Abstract

Fourier transform techniques have been used to map the complex fringe visibility in several types of interferogram. A Gerchberg-type iterative technique is used to eliminate edge effects. Results are presented for two specific cases: seeing measurements and interferometric tests.

© 1987 Optical Society of America

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References

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  1. C. Roddier, “Enregistrement holographique d’images astronomiques degradées par la turbulence atmosphérique,” ATP traitement des images du CNRS, contract 3109, final report (1979).
  2. F. Roddier, C. Roddier, “Imaging with a Multi-Mirror Telescope,” in Proceedings, ESO Conference on Optical Telescopes of the Future, Geneva (1977), F.Pacini Pacini, W. Richter, R. N. Wilson, Eds., Geneva 23: ESO c/o CERN (1978).
  3. C. Roddier, F. Roddier, “Imaging with a Coherence Interferometer in Optical Astronomy” in Proceedings, IAU Colloquium 49 on Formation of images from spatial coherence functions in astronomy, Groningen, 1978, C. van Schooneveld, Ed. (Reidel, Norwell, MA, 1979).
  4. F. Roddier, “Rotation Shearing Interferometry,” in Proceedings, IAU Colloquium 50 on High Angular Resolution Stellar Interferometry, U. Maryland, 1978, J. Davis, W. J. Tango, Ed. (Chatterton Astronomy Dept., U. Sydney, 1979).
  5. C. Roddier, F. Roddier, “High Angular Resolution Observations of Alpha Orionis with a Rotation Shearing Interferometer,” Astrophys. J. 270, L23 (1983).
    [CrossRef]
  6. F. Roddier, C. Roddier, “An Image Reconstruction of Alpha Orionis,” Astrophys. J. 295, L21 (1985).
    [CrossRef]
  7. 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 (1982).
    [CrossRef]
  8. W. W. Macy, “Two-Dimensional Fringe-Pattern Analysis,” Appl. Opt. 22, 3898 (1983).
    [CrossRef] [PubMed]
  9. K. A. Nugent, “Interferogram Analysis Using an Accurate Fully Automatic Algorithm,” Appl. Opt. 24, 3103 (1985).
    [CrossRef]
  10. D. J. Bone, H.-A. Bachor, R. J. Sandeman, “Fringe-Pattern Analysis Using a 2-D Fourier Transform,” Appl. Opt. 25, 1653 (1986).
    [CrossRef] [PubMed]
  11. J. M. Burch, “Scatter Fringes of Equal Thickness,” Nature London 171, 889 (1953).
    [CrossRef]
  12. R. M. Scott, “Scatter Plate Interferometry,” Appl. Opt. 8, 531 (1969).
  13. R. W. Gerchberg, “Super-Resolution Through Error Energy Reduction,” Opt. Acta 21, 709 (1974).
    [CrossRef]
  14. B. L. McGlamery, “Image Restoration Techniques Applied to Astronomical Photography,” NASA Tech. Rep. SP-256 (1971).
  15. K. Itoh, “Analysis of the Phase Unwrapping Algorithm,” Appl. Opt. 21, 2470 (1982).
    [CrossRef] [PubMed]
  16. K. Itoh, Y. Ohtsuka, “Phase Estimation Based on the Maximum Likelihood Criterion,” Appl. Opt. 22, 3054 (1983).
    [CrossRef] [PubMed]
  17. K. Itoh, Y. Ohtsuka, “Interferometric Imaging of a Thermally Luminous Two-Dimensional Object,” Opt. Commun. 48, 75 (1983).
    [CrossRef]
  18. L. N. Mertz, “Phase Estimation with Few Photons,” Appl. Opt. 23, 1638 (1984).
    [CrossRef] [PubMed]
  19. L. N. Mertz, “Multichannel Seeing Compensation via Software,” Appl. Opt. 24, 2898 (1985).
    [CrossRef] [PubMed]
  20. D. L. Fried, “Least-Squares Fitting a Wave-Front Distortion Estimate to an Array of Phase-Difference Measurements,” J. Opt. Soc. Am. 67, 370 (1977).
    [CrossRef]
  21. R. H. Hudgin, “Wave-Front Reconstruction for Compensated Imaging,” J. Opt. Soc. Am. 67, 375 (1977).
    [CrossRef]
  22. B. R. Hunt, “Matrix Formulation of the Reconstruction of Phase Values from Phase differences,” J. Opt. Soc. Am. 69, 393 (1979).
    [CrossRef]
  23. J. M. Tribolet, “A New Phase Unwrapping Algorithm,” IEEE Trans. Acoust. Speech Signal Process. ASSP-25, 170 (1977).
    [CrossRef]
  24. D. C. Ghiglia, G. A. Mastin, L. A. Romero, “Cellular Automata Method for Phase Unwrapping,” J. Opt. Soc. Am. A 4, 267 (1987).
    [CrossRef]
  25. C. Roddier, “Measurements of the Atmospheric Attenuation of the Spectral Components of AstronomicalImages,” J. Opt. Soc. Am. 66, 478 (1976).
    [CrossRef]
  26. C. Roddier, J. Vernin, “Relative contribution of upper and lower atmosphere to integrated refractive-index profiles,” Appl. Opt. 16, 2252 (1977).
    [CrossRef] [PubMed]
  27. F. Roddier, “The Effects of Atmospheric Turbulence in Optical Astronomy,” Prog. Opt. 19, 281 (1981).
    [CrossRef]

