Abstract

The temporal behavior of stellar speckle patterns is statistically analyzed. The time-only power spectrum is shown to be the sum of two exponentially decreasing functions defining two characteristic time constants. The corresponding correlation is the sum of two Lorentzian functions. This is consistent with the first-order expansion of the power spectrum deduced from the multiple-layer model for atmospheric turbulence. However, this model fails to account for the experimental data that show a strong correlation between the spatial structure of a speckle pattern and its temporal behavior. This leads to the introduction of a new empirical model, called the randomly jittered speckle pattern model, which gives a preponderant place to image motion. The speckle lifetime then appears to be substantially longer than the corresponding measured time constant. As a consequence, a preliminary compensation of the image motion appears to be particularly interesting in speckle interferometry or active optics experiments.

© 1986 Optical Society of America

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References

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  1. A. Labeyrie, “Attainment of diffraction-limited resolution in large telescopes by Fourier analysing speckle patterns in star images,” Astron. Astrophys. 6, 85–87 (1970).
  2. D. Bonneau, R. Foy, “Intérférométrie au 3.60 m C.F.H. résolution du système Pluton–Charon,” Astron. Astrophys. 92, L1–L4 (1980).
  3. G. Ricort, C. Aime, “Solar seeing and the statistical properties of the photospheric solar granulation. III. Solar speckle interferometry,” Astron. Astrophys. 76, 324–335 (1979).
  4. A. Chelli, P. Lena, F. Sibille, “Angular dimensions of accreting young stars,” Nature 278, 143–146 (1979).
    [Crossref]
  5. C. Aime, S. Kadiri, G. Ricort, “The influence of scanning rate in sequential analysis of a speckle pattern. Application to speckle boiling,” Opt. Commun. 35, 169–174 (1980).
    [Crossref]
  6. F. Roddier, “The effects of atmospheric turbulence in optical astronomy,” in Progress in Optics XIX, E. Wolf, ed. (North-Holland, Amsterdam, 1981).
    [Crossref]
  7. R. J. Scaddan, J. G. Walker, “Statistics of stellar speckle patterns,” Appl. opt. 17, 3779–3784 (1978).
    [Crossref] [PubMed]
  8. D. P. Karo, A. M. Scheiderman, “Speckle interferometry at finite bandwidths and exposure times,” J. Opt. Soc. Am. 68, 480–485 (1978).
    [Crossref]
  9. A. W. Lohmann, G. P. Weigelt, “Astronomical speckle interferometry: Measurements of isoplanicity and of temporal correlation,” Optik 53, 167–180 (1979).
  10. R. Petrov, S. Kadiri, F. Martin, G. Ricort, C. Aime, “Application de l’analyse interspectrale à la speckle intérférométrie,” J. Opt. (Paris) 13, 331–337 (1982).
    [Crossref]
  11. G. Parry, J. G. Walker, R. J. Scaddan, “On the statistics on stellar speckle patterns and pupil plane scintillation,” Opt. Acta 26, 563–574 (1979).
    [Crossref]
  12. J. C. Dainty, D. R. Hennings, K. A. O’Donnell, “Space–time correlation of stellar speckle patterns,” J. Opt. Soc. Am. 71, 490–492 (1981).
    [Crossref]
  13. F. Roddier, J. M. Gilli, G. Lund, “On the origin of speckle boiling and its effects in stellar speckle interferometry,” J. Opt. (Paris) 13, 263–271 (1982).
    [Crossref]
  14. R. Petrov, “Application de l’analyse interspectrale à la speckle interférométrie différentielle,” Thèse de spécialité (Nice University, Nice, France, 1983).
  15. C. Aime, S. Kadiri, F. Martin, G. Ricort, “Temporal autocorrelation functions of solar speckle pattern,” Opt. Commun. 39, 287–292 (1981).
    [Crossref]
  16. D. L. Fried, “Optical resolution through a randomly inhomogeneous medium for very long and very short exposures,” J. Opt. Soc. Am. 56, 1372–1379 (1966).
    [Crossref]
  17. F. Roddier, J. M. Gilli, J. Vernin, “On the isoplanatic patch size in stellar speckle interferometry,” J. Opt. (Paris) 13, 63–70 (1982).
    [Crossref]
  18. V. I. Tatarski, The Effects of the Turbulent Atmosphere on Wave Propagation (Israel Program for Scientific Translations, Jerusalem, 1971).
  19. J. C. Dainty, “Stellar speckle interferometry,” in Laser Speckle and Related Phenomena, Vol. IX of Topics in Applied Optics, J. C. Dainty, ed. (Springer-Verlag, Berlin, 1975).
    [Crossref]
  20. S. Kadiri, “Etude spatio-temporelle des effets de la turbulence atmosphérique sur les observations astronomiques par analyse interspectrale,” Thèse d’Etat (Nice University, Nice, France, 1983).
  21. C. Aime, G. Ricort, “Sur l’analyse statistique de la granulation solaire,” Astron. Astrophys. 39, 319–324 (1975).
  22. A. Rose, Vision—Human and Electronic (Plenum, New York, 1973), p. 32.
  23. T. N. Cornsweet, H. D. Crane, “Accurate two-dimensional eye tracker using first and fourth Purkinje images,” J. Opt. Soc. Am. 63, 921–928 (1973).
    [Crossref] [PubMed]

