Abstract

We establish a theoretical model of the Doppler effect in absolute distance measurements using frequency scanning interferometry (FSI) and propose a novel FSI absolute distance measurement system. This system incorporates a basic FSI system and a laser Doppler velocimeter (LDV). The LDV results are used to correct for the Doppler effect in the absolute distance measurement signal obtained by the basic FSI system. In the measurement of a target located at 16 m, a measurement resolution of 65.5 μm is obtained, which is close to the theoretical resolution, and a standard deviation of 3.15 μm is obtained. The theoretical measurement uncertainty is 8.6 μm + 0.16 μm/m Rm (k = 2) within a distance range of 1 m to 24 m neglecting the influence of air refractive index, which has been verified with experiments.

© 2016 Optical Society of America

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References

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  1. D. N. Keep, “Frequency-modulation radar for use in the mercantile marine,” Proc. IEEE: Radio Electron. Eng. 103, 519–523 (1956).
    [Crossref]
  2. B. Culshaw and I. P. Giles, “Frequency modulated heterodyne optical fiber Sagnac interferometer,” IEEE Trans. Microw. Theory Tech. 30(4), 536–539 (1982).
    [Crossref]
  3. W. Eickhoff and R. Ulrich, “Optical frequency domain reflectometry in single-mode fiber,” Appl. Phys. Lett. 39(9), 693–695 (1981).
    [Crossref]
  4. S. A. Kingsley and D. E. N. Davies, “OFDR diagnostics for fibre and integrated-optic systems,” Electron. Lett. 21(10), 434–435 (1985).
    [Crossref]
  5. K. Shimizu, T. Horiguchi, and Y. Koyamada, “Measurement of Rayleigh backscattering in single-mode fibers based on coherent OFDR employing a DFB laser diode,” IEEE Photonics Technol. Lett. 3(11), 1039–1041 (1991).
    [Crossref]
  6. R. Passy, N. Gisin, and J. P. von der Weid, “High-sensitivity-coherent optical frequency-domain reflectometry for characterization of fiber-optic network components,” IEEE Photonics Technol. Lett. 7(6), 667–669 (1995).
    [Crossref]
  7. D. K. Gifford, B. J. Soller, M. S. Wolfe, and M. E. Froggatt, “Optical vector network analyzer for single-scan measurements of loss, group delay, and polarization mode dispersion,” Appl. Opt. 44(34), 7282–7286 (2005).
    [Crossref] [PubMed]
  8. A. K. Sang, D. K. Gifford, B. D. Dickerson, B. F. Fielder, and M. E. Froggatt, “One centimeter spatial resolution temperature measurements in a nuclear reactor using Rayleigh scatter in optical fiber,” Proc. SPIE 6619, 66193D (2007).
    [Crossref]
  9. M. Jiang, D. Chen, and S. He, “Multiplexing scheme of Long-period grating sensors based on a modified optical frequency domain reflectometry,” IEEE Photonics Technol. Lett. 20(23), 1962–1964 (2008).
    [Crossref]
  10. E. C. Burrows and K.-Y. Liou, “High resolution laser LIDAR utilising two-section distributed feedback semiconductor laser as a coherent source,” Electron. Lett. 26(9), 577–579 (1990).
    [Crossref]
  11. A. Dieckmann, “FMCW-LIDAR with tunable twin-guide laser diode,” Electron. Lett. 30(4), 308–309 (1994).
    [Crossref]
  12. E. D. Moore and R. R. McLeod, “Phase-sensitive swept-source interferometry for absolute ranging with application to measurements of group refractive index and thickness,” Opt. Express 19(9), 8117–8126 (2011).
    [Crossref] [PubMed]
  13. M. S. Warden, “Precision of frequency scanning interferometry distance measurements in the presence of noise,” Appl. Opt. 53(25), 5800–5806 (2014).
    [Crossref] [PubMed]
  14. S. Yun, G. Tearney, J. de Boer, N. Iftimia, and B. Bouma, “High-speed optical frequency-domain imaging,” Opt. Express 11(22), 2953–2963 (2003).
    [Crossref] [PubMed]
  15. I. Yamaguchi, A. Yamamoto, and M. Yano, “Surface topography by wavelength scanning interferometry,” Opt. Eng. 39(1), 40–46 (2000).
    [Crossref]
  16. M. Wojtkowski, V. Srinivasan, T. Ko, J. Fujimoto, A. Kowalczyk, and J. Duker, “Ultrahigh-resolution, high-speed, Fourier domain optical coherence tomography and methods for dispersion compensation,” Opt. Express 12(11), 2404–2422 (2004).
    [Crossref] [PubMed]
  17. N. Lippok, S. Coen, P. Nielsen, and F. Vanholsbeeck, “Dispersion compensation in Fourier domain optical coherence tomography using the fractional Fourier transform,” Opt. Express 20(21), 23398–23413 (2012).
    [Crossref] [PubMed]
  18. K. Asaka and K. Ohbayashi, “Dispersion matching of sample and reference arms in optical frequency domain reflectometry-optical coherence tomography using a dispersion-shifted fiber,” Opt. Express 15(8), 5030–5042 (2007).
    [Crossref] [PubMed]
  19. J. Dale, B. Hughes, A. J. Lancaster, A. J. Lewis, A. J. H. Reichold, and M. S. Warden, “Multi-channel absolute distance measurement system with sub ppm-accuracy and 20 m range using frequency scanning interferometry and gas absorption cells,” Opt. Express 22(20), 24869–24893 (2014).
    [Crossref] [PubMed]
  20. R. Schneider, P. Thuermel, and M. Stockmann, “Distance measurement of moving objects by frequency modulated laser radar,” Opt. Eng. 40(1), 33–37 (2001).
    [Crossref]
  21. M. Rezk and A. Slotwinski, “Compact fiber optic geometry for a counter-chirp FMCW coherent laser radar,” U.S. Patent No. 8,687,173 (2014
  22. M. Warden, “Absolute distance metrology using frequency swept lasers,” Ph.D. thesis, University of Oxford (2011).
  23. C. Lu, G. Liu, B. Liu, F. Chen, T. Hu, Z. Zhuang, X. Xu, and Y. Gan, “Method based on chirp decomposition for dispersion mismatch compensation in precision absolute distance measurement using swept-wavelength interferometry,” Opt. Express 23(25), 31662–31671 (2015).
    [Crossref] [PubMed]

