Abstract

A novel temporal phase-analysis technique that is based on wavelet analysis and a temporal carrier is presented. To measure displacement on a vibrating object by using electronic speckle pattern interferometry, one captures a series of speckle patterns, using a high-speed CCD camera. To avoid ambiguity in phase estimation, a temporal carrier is generated by a piezoelectric transducer stage in the reference beam of the interferometer. The intensity variation of each pixel on recorded images is then analyzed along the time axis by a robust mathematical tool, i.e., a complex Morlet wavelet transform. After the temporal carrier is removed, the absolute displacement of a vibrating object is obtained without the need for temporal or spatial phase unwrapping. The results obtained by a wavelet transform are compared with those from a temporal Fourier transform.

© 2005 Optical Society of America

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References

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  1. J. M. Huntley, G. H. Kaufmann, D. Kerr, “Phase-shifted dynamic speckle pattern interferometry at 1 kHz,” Appl. Opt. 38, 6556–6563 (1999).
    [CrossRef]
  2. D. C. Ghilia, M. D. Pritt, Two-Dimensional Phase Unwrapping (Wiley, New York, 1998).
  3. H. J. Tiziani, “Spectral and temporal phase evaluation for interferometry and speckle applications,” in Trends in Optical Nondestructive Testing and Inspection, P. K. Rastogi, D. Inaudi, eds. (Elsevier, Amsterdam, 2000), pp. 323–343.
  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, “Challenges in phase unwrapping,” in Trends in Optical Nondestructive Testing and Inspection, P. K. Rastogi, D. Inaudi, eds., (Elsevier, Amsterdam, 2000), pp. 37–44.
  6. G. H. Kaufmann, G. E. Galizzi, “Phase measurement in temporal speckle pattern interferometry: comparison between the phase-shifting and the Fourier transform methods,” Appl. Opt. 41, 7254–7263 (2002).
    [CrossRef] [PubMed]
  7. X. Colonna de Lega, P. Jacquot, “Interferometric deformation measurement using object induced dynamic phase shifting,” in Optical Inspection and Micromeasurements, C. Goreuki, ed., Proc. SPIE2782, 169–179 (1996).
    [CrossRef]
  8. X. Colonna de Lega, P. Jacquot, “Deformation measurement with object-induced dynamic phase shifting,” Appl. Opt. 35, 5115–5121 (1996).
    [CrossRef] [PubMed]
  9. C. Joenathan, B. Franze, P. Harble, H. J. Tiziani, “Speckle interferometry with temporal phase evaluation for measuring large-object deformation,” Appl. Opt. 37, 2608–2614 (1998).
    [CrossRef]
  10. C. Joenathan, B. Franze, P. Harble, 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. Harble, H. J. Tiziani, “Novel temporal Fourier transform speckle pattern shearing interferometer,” Opt. Eng. 37, 1790–1795 (1998).
    [CrossRef]
  12. X. Li, G. Tao, Y. Yang, “Continual deformation analysis with scanning phase method and time sequence phase method in temporal speckle pattern interferomtry,” Opt. Laser Technol. 33, 53–59 (2001).
    [CrossRef]
  13. 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]
  14. G. H. Kaufmann, “Phase measurement in temporal speckle pattern interferometry using the Fouier transform method with and without a temporal carrier,” Opt. Commmun. 217, 141–149 (2003).
    [CrossRef]
  15. I. Daubechies, Ten Lectures on Wavelets (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1992).
    [CrossRef]
  16. X. Colonna de Lega, “Continuous deformation measurement using dynamic phase-shifting and wavelet transform,” in Applied Optics and Optoeletronics 1996, K. T. V. Grattan, ed., (Institute of Physics Publishing, Bristol, U.K., 1996), pp. 261–267.
  17. X. Colonna de Lega, “Processing of non-stationary interference patterns: adapted phase shifting algorithms and wavelet analysis. Application to dynamic deformation measurements by holographic and speckle interferometry,” Thesis 1666, (Swiss Federal Institute of Technology, Lausanne, Switzerland, 1997).
  18. M. Cherbuliez, P. Jacquot, “Phase computation through wavelet analysis: yesterday and nowadays,” in Fringe 2001, W. Osten, W. Juptner, eds. (Elsevier, Paris, 2001), pp. 154–162.
  19. M. Cherbuliez, “Wavelet analysis of interference patterns and signals: development of fast and efficient processing techniques,” Thesis 2377 (Swiss Federal Institute of Technology, Lausanne, Switzerland, (2001).
  20. C. J. Tay, C. Quan, Y. Fu, Y. Huang, “Instantaneous velocity displacement and contour measurement by use of shadow moire and temporal wavelet analysis,” Appl. Opt. 43, 4164–4171 (2004).
    [CrossRef] [PubMed]
  21. Y. Fu, C. J. Tay, C. Quan, L. J. Chen, “Temporal wavelet analysis for deformation and velocity measurement in speckle interferometry,” Opt. Eng. 43, 2780–2787 (2004).
    [CrossRef]
  22. C. A. Sciammarella, T. Kim, “Determination of strains from fringe patterns using space-frequency representations,” Opt. Eng. 42, 3182–3193 (2003).
    [CrossRef]
  23. W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in fortran, 2nd ed. (Cambridge U. Press, Cambridge, 1992), Chap. 13.
  24. S. Mallat, A Wavelet Tour of Signal Processing (Academic, San Diego, Calif., 1998).

