber optic gas sensors,” in Chemical, Biochemical, and Environmental Fiber Sensors IX, R. A. Lieberman, ed., Proc. SPIE3105, 266–274 (1997).
[CrossRef]

Vannucci, A.

J. Dunphy, G. Ball, F. D’Amato, P. Ferraro, S. Inserra, A. Vannucci, M. Varasi, “Instrumentation development in support of fiber grating sensor arrays,” in Distributed and Multiplexed Fiber-Optic Sensors III, A. D. Kersey, J. P. Dakin, eds., Proc. SPIE2071, 2–11 (1993).
[CrossRef]

Varasi, M.

J. Dunphy, G. Ball, F. D’Amato, P. Ferraro, S. Inserra, A. Vannucci, M. Varasi, “Instrumentation development in support of fiber grating sensor arrays,” in Distributed and Multiplexed Fiber-Optic Sensors III, A. D. Kersey, J. P. Dakin, eds., Proc. SPIE2071, 2–11 (1993).
[CrossRef]

Xu, M. G.

M. G. Xu, H. Geiger, J. P. Dakin, “Modeling and performance analysis of a fiber Bragg grating interrogation system using an acousto-optic tunable filter,” J. Lightwave Technol. 14, 391–396 (1996).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. Lett. (1)

K. O. Hill, F. Fujii, D. C. Johnson, B. S. Kawasaki, “Photosensitivity on optical fiber waveguide: application to reflection filter fabrication,” Appl. Phys. Lett. 32, 647–649 (1978).
[CrossRef]

J. Lightwave Technol. (2)

B. Moslehi, “Noise power spectra of optical two-beam interferometers induced by laser phase noise,” J. Lightwave Technol. LT-4, 1704–1710 (1986).
[CrossRef]

M. G. Xu, H. Geiger, J. P. Dakin, “Modeling and performance analysis of a fiber Bragg grating interrogation system using an acousto-optic tunable filter,” J. Lightwave Technol. 14, 391–396 (1996).
[CrossRef]

Meas. Sci. Technol. (1)

Y. J. Rao, “In-fiber Bragg grating sensor,” Meas. Sci. Technol. 8, 355–375 (1997).
[CrossRef]

Opt. Lett. (1)

Other (6)

P. St. J. Russel and J. L. Archambault, “Fiber gratings,” in Optical Fiber Sensors: Components and Subsystems, Vol. 3, J. Dakin, B. Culshaw, eds. (Artech House, Norwood, Mass., 1996), pp. 15–16.

W. Jin, M. S. Demokan, G. Stewart, W. Philp, B. Culshaw, “Coherent backscatter noise in fiber optic gas sensors,” in Chemical, Biochemical, and Environmental Fiber Sensors IX, R. A. Lieberman, ed., Proc. SPIE3105, 266–274 (1997).
[CrossRef]

A. D. Kersey, “Multiplexing techniques for fiber-optic sensors,” in Optical Fiber Sensors: Applications, Analysis, and Future Trends, J. Dakin, B. Culshaw, eds. (Artech House, Norwood, Mass., 1997), pp. 373–376.

J. Dunphy, G. Ball, F. D’Amato, P. Ferraro, S. Inserra, A. Vannucci, M. Varasi, “Instrumentation development in support of fiber grating sensor arrays,” in Distributed and Multiplexed Fiber-Optic Sensors III, A. D. Kersey, J. P. Dakin, eds., Proc. SPIE2071, 2–11 (1993).
[CrossRef]

L. Bjerkan, D. R. Hjelme, K. Johannessen, “Bragg grating sensor demodulation scheme using semiconductor laser for measuring slamming forces of marine vehicle models,” in Proceedings of Eleventh International Conference on Optical Fiber Sensors (Japan Society of Applied Physics, Sapporo, Japan, 1996), pp. 236–239.

W. W. Morey, G. Meltz, W. H. Glenn, “Fiber optic Bragg grating sensors,” in Fiber Optic and Laser Sensors VII, R. P. Depaula, E. Udd, eds., Proc. SPIE1169, 98–107 (1989).
[CrossRef]

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Figures (7)

Fig. 1
Fig. 1

Signal and reflected waves in a reflective-type sensor: (a) signal wave, (b) first-order reflected waves from a point before the grating, (c) first-order reflected wave from a point behind the grating, (d) example of higher-order reflected wave.

Fig. 2
Fig. 2

Signal and reflected waves in a transmission-type sensor: (a) signal wave, (b) first-order reflected wave from a point before the grating, (c) first-order reflected wave from a point behind the grating.

