Abstract

A new technique (speckle interferometry) has been developed by Gezari, Labeyrie, and Stachnik, which allows the measurement of stellar diameters from a series of photographs obtained from large-aperture ground-based telescopes. The series of photographs is processed to obtain the Weiner spectrum of the photographic image, i.e., the ensemble-averaged modulus-squared Fourier transform obtained from the series of images. Gezari, Labeyrie, and Stachnik have measured stellar diameters as small as 0″.05, about 20 times betterthan is usually possible. In this paper, mathematical expressions are obtained for the Wiener spectrum of the image of a point source. As is well known, the Wiener spectrum of the image of an extended, incoherently radiating object, is expressible as a product of this point-source spectrum and the object spectrum. Calculations are performed using the Rytov approximation and assuming that the underlying atmospheric turbulence is describable by a Kolmogorov spectrum. Asymptotic closed-form expressions are obtained for angular frequencies much less than, and much greater than, the conventional seeing limit. In the latter case, the Wiener spectrum is found to be proportional to the optical transfer function.

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  1. D. Gezari, A. Labeyrie, and R. Stachnik, Bull. Am. Astron. Soc. 3, 244 (1971).
  2. D. Fried, J. Opt. Soc. Am. 56, 1372 (1966).
  3. The seeing angle can be estimated from Fried's definition (Ref. 2) of infinite-aperture long-exposure resolution and is typically of the order of 1 arc second for zenith viewing.
  4. It may be shown that <τ(0)> = 1, because the atmosphere does not absorb light, but only redistributes it. Hence, IT = <∫ I(x) dx> (i.e., the averaged integrated irradiance of the image) is unaffected by the turbulence. If <|τ(0)|2> = 1, so that the normalized <|τ(f)|2> equals the unnormalized <|τ(f)|2>, this would imply that <I2T> is unaffected by the turbulence. This statement is, in general, not true, because <I2T> ≠ <IT>2—i.e., the atmosphere produces scintillation. Nevertheless, it is approximately true if geometric optics is valid and/or the aperture averaging effectively eliminates the scintillation. This latter condition will, in general, be satisfied for values of D much greater than r0.
  5. Cf. M. Born and E. Wolf, Principles of Optics, 2nd ed. (Pergamon, New York, 1964).
  6. L. S. Taylor, J. Opt. Soc. Am. 58, 705 (1968).
  7. V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation (U.S. Dept. of Commerce, NTIS, Springfield, Va., 1971), p. 235.
  8. R. Hufnagel and N. R. Stanley, J. Opt. Soc. Am. 54, 52 (1964), p. 56.
  9. Reference 2, p. 1374.
  10. Reference 7, p. 239, Eqs. 36a and 37.
  11. Reference 2, p. 1377.
  12. Reference 2, p. 1366.
  13. Reference 7, pp. 74–102.
  14. R. Hufnagel, Restoration of Atmospherically Degraded Images Woods Hole Summer Study, Vol. 2, App. 3 (Defense Documentation Center, Alexandria, Va., 1966).
  15. Reference 5, Sec. 10.4.
  16. D. Korff, G. Dryden, and M. Miller, Opt. Commun. 5 (3), 187 (1972).
  17. D. Gezari, A. Labeyrie, and R. Stachnik, Astrophys. J. 173, L1-L5 (1 April 1972).

Born, M.

Cf. M. Born and E. Wolf, Principles of Optics, 2nd ed. (Pergamon, New York, 1964).

Dryden, G.

D. Korff, G. Dryden, and M. Miller, Opt. Commun. 5 (3), 187 (1972).

Fried, D.

D. Fried, J. Opt. Soc. Am. 56, 1372 (1966).

Gezari, D.

D. Gezari, A. Labeyrie, and R. Stachnik, Bull. Am. Astron. Soc. 3, 244 (1971).

D. Gezari, A. Labeyrie, and R. Stachnik, Astrophys. J. 173, L1-L5 (1 April 1972).

Hufnagel, R.