1982 (3)

R. Petrov, S. Kadiri, F. Martin, G. Ricort, C. Aime, “Application de l’analyse interspectrale à la speckle intérférométrie,” J. Opt. (Paris) 13, 331–337 (1982).
[Crossref]

F. Roddier, J. M. Gilli, G. Lund, “On the origin of speckle boiling and its effects in stellar speckle interferometry,” J. Opt. (Paris) 13, 263–271 (1982).
[Crossref]

F. Roddier, J. M. Gilli, J. Vernin, “On the isoplanatic patch size in stellar speckle interferometry,” J. Opt. (Paris) 13, 63–70 (1982).
[Crossref]

1981 (2)

C. Aime, S. Kadiri, F. Martin, G. Ricort, “Temporal autocorrelation functions of solar speckle pattern,” Opt. Commun. 39, 287–292 (1981).
[Crossref]

J. C. Dainty, D. R. Hennings, K. A. O’Donnell, “Space–time correlation of stellar speckle patterns,” J. Opt. Soc. Am. 71, 490–492 (1981).
[Crossref]

1980 (2)

D. Bonneau, R. Foy, “Intérférométrie au 3.60 m C.F.H. résolution du système Pluton–Charon,” Astron. Astrophys. 92, L1–L4 (1980).

C. Aime, S. Kadiri, G. Ricort, “The influence of scanning rate in sequential analysis of a speckle pattern. Application to speckle boiling,” Opt. Commun. 35, 169–174 (1980).
[Crossref]

1979 (4)

G. Ricort, C. Aime, “Solar seeing and the statistical properties of the photospheric solar granulation. III. Solar speckle interferometry,” Astron. Astrophys. 76, 324–335 (1979).

A. Chelli, P. Lena, F. Sibille, “Angular dimensions of accreting young stars,” Nature 278, 143–146 (1979).
[Crossref]

G. Parry, J. G. Walker, R. J. Scaddan, “On the statistics on stellar speckle patterns and pupil plane scintillation,” Opt. Acta 26, 563–574 (1979).
[Crossref]

A. W. Lohmann, G. P. Weigelt, “Astronomical speckle interferometry: Measurements of isoplanicity and of temporal correlation,” Optik 53, 167–180 (1979).

1978 (2)

R. J. Scaddan, J. G. Walker, “Statistics of stellar speckle patterns,” Appl. opt. 17, 3779–3784 (1978).
[Crossref] [PubMed]

D. P. Karo, A. M. Scheiderman, “Speckle interferometry at finite bandwidths and exposure times,” J. Opt. Soc. Am. 68, 480–485 (1978).
[Crossref]

1975 (1)

C. Aime, G. Ricort, “Sur l’analyse statistique de la granulation solaire,” Astron. Astrophys. 39, 319–324 (1975).

1973 (1)

T. N. Cornsweet, H. D. Crane, “Accurate two-dimensional eye tracker using first and fourth Purkinje images,” J. Opt. Soc. Am. 63, 921–928 (1973).
[Crossref] [PubMed]

1970 (1)

A. Labeyrie, “Attainment of diffraction-limited resolution in large telescopes by Fourier analysing speckle patterns in star images,” Astron. Astrophys. 6, 85–87 (1970).

1966 (1)

Aime, C.

