Abstract

The normal model of light propagation through a turbulent atmosphere is used to investigate the telescope–atmosphere transfer function in stellar speckle interferometry. The effect of the finite bandwidth of a spectral filter on the transfer function is considered. The permissible focusing errors for successful processing in stellar speckle interferometry are obtained based on the normal model. Some observational results are presented and compared with theory. The normal model leads to an analytically simple solution and properly describes light scattering from turbulent atmosphere. The results for the normal model are compared with those for the log-normal model.

© 1985 Optical Society of America

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References

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  1. F. Roddier, “The effects of atmospheric turbulence in optical astronomy,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1981).
    [CrossRef]
  2. A. Labeyrie, “Attainment of diffraction limited resolution in large telescopes by Fourier analyzing speckle patterns in star image,” Astron. Astrophys. 6, 85 (1970).
  3. D. Korff, G. Dryden, M. G. Millen, “Information retrieval from atmospheric induced speckle patterns,” Opt. Commun. 5, 187 (1972).
    [CrossRef]
  4. D. Korff, “Analysis of a method for obtaining near-diffraction-limited information in the presence of atmospheric turbulence,”J. Opt. Soc. Am. 63, 971 (1973).
    [CrossRef]
  5. J. C. Dainty, “Diffraction-limited imaging of stellar object using telescopes of low optical quality,” Opt. Commun. 7, 129 (1973).
    [CrossRef]
  6. J. C. Dainty, “The transfer function, signal-to-noise ratio and limiting magnitude in stellar speckle interferometry,” Mon. Not. R. Astron. Soc. 169, 631 (1974).
  7. R. Barakat, P. Nisenson, “Influence of the wave-front correlation function and deterministic wave-front aberrations on the speckle image-reconstruction problem in the high-light-level regime,”J. Opt. Soc. Am. 71, 1390 (1981).
  8. R. Barakat, P. Nisenson, “Finite exposure time, astronomical speckle transfer function,” Opt. Acta 30, 1405 (1983).
    [CrossRef]
  9. F. Roddier, G. Ricort, C. Roddier, “Defocusing effects in astronomical speckle interferometry,” Opt. Commun. 24, 281 (1978).
    [CrossRef]
  10. C. Aime, S. Kadiri, G. Ricort, C. Roddier, J. Vernin, “Measurements of stellar speckle interferometry lens-atmosphere modulation transfer function,” Opt. Acta 26, 575 (1979).
    [CrossRef]
  11. A. Chelli, P. Lena, C. Roddier, F. Roddier, F. Sibille, “Modulation transfer function for infra-red steller speckle interferometry, evidence for a log-normal statistics,” Opt. Acta 26, 583 (1979).
    [CrossRef]
  12. D. P. Karo, A. M. Schneiderman, “Speckle interferometry lens-atmosphere MTF measurements,”J. Opt. Soc. Am. 66, 1252 (1976).
    [CrossRef]
  13. D. P. Karo, A. M. Schneiderman, “Speckle interferometry with severely aberrated telescope,”J. Opt. Soc. Am. 67, 1277 (1977).
    [CrossRef]
  14. D. P. Karo, A. M. Schneiderman, “Transfer function, correlation scales and phase retrieval in speckle interferometry,”J. Opt. Soc. Am. 67, 1583 (1977).
    [CrossRef]
  15. D. P. Karo, A. M. Schneiderman, “Speckle interferometry at finite spectral band widths and exposure time,”J. Opt. Soc. Am. 68, 480 (1978).
    [CrossRef]
  16. G. N. Watson, Theory of Bessel Functions (Cambridge U. Press, Cambridge, 1966).
  17. J. Ohtsubo, T. Eiju, T. Kohno, K. Tomita, M. Noguchi, “Stellar speckle interferometry using Dodaira 91 cm telescope,” Tokyo Tenmondaihou 20, 255 (1984).
  18. G. W. Simon, “A practical solution of the atmospheric dispersion problem,” Astron. J. 71, 190 (1966).
    [CrossRef]

1984 (1)

J. Ohtsubo, T. Eiju, T. Kohno, K. Tomita, M. Noguchi, “Stellar speckle interferometry using Dodaira 91 cm telescope,” Tokyo Tenmondaihou 20, 255 (1984).