1987

1986

1985

K. A. Nugent, “Interferogram Analysis Using an Accurate Fully Automatic Algorithm,” Appl. Opt. 24, 3103 (1985).
[CrossRef]

F. Roddier, C. Roddier, “An Image Reconstruction of Alpha Orionis,” Astrophys. J. 295, L21 (1985).
[CrossRef]

L. N. Mertz, “Multichannel Seeing Compensation via Software,” Appl. Opt. 24, 2898 (1985).
[CrossRef] [PubMed]

1984

1983

K. Itoh, Y. Ohtsuka, “Phase Estimation Based on the Maximum Likelihood Criterion,” Appl. Opt. 22, 3054 (1983).
[CrossRef] [PubMed]

K. Itoh, Y. Ohtsuka, “Interferometric Imaging of a Thermally Luminous Two-Dimensional Object,” Opt. Commun. 48, 75 (1983).
[CrossRef]

C. Roddier, F. Roddier, “High Angular Resolution Observations of Alpha Orionis with a Rotation Shearing Interferometer,” Astrophys. J. 270, L23 (1983).
[CrossRef]

W. W. Macy, “Two-Dimensional Fringe-Pattern Analysis,” Appl. Opt. 22, 3898 (1983).
[CrossRef] [PubMed]

1982

1981

F. Roddier, “The Effects of Atmospheric Turbulence in Optical Astronomy,” Prog. Opt. 19, 281 (1981).
[CrossRef]

1979

1977

1976

1974

R. W. Gerchberg, “Super-Resolution Through Error Energy Reduction,” Opt. Acta 21, 709 (1974).
[CrossRef]

1969

R. M. Scott, “Scatter Plate Interferometry,” Appl. Opt. 8, 531 (1969).

1953

J. M. Burch, “Scatter Fringes of Equal Thickness,” Nature London 171, 889 (1953).
[CrossRef]

Bachor, H.-A.

Bone, D. J.

Burch, J. M.

J. M. Burch, “Scatter Fringes of Equal Thickness,” Nature London 171, 889 (1953).
[CrossRef]

Fried, D. L.

Gerchberg, R. W.

R. W. Gerchberg, “Super-Resolution Through Error Energy Reduction,” Opt. Acta 21, 709 (1974).
[CrossRef]

Ghiglia, D. C.

Hideki, I.

Hudgin, R. H.

Hunt, B. R.

Itoh, K.

Kobayashi, S.

Macy, W. W.

Mastin, G. A.

McGlamery, B. L.

B. L. McGlamery, “Image Restoration Techniques Applied to Astronomical Photography,” NASA Tech. Rep. SP-256 (1971).

Mertz, L. N.

Nugent, K. A.

K. A. Nugent, “Interferogram Analysis Using an Accurate Fully Automatic Algorithm,” Appl. Opt. 24, 3103 (1985).
[CrossRef]

Ohtsuka, Y.

