Abstract

The standard method of stellar speckle interferometry, in which short exposure photographs are individually analyzed, is not the most general method of extracting object information from the time-varying image intensity. We introduce a space-time analysis in which both spatial and temporal fluctuations are taken into account; the aim is to measure the power spectrum of the image with an increased signal to noise ratio. Surprisingly, our more general space-time analysis does not yield an improved signal to noise ratio at very low light levels.

© 1980 Optical Society of America

PDF Article

References

  • View by:
  • |
  • |
  • |

  1. A. Labeyrie, "High resolution techniques in optical astronomy," in Progress in Optics, edited by E. Wolf (North-Holland, Amsterdam, 1976), Vol. 14, pp. 47–87.
  2. J. C. Dainty, "Stellar speckle interferometry," in Laser Speckle and Related Phenomena, Topics in Applied Physics, Vol. 9, edited by J. C. Dainty (Springer-Verlag, New York, 1975), pp. 255–280.
  3. S. P. Worden, "Astronomical image reconstruction in astronomy," Vistas Astron. 20, 301–317 (1977).
  4. D. Korff, G. Dryden, and M. G. Miller, "Information retrival from atmospheric induced speckle patterns," Opt. Commun. 5, 187–192 (1972).
  5. D. Korff, "Analysis of a method for obtaining near-diffraction-limited information in the presence of atmospheric turbulence," J. Opt. Soc. Am. 63, 971–980 (1973).
  6. J. C. Dainty, "Diffraction limited imaging of stellar objects using telescopes of low optical quality," Opt. Commun. 7, 129–134 (1973).
  7. J. W. Goodman and J. F. Belsher, "Photon limited images and their restoration," Technical Report RADC-TR-76-50 (ARPA Order No. 2646), Rome Air Development Center, Griffiss AFB, New York 13441 (unpublished).
  8. J. C. Dainty and A. H. Greenaway, "Estimation of spatial power spectra in speckle interferometry," J. Opt. Soc. Am 69, 786–790 (1979).
  9. M. G. Miller, "Noise considerations in stellar speckle interferometry," J. Opt. Soc. Am. 67, 1176–1184 (1977).
  10. F. Roddier, "Signal to noise ratio in speckle interferometry," Imaging in Astronomy, Conference Proceedings, Boston, June 1975 (unpublished).
  11. D. L. Fried, "Analysis of techniques for imaging through the atmosphere," Technical Report RADC-TR-79-190, Rome Air Development Center, Griffiss AFB, New York 13441 (unpublished).
  12. The quantity Α(0,t) does in fact fluctuate owing to scintillation and this effect is small if Dr0. Because of this (small) fluctuation ømeas (0) does not exactly equal the square of the power; this is discussed by Korff.5.
  13. Details of this transformation may be found in A. Papoulis, Probability, Random Variables and Stochastic Processes, (McGraw-Hill, New York, 1965), p. 325.
  14. C. Roddier and F. Roddier, "Influence of exposure time on the spectral properties of turbulence-degraded astronomical images," J. Opt. Soc. Am. 65, 664–667 (1975).
  15. R. J. Scaddan and J. G. Walker, "Statistics of stellar speckle patterns," Appl. Opt. 17, 3779–3784 (1978).
  16. G. Parry, J. G. Walker, and R. J. Scaddan, "On the statistics of stellar speckle patterns and pupil plane scintillation," Opt. Acta 62, 563–574 (1979).
  17. J. G. Walker, "Optimum exposure time and filter bandwidth in speckle interferometry," IAU Colloquium No. 50, High Angular Resolution Stellar Interferometry, Maryland, Aug.–Sept. 1978 (unpublished).
  18. We are not aware of any rigorous proof that Λ(u,t) is a circular complex Gaussian random process; it can be shown to be a circular complex Gaussian random variable, for frequencies greater than the reciprocal of the width of the point-spread function, by the following heuristic argument. Using the autocorrelation theorem of Fourier transform theory, Λ(u,t) α ∫DA*(ξ) A (ξ + λƒu) d ξ where A(ξ) is the complex amplitude within the telescope pupil. For atmospheric turbulence, the phase of A*(ξ) is distributed uniformly between -π and π and so is that of the product A*(ξ)A(ξ + λƒu) for the case λƒu > r0. Provided that Dr0, we can thus invoke the central limit theorem to show that Λ(u,t) is a circular complex Gaussian random variable.
  19. L. Mertz, "Speckle imaging, photon by photon," Appl. Opt. 18, 611–614 (1979).
  20. L. Mandel, "Fluctuations of photon beams and their correlations," Proc. Phys. Soc. 72, 1037–1048 (1958).