R. Petrov, S. Kadiri, F. Martin, G. Ricort, C. Aime, “Application de l’analyse interspectrale à la speckle intérférométrie,” J. Opt. (Paris) 13, 331–337 (1982).
[Crossref]

C. Aime, S. Kadiri, F. Martin, G. Ricort, “Temporal autocorrelation functions of solar speckle pattern,” Opt. Commun. 39, 287–292 (1981).
[Crossref]

C. Aime, S. Kadiri, G. Ricort, “The influence of scanning rate in sequential analysis of a speckle pattern. Application to speckle boiling,” Opt. Commun. 35, 169–174 (1980).
[Crossref]

G. Ricort, C. Aime, “Solar seeing and the statistical properties of the photospheric solar granulation. III. Solar speckle interferometry,” Astron. Astrophys. 76, 324–335 (1979).

C. Aime, G. Ricort, “Sur l’analyse statistique de la granulation solaire,” Astron. Astrophys. 39, 319–324 (1975).

Bonneau, D.

D. Bonneau, R. Foy, “Intérférométrie au 3.60 m C.F.H. résolution du système Pluton–Charon,” Astron. Astrophys. 92, L1–L4 (1980).

Chelli, A.

A. Chelli, P. Lena, F. Sibille, “Angular dimensions of accreting young stars,” Nature 278, 143–146 (1979).
[Crossref]

Cornsweet, T. N.

T. N. Cornsweet, H. D. Crane, “Accurate two-dimensional eye tracker using first and fourth Purkinje images,” J. Opt. Soc. Am. 63, 921–928 (1973).
[Crossref] [PubMed]

Crane, H. D.

T. N. Cornsweet, H. D. Crane, “Accurate two-dimensional eye tracker using first and fourth Purkinje images,” J. Opt. Soc. Am. 63, 921–928 (1973).
[Crossref] [PubMed]

Dainty, J. C.

J. C. Dainty, D. R. Hennings, K. A. O’Donnell, “Space–time correlation of stellar speckle patterns,” J. Opt. Soc. Am. 71, 490–492 (1981).
[Crossref]

J. C. Dainty, “Stellar speckle interferometry,” in Laser Speckle and Related Phenomena, Vol. IX of Topics in Applied Optics, J. C. Dainty, ed. (Springer-Verlag, Berlin, 1975).
[Crossref]

Foy, R.

D. Bonneau, R. Foy, “Intérférométrie au 3.60 m C.F.H. résolution du système Pluton–Charon,” Astron. Astrophys. 92, L1–L4 (1980).

Fried, D. L.

Gilli, J. M.

F. Roddier, J. M. Gilli, G. Lund, “On the origin of speckle boiling and its effects in stellar speckle interferometry,” J. Opt. (Paris) 13, 263–271 (1982).
[Crossref]

F. Roddier, J. M. Gilli, J. Vernin, “On the isoplanatic patch size in stellar speckle interferometry,” J. Opt. (Paris) 13, 63–70 (1982).
[Crossref]

Hennings, D. R.

J. C. Dainty, D. R. Hennings, K. A. O’Donnell, “Space–time correlation of stellar speckle patterns,” J. Opt. Soc. Am. 71, 490–492 (1981).
[Crossref]

Kadiri, S.

R. Petrov, S. Kadiri, F. Martin, G. Ricort, C. Aime, “Application de l’analyse interspectrale à la speckle intérférométrie,” J. Opt. (Paris) 13, 331–337 (1982).
[Crossref]

C. Aime, S. Kadiri, F. Martin, G. Ricort, “Temporal autocorrelation functions of solar speckle pattern,” Opt. Commun. 39, 287–292 (1981).
[Crossref]

C. Aime, S. Kadiri, G. Ricort, “The influence of scanning rate in sequential analysis of a speckle pattern. Application to speckle boiling,” Opt. Commun. 35, 169–174 (1980).
[Crossref]

S. Kadiri, “Etude spatio-temporelle des effets de la turbulence atmosphérique sur les observations astronomiques par analyse interspectrale,” Thèse d’Etat (Nice University, Nice, France, 1983).

Karo, D. P.

D. P. Karo, A. M. Scheiderman, “Speckle interferometry at finite bandwidths and exposure times,” J. Opt. Soc. Am. 68, 480–485 (1978).
[Crossref]

Labeyrie, A.