K. Itoh, Y. Ohtsuka, “Interferometric Imaging of a Thermally Luminous Two-Dimensional Object,” Opt. Commun. 48, 75 (1983).
[CrossRef]

K. Itoh, Y. Ohtsuka, “Phase Estimation Based on the Maximum Likelihood Criterion,” Appl. Opt. 22, 3054 (1983).
[CrossRef] [PubMed]

Roddier, C.

F. Roddier, C. Roddier, “An Image Reconstruction of Alpha Orionis,” Astrophys. J. 295, L21 (1985).
[CrossRef]

C. Roddier, F. Roddier, “High Angular Resolution Observations of Alpha Orionis with a Rotation Shearing Interferometer,” Astrophys. J. 270, L23 (1983).
[CrossRef]

C. Roddier, J. Vernin, “Relative contribution of upper and lower atmosphere to integrated refractive-index profiles,” Appl. Opt. 16, 2252 (1977).
[CrossRef] [PubMed]

C. Roddier, “Measurements of the Atmospheric Attenuation of the Spectral Components of AstronomicalImages,” J. Opt. Soc. Am. 66, 478 (1976).
[CrossRef]

C. Roddier, “Enregistrement holographique d’images astronomiques degradées par la turbulence atmosphérique,” ATP traitement des images du CNRS, contract 3109, final report (1979).

F. Roddier, C. Roddier, “Imaging with a Multi-Mirror Telescope,” in Proceedings, ESO Conference on Optical Telescopes of the Future, Geneva (1977), F.Pacini Pacini, W. Richter, R. N. Wilson, Eds., Geneva 23: ESO c/o CERN (1978).

C. Roddier, F. Roddier, “Imaging with a Coherence Interferometer in Optical Astronomy” in Proceedings, IAU Colloquium 49 on Formation of images from spatial coherence functions in astronomy, Groningen, 1978, C. van Schooneveld, Ed. (Reidel, Norwell, MA, 1979).

Roddier, F.

F. Roddier, C. Roddier, “An Image Reconstruction of Alpha Orionis,” Astrophys. J. 295, L21 (1985).
[CrossRef]

C. Roddier, F. Roddier, “High Angular Resolution Observations of Alpha Orionis with a Rotation Shearing Interferometer,” Astrophys. J. 270, L23 (1983).
[CrossRef]

F. Roddier, “The Effects of Atmospheric Turbulence in Optical Astronomy,” Prog. Opt. 19, 281 (1981).
[CrossRef]

F. Roddier, C. Roddier, “Imaging with a Multi-Mirror Telescope,” in Proceedings, ESO Conference on Optical Telescopes of the Future, Geneva (1977), F.Pacini Pacini, W. Richter, R. N. Wilson, Eds., Geneva 23: ESO c/o CERN (1978).

C. Roddier, F. Roddier, “Imaging with a Coherence Interferometer in Optical Astronomy” in Proceedings, IAU Colloquium 49 on Formation of images from spatial coherence functions in astronomy, Groningen, 1978, C. van Schooneveld, Ed. (Reidel, Norwell, MA, 1979).

F. Roddier, “Rotation Shearing Interferometry,” in Proceedings, IAU Colloquium 50 on High Angular Resolution Stellar Interferometry, U. Maryland, 1978, J. Davis, W. J. Tango, Ed. (Chatterton Astronomy Dept., U. Sydney, 1979).

Romero, L. A.

Sandeman, R. J.

Scott, R. M.

R. M. Scott, “Scatter Plate Interferometry,” Appl. Opt. 8, 531 (1969).

Takeda, M.

Tribolet, J. M.

J. M. Tribolet, “A New Phase Unwrapping Algorithm,” IEEE Trans. Acoust. Speech Signal Process. ASSP-25, 170 (1977).
[CrossRef]

Vernin, J.

Appl. Opt.

Astrophys. J.

C. Roddier, F. Roddier, “High Angular Resolution Observations of Alpha Orionis with a Rotation Shearing Interferometer,” Astrophys. J. 270, L23 (1983).
[CrossRef]

F. Roddier, C. Roddier, “An Image Reconstruction of Alpha Orionis,” Astrophys. J. 295, L21 (1985).
[CrossRef]

IEEE Trans. Acoust. Speech Signal Process.