2004 (2)

Y. Fu, C. J. Tay, C. Quan, L. J. Chen, “Temporal wavelet analysis for deformation and velocity measurement in speckle interferometry,” Opt. Eng. 43, 2780–2787 (2004).
[CrossRef]

C. J. Tay, C. Quan, Y. Fu, Y. Huang, “Instantaneous velocity displacement and contour measurement by use of shadow moire and temporal wavelet analysis,” Appl. Opt. 43, 4164–4171 (2004).
[CrossRef] [PubMed]

2003 (2)

G. H. Kaufmann, “Phase measurement in temporal speckle pattern interferometry using the Fouier transform method with and without a temporal carrier,” Opt. Commmun. 217, 141–149 (2003).
[CrossRef]

C. A. Sciammarella, T. Kim, “Determination of strains from fringe patterns using space-frequency representations,” Opt. Eng. 42, 3182–3193 (2003).
[CrossRef]

2002 (1)

2001 (1)

X. Li, G. Tao, Y. Yang, “Continual deformation analysis with scanning phase method and time sequence phase method in temporal speckle pattern interferomtry,” Opt. Laser Technol. 33, 53–59 (2001).
[CrossRef]

1999 (1)

1998 (3)

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

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

1996 (1)

1993 (1)

1982 (1)

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]

Chen, L. J.

Y. Fu, C. J. Tay, C. Quan, L. J. Chen, “Temporal wavelet analysis for deformation and velocity measurement in speckle interferometry,” Opt. Eng. 43, 2780–2787 (2004).
[CrossRef]

Cherbuliez, M.

M. Cherbuliez, P. Jacquot, “Phase computation through wavelet analysis: yesterday and nowadays,” in Fringe 2001, W. Osten, W. Juptner, eds. (Elsevier, Paris, 2001), pp. 154–162.

M. Cherbuliez, “Wavelet analysis of interference patterns and signals: development of fast and efficient processing techniques,” Thesis 2377 (Swiss Federal Institute of Technology, Lausanne, Switzerland, (2001).

Colonna de Lega, X.

