Abstract

The measurement of dynamic displacements by use of speckle pattern interferometry and temporal phase unwrapping allows for the evaluation of large-object displacement fields without the propagation of spatial unwrapping errors. If a temporal carrier is introduced in one of the beams of the interferometer, phase data for whole-object displacement can be retrieved by use of a temporal phase-shifting method or a temporal Fourier transformation approach. We present a comparison between both methods of temporal phase measurement in terms of precision and execution speed. We performed the analysis by using computer-simulated speckle interferograms, an approach that allowed us to know precisely the original phase distribution and also to determine the spatial rms phase error as a function of the phase change introduced between two consecutive speckle interferograms. The performance of both methods to process experimental data is also illustrated by use of the results from a high-speed speckle interferometry study of a carbon fiber panel.

© 2002 Optical Society of America

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References

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  1. R. S. Sirohi, Speckle Metrology (Marcel Dekker, New York, 1993).
  2. J. M. Huntley, “Automated analysis of speckle interferograms,” in Digital Speckle Pattern Interferometry and Related Techniques, P. K. Rastogi, ed. (Wiley, Chichester, England, 2001), Chap. 2, pp. 59–139.
  3. D. C. Ghiglia, M. D. Pritt, Two-Dimensional Phase Unwrapping (Wiley, New York, 1998).
  4. J. M. Huntley, H. Saldner, “Temporal phase-unwrapping algorithm for automated interferogram analysis,” Appl. Opt. 32, 3047–3052 (1993).
    [CrossRef] [PubMed]
  5. J. M. Huntley, G. H. Kaufmann, D. Kerr, “Phase-shifted dynamic speckle pattern interferometry at 1 kHz,” Appl. Opt. 38, 6556–6563 (1999).
    [CrossRef]
  6. P. D. Ruiz, J. M. Huntley, Y. Shen, C. R. Coggrave, G. H. Kaufmann, “Vibration-induced phase errors in high-speed phase-shifting speckle pattern interferometry,” Appl. Opt. 40, 2117–2125 (2001).
    [CrossRef]
  7. P. D. Ruiz, J. M. Huntley, Y. Shen, C. R. Coggrave, G. H. Kaufmann, “Phase errors in low-frequency vibration measurement with high-speed phase-shifting speckle pattern interferometry,” Opt. Eng. 40, 1984–1992 (2001).
    [CrossRef]
  8. P. D. Ruiz, J. M. Huntley, Y. Shen, C. R. Coggrave, G. H. Kaufmann, “Effects of random vibration in high-speed phase-shifting speckle pattern interferometry,” Appl. Opt. 41, 3941–3949 (2002).
    [CrossRef] [PubMed]
  9. A. Davila, P. D. Ruiz, G. H. Kaufmann, J. M. Huntley, “Measurement of sub-surface delaminations in carbon fibre composites using high-speed phase-shifted speckle interferometry and temporal phase unwrapping,” Opt. Lasers Eng. (to be published).
  10. C. Joenathan, B. Franze, P. Haible, H. J. Tiziani, “Large in-plane displacement measurement in dual-beam speckle interferometry using temporal phase measurement,” J. Mod. Opt. 45, 1975–1984 (1998).
    [CrossRef]
  11. C. Joenathan, B. Franze, P. Haible, H. J. Tiziani, “Speckle interferometry with temporal phase evaluation for measuring large-object deformation,” Appl. Opt. 37, 2608–2614 (1998).
    [CrossRef]
  12. C. Joenathan, B. Franze, P. Haible, H. J. Tiziani, “Novel temporal Fourier transform speckle pattern shearing interferometer,” Opt. Eng. 37, 1790–1795 (1998).
    [CrossRef]
  13. C. Joenathan, P. Haible, H. J. Tiziani, “Speckle interferometry with temporal phase evaluation: influence of decorrelation, speckle size, and nonlinearity of the camera,” Appl. Opt. 38, 1169–1178 (1999).
    [CrossRef]
  14. P. Haible, M. P. Kothiyal, H. J. Tiziani, “Heterodyne temporal speckle-pattern interferometry,” Appl. Opt. 39, 114–117 (2000).
    [CrossRef]
  15. I. Yamaguchi, “Speckle displacement and decorrelation in the diffraction and image fields for small object deformations,” Opt. Acta 28, 1359–1376 (1981).
    [CrossRef]
  16. K. Creath, “Temporal phase measurement methods,” in Interferometry Analysis, D. W. Robinson, G. T. Reid, eds. (Institute of Physics Publishing, Bristol, UK, 1993), Chap. 4, pp. 94–140.
  17. M. Takeda, H. Ina, S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. 72, 156–160 (1982).
    [CrossRef]
  18. A. Davila, G. H. Kaufmann, D. Kerr, “Digital processing of ESPI addition fringes,” in Proceedings of Fringe’93, W. Juptner, W. Osten, eds. (Akademie Verlag, Berlin, 1993), pp. 339–346.
  19. A. Davila, G. H. Kaufmann, D. Kerr, “An evaluation of synthetic aperture radar noise reduction techniques for the smoothing of electronic speckle pattern interferometer fringes,” J. Mod. Opt. 42, 1795–1804 (1995).
    [CrossRef]
  20. G. H. Kaufmann, G. E. Galizzi, “Speckle noise reduction in television holography fringes using wavelet thresholding,” Opt. Eng. 35, 1–9 (1996).
    [CrossRef]
  21. A. Fernández, G. H. Kaufmann, A. F. Doval, J. Blanco-García, J. L. Fernández, “Comparison of carrier removal methods in the analysis of TV holography fringes by the Fourier transform method,” Opt. Eng. 37, 2899–2905 (1998).
    [CrossRef]
  22. MATLAB, a technical computer language, is a trademark of MathWorks, Inc., 3 Apple Hill Drive, Natick, Mass.