A. Labeyrie, “Attainment of diffraction-limited resolution in large telescopes by Fourier analysing speckle patterns in star images,” Astron. Astrophys. 6, 85–87 (1970).

Lena, P.

A. Chelli, P. Lena, F. Sibille, “Angular dimensions of accreting young stars,” Nature 278, 143–146 (1979).
[Crossref]

Lohmann, A. W.

A. W. Lohmann, G. P. Weigelt, “Astronomical speckle interferometry: Measurements of isoplanicity and of temporal correlation,” Optik 53, 167–180 (1979).

Lund, G.

F. Roddier, J. M. Gilli, G. Lund, “On the origin of speckle boiling and its effects in stellar speckle interferometry,” J. Opt. (Paris) 13, 263–271 (1982).
[Crossref]

Martin, F.

R. Petrov, S. Kadiri, F. Martin, G. Ricort, C. Aime, “Application de l’analyse interspectrale à la speckle intérférométrie,” J. Opt. (Paris) 13, 331–337 (1982).
[Crossref]

C. Aime, S. Kadiri, F. Martin, G. Ricort, “Temporal autocorrelation functions of solar speckle pattern,” Opt. Commun. 39, 287–292 (1981).
[Crossref]

O’Donnell, K. A.

J. C. Dainty, D. R. Hennings, K. A. O’Donnell, “Space–time correlation of stellar speckle patterns,” J. Opt. Soc. Am. 71, 490–492 (1981).
[Crossref]

Parry, G.

G. Parry, J. G. Walker, R. J. Scaddan, “On the statistics on stellar speckle patterns and pupil plane scintillation,” Opt. Acta 26, 563–574 (1979).
[Crossref]

Petrov, R.

R. Petrov, S. Kadiri, F. Martin, G. Ricort, C. Aime, “Application de l’analyse interspectrale à la speckle intérférométrie,” J. Opt. (Paris) 13, 331–337 (1982).
[Crossref]

R. Petrov, “Application de l’analyse interspectrale à la speckle interférométrie différentielle,” Thèse de spécialité (Nice University, Nice, France, 1983).

Ricort, G.

R. Petrov, S. Kadiri, F. Martin, G. Ricort, C. Aime, “Application de l’analyse interspectrale à la speckle intérférométrie,” J. Opt. (Paris) 13, 331–337 (1982).
[Crossref]

C. Aime, S. Kadiri, F. Martin, G. Ricort, “Temporal autocorrelation functions of solar speckle pattern,” Opt. Commun. 39, 287–292 (1981).
[Crossref]

C. Aime, S. Kadiri, G. Ricort, “The influence of scanning rate in sequential analysis of a speckle pattern. Application to speckle boiling,” Opt. Commun. 35, 169–174 (1980).
[Crossref]

G. Ricort, C. Aime, “Solar seeing and the statistical properties of the photospheric solar granulation. III. Solar speckle interferometry,” Astron. Astrophys. 76, 324–335 (1979).

C. Aime, G. Ricort, “Sur l’analyse statistique de la granulation solaire,” Astron. Astrophys. 39, 319–324 (1975).

Roddier, F.

F. Roddier, J. M. Gilli, J. Vernin, “On the isoplanatic patch size in stellar speckle interferometry,” J. Opt. (Paris) 13, 63–70 (1982).
[Crossref]

F. Roddier, J. M. Gilli, G. Lund, “On the origin of speckle boiling and its effects in stellar speckle interferometry,” J. Opt. (Paris) 13, 263–271 (1982).
[Crossref]

F. Roddier, “The effects of atmospheric turbulence in optical astronomy,” in Progress in Optics XIX, E. Wolf, ed. (North-Holland, Amsterdam, 1981).
[Crossref]

Rose, A.

A. Rose, Vision—Human and Electronic (Plenum, New York, 1973), p. 32.

Scaddan, R. J.

G. Parry, J. G. Walker, R. J. Scaddan, “On the statistics on stellar speckle patterns and pupil plane scintillation,” Opt. Acta 26, 563–574 (1979).
[Crossref]

R. J. Scaddan, J. G. Walker, “Statistics of stellar speckle patterns,” Appl. opt. 17, 3779–3784 (1978).
[Crossref] [PubMed]

Scheiderman, A. M.

D. P. Karo, A. M. Scheiderman, “Speckle interferometry at finite bandwidths and exposure times,” J. Opt. Soc. Am. 68, 480–485 (1978).
[Crossref]

Sibille, F.