1983 (1)

R. Barakat, P. Nisenson, “Finite exposure time, astronomical speckle transfer function,” Opt. Acta 30, 1405 (1983).
[CrossRef]

1981 (1)

1979 (2)

C. Aime, S. Kadiri, G. Ricort, C. Roddier, J. Vernin, “Measurements of stellar speckle interferometry lens-atmosphere modulation transfer function,” Opt. Acta 26, 575 (1979).
[CrossRef]

A. Chelli, P. Lena, C. Roddier, F. Roddier, F. Sibille, “Modulation transfer function for infra-red steller speckle interferometry, evidence for a log-normal statistics,” Opt. Acta 26, 583 (1979).
[CrossRef]

1978 (2)

F. Roddier, G. Ricort, C. Roddier, “Defocusing effects in astronomical speckle interferometry,” Opt. Commun. 24, 281 (1978).
[CrossRef]

D. P. Karo, A. M. Schneiderman, “Speckle interferometry at finite spectral band widths and exposure time,”J. Opt. Soc. Am. 68, 480 (1978).
[CrossRef]

1977 (2)

1976 (1)

1974 (1)

J. C. Dainty, “The transfer function, signal-to-noise ratio and limiting magnitude in stellar speckle interferometry,” Mon. Not. R. Astron. Soc. 169, 631 (1974).

1973 (2)

D. Korff, “Analysis of a method for obtaining near-diffraction-limited information in the presence of atmospheric turbulence,”J. Opt. Soc. Am. 63, 971 (1973).
[CrossRef]

J. C. Dainty, “Diffraction-limited imaging of stellar object using telescopes of low optical quality,” Opt. Commun. 7, 129 (1973).
[CrossRef]

1972 (1)

D. Korff, G. Dryden, M. G. Millen, “Information retrieval from atmospheric induced speckle patterns,” Opt. Commun. 5, 187 (1972).
[CrossRef]

1970 (1)

A. Labeyrie, “Attainment of diffraction limited resolution in large telescopes by Fourier analyzing speckle patterns in star image,” Astron. Astrophys. 6, 85 (1970).

1966 (1)

G. W. Simon, “A practical solution of the atmospheric dispersion problem,” Astron. J. 71, 190 (1966).
[CrossRef]

Aime, C.

C. Aime, S. Kadiri, G. Ricort, C. Roddier, J. Vernin, “Measurements of stellar speckle interferometry lens-atmosphere modulation transfer function,” Opt. Acta 26, 575 (1979).
[CrossRef]

Barakat, R.

Chelli, A.

A. Chelli, P. Lena, C. Roddier, F. Roddier, F. Sibille, “Modulation transfer function for infra-red steller speckle interferometry, evidence for a log-normal statistics,” Opt. Acta 26, 583 (1979).
[CrossRef]

Dainty, J. C.

J. C. Dainty, “The transfer function, signal-to-noise ratio and limiting magnitude in stellar speckle interferometry,” Mon. Not. R. Astron. Soc. 169, 631 (1974).

J. C. Dainty, “Diffraction-limited imaging of stellar object using telescopes of low optical quality,” Opt. Commun. 7, 129 (1973).
[CrossRef]

Dryden, G.

D. Korff, G. Dryden, M. G. Millen, “Information retrieval from atmospheric induced speckle patterns,” Opt. Commun. 5, 187 (1972).
[CrossRef]

Eiju, T.

J. Ohtsubo, T. Eiju, T. Kohno, K. Tomita, M. Noguchi, “Stellar speckle interferometry using Dodaira 91 cm telescope,” Tokyo Tenmondaihou 20, 255 (1984).

Kadiri, S.

C. Aime, S. Kadiri, G. Ricort, C. Roddier, J. Vernin, “Measurements of stellar speckle interferometry lens-atmosphere modulation transfer function,” Opt. Acta 26, 575 (1979).
[CrossRef]

Karo, D. P.

Kohno, T.

J. Ohtsubo, T. Eiju, T. Kohno, K. Tomita, M. Noguchi, “Stellar speckle interferometry using Dodaira 91 cm telescope,” Tokyo Tenmondaihou 20, 255 (1984).

Korff, D.

D. Korff, “Analysis of a method for obtaining near-diffraction-limited information in the presence of atmospheric turbulence,”J. Opt. Soc. Am. 63, 971 (1973).
[CrossRef]

D. Korff, G. Dryden, M. G. Millen, “Information retrieval from atmospheric induced speckle patterns,” Opt. Commun. 5, 187 (1972).
[CrossRef]

Labeyrie, A.