R. Hufnagel, Restoration of Atmospherically Degraded Images Woods Hole Summer Study, Vol. 2, App. 3 (Defense Documentation Center, Alexandria, Va., 1966).

R. Hufnagel and N. R. Stanley, J. Opt. Soc. Am. 54, 52 (1964), p. 56.

Korff, D.

D. Korff, G. Dryden, and M. Miller, Opt. Commun. 5 (3), 187 (1972).

Labeyrie, A.

D. Gezari, A. Labeyrie, and R. Stachnik, Astrophys. J. 173, L1-L5 (1 April 1972).

D. Gezari, A. Labeyrie, and R. Stachnik, Bull. Am. Astron. Soc. 3, 244 (1971).

Miller, M.

D. Korff, G. Dryden, and M. Miller, Opt. Commun. 5 (3), 187 (1972).

Stachnik, R.

D. Gezari, A. Labeyrie, and R. Stachnik, Astrophys. J. 173, L1-L5 (1 April 1972).

D. Gezari, A. Labeyrie, and R. Stachnik, Bull. Am. Astron. Soc. 3, 244 (1971).

Stanley, N. R.

R. Hufnagel and N. R. Stanley, J. Opt. Soc. Am. 54, 52 (1964), p. 56.

Tatarskii, V. I.

V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation (U.S. Dept. of Commerce, NTIS, Springfield, Va., 1971), p. 235.

Taylor, L. S.

L. S. Taylor, J. Opt. Soc. Am. 58, 705 (1968).

Wolf, E.

Cf. M. Born and E. Wolf, Principles of Optics, 2nd ed. (Pergamon, New York, 1964).

Other (17)

D. Gezari, A. Labeyrie, and R. Stachnik, Bull. Am. Astron. Soc. 3, 244 (1971).

D. Fried, J. Opt. Soc. Am. 56, 1372 (1966).

The seeing angle can be estimated from Fried's definition (Ref. 2) of infinite-aperture long-exposure resolution and is typically of the order of 1 arc second for zenith viewing.

It may be shown that <τ(0)> = 1, because the atmosphere does not absorb light, but only redistributes it. Hence, IT = <∫ I(x) dx> (i.e., the averaged integrated irradiance of the image) is unaffected by the turbulence. If <|τ(0)|2> = 1, so that the normalized <|τ(f)|2> equals the unnormalized <|τ(f)|2>, this would imply that <I2T> is unaffected by the turbulence. This statement is, in general, not true, because <I2T> ≠ <IT>2—i.e., the atmosphere produces scintillation. Nevertheless, it is approximately true if geometric optics is valid and/or the aperture averaging effectively eliminates the scintillation. This latter condition will, in general, be satisfied for values of D much greater than r0.

Cf. M. Born and E. Wolf, Principles of Optics, 2nd ed. (Pergamon, New York, 1964).

L. S. Taylor, J. Opt. Soc. Am. 58, 705 (1968).

V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation (U.S. Dept. of Commerce, NTIS, Springfield, Va., 1971), p. 235.

R. Hufnagel and N. R. Stanley, J. Opt. Soc. Am. 54, 52 (1964), p. 56.

Reference 2, p. 1374.

Reference 7, p. 239, Eqs. 36a and 37.

Reference 2, p. 1377.

Reference 2, p. 1366.

Reference 7, pp. 74–102.

R. Hufnagel, Restoration of Atmospherically Degraded Images Woods Hole Summer Study, Vol. 2, App. 3 (Defense Documentation Center, Alexandria, Va., 1966).

Reference 5, Sec. 10.4.

D. Korff, G. Dryden, and M. Miller, Opt. Commun. 5 (3), 187 (1972).

D. Gezari, A. Labeyrie, and R. Stachnik, Astrophys. J. 173, L1-L5 (1 April 1972).

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