Abstract

I systematically analyze the theory of optical frequency-modulated continuous-wave (FMCW) interference. There are three different versions of optical FMCW interference, discussed in detail: sawtooth-wave optical FMCW interference, triangular-wave optical FMCW interference, and sinusoidal-wave optical FMCW interference. The essential concepts and technical terms are clearly defined, the necessary simplifications are introduced according to the characteristics of optical waves, and the formulas used to calculate the signal intensities under two different situations (static and dynamic) are properly derived. Advantages and limitations of each version of optical FMCW interference are also discussed.

© 2004 Optical Society of America

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  1. A. J. Hymans, J. Lait, “Analysis of a frequency-modulated continuous-wave ranging system,” Proc. IEEE 107, 365–372 (1960).
  2. M. I. Skolnik, Introduction to Radar Systems (McGraw-Hill, New York, 1962).
  3. B. Culshaw, I. P. Giles, “Frequency modulated heterodyne optical Sagnac interferometer,” IEEE J. Quantum Electron. QE-18, 690–693 (1982).
    [CrossRef]
  4. D. A. Jackson, A. D. Kersey, M. Corke, J. D. C. Jones, “Pseudoheterodyne detection scheme for optical interferometers,” Electron. Lett. 18, 1081–1083 (1982).
    [CrossRef]
  5. M. Corke, A. D. Kersey, D. A. Jackson, J. D. C. Jones, “All-fiber Michelson thermometer,” Electron. Lett. 19, 471–472 (1983).
    [CrossRef]
  6. D. Uttam, B. Culshaw, “Precision time domain reflectometry in optical fiber systems using a frequency modulated continuous wave ranging technique,” J. Lightwave Technol. LT-3, 971–976 (1985).
    [CrossRef]
  7. R. B. Franks, W. Torruellas, R. C. Youngquist, “Birefringent stress location sensor,” in Instrumentation for Optical Remote Sensing from Space, J. W. Lear, M. Monfils, S. L. Russak, J. S. Seeley, eds., Proc. SPIE589, 84–89 (1985).
  8. P. A. Leilabady, “Optical fiber point temperature sensor,” in Fiber Optic and Laser Sensors V, R. P. DePaula, E. Udd, eds., Proc. SPIE838, 231–237 (1987).
    [CrossRef]
  9. G. Zheng, Q. Tian, J. W. Liang, “Multifunction multichannel remote-reading optical fiber sensor system,” in International Conference on Optical Fibre Sensors in China, B. Culshaw, Y. Lian, eds., Proc. SPIE, 1572, 299–303 (1991).
    [CrossRef]
  10. T. A. Berkoff, A. D. Kersey, “Reflectometric two-mode elliptical-core fiber strain sensor with remote interrogation,” Electron. Lett. 28, 562–564 (1992).
    [CrossRef]
  11. G. Zheng, M. Campbell, P. A. Wallace, “Length-division-sensitive birefringent fiber FMCW remote strain sensor,” in Micro-optical Technologies for Measurement, Sensors, and Microsystems, O. M. Parriaux, ed., Proc. SPIE2783, 307–311 (1996).
    [CrossRef]
  12. G. Zheng, M. Campbell, P. A. Wallace, “Reflectometric frequency-modulation continuous-wave distributed fiber-optic stress sensor with forward coupled beams,” Appl. Opt. 35, 5722–5726 (1996).
    [CrossRef] [PubMed]
  13. M. Campbell, G. Zheng, P. A. Wallace, A. S. Holmes-Smith, “Reflectometric birefringent fiber absolute and relative strain sensor with environment-insensitive lead-in/lead-out fiber,” in Fiber Optic Sensors V, K. D. Bennett, B. Y. Kim, Y. Liao, eds., Proc. SPIE2895, 222–227 (1996).
    [CrossRef]

1996 (1)

1992 (1)

T. A. Berkoff, A. D. Kersey, “Reflectometric two-mode elliptical-core fiber strain sensor with remote interrogation,” Electron. Lett. 28, 562–564 (1992).
[CrossRef]

1985 (1)