X. Colonna de Lega, P. Jacquot, “Deformation measurement with object-induced dynamic phase shifting,” Appl. Opt. 35, 5115–5121 (1996).
[CrossRef] [PubMed]

X. Colonna de Lega, P. Jacquot, “Interferometric deformation measurement using object induced dynamic phase shifting,” in Optical Inspection and Micromeasurements, C. Goreuki, ed., Proc. SPIE2782, 169–179 (1996).
[CrossRef]

X. Colonna de Lega, “Continuous deformation measurement using dynamic phase-shifting and wavelet transform,” in Applied Optics and Optoeletronics 1996, K. T. V. Grattan, ed., (Institute of Physics Publishing, Bristol, U.K., 1996), pp. 261–267.

X. Colonna de Lega, “Processing of non-stationary interference patterns: adapted phase shifting algorithms and wavelet analysis. Application to dynamic deformation measurements by holographic and speckle interferometry,” Thesis 1666, (Swiss Federal Institute of Technology, Lausanne, Switzerland, 1997).

Daubechies, I.

I. Daubechies, Ten Lectures on Wavelets (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1992).
[CrossRef]

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in fortran, 2nd ed. (Cambridge U. Press, Cambridge, 1992), Chap. 13.

Franze, B.

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

C. Joenathan, B. Franze, P. Harble, 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. Harble, 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]

Fu, Y.

C. J. Tay, C. Quan, Y. Fu, Y. Huang, “Instantaneous velocity displacement and contour measurement by use of shadow moire and temporal wavelet analysis,” Appl. Opt. 43, 4164–4171 (2004).
[CrossRef] [PubMed]

Y. Fu, C. J. Tay, C. Quan, L. J. Chen, “Temporal wavelet analysis for deformation and velocity measurement in speckle interferometry,” Opt. Eng. 43, 2780–2787 (2004).
[CrossRef]

Galizzi, G. E.

Ghilia, D. C.

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

Harble, P.

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

C. Joenathan, B. Franze, P. Harble, 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]

Huang, Y.

Huntley, J. M.

Ina, H.

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]

Jacquot, P.

X. Colonna de Lega, P. Jacquot, “Deformation measurement with object-induced dynamic phase shifting,” Appl. Opt. 35, 5115–5121 (1996).
[CrossRef] [PubMed]

X. Colonna de Lega, P. Jacquot, “Interferometric deformation measurement using object induced dynamic phase shifting,” in Optical Inspection and Micromeasurements, C. Goreuki, ed., Proc. SPIE2782, 169–179 (1996).
[CrossRef]

M. Cherbuliez, P. Jacquot, “Phase computation through wavelet analysis: yesterday and nowadays,” in Fringe 2001, W. Osten, W. Juptner, eds. (Elsevier, Paris, 2001), pp. 154–162.

Joenathan, C.

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

Kaufmann, G. H.

Kerr, D.

Kim, T.

C. A. Sciammarella, T. Kim, “Determination of strains from fringe patterns using space-frequency representations,” Opt. Eng. 42, 3182–3193 (2003).
[CrossRef]

Kobayashi, S.

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]

Li, X.

X. Li, G. Tao, Y. Yang, “Continual deformation analysis with scanning phase method and time sequence phase method in temporal speckle pattern interferomtry,” Opt. Laser Technol. 33, 53–59 (2001).
[CrossRef]

Mallat, S.

S. Mallat, A Wavelet Tour of Signal Processing (Academic, San Diego, Calif., 1998).

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in fortran, 2nd ed. (Cambridge U. Press, Cambridge, 1992), Chap. 13.

Pritt, M. D.

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

Quan, C.

C. J. Tay, C. Quan, Y. Fu, Y. Huang, “Instantaneous velocity displacement and contour measurement by use of shadow moire and temporal wavelet analysis,” Appl. Opt. 43, 4164–4171 (2004).
[CrossRef] [PubMed]

Y. Fu, C. J. Tay, C. Quan, L. J. Chen, “Temporal wavelet analysis for deformation and velocity measurement in speckle interferometry,” Opt. Eng. 43, 2780–2787 (2004).
[CrossRef]

Saldner, H.

Sciammarella, C. A.