A. Chelli, P. Lena, F. Sibille, “Angular dimensions of accreting young stars,” Nature 278, 143–146 (1979).
[Crossref]

Tatarski, V. I.

V. I. Tatarski, The Effects of the Turbulent Atmosphere on Wave Propagation (Israel Program for Scientific Translations, Jerusalem, 1971).

Vernin, J.

F. Roddier, J. M. Gilli, J. Vernin, “On the isoplanatic patch size in stellar speckle interferometry,” J. Opt. (Paris) 13, 63–70 (1982).
[Crossref]

Walker, J. G.

G. Parry, J. G. Walker, R. J. Scaddan, “On the statistics on stellar speckle patterns and pupil plane scintillation,” Opt. Acta 26, 563–574 (1979).
[Crossref]

R. J. Scaddan, J. G. Walker, “Statistics of stellar speckle patterns,” Appl. opt. 17, 3779–3784 (1978).
[Crossref] [PubMed]

Weigelt, G. P.

A. W. Lohmann, G. P. Weigelt, “Astronomical speckle interferometry: Measurements of isoplanicity and of temporal correlation,” Optik 53, 167–180 (1979).

Appl. opt. (1)

Astron. Astrophys. (2)

D. Bonneau, R. Foy, “Intérférométrie au 3.60 m C.F.H. résolution du système Pluton–Charon,” Astron. Astrophys. 92, L1–L4 (1980).

C. Aime, G. Ricort, “Sur l’analyse statistique de la granulation solaire,” Astron. Astrophys. 39, 319–324 (1975).

Astron. Astrophys. (2)

G. Ricort, C. Aime, “Solar seeing and the statistical properties of the photospheric solar granulation. III. Solar speckle interferometry,” Astron. Astrophys. 76, 324–335 (1979).

A. Labeyrie, “Attainment of diffraction-limited resolution in large telescopes by Fourier analysing speckle patterns in star images,” Astron. Astrophys. 6, 85–87 (1970).

J. Opt. (Paris) (1)

R. Petrov, S. Kadiri, F. Martin, G. Ricort, C. Aime, “Application de l’analyse interspectrale à la speckle intérférométrie,” J. Opt. (Paris) 13, 331–337 (1982).
[Crossref]

J. Opt. Soc. Am. (1)

T. N. Cornsweet, H. D. Crane, “Accurate two-dimensional eye tracker using first and fourth Purkinje images,” J. Opt. Soc. Am. 63, 921–928 (1973).
[Crossref] [PubMed]

J. Opt. (Paris) (1)

F. Roddier, J. M. Gilli, J. Vernin, “On the isoplanatic patch size in stellar speckle interferometry,” J. Opt. (Paris) 13, 63–70 (1982).
[Crossref]

J. Opt. (Paris) (1)

F. Roddier, J. M. Gilli, G. Lund, “On the origin of speckle boiling and its effects in stellar speckle interferometry,” J. Opt. (Paris) 13, 263–271 (1982).
[Crossref]

J. Opt. Soc. Am. (2)

J. C. Dainty, D. R. Hennings, K. A. O’Donnell, “Space–time correlation of stellar speckle patterns,” J. Opt. Soc. Am. 71, 490–492 (1981).
[Crossref]

D. P. Karo, A. M. Scheiderman, “Speckle interferometry at finite bandwidths and exposure times,” J. Opt. Soc. Am. 68, 480–485 (1978).
[Crossref]

J. Opt. Soc. Am. (1)

Nature (1)

A. Chelli, P. Lena, F. Sibille, “Angular dimensions of accreting young stars,” Nature 278, 143–146 (1979).
[Crossref]

Opt. Commun. (1)

C. Aime, S. Kadiri, F. Martin, G. Ricort, “Temporal autocorrelation functions of solar speckle pattern,” Opt. Commun. 39, 287–292 (1981).
[Crossref]

Opt. Acta (1)

G. Parry, J. G. Walker, R. J. Scaddan, “On the statistics on stellar speckle patterns and pupil plane scintillation,” Opt. Acta 26, 563–574 (1979).
[Crossref]

Opt. Commun. (1)

C. Aime, S. Kadiri, G. Ricort, “The influence of scanning rate in sequential analysis of a speckle pattern. Application to speckle boiling,” Opt. Commun. 35, 169–174 (1980).
[Crossref]

Optik (1)

A. W. Lohmann, G. P. Weigelt, “Astronomical speckle interferometry: Measurements of isoplanicity and of temporal correlation,” Optik 53, 167–180 (1979).

