Abstract

Due to the turbulent atmosphere the resolution of conventional astrophotography is limited to ∼1 sec of arc. However, the speckle-masking method can yield diffraction-limited resolution, i.e., 0.03 sec of arc with a 3.6-m telescope. Speckle masking yields true images of general astronomical objects. No point source is required in the isoplanatic field of the object. We present the theory of speckle masking; it makes use of triple correlations and their Fourier counterparts, the bispectra. We show algorithms for the recovery of the object from genuine astronomical bispectra data.

© 1983 Optical Society of America

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References

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  1. A. Labeyrie, Astron. Astrophys. 6, 85 (1970).
  2. C. Y. C. Liu, A. W. Lohmann, Opt. Commun. 8, 372 (1973).
    [CrossRef]
  3. R. H. T. Bates, P. T. Gough, P. J. Napier, Astron. Astrophys. 22, 319 (1973).
  4. G. P. Weigelt, Appl. Opt. 17, 2660 (1978).
    [CrossRef] [PubMed]
  5. C. R. Lynds, S. P. Worden, J. W. Harvey, Astrophys. J. 207, 174 (1976).
    [CrossRef]
  6. K. T. Knox, B. J. Thompson, Astrophys. J. Lett. 193, L45 (1974).
    [CrossRef]
  7. D. C. Ehn, P. Nisenson, J. Opt. Soc. Am. 65, 1196 (1975).
  8. G. P. Weigelt, Opt. Commun. 21, 55 (1977).
    [CrossRef]
  9. R. H. T. Bates, M. O. Milner, G. I. Lund, A. D. Seager, Opt. Commun. 26, 22 (1978).
    [CrossRef]
  10. K. von der Heide, Astron. Astrophys. 70, 777 (1978).
  11. J. R. Fienup, Opt. Lett. 3, 27 (1978).
    [CrossRef] [PubMed]
  12. B. J. Brames, J. C. Dainty, J. Opt. Soc. Am. 71, 1542 (1981).
    [CrossRef]
  13. J. G. Walker, Appl. Opt. 21, 3132 (1982).
    [CrossRef] [PubMed]
  14. L. N. Mertz, Appl. Opt. 18, 611 (1979).
    [CrossRef] [PubMed]
  15. W. J. Cocke, Proc. Soc. Photo-Opt. Instrum. Eng. 231, 11 (1980).
  16. C. Roddier, F. Roddier, J. Vernin, in Proceedings, ESO Conference on Scientific Importance of High Angular Resolution at IR and Optical Wavelengths, M. H. Ulrich, K. Kjär, Eds. (ESO, Garching, Germany, 1981), p. 165.
  17. A. M. J. Huiser, Opt. Commun. 42, 226 (1982).
    [CrossRef]
  18. A. H. Greenaway, Opt. Commun. 42, 157 (1982).
    [CrossRef]
  19. R. H. T. Bates, F. M. Cady, Opt. Commun. 32, 365 (1980).
    [CrossRef]
  20. G. P. Weigelt, B. Wirnitzer, Opt. Lett. 8, 389 (1983).
    [CrossRef] [PubMed]
  21. A. W. Lohmann, G. Weigelt, in Proceedings, ESA/ESO Workshop on Astronomical Uses of the Space Telescope,F. Macchetto, F. Pacini, M. Tarenghi, Eds. (ESO, Geneva, 1979), p. 353.
  22. G. Weigelt, Proc. Soc. Photo-Opt. Instrum. Eng. 332, 284 (1982).
  23. G. Weigelt, in Proceedings, Conference on Image Processing in Astronomy,G. Sedmak, M. Capaccioli, R. J. Allen, Eds. (Osservatorio Astronomico di Trieste, Trieste, Italy, 1979), p. 422.
  24. J. C. Dainty, Opt. Commun. 7, 129 (1973).
    [CrossRef]
  25. G. Weigelt, in Proceedings, ESO Conference on Scientific Importance of High Angular Resolution at IR and Optical Wavelengths,M. H. Ulrich, K. Kjar, Eds. (ESO, Garching, Germany, 1981), p. 95.
  26. T. R. Bader, Proc. Soc. Photo-Opt. Instrum. Eng. 185, 140 (1979).
  27. G. Baier, G. Weigelt, Astron. Astrophys. 121, 137 (1983).
  28. J. C. Dainty, in Laser Speckle and Related Phenomena,J. C. Dainty, Ed. (Springer, Berlin, 1975).
  29. H. H. Barrett, W. Swindell, Radiological Imaging: Theory and Image Formation, Detection and Processing (Academic, New York, 1981), Vols. 1 and 2.
  30. K. Sasaki, T. Sato, Y. Namamura, IEEE Trans. Sonics Ultrasón. SU-24, 193 (1977).
    [CrossRef]
  31. H. Gamo, J. Appl. Phys. 34, 875 (1963).
    [CrossRef]
  32. D. Casasent, G. Silbershatz, Appl. Opt. 21, 2076 (1982).
    [CrossRef] [PubMed]
  33. J. Cohen, Proc. Soc. Photo-Opt. Instrum. Eng. 180, 134 (1979).
  34. P. Kellman, Opt. Eng. 19, 370 (1980).
    [CrossRef]
  35. J. W. Goodman, Aust. J. Electr. Electron. Eng. 2, 140 (1982).
  36. I. S. Reed, IRE Trans. Inf. Theory IT-8, 194 (1964).

