Abstract

We report on the use of a frequency-modulated continuous-wave technique for multiplexing optical fiber gas sensors. The sensor network is of a ladder topology and is interrogated by a tunable laser. The system performance in terms of detection sensitivity and cross talk between sensors was investigated and found to be limited by coherent mixing between signals from different channels. The system performance can be improved significantly by use of appropriate wavelength modulation–scanning coupled with low-pass filtering. Computer simulation shows that an array of 37 acetylene sensors with a detection accuracy of 2000 parts in 106 for each sensor may be realized. A two-sensor acetylene detection system was experimentally demonstrated that had a detection sensitivity of 165 parts in 106 for 2.5-cm gas cells (or a minimum detectable absorbance of 2.1 × 10-4) and a cross talk of -25 dB.

© 2001 Optical Society of America

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References

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  1. G. Stewart, “Fibre optic sensors,” in Sensor Technologies, Vol. 1 of Sensor Systems for Environmental Monitoring, M. Campbell, ed. (Chapman & Hall, London, 1997), Chap. 1, pp. 1–40.
  2. J. P. Dakin, C. A. Wade, D. Pinchbeck, J. S. Wykes, “A novel optical fibre methane sensor,” J. Opt. Sensors 2, 261–267 (1987).
  3. K. Chan, H. Ito, H. Inaba, T. Furuya, “10 km-long fiber optic remote sensing of CH4 gas by near infrared absorption,” Appl. Phys. B 38, 11–15 (1985).
    [CrossRef]
  4. D. M. Bruce, D. T. Cassidy, “Detection of oxygen using short external cavity GaAs semiconductor diode lasers,” Appl. Opt. 29, 1327–1332 (1990).
    [CrossRef] [PubMed]
  5. D. B. Oh, D. C. Hovde, “Wavelength-modulation detection of acetylene with near-infrared external-cavity diode laser,” Appl. Opt. 34, 7002–7005 (1995).
    [CrossRef] [PubMed]
  6. X. Zhu, D. T. Cassidy, “Electronic subtracter for trace-gas detection with InGaAsP diode laser,” Appl. Opt. 34, 8303–8308 (1995).
    [CrossRef] [PubMed]
  7. J. A. Silver, “Frequency-modulation spectroscopy for trace species detection: theory and comparison among experimental methods,” Appl. Opt. 31, 707–717 (1992).
    [CrossRef] [PubMed]
  8. D. S. Bomse, A. C. Stanton, J. A. Silver, “Frequency-modulation and wavelength modulation spectroscopies: comparison of experimental methods using a lead-salt diode laser,” Appl. Opt. 31, 718–731 (1992).
    [CrossRef] [PubMed]
  9. W. Jin, Y. Z. Xu, M. S. Demokan, G. Stewart, “Investigation of interferometric noise in fiber-optic gas sensors with use of wavelength modulation spectroscopy,” Appl. Opt. 36, 7239–7246 (1997).
    [CrossRef]
  10. J. Reid, D. Labrie, “Second-harmonic detection with tunable diode lasers—comparison of experiment and theory,” Appl. Phys. B 26, 203–210 (1981).
    [CrossRef]
  11. B. F. Ventrudo, D. T. Cassidy, “Operating characteristics of a tunable diode laser absorption spectrometer using short-external-cavity and DFB laser diodes,” Appl. Opt. 29, 5007–5013 (1990).
    [CrossRef] [PubMed]
  12. Y. Shimose, T. Okamoto, A. Maruyama, M. Aizawa, H. Nagai, “Remote sensing of methane gas by differential absorption measurement using a wavelength-tunable DFB LD,” IEEE Photon. Technol. Lett. 3, 86–87 (1991).
    [CrossRef]
  13. G. Stewart, C. Tandy, D. Moodie, M. A. Morante, F. Dong, “Design of a fiber optic multipoint sensor for gas detection,” Sensors Actuators B 51, 227–232 (1998).
    [CrossRef]
  14. W. Jin, “Performance analysis of a time-division-multiplexed fiber-optic gas-sensor array by wavelength modulation of a distributed-feedback laser,” Appl. Opt. 38, 5290–5297 (1999).
    [CrossRef]
  15. J. P. Dakin, “Multiplexed and distributed optical fiber sensors,” J. Phys. E 20, 954–967 (1987).
    [CrossRef]
  16. J. L. Brooks, B. Moslehi, B. Y. Kim, H. J. Shaw, “Time-domain addressing of remote fiber-optic interferometric sensor arrays,” J. Lightwave Technol. LT-5, 1014–1023 (1987).
    [CrossRef]
  17. P. K. C. Chan, W. Jin, J. M. Gong, M. S. Demokan, “Multiplexing of fiber Bragg grating sensors using an FMCW technique,” IEEE Photon. Technol. Lett. 11, 1470–1472 (1999).
    [CrossRef]
  18. P. K. C. Chan, “Multiplexing of fiber optic bragg grating sensors,” Ph.D. dissertation (Hong Kong Polytechnic University, Hong Kong, 2000).
  19. H. Riris, C. B. Carlisle, R. E. Warren, D. E. Cooper, “Signal-to-noise ratio enhancement in frequency-modulation spectrometers by digital signal processing,” Opt. Lett. 19, 144–146 (1994).
    [CrossRef] [PubMed]
  20. C. C. Chan, “Interrogation of fiber Bragg grating sensors with a tunable laser,” Ph.D. dissertation (Hong Kong Polytechnic University, Hong Kong, 2000).
  21. W. R. Philp, W. Jin, A. Mencaglia, G. Stewart, B. Culshaw, “Interferometric noise in frequency modulated optical gas sensors,” in Proceedings of Twenty-First Australian Conference on Optical Fibre Technology (Institute of Radio and Electronics Engineers Society, Milsons Point, Australia, 1986), pp. 185–188.

