Abstract

A modified Fourier transform method for interferogram fringe pattern analysis is proposed. While it retains most of the advantages of the Fourier transform method, the new method overcomes some drawbacks of the previous method. It eliminates the assumptions of slowly varying phase variation in the test section and the constant spatial carrier frequency. It also extends the frequency bandwidth and avoids phase distortion caused by discreteness of the sampling frequency. Both numerical simulation and experimental examination are performed to evaluate the performance of the method.

© 1997 Optical Society of America

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References

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  1. D. Malacara, “Optical testing,” in Handbook of Optics, M. Bass, E. W. Van Stryland, D. R. Williams, W. L. Wolfe, eds. (McGraw–Hill, New York, 1995), Chap. 3.
  2. M. Takeda, I. Hideki, S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. 72, 156–160 (1982).
    [CrossRef]
  3. C. Roddier, F. Roddier, “Interferogram analysis using Fourier transform techniques,” Appl. Opt. 26, 1668–1673 (1987).
    [CrossRef] [PubMed]
  4. C. Roddier, “Enregistrement holographique d’images astronomiques degradees par la turbulence atmospherique,” (1979).
  5. F. Roddier, C. Roddier, “Imaging with a multi-mirror telescope,” in Proceedings of the ESO Conference on Optical Telescopes of the Future, F. Pacini, W. Richter, R. N. Wilson, eds. (CERN for the European Southern Observatory, Geneva, 1978).
  6. C. Roddier, “Rotation shearing interferometry,” in Proceedings of the IAU Colloquium 50 on High Angular Resolution Stellar Interferometry, J. Davis, W. J. Tango, eds. (Chatterton Astronomy Dept., U. Sydney, Australia, 1979), p. 32.
  7. C. Roddier, F. Roddier, “High angular resolution observation of Alpha Orionis with a rotation shearing interferometer,” Astrophys. J. 270, L23–L26 (1983).
    [CrossRef]
  8. F. Roddier, C. Roddier, “An image reconstruction of Alpha Orionis,” Astrophy. J. 295, L21–L23 (1985).
    [CrossRef]
  9. K. A. Nugent, “Interferogram analysis using an accurate fully automatic algorithm,” Appl. Opt. 24, 3101–3105 (1985).
    [CrossRef] [PubMed]
  10. W. W. Macy, “Two-dimensional fringe-pattern analysis,” Appl. Opt. 22, 3898–3901 (1983).
    [CrossRef] [PubMed]
  11. K. H. Womack, “Frequency domain description of interferogram analysis,” Opt. Eng. 23, 396–400 (1984).
    [CrossRef]
  12. M. Kujawinska, “Spatial phase measurement methods,” in Interferogram Analysis, Digital Fringe Pattern Measurement Techniques, D. W. Robinson, G. D. Reid, eds. (Institute of Physics, Bristol, 1993), pp. 141–193.
  13. M. Kujawinska, J. Wojciak, “Spatial phase-shifting techniques of fringe pattern analysis in photomechanics,” in Moire Techniques, Holographic Interferometry, Optical NDT and Applications to Fluid Mechanics, Proc. SPIE1554B, 503–513 (1991).
  14. M. Pirga, M. Kujawinska, “Two directional spatial-carrier phase-shifting method for analysis of crossed and closed fringe patterns,” Opt. Eng. 34, 2459–2466 (1995).
    [CrossRef]
  15. J. Li, X. Y. Su, L. R. Guo, “Improved Fourier transform profilometry for the automatic measurement of three-dimensional object shapes,” Opt. Eng. 29, 1439–1444 (1990).
    [CrossRef]
  16. F. J. Weinberg, Optics of Flames (Butterworths, London, 1963), Chap. 4, pp. 116–149.
  17. S. M. Pandit, N. Jordache, “Data-dependent-systems and Fourier-transform methods for single-interferogram analysis,” Appl. Opt. 34, 5945–5951 (1995).
    [CrossRef] [PubMed]
  18. J. Schmit, K. Creath, M. Kujawinska, “Spatial and temporal phase-measurement techniques: a comparison of major error sources in one dimension,” in Interferometry: Techniques and Analysis, G. M. Brown, M. Kujawinska, O. Y. Kwon, G. T. Reid, eds., Proc. SPIE1755, 202–211 (1992).
  19. M. P. Rimmer, “Methods for evaluating lateral shearing interferograms,” Appl. Opt. 13, 623–629 (1974).
    [CrossRef] [PubMed]
  20. D. J. Bone, H. A. Bachor, R. J. Sandeman, “Fringe-pattern analysis using a 2-D Fourier transform,” Appl. Opt. 25, 1653–1660 (1986).
    [CrossRef] [PubMed]