1979

J. C. Dainty and A. H. Greenaway, "Estimation of spatial power spectra in speckle interferometry," J. Opt. Soc. Am 69, 786–790 (1979).

G. Parry, J. G. Walker, and R. J. Scaddan, "On the statistics of stellar speckle patterns and pupil plane scintillation," Opt. Acta 62, 563–574 (1979).

L. Mertz, "Speckle imaging, photon by photon," Appl. Opt. 18, 611–614 (1979).

1978

1977

M. G. Miller, "Noise considerations in stellar speckle interferometry," J. Opt. Soc. Am. 67, 1176–1184 (1977).

S. P. Worden, "Astronomical image reconstruction in astronomy," Vistas Astron. 20, 301–317 (1977).

1975

1973

D. Korff, "Analysis of a method for obtaining near-diffraction-limited information in the presence of atmospheric turbulence," J. Opt. Soc. Am. 63, 971–980 (1973).

J. C. Dainty, "Diffraction limited imaging of stellar objects using telescopes of low optical quality," Opt. Commun. 7, 129–134 (1973).

1972

D. Korff, G. Dryden, and M. G. Miller, "Information retrival from atmospheric induced speckle patterns," Opt. Commun. 5, 187–192 (1972).

1958

L. Mandel, "Fluctuations of photon beams and their correlations," Proc. Phys. Soc. 72, 1037–1048 (1958).

Belsher, J. F.

J. W. Goodman and J. F. Belsher, "Photon limited images and their restoration," Technical Report RADC-TR-76-50 (ARPA Order No. 2646), Rome Air Development Center, Griffiss AFB, New York 13441 (unpublished).

Dainty, J. C.

J. C. Dainty and A. H. Greenaway, "Estimation of spatial power spectra in speckle interferometry," J. Opt. Soc. Am 69, 786–790 (1979).

J. C. Dainty, "Diffraction limited imaging of stellar objects using telescopes of low optical quality," Opt. Commun. 7, 129–134 (1973).

J. C. Dainty, "Stellar speckle interferometry," in Laser Speckle and Related Phenomena, Topics in Applied Physics, Vol. 9, edited by J. C. Dainty (Springer-Verlag, New York, 1975), pp. 255–280.

Dryden, G.

D. Korff, G. Dryden, and M. G. Miller, "Information retrival from atmospheric induced speckle patterns," Opt. Commun. 5, 187–192 (1972).

Fried, D. L.

D. L. Fried, "Analysis of techniques for imaging through the atmosphere," Technical Report RADC-TR-79-190, Rome Air Development Center, Griffiss AFB, New York 13441 (unpublished).

Goodman, J. W.

J. W. Goodman and J. F. Belsher, "Photon limited images and their restoration," Technical Report RADC-TR-76-50 (ARPA Order No. 2646), Rome Air Development Center, Griffiss AFB, New York 13441 (unpublished).

Greenaway, A. H.

J. C. Dainty and A. H. Greenaway, "Estimation of spatial power spectra in speckle interferometry," J. Opt. Soc. Am 69, 786–790 (1979).

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, 971–980 (1973).

D. Korff, G. Dryden, and M. G. Miller, "Information retrival from atmospheric induced speckle patterns," Opt. Commun. 5, 187–192 (1972).

Labeyrie, A.

A. Labeyrie, "High resolution techniques in optical astronomy," in Progress in Optics, edited by E. Wolf (North-Holland, Amsterdam, 1976), Vol. 14, pp. 47–87.

Mandel, L.

L. Mandel, "Fluctuations of photon beams and their correlations," Proc. Phys. Soc. 72, 1037–1048 (1958).

Mertz, L.

Miller, M. G.

M. G. Miller, "Noise considerations in stellar speckle interferometry," J. Opt. Soc. Am. 67, 1176–1184 (1977).

D. Korff, G. Dryden, and M. G. Miller, "Information retrival from atmospheric induced speckle patterns," Opt. Commun. 5, 187–192 (1972).

Papoulis, A.

Details of this transformation may be found in A. Papoulis, Probability, Random Variables and Stochastic Processes, (McGraw-Hill, New York, 1965), p. 325.

Parry, G.

G. Parry, J. G. Walker, and R. J. Scaddan, "On the statistics of stellar speckle patterns and pupil plane scintillation," Opt. Acta 62, 563–574 (1979).

Roddier, C.

Roddier, F.

C. Roddier and F. Roddier, "Influence of exposure time on the spectral properties of turbulence-degraded astronomical images," J. Opt. Soc. Am. 65, 664–667 (1975).

