Abstract

Monte Carlo simulations of an atmospheric phase screen, based on a Kolmogorov spectrum of phase fluctuations, were performed. Speckle patterns produced from the phase screens were used to derive statistical properties of power spectra and bispectra of speckle interferograms. We present the bispectral modulation transfer function and its signal-to-noise ratio at high light levels. The results confirm the validity of a heuristic treatment based on an interferometric picture of speckle pattern formation in deriving the attenuation factor and the signal-to-noise ratio of the bispectral modulation transfer function in the mid-spatial-frequency range. The derived modulation transfer function is also interpreted in terms of the signal-to-noise ratio at low light levels. A general expression of the signal-to-noise ratio of the bispectrum is derived as a function of the transfer functions of the telescope, the number of speckles, and the mean photon counts in the mid-spatial-frequency range.

© 1988 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. K. T. Knox, B. J. Thompson, “Recovery of images from atmospherically degraded short-exposure photographs,” Astrophys. J. 193, L45–L48 (1974).
    [CrossRef]
  3. A. C. S. Readhead, P. N. Wilkinson, “The mapping of compact radio sources from VLBI data,” Astrophys. J. 223, 25–36 (1978).
    [CrossRef]
  4. A. W. Lohmann, G. Weigelt, B. Wirnitzer, “Speckle masking in astronomy: triple correlation theory and applications,” Appl. Opt. 22, 4028–4037 (1983).
    [CrossRef] [PubMed]
  5. K.-H. Hofmann, G. Weigelt, “Speckle masking observation of the central object in the giant HII region NGC 3603,” Astron. Astrophys. 167, L15–L16 (1986).
  6. G. L. Rogers, “The process of image formation as the re-transformation of the partial coherence pattern of the object,” Proc. Phys. Soc. 81, 323–331 (1963).
    [CrossRef]
  7. 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).
    [CrossRef]
  8. F. Roddier, “Triple correlation as a phase closure technique,” Opt. Commun. 60, 145–149 (1986).
    [CrossRef]
  9. S. N. Karbelkar, R. Nityananda, “Atmospheric noise on the bispectrum in optical speckle interferometry,” J. Astrophys. Astr. 8, 271–274 (1987).
    [CrossRef]
  10. A. C. S. Readhead, T. Nakajima, T. J. Pearson, G. Neugebauer, J. B. Oke, W. L. W. Sargent, “Diffraction limited imaging with ground-based optical telescopes,” Astron. J. 95, 1278–1296 (1988).
    [CrossRef]
  11. J. W. Goodman, J. F. Belsher, “Photon limited images and their restoration,” Tech. Rep. RADC-TR-76-50 (Rome Air Development Center, New York, 1976).
  12. J. W. Goodman, J. F. Belsher, “Precompensation and post-compensation of photon limited degraded images,” Tech. Rep. RADC-TR-76-382 (Rome Air Development Center, New York, 1976).
  13. J. W. Goodman, J. F. Belsher, “Photon limitations in imaging and image restoration,” Tech. Rep. RADC-TR-77-175 (Rome Air Development Center, New York, 1977).
  14. J. C. Dainty, A. H. Greenaway, “Estimation of spatial power spectra in speckle interferometry,” J. Opt. Soc. Am. 69, 786–790 (1979).
    [CrossRef]
  15. B. Wirnitzer, “Bispectral analysis at low light levels and astronomical speckle masking,” J. Opt. Soc. Am. A 2, 14–21 (1985).
    [CrossRef]
  16. A. N. Kolmogorov, “The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers,” in Turbulence, Classical Papers on Statistical Theory, S. K. Friedlander, L. Topper, eds. (Wiley-Interscience, New York, 1961).
  17. V. I. Tatarskii, Wave Propagation in a Turbulent Medium (McGraw-Hill, New York, 1961).
  18. V. I. Tatarskii, The Effects on the Turbulent Atmosphere on Wave Propagation (Nauka, Moscow, 1967). (Translated by Israel Program for Scientific Translations, Keter, Jerusalem, 1971.)
  19. J. Vernin, “Scidar measurements and model forecasting of free atmosphere turbulence,” in Proceedings of the Second Workshop on ESO’s Very Large Telescope, S. D’Odorico, J. P. Swings, eds. (European Southern Observatory, Garching, 1986), pp. 279–288.
  20. C. Roddier, F. Roddier, “Interferometric seeing measurements at La Silla,” in Proceedings of the Second Workshop on ESO’s Very Large Telescope, S. D’Odorico, J. P. Swings, eds. (European Southern Observatory, Garching, 1986), pp. 269–278.
  21. G. I. Taylor, “Statistical theory of turbulence,” in Turbulence, Classical Papers on Statistical Theory, S. K. Friedlander, L. Topper, eds. (Wiley-Interscience, New York, 1961).
  22. F. Roddier, “Atmospheric limitations to high angular resolution imaging,” in Proceedings of the ESO Conference on The Scientific Importance of High Angular Resolution at Infrared and Optical Wavelengths, M. H. Ulrich, K. Kjar, eds. (European Southern Observatory, Garching, 1981), pp. 5–23.
  23. J. W. Goodman, Statistical Optics (Wiley, New York, 1985).
  24. J. E. Baldwin, C. A. Haniff, C. D. Mackay, P. J. Warner, “Closure phase in high-resolution optical imaging,” Nature 320, 595–597 (1986).
    [CrossRef]
  25. C. A. Haniff, C. D. Mackay, D. J. Titterington, D. Sivia, J. E. Baldwin, P. J. Warner, “The first images from optical aperture synthesis,” Nature 328, 694–696 (1987).
    [CrossRef]
  26. S. R. Kulkarni, California Institute of Technology, Pasadena, California 91126 (personal communication, 1987).
  27. G. R. Ayers, M. J. Northcott, J. C. Dainty, “Knox-Thompson and triple-correlation imaging through atmospheric turbulence,” J. Opt. Soc. Am. A 5, 963–985 (1988).
    [CrossRef]

1988

A. C. S. Readhead, T. Nakajima, T. J. Pearson, G. Neugebauer, J. B. Oke, W. L. W. Sargent, “Diffraction limited imaging with ground-based optical telescopes,” Astron. J. 95, 1278–1296 (1988).
[CrossRef]

G. R. Ayers, M. J. Northcott, J. C. Dainty, “Knox-Thompson and triple-correlation imaging through atmospheric turbulence,” J. Opt. Soc. Am. A 5, 963–985 (1988).
[CrossRef]

1987

C. A. Haniff, C. D. Mackay, D. J. Titterington, D. Sivia, J. E. Baldwin, P. J. Warner, “The first images from optical aperture synthesis,” Nature 328, 694–696 (1987).
[CrossRef]

S. N. Karbelkar, R. Nityananda, “Atmospheric noise on the bispectrum in optical speckle interferometry,” J. Astrophys. Astr. 8, 271–274 (1987).
[CrossRef]

1986

J. E. Baldwin, C. A. Haniff, C. D. Mackay, P. J. Warner, “Closure phase in high-resolution optical imaging,” Nature 320, 595–597 (1986).
[CrossRef]

K.-H. Hofmann, G. Weigelt, “Speckle masking observation of the central object in the giant HII region NGC 3603,” Astron. Astrophys. 167, L15–L16 (1986).

