Abstract

Stellar speckle interferometry of unresolved objects yields the short-exposure lens–atmosphere MTF. Using Korff’s calculations, a value of the wave correlation scale r0 can be obtained from the MTF. We demonstrate this, and also show how the power spectra can be corrected so that r0 values can be extracted from resolvable object data. Assuming the validity of Korff’s calculations for speckle interferometry, unresolved and resolved object data can be then used to test calculations for short-and long-exposure imaging. Tests with field data show only a weak distinction in speckle interferometry between short- and long-exposure imaging. A Gaussian image shape does not agree with the data.

© 1978 Optical Society of America

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References

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  1. R. E. Hufnagel and N. R. Stanley, “Modulation transfer function associated with image transmission through turbulent media,” J. Opt. Soc. Am. 54, 52–61 (1964).
    [Crossref]
  2. D. L. Fried, “Optical resolution through a randomly inhomogeneous medium for very long and very short exposures,” J. Opt. Soc. Am. 56, 1372–79 (1966).
    [Crossref]
  3. V. I. Tatarski, Wave Propagation in a Turbulent Medium (Dover, New York, 1961).
  4. A. Labeyrie, “Attainment of diffraction limited resolution in large telescopes by Fourier analyzing speckle patterns in star images,” Astron. Astrophys. 6, 85–87 (1970).
  5. D. Korff, “Analysis of a method for obtaining near-diffraction-limited information in the presence of atmospheric turbulence,” J. Opt. Soc. Am. 63, 971–80 (1973).
    [Crossref]
  6. A. M. Schneiderman and D. P. Karo, “Double star speckle interferometry measurements of atomospheric non-isoplanicity,” AERL RR 441 (May3, 1977). Also “Speckle interferometry measurements of atmospheric nonisoplanicity using double stars,” J. Opt. Soc. Am.,  68, 338 (1978) (this issue).
  7. A. M. Schneiderman and D. P. Karo, “How to build a speckle interferometer,” Opt. Engin. 16, 72–79 (1977).
    [Crossref]
  8. D. P. Karo and A. M. Schneiderman, “Speckle interferometry lens–atmosphere MTF measurements,” J. Opt. Soc. Am. 66, 1252–56 (1976).
    [Crossref]

1977 (2)

A. M. Schneiderman and D. P. Karo, “Double star speckle interferometry measurements of atomospheric non-isoplanicity,” AERL RR 441 (May3, 1977). Also “Speckle interferometry measurements of atmospheric nonisoplanicity using double stars,” J. Opt. Soc. Am.,  68, 338 (1978) (this issue).

A. M. Schneiderman and D. P. Karo, “How to build a speckle interferometer,” Opt. Engin. 16, 72–79 (1977).
[Crossref]

1976 (1)

1973 (1)

1970 (1)

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

1966 (1)

1964 (1)

Fried, D. L.

Hufnagel, R. E.

Karo, D. P.

A. M. Schneiderman and D. P. Karo, “Double star speckle interferometry measurements of atomospheric non-isoplanicity,” AERL RR 441 (May3, 1977). Also “Speckle interferometry measurements of atmospheric nonisoplanicity using double stars,” J. Opt. Soc. Am.,  68, 338 (1978) (this issue).

A. M. Schneiderman and D. P. Karo, “How to build a speckle interferometer,” Opt. Engin. 16, 72–79 (1977).
[Crossref]

D. P. Karo and A. M. Schneiderman, “Speckle interferometry lens–atmosphere MTF measurements,” J. Opt. Soc. Am. 66, 1252–56 (1976).
[Crossref]

Korff, D.

Labeyrie, A.

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

Schneiderman, A. M.

A. M. Schneiderman and D. P. Karo, “Double star speckle interferometry measurements of atomospheric non-isoplanicity,” AERL RR 441 (May3, 1977). Also “Speckle interferometry measurements of atmospheric nonisoplanicity using double stars,” J. Opt. Soc. Am.,  68, 338 (1978) (this issue).