1999 (2)

W. Jin, “Performance analysis of a time-division-multiplexed fiber-optic gas-sensor array by wavelength modulation of a distributed-feedback laser,” Appl. Opt. 38, 5290–5297 (1999).
[CrossRef]

P. K. C. Chan, W. Jin, J. M. Gong, M. S. Demokan, “Multiplexing of fiber Bragg grating sensors using an FMCW technique,” IEEE Photon. Technol. Lett. 11, 1470–1472 (1999).
[CrossRef]

1998 (1)

G. Stewart, C. Tandy, D. Moodie, M. A. Morante, F. Dong, “Design of a fiber optic multipoint sensor for gas detection,” Sensors Actuators B 51, 227–232 (1998).
[CrossRef]

1997 (1)

1995 (2)

1994 (1)

1992 (2)

1991 (1)

Y. Shimose, T. Okamoto, A. Maruyama, M. Aizawa, H. Nagai, “Remote sensing of methane gas by differential absorption measurement using a wavelength-tunable DFB LD,” IEEE Photon. Technol. Lett. 3, 86–87 (1991).
[CrossRef]

1990 (2)

1987 (3)

J. P. Dakin, “Multiplexed and distributed optical fiber sensors,” J. Phys. E 20, 954–967 (1987).
[CrossRef]

J. L. Brooks, B. Moslehi, B. Y. Kim, H. J. Shaw, “Time-domain addressing of remote fiber-optic interferometric sensor arrays,” J. Lightwave Technol. LT-5, 1014–1023 (1987).
[CrossRef]

J. P. Dakin, C. A. Wade, D. Pinchbeck, J. S. Wykes, “A novel optical fibre methane sensor,” J. Opt. Sensors 2, 261–267 (1987).

1985 (1)

K. Chan, H. Ito, H. Inaba, T. Furuya, “10 km-long fiber optic remote sensing of CH4 gas by near infrared absorption,” Appl. Phys. B 38, 11–15 (1985).
[CrossRef]

1981 (1)

J. Reid, D. Labrie, “Second-harmonic detection with tunable diode lasers—comparison of experiment and theory,” Appl. Phys. B 26, 203–210 (1981).
[CrossRef]

Aizawa, M.

Y. Shimose, T. Okamoto, A. Maruyama, M. Aizawa, H. Nagai, “Remote sensing of methane gas by differential absorption measurement using a wavelength-tunable DFB LD,” IEEE Photon. Technol. Lett. 3, 86–87 (1991).
[CrossRef]

Bomse, D. S.

Brooks, J. L.

J. L. Brooks, B. Moslehi, B. Y. Kim, H. J. Shaw, “Time-domain addressing of remote fiber-optic interferometric sensor arrays,” J. Lightwave Technol. LT-5, 1014–1023 (1987).
[CrossRef]

Bruce, D. M.