F. Roddier, “Triple correlation as a phase closure technique,” Opt. Commun. 60, 145–149 (1986).
[CrossRef]

1985

1983

1979

1978

A. C. S. Readhead, P. N. Wilkinson, “The mapping of compact radio sources from VLBI data,” Astrophys. J. 223, 25–36 (1978).
[CrossRef]

1974

K. T. Knox, B. J. Thompson, “Recovery of images from atmospherically degraded short-exposure photographs,” Astrophys. J. 193, L45–L48 (1974).
[CrossRef]

1973

1970

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

1963

G. L. Rogers, “The process of image formation as the re-transformation of the partial coherence pattern of the object,” Proc. Phys. Soc. 81, 323–331 (1963).
[CrossRef]

Ayers, G. R.

Baldwin, J. E.

C. A. Haniff, C. D. Mackay, D. J. Titterington, D. Sivia, J. E. Baldwin, P. J. Warner, “The first images from optical aperture synthesis,” Nature 328, 694–696 (1987).
[CrossRef]

J. E. Baldwin, C. A. Haniff, C. D. Mackay, P. J. Warner, “Closure phase in high-resolution optical imaging,” Nature 320, 595–597 (1986).
[CrossRef]

Belsher, J. F.

J. W. Goodman, J. F. Belsher, “Photon limitations in imaging and image restoration,” Tech. Rep. RADC-TR-77-175 (Rome Air Development Center, New York, 1977).

J. W. Goodman, J. F. Belsher, “Photon limited images and their restoration,” Tech. Rep. RADC-TR-76-50 (Rome Air Development Center, New York, 1976).

J. W. Goodman, J. F. Belsher, “Precompensation and post-compensation of photon limited degraded images,” Tech. Rep. RADC-TR-76-382 (Rome Air Development Center, New York, 1976).

Dainty, J. C.

Goodman, J. W.

J. W. Goodman, J. F. Belsher, “Photon limitations in imaging and image restoration,” Tech. Rep. RADC-TR-77-175 (Rome Air Development Center, New York, 1977).

J. W. Goodman, Statistical Optics (Wiley, New York, 1985).

J. W. Goodman, J. F. Belsher, “Photon limited images and their restoration,” Tech. Rep. RADC-TR-76-50 (Rome Air Development Center, New York, 1976).

J. W. Goodman, J. F. Belsher, “Precompensation and post-compensation of photon limited degraded images,” Tech. Rep. RADC-TR-76-382 (Rome Air Development Center, New York, 1976).

Greenaway, A. H.

Haniff, C. A.

C. A. Haniff, C. D. Mackay, D. J. Titterington, D. Sivia, J. E. Baldwin, P. J. Warner, “The first images from optical aperture synthesis,” Nature 328, 694–696 (1987).
[CrossRef]

J. E. Baldwin, C. A. Haniff, C. D. Mackay, P. J. Warner, “Closure phase in high-resolution optical imaging,” Nature 320, 595–597 (1986).
[CrossRef]

Hofmann, K.-H.

K.-H. Hofmann, G. Weigelt, “Speckle masking observation of the central object in the giant HII region NGC 3603,” Astron. Astrophys. 167, L15–L16 (1986).

Karbelkar, S. N.

S. N. Karbelkar, R. Nityananda, “Atmospheric noise on the bispectrum in optical speckle interferometry,” J. Astrophys. Astr. 8, 271–274 (1987).
[CrossRef]

Knox, K. T.

K. T. Knox, B. J. Thompson, “Recovery of images from atmospherically degraded short-exposure photographs,” Astrophys. J. 193, L45–L48 (1974).
[CrossRef]

Kolmogorov, A. N.

A. N. Kolmogorov, “The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers,” in Turbulence, Classical Papers on Statistical Theory, S. K. Friedlander, L. Topper, eds. (Wiley-Interscience, New York, 1961).

Korff, D.

Kulkarni, S. R.

S. R. Kulkarni, California Institute of Technology, Pasadena, California 91126 (personal communication, 1987).

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).

Lohmann, A. W.

Mackay, C. D.

C. A. Haniff, C. D. Mackay, D. J. Titterington, D. Sivia, J. E. Baldwin, P. J. Warner, “The first images from optical aperture synthesis,” Nature 328, 694–696 (1987).
[CrossRef]

J. E. Baldwin, C. A. Haniff, C. D. Mackay, P. J. Warner, “Closure phase in high-resolution optical imaging,” Nature 320, 595–597 (1986).
[CrossRef]

Nakajima, T.

A. C. S. Readhead, T. Nakajima, T. J. Pearson, G. Neugebauer, J. B. Oke, W. L. W. Sargent, “Diffraction limited imaging with ground-based optical telescopes,” Astron. J. 95, 1278–1296 (1988).
[CrossRef]

Neugebauer, G.

A. C. S. Readhead, T. Nakajima, T. J. Pearson, G. Neugebauer, J. B. Oke, W. L. W. Sargent, “Diffraction limited imaging with ground-based optical telescopes,” Astron. J. 95, 1278–1296 (1988).
[CrossRef]

Nityananda, R.

S. N. Karbelkar, R. Nityananda, “Atmospheric noise on the bispectrum in optical speckle interferometry,” J. Astrophys. Astr. 8, 271–274 (1987).
[CrossRef]

Northcott, M. J.

Oke, J. B.

A. C. S. Readhead, T. Nakajima, T. J. Pearson, G. Neugebauer, J. B. Oke, W. L. W. Sargent, “Diffraction limited imaging with ground-based optical telescopes,” Astron. J. 95, 1278–1296 (1988).
[CrossRef]

Pearson, T. J.

A. C. S. Readhead, T. Nakajima, T. J. Pearson, G. Neugebauer, J. B. Oke, W. L. W. Sargent, “Diffraction limited imaging with ground-based optical telescopes,” Astron. J. 95, 1278–1296 (1988).
[CrossRef]

Readhead, A. C. S.

A. C. S. Readhead, T. Nakajima, T. J. Pearson, G. Neugebauer, J. B. Oke, W. L. W. Sargent, “Diffraction limited imaging with ground-based optical telescopes,” Astron. J. 95, 1278–1296 (1988).
[CrossRef]

A. C. S. Readhead, P. N. Wilkinson, “The mapping of compact radio sources from VLBI data,” Astrophys. J. 223, 25–36 (1978).
[CrossRef]

Roddier, C.

C. Roddier, F. Roddier, “Interferometric seeing measurements at La Silla,” in Proceedings of the Second Workshop on ESO’s Very Large Telescope, S. D’Odorico, J. P. Swings, eds. (European Southern Observatory, Garching, 1986), pp. 269–278.

Roddier, F.

F. Roddier, “Triple correlation as a phase closure technique,” Opt. Commun. 60, 145–149 (1986).
[CrossRef]

C. Roddier, F. Roddier, “Interferometric seeing measurements at La Silla,” in Proceedings of the Second Workshop on ESO’s Very Large Telescope, S. D’Odorico, J. P. Swings, eds. (European Southern Observatory, Garching, 1986), pp. 269–278.