D. Uttam, B. Culshaw, “Precision time domain reflectometry in optical fiber systems using a frequency modulated continuous wave ranging technique,” J. Lightwave Technol. LT-3, 971–976 (1985).
[CrossRef]

1983 (1)

M. Corke, A. D. Kersey, D. A. Jackson, J. D. C. Jones, “All-fiber Michelson thermometer,” Electron. Lett. 19, 471–472 (1983).
[CrossRef]

1982 (2)

B. Culshaw, I. P. Giles, “Frequency modulated heterodyne optical Sagnac interferometer,” IEEE J. Quantum Electron. QE-18, 690–693 (1982).
[CrossRef]

D. A. Jackson, A. D. Kersey, M. Corke, J. D. C. Jones, “Pseudoheterodyne detection scheme for optical interferometers,” Electron. Lett. 18, 1081–1083 (1982).
[CrossRef]

1960 (1)

A. J. Hymans, J. Lait, “Analysis of a frequency-modulated continuous-wave ranging system,” Proc. IEEE 107, 365–372 (1960).

Berkoff, T. A.

T. A. Berkoff, A. D. Kersey, “Reflectometric two-mode elliptical-core fiber strain sensor with remote interrogation,” Electron. Lett. 28, 562–564 (1992).
[CrossRef]

Campbell, M.

G. Zheng, M. Campbell, P. A. Wallace, “Reflectometric frequency-modulation continuous-wave distributed fiber-optic stress sensor with forward coupled beams,” Appl. Opt. 35, 5722–5726 (1996).
[CrossRef] [PubMed]

M. Campbell, G. Zheng, P. A. Wallace, A. S. Holmes-Smith, “Reflectometric birefringent fiber absolute and relative strain sensor with environment-insensitive lead-in/lead-out fiber,” in Fiber Optic Sensors V, K. D. Bennett, B. Y. Kim, Y. Liao, eds., Proc. SPIE2895, 222–227 (1996).
[CrossRef]

G. Zheng, M. Campbell, P. A. Wallace, “Length-division-sensitive birefringent fiber FMCW remote strain sensor,” in Micro-optical Technologies for Measurement, Sensors, and Microsystems, O. M. Parriaux, ed., Proc. SPIE2783, 307–311 (1996).
[CrossRef]

Corke, M.

M. Corke, A. D. Kersey, D. A. Jackson, J. D. C. Jones, “All-fiber Michelson thermometer,” Electron. Lett. 19, 471–472 (1983).
[CrossRef]

D. A. Jackson, A. D. Kersey, M. Corke, J. D. C. Jones, “Pseudoheterodyne detection scheme for optical interferometers,” Electron. Lett. 18, 1081–1083 (1982).
[CrossRef]

Culshaw, B.

D. Uttam, B. Culshaw, “Precision time domain reflectometry in optical fiber systems using a frequency modulated continuous wave ranging technique,” J. Lightwave Technol. LT-3, 971–976 (1985).
[CrossRef]

B. Culshaw, I. P. Giles, “Frequency modulated heterodyne optical Sagnac interferometer,” IEEE J. Quantum Electron. QE-18, 690–693 (1982).
[CrossRef]

Franks, R. B.

R. B. Franks, W. Torruellas, R. C. Youngquist, “Birefringent stress location sensor,” in Instrumentation for Optical Remote Sensing from Space, J. W. Lear, M. Monfils, S. L. Russak, J. S. Seeley, eds., Proc. SPIE589, 84–89 (1985).

Giles, I. P.

B. Culshaw, I. P. Giles, “Frequency modulated heterodyne optical Sagnac interferometer,” IEEE J. Quantum Electron. QE-18, 690–693 (1982).
[CrossRef]

Holmes-Smith, A. S.

M. Campbell, G. Zheng, P. A. Wallace, A. S. Holmes-Smith, “Reflectometric birefringent fiber absolute and relative strain sensor with environment-insensitive lead-in/lead-out fiber,” in Fiber Optic Sensors V, K. D. Bennett, B. Y. Kim, Y. Liao, eds., Proc. SPIE2895, 222–227 (1996).
[CrossRef]

Hymans, A. J.