C. A. Sciammarella, T. Kim, “Determination of strains from fringe patterns using space-frequency representations,” Opt. Eng. 42, 3182–3193 (2003).
[CrossRef]

Takeda, M.

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]

Tao, G.

X. Li, G. Tao, Y. Yang, “Continual deformation analysis with scanning phase method and time sequence phase method in temporal speckle pattern interferomtry,” Opt. Laser Technol. 33, 53–59 (2001).
[CrossRef]

Tay, C. J.

C. J. Tay, C. Quan, Y. Fu, Y. Huang, “Instantaneous velocity displacement and contour measurement by use of shadow moire and temporal wavelet analysis,” Appl. Opt. 43, 4164–4171 (2004).
[CrossRef] [PubMed]

Y. Fu, C. J. Tay, C. Quan, L. J. Chen, “Temporal wavelet analysis for deformation and velocity measurement in speckle interferometry,” Opt. Eng. 43, 2780–2787 (2004).
[CrossRef]

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in fortran, 2nd ed. (Cambridge U. Press, Cambridge, 1992), Chap. 13.

Tiziani, H. J.

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

C. Joenathan, B. Franze, P. Harble, 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. Harble, 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]

H. J. Tiziani, “Spectral and temporal phase evaluation for interferometry and speckle applications,” in Trends in Optical Nondestructive Testing and Inspection, P. K. Rastogi, D. Inaudi, eds. (Elsevier, Amsterdam, 2000), pp. 323–343.

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in fortran, 2nd ed. (Cambridge U. Press, Cambridge, 1992), Chap. 13.

Yang, Y.

X. Li, G. Tao, Y. Yang, “Continual deformation analysis with scanning phase method and time sequence phase method in temporal speckle pattern interferomtry,” Opt. Laser Technol. 33, 53–59 (2001).
[CrossRef]

Appl. Opt. (6)

J Opt. Soc. Am. (1)

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]

J. Mod. Opt. (1)

C. Joenathan, B. Franze, P. Harble, 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]

Opt. Commmun. (1)

G. H. Kaufmann, “Phase measurement in temporal speckle pattern interferometry using the Fouier transform method with and without a temporal carrier,” Opt. Commmun. 217, 141–149 (2003).
[CrossRef]

Opt. Eng. (3)

Y. Fu, C. J. Tay, C. Quan, L. J. Chen, “Temporal wavelet analysis for deformation and velocity measurement in speckle interferometry,” Opt. Eng. 43, 2780–2787 (2004).
[CrossRef]

C. A. Sciammarella, T. Kim, “Determination of strains from fringe patterns using space-frequency representations,” Opt. Eng. 42, 3182–3193 (2003).
[CrossRef]

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

Opt. Laser Technol. (1)

X. Li, G. Tao, Y. Yang, “Continual deformation analysis with scanning phase method and time sequence phase method in temporal speckle pattern interferomtry,” Opt. Laser Technol. 33, 53–59 (2001).
[CrossRef]

Other (11)

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in fortran, 2nd ed. (Cambridge U. Press, Cambridge, 1992), Chap. 13.

S. Mallat, A Wavelet Tour of Signal Processing (Academic, San Diego, Calif., 1998).

I. Daubechies, Ten Lectures on Wavelets (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1992).
[CrossRef]

X. Colonna de Lega, “Continuous deformation measurement using dynamic phase-shifting and wavelet transform,” in Applied Optics and Optoeletronics 1996, K. T. V. Grattan, ed., (Institute of Physics Publishing, Bristol, U.K., 1996), pp. 261–267.

X. Colonna de Lega, “Processing of non-stationary interference patterns: adapted phase shifting algorithms and wavelet analysis. Application to dynamic deformation measurements by holographic and speckle interferometry,” Thesis 1666, (Swiss Federal Institute of Technology, Lausanne, Switzerland, 1997).

M. Cherbuliez, P. Jacquot, “Phase computation through wavelet analysis: yesterday and nowadays,” in Fringe 2001, W. Osten, W. Juptner, eds. (Elsevier, Paris, 2001), pp. 154–162.