A. Labeyrie, “Attainment of diffraction limited resolution in large telescopes by Fourier analyzing speckle patterns in star image,” Astron. Astrophys. 6, 85 (1970).

Lena, P.

A. Chelli, P. Lena, C. Roddier, F. Roddier, F. Sibille, “Modulation transfer function for infra-red steller speckle interferometry, evidence for a log-normal statistics,” Opt. Acta 26, 583 (1979).
[CrossRef]

Millen, M. G.

D. Korff, G. Dryden, M. G. Millen, “Information retrieval from atmospheric induced speckle patterns,” Opt. Commun. 5, 187 (1972).
[CrossRef]

Nisenson, P.

Noguchi, M.

J. Ohtsubo, T. Eiju, T. Kohno, K. Tomita, M. Noguchi, “Stellar speckle interferometry using Dodaira 91 cm telescope,” Tokyo Tenmondaihou 20, 255 (1984).

Ohtsubo, J.

J. Ohtsubo, T. Eiju, T. Kohno, K. Tomita, M. Noguchi, “Stellar speckle interferometry using Dodaira 91 cm telescope,” Tokyo Tenmondaihou 20, 255 (1984).

Ricort, G.

C. Aime, S. Kadiri, G. Ricort, C. Roddier, J. Vernin, “Measurements of stellar speckle interferometry lens-atmosphere modulation transfer function,” Opt. Acta 26, 575 (1979).
[CrossRef]

F. Roddier, G. Ricort, C. Roddier, “Defocusing effects in astronomical speckle interferometry,” Opt. Commun. 24, 281 (1978).
[CrossRef]

Roddier, C.

C. Aime, S. Kadiri, G. Ricort, C. Roddier, J. Vernin, “Measurements of stellar speckle interferometry lens-atmosphere modulation transfer function,” Opt. Acta 26, 575 (1979).
[CrossRef]

A. Chelli, P. Lena, C. Roddier, F. Roddier, F. Sibille, “Modulation transfer function for infra-red steller speckle interferometry, evidence for a log-normal statistics,” Opt. Acta 26, 583 (1979).
[CrossRef]

F. Roddier, G. Ricort, C. Roddier, “Defocusing effects in astronomical speckle interferometry,” Opt. Commun. 24, 281 (1978).
[CrossRef]

Roddier, F.

A. Chelli, P. Lena, C. Roddier, F. Roddier, F. Sibille, “Modulation transfer function for infra-red steller speckle interferometry, evidence for a log-normal statistics,” Opt. Acta 26, 583 (1979).
[CrossRef]

F. Roddier, G. Ricort, C. Roddier, “Defocusing effects in astronomical speckle interferometry,” Opt. Commun. 24, 281 (1978).
[CrossRef]

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

Schneiderman, A. M.

Sibille, F.

A. Chelli, P. Lena, C. Roddier, F. Roddier, F. Sibille, “Modulation transfer function for infra-red steller speckle interferometry, evidence for a log-normal statistics,” Opt. Acta 26, 583 (1979).
[CrossRef]

Simon, G. W.

G. W. Simon, “A practical solution of the atmospheric dispersion problem,” Astron. J. 71, 190 (1966).
[CrossRef]

Tomita, K.

J. Ohtsubo, T. Eiju, T. Kohno, K. Tomita, M. Noguchi, “Stellar speckle interferometry using Dodaira 91 cm telescope,” Tokyo Tenmondaihou 20, 255 (1984).

Vernin, J.

C. Aime, S. Kadiri, G. Ricort, C. Roddier, J. Vernin, “Measurements of stellar speckle interferometry lens-atmosphere modulation transfer function,” Opt. Acta 26, 575 (1979).
[CrossRef]

Watson, G. N.

G. N. Watson, Theory of Bessel Functions (Cambridge U. Press, Cambridge, 1966).

Astron. Astrophys. (1)

A. Labeyrie, “Attainment of diffraction limited resolution in large telescopes by Fourier analyzing speckle patterns in star image,” Astron. Astrophys. 6, 85 (1970).

Astron. J. (1)

G. W. Simon, “A practical solution of the atmospheric dispersion problem,” Astron. J. 71, 190 (1966).
[CrossRef]

J. Opt. Soc. Am. (6)

Mon. Not. R. Astron. Soc. (1)

J. C. Dainty, “The transfer function, signal-to-noise ratio and limiting magnitude in stellar speckle interferometry,” Mon. Not. R. Astron. Soc. 169, 631 (1974).