J. M. Tribolet, “A New Phase Unwrapping Algorithm,” IEEE Trans. Acoust. Speech Signal Process. ASSP-25, 170 (1977).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Nature London

J. M. Burch, “Scatter Fringes of Equal Thickness,” Nature London 171, 889 (1953).
[CrossRef]

Opt. Acta

R. W. Gerchberg, “Super-Resolution Through Error Energy Reduction,” Opt. Acta 21, 709 (1974).
[CrossRef]

Opt. Commun.

K. Itoh, Y. Ohtsuka, “Interferometric Imaging of a Thermally Luminous Two-Dimensional Object,” Opt. Commun. 48, 75 (1983).
[CrossRef]

Prog. Opt.

F. Roddier, “The Effects of Atmospheric Turbulence in Optical Astronomy,” Prog. Opt. 19, 281 (1981).
[CrossRef]

Other

B. L. McGlamery, “Image Restoration Techniques Applied to Astronomical Photography,” NASA Tech. Rep. SP-256 (1971).

C. Roddier, “Enregistrement holographique d’images astronomiques degradées par la turbulence atmosphérique,” ATP traitement des images du CNRS, contract 3109, final report (1979).

F. Roddier, C. Roddier, “Imaging with a Multi-Mirror Telescope,” in Proceedings, ESO Conference on Optical Telescopes of the Future, Geneva (1977), F.Pacini Pacini, W. Richter, R. N. Wilson, Eds., Geneva 23: ESO c/o CERN (1978).

C. Roddier, F. Roddier, “Imaging with a Coherence Interferometer in Optical Astronomy” in Proceedings, IAU Colloquium 49 on Formation of images from spatial coherence functions in astronomy, Groningen, 1978, C. van Schooneveld, Ed. (Reidel, Norwell, MA, 1979).

F. Roddier, “Rotation Shearing Interferometry,” in Proceedings, IAU Colloquium 50 on High Angular Resolution Stellar Interferometry, U. Maryland, 1978, J. Davis, W. J. Tango, Ed. (Chatterton Astronomy Dept., U. Sydney, 1979).

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Figures (7)

Fig. 1
Fig. 1

Examples of interferograms: (a) stellar interferogram, (b) interferometric seeing measurement, (c) interferometric test.

Fig. 2
Fig. 2

Seeing measurements; correction for the fluctuations of the continuum: (a) raw data, (b) estimated continuum, (c) corrected data.

Fig. 3
Fig. 3

Processing of the interferometric test: (a) after sampling and correction for the continuum, (b) after extrapolation of the fringes, (c) same as (b) but shifted by half a period.

Fig. 4
Fig. 4

Gerchberg’s extrapolation algorithm.

Fig. 5
Fig. 5

Seeing measurements: plot of the log–log visibility vs log base line (dots) and 5/3 slope best fit (full line).

Fig. 6
Fig. 6

(a) Wavefront slopes derived from the interferogram of Fig. 1(c) and (b) knife-edge test on the same mirror.

Fig. 7
Fig. 7

Profile of the reconstructed wavefront displayed in Fig. 6(a). Horizontal cut near the center.

Equations (8)

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I ( r ) = D ( r ) A ( r ) { 1 + V ( r ) cos [ 2 π f 0 · r - ϕ ( r ) ] } ;
D ( r ) = { 1 inside the inerferogram , 0 outside the interferogram .
I ( r ) = D ( r ) A ( r ) [ 1 + ReC ( r ) exp ( 2 i π f 0 · r ) ] = D ( r ) A ( r ) [ 1 + ½ C ( r ) exp ( 2 i π f 0 · r ) + ½ C ( r ) exp ( - 2 i π f 0 · r ) ] ,
C ( r ) = V ( r ) exp [ - i ϕ ( r ) ] .
I ^ ( f ) = D ^ ( f ) * A ^ ( f ) * [ δ ( f ) + ½ C ^ ( f - f 0 ) + ½ C ^ * ( - f - f 0 ) ] ,
J ( r ) = I ( r ) A ( r ) - D ( r ) = D ( r ) V ( r ) cos [ 2 π f 0 · r - ϕ ( r ) ] ,
J ^ ( f ) = D ^ ( f ) * [ ½ C ^ ( f - f 0 ) + ½ C ^ * ( - f - f 0 ) ] .
V ( r ) = exp [ - 3.44 ( r / r 0 ) 5 / 3 ] ,

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