2015 (1)

2014 (2)

2012 (1)

2011 (1)

2008 (1)

M. Jiang, D. Chen, and S. He, “Multiplexing scheme of Long-period grating sensors based on a modified optical frequency domain reflectometry,” IEEE Photonics Technol. Lett. 20(23), 1962–1964 (2008).
[Crossref]

2007 (2)

K. Asaka and K. Ohbayashi, “Dispersion matching of sample and reference arms in optical frequency domain reflectometry-optical coherence tomography using a dispersion-shifted fiber,” Opt. Express 15(8), 5030–5042 (2007).
[Crossref] [PubMed]

A. K. Sang, D. K. Gifford, B. D. Dickerson, B. F. Fielder, and M. E. Froggatt, “One centimeter spatial resolution temperature measurements in a nuclear reactor using Rayleigh scatter in optical fiber,” Proc. SPIE 6619, 66193D (2007).
[Crossref]

2005 (1)

2004 (1)

2003 (1)

2001 (1)

R. Schneider, P. Thuermel, and M. Stockmann, “Distance measurement of moving objects by frequency modulated laser radar,” Opt. Eng. 40(1), 33–37 (2001).
[Crossref]

2000 (1)

I. Yamaguchi, A. Yamamoto, and M. Yano, “Surface topography by wavelength scanning interferometry,” Opt. Eng. 39(1), 40–46 (2000).
[Crossref]

1995 (1)

R. Passy, N. Gisin, and J. P. von der Weid, “High-sensitivity-coherent optical frequency-domain reflectometry for characterization of fiber-optic network components,” IEEE Photonics Technol. Lett. 7(6), 667–669 (1995).
[Crossref]