Opt. Acta (3)

R. Barakat, P. Nisenson, “Finite exposure time, astronomical speckle transfer function,” Opt. Acta 30, 1405 (1983).
[CrossRef]

C. Aime, S. Kadiri, G. Ricort, C. Roddier, J. Vernin, “Measurements of stellar speckle interferometry lens-atmosphere modulation transfer function,” Opt. Acta 26, 575 (1979).
[CrossRef]

A. Chelli, P. Lena, C. Roddier, F. Roddier, F. Sibille, “Modulation transfer function for infra-red steller speckle interferometry, evidence for a log-normal statistics,” Opt. Acta 26, 583 (1979).
[CrossRef]

Opt. Commun. (3)

F. Roddier, G. Ricort, C. Roddier, “Defocusing effects in astronomical speckle interferometry,” Opt. Commun. 24, 281 (1978).
[CrossRef]

D. Korff, G. Dryden, M. G. Millen, “Information retrieval from atmospheric induced speckle patterns,” Opt. Commun. 5, 187 (1972).
[CrossRef]

J. C. Dainty, “Diffraction-limited imaging of stellar object using telescopes of low optical quality,” Opt. Commun. 7, 129 (1973).
[CrossRef]

Tokyo Tenmondaihou (1)

J. Ohtsubo, T. Eiju, T. Kohno, K. Tomita, M. Noguchi, “Stellar speckle interferometry using Dodaira 91 cm telescope,” Tokyo Tenmondaihou 20, 255 (1984).

Other (2)

G. N. Watson, Theory of Bessel Functions (Cambridge U. Press, Cambridge, 1966).

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

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Figures (6)

Fig. 1
Fig. 1

Dependence of the focusing error on the transfer function T2. n = 3, N = 400, and M = 20. In this and Figs. 2, 5, and 6 relating to the transfer function, the frequency axis is normalized by the cut-off frequency of the telescope aperture, and the peak power of the transfer function is also normalized as unity.

Fig. 2
Fig. 2

Dependence of the spectral bandwidth on the transfer function T2. m= 0, n = 3, and N = 400.

Fig. 3
Fig. 3

Averaged image of a single star, 37 LMi, without the dancing effect.

Fig. 4
Fig. 4

Center of the gravity of each short-exposure image of 37 LMi (crosses) and spread of the averaged image (circle).

Fig. 5
Fig. 5

Telescope–atmosphere MTF. Observational data (crosses) and theoretical prediction of Eq. (24) (solid curve).

Fig. 6
Fig. 6

Comparison of the MTF’s of the normal and log-normal models. m = 0 and Δλ = 0.

Tables (1)

Tables Icon

Table 1 Square Root of the Number of Correlation Cells and Corresponding Seeing Size

Equations (37)