A. J. Hymans, J. Lait, “Analysis of a frequency-modulated continuous-wave ranging system,” Proc. IEEE 107, 365–372 (1960).

Jackson, D. A.

M. Corke, A. D. Kersey, D. A. Jackson, J. D. C. Jones, “All-fiber Michelson thermometer,” Electron. Lett. 19, 471–472 (1983).
[CrossRef]

D. A. Jackson, A. D. Kersey, M. Corke, J. D. C. Jones, “Pseudoheterodyne detection scheme for optical interferometers,” Electron. Lett. 18, 1081–1083 (1982).
[CrossRef]

Jones, J. D. C.

M. Corke, A. D. Kersey, D. A. Jackson, J. D. C. Jones, “All-fiber Michelson thermometer,” Electron. Lett. 19, 471–472 (1983).
[CrossRef]

D. A. Jackson, A. D. Kersey, M. Corke, J. D. C. Jones, “Pseudoheterodyne detection scheme for optical interferometers,” Electron. Lett. 18, 1081–1083 (1982).
[CrossRef]

Kersey, A. D.

T. A. Berkoff, A. D. Kersey, “Reflectometric two-mode elliptical-core fiber strain sensor with remote interrogation,” Electron. Lett. 28, 562–564 (1992).
[CrossRef]

M. Corke, A. D. Kersey, D. A. Jackson, J. D. C. Jones, “All-fiber Michelson thermometer,” Electron. Lett. 19, 471–472 (1983).
[CrossRef]

D. A. Jackson, A. D. Kersey, M. Corke, J. D. C. Jones, “Pseudoheterodyne detection scheme for optical interferometers,” Electron. Lett. 18, 1081–1083 (1982).
[CrossRef]

Lait, J.

A. J. Hymans, J. Lait, “Analysis of a frequency-modulated continuous-wave ranging system,” Proc. IEEE 107, 365–372 (1960).

Leilabady, P. A.

P. A. Leilabady, “Optical fiber point temperature sensor,” in Fiber Optic and Laser Sensors V, R. P. DePaula, E. Udd, eds., Proc. SPIE838, 231–237 (1987).
[CrossRef]

Liang, J. W.

G. Zheng, Q. Tian, J. W. Liang, “Multifunction multichannel remote-reading optical fiber sensor system,” in International Conference on Optical Fibre Sensors in China, B. Culshaw, Y. Lian, eds., Proc. SPIE, 1572, 299–303 (1991).
[CrossRef]

Skolnik, M. I.

M. I. Skolnik, Introduction to Radar Systems (McGraw-Hill, New York, 1962).

Tian, Q.

G. Zheng, Q. Tian, J. W. Liang, “Multifunction multichannel remote-reading optical fiber sensor system,” in International Conference on Optical Fibre Sensors in China, B. Culshaw, Y. Lian, eds., Proc. SPIE, 1572, 299–303 (1991).
[CrossRef]

Torruellas, W.

R. B. Franks, W. Torruellas, R. C. Youngquist, “Birefringent stress location sensor,” in Instrumentation for Optical Remote Sensing from Space, J. W. Lear, M. Monfils, S. L. Russak, J. S. Seeley, eds., Proc. SPIE589, 84–89 (1985).

Uttam, D.

D. Uttam, B. Culshaw, “Precision time domain reflectometry in optical fiber systems using a frequency modulated continuous wave ranging technique,” J. Lightwave Technol. LT-3, 971–976 (1985).
[CrossRef]

Wallace, P. A.

G. Zheng, M. Campbell, P. A. Wallace, “Reflectometric frequency-modulation continuous-wave distributed fiber-optic stress sensor with forward coupled beams,” Appl. Opt. 35, 5722–5726 (1996).
[CrossRef] [PubMed]

M. Campbell, G. Zheng, P. A. Wallace, A. S. Holmes-Smith, “Reflectometric birefringent fiber absolute and relative strain sensor with environment-insensitive lead-in/lead-out fiber,” in Fiber Optic Sensors V, K. D. Bennett, B. Y. Kim, Y. Liao, eds., Proc. SPIE2895, 222–227 (1996).
[CrossRef]

G. Zheng, M. Campbell, P. A. Wallace, “Length-division-sensitive birefringent fiber FMCW remote strain sensor,” in Micro-optical Technologies for Measurement, Sensors, and Microsystems, O. M. Parriaux, ed., Proc. SPIE2783, 307–311 (1996).
[CrossRef]

Youngquist, R. C.