2002

2001

P. D. Ruiz, J. M. Huntley, Y. Shen, C. R. Coggrave, G. H. Kaufmann, “Vibration-induced phase errors in high-speed phase-shifting speckle pattern interferometry,” Appl. Opt. 40, 2117–2125 (2001).
[CrossRef]

P. D. Ruiz, J. M. Huntley, Y. Shen, C. R. Coggrave, G. H. Kaufmann, “Phase errors in low-frequency vibration measurement with high-speed phase-shifting speckle pattern interferometry,” Opt. Eng. 40, 1984–1992 (2001).
[CrossRef]

2000

1999

1998

A. Fernández, G. H. Kaufmann, A. F. Doval, J. Blanco-García, J. L. Fernández, “Comparison of carrier removal methods in the analysis of TV holography fringes by the Fourier transform method,” Opt. Eng. 37, 2899–2905 (1998).
[CrossRef]

C. Joenathan, B. Franze, P. Haible, H. J. Tiziani, “Speckle interferometry with temporal phase evaluation for measuring large-object deformation,” Appl. Opt. 37, 2608–2614 (1998).
[CrossRef]

C. Joenathan, B. Franze, P. Haible, H. J. Tiziani, “Large in-plane displacement measurement in dual-beam speckle interferometry using temporal phase measurement,” J. Mod. Opt. 45, 1975–1984 (1998).
[CrossRef]

C. Joenathan, B. Franze, P. Haible, H. J. Tiziani, “Novel temporal Fourier transform speckle pattern shearing interferometer,” Opt. Eng. 37, 1790–1795 (1998).
[CrossRef]

1996

G. H. Kaufmann, G. E. Galizzi, “Speckle noise reduction in television holography fringes using wavelet thresholding,” Opt. Eng. 35, 1–9 (1996).
[CrossRef]

1995

A. Davila, G. H. Kaufmann, D. Kerr, “An evaluation of synthetic aperture radar noise reduction techniques for the smoothing of electronic speckle pattern interferometer fringes,” J. Mod. Opt. 42, 1795–1804 (1995).
[CrossRef]

1993

1982

1981

I. Yamaguchi, “Speckle displacement and decorrelation in the diffraction and image fields for small object deformations,” Opt. Acta 28, 1359–1376 (1981).
[CrossRef]

Blanco-García, J.

A. Fernández, G. H. Kaufmann, A. F. Doval, J. Blanco-García, J. L. Fernández, “Comparison of carrier removal methods in the analysis of TV holography fringes by the Fourier transform method,” Opt. Eng. 37, 2899–2905 (1998).
[CrossRef]

Coggrave, C. R.

Creath, K.

K. Creath, “Temporal phase measurement methods,” in Interferometry Analysis, D. W. Robinson, G. T. Reid, eds. (Institute of Physics Publishing, Bristol, UK, 1993), Chap. 4, pp. 94–140.

Davila, A.