M. Cherbuliez, “Wavelet analysis of interference patterns and signals: development of fast and efficient processing techniques,” Thesis 2377 (Swiss Federal Institute of Technology, Lausanne, Switzerland, (2001).

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

H. J. Tiziani, “Spectral and temporal phase evaluation for interferometry and speckle applications,” in Trends in Optical Nondestructive Testing and Inspection, P. K. Rastogi, D. Inaudi, eds. (Elsevier, Amsterdam, 2000), pp. 323–343.

J. M. Huntley, “Challenges in phase unwrapping,” in Trends in Optical Nondestructive Testing and Inspection, P. K. Rastogi, D. Inaudi, eds., (Elsevier, Amsterdam, 2000), pp. 37–44.

X. Colonna de Lega, P. Jacquot, “Interferometric deformation measurement using object induced dynamic phase shifting,” in Optical Inspection and Micromeasurements, C. Goreuki, ed., Proc. SPIE2782, 169–179 (1996).
[CrossRef]

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Figures (8)

Fig. 1
Fig. 1

Schematic layout of the ESPI system.

Fig. 2
Fig. 2

(a) Reference block, cantilevered beam, and loading device. (b) Typical speckle pattern on a reference block and a cantilever beam and area of interest.

Fig. 3
Fig. 3

(a) Temporal intensity variation of points R on the reference block. (b) Plot of the modulus of the Morlet wavelet transform at point R. (c) The average ridge detected on the reference block.

Fig. 4
Fig. 4

(a) Temporal intensity variation of points A on the cantilever beam. (b) Plot of the modulus of the Morlet wavelet transform at point A.

Fig. 5
Fig. 5

(a) Temporal intensity variations of points B on a cantilever beam. (b) Plot of the modulus of Morlet wavelet transform at point B.

Fig. 6
Fig. 6

(a) Phase variation retrieved on the reference block and at point B. (b) Out-of-plane displacement obtained at point B.

Fig. 7
Fig. 7

Displacement distribution on cross section C–C at different time intervals obtained by (a) temporal wavelet transform and (b) temporal Fourier transform.

Fig. 8
Fig. 8

Displacement distribution T1T0 on a cantilever beam obtained by (a) a temporal wavelet transform and (b) a temporal Fourier transform.

Equations (10)

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

I x y ( t ) = I 0 x y ( t ) + A x y ( t ) cos [ φ x y ( t ) ] = I 0 x y ( t ) + A x y ( t ) cos [ ϕ c ( t ) + ϕ x y ( t ) ] = I 0 x y ( t ) { 1 + V cos [ 2 π f c t + ϕ 0 x y + 4 π z x y ( t ) λ ] } ,
W s ( a ,     b ) = - + s ( t ) ψ a ,     b * ( t ) d t ,
ψ a ,     b ( t ) = 1 a ψ ( t - b a ) ,             a ,             b R ,             a > 0.
s ( t ) = 1 C Ψ - + 0 + W s ( a ,     b ) ψ ( t - b a ) d a a 2 d b ,
C Ψ = - + ψ ^ ( ω ) 2 ω d ω < +
ψ ( t ) = g ( t ) exp ( i ω 0 t ) ,             g ( t ) = exp ( - t 2 2 ) .
W x y ( a ,     b ) = a 2 A x y ( b ) ( g ^ { a [ ζ - φ x y ( b ) ] } + ɛ ( b ,     ω 0 a ) ) { exp [ i φ x y ( b ) ] } ,
φ x y ( b ) = ζ r b = ω 0 / a r b ,
W x y ( a r b ,     b ) a r b 2 A x y ( b ) g ^ ( 0 ) exp [ i φ x y ( b ) ] .
Δ φ = φ x y T 2 - φ x y T 1 = 2 π f c ( T 2 - T 1 ) + 4 π z ( T 2 - T 1 ) λ ,

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