F. Roddier, “Atmospheric limitations to high angular resolution imaging,” in Proceedings of the ESO Conference on The Scientific Importance of High Angular Resolution at Infrared and Optical Wavelengths, M. H. Ulrich, K. Kjar, eds. (European Southern Observatory, Garching, 1981), pp. 5–23.

Rogers, G. L.

G. L. Rogers, “The process of image formation as the re-transformation of the partial coherence pattern of the object,” Proc. Phys. Soc. 81, 323–331 (1963).
[CrossRef]

Sargent, W. L. W.

A. C. S. Readhead, T. Nakajima, T. J. Pearson, G. Neugebauer, J. B. Oke, W. L. W. Sargent, “Diffraction limited imaging with ground-based optical telescopes,” Astron. J. 95, 1278–1296 (1988).
[CrossRef]

Sivia, D.

C. A. Haniff, C. D. Mackay, D. J. Titterington, D. Sivia, J. E. Baldwin, P. J. Warner, “The first images from optical aperture synthesis,” Nature 328, 694–696 (1987).
[CrossRef]

Tatarskii, V. I.

V. I. Tatarskii, Wave Propagation in a Turbulent Medium (McGraw-Hill, New York, 1961).

V. I. Tatarskii, The Effects on the Turbulent Atmosphere on Wave Propagation (Nauka, Moscow, 1967). (Translated by Israel Program for Scientific Translations, Keter, Jerusalem, 1971.)

Taylor, G. I.

G. I. Taylor, “Statistical theory of turbulence,” in Turbulence, Classical Papers on Statistical Theory, S. K. Friedlander, L. Topper, eds. (Wiley-Interscience, New York, 1961).

Thompson, B. J.

K. T. Knox, B. J. Thompson, “Recovery of images from atmospherically degraded short-exposure photographs,” Astrophys. J. 193, L45–L48 (1974).
[CrossRef]

Titterington, D. J.

C. A. Haniff, C. D. Mackay, D. J. Titterington, D. Sivia, J. E. Baldwin, P. J. Warner, “The first images from optical aperture synthesis,” Nature 328, 694–696 (1987).
[CrossRef]

Vernin, J.

J. Vernin, “Scidar measurements and model forecasting of free atmosphere turbulence,” in Proceedings of the Second Workshop on ESO’s Very Large Telescope, S. D’Odorico, J. P. Swings, eds. (European Southern Observatory, Garching, 1986), pp. 279–288.

Warner, P. J.

C. A. Haniff, C. D. Mackay, D. J. Titterington, D. Sivia, J. E. Baldwin, P. J. Warner, “The first images from optical aperture synthesis,” Nature 328, 694–696 (1987).
[CrossRef]

J. E. Baldwin, C. A. Haniff, C. D. Mackay, P. J. Warner, “Closure phase in high-resolution optical imaging,” Nature 320, 595–597 (1986).
[CrossRef]

Weigelt, G.

K.-H. Hofmann, G. Weigelt, “Speckle masking observation of the central object in the giant HII region NGC 3603,” Astron. Astrophys. 167, L15–L16 (1986).

A. W. Lohmann, G. Weigelt, B. Wirnitzer, “Speckle masking in astronomy: triple correlation theory and applications,” Appl. Opt. 22, 4028–4037 (1983).
[CrossRef] [PubMed]

Wilkinson, P. N.

A. C. S. Readhead, P. N. Wilkinson, “The mapping of compact radio sources from VLBI data,” Astrophys. J. 223, 25–36 (1978).
[CrossRef]

Wirnitzer, B.

Appl. Opt.

Astron. Astrophys.

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

K.-H. Hofmann, G. Weigelt, “Speckle masking observation of the central object in the giant HII region NGC 3603,” Astron. Astrophys. 167, L15–L16 (1986).

Astron. J.

A. C. S. Readhead, T. Nakajima, T. J. Pearson, G. Neugebauer, J. B. Oke, W. L. W. Sargent, “Diffraction limited imaging with ground-based optical telescopes,” Astron. J. 95, 1278–1296 (1988).
[CrossRef]

Astrophys. J.

K. T. Knox, B. J. Thompson, “Recovery of images from atmospherically degraded short-exposure photographs,” Astrophys. J. 193, L45–L48 (1974).
[CrossRef]

A. C. S. Readhead, P. N. Wilkinson, “The mapping of compact radio sources from VLBI data,” Astrophys. J. 223, 25–36 (1978).
[CrossRef]

J. Astrophys. Astr.

S. N. Karbelkar, R. Nityananda, “Atmospheric noise on the bispectrum in optical speckle interferometry,” J. Astrophys. Astr. 8, 271–274 (1987).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Nature

J. E. Baldwin, C. A. Haniff, C. D. Mackay, P. J. Warner, “Closure phase in high-resolution optical imaging,” Nature 320, 595–597 (1986).
[CrossRef]

C. A. Haniff, C. D. Mackay, D. J. Titterington, D. Sivia, J. E. Baldwin, P. J. Warner, “The first images from optical aperture synthesis,” Nature 328, 694–696 (1987).
[CrossRef]

Opt. Commun.

F. Roddier, “Triple correlation as a phase closure technique,” Opt. Commun. 60, 145–149 (1986).
[CrossRef]

Proc. Phys. Soc.

G. L. Rogers, “The process of image formation as the re-transformation of the partial coherence pattern of the object,” Proc. Phys. Soc. 81, 323–331 (1963).
[CrossRef]

Other

S. R. Kulkarni, California Institute of Technology, Pasadena, California 91126 (personal communication, 1987).

J. W. Goodman, J. F. Belsher, “Photon limited images and their restoration,” Tech. Rep. RADC-TR-76-50 (Rome Air Development Center, New York, 1976).

J. W. Goodman, J. F. Belsher, “Precompensation and post-compensation of photon limited degraded images,” Tech. Rep. RADC-TR-76-382 (Rome Air Development Center, New York, 1976).

J. W. Goodman, J. F. Belsher, “Photon limitations in imaging and image restoration,” Tech. Rep. RADC-TR-77-175 (Rome Air Development Center, New York, 1977).

A. N. Kolmogorov, “The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers,” in Turbulence, Classical Papers on Statistical Theory, S. K. Friedlander, L. Topper, eds. (Wiley-Interscience, New York, 1961).

V. I. Tatarskii, Wave Propagation in a Turbulent Medium (McGraw-Hill, New York, 1961).

V. I. Tatarskii, The Effects on the Turbulent Atmosphere on Wave Propagation (Nauka, Moscow, 1967). (Translated by Israel Program for Scientific Translations, Keter, Jerusalem, 1971.)

J. Vernin, “Scidar measurements and model forecasting of free atmosphere turbulence,” in Proceedings of the Second Workshop on ESO’s Very Large Telescope, S. D’Odorico, J. P. Swings, eds. (European Southern Observatory, Garching, 1986), pp. 279–288.

C. Roddier, F. Roddier, “Interferometric seeing measurements at La Silla,” in Proceedings of the Second Workshop on ESO’s Very Large Telescope, S. D’Odorico, J. P. Swings, eds. (European Southern Observatory, Garching, 1986), pp. 269–278.