Other (6)

F. Roddier, “The effects of atmospheric turbulence in optical astronomy,” in Progress in Optics XIX, E. Wolf, ed. (North-Holland, Amsterdam, 1981).
[Crossref]

R. Petrov, “Application de l’analyse interspectrale à la speckle interférométrie différentielle,” Thèse de spécialité (Nice University, Nice, France, 1983).

V. I. Tatarski, The Effects of the Turbulent Atmosphere on Wave Propagation (Israel Program for Scientific Translations, Jerusalem, 1971).

J. C. Dainty, “Stellar speckle interferometry,” in Laser Speckle and Related Phenomena, Vol. IX of Topics in Applied Optics, J. C. Dainty, ed. (Springer-Verlag, Berlin, 1975).
[Crossref]

S. Kadiri, “Etude spatio-temporelle des effets de la turbulence atmosphérique sur les observations astronomiques par analyse interspectrale,” Thèse d’Etat (Nice University, Nice, France, 1983).

A. Rose, Vision—Human and Electronic (Plenum, New York, 1973), p. 32.

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Figures (9)

Fig. 1
Fig. 1

Schematic outline of the speckle interferometer. The objective O images the speckle pattern onto the analyzing aperture A, which is the entrance of a Czerny–Turner spectrograph. A, which can be either a pinhole or a slit, scans the image by means of the tilting mirror T.M. when the spatial properties of the image are studied. This mirror is held fixed when the temporal behavior of this part of the image is investigated. A semitransparent plate S set at the exit slit of the spectrograph permits the two photomultipliers PM1 and PM2 to analyze the speckle pattern at the same given wavelength. The dual-channel Fourier analyzer F.A. computes and averages the cross spectrum of the signals produced by PM1 and PM2.

Fig. 2
Fig. 2

Estimated time-only cross spectrum of the speckle pattern obtained from the 193-cm telescope of the Haute Provence Observatory by using a slit as a detector aperture. A, Real part (α) and imaginary part (β) of the cross spectrum versus frequency; the Nyquist plot (γ) shows the imaginary part versus the real part. B, Magnitude (α) and phase (β) of the cross spectrum versus frequency; the Nichols plot (γ) shows the phase versus the modulus. For the logarithmic plots, the conventions are as follows: For the Y axis, any value y such as |y| < 10−5 has been set equal to 10−5; any positive value y has been represented by log (y) in the upper part of each plot; any negative value y has been represented by log (−y) in the lower part of each plot. The same convention has been used in the Nyquist plot for the X axis.

Fig. 3
Fig. 3

Magnitude of the time-only power spectrum, obtained from the 80-cm telescope of the Haute Provence Observatory by using a pinhole as a detector aperture.

Fig. 4
Fig. 4

Examples of observed time-only power spectra (dots) fitted with sums of two exponential functions (solid line). Corresponding time constants are A, τ = 7 msec, T = 190 msec; B, τ = 13 msec, T = 200 msec.

Fig. 5
Fig. 5

Variations of the time constants τ and T with time for the nights of January 1 and 2, 1983.

Fig. 6
Fig. 6

Comparison between experiments performed with a slit (S) and a pinhole (P). A, Experimental time-only power spectra. B, Theoretical spatial power spectra. fc is the cutoff frequency of the telescope. Note the similarity between the shapes of the space and time functions.

Fig. 7
Fig. 7

Evolution of the space-only power spectra, obtained by using decreasing scanning speeds, toward the time-only power spectrum. The speckle pattern is analyzed 12.5 (●), 5 (▼), 1.25 (+) times by second. Time-only power spectrum (solid line).

Fig. 8
Fig. 8

Comparison between estimations of the time constants obtained by using a slit τs (+) and a pinhole τp (●). The two sets of values are clearly separated, and the ratio between the mean values of τs and τp is 1.47.

Fig. 9
Fig. 9

Spatiotemporal analysis of the speckle patterns obtained by observing the double star, Capella, with the 193-cm telescope of the Haute Provence Observatory. A, Two-dimensional spatial power spectrum shown by means of contour lines (solid lines). The observed anistropy gives the direction defined by the two components of the star. The dotted circles are contour lines of the telescope transfer function; fc is the spatial-frequency cutoff. B, Corresponding time constants τs obtained simultaneously with a slit as a detector aperture, in different directions crossing the star. The arrow shows the direction of the double star (separation of about 50 msec of arc).