Carlisle, C. B.

Cassidy, D. T.

Chan, C. C.

C. C. Chan, “Interrogation of fiber Bragg grating sensors with a tunable laser,” Ph.D. dissertation (Hong Kong Polytechnic University, Hong Kong, 2000).

Chan, K.

K. Chan, H. Ito, H. Inaba, T. Furuya, “10 km-long fiber optic remote sensing of CH4 gas by near infrared absorption,” Appl. Phys. B 38, 11–15 (1985).
[CrossRef]

Chan, P. K. C.

P. K. C. Chan, W. Jin, J. M. Gong, M. S. Demokan, “Multiplexing of fiber Bragg grating sensors using an FMCW technique,” IEEE Photon. Technol. Lett. 11, 1470–1472 (1999).
[CrossRef]

P. K. C. Chan, “Multiplexing of fiber optic bragg grating sensors,” Ph.D. dissertation (Hong Kong Polytechnic University, Hong Kong, 2000).

Cooper, D. E.

Culshaw, B.

W. R. Philp, W. Jin, A. Mencaglia, G. Stewart, B. Culshaw, “Interferometric noise in frequency modulated optical gas sensors,” in Proceedings of Twenty-First Australian Conference on Optical Fibre Technology (Institute of Radio and Electronics Engineers Society, Milsons Point, Australia, 1986), pp. 185–188.

Dakin, J. P.

J. P. Dakin, “Multiplexed and distributed optical fiber sensors,” J. Phys. E 20, 954–967 (1987).
[CrossRef]

J. P. Dakin, C. A. Wade, D. Pinchbeck, J. S. Wykes, “A novel optical fibre methane sensor,” J. Opt. Sensors 2, 261–267 (1987).

Demokan, M. S.

P. K. C. Chan, W. Jin, J. M. Gong, M. S. Demokan, “Multiplexing of fiber Bragg grating sensors using an FMCW technique,” IEEE Photon. Technol. Lett. 11, 1470–1472 (1999).
[CrossRef]

W. Jin, Y. Z. Xu, M. S. Demokan, G. Stewart, “Investigation of interferometric noise in fiber-optic gas sensors with use of wavelength modulation spectroscopy,” Appl. Opt. 36, 7239–7246 (1997).
[CrossRef]

Dong, F.

G. Stewart, C. Tandy, D. Moodie, M. A. Morante, F. Dong, “Design of a fiber optic multipoint sensor for gas detection,” Sensors Actuators B 51, 227–232 (1998).
[CrossRef]

Furuya, T.

K. Chan, H. Ito, H. Inaba, T. Furuya, “10 km-long fiber optic remote sensing of CH4 gas by near infrared absorption,” Appl. Phys. B 38, 11–15 (1985).
[CrossRef]

Gong, J. M.

P. K. C. Chan, W. Jin, J. M. Gong, M. S. Demokan, “Multiplexing of fiber Bragg grating sensors using an FMCW technique,” IEEE Photon. Technol. Lett. 11, 1470–1472 (1999).
[CrossRef]

Hovde, D. C.

Inaba, H.

K. Chan, H. Ito, H. Inaba, T. Furuya, “10 km-long fiber optic remote sensing of CH4 gas by near infrared absorption,” Appl. Phys. B 38, 11–15 (1985).
[CrossRef]

Ito, H.

K. Chan, H. Ito, H. Inaba, T. Furuya, “10 km-long fiber optic remote sensing of CH4 gas by near infrared absorption,” Appl. Phys. B 38, 11–15 (1985).
[CrossRef]

Jin, W.

P. K. C. Chan, W. Jin, J. M. Gong, M. S. Demokan, “Multiplexing of fiber Bragg grating sensors using an FMCW technique,” IEEE Photon. Technol. Lett. 11, 1470–1472 (1999).
[CrossRef]

W. Jin, “Performance analysis of a time-division-multiplexed fiber-optic gas-sensor array by wavelength modulation of a distributed-feedback laser,” Appl. Opt. 38, 5290–5297 (1999).
[CrossRef]

W. Jin, Y. Z. Xu, M. S. Demokan, G. Stewart, “Investigation of interferometric noise in fiber-optic gas sensors with use of wavelength modulation spectroscopy,” Appl. Opt. 36, 7239–7246 (1997).
[CrossRef]

W. R. Philp, W. Jin, A. Mencaglia, G. Stewart, B. Culshaw, “Interferometric noise in frequency modulated optical gas sensors,” in Proceedings of Twenty-First Australian Conference on Optical Fibre Technology (Institute of Radio and Electronics Engineers Society, Milsons Point, Australia, 1986), pp. 185–188.