G. I. Taylor, “Statistical theory of turbulence,” in Turbulence, Classical Papers on Statistical Theory, S. K. Friedlander, L. Topper, eds. (Wiley-Interscience, New York, 1961).

F. Roddier, “Atmospheric limitations to high angular resolution imaging,” in Proceedings of the ESO Conference on The Scientific Importance of High Angular Resolution at Infrared and Optical Wavelengths, M. H. Ulrich, K. Kjar, eds. (European Southern Observatory, Garching, 1981), pp. 5–23.

J. W. Goodman, Statistical Optics (Wiley, New York, 1985).

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Figures (10)

Fig. 1
Fig. 1

Cross section of a simulated phase screen, plotted as a function of the x coordinate on the aperture plane.

Fig. 2
Fig. 2

Two-dimensional coordinate system (ζ, η), adopted to represent a two-dimensional cross section of the four-dimensional bispectrum b ˜ ( 3 ) ( ζ , 0 , 0 , η ), and that of the SNR. u1 = (ζ, 0) and u2 = (0, η) are perpendicular to each other, and the third spatial frequency −u1u2 = (−ζ, −η) has the largest modulus (ζ2 + η2)1/2. Therefore the circle ζ2 + η2 = (D/λ)2 forms the boundary, where D/λ is the telescope-cutoff frequency.

Fig. 3
Fig. 3

Contour map of the normalized bispectral MTF b ˜ ( 3 ) ( ζ , 0 , 0 , η ) for a 2-m telescope, drawn on the ζη plane in a logarithmic scale. The numbers labeling contours indicate powers of 10.

Fig. 4
Fig. 4

Contour map of the normalized OTF of the telescope, t ˜ ( 3 ) ( ζ , 0 , 0 , η ), drawn on the ζη plane in a logarithmic scale. The numbers labeling contours indicate powers of 10.

Fig. 5
Fig. 5

Contour map of the attenuation factor, ã(3)(ζ, 0, 0, η) for a 2-m telescope, drawn on the ζη plane in a logarithmic scale. The numbers labeling contours indicate powers of 10.

Fig. 6
Fig. 6

Contour map of the SNR of the bispectral MTF SNR [ b ˜ ( 3 ) ( ζ , 0 , 0 , η ) ] for a single frame obtained with a 2-m telescope, drawn on the ζη plane. This map shows the saturated SNR at high light levels.

Fig. 7
Fig. 7

Power-spectrum atmospheric transfer function |ã(u)|2, plotted as a function of the modulus of the radial spatial frequency |u|.

Fig. 8
Fig. 8

Contour map of the SNR for a V = 12.3 magnitude star obtained with a 2-m telescope after integrating 104 frames, assuming a 10% observing efficiency, a 10% fractional bandwidth and a 10-msec integration time. These brightness and observing efficiencies correspond to 1 photon per speckle.

Fig. 9
Fig. 9

Behavior of 3σ contours, plotted according to the light levels. Magnitudes are calculated for the same conditions as for Fig. 8.

Fig. 10
Fig. 10

Light level and spatial-frequency dependence of the SNR n s 1 / 2 · SNR [ Q ˜ ( 3 ) ( x · D / λ , 0 , 0 , x · D / λ ) ], which is independent of the size of the telescope, plotted as a function of the mean photon counts per speckle n ¯ and the normalized spatial frequency x. Since ns1/2 = D/rc, the SNR for a telescope with a diameter of D meters can be obtained by lowering the whole plot by log(D/rc), as indicated by the arrow.

Equations (112)