R. B. Franks, W. Torruellas, R. C. Youngquist, “Birefringent stress location sensor,” in Instrumentation for Optical Remote Sensing from Space, J. W. Lear, M. Monfils, S. L. Russak, J. S. Seeley, eds., Proc. SPIE589, 84–89 (1985).

Zheng, G.

G. Zheng, M. Campbell, P. A. Wallace, “Reflectometric frequency-modulation continuous-wave distributed fiber-optic stress sensor with forward coupled beams,” Appl. Opt. 35, 5722–5726 (1996).
[CrossRef] [PubMed]

M. Campbell, G. Zheng, P. A. Wallace, A. S. Holmes-Smith, “Reflectometric birefringent fiber absolute and relative strain sensor with environment-insensitive lead-in/lead-out fiber,” in Fiber Optic Sensors V, K. D. Bennett, B. Y. Kim, Y. Liao, eds., Proc. SPIE2895, 222–227 (1996).
[CrossRef]

G. Zheng, M. Campbell, P. A. Wallace, “Length-division-sensitive birefringent fiber FMCW remote strain sensor,” in Micro-optical Technologies for Measurement, Sensors, and Microsystems, O. M. Parriaux, ed., Proc. SPIE2783, 307–311 (1996).
[CrossRef]

G. Zheng, Q. Tian, J. W. Liang, “Multifunction multichannel remote-reading optical fiber sensor system,” in International Conference on Optical Fibre Sensors in China, B. Culshaw, Y. Lian, eds., Proc. SPIE, 1572, 299–303 (1991).
[CrossRef]

Appl. Opt. (1)

Electron. Lett. (3)

D. A. Jackson, A. D. Kersey, M. Corke, J. D. C. Jones, “Pseudoheterodyne detection scheme for optical interferometers,” Electron. Lett. 18, 1081–1083 (1982).
[CrossRef]

M. Corke, A. D. Kersey, D. A. Jackson, J. D. C. Jones, “All-fiber Michelson thermometer,” Electron. Lett. 19, 471–472 (1983).
[CrossRef]

T. A. Berkoff, A. D. Kersey, “Reflectometric two-mode elliptical-core fiber strain sensor with remote interrogation,” Electron. Lett. 28, 562–564 (1992).
[CrossRef]

IEEE J. Quantum Electron. (1)

B. Culshaw, I. P. Giles, “Frequency modulated heterodyne optical Sagnac interferometer,” IEEE J. Quantum Electron. QE-18, 690–693 (1982).
[CrossRef]

J. Lightwave Technol. (1)

D. Uttam, B. Culshaw, “Precision time domain reflectometry in optical fiber systems using a frequency modulated continuous wave ranging technique,” J. Lightwave Technol. LT-3, 971–976 (1985).
[CrossRef]

Proc. IEEE (1)

A. J. Hymans, J. Lait, “Analysis of a frequency-modulated continuous-wave ranging system,” Proc. IEEE 107, 365–372 (1960).

Other (6)

M. I. Skolnik, Introduction to Radar Systems (McGraw-Hill, New York, 1962).

G. Zheng, M. Campbell, P. A. Wallace, “Length-division-sensitive birefringent fiber FMCW remote strain sensor,” in Micro-optical Technologies for Measurement, Sensors, and Microsystems, O. M. Parriaux, ed., Proc. SPIE2783, 307–311 (1996).
[CrossRef]

M. Campbell, G. Zheng, P. A. Wallace, A. S. Holmes-Smith, “Reflectometric birefringent fiber absolute and relative strain sensor with environment-insensitive lead-in/lead-out fiber,” in Fiber Optic Sensors V, K. D. Bennett, B. Y. Kim, Y. Liao, eds., Proc. SPIE2895, 222–227 (1996).
[CrossRef]

R. B. Franks, W. Torruellas, R. C. Youngquist, “Birefringent stress location sensor,” in Instrumentation for Optical Remote Sensing from Space, J. W. Lear, M. Monfils, S. L. Russak, J. S. Seeley, eds., Proc. SPIE589, 84–89 (1985).