1983 (2)

G. Baier, G. Weigelt, Astron. Astrophys. 121, 137 (1983).

G. P. Weigelt, B. Wirnitzer, Opt. Lett. 8, 389 (1983).
[CrossRef] [PubMed]

1982 (6)

D. Casasent, G. Silbershatz, Appl. Opt. 21, 2076 (1982).
[CrossRef] [PubMed]

J. G. Walker, Appl. Opt. 21, 3132 (1982).
[CrossRef] [PubMed]

J. W. Goodman, Aust. J. Electr. Electron. Eng. 2, 140 (1982).

A. M. J. Huiser, Opt. Commun. 42, 226 (1982).
[CrossRef]

A. H. Greenaway, Opt. Commun. 42, 157 (1982).
[CrossRef]

G. Weigelt, Proc. Soc. Photo-Opt. Instrum. Eng. 332, 284 (1982).

1981 (1)

1980 (3)

P. Kellman, Opt. Eng. 19, 370 (1980).
[CrossRef]

R. H. T. Bates, F. M. Cady, Opt. Commun. 32, 365 (1980).
[CrossRef]

W. J. Cocke, Proc. Soc. Photo-Opt. Instrum. Eng. 231, 11 (1980).

1979 (3)

T. R. Bader, Proc. Soc. Photo-Opt. Instrum. Eng. 185, 140 (1979).

J. Cohen, Proc. Soc. Photo-Opt. Instrum. Eng. 180, 134 (1979).

L. N. Mertz, Appl. Opt. 18, 611 (1979).
[CrossRef] [PubMed]

1978 (4)

G. P. Weigelt, Appl. Opt. 17, 2660 (1978).
[CrossRef] [PubMed]

J. R. Fienup, Opt. Lett. 3, 27 (1978).
[CrossRef] [PubMed]

R. H. T. Bates, M. O. Milner, G. I. Lund, A. D. Seager, Opt. Commun. 26, 22 (1978).
[CrossRef]

K. von der Heide, Astron. Astrophys. 70, 777 (1978).

1977 (2)

G. P. Weigelt, Opt. Commun. 21, 55 (1977).
[CrossRef]

K. Sasaki, T. Sato, Y. Namamura, IEEE Trans. Sonics Ultrasón. SU-24, 193 (1977).
[CrossRef]

1976 (1)

C. R. Lynds, S. P. Worden, J. W. Harvey, Astrophys. J. 207, 174 (1976).
[CrossRef]

1975 (1)

D. C. Ehn, P. Nisenson, J. Opt. Soc. Am. 65, 1196 (1975).

1974 (1)

K. T. Knox, B. J. Thompson, Astrophys. J. Lett. 193, L45 (1974).
[CrossRef]

1973 (3)

C. Y. C. Liu, A. W. Lohmann, Opt. Commun. 8, 372 (1973).
[CrossRef]

R. H. T. Bates, P. T. Gough, P. J. Napier, Astron. Astrophys. 22, 319 (1973).

J. C. Dainty, Opt. Commun. 7, 129 (1973).
[CrossRef]

1970 (1)

A. Labeyrie, Astron. Astrophys. 6, 85 (1970).

1964 (1)

I. S. Reed, IRE Trans. Inf. Theory IT-8, 194 (1964).

1963 (1)

H. Gamo, J. Appl. Phys. 34, 875 (1963).
[CrossRef]

Bader, T. R.