A. Davila, G. H. Kaufmann, D. Kerr, “An evaluation of synthetic aperture radar noise reduction techniques for the smoothing of electronic speckle pattern interferometer fringes,” J. Mod. Opt. 42, 1795–1804 (1995).
[CrossRef]

A. Davila, G. H. Kaufmann, D. Kerr, “Digital processing of ESPI addition fringes,” in Proceedings of Fringe’93, W. Juptner, W. Osten, eds. (Akademie Verlag, Berlin, 1993), pp. 339–346.

A. Davila, P. D. Ruiz, G. H. Kaufmann, J. M. Huntley, “Measurement of sub-surface delaminations in carbon fibre composites using high-speed phase-shifted speckle interferometry and temporal phase unwrapping,” Opt. Lasers Eng. (to be published).

Doval, A. F.

A. Fernández, G. H. Kaufmann, A. F. Doval, J. Blanco-García, J. L. Fernández, “Comparison of carrier removal methods in the analysis of TV holography fringes by the Fourier transform method,” Opt. Eng. 37, 2899–2905 (1998).
[CrossRef]

Fernández, A.

A. Fernández, G. H. Kaufmann, A. F. Doval, J. Blanco-García, J. L. Fernández, “Comparison of carrier removal methods in the analysis of TV holography fringes by the Fourier transform method,” Opt. Eng. 37, 2899–2905 (1998).
[CrossRef]

Fernández, J. L.

A. Fernández, G. H. Kaufmann, A. F. Doval, J. Blanco-García, J. L. Fernández, “Comparison of carrier removal methods in the analysis of TV holography fringes by the Fourier transform method,” Opt. Eng. 37, 2899–2905 (1998).
[CrossRef]

Franze, B.

C. Joenathan, B. Franze, P. Haible, H. J. Tiziani, “Novel temporal Fourier transform speckle pattern shearing interferometer,” Opt. Eng. 37, 1790–1795 (1998).
[CrossRef]

C. Joenathan, B. Franze, P. Haible, H. J. Tiziani, “Large in-plane displacement measurement in dual-beam speckle interferometry using temporal phase measurement,” J. Mod. Opt. 45, 1975–1984 (1998).
[CrossRef]

C. Joenathan, B. Franze, P. Haible, H. J. Tiziani, “Speckle interferometry with temporal phase evaluation for measuring large-object deformation,” Appl. Opt. 37, 2608–2614 (1998).
[CrossRef]

Galizzi, G. E.

G. H. Kaufmann, G. E. Galizzi, “Speckle noise reduction in television holography fringes using wavelet thresholding,” Opt. Eng. 35, 1–9 (1996).
[CrossRef]

Ghiglia, D. C.

D. C. Ghiglia, M. D. Pritt, Two-Dimensional Phase Unwrapping (Wiley, New York, 1998).

Haible, P.

Huntley, J. M.

P. D. Ruiz, J. M. Huntley, Y. Shen, C. R. Coggrave, G. H. Kaufmann, “Effects of random vibration in high-speed phase-shifting speckle pattern interferometry,” Appl. Opt. 41, 3941–3949 (2002).
[CrossRef] [PubMed]

P. D. Ruiz, J. M. Huntley, Y. Shen, C. R. Coggrave, G. H. Kaufmann, “Vibration-induced phase errors in high-speed phase-shifting speckle pattern interferometry,” Appl. Opt. 40, 2117–2125 (2001).
[CrossRef]

P. D. Ruiz, J. M. Huntley, Y. Shen, C. R. Coggrave, G. H. Kaufmann, “Phase errors in low-frequency vibration measurement with high-speed phase-shifting speckle pattern interferometry,” Opt. Eng. 40, 1984–1992 (2001).
[CrossRef]

J. M. Huntley, G. H. Kaufmann, D. Kerr, “Phase-shifted dynamic speckle pattern interferometry at 1 kHz,” Appl. Opt. 38, 6556–6563 (1999).
[CrossRef]

J. M. Huntley, H. Saldner, “Temporal phase-unwrapping algorithm for automated interferogram analysis,” Appl. Opt. 32, 3047–3052 (1993).
[CrossRef] [PubMed]

J. M. Huntley, “Automated analysis of speckle interferograms,” in Digital Speckle Pattern Interferometry and Related Techniques, P. K. Rastogi, ed. (Wiley, Chichester, England, 2001), Chap. 2, pp. 59–139.