1995

M. Pirga, M. Kujawinska, “Two directional spatial-carrier phase-shifting method for analysis of crossed and closed fringe patterns,” Opt. Eng. 34, 2459–2466 (1995).
[CrossRef]

S. M. Pandit, N. Jordache, “Data-dependent-systems and Fourier-transform methods for single-interferogram analysis,” Appl. Opt. 34, 5945–5951 (1995).
[CrossRef] [PubMed]

1990

J. Li, X. Y. Su, L. R. Guo, “Improved Fourier transform profilometry for the automatic measurement of three-dimensional object shapes,” Opt. Eng. 29, 1439–1444 (1990).
[CrossRef]

1987

1986

1985

F. Roddier, C. Roddier, “An image reconstruction of Alpha Orionis,” Astrophy. J. 295, L21–L23 (1985).
[CrossRef]

K. A. Nugent, “Interferogram analysis using an accurate fully automatic algorithm,” Appl. Opt. 24, 3101–3105 (1985).
[CrossRef] [PubMed]

1984

K. H. Womack, “Frequency domain description of interferogram analysis,” Opt. Eng. 23, 396–400 (1984).
[CrossRef]

1983

W. W. Macy, “Two-dimensional fringe-pattern analysis,” Appl. Opt. 22, 3898–3901 (1983).
[CrossRef] [PubMed]

C. Roddier, F. Roddier, “High angular resolution observation of Alpha Orionis with a rotation shearing interferometer,” Astrophys. J. 270, L23–L26 (1983).
[CrossRef]

1982

1974

Bachor, H. A.

Bone, D. J.

Creath, K.

J. Schmit, K. Creath, M. Kujawinska, “Spatial and temporal phase-measurement techniques: a comparison of major error sources in one dimension,” in Interferometry: Techniques and Analysis, G. M. Brown, M. Kujawinska, O. Y. Kwon, G. T. Reid, eds., Proc. SPIE1755, 202–211 (1992).

Guo, L. R.

J. Li, X. Y. Su, L. R. Guo, “Improved Fourier transform profilometry for the automatic measurement of three-dimensional object shapes,” Opt. Eng. 29, 1439–1444 (1990).
[CrossRef]

Hideki, I.

Jordache, N.

Kobayashi, S.

Kujawinska, M.

M. Pirga, M. Kujawinska, “Two directional spatial-carrier phase-shifting method for analysis of crossed and closed fringe patterns,” Opt. Eng. 34, 2459–2466 (1995).
[CrossRef]

M. Kujawinska, “Spatial phase measurement methods,” in Interferogram Analysis, Digital Fringe Pattern Measurement Techniques, D. W. Robinson, G. D. Reid, eds. (Institute of Physics, Bristol, 1993), pp. 141–193.

M. Kujawinska, J. Wojciak, “Spatial phase-shifting techniques of fringe pattern analysis in photomechanics,” in Moire Techniques, Holographic Interferometry, Optical NDT and Applications to Fluid Mechanics, Proc. SPIE1554B, 503–513 (1991).

J. Schmit, K. Creath, M. Kujawinska, “Spatial and temporal phase-measurement techniques: a comparison of major error sources in one dimension,” in Interferometry: Techniques and Analysis, G. M. Brown, M. Kujawinska, O. Y. Kwon, G. T. Reid, eds., Proc. SPIE1755, 202–211 (1992).

Li, J.

J. Li, X. Y. Su, L. R. Guo, “Improved Fourier transform profilometry for the automatic measurement of three-dimensional object shapes,” Opt. Eng. 29, 1439–1444 (1990).
[CrossRef]

Macy, W. W.

Malacara, D.

D. Malacara, “Optical testing,” in Handbook of Optics, M. Bass, E. W. Van Stryland, D. R. Williams, W. L. Wolfe, eds. (McGraw–Hill, New York, 1995), Chap. 3.

Nugent, K. A.

Pandit, S. M.

Pirga, M.

M. Pirga, M. Kujawinska, “Two directional spatial-carrier phase-shifting method for analysis of crossed and closed fringe patterns,” Opt. Eng. 34, 2459–2466 (1995).
[CrossRef]

Rimmer, M. P.

Roddier, C.