Kim, B. Y.

J. L. Brooks, B. Moslehi, B. Y. Kim, H. J. Shaw, “Time-domain addressing of remote fiber-optic interferometric sensor arrays,” J. Lightwave Technol. LT-5, 1014–1023 (1987).
[CrossRef]

Labrie, D.

J. Reid, D. Labrie, “Second-harmonic detection with tunable diode lasers—comparison of experiment and theory,” Appl. Phys. B 26, 203–210 (1981).
[CrossRef]

Maruyama, A.

Y. Shimose, T. Okamoto, A. Maruyama, M. Aizawa, H. Nagai, “Remote sensing of methane gas by differential absorption measurement using a wavelength-tunable DFB LD,” IEEE Photon. Technol. Lett. 3, 86–87 (1991).
[CrossRef]

Mencaglia, A.

W. R. Philp, W. Jin, A. Mencaglia, G. Stewart, B. Culshaw, “Interferometric noise in frequency modulated optical gas sensors,” in Proceedings of Twenty-First Australian Conference on Optical Fibre Technology (Institute of Radio and Electronics Engineers Society, Milsons Point, Australia, 1986), pp. 185–188.

Moodie, D.

G. Stewart, C. Tandy, D. Moodie, M. A. Morante, F. Dong, “Design of a fiber optic multipoint sensor for gas detection,” Sensors Actuators B 51, 227–232 (1998).
[CrossRef]

Morante, M. A.

G. Stewart, C. Tandy, D. Moodie, M. A. Morante, F. Dong, “Design of a fiber optic multipoint sensor for gas detection,” Sensors Actuators B 51, 227–232 (1998).
[CrossRef]

Moslehi, B.

J. L. Brooks, B. Moslehi, B. Y. Kim, H. J. Shaw, “Time-domain addressing of remote fiber-optic interferometric sensor arrays,” J. Lightwave Technol. LT-5, 1014–1023 (1987).
[CrossRef]

Nagai, H.

Y. Shimose, T. Okamoto, A. Maruyama, M. Aizawa, H. Nagai, “Remote sensing of methane gas by differential absorption measurement using a wavelength-tunable DFB LD,” IEEE Photon. Technol. Lett. 3, 86–87 (1991).
[CrossRef]

Oh, D. B.

Okamoto, T.

Y. Shimose, T. Okamoto, A. Maruyama, M. Aizawa, H. Nagai, “Remote sensing of methane gas by differential absorption measurement using a wavelength-tunable DFB LD,” IEEE Photon. Technol. Lett. 3, 86–87 (1991).
[CrossRef]

Philp, W. R.

W. R. Philp, W. Jin, A. Mencaglia, G. Stewart, B. Culshaw, “Interferometric noise in frequency modulated optical gas sensors,” in Proceedings of Twenty-First Australian Conference on Optical Fibre Technology (Institute of Radio and Electronics Engineers Society, Milsons Point, Australia, 1986), pp. 185–188.

Pinchbeck, D.

J. P. Dakin, C. A. Wade, D. Pinchbeck, J. S. Wykes, “A novel optical fibre methane sensor,” J. Opt. Sensors 2, 261–267 (1987).

Reid, J.

J. Reid, D. Labrie, “Second-harmonic detection with tunable diode lasers—comparison of experiment and theory,” Appl. Phys. B 26, 203–210 (1981).
[CrossRef]

Riris, H.

Shaw, H. J.

J. L. Brooks, B. Moslehi, B. Y. Kim, H. J. Shaw, “Time-domain addressing of remote fiber-optic interferometric sensor arrays,” J. Lightwave Technol. LT-5, 1014–1023 (1987).
[CrossRef]

Shimose, Y.

Y. Shimose, T. Okamoto, A. Maruyama, M. Aizawa, H. Nagai, “Remote sensing of methane gas by differential absorption measurement using a wavelength-tunable DFB LD,” IEEE Photon. Technol. Lett. 3, 86–87 (1991).
[CrossRef]

Silver, J. A.