A. Davila, P. D. Ruiz, G. H. Kaufmann, J. M. Huntley, “Measurement of sub-surface delaminations in carbon fibre composites using high-speed phase-shifted speckle interferometry and temporal phase unwrapping,” Opt. Lasers Eng. (to be published).

Ina, H.

Joenathan, C.

C. Joenathan, P. Haible, H. J. Tiziani, “Speckle interferometry with temporal phase evaluation: influence of decorrelation, speckle size, and nonlinearity of the camera,” Appl. Opt. 38, 1169–1178 (1999).
[CrossRef]

C. Joenathan, B. Franze, P. Haible, H. J. Tiziani, “Speckle interferometry with temporal phase evaluation for measuring large-object deformation,” Appl. Opt. 37, 2608–2614 (1998).
[CrossRef]

C. Joenathan, B. Franze, P. Haible, H. J. Tiziani, “Large in-plane displacement measurement in dual-beam speckle interferometry using temporal phase measurement,” J. Mod. Opt. 45, 1975–1984 (1998).
[CrossRef]

C. Joenathan, B. Franze, P. Haible, H. J. Tiziani, “Novel temporal Fourier transform speckle pattern shearing interferometer,” Opt. Eng. 37, 1790–1795 (1998).
[CrossRef]

Kaufmann, G. H.

P. D. Ruiz, J. M. Huntley, Y. Shen, C. R. Coggrave, G. H. Kaufmann, “Effects of random vibration in high-speed phase-shifting speckle pattern interferometry,” Appl. Opt. 41, 3941–3949 (2002).
[CrossRef] [PubMed]

P. D. Ruiz, J. M. Huntley, Y. Shen, C. R. Coggrave, G. H. Kaufmann, “Vibration-induced phase errors in high-speed phase-shifting speckle pattern interferometry,” Appl. Opt. 40, 2117–2125 (2001).
[CrossRef]

P. D. Ruiz, J. M. Huntley, Y. Shen, C. R. Coggrave, G. H. Kaufmann, “Phase errors in low-frequency vibration measurement with high-speed phase-shifting speckle pattern interferometry,” Opt. Eng. 40, 1984–1992 (2001).
[CrossRef]

J. M. Huntley, G. H. Kaufmann, D. Kerr, “Phase-shifted dynamic speckle pattern interferometry at 1 kHz,” Appl. Opt. 38, 6556–6563 (1999).
[CrossRef]

A. Fernández, G. H. Kaufmann, A. F. Doval, J. Blanco-García, J. L. Fernández, “Comparison of carrier removal methods in the analysis of TV holography fringes by the Fourier transform method,” Opt. Eng. 37, 2899–2905 (1998).
[CrossRef]

G. H. Kaufmann, G. E. Galizzi, “Speckle noise reduction in television holography fringes using wavelet thresholding,” Opt. Eng. 35, 1–9 (1996).
[CrossRef]

A. Davila, G. H. Kaufmann, D. Kerr, “An evaluation of synthetic aperture radar noise reduction techniques for the smoothing of electronic speckle pattern interferometer fringes,” J. Mod. Opt. 42, 1795–1804 (1995).
[CrossRef]

A. Davila, G. H. Kaufmann, D. Kerr, “Digital processing of ESPI addition fringes,” in Proceedings of Fringe’93, W. Juptner, W. Osten, eds. (Akademie Verlag, Berlin, 1993), pp. 339–346.

A. Davila, P. D. Ruiz, G. H. Kaufmann, J. M. Huntley, “Measurement of sub-surface delaminations in carbon fibre composites using high-speed phase-shifted speckle interferometry and temporal phase unwrapping,” Opt. Lasers Eng. (to be published).

Kerr, D.

J. M. Huntley, G. H. Kaufmann, D. Kerr, “Phase-shifted dynamic speckle pattern interferometry at 1 kHz,” Appl. Opt. 38, 6556–6563 (1999).
[CrossRef]

A. Davila, G. H. Kaufmann, D. Kerr, “An evaluation of synthetic aperture radar noise reduction techniques for the smoothing of electronic speckle pattern interferometer fringes,” J. Mod. Opt. 42, 1795–1804 (1995).
[CrossRef]

A. Davila, G. H. Kaufmann, D. Kerr, “Digital processing of ESPI addition fringes,” in Proceedings of Fringe’93, W. Juptner, W. Osten, eds. (Akademie Verlag, Berlin, 1993), pp. 339–346.