1994 (1)

A. Dieckmann, “FMCW-LIDAR with tunable twin-guide laser diode,” Electron. Lett. 30(4), 308–309 (1994).
[Crossref]

1991 (1)

K. Shimizu, T. Horiguchi, and Y. Koyamada, “Measurement of Rayleigh backscattering in single-mode fibers based on coherent OFDR employing a DFB laser diode,” IEEE Photonics Technol. Lett. 3(11), 1039–1041 (1991).
[Crossref]

1990 (1)

E. C. Burrows and K.-Y. Liou, “High resolution laser LIDAR utilising two-section distributed feedback semiconductor laser as a coherent source,” Electron. Lett. 26(9), 577–579 (1990).
[Crossref]

1985 (1)

S. A. Kingsley and D. E. N. Davies, “OFDR diagnostics for fibre and integrated-optic systems,” Electron. Lett. 21(10), 434–435 (1985).
[Crossref]

1982 (1)

B. Culshaw and I. P. Giles, “Frequency modulated heterodyne optical fiber Sagnac interferometer,” IEEE Trans. Microw. Theory Tech. 30(4), 536–539 (1982).
[Crossref]

1981 (1)

W. Eickhoff and R. Ulrich, “Optical frequency domain reflectometry in single-mode fiber,” Appl. Phys. Lett. 39(9), 693–695 (1981).
[Crossref]

Asaka, K.

Bouma, B.

Burrows, E. C.

E. C. Burrows and K.-Y. Liou, “High resolution laser LIDAR utilising two-section distributed feedback semiconductor laser as a coherent source,” Electron. Lett. 26(9), 577–579 (1990).
[Crossref]

Chen, D.

M. Jiang, D. Chen, and S. He, “Multiplexing scheme of Long-period grating sensors based on a modified optical frequency domain reflectometry,” IEEE Photonics Technol. Lett. 20(23), 1962–1964 (2008).
[Crossref]

Chen, F.

Coen, S.

Culshaw, B.

B. Culshaw and I. P. Giles, “Frequency modulated heterodyne optical fiber Sagnac interferometer,” IEEE Trans. Microw. Theory Tech. 30(4), 536–539 (1982).
[Crossref]

Dale, J.

Davies, D. E. N.

S. A. Kingsley and D. E. N. Davies, “OFDR diagnostics for fibre and integrated-optic systems,” Electron. Lett. 21(10), 434–435 (1985).
[Crossref]

de Boer, J.

Dickerson, B. D.

A. K. Sang, D. K. Gifford, B. D. Dickerson, B. F. Fielder, and M. E. Froggatt, “One centimeter spatial resolution temperature measurements in a nuclear reactor using Rayleigh scatter in optical fiber,” Proc. SPIE 6619, 66193D (2007).
[Crossref]

Dieckmann, A.

A. Dieckmann, “FMCW-LIDAR with tunable twin-guide laser diode,” Electron. Lett. 30(4), 308–309 (1994).
[Crossref]

Duker, J.

Eickhoff, W.

W. Eickhoff and R. Ulrich, “Optical frequency domain reflectometry in single-mode fiber,” Appl. Phys. Lett. 39(9), 693–695 (1981).
[Crossref]

Fielder, B. F.

A. K. Sang, D. K. Gifford, B. D. Dickerson, B. F. Fielder, and M. E. Froggatt, “One centimeter spatial resolution temperature measurements in a nuclear reactor using Rayleigh scatter in optical fiber,” Proc. SPIE 6619, 66193D (2007).
[Crossref]

Froggatt, M. E.

A. K. Sang, D. K. Gifford, B. D. Dickerson, B. F. Fielder, and M. E. Froggatt, “One centimeter spatial resolution temperature measurements in a nuclear reactor using Rayleigh scatter in optical fiber,” Proc. SPIE 6619, 66193D (2007).
[Crossref]

D. K. Gifford, B. J. Soller, M. S. Wolfe, and M. E. Froggatt, “Optical vector network analyzer for single-scan measurements of loss, group delay, and polarization mode dispersion,” Appl. Opt. 44(34), 7282–7286 (2005).
[Crossref] [PubMed]

Fujimoto, J.