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

F S ( κ r ) = 0.0332 π k 2 [ C n 2 ( L ) d L ] κ r 11 / 3 ,
B ˜ j ( 3 ) ( u 1 , u 2 ) = I ˜ j ( u 1 ) I ˜ j ( u 2 ) I ˜ j ( u 1 u 2 ) ,
j = 1 n B ˜ j ( 3 ) ( u 1 , u 2 ) ,
j = 1 n | B ˜ j ( 3 ) ( u 1 , u 2 ) | 2 ,
B ˜ ( 3 ) ( u 1 , u 2 ) = j = 1 n B ˜ j ( 3 ) ( u 1 , u 2 ) n ,
σ 2 [ B ˜ 3 ( u 1 , u 2 ) ] = j = 1 n | B ˜ j ( 3 ) ( u 1 , u 2 ) | 2 n | B ˜ 3 ( u 1 , u 2 ) | 2 n 1 .
SNR [ B ˜ ( 3 ) ( u 1 , u 2 ) ] = Re [ B ˜ ( 3 ) ( u 1 , u 2 ) ] { σ 2 [ B ˜ ( 3 ) ( u 1 , u 2 ) ] } 1 / 2 .
C n 2 ( L ) d L = 5 × 10 13 m 1 / 3 ,
B ˜ ( 3 ) ( ζ , 0 , 0 , η ) = I ˜ ( ζ , 0 ) I ˜ ( 0 , η ) I ˜ ( ζ , η ) .
ζ 2 + η 2 = ( D / λ ) 2 ,
b ˜ ( 3 ) ( u 1 , u 2 ) = B ˜ ( 3 ) ( u 1 , u 2 ) B ˜ ( 3 ) ( 0 , 0 ) .
B ˜ ( 3 ) ( 0 , 0 ) = I ˜ ( 0 ) 3 = const .,
b ˜ ( 3 ) ( u 1 , u 2 ) = i ˜ ( u 1 ) i ˜ ( u 2 ) i ˜ ( u 1 u 2 ) ,
SNR [ b ˜ ( 3 ) ( u 1 , u 2 ) ] = SNR [ B ˜ ( 3 ) ( u 1 , u 2 ) ] .
b ˜ ( 3 ) ( ζ , 0 , 0 , 0 ) = i ˜ ( ζ , 0 ) i ˜ ( 0 , 0 ) i ˜ ( ζ , 0 ) = i ˜ ( ζ , 0 ) 1 i ˜ ( ζ , 0 ) * = | i ˜ ( ζ , 0 ) | 2 ,
b ˜ ( 3 ) ( 0 , 0 , 0 , η ) = | i ˜ ( 0 , η ) | 2 .
a ˜ ( 3 ) ( ζ , 0 , 0 , η ) = b ˜ ( 3 ) ( ζ , 0 , 0 , η ) t ˜ ( 3 ) ( ζ , 0 , 0 , η ) ,
a ˜ ( 3 ) ( u 1 , u 2 ) { SNR [ b ˜ ( 3 ) ( u 1 , u 2 ) ] } 4 .
b ˜ ( 3 ) ( u 1 , u 2 ) × N ¯ 3 / 2 ,
D j ( x ) = k = 1 N j δ ( x x k ) ,
D ˜ j ( u ) = k = 1 N j δ ( x x k ) exp ( i ux ) d x = k = 1 N j exp ( i ux k ) .
D ˜ j ( 3 ) ( u 1 , u 2 ) = D ˜ j ( u 1 ) D ˜ j ( u 2 ) D ˜ j ( u 1 u 2 ) = k = 1 N j l = 1 N j m = 1 N j exp { i [ u 1 ( x k x m ) + u 2 ( x l x m ) ] } .
p j ( x ) = λ j ( x ) λ j ( x ) d x = I j ( x ) I j ( x ) d x .
p ˜ j ( u ) = p j ( x ) exp ( i ux ) d x = I j ( x ) exp ( i ux ) d x I j ( x ) d x = I ˜ j ( u ) I ˜ j ( 0 ) = i ˜ j ( u ) .
E k l m [ D ˜ j ( 3 ) ( u 1 , u 2 ) ] = E k l m ( k = 1 N j l = 1 N j m = 1 N j exp { i [ u 1 ( x k x m ) + u 2 ( x l x m ) ] } ) = k = 1 N j l = 1 N j m = 1 N j E k l m ( exp { i [ u 1 ( x k x m ) + u 2 ( x l x m ) ] } ) ,
E k l m ( 1 ) = p j ( x k ) d x k = 1.
E k l m { exp [ i u 1 ( x k k l ) ] } = exp [ i u 1 ( x k x l ) ] p j ( x k ) p j ( x l ) d x k d x l = [ p j ( x k ) exp ( i u 1 x k ) d x k ] [ p j ( x l ) exp ( i u 1 x l ) d x l ] = i ˜ j ( u 1 ) i ˜ j ( u 1 ) = | i ˜ j ( u 1 ) | 2 ,
E k l m { exp [ i u 1 ( x l x m ) ] } = | i ˜ j ( u 2 ) | 2 .
E k l m { exp [ i ( u 1 u 2 ) ( x l x m ) ] } = | i ˜ j ( u 1 u 2 ) | 2 .
E k l m ( exp { i [ u 1 ( x k x m ) + u 2 ( x l x m ) ] } ) = [ p j ( x k ) exp ( i u 1 x k ) d x k ] [ p j ( x l ) exp ( i u 2 x l ) d x l ] × ( p j ( x m ) exp [ i ( u 1 u 2 ) x m ] d x m ) = i ˜ j ( u 1 ) i ˜ j ( u 2 ) i ˜ j ( u 1 u 2 ) = b ˜ j ( 3 ) ( u 1 , u 2 ) ,
E k l m [ D ˜ j ( 3 ) ( u 1 , u 2 ) ] = N j + N j ( N j 1 ) [ | i ˜ j ( u 1 ) | 2 + | i ˜ j ( u 2 ) | 2 + | i ˜ j ( u 1 u 2 ) | 2 ] + N j ( N j 1 ) ( N j 2 ) × b ˜ j ( 3 ) ( u 1 , u 2 ) .
E [ N j ( N j 1 ) ( N j r + 1 ) ] = N ¯ j r ,
E k l m , N j [ D ˜ j ( 3 ) ( u 1 , u 2 ) ] = N ¯ j + N ¯ j 2 [ | i ˜ j ( u 1 ) | 2 + | i ˜ j ( u 2 ) | 2 + | i ˜ j ( u 1 u 2 ) | 2 ] + N ¯ j 3 b ˜ j ( 3 ) ( u 1 , u 2 ) .
E [ D ˜ ( 3 ) ( u 1 , u 2 ) ] = N ¯ j + N ¯ j 2 [ | i ˜ ( u 1 ) | 2 + | i ˜ ( u 2 ) | 2 + | i ˜ ( u 1 u 2 ) | 2 ] + N ¯ j 3 b ˜ ( 3 ) ( u 1 , u 2 ) .
E ( N ¯ j r ) = E ( N j ) r = N ¯ r ,
E [ D ˜ ( 3 ) ( u 1 , u 2 ) ] = N ¯ + N ¯ 2 [ | i ˜ ( u 1 ) | 2 + | i ˜ ( u 2 ) | 2 + | i ˜ ( u 1 u 2 ) | 2 ] + N ¯ 3 b ˜ ( 3 ) ( u 1 , u 2 ) .
E [ | D ˜ ( u ) | 2 ] = N ¯ + N ¯ 2 | i ˜ ( u ) | 2
N ¯ 3 b ˜ ( 3 ) ( u 1 , u 2 ) = E { D ˜ ( 3 ) ( u 1 , u 2 ) [ | D ˜ ( u 1 ) | 2 + | D ˜ ( u 2 ) | 2 + | D ˜ ( u 1 u 2 ) | 2 2 N ¯ ] } .