Fig. 3
Fig. 3

(a) Spectrum response of a reflective-type sensor; (b) measurement error when the direct detection technique is used.

Fig. 4
Fig. 4

(a) Spectrum response of a transmission-type sensor; (b) measurement error when the direct detection technique is used.

Fig. 5
Fig. 5

Wavelength detection errors as functions of R G for the direct scanning technique: 1, reflective-type sensor, reflection before the grating; 2, transmission-type sensor reflection before or behind the grating; or reflection-type sensor, reflection behind the grating.

Fig. 6
Fig. 6

Wavelength detection errors as functions of R G and τ for the first derivative technique: γ m = 0.05, reflective-type sensor, reflection before the grating: (a) f = 100 MHz; (b) f = 10 kHz.

Fig. 7
Fig. 7

Wavelength detection errors as a function of R G and τ for the first derivative technique. γ m = 0.05. Transmission-type sensor, reflection before or behind the grating; or reflection-type sensor, reflection after the grating: (a) f = 100 MHz, (b) f = 10 kHz.

Equations (46)

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I sr λ = I 0 4   R λ
I st λ = I 0 1 - R λ
R λ = R G exp - 4   ln   2 λ - λ B B G 2 ,
| Δ I | = I 0 4   | R λ B - R λ B ± Δ λ |
| Δ I | = I 0 | R λ B - R λ B ± Δ λ |
| Δ I | 4 Γ   ln   2 I 0 R G Δ λ B G 2 ,
| Δ λ | B G 1 4 Γ   ln   2 R G × | Δ I | I 0 1 / 2 .
λ = λ 0 + λ m   cos   ω t ,
R λ t = R G exp - 4   ln   2 γ + γ m   cos   ω t 2 = R G exp - 4   ln   2 γ 2 exp - 4   ln   2 2 γ γ m × cos   ω t + γ m 2 cos 2   ω t ,
γ = λ 0 - λ B B G
γ m = λ m B G
R λ t R G exp - 4   ln   2 γ 2 1 - 8   ln   2 γ γ m   cos   ω t ,
R ω - 8   ln   2 R G exp - 4   ln   2 γ 2 γ γ m .
R ω - 8   ln   2 R G γ m γ .
I ω - 8 Γ   ln   2 I 0 R G γ m γ ,
| Δ λ | B G 1 8   ln   2 Γ R G γ m | Δ I ω | I 0 .
ν t = c λ 0 + λ m   cos   ω t c λ 0 - c λ m λ 0 2 cos   ω t ,
E sr t = I 0 4 1 / 2 R λ t 1 / 2 × exp j 2 π   0 t ν t d t + ϕ λ t = I 0 4 1 / 2 R λ t 1 / 2 × exp j 2 π c λ 0   t - 2 π c λ m ω λ 0 2 sin   ω t + ϕ λ t ,
E r 1 , 1 t = I 0 4 1 / 2 α 1 exp j 2 π c λ 0 t + τ 1 , 1 - 2 π c λ m ω λ 0 2 sin   ω t + τ 1 , 1 ,
E r 1 , 2 t = I 0 4 1 / 2 R λ t - τ 1 , 2 α 1 × exp j 2 π c λ 0 t - τ 1 , 2 - 2 π c λ m ω λ 0 2 sin   ω t - τ 1 , 2 + ϕ λ t + ϕ λ t - τ 1 , 2 ,
Δ I t = 2   Re E sr t E r 1 , 1 t + E r 1 , 2 t * = I 0 2   α 1 R λ t 1 / 2 cos 2 π c λ 0   τ 1 - 4 π c λ m ω λ 0 2 × sin ω τ 1 2 cos   ω t + τ 1 2 - ϕ λ t + R λ t - τ 1 cos 2 π c λ 0   τ 1 - 4 π c λ m ω λ 0 2 × sin ω τ 1 2 cos   ω t - τ 1 2 - ϕ λ t - τ 1 .