Gan, Y.

Gifford, D. K.

A. K. Sang, D. K. Gifford, B. D. Dickerson, B. F. Fielder, and M. E. Froggatt, “One centimeter spatial resolution temperature measurements in a nuclear reactor using Rayleigh scatter in optical fiber,” Proc. SPIE 6619, 66193D (2007).
[Crossref]

D. K. Gifford, B. J. Soller, M. S. Wolfe, and M. E. Froggatt, “Optical vector network analyzer for single-scan measurements of loss, group delay, and polarization mode dispersion,” Appl. Opt. 44(34), 7282–7286 (2005).
[Crossref] [PubMed]

Giles, I. P.

B. Culshaw and I. P. Giles, “Frequency modulated heterodyne optical fiber Sagnac interferometer,” IEEE Trans. Microw. Theory Tech. 30(4), 536–539 (1982).
[Crossref]

Gisin, N.

R. Passy, N. Gisin, and J. P. von der Weid, “High-sensitivity-coherent optical frequency-domain reflectometry for characterization of fiber-optic network components,” IEEE Photonics Technol. Lett. 7(6), 667–669 (1995).
[Crossref]

He, S.

M. Jiang, D. Chen, and S. He, “Multiplexing scheme of Long-period grating sensors based on a modified optical frequency domain reflectometry,” IEEE Photonics Technol. Lett. 20(23), 1962–1964 (2008).
[Crossref]

Horiguchi, T.

K. Shimizu, T. Horiguchi, and Y. Koyamada, “Measurement of Rayleigh backscattering in single-mode fibers based on coherent OFDR employing a DFB laser diode,” IEEE Photonics Technol. Lett. 3(11), 1039–1041 (1991).
[Crossref]

Hu, T.

Hughes, B.

Iftimia, N.

Jiang, M.

M. Jiang, D. Chen, and S. He, “Multiplexing scheme of Long-period grating sensors based on a modified optical frequency domain reflectometry,” IEEE Photonics Technol. Lett. 20(23), 1962–1964 (2008).
[Crossref]

Kingsley, S. A.

S. A. Kingsley and D. E. N. Davies, “OFDR diagnostics for fibre and integrated-optic systems,” Electron. Lett. 21(10), 434–435 (1985).
[Crossref]

Ko, T.

Kowalczyk, A.

Koyamada, Y.

K. Shimizu, T. Horiguchi, and Y. Koyamada, “Measurement of Rayleigh backscattering in single-mode fibers based on coherent OFDR employing a DFB laser diode,” IEEE Photonics Technol. Lett. 3(11), 1039–1041 (1991).
[Crossref]

Lancaster, A. J.

Lewis, A. J.

Liou, K.-Y.

E. C. Burrows and K.-Y. Liou, “High resolution laser LIDAR utilising two-section distributed feedback semiconductor laser as a coherent source,” Electron. Lett. 26(9), 577–579 (1990).
[Crossref]

Lippok, N.

Liu, B.

Liu, G.

Lu, C.

McLeod, R. R.

Moore, E. D.

Nielsen, P.

Ohbayashi, K.

Passy, R.

R. Passy, N. Gisin, and J. P. von der Weid, “High-sensitivity-coherent optical frequency-domain reflectometry for characterization of fiber-optic network components,” IEEE Photonics Technol. Lett. 7(6), 667–669 (1995).
[Crossref]

Reichold, A. J. H.

Sang, A. K.

A. K. Sang, D. K. Gifford, B. D. Dickerson, B. F. Fielder, and M. E. Froggatt, “One centimeter spatial resolution temperature measurements in a nuclear reactor using Rayleigh scatter in optical fiber,” Proc. SPIE 6619, 66193D (2007).
[Crossref]

Schneider, R.