Stanton, A. C.

Stewart, G.

G. Stewart, C. Tandy, D. Moodie, M. A. Morante, F. Dong, “Design of a fiber optic multipoint sensor for gas detection,” Sensors Actuators B 51, 227–232 (1998).
[CrossRef]

W. Jin, Y. Z. Xu, M. S. Demokan, G. Stewart, “Investigation of interferometric noise in fiber-optic gas sensors with use of wavelength modulation spectroscopy,” Appl. Opt. 36, 7239–7246 (1997).
[CrossRef]

W. R. Philp, W. Jin, A. Mencaglia, G. Stewart, B. Culshaw, “Interferometric noise in frequency modulated optical gas sensors,” in Proceedings of Twenty-First Australian Conference on Optical Fibre Technology (Institute of Radio and Electronics Engineers Society, Milsons Point, Australia, 1986), pp. 185–188.

G. Stewart, “Fibre optic sensors,” in Sensor Technologies, Vol. 1 of Sensor Systems for Environmental Monitoring, M. Campbell, ed. (Chapman & Hall, London, 1997), Chap. 1, pp. 1–40.

Tandy, C.

G. Stewart, C. Tandy, D. Moodie, M. A. Morante, F. Dong, “Design of a fiber optic multipoint sensor for gas detection,” Sensors Actuators B 51, 227–232 (1998).
[CrossRef]

Ventrudo, B. F.

Wade, C. A.

J. P. Dakin, C. A. Wade, D. Pinchbeck, J. S. Wykes, “A novel optical fibre methane sensor,” J. Opt. Sensors 2, 261–267 (1987).

Warren, R. E.

Wykes, J. S.

J. P. Dakin, C. A. Wade, D. Pinchbeck, J. S. Wykes, “A novel optical fibre methane sensor,” J. Opt. Sensors 2, 261–267 (1987).

Xu, Y. Z.

Zhu, X.

Appl. Opt. (8)

D. M. Bruce, D. T. Cassidy, “Detection of oxygen using short external cavity GaAs semiconductor diode lasers,” Appl. Opt. 29, 1327–1332 (1990).
[CrossRef] [PubMed]

D. B. Oh, D. C. Hovde, “Wavelength-modulation detection of acetylene with near-infrared external-cavity diode laser,” Appl. Opt. 34, 7002–7005 (1995).
[CrossRef] [PubMed]

X. Zhu, D. T. Cassidy, “Electronic subtracter for trace-gas detection with InGaAsP diode laser,” Appl. Opt. 34, 8303–8308 (1995).
[CrossRef] [PubMed]

J. A. Silver, “Frequency-modulation spectroscopy for trace species detection: theory and comparison among experimental methods,” Appl. Opt. 31, 707–717 (1992).
[CrossRef] [PubMed]

D. S. Bomse, A. C. Stanton, J. A. Silver, “Frequency-modulation and wavelength modulation spectroscopies: comparison of experimental methods using a lead-salt diode laser,” Appl. Opt. 31, 718–731 (1992).
[CrossRef] [PubMed]

W. Jin, Y. Z. Xu, M. S. Demokan, G. Stewart, “Investigation of interferometric noise in fiber-optic gas sensors with use of wavelength modulation spectroscopy,” Appl. Opt. 36, 7239–7246 (1997).
[CrossRef]

B. F. Ventrudo, D. T. Cassidy, “Operating characteristics of a tunable diode laser absorption spectrometer using short-external-cavity and DFB laser diodes,” Appl. Opt. 29, 5007–5013 (1990).
[CrossRef] [PubMed]

W. Jin, “Performance analysis of a time-division-multiplexed fiber-optic gas-sensor array by wavelength modulation of a distributed-feedback laser,” Appl. Opt. 38, 5290–5297 (1999).
[CrossRef]

Appl. Phys. B (2)

K. Chan, H. Ito, H. Inaba, T. Furuya, “10 km-long fiber optic remote sensing of CH4 gas by near infrared absorption,” Appl. Phys. B 38, 11–15 (1985).
[CrossRef]

J. Reid, D. Labrie, “Second-harmonic detection with tunable diode lasers—comparison of experiment and theory,” Appl. Phys. B 26, 203–210 (1981).
[CrossRef]