F. Roddier, "Signal to noise ratio in speckle interferometry," Imaging in Astronomy, Conference Proceedings, Boston, June 1975 (unpublished).

Scaddan, R. J.

G. Parry, J. G. Walker, and R. J. Scaddan, "On the statistics of stellar speckle patterns and pupil plane scintillation," Opt. Acta 62, 563–574 (1979).

R. J. Scaddan and J. G. Walker, "Statistics of stellar speckle patterns," Appl. Opt. 17, 3779–3784 (1978).

Walker, J. G.

G. Parry, J. G. Walker, and R. J. Scaddan, "On the statistics of stellar speckle patterns and pupil plane scintillation," Opt. Acta 62, 563–574 (1979).

R. J. Scaddan and J. G. Walker, "Statistics of stellar speckle patterns," Appl. Opt. 17, 3779–3784 (1978).

J. G. Walker, "Optimum exposure time and filter bandwidth in speckle interferometry," IAU Colloquium No. 50, High Angular Resolution Stellar Interferometry, Maryland, Aug.–Sept. 1978 (unpublished).

Worden, S. P.

S. P. Worden, "Astronomical image reconstruction in astronomy," Vistas Astron. 20, 301–317 (1977).

Appl. Opt.

J. Opt. Soc. Am

J. C. Dainty and A. H. Greenaway, "Estimation of spatial power spectra in speckle interferometry," J. Opt. Soc. Am 69, 786–790 (1979).

J. Opt. Soc. Am.

Opt. Acta

G. Parry, J. G. Walker, and R. J. Scaddan, "On the statistics of stellar speckle patterns and pupil plane scintillation," Opt. Acta 62, 563–574 (1979).

Opt. Commun.

J. C. Dainty, "Diffraction limited imaging of stellar objects using telescopes of low optical quality," Opt. Commun. 7, 129–134 (1973).

D. Korff, G. Dryden, and M. G. Miller, "Information retrival from atmospheric induced speckle patterns," Opt. Commun. 5, 187–192 (1972).

Proc. Phys. Soc.

L. Mandel, "Fluctuations of photon beams and their correlations," Proc. Phys. Soc. 72, 1037–1048 (1958).

Vistas Astron.

S. P. Worden, "Astronomical image reconstruction in astronomy," Vistas Astron. 20, 301–317 (1977).

Other

A. Labeyrie, "High resolution techniques in optical astronomy," in Progress in Optics, edited by E. Wolf (North-Holland, Amsterdam, 1976), Vol. 14, pp. 47–87.

J. C. Dainty, "Stellar speckle interferometry," in Laser Speckle and Related Phenomena, Topics in Applied Physics, Vol. 9, edited by J. C. Dainty (Springer-Verlag, New York, 1975), pp. 255–280.

F. Roddier, "Signal to noise ratio in speckle interferometry," Imaging in Astronomy, Conference Proceedings, Boston, June 1975 (unpublished).

D. L. Fried, "Analysis of techniques for imaging through the atmosphere," Technical Report RADC-TR-79-190, Rome Air Development Center, Griffiss AFB, New York 13441 (unpublished).

The quantity Α(0,t) does in fact fluctuate owing to scintillation and this effect is small if Dr0. Because of this (small) fluctuation ømeas (0) does not exactly equal the square of the power; this is discussed by Korff.5.

Details of this transformation may be found in A. Papoulis, Probability, Random Variables and Stochastic Processes, (McGraw-Hill, New York, 1965), p. 325.

J. W. Goodman and J. F. Belsher, "Photon limited images and their restoration," Technical Report RADC-TR-76-50 (ARPA Order No. 2646), Rome Air Development Center, Griffiss AFB, New York 13441 (unpublished).

J. G. Walker, "Optimum exposure time and filter bandwidth in speckle interferometry," IAU Colloquium No. 50, High Angular Resolution Stellar Interferometry, Maryland, Aug.–Sept. 1978 (unpublished).

We are not aware of any rigorous proof that Λ(u,t) is a circular complex Gaussian random process; it can be shown to be a circular complex Gaussian random variable, for frequencies greater than the reciprocal of the width of the point-spread function, by the following heuristic argument. Using the autocorrelation theorem of Fourier transform theory, Λ(u,t) α ∫DA*(ξ) A (ξ + λƒu) d ξ where A(ξ) is the complex amplitude within the telescope pupil. For atmospheric turbulence, the phase of A*(ξ) is distributed uniformly between -π and π and so is that of the product A*(ξ)A(ξ + λƒu) for the case λƒu > r0. Provided that Dr0, we can thus invoke the central limit theorem to show that Λ(u,t) is a circular complex Gaussian random variable.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.