Q ˜ j ( 3 ) ( u 1 , u 2 ) = D ˜ j ( 3 ) ( u 1 , u 2 ) [ | D ˜ j ( u 1 ) | 2 + | D ˜ j ( u 2 ) | 2 + | D ˜ j ( u 1 u 2 ) | 2 2 N ¯ j ] ,
Q ˜ j ( 3 ) ( u 1 , u 2 ) = k l m exp { i [ u 1 ( x k x m ) + u 2 ( x l x m ) ] } .
σ 2 [ Q ˜ j ( 3 ) ( u 1 , u 2 ) ] = N ¯ 3 [ 1 + | i ˜ ( u 1 u 2 ) | 2 + | i ˜ ( 2 u 1 + u 2 ) | 2 + | i ˜ ( u 1 + 2 u 2 ) | 2 + b ˜ ( 3 ) ( u 1 u 2 , u 1 + 2 u 2 ) + b ˜ ( 3 ) ( 2 u 1 + u 2 , u 1 + u 2 ) ] + N ¯ 4 [ | i ˜ ( u 1 ) | 2 + | i ˜ ( u 2 ) | 2 + | i ˜ ( u 1 u 2 ) | 2 + b ˜ ( 3 ) ( u 1 , u 2 ) + c . c . + b ˜ ( 3 ) ( u 1 + u 2 , u 1 ) + c . c . + b ˜ ( 3 ) ( u 2 , u 1 + u 2 ) + c . c . + | i ˜ ( u 1 ) | 2 | i ˜ ( u 1 + 2 u 2 ) | 2 + | i ˜ ( u 2 ) | 2 | i ˜ ( 2 u 1 + u 2 ) | 2 + | i ˜ ( u 1 + u 2 ) | 2 | i ˜ ( u 1 u 2 ) | 2 + f ˜ ( 4 ) ( u 1 u 2 , u 1 + 2 u 2 , u 1 u 2 ) + c . c . + f ˜ ( 4 ) ( u 1 , u 1 + 2 u 2 , 2 u 1 u 2 ) + c . c . + f ˜ ( 4 ) ( 2 u 1 + u 2 , u 1 + u 2 , u 1 u 2 ) + c . c . ] + N ¯ 5 { | i ˜ ( u 1 ) | 2 | i ˜ ( u 1 u 2 ) | 2 + | i ˜ ( u 2 ) | 2 | i ˜ ( u 1 u 2 ) | 2 + | i ˜ ( u 1 ) | 2 | i ˜ ( u 2 ) | 2 + | i ˜ ( u 1 ) | 2 [ b ˜ ( 3 ) ( u 2 , u 1 + u 2 ) + c . c . ] + | i ˜ ( u 2 ) | 2 [ b ˜ ( 3 ) ( u 1 , u 1 + u 2 ) + c . c . ] + | i ˜ ( u 1 u 2 ) | 2 [ b ˜ ( 3 ) ( u 1 , u 2 ) + c . c . ] } + N ¯ 6 [ | b ˜ ( 3 ) ( u 1 , u 2 ) | 2 | b ˜ ( 3 ) ( u 1 , u 2 ) | 2 ] ,
f ˜ ( 4 ) ( u 1 , u 2 , u 3 ) = i ˜ ( u 1 ) i ˜ ( u 2 ) i ˜ ( u 3 ) i ˜ ( u 1 u 2 u 3 )
SNR [ Q ˜ ( 3 ) ( u 1 , u 2 ) ] = N ¯ 3 b ˜ ( 3 ) ( u 1 , u 2 ) { σ 2 [ Q ˜ ( 3 ) ( u 1 , u 2 ) ] } 1 / 2 .
| i ˜ ( u ) | 2 = n s 1 | t ˜ ( u ) | 2 ,
b ˜ ( 3 ) ( u 1 , u 2 ) = n s 2 t ˜ ( 3 ) ( u 1 , u 2 ) ,
f ˜ ( 4 ) ( u 1 , u 2 , u 3 ) = n s 3 t ˜ ( 4 ) ( u 1 , u 2 , u 3 ) ,
| i ˜ ( u 1 ) | 2 | i ˜ ( u 2 ) | 2 = n s 2 | t ˜ ( u 1 ) | 2 | t ˜ ( u 2 ) | 2 ,
| i ˜ ( u 1 ) | 2 b ˜ ( 3 ) ( u 2 , u 3 ) = n s 3 | t ˜ ( u 1 ) | 2 t ˜ ( 3 ) ( u 2 , u 3 ) ,
| b ˜ ( 3 ) ( u 1 , u 2 ) | 2 = n s 3 | t ˜ ( 3 ) ( u 1 , u 2 ) | 2 ,
| b ˜ ( 3 ) ( u 1 , u 2 ) | 2 = n s 4 | t ˜ ( 3 ) ( u 1 , u 2 ) | 2 .
σ 2 [ Q ˜ ( 3 ) ( u 1 , u 2 ) ] = N ¯ 3 + N ¯ 4 n s 1 × [ | t ˜ ( u 1 ) | 2 + | t ˜ ( u 2 ) | 2 + | t ˜ ( u 1 u 2 ) | 2 ] + N ¯ 5 n s 2 [ | t ˜ ( u 1 ) | 2 | t ˜ ( u 1 u 2 ) | 2 + | t ˜ ( u 2 ) | 2 | t ˜ ( u 1 u 2 ) | 2 + | t ˜ ( u 1 ) | 2 | t ˜ ( u 2 ) | 2 ] + N ¯ 6 n s 3 t ˜ ( 3 ) ( u 1 , u 2 ) .
SNR [ Q ˜ ( 3 ) ( u 1 , u 2 ) ] = N ¯ 3 n s 2 t ˜ ( 3 ) ( u 1 , u 2 ) { σ 2 [ Q ˜ ( 3 ) ( u 1 , u 2 ) ] } 1 / 2 .
SNR [ Q ˜ ( 3 ) ( u 1 , u 2 ) ] = n s 1 / 2 × n ¯ 3 / 2 × t ˜ ( 3 ) ( u 1 , u 2 ) A 1 / 2 ,
A = 1 + n ¯ ( | t ˜ ( u 1 ) | 2 + | t ˜ ( u 2 ) | 2 + | t ˜ ( u 1 u 2 ) | 2 ) + n ¯ 2 ( | t ˜ ( u 1 ) | 2 | t ˜ ( u 2 ) | 2 + | t ˜ ( u 1 ) | 2 | t ˜ ( u 1 u 2 ) | 2 + | t ˜ ( u 2 ) | 2 × | t ˜ ( u 1 u 2 ) | 2 ) + n ¯ 3 | t ˜ ( 3 ) ( u 1 , u 2 ) | 2
SNR ( map ) = ( π 2 32 n s 2 ) 1 / 2 × SNR [ Q ˜ ( 3 ) ( u 1 , u 2 ) ] ¯ Z 1 / 2 ,
SNR [ Q ˜ ( 3 ) ( u 1 , u 2 ) ] n s 1 / 2 n ¯ 3 / 2 t ˜ ( 3 ) ( u 1 , u 2 ) ,
SNR ( map ) = ( π 2 32 ) 1 / 2 ( n s n ¯ 3 ) 1 / 2 t ˜ ( 3 ) ( u 1 , u 2 ) ¯ × Z 1 / 2 0.027 ( n s n ¯ 3 ) 1 / 2 Z 1 / 2 ,
m lim = 13.3 + 2.5 { log ( Δ λ / λ 0.1 ) + log ( η 0.1 ) + log ( Δ τ 10 msec ) + 4 3 log ( r c 14 cm ) + 2 3 log ( D 1 m ) + 1 3 log ( Z 10 4 ) 2 3 log [ SNR ( map ) 10 ] } .
I ˜ ( u 1 ) = k = 1 N ( u 1 ) exp ( i Φ k ) ,
| I ˜ ( u 1 ) | 2 = k = 1 N ( u 1 ) l = 1 N ( u 1 ) exp [ i ( Φ k Φ l ) ] = k = 1 N ( u 1 ) l = 1 N ( u 1 ) exp [ i ( Φ k Φ l ) ] .
exp [ i ( Φ k Φ l ) ] = 1 , k = l , with N ( u 1 ) terms = 0 , k l , with N ( u 1 ) [ N   ( u 1 ) 1 ] terms ;
| I ˜ ( u 1 ) | 2 = N ( u 1 ) .
| I ˜ ( u 1 ) | 2 = N ( u 1 ) 2 .