P. A. Leilabady, “Optical fiber point temperature sensor,” in Fiber Optic and Laser Sensors V, R. P. DePaula, E. Udd, eds., Proc. SPIE838, 231–237 (1987).
[CrossRef]

G. Zheng, Q. Tian, J. W. Liang, “Multifunction multichannel remote-reading optical fiber sensor system,” in International Conference on Optical Fibre Sensors in China, B. Culshaw, Y. Lian, eds., Proc. SPIE, 1572, 299–303 (1991).
[CrossRef]

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Figures (12)

Fig. 1
Fig. 1

Angular-frequency relationships between the interfering waves and the produced beat signal in sawtooth-wave FMCW interference.

Fig. 2
Fig. 2

Waveform of the beat signal produced by two coherent sawtooth-wave FMCW waves.

Fig. 3
Fig. 3

Waveforms of signals from a real optical sawtooth-wave FMCW interferometer. The upper trace is the waveform of the modulation signal; the lower trace is the waveform of the beat signal.

Fig. 4
Fig. 4

Angular-frequency relationships between the interfering waves and the detected signal in sawtooth-wave FMCW interference when the delay time is changing.

Fig. 5
Fig. 5

Angular-frequency relationships between the interfering waves and the beat signal in triangular-wave FMCW interference.

Fig. 6
Fig. 6

Waveform of the beat signal produced by two coherent triangular-wave FMCW waves.

Fig. 7
Fig. 7

Waveforms of signals from a real optical triangular-wave FMCW interferometer. The upper trace is the waveform of the frequency modulation signal; the lower trace is the waveform of the produced beat signal.

Fig. 8
Fig. 8

Angular-frequency relationship between the interfering waves and the beat signal in triangular-wave FMCW interference when the delay time is changing.

Fig. 9
Fig. 9

Angular-frequency relationship between the interfering waves and the produced beat signal in sinusoidal-wave FMCW interference.

Fig. 10
Fig. 10

Waveform of the beat signal produced by two coherent sinusoidal-wave FMCW waves.

Fig. 11
Fig. 11

Beat signal produced by two coherent optical sinusoidal-wave FMCW waves. The upper trace is the waveform of the frequency modulation signal; the lower trace is the waveform of the produced beat signal.

Fig. 12
Fig. 12

Angular-frequency relationship between the interfering waves and the produced beat signal in sinusoidal-wave FMCW interference when the delay time is changing.

Equations (72)