R. Schneider, P. Thuermel, and M. Stockmann, “Distance measurement of moving objects by frequency modulated laser radar,” Opt. Eng. 40(1), 33–37 (2001).
[Crossref]

Shimizu, K.

K. Shimizu, T. Horiguchi, and Y. Koyamada, “Measurement of Rayleigh backscattering in single-mode fibers based on coherent OFDR employing a DFB laser diode,” IEEE Photonics Technol. Lett. 3(11), 1039–1041 (1991).
[Crossref]

Soller, B. J.

Srinivasan, V.

Stockmann, M.

R. Schneider, P. Thuermel, and M. Stockmann, “Distance measurement of moving objects by frequency modulated laser radar,” Opt. Eng. 40(1), 33–37 (2001).
[Crossref]

Tearney, G.

Thuermel, P.

R. Schneider, P. Thuermel, and M. Stockmann, “Distance measurement of moving objects by frequency modulated laser radar,” Opt. Eng. 40(1), 33–37 (2001).
[Crossref]

Ulrich, R.

W. Eickhoff and R. Ulrich, “Optical frequency domain reflectometry in single-mode fiber,” Appl. Phys. Lett. 39(9), 693–695 (1981).
[Crossref]

Vanholsbeeck, F.

von der Weid, J. P.

R. Passy, N. Gisin, and J. P. von der Weid, “High-sensitivity-coherent optical frequency-domain reflectometry for characterization of fiber-optic network components,” IEEE Photonics Technol. Lett. 7(6), 667–669 (1995).
[Crossref]

Warden, M. S.

Wojtkowski, M.

Wolfe, M. S.

Xu, X.

Yamaguchi, I.

I. Yamaguchi, A. Yamamoto, and M. Yano, “Surface topography by wavelength scanning interferometry,” Opt. Eng. 39(1), 40–46 (2000).
[Crossref]

Yamamoto, A.

I. Yamaguchi, A. Yamamoto, and M. Yano, “Surface topography by wavelength scanning interferometry,” Opt. Eng. 39(1), 40–46 (2000).
[Crossref]

Yano, M.

I. Yamaguchi, A. Yamamoto, and M. Yano, “Surface topography by wavelength scanning interferometry,” Opt. Eng. 39(1), 40–46 (2000).
[Crossref]

Yun, S.

Zhuang, Z.

Appl. Opt. (2)

Appl. Phys. Lett. (1)

W. Eickhoff and R. Ulrich, “Optical frequency domain reflectometry in single-mode fiber,” Appl. Phys. Lett. 39(9), 693–695 (1981).
[Crossref]

Electron. Lett. (3)

S. A. Kingsley and D. E. N. Davies, “OFDR diagnostics for fibre and integrated-optic systems,” Electron. Lett. 21(10), 434–435 (1985).
[Crossref]

E. C. Burrows and K.-Y. Liou, “High resolution laser LIDAR utilising two-section distributed feedback semiconductor laser as a coherent source,” Electron. Lett. 26(9), 577–579 (1990).
[Crossref]

A. Dieckmann, “FMCW-LIDAR with tunable twin-guide laser diode,” Electron. Lett. 30(4), 308–309 (1994).
[Crossref]

IEEE Photonics Technol. Lett. (3)

K. Shimizu, T. Horiguchi, and Y. Koyamada, “Measurement of Rayleigh backscattering in single-mode fibers based on coherent OFDR employing a DFB laser diode,” IEEE Photonics Technol. Lett. 3(11), 1039–1041 (1991).
[Crossref]

R. Passy, N. Gisin, and J. P. von der Weid, “High-sensitivity-coherent optical frequency-domain reflectometry for characterization of fiber-optic network components,” IEEE Photonics Technol. Lett. 7(6), 667–669 (1995).
[Crossref]

M. Jiang, D. Chen, and S. He, “Multiplexing scheme of Long-period grating sensors based on a modified optical frequency domain reflectometry,” IEEE Photonics Technol. Lett. 20(23), 1962–1964 (2008).
[Crossref]