C. Roddier, F. Roddier, “Interferogram analysis using Fourier transform techniques,” Appl. Opt. 26, 1668–1673 (1987).
[CrossRef] [PubMed]

F. Roddier, C. Roddier, “An image reconstruction of Alpha Orionis,” Astrophy. J. 295, L21–L23 (1985).
[CrossRef]

C. Roddier, F. Roddier, “High angular resolution observation of Alpha Orionis with a rotation shearing interferometer,” Astrophys. J. 270, L23–L26 (1983).
[CrossRef]

F. Roddier, C. Roddier, “Imaging with a multi-mirror telescope,” in Proceedings of the ESO Conference on Optical Telescopes of the Future, F. Pacini, W. Richter, R. N. Wilson, eds. (CERN for the European Southern Observatory, Geneva, 1978).

C. Roddier, “Rotation shearing interferometry,” in Proceedings of the IAU Colloquium 50 on High Angular Resolution Stellar Interferometry, J. Davis, W. J. Tango, eds. (Chatterton Astronomy Dept., U. Sydney, Australia, 1979), p. 32.

C. Roddier, “Enregistrement holographique d’images astronomiques degradees par la turbulence atmospherique,” (1979).

Roddier, F.

C. Roddier, F. Roddier, “Interferogram analysis using Fourier transform techniques,” Appl. Opt. 26, 1668–1673 (1987).
[CrossRef] [PubMed]

F. Roddier, C. Roddier, “An image reconstruction of Alpha Orionis,” Astrophy. J. 295, L21–L23 (1985).
[CrossRef]

C. Roddier, F. Roddier, “High angular resolution observation of Alpha Orionis with a rotation shearing interferometer,” Astrophys. J. 270, L23–L26 (1983).
[CrossRef]

F. Roddier, C. Roddier, “Imaging with a multi-mirror telescope,” in Proceedings of the ESO Conference on Optical Telescopes of the Future, F. Pacini, W. Richter, R. N. Wilson, eds. (CERN for the European Southern Observatory, Geneva, 1978).

Sandeman, R. J.

Schmit, J.

J. Schmit, K. Creath, M. Kujawinska, “Spatial and temporal phase-measurement techniques: a comparison of major error sources in one dimension,” in Interferometry: Techniques and Analysis, G. M. Brown, M. Kujawinska, O. Y. Kwon, G. T. Reid, eds., Proc. SPIE1755, 202–211 (1992).

Su, X. Y.

J. Li, X. Y. Su, L. R. Guo, “Improved Fourier transform profilometry for the automatic measurement of three-dimensional object shapes,” Opt. Eng. 29, 1439–1444 (1990).
[CrossRef]

Takeda, M.

Weinberg, F. J.

F. J. Weinberg, Optics of Flames (Butterworths, London, 1963), Chap. 4, pp. 116–149.

Wojciak, J.

M. Kujawinska, J. Wojciak, “Spatial phase-shifting techniques of fringe pattern analysis in photomechanics,” in Moire Techniques, Holographic Interferometry, Optical NDT and Applications to Fluid Mechanics, Proc. SPIE1554B, 503–513 (1991).

Womack, K. H.

K. H. Womack, “Frequency domain description of interferogram analysis,” Opt. Eng. 23, 396–400 (1984).
[CrossRef]

Appl. Opt.

Astrophy. J.

F. Roddier, C. Roddier, “An image reconstruction of Alpha Orionis,” Astrophy. J. 295, L21–L23 (1985).
[CrossRef]

Astrophys. J.

C. Roddier, F. Roddier, “High angular resolution observation of Alpha Orionis with a rotation shearing interferometer,” Astrophys. J. 270, L23–L26 (1983).
[CrossRef]

J. Opt. Soc. Am.

Opt. Eng.

M. Pirga, M. Kujawinska, “Two directional spatial-carrier phase-shifting method for analysis of crossed and closed fringe patterns,” Opt. Eng. 34, 2459–2466 (1995).
[CrossRef]

J. Li, X. Y. Su, L. R. Guo, “Improved Fourier transform profilometry for the automatic measurement of three-dimensional object shapes,” Opt. Eng. 29, 1439–1444 (1990).
[CrossRef]

K. H. Womack, “Frequency domain description of interferogram analysis,” Opt. Eng. 23, 396–400 (1984).
[CrossRef]

Other

M. Kujawinska, “Spatial phase measurement methods,” in Interferogram Analysis, Digital Fringe Pattern Measurement Techniques, D. W. Robinson, G. D. Reid, eds. (Institute of Physics, Bristol, 1993), pp. 141–193.