A. M. Schneiderman and D. P. Karo, “How to build a speckle interferometer,” Opt. Engin. 16, 72–79 (1977).
[Crossref]

D. P. Karo and A. M. Schneiderman, “Speckle interferometry lens–atmosphere MTF measurements,” J. Opt. Soc. Am. 66, 1252–56 (1976).
[Crossref]

Stanley, N. R.

Tatarski, V. I.

V. I. Tatarski, Wave Propagation in a Turbulent Medium (Dover, New York, 1961).

AERL RR 441 (1)

A. M. Schneiderman and D. P. Karo, “Double star speckle interferometry measurements of atomospheric non-isoplanicity,” AERL RR 441 (May3, 1977). Also “Speckle interferometry measurements of atmospheric nonisoplanicity using double stars,” J. Opt. Soc. Am.,  68, 338 (1978) (this issue).

Astron. Astrophys. (1)

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

J. Opt. Soc. Am. (4)

Opt. Engin. (1)

A. M. Schneiderman and D. P. Karo, “How to build a speckle interferometer,” Opt. Engin. 16, 72–79 (1977).
[Crossref]

Other (1)

V. I. Tatarski, Wave Propagation in a Turbulent Medium (Dover, New York, 1961).

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Figures (5)

FIG. 1
FIG. 1

Test of short-exposure theory [Eq. (9)] with data from an unresolved star. The slope of the straight-line (theory) fit through the data points is related to the value of r0. The approximate seeing limited spatial frequency corresponds to q = r0/D.

FIG. 2
FIG. 2

Test of long-exposure theory [Eq. (8)] with the same data of Fig. 1.

FIG. 3
FIG. 3

Test of assumed Gaussian image structure [Eq. (16)] with the data of Fig. 1.

FIG. 4
FIG. 4

Extraction of r0 estimate from data on a 1.55 arc sec double star. The data points are corrected from values lying on the 〈|Ĩ(k)|2〉 curve as explained in the text. The slope of the straight line through selected data points is related to the r0 value in the short-exposure theory.

FIG. 5
FIG. 5

Time history of four r0 determinations (corrected to zenith) made from data on double stars with separations of 0.22 to 1.55 arc sec which was corrected as indicated in the text. The length of the data marker indicates the time period of data collection.

Equations (20)

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R ( ξ ) = ũ ( x ) ũ * ( x + ξ ) ,
C ( ξ ) = R ( ξ ) - R ( ) R ( 0 ) - R ( ) .
C ( 0 ) = 1 ,
C ( ) = 0 ,
    C ( ξ ) 1 ,
π r c 2 4 = 0 C ( ξ ) d ξ .
τ ˜ ( k ) LE = C ( λ f k ) τ 0 ( k ) ,
C ( ξ ) = exp ( - D ( ξ ) / 2 )
D ( ξ ) = 6.88 ( ξ / r 0 ) 5 / 3 ,
τ ˜ ( q ) LE = τ 0 ( q ) exp [ - 3.44 ( D / r 0 ) 5 / 3 q 5 / 3 ] ,
τ ˜ ( q ) SE = τ 0 ( q ) exp [ - 3.44 ( D / r 0 ) 5 / 3 ( q 5 / 3 - q 2 ) ] .
Ĩ ( k ) 2 = τ ˜ ( k ) 2 .
τ ˜ ( k ) 2 τ ˜ ( k ) SE 2 .
Ĩ ( k ) 2 = τ ˜ ( k ) 2 O ( k ) 2 ,
O ( q ) 2 J 1 2 ( q a λ / D ) / q 2 , for q λ / D a
Ĩ ( k ) 2 = 1 + α 2 ( 1 + α ) 2 τ ˜ ( k ) 2 × { 1 + 2 α 1 + α 2 T ( k , a ) cos ( k · 2 a ) } ,
ln τ ˜ ( q ) SE τ 0 ( q ) = - 3.44 ( D / r 0 ) 5 / 3 ( q 5 / 3 - q 2 ) ,
q < r 0 / D and τ ˜ ( q ) SE > ( r 0 / D ) 2 .
I ( r ) = I ( 0 ) e - β r 2 .
ln τ ( k ) Gaussian - 1 4 β D 2 λ 2 q 2 .