I ˜ ( 3 ) ( u 1 , u 2 ) = k = 1 N ( u 1 ) exp ( i Φ 12 , k ) l = 1 N ( u 2 ) exp ( i Φ 23 , l ) m = 1 N ( u 1 u 2 ) exp ( i Φ 31 , m ) = k = 1 N ( u 1 ) l = 1 N ( u 2 ) m = 1 N ( u 1 u 2 ) exp [ i ( Φ 12 , k + Φ 23 , l + Φ 31 , m ) ] ,
I ˜ ( 3 ) ( u 1 , u 2 ) = Min [ N ( u 1 ) , N ( u 2 ) , N ( u 1 u 2 ) ] ,
I ˜ ( 3 ) ( u 1 , u 2 ) = N ( u 1 ) N ( u 2 ) N ( u 1 u 2 ) ;
Min [ N ( u 1 ) , N ( u 2 ) , N ( u 1 u 2 ) ] N ( u 1 ) N ( u 2 ) N ( u 1 u 2 ) .
Min [ N ( u 1 ) , N ( u 2 ) , N ( u 1 u 2 ) ] [ N ( u 1 ) N ( u 2 ) N ( u 1 u 2 ) ] 1 / 2 .
a ˜ ( 4 ) ( u 1 , u 2 , u 3 ) = k = 1 N ( u 1 ) l = 1 N ( u 2 ) m = 1 N ( u 3 ) n = 1 N ( u 1 u 2 u 3 ) exp [ i ˜ ( Φ 12 , k + Φ 23 , l + Φ 34 , m + Φ 41 , n ) ] N ( u 1 ) N ( u 2 ) N ( u 3 ) N ( u 1 u 2 u 3 ) = Min [ N ( u 1 ) , N ( u 2 ) , N ( u 3 ) , N ( u 1 u 2 u 3 ) ] N ( u 1 ) N ( u 2 ) N ( u 3 ) N ( u 1 u 2 u 3 ) n s 3 ,
Φ 12 , k + Φ 23 , k + Φ 34 , k + Φ 41 , k = ( Φ 12 , k + Φ 23 , k + Φ 31 , k ) + ( Φ 13 , k + Φ 34 , k + Φ 41 , k ) = 0.
| a ˜ ( u 1 ) | 2 | a ˜ ( u 2 ) | 2 = k = 1 N ( u 1 ) l = 1 N ( u 1 ) m = 1 N ( u 2 ) n = 1 N ( u 2 ) exp [ i ( Φ 12 , k Φ 12 , l + Φ 34 , m Φ 34 , n ) ] N ( u 1 ) 2 N ( u 2 ) 2 = N ( u 1 ) N ( u 2 ) N ( u 1 ) 2 N ( u 2 ) 2 n s 2 ,
exp [ i ( Φ 12 , k Φ 12 , l + Φ 34 , m Φ 34 , n ) ] = 1 ;
| a ˜ ( u 1 ) | 2 a ˜ ( 3 ) ( u 2 , u 3 ) = k = 1 N ( u 1 ) l = 1 N ( u 1 ) m = 1 N ( u 2 ) n = 1 N ( u 3 ) o = 1 N ( u 2 u 3 ) exp [ i ( Φ 12 , k Φ 12 , l + Φ 34 , m + Φ 45 , n + Φ 53 , o ) ] / N ( u 1 ) 2 N ( u 2 ) N ( u 3 ) N ( u 2 u 3 ) = N ( u 1 ) × Min [ N ( u 2 ) , N ( u 3 ) , N ( u 2 u 3 ) ] N ( u 1 ) 2 N ( u 2 ) N ( u 3 ) N ( u 2 u 3 ) n s 3 ;
exp [ i ( Φ 23 , k Φ 12 , l + Φ 34 , m + Φ 45 , n + Φ 53 , o ) ] = 1 ,
| a ˜ ( 3 ) ( u 1 , u 2 ) | 2 = k = 1 N ( u 1 ) l = 1 N ( u 2 ) m = 1 N ( u 1 u 2 ) n = 1 N ( u 1 ) o = 1 N ( u 2 ) p = 1 N ( u 1 u 2 ) exp [ i ( Φ 12 , k + Φ 23 , l + Φ 31 , m Φ 12 , n Φ 23 , o Φ 31 , p ) ] / N ( u 1 ) 2 N ( u 2 ) 2 N ( u 1 u 2 ) 2 = N ( u 1 ) N ( u 2 ) N ( u 1 u 2 ) + Min [ N ( u 2 ) , N ( u 3 ) , N ( u 2 u 3 ) ] 2 N ( u 1 ) 2 N ( u 3 ) 2 N ( u 1 u 2 ) 2 n s 3 ;
exp [ i ( Φ 12 , k + Φ 23 , l + Φ 31 , m Φ 12 , n Φ 23 , o Φ 31 , p ) ] = 1.
E α β γ δ ζ ( α β γ δ ζ exp { i [ u 1 ( x α x γ x δ + x ζ ) + u 2 ( x β x γ x + x ζ ) ] } ) = α β γ δ ζ E α β γ δ ζ ( exp { i [ u 1 ( x α x γ x δ + x ζ ) + u 2 ( x β x γ x + x ζ ) ] } ) ,
E α β γ δ ζ ( 1 ) = 1.
E α β γ δ ζ { exp [ i ( u 1 u 2 ) ( x α x β ) ] } = | i ˜ j ( u 1 u 2 ) | 2 .
E α β γ δ ζ { exp [ i ( 2 u 1 + u 2 ) ( x α x γ ) ] } = | i ˜ j ( 2 u 1 + u 2 ) | 2 .
E α β γ δ ζ { exp [ i ( u 1 + 2 u 2 ) ( x β x γ ) ] } = | i ˜ j ( u 1 + 2 u 2 ) | 2 .
E α β γ δ ζ ( exp { i [ ( u 1 u 2 ) x α + ( u 1 + 2 u 2 ) x β + ( 2 u 1 u 2 ) x γ ] } ) = b ˜ j ( 3 ) ( u 1 u 2 , u 1 + 2 u 2 ) .
E α β γ δ ζ ( exp { i [ ( 2 u 1 + u 2 ) x α + ( u 1 + u 2 ) x β + ( u 1 2 u 2 ) x γ ] } ) = b ˜ j ( 3 ) ( 2 u 1 + u 2 , u 1 + u 2 ) .
E α β γ δ ζ { exp [ i u 1 ( x α x δ ) ] } = | i ˜ j ( u 1 ) | 2 .
E α β γ δ ζ { exp [ i u 2 ( x β x ) ] } = | i ˜ j ( u 2 ) | 2 .
E α β γ δ ζ { exp [ i ( u 1 u 2 ) ( x γ x ζ ) ] } = | i ˜ j ( u 1 u 2 ) | 2 .
E α β γ δ ζ ( exp { i [ u 1 x α + ( u 1 + u 2 ) x β u 2 x ] } ) = b ˜ j ( 3 ) ( u 1 , u 2 ) .
E α β γ δ ζ ( exp { i [ ( u 1 u 2 ) x α + u 2 x β u 1 x δ ] } ) = b ˜ j ( 3 ) ( u 1 , u 2 ) = b ˜ j ( 3 ) ( u 1 , u 2 ) * .
E α β γ δ ζ ( exp { i [ u 2 x β ( u 1 + 2 u 2 ) x γ + ( u 1 + u 2 ) x ζ ] } ) = b ˜ j ( 3 ) ( u 1 + u 2 , u 2 ) .
E α β γ δ ζ ( exp { i [ ( u 1 + 2 u 2 ) x β ( u 1 + u 2 ) x γ + u 2 x ] } ) = b ˜ j ( 3 ) ( u 1 u 2 , u 2 ) = b ˜ j ( 3 ) ( u 1 + u 2 , u 2 ) * .