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S ( x , k ) = k 2 | d 2 ξ U ( ξ , k ) H ( ξ , k ) exp ( - i k f ξ · x ) | 2 ,
S ( x ) = d k P ( k ) S ( x , k ) ,
T ( u ) = d k P ( k ) d 2 ξ U ( ξ , k ) × U * ( ξ + u , k ) H ( ξ , k ) H * ( ξ + u , k ) ,
u = ( λ ¯ f p , λ ¯ f q ) ;
U ( ξ , k ) = exp [ i k ϕ ( ξ ) ] ,
T ( u ) 2 = T 1 + T 2 ,
T 1 = d k 1 d k 2 P ( k 1 ) P ( k 2 ) C ( u , k 1 , k 2 ) C * ( u , k 2 , k 2 ) × d 2 ξ 1 H ( ξ 1 , k 1 ) H * ( ξ 1 + u , k 1 ) × d 2 ξ 2 H ( ξ 2 , k 2 ) H * ( ξ 2 + u , k 2 ) ,
T 2 = d k 1 d k 2 P ( k 1 ) P ( k 2 ) d 2 ξ 1 d 2 ξ 2 C ( ξ 1 - ξ 2 , k 1 , k 2 ) 2 × H ( ξ 1 , k 1 ) H * ( ξ 2 , k 2 ) H * ( ξ 1 + u , k 1 ) H ( ξ 2 + u , k 2 ) .
C ( ξ , k 1 , k 2 ) = exp [ i k 1 ϕ ( ξ ) - i k 2 ϕ ( ξ + ξ ) ] = exp [ - ( k 1 - k 2 ) 2 σ ϕ 2 2 ] exp { - k 1 k 2 σ ϕ 2 [ 1 - ρ ( ξ ) ] } ,
ρ ( ξ ) = exp ( - ξ 2 r 0 2 ) .
C ( ξ , k 1 , k 2 ) = exp [ - ( k 1 - k 2 ) 2 σ ϕ 2 2 - k 1 k 2 σ ϕ 2 r 0 2 ξ 2 ] .
H ( ξ , k ) = exp [ - ξ 2 ( D 2 ) 2 + i W ( ξ , k ) ] ,
W ( ξ , k ) = k z 2 f 2 ξ 2 ,
[ 2 J 1 ( x ) x ] 2 exp ( - x 2 4 ) ,
D = D 0 / 2 .
P ( k ) = 1 π w 0 exp [ - ( k - k ¯ ) 2 w 0 2 ] ,
w 0 = 2 π Δ λ λ ¯ 2 .
T 1 = π 256 D 0 4 d k 1 d k 2 P ( k 1 ) P ( k 2 ) g ( u , k 1 , k 2 ) ,
g ( u , k 1 , k 2 ) = exp [ - 4 k 1 2 + k 2 2 k ¯ 2 N X 2 - 1 2 ( k ¯ z F 2 ) 2 k 1 2 + k 2 2 k ¯ 2 X 2 - 8 X 2 ] ,
N = D 0 2 ( 2 r 0 k ¯ σ ϕ ) 2 .
M = k ¯ w 0 = λ ¯ Δ λ .
g ( u , k 1 , k 2 ) = exp ( - 4 k 1 2 + k 2 2 k ¯ 2 N X 2 ) .
T 1 = π 2 D 0 2 256 1 1 + ( 2.83 N ) 2 2 M 2 X 2 × exp [ - ( 2.83 N ) 2 X 2 1 + ( 2.83 N ) 2 2 M 2 X 2 ] π 2 D 0 2 256 exp [ - ( 2.83 N ) 2 X 2 ] .
T 2 = π 2 D 0 2 256 1 N exp ( - 8 X 2 ) × d k 1 d k 2 P ( k 1 ) P ( k 2 ) h ( u , k 1 , k 2 ) ,
h ( u , k 1 , k 2 ) = k ¯ 2 k 1 k 2 exp [ - ( k 1 - k 2 ) 2 σ ϕ 2 - 1 128 ( k ¯ z F 2 ) ( k 1 - k 2 ) 2 k ¯ 2 X 2 - 1 32 ( 1 N k ¯ z F 2 ) k 1 2 + k 2 2 k ¯ 2 X 2 ] .
1 128 ( k ¯ z F 2 ) 2 ( k 1 - k 2 ) 2 k ¯ 2 1 32 ( w 0 z F 2 ) 2 = π 2 8 ( m M ) 2 1
1 32 ( 1 N k ¯ z F 2 ) 2 k 1 2 + k 2 2 k ¯ 2 ~ π 2 8 ( m N ) 2 1
h ( u , k 1 , k 2 ) = k ¯ 2 k 1 k 2 exp [ - ( k 1 - k 2 ) 2 σ ϕ 2 ] .
T 2 = π 2 D 0 2 256 1 N 1 ( 1 + 2 w 0 2 σ ϕ 2 ) 1 / 2 exp ( - 8 X 2 ) .
d 2 ξ H ( ξ ) 2 H ( ξ + u ) 2 = π 2 D 0 2 256 exp ( - 8 X 2 ) .
m N ,
m M .
T ( u ) 2 = exp [ - ( 2.83 N X ) 2 ] + 1 N 1 ( 1 + 2 w 0 2 σ ϕ 2 ) 1 / 2 exp ( - 8 X 2 ) .
T ( u ) 2 log - normal = exp [ - ( 3.18 N X ) 5 / 3 ] + 0.435 N T D ( u ) ,
T D ( u ) = d 2 ξ H ( ξ ) 2 H ( ξ + u ) 2 = 2 π [ cos - 1 X - X 2 ( 1 - X 2 ) 1 / 2 ] .
d n = 2 r 0 k ¯ σ ϕ .
N log - normal = ( D 0 d ) 2 .

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