Kobayashi, S.

Kothiyal, M. P.

Pritt, M. D.

D. C. Ghiglia, M. D. Pritt, Two-Dimensional Phase Unwrapping (Wiley, New York, 1998).

Ruiz, P. D.

P. D. Ruiz, J. M. Huntley, Y. Shen, C. R. Coggrave, G. H. Kaufmann, “Effects of random vibration in high-speed phase-shifting speckle pattern interferometry,” Appl. Opt. 41, 3941–3949 (2002).
[CrossRef] [PubMed]

P. D. Ruiz, J. M. Huntley, Y. Shen, C. R. Coggrave, G. H. Kaufmann, “Vibration-induced phase errors in high-speed phase-shifting speckle pattern interferometry,” Appl. Opt. 40, 2117–2125 (2001).
[CrossRef]

P. D. Ruiz, J. M. Huntley, Y. Shen, C. R. Coggrave, G. H. Kaufmann, “Phase errors in low-frequency vibration measurement with high-speed phase-shifting speckle pattern interferometry,” Opt. Eng. 40, 1984–1992 (2001).
[CrossRef]

A. Davila, P. D. Ruiz, G. H. Kaufmann, J. M. Huntley, “Measurement of sub-surface delaminations in carbon fibre composites using high-speed phase-shifted speckle interferometry and temporal phase unwrapping,” Opt. Lasers Eng. (to be published).

Saldner, H.

Shen, Y.

Sirohi, R. S.

R. S. Sirohi, Speckle Metrology (Marcel Dekker, New York, 1993).

Takeda, M.

Tiziani, H. J.

Yamaguchi, I.

I. Yamaguchi, “Speckle displacement and decorrelation in the diffraction and image fields for small object deformations,” Opt. Acta 28, 1359–1376 (1981).
[CrossRef]

Appl. Opt.

J. Mod. Opt.

A. Davila, G. H. Kaufmann, D. Kerr, “An evaluation of synthetic aperture radar noise reduction techniques for the smoothing of electronic speckle pattern interferometer fringes,” J. Mod. Opt. 42, 1795–1804 (1995).
[CrossRef]

C. Joenathan, B. Franze, P. Haible, H. J. Tiziani, “Large in-plane displacement measurement in dual-beam speckle interferometry using temporal phase measurement,” J. Mod. Opt. 45, 1975–1984 (1998).
[CrossRef]

J. Opt. Soc. Am.

Opt. Acta

I. Yamaguchi, “Speckle displacement and decorrelation in the diffraction and image fields for small object deformations,” Opt. Acta 28, 1359–1376 (1981).
[CrossRef]

Opt. Eng.

C. Joenathan, B. Franze, P. Haible, H. J. Tiziani, “Novel temporal Fourier transform speckle pattern shearing interferometer,” Opt. Eng. 37, 1790–1795 (1998).
[CrossRef]

P. D. Ruiz, J. M. Huntley, Y. Shen, C. R. Coggrave, G. H. Kaufmann, “Phase errors in low-frequency vibration measurement with high-speed phase-shifting speckle pattern interferometry,” Opt. Eng. 40, 1984–1992 (2001).
[CrossRef]

G. H. Kaufmann, G. E. Galizzi, “Speckle noise reduction in television holography fringes using wavelet thresholding,” Opt. Eng. 35, 1–9 (1996).
[CrossRef]

A. Fernández, G. H. Kaufmann, A. F. Doval, J. Blanco-García, J. L. Fernández, “Comparison of carrier removal methods in the analysis of TV holography fringes by the Fourier transform method,” Opt. Eng. 37, 2899–2905 (1998).
[CrossRef]

Other

MATLAB, a technical computer language, is a trademark of MathWorks, Inc., 3 Apple Hill Drive, Natick, Mass.

A. Davila, P. D. Ruiz, G. H. Kaufmann, J. M. Huntley, “Measurement of sub-surface delaminations in carbon fibre composites using high-speed phase-shifted speckle interferometry and temporal phase unwrapping,” Opt. Lasers Eng. (to be published).

R. S. Sirohi, Speckle Metrology (Marcel Dekker, New York, 1993).

J. M. Huntley, “Automated analysis of speckle interferograms,” in Digital Speckle Pattern Interferometry and Related Techniques, P. K. Rastogi, ed. (Wiley, Chichester, England, 2001), Chap. 2, pp. 59–139.