IEEE Trans. Microw. Theory Tech. (1)

B. Culshaw and I. P. Giles, “Frequency modulated heterodyne optical fiber Sagnac interferometer,” IEEE Trans. Microw. Theory Tech. 30(4), 536–539 (1982).
[Crossref]

Opt. Eng. (2)

I. Yamaguchi, A. Yamamoto, and M. Yano, “Surface topography by wavelength scanning interferometry,” Opt. Eng. 39(1), 40–46 (2000).
[Crossref]

R. Schneider, P. Thuermel, and M. Stockmann, “Distance measurement of moving objects by frequency modulated laser radar,” Opt. Eng. 40(1), 33–37 (2001).
[Crossref]

Opt. Express (7)

S. Yun, G. Tearney, J. de Boer, N. Iftimia, and B. Bouma, “High-speed optical frequency-domain imaging,” Opt. Express 11(22), 2953–2963 (2003).
[Crossref] [PubMed]

M. Wojtkowski, V. Srinivasan, T. Ko, J. Fujimoto, A. Kowalczyk, and J. Duker, “Ultrahigh-resolution, high-speed, Fourier domain optical coherence tomography and methods for dispersion compensation,” Opt. Express 12(11), 2404–2422 (2004).
[Crossref] [PubMed]

K. Asaka and K. Ohbayashi, “Dispersion matching of sample and reference arms in optical frequency domain reflectometry-optical coherence tomography using a dispersion-shifted fiber,” Opt. Express 15(8), 5030–5042 (2007).
[Crossref] [PubMed]

E. D. Moore and R. R. McLeod, “Phase-sensitive swept-source interferometry for absolute ranging with application to measurements of group refractive index and thickness,” Opt. Express 19(9), 8117–8126 (2011).
[Crossref] [PubMed]

N. Lippok, S. Coen, P. Nielsen, and F. Vanholsbeeck, “Dispersion compensation in Fourier domain optical coherence tomography using the fractional Fourier transform,” Opt. Express 20(21), 23398–23413 (2012).
[Crossref] [PubMed]

J. Dale, B. Hughes, A. J. Lancaster, A. J. Lewis, A. J. H. Reichold, and M. S. Warden, “Multi-channel absolute distance measurement system with sub ppm-accuracy and 20 m range using frequency scanning interferometry and gas absorption cells,” Opt. Express 22(20), 24869–24893 (2014).
[Crossref] [PubMed]

C. Lu, G. Liu, B. Liu, F. Chen, T. Hu, Z. Zhuang, X. Xu, and Y. Gan, “Method based on chirp decomposition for dispersion mismatch compensation in precision absolute distance measurement using swept-wavelength interferometry,” Opt. Express 23(25), 31662–31671 (2015).
[Crossref] [PubMed]

Proc. SPIE (1)

A. K. Sang, D. K. Gifford, B. D. Dickerson, B. F. Fielder, and M. E. Froggatt, “One centimeter spatial resolution temperature measurements in a nuclear reactor using Rayleigh scatter in optical fiber,” Proc. SPIE 6619, 66193D (2007).
[Crossref]

Other (3)

M. Rezk and A. Slotwinski, “Compact fiber optic geometry for a counter-chirp FMCW coherent laser radar,” U.S. Patent No. 8,687,173 (2014

M. Warden, “Absolute distance metrology using frequency swept lasers,” Ph.D. thesis, University of Oxford (2011).

D. N. Keep, “Frequency-modulation radar for use in the mercantile marine,” Proc. IEEE: Radio Electron. Eng. 103, 519–523 (1956).
[Crossref]

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Figures (10)