E α β γ δ ζ ( exp { i [ u 1 x α ( 2 u 1 + u 2 ) x γ + ( u 1 + u 2 ) x ζ ] } ) = b ˜ j ( 3 ) ( u 1 + u 2 , u 1 ) .
E α β γ δ ζ ( exp { i [ ( 2 u 1 + u 2 ) x α ( u 1 + u 2 ) x γ u 1 x δ ] } ) = b ˜ j ( 3 ) ( u 1 u 2 , u 1 ) = b ˜ j ( 3 ) ( u 1 + u 2 , u 1 ) * .
E α β γ δ ζ ( exp { i [ u 1 ( x α x δ ) + ( u 1 + 2 u 2 ) ( x β x γ ) ] } ) = | i ˜ j ( u 1 ) | 2 | i ˜ j ( u 1 + 2 u 2 ) | 2 .
E α β γ δ ζ ( exp { i [ u 2 ( x β x ) + ( 2 u 1 + u 2 ) ( x α x γ ) ] } ) = | i ˜ j ( u 2 ) | 2 | i ˜ j ( 2 u 1 + u 2 ) | 2 .
E α β γ δ ζ ( exp { i [ ( u 1 u 2 ) ( x α x β ) ( u 1 + u 2 ) ( x γ x ζ ) ] } ) = | i ˜ j ( u 1 u 2 ) | 2 | i ˜ j ( u 1 + u 2 ) | 2 .
E α β γ δ ζ ( exp { i [ ( u 1 u 2 ) x α + ( u 1 + 2 u 2 ) x β ( u 1 + u 2 ) x γ u 1 x δ ] } ) = i ˜ j ( u 1 u 2 ) i ˜ j ( u 1 + 2 u 2 ) i ˜ j ( u 1 u 2 ) i ˜ j ( u 1 ) = f ˜ j ( 4 ) ( u 1 u 2 , u 1 + 2 u 2 , u 1 u 2 ) .
E α β γ δ ζ ( exp { i [ u 1 x α + ( u 1 + u 2 ) x β ( u 1 + 2 u 2 ) x γ + ( u 1 + u 2 ) x δ ] } ) = i ˜ j ( u 1 ) i ˜ j ( u 1 + u 2 ) i ˜ j ( u 1 2 u 2 ) i ˜ j ( u 1 + u 2 ) = f ˜ j ( 4 ) ( u 1 u 2 , u 1 + 2 u 2 , u 1 u 2 ) * .
E α β γ δ ζ ( exp { i [ u 1 x α + ( u 1 + 2 u 2 ) x β ( 2 u 1 + u 2 ) x γ u 2 x ] } ) = f ˜ j ( 4 ) ( u 1 , u 1 + 2 u 2 , 2 u 1 u 2 ) .
E α β γ δ ζ ( exp { i [ ( 2 u 1 + u 2 ) x α + u 2 x β ( u 1 + 2 u 2 ) x γ u 1 x δ ] } ) = f ˜ j ( 4 ) ( u 1 , u 1 + 2 u 2 , 2 u 1 u 2 ) * .
E α β γ δ ζ ( exp { i [ ( 2 u 1 + u 2 ) x α + ( u 1 + u 2 ) x β ( u 1 + u 2 ) x γ u 2 x ] } ) = f ˜ j ( 4 ) ( 2 u 1 + u 2 , u 1 + u 2 , u 1 u 2 ) .
E α β γ δ ζ ( exp { i [ ( u 1 u 2 ) x α + u 2 x β + ( 2 u 1 + u 2 ) x γ + ( u 1 + u 2 ) x ] } ) = f ˜ j ( 4 ) ( 2 u 1 + u 2 , u 1 + u 2 , u 1 u 2 ) * .
E α β γ δ ζ ( exp { i [ u 1 ( x α x δ ) + u 2 ( x β x ) ] } ) = | i ˜ j ( u 1 ) | 2 | i ˜ j ( u 2 ) | 2 .
E α β γ δ ζ ( exp { i [ u 1 ( x α x δ ) ( u 1 + u 2 ) ( x γ x ζ ) ] } ) = | i ˜ j ( u 1 ) | 2 | i ˜ j ( u 1 u 2 ) | 2 .
E α β γ δ ζ ( exp { i [ u 2 ( x β x ) ( u 1 + u 2 ) ( x γ x ζ ) ] } ) = | i ˜ j ( u 2 ) | 2 | i ˜ j ( u 1 u 2 ) | 2 .
E α β γ δ ζ ( exp { i [ u 1 ( x α x δ ) + u 2 ( x β x γ ) + ( u 1 + u 2 ) ( x ζ x γ ) ] } ) = | i ˜ j ( u 1 ) | 2 b ˜ j ( 3 ) ( u 2 , u 1 + u 2 ) .
E α β γ δ ζ ( exp { i [ u 1 ( x α x δ ) u 2 ( x x β ) ( u 1 + u 2 ) ( x γ x β ) ] } ) = | i ˜ j ( u 1 ) | 2 b ˜ j ( 3 ) ( u 2 , u 1 + u 2 ) *
E α β γ δ ζ ( exp { i [ u 2 ( x β x ) + u 1 ( x α x γ ) + ( u 1 + u 2 ) ( x ζ x γ ) ] } ) = | i ˜ j ( u 2 ) | 2 b ˜ j ( 3 ) ( u 1 , u 1 + u 2 ) .
E α β γ δ ζ ( exp { i [ u 2 ( x β x ) u 1 ( x δ x α ) ( u 1 + u 2 ) ( x γ x α ) ] } ) = | i ˜ j ( u 2 ) | 2 b ˜ j ( 3 ) ( u 1 , u 1 + u 2 ) * .
E α β γ δ ζ ( exp { i [ ( u 1 + u 2 ) ( x γ x ζ ) + u 1 ( x α x β ) u 2 ( x x β ) ] } ) = | i ˜ j ( u 1 u 2 ) | 2 b ˜ j ( 3 ) ( u 1 , u 2 ) .
E α β γ δ ζ ( exp { i [ ( u 1 + u 2 ) ( x γ x ζ ) u 1 ( x δ x α ) + u 2 ( x β x α ) ] } ) = | i ˜ j ( u 1 u 2 ) | 2 b ˜ j ( 3 ) ( u 1 , u 2 ) * .
E α β γ δ ζ ( exp { i [ u 1 ( x α x δ ) + u 2 ( x β x ) ( u 1 + u 2 ) ( x γ x ζ ) ] } ) = | i ˜ j ( u 1 ) | 2 | i ˜ j ( u 2 ) | 2 | i ˜ j ( u 1 u 2 ) | 2 = | b ˜ j ( 3 ) ( u 1 , u 2 ) | 2
N ¯ + N ¯ 2 + 2 ( N ¯ 3 + 2 N ¯ 2 ) [ | ( u 2 ) | 2 + | i ˜ ( u 2 ) | 2 + | i ˜ ( u 1 u 2 ) | 2 ] + ( N ¯ 4 + 3 N ¯ 3 ) [ b ˜ ( 3 ) ( u 1 , u 2 ) + c . c . ] + ( N ¯ 4 + 4 N ¯ 3 + 2 N ¯ 2 ) [ | i ˜ ( u 1 ) | 4 + | i ˜ ( u 2 ) | 4 + | i ˜ ( u 1 u 2 ) | 4 + 2 | i ˜ ( u 1 ) | 2 | i ˜ ( u 2 ) | 2 + | i ˜ ( u 2 ) | 2 | i ˜ ( u 1 u 2 ) | 2 + | i ˜ ( u 1 u 2 ) | 2 | i ˜ ( u 2 ) | 2 ] + ( N ¯ 5 + 6 N ¯ 4 + 6 N ¯ 3 ) × [ b ˜ ( 3 ) ( u 1 , u 2 ) + c . c . ] [ | i ˜ ( u 1 ) | 2 + | i ˜ ( u 2 ) | 2 + | i ˜ ( u 1 u 2 ) | 2 ] .

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