D. C. Ghiglia, M. D. Pritt, Two-Dimensional Phase Unwrapping (Wiley, New York, 1998).

K. Creath, “Temporal phase measurement methods,” in Interferometry Analysis, D. W. Robinson, G. T. Reid, eds. (Institute of Physics Publishing, Bristol, UK, 1993), Chap. 4, pp. 94–140.

A. Davila, G. H. Kaufmann, D. Kerr, “Digital processing of ESPI addition fringes,” in Proceedings of Fringe’93, W. Juptner, W. Osten, eds. (Akademie Verlag, Berlin, 1993), pp. 339–346.

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Figures (7)

Fig. 1
Fig. 1

Computer simulation generated for k = 0.5 for the first speckle interferogram of the sequence.

Fig. 2
Fig. 2

Computer simulation generated for k = 0.5 for the intensity variation at one pixel as a function of time with only 128 frames shown for clarity.

Fig. 3
Fig. 3

Results obtained when a sequence of consecutive tilts generated for k = 0.5 was analyzed by means of the four-frame phase-shifting algorithm for (a) the wrapped phase change Δϕ w (508, 0) and (b) the unwrapped phase change Δϕ u (508, 0).

Fig. 4
Fig. 4

Profiles of the unwrapped phase change Δϕ u (508, 0) that was obtained along a horizontal line that crosses the center of the image without rereferencing the temporal unwrapping algorithm (τ r = ∞) and by rereferencing the algorithm every 100 frames (τ r = 100).

Fig. 5
Fig. 5

Results obtained when a sequence of consecutive tilts generated for k = 0.5 was analyzed by means of the temporal Fourier transform (FT) method: (a) Fourier spectrum of the temporal intensity variation at one pixel, (b) wrapped phase ϕ w (t) as a function of time with only 128 frames shown for clarity, (c) profile of the unwrapped phase change Δϕ u (508, 0) along a horizontal line that crosses the center of the image.

Fig. 6
Fig. 6

Results obtained from a high-speed speckle interferometry study of a carbon fiber panel with an internal delamination for the wrapped phase change Δϕ w (507, 0) obtained with the five-frame phase-shifting algorithm.

Fig. 7
Fig. 7

Profiles of the unwrapped phase change Δϕ u (507, 0) obtained along a horizontal line above the delamination by use of the five-frame phase-shifting algorithm with τ r = 60 and the temporal Fourier transform (FT) method.

Tables (1)

Tables Icon

Table 1 Rms Phase Error σ Obtained for Two Phase-Shifting Algorithms and the Temporal Fourier Transform Method with a Sequence of Consecutive Tilts at Different Speeds

Equations (19)

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

Im, n, t=I0m, n, t+IMm, n, t×cosϕm, n, t+ψt,
Δϕwt2, t1=arctanNt2Dt1-Dt2Nt1Dt2Dt1+Nt2Nt1,
Nt=Imzt, Dt=Rezt,
zt=t=0M-1atIt+t+i t=0M-1btIt+t×exp-iπt/2,
at+ibt=wtexp-iπt/2, t=0, 1, 2,,M-1,
Ĩkt, t=t=0M-1It+twtexp-2πiktt/N.
dt=NINTΔϕwt, 0-Δϕwt-1, 0/2π, t=2, 3,Nt-M,
vt=t=2tdt, t=2, 3,Nt-M, v1=0,
Δϕut, 0=Δϕwt, 0-2πvt, t=1, 2,Nt-M.
Δϕut, 0=Δϕut, tκ+k=2κΔϕutk, tk-1+Δϕut1, 0.
It=I0t+IMt2expiϕt+ψt+IMt2exp-iϕt+ψt.
IFt=IMt2expiϕt+ψt.
ϕwt+ψt=arctanImIFtReIFt.
Im, n, t=|R expiα+-1Hu, vexpiϕ0m, n, t+iθtUm, n|2,
θt=π2REMt+3, 4,
s=Nm/2r.
ϕ0m, n, t=kπ2mNm-1t,
σ=1LL-1m=0Nm-1n=0Nn-1|Δϕum, n, Nt-M, 0-Δϕ0m, n, Nt-M, 0|-ΔϕuNt-M, 021/2,
ΔϕuNt-M, 0=1Lm=0Nm-1n=0Nn-1|Δϕum, n, Nt-M, 0-Δϕ0m, n, Nt-M, 0|,

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