M. Kujawinska, J. Wojciak, “Spatial phase-shifting techniques of fringe pattern analysis in photomechanics,” in Moire Techniques, Holographic Interferometry, Optical NDT and Applications to Fluid Mechanics, Proc. SPIE1554B, 503–513 (1991).

F. J. Weinberg, Optics of Flames (Butterworths, London, 1963), Chap. 4, pp. 116–149.

J. Schmit, K. Creath, M. Kujawinska, “Spatial and temporal phase-measurement techniques: a comparison of major error sources in one dimension,” in Interferometry: Techniques and Analysis, G. M. Brown, M. Kujawinska, O. Y. Kwon, G. T. Reid, eds., Proc. SPIE1755, 202–211 (1992).

D. Malacara, “Optical testing,” in Handbook of Optics, M. Bass, E. W. Van Stryland, D. R. Williams, W. L. Wolfe, eds. (McGraw–Hill, New York, 1995), Chap. 3.

C. Roddier, “Enregistrement holographique d’images astronomiques degradees par la turbulence atmospherique,” (1979).

F. Roddier, C. Roddier, “Imaging with a multi-mirror telescope,” in Proceedings of the ESO Conference on Optical Telescopes of the Future, F. Pacini, W. Richter, R. N. Wilson, eds. (CERN for the European Southern Observatory, Geneva, 1978).

C. Roddier, “Rotation shearing interferometry,” in Proceedings of the IAU Colloquium 50 on High Angular Resolution Stellar Interferometry, J. Davis, W. J. Tango, eds. (Chatterton Astronomy Dept., U. Sydney, Australia, 1979), p. 32.

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Figures (6)

Fig. 1
Fig. 1

Procedure of processing an interferogram and comparison between the FT method and the MFT method. a, Assumed continuum a(x, y) with Hanning window applied; b, assumed amplitude of modulation b(x, y) with Hanning window applied; c, assumed background phase distribution c(x, y); d, assumed phase variation under disturbed conditions d(x, y); e, composed fringe pattern of an interferogram under the undisturbed condition; f, composed fringe pattern of an interferogram under the disturbed condition.

Fig. 2
Fig. 2

Processing procedure of an interferogram and the comparison between the FT method and the MFT method. a, Fourier transform of the fringe pattern under the undisturbed condition; b, Fourier transform of the fringe pattern under the disturbed condition; c, discrepancy of recovered background phase distribution by the FT method; d, discrepancy of recovered function [2πf0 x + c(x, y)]rec by the MFT method; e, retrieved tested phase variation d(x, y)rec by the MFT method; f, discrepancy of the retrieved tested phase variation by the MFT method.

Fig. 3
Fig. 3

Processing procedure of interferogram with nonconstant space carrier frequency and comparison between the FT method and the MFT method. a, Space carrier frequency versus distance; b, discrepancy of recovered background phase distribution by the FT method; c, discrepancy of recovered function [2πf0 x + c(x, y)]rec by the MFT method.

Fig. 4
Fig. 4

Experimental apparatus block diagram.

Fig. 5
Fig. 5

Interferogram of a candle flame. The data analyzed (Fig. 6) were acquired along the horizontal white line.

Fig. 6
Fig. 6

Recovered shearing phase difference distributions of the candle flame and nearby flow determined by the FT and the MFT methods.

Equations (16)

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

gx, y=ax, y+bx, ycos2πf0x+cx, y+dx, y,
Gf, y=Af, y+Cf-f0, y+C*f+f0, y,
g0x, y=ax, y+bx, ycos2πf0x+cx, y,
gx, y=ax, y+bx, y×cos2πf0x+cx, y+dx, y.
2πf0x+cx, y+dx, y=Arccos Φ,
Φ=gx, y-ax, y/bx, y.
Arccos Φ=2nπ±arccos Φ, n=0, ±1, ±2, ,
0arccos Φπ.
Arccos Φk,n,m=2nπ+m arccosΦk, n=0, ±1, ±2, , k=1, 2, K, m=±1,
d1=arccos Φ1-2πf0x1+c1rec,
d2=arccos Φ2-2πf0x2+c2rec.
dk,est=2dk-1-dk-2, k=3, 4,, K.
dk,n,m=Arccos Φk,n-2πf0x+ckrec=2nπ+m arccos Φk-2πf0x+ckrec, n=0, ±1, ±2, ±K/2, k=3, 4, , K, m=±1.
mindk,n,m-dk,est=dk-dk,est.
Hk=0.51-cos2πk/K+1, k=1, 2,, K,
dx, y=15 exp-x-1352/102+20 exp-x-1802/152.

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