Δ I t I 0 2   α 1 R G cos 2 π c λ 0   τ 1 - 4 π c λ m ω λ 0 2 sin ω τ 1 2 × cos   ω t + τ 1 2 + R G cos 2 π c λ 0   τ 1 - 4 π c λ m ω λ 0 2 × sin ω τ 1 2 cos   ω t - τ 1 2 ,
Δ I ω = I 0 2   α 1 R G × sin 2 π c λ 0   τ 1 J 1 4 π c λ m ω λ 0 2 sin ω τ 1 2 × 1 + R G 2 cos 2 ω τ 1 2 + 1 - R G 2 sin 2 ω τ 1 2 1 / 2 × sin ω t + ψ
ψ = tan - 1 - 1 + R G 1 - R G tan   ω τ 1 .
| Δ I ω | I 0 = 1 2   α 1 R G J 1 4 π c λ m ω λ 0 2 sin ω τ 1 2 × 1 + R G 2 cos 2 ω τ 1 2 + 1 - R G 2 sin 2 ω τ 1 2 1 / 2 .
| Δ λ | B G α 1 4   ln   2 R G   γ m J 1 4 π c λ m ω λ 0 2 sin ω τ 1 2 × 1 + R G 2 cos 2 ω τ 1 2 + 1 - R G 2 sin 2 ω τ 1 2 1 / 2 .
Δ I = I 0 2   α 1 R λ 0 1 / 2 cos 2 π c λ 0   τ 1 - ϕ λ 0 + R λ 0 cos 2 π c λ 0   τ 1 - ϕ λ 0 = I 0 2   α 1 R λ 0 1 / 2 1 + R λ 0 cos 2 π c λ 0   τ 1 - ϕ λ 0 .
| Δ I | I 0 = 1 2   α 1 R λ 0 1 / 2 1 + R λ 0 .
| Δ λ | B G = α 1 2   ln   2 1 R G + R G 1 / 2 .
E r 2 t = I 0 4 1 / 2 1 - R λ t - τ 2 α 2 exp j 2 π c λ 0 t - τ 2 - 2 π c λ m ω λ 0 2 sin   ω t - τ 2 ;
Δ I t = 2   Re E sr t E r 2 * t = I 0 2 1 - R λ t - τ 2 R λ t 1 / 2 α 2 cos 2 π c λ 0   τ 2 - 4 π c λ m ω λ 0 2 sin ω τ 2 2 cos   ω t - τ 2 2 + ϕ λ t .
| Δ I ω | I 0 = 1 2 1 - R G R G   α 2 J 1 4 π c λ m ω λ 0 2 sin ω τ 2 2 ,
| Δ λ | B G = α 2 4   ln   2 γ m 1 R G - R G J 1 4 π c λ m ω λ 0 2 sin ω τ 2 2 .
| Δ I | I 0 = 1 2 1 - R λ R λ 1 / 2 α 2 ,
| Δ λ | B G = α 2 2   ln   2 1 R G - R G 1 / 2 .
E st t = I 0 1 - R λ t 1 / 2   exp j 2 π c λ 0   t - 2 π c λ m ω λ 0 sin   ω t ,
E r 1 t = I 0 R λ t - τ 1 1 - R λ t 1 / 2 α 1 × exp j 2 π c λ 0 t - τ 1 - 2 π c λ m ω λ 0 2 sin   ω t - τ 1 + ϕ λ t - τ 1
E r 2 t = I 0 R λ t 1 - R λ t - τ 2 1 / 2 α 2 × exp j 2 π c λ 0 t - τ 2 - 2 π c λ m ω λ 0 2 sin   ω t - τ 2 + ϕ λ t
Δ I 1 = 2   Re E st t E r 1 * t = 2 I 0 α 1 R λ t - τ 1 1 / 2 1 - R λ t × cos 2 π c λ 0   τ 1 - 4 π c λ m ω λ 0 2 sin ω τ 1 2 cos   ω t - τ 1 2 - ϕ λ t - τ 1
Δ I 2 = 2   Re E st t E r 2 * t = 2 I 0 α 2 R λ t 1 / 2 1 - R λ t 1 - R λ t - τ 2 1 / 2 × cos 2 π c λ 0   τ 2 - 4 π c λ m ω λ 0 2 sin ω τ 2 2 cos   ω t - τ 2 2 - ϕ λ t
| Δ I ω , i | I 0 = 2 α i R G 1 - R G J 1 4 π c λ m ω λ 0 2 sin ω τ i 2 ,
| Δ I i | I 0 = 2 α i R G 1 - R G ,
| Δ I | = 2 qBR G ρ I 0 4 1 / 2 = I 0 qBR G 2 ρ 1 / 2 ,
| Δ I | I 0 = qBR G 2 ρ I 0 1 / 2 .
| Δ λ | B G 1 ln   2 1 / 2 qB 2 ρ R G I 0 1 / 4 .
| Δ λ | B G 1 ln   2 γ m qB 2 ρ R G I 0 1 / 2 .

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