IEEE Photon. Technol. Lett. (2)

Y. Shimose, T. Okamoto, A. Maruyama, M. Aizawa, H. Nagai, “Remote sensing of methane gas by differential absorption measurement using a wavelength-tunable DFB LD,” IEEE Photon. Technol. Lett. 3, 86–87 (1991).
[CrossRef]

P. K. C. Chan, W. Jin, J. M. Gong, M. S. Demokan, “Multiplexing of fiber Bragg grating sensors using an FMCW technique,” IEEE Photon. Technol. Lett. 11, 1470–1472 (1999).
[CrossRef]

J. Lightwave Technol. (1)

J. L. Brooks, B. Moslehi, B. Y. Kim, H. J. Shaw, “Time-domain addressing of remote fiber-optic interferometric sensor arrays,” J. Lightwave Technol. LT-5, 1014–1023 (1987).
[CrossRef]

J. Opt. Sensors (1)

J. P. Dakin, C. A. Wade, D. Pinchbeck, J. S. Wykes, “A novel optical fibre methane sensor,” J. Opt. Sensors 2, 261–267 (1987).

J. Phys. E (1)

J. P. Dakin, “Multiplexed and distributed optical fiber sensors,” J. Phys. E 20, 954–967 (1987).
[CrossRef]

Opt. Lett. (1)

Sensors Actuators B (1)

G. Stewart, C. Tandy, D. Moodie, M. A. Morante, F. Dong, “Design of a fiber optic multipoint sensor for gas detection,” Sensors Actuators B 51, 227–232 (1998).
[CrossRef]

Other (4)

G. Stewart, “Fibre optic sensors,” in Sensor Technologies, Vol. 1 of Sensor Systems for Environmental Monitoring, M. Campbell, ed. (Chapman & Hall, London, 1997), Chap. 1, pp. 1–40.

C. C. Chan, “Interrogation of fiber Bragg grating sensors with a tunable laser,” Ph.D. dissertation (Hong Kong Polytechnic University, Hong Kong, 2000).

W. R. Philp, W. Jin, A. Mencaglia, G. Stewart, B. Culshaw, “Interferometric noise in frequency modulated optical gas sensors,” in Proceedings of Twenty-First Australian Conference on Optical Fibre Technology (Institute of Radio and Electronics Engineers Society, Milsons Point, Australia, 1986), pp. 185–188.

P. K. C. Chan, “Multiplexing of fiber optic bragg grating sensors,” Ph.D. dissertation (Hong Kong Polytechnic University, Hong Kong, 2000).

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Figures (6)

Fig. 1
Fig. 1

FMCW multiplexed ladder gas-sensor array.

Fig. 2
Fig. 2

M s,12/k 0, M 0,12/k 0, and M c,12/k 0 as functions of ν Lm .

Fig. 3
Fig. 3

dc component of cos ξ ij as a function of time-delay difference τ ji between sensors.

Fig. 4
Fig. 4

Minimum detectable gas concentration versus sensor number: curve a, the interferometric effect; curves b–e, the effect of the sideline of FMCW.

Fig. 5
Fig. 5

Output signal spectrum of a two-sensor system. ω m = 500 Hz. (a) Small modulation amplitude ν Lm ≈ 100 MHz, (b) large modulation amplitude ν Lm ≈ 22 GHz.

Fig. 6
Fig. 6

Second-harmonic output of sensor 1.

Equations (52)