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ωt=dϕtdt,
ϕt=0tωtdt+ϕ0,
Et=E0 expjϕt,
El, t=E0lexpjϕt-lυ,
Eτ, t=E0lexpjϕt-τ,
τ=lυ=nlc,
I=Eτ, t2 |Eτ, t|2,
Iτ1, τm, t=i=1m Eiτi, t2 i=1m Eiτi, t2.
Iτ1, τ2, t=|E1τ1, t+E2τ2, t|2=E1τ1, t+E2τ2, tE1τ1, t+E2τ2, t*=E1τ1, tE1*τ1, t+E2τ2, tE2*τ2, t+E1τ1, tE2*τ2, t+E1*τ1, tE2τ2, t=E012+E022+2E01E02 cosϕt-τ1-ϕt-τ2=I1+I2+2I1I2 cosϕt-τ1-ϕt-τ2,
Iτ1, τ2, t=I01+V cosϕt-τ1-ϕt-τ2,
V=2I1I2I1+I2.
Iτ, t=I01+V cosϕt-ϕt-τ,
τ=τ2-τ1.
ω1t=αt+ω0,
α=ΔωTm,
ϕ1t=12 αt2+ω0t+ϕ0,
E1t=E01 expj12 αt2+ω0t+ϕ0,
ω2τ, t=αt-τ+ω0,
ϕ2τ, t=12 αt-τ2+ω0t-τ+ϕ0,
E2τ, t=E02 expj12 αt-τ2+ω0t-τ+ϕ0,
Iτ, t=|E1t+E2τ, t|2=E1t+E2τ, tE1t+E2τ, t*=E1tE1*t+E2τ, tE2*τ, t+E1tE2*τ, t+E1*tE2τ, t=E012+E022+2E01E02 cosατt+ω0τ-ατ22=I1+I2+2I1I2 cosατt+ω0τ-ατ22=I01+V cosατt+ω0τ-ατ22,
Iτ, t=I01+V cosατt+ω0τ=I01+V cosωbt+ϕb0,
ωb=ατ,
ϕb0=ω0τ.
ω2τ, t=αt-τ-Tm+ω0,
ϕ2τ, t=12 αt-τ-Tm2+ω0t-(τ-Tm+ϕ0,
E2τ, t=E02 expj12 αt-τ-Tm2+ω0t-(τ-Tm+ϕ0
Iτ, t=I01+V cosατ-Tmt+ω0τ-Tm- 12 ατ-Tm2.
ωb=ατ-Tm.
Iτ, t=I01+V cosατt+ω0τ=I01+V cosωbt+ϕb0.
Iτ, t=I01+VTm cosατt+ω0τrectTmtn=- δt-nTm=I01+VTm cosωbt+ϕb0rectTmt n=- δt-nTm,
rectTt=1/T|t|T/20|t|>T/2.
IOPD, t=I01+V cos2πΔννmOPDc t+2πλ0OPD=I01+V cos2πνbt+ϕb0,
νb=αOPD2πc=ΔννmOPDc,
ϕb0=ω0OPDc=k0OPD=2πOPDλ0,
OPD=cνbΔννm,
δOPD=λ0δϕb02π.
Iτ, t=I01+V cosατtt+ω0τt.
ωb=ddtατtt+ω0τt=ατt+ω0+αtdτtdt=ωb+ωD,
ωD=ω0+αtdτtdt.
ωD¯=ω0dτtdt.
νD¯=1λ0dOPDtdt,
νD¯=2nλ0 s,
Irτ,t=I01+V cosατt+ω0τ,
ωbr=ατ,
ϕb0r=ω0τ,
Ifτ, t=I01+V cos-ατt+ω0τ,
ωbf=-ατ,
ϕb0f=ω0τ
ωbr¯=ωb+ωD¯,
ωbf¯=ωD¯-ωb.
|ωbf¯|=ωb-ωD¯.
ωb=12ωbr¯+|ωbf¯|.
ωD¯=12ωbr¯-|ωbf¯|.
s=λ02nνD¯,
ω1t=ω0+Δω2sinωmt,
ϕ1t=ω0t- Δω2ωmcosωmt+ϕ0,
E1t=E01 expjω0t- Δω2ωmcosωmt+ϕ0,
ω2τ, t=ω0+Δω2sinωmt-τ,
ϕ2τ, t=ω0t-τ- Δω2ωmcosωmt-τ+ϕ0,
E2τ, t=E02 expjω0t-τ- Δω2ωm×cosωmt-τ+ϕ0,
Iτ, t=|E1t+E2τ, t|2=E1t+E2τ, tE1t+E2τ, t*=E1tE1*t+E2τ, tE2*τ, t+E1tE2*τ, t+E1*tE2τ, t=E012+E022+2E01E02 cosΔω2ωmcos ωmt-τ-cos ωmt+ω0τ=I1+I2+2I1I2 cosΔω2ωmcos ωmt-τ-cos ωmt+ω0τ,
Iτ, t=I01+V cosΔωωmsinωmτ2sin ωmt-12 τ+ω0τ,
Iτ, t=I01+V cosω0τ+Δωτ2sinωmt.
ωb=Δωωmτ2cosωmt,
ϕb0=ω0τ.
|ωb¯|=Δωωmτπ.
|ωb¯|=ΔωτTm=ατ,
ωbr¯=ωb¯+ωD¯,
|ωbf¯|=ωb¯-ωD¯.
ωb¯ =12ωbr¯+|ωbf¯|.
ωD¯=12ωbr¯-|ωbf¯|.

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