Equations (26)

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

a ( t ) = s ( t ) + n 1 ( t ) , b ( t ) = s ( t ) + n 2 ( t ) .
W k ( f ) = ( 1 / k ) a i * ( f ) b i ( f ) = ( 1 / k ) | s i ( f ) | 2 + ( 1 / k ) [ s i * ( f ) n 2 i ( f ) + s i ( f ) n 1 i * ( f ) + n 1 i * ( f ) n 2 i ( f ) ] ,
W ( f ) = | s ( f ) | 2 ,
E ( f ) = a exp ( b | f | )
C ( t ) = 2 a b / ( b 2 + 4 π 2 t 2 ) .
τ = b / 2 ,
E ( f ) = a exp ( b | f | ) + a exp ( b | f | )
τ g = 0.64 r 0 / Δ υ ,
C g ( τ ) 0.435 ( r 0 / D ) 2 exp [ 7.65 ( Δ υ / r 0 ) 2 τ 2 ] .
C g ( τ ) 0.435 ( r 0 / D ) 2 [ 1 7.65 ( Δ υ / r 0 ) 2 τ 2 ] .
C l ( τ ) 0.435 ( r 0 / D ) 2 / [ 1 + 7.65 ( Δ υ / r 0 ) 2 τ 2 ] .
τ 1 = 1.135 r 0 / Δ υ = 1.77 τ g .
τ p s T m / σ m ,
τ s 2 τ p ,
τ ( α ) = s ( α ) T m 2 / σ m .
σ ( θ ) 0 + d r r exp ( K m r 5 / 3 ) × 0 π d φ exp [ ( 5 / 18 ) K m δ u 2 r 1 / 3 ( 3 cos 2 φ ) ] ,
K m = 6.88 ( r 0 / λ ) 5 / 3
σ ( θ ) 2 0 + d r r exp ( K m r 5 / 3 ) exp [ ( 25 / 36 ) K m δ u 2 r 1 / 3 ] × 0 π d α exp { [ ( 5 / 36 ) K m δ u 2 r 1 / 3 ] cos α } .
σ ( θ ) 2 π 0 + r d r exp ( K m r 5 / 3 ) exp [ ( 25 / 36 ) K m δ u 2 r 1 / 3 ] × J 0 ( i ( 5 / 36 ) K m δ u 2 r 1 / 3 ) .
σ ( θ ) 2 π 0 + r d r exp ( K m r 5 / 3 ) exp [ ( 25 / 36 ) K m δ u 2 r 1 / 3 ] × n ( 1 / n ! ) 2 [ ( 5 / 72 ) K m δ u 2 r 1 / 3 ] 2 n ,
σ ( θ ) ( 6 π / 5 ) ( K m ) 6 / 5 0 + d t t 1 / 5 exp ( t ) exp ( 10 B t 1 / 5 ) × n ( 1 / n ! ) 2 ( B t 1 / 5 ) 2 n
σ ( θ ) ( 6 π / 5 ) ( K m ) 6 / 5 0 + d t t 1 / 5 exp ( t ) × p [ ( 10 ) p / p ! ] ( B t 1 / 5 ) p n ( 1 / n ! ) 2 ( B t 1 / 5 ) 2 n ,
σ ( θ ) ( 6 π / 5 ) ( K m ) 6 / 5 p n { [ ( 10 ) p / p ! ] ( 1 / n ! ) 2 B p + 2 n × 0 + d t t ( 1 p 2 n ) / 5 exp ( t ) } ,
Γ ( 1 + x ) = 0 + d t t x exp ( t ) ,
σ ( θ ) ( 6 π / 5 ) p n [ ( 10 ) p / p ! ] ( 1 / n ! ) 2 ( 6.88 ) 6 ( p + 2 n 1 ) / 5 × ( r 0 / λ ) 2 ( p + 2 n 1 ) [ ( 5 / 72 ) δ u 2 ] p + 2 n Γ [ 1 + ( 1 p 2 n ) / 5 ] .
0.342 ( r 0 / λ ) 2 ( 5 π / 6 ) δ u 2 .

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