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

Δωτi=kπ,
ω0τi=kπ2+lπ,
Viωi=kωsχIA exp-2ανCiL,
Vi,nωi=kωs=χIAj=1,j1Nexp-ανCi+CjL×cos ΔψijB,
νt=νL0+νLm sin ωmt,
IAt=I01+η sin ωmt,
Viωi=kωs=χI01+η sinωmt×exp-2ανL0+νLm sin ωmtCiL.
αν=α01+ν-νg/δν)2,
Vi,2ωm-2χI0k0α0CiL,
k0=1-1+x22x21+x2,
x=νLm/δν.
Vi,n,2ωmωi=kωs=χI0j=1,jiNcos ξij×Ms,ij+η sin ξij×M0,ij+cos ξij×α0CiL×Mc,ij+cos ξij×α0CjL×Mc,ij,
Mc,ij=-x2J2ζij1-1+x22-J0ζij+J4ζij-J2ζij+J6ζij2x2-121-1+x22,
M0,ij=-J1ζij-J3ζijx21+x21-1+x22,
Ms,ij=-J2ζijx21+x21-1+x22,
ξij=2π t-τjitνL0tdt+Δϕij,
ζij=4πνLmωmsinωmτji/22πνLmτji,
Ci,min1=j=1,jiNMs,ij cos ξij+M0,ijη sin ξij2α0Lk0j=1,jiN Ms,ij cos ξij2α0Lk0,
Ci,min1rms2 j=1,jiNMs,ij21/24α0Lk02N-1 Ms,max4α0Lk0,
Ci,min2i=1,jiN CjMc,ij cos ξij2k0;
Ci,min2rms2 i=1,ijN Cj2Mc,ij21/24k02N-1 Mc,maxCmax4k0,
ξij=2π t-τjitνL0tdt+Δϕij=2πνL0τji+Δϕij.
νL0t=νLa+BLbt,  t0, T,
ξij=2π t-τjitνL0tdt+Δϕij=2πBLbτjit+2πνLa-½BLbτjiτji+Δϕij,
cos ξij=cos2πBLbτjit+ϕ0+Δϕij,
Ci,min1j=1,jiN Ms,ijcos ξijdc2α0Lk0N-1Ms,maxcos ξijdc,max2α0Lk0,
Ci,min2j=1,jiN CjMc,ijcos ξijdc2k0N-1CmaxMc,maxcos ξijdc,max2k0,
Ci,crossNmin=2 j=1,jiN MjCjTs2Tsj=1,jiN MjCj2Ts×maxj=1,jiN MjCj,
Mj=-sinΔω/Tsτi-τj½Ts-τjΔω/Tsτi-τjcosω0τj-½Δωτi,
EBt=IA1+m cos φt1/2 expj2π 0tνtdt,
φt= ωtdt.
ωt=ω0+Δω2-2ΔωTst-nTs-Ts2, tnTs, n+1T,
Ec=i=1N Eci,
Ecit=kIA1/21+m cos φti1/2×exp-ανtiCiLexpj2π 0tiνtidt+ϕνti
Ic=EcEc*=i=1N |Eci|2+2 j=1,j>iNReEiEj*,
Ic=i=1N |Eci|2=kIAi=1Nexp-2ανCiL×1+m cos φti.
Vr=V0 cosφtr,
Vt=½KkmGV0i=1N IAtiexp-2ανCiL×cosφti-φtr,
VFω=i=1N Viω,
Viω½kKGmV0IA exp-2ανCiLk=0+ δω-kωs4×sinω-2Δω/Tsτi½½Ts-τiω-2Δω/Tsτi×cosω0τi-¼ωTsexpj½ωτi,
Vikωs=kKGmV0IA exp-2ανCiLTs2-τi½kKGmV0TsIA exp-2ανCiL,
Ic,nt=2 i=1,j>iNReEciEcj*=2 i=1,j>iNIiIj1/21+m cos φti1/2×1+m cos φtj1/2 cosΔψij,
Il=kIA1/2 exp-ανtlClL,
Δψij=2π tjtiνtdt+Δϕij,
Ic,nt=2 i=1,j>iNIiIj1/21+½mcos φti+cos φtjcosΔψij,
1+m cos φt1/21+½m cos φt
Vnt=½KmGV0i=1,j>iNIiIj1/2 cosΔψlm×l=i,jcosφtl-φtr.
Vi,nωi=kωs=½KmGV0 TsIAj=1,jiNexp-αν×Ci+CjLcos Δψi,jB.
Δψij=2π tjtiνL0tdt+2πνLmtjtisin ωmtdt+Δϕ=ξij+ζij sin ωmt-τji/2,
Vi,nkωs=½kKGmV0TsI0j=1,jiN1+η sin ωmti1+η sin ωmtj1/2exp-l=i,j ανtlClL×cosξij+ζij sin ωmt-τji/2B.
V2ωmωi=kωsj=-2k0kKGmV0I0×sinΔω/Tsτi-τj½Ts-τjΔω/Tsτi-τj×cosω0τj-½Δωτiα0CjL.
V2ωmNωij=j=1,jiN V2ωmωij=2k0kKGmV0I0α0L j=1,jiN MjCj,

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