Fig. 1
Fig. 1 Schematic diagram of the FSI-based absolute distance measurement system.
Fig. 2
Fig. 2 The OPD variation simulation of measurement interferometer.
Fig. 3
Fig. 3 The distance spectrum of a target located at 10 m with an OPD variation as in Fig. 2.
Fig. 4
Fig. 4 Schematic diagram of the novel FSI absolute distance measurement system
Fig. 5
Fig. 5 The Fourier transform of the measurement interferometer signal with a target located at 16 m. The signal in the red rectangle is used for target distance measurement. The signal in the green rectangle is used to measure the OPD variation of the measurement interferometer.
Fig. 6
Fig. 6 The distance spectrum of a target located at 16 m. The blue line is the distance spectrum without the Doppler effect correction. The green line is the simulation result of an ideal signal.
Fig. 7
Fig. 7 The OPD variation measurement results of our novel system, which is used to correct the Doppler effect in our distance measurement signal.
Fig. 8
Fig. 8 The distance spectrum with the Doppler effect correction of a target located at 16 m. The blue line is the distance spectrum of our novel system, which is corrected with the OPD variation measurement result shown in Fig. 7. The green dotted line is the simulation result of an ideal signal.
Fig. 9
Fig. 9 The comparison setup to verify the uncertainty of our system.
Fig. 10
Fig. 10 The measurement length residual between our absolute distance measurement system and the laser interferometer (Renishaw ML10). The dotted lines indicate a linear estimate of the error of the experiments based on a graphic analysis. The dashed line represents the theoretical measurement uncertainty.

Equations (16)

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

I 0 ( f ) = A 0 cos ( 2 π f τ 0 ) .
2 π Δ f ( k ) τ 0 = 2 π k ,
Δ f ( k ) = k τ 0 .
I 1 ( f ) = A 1 cos ( 2 π f τ m ) = A 1 cos ( 2 π Δ f τ m + 2 π f 0 τ m ) ,
I 1 ( k ) = A 1 cos ( 2 π τ m τ 0 k + 2 π f 0 τ m ) .
I 1 ( k ) = A 1 cos [ 2 π τ m ( k ) τ 0 k + 2 π f 0 τ m ( k ) ] = A 1 cos [ 2 π τ m 0 τ 0 k + 2 π f 0 Δ τ m ( k ) + 2 π f 0 τ m 0 +2 π k τ 0 Δ τ m ( k ) ] ,
E D r ( t ) = A D r exp [ j 2 π ( f 1 + f A O M 2 ) t + j 2 π ( f 1 + f A O M 2 ) τ D r + j φ 0 ] ,
E D m ( t ) = A D m exp [ j 2 π ( f 1 + f A O M 1 ) t + j 2 π ( f 1 + f A O M 1 ) τ D m ( t ) + j φ 0 ] .
I 2 ( t ) = A 2 cos [ 2 π Δ f A O M t + 2 π ( f 1 + f A O M 1 ) τ D m ( t ) 2 π ( f 1 + f A O M 2 ) τ D r ] = A 2 cos [ 2 π Δ f A O M t + 2 π ( f 1 + f A O M 1 ) Δ τ m ( t ) + φ D ] .
I 2 ( k ) = A 2 cos [ 2 π Δ f A O M t ( k ) + 2 π ( f 1 + f A O M 1 ) Δ τ m ( k ) + φ D ] .
I 3 ( k ) = A 3 cos [ 2 π Δ f A O M t ( k ) + φ D 2 ] .
Δ φ 1 ( k ) = 2 π τ m 0 τ 0 k + 2 π f 0 Δ τ m ( k ) +2 π k τ 0 Δ τ m ( k ) ,
Δ φ 2 ( k ) = 2 π Δ f A O M t ( k ) + 2 π ( f 1 + f A O M 1 ) Δ τ m ( k ) ,
Δ φ 3 ( k ) = 2 π Δ f A O M t ( k ) .
Δ τ m ( k ) = [ Δ φ 2 ( k ) Δ φ 3 ( k ) ] 2 π ( f 1 + f A O M 1 ) .
R m ( k ) = c τ m ( k ) 2 n = c [ τ m 0 + Δ τ m ( k ) ] 2 n = c 2 n τ 0 2 π k [ Δ φ 1 ( k ) 2 π f 0 Δ τ m ( k ) ] .

Metrics