T. R. Bader, Proc. Soc. Photo-Opt. Instrum. Eng. 185, 140 (1979).

Baier, G.

G. Baier, G. Weigelt, Astron. Astrophys. 121, 137 (1983).

Barrett, H. H.

H. H. Barrett, W. Swindell, Radiological Imaging: Theory and Image Formation, Detection and Processing (Academic, New York, 1981), Vols. 1 and 2.

Bates, R. H. T.

R. H. T. Bates, F. M. Cady, Opt. Commun. 32, 365 (1980).
[CrossRef]

R. H. T. Bates, M. O. Milner, G. I. Lund, A. D. Seager, Opt. Commun. 26, 22 (1978).
[CrossRef]

R. H. T. Bates, P. T. Gough, P. J. Napier, Astron. Astrophys. 22, 319 (1973).

Brames, B. J.

Cady, F. M.

R. H. T. Bates, F. M. Cady, Opt. Commun. 32, 365 (1980).
[CrossRef]

Casasent, D.

Cocke, W. J.

W. J. Cocke, Proc. Soc. Photo-Opt. Instrum. Eng. 231, 11 (1980).

Cohen, J.

J. Cohen, Proc. Soc. Photo-Opt. Instrum. Eng. 180, 134 (1979).

Dainty, J. C.

B. J. Brames, J. C. Dainty, J. Opt. Soc. Am. 71, 1542 (1981).
[CrossRef]

J. C. Dainty, Opt. Commun. 7, 129 (1973).
[CrossRef]

J. C. Dainty, in Laser Speckle and Related Phenomena,J. C. Dainty, Ed. (Springer, Berlin, 1975).

Ehn, D. C.

D. C. Ehn, P. Nisenson, J. Opt. Soc. Am. 65, 1196 (1975).

Fienup, J. R.

Gamo, H.

H. Gamo, J. Appl. Phys. 34, 875 (1963).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Aust. J. Electr. Electron. Eng. 2, 140 (1982).

Gough, P. T.

R. H. T. Bates, P. T. Gough, P. J. Napier, Astron. Astrophys. 22, 319 (1973).

Greenaway, A. H.

A. H. Greenaway, Opt. Commun. 42, 157 (1982).
[CrossRef]

Harvey, J. W.

C. R. Lynds, S. P. Worden, J. W. Harvey, Astrophys. J. 207, 174 (1976).
[CrossRef]

Huiser, A. M. J.

A. M. J. Huiser, Opt. Commun. 42, 226 (1982).
[CrossRef]

Kellman, P.

P. Kellman, Opt. Eng. 19, 370 (1980).
[CrossRef]

Knox, K. T.

K. T. Knox, B. J. Thompson, Astrophys. J. Lett. 193, L45 (1974).
[CrossRef]

Labeyrie, A.

A. Labeyrie, Astron. Astrophys. 6, 85 (1970).

Liu, C. Y. C.

C. Y. C. Liu, A. W. Lohmann, Opt. Commun. 8, 372 (1973).
[CrossRef]

Lohmann, A. W.

C. Y. C. Liu, A. W. Lohmann, Opt. Commun. 8, 372 (1973).
[CrossRef]

A. W. Lohmann, G. Weigelt, in Proceedings, ESA/ESO Workshop on Astronomical Uses of the Space Telescope,F. Macchetto, F. Pacini, M. Tarenghi, Eds. (ESO, Geneva, 1979), p. 353.

Lund, G. I.

R. H. T. Bates, M. O. Milner, G. I. Lund, A. D. Seager, Opt. Commun. 26, 22 (1978).
[CrossRef]

Lynds, C. R.

C. R. Lynds, S. P. Worden, J. W. Harvey, Astrophys. J. 207, 174 (1976).
[CrossRef]

Mertz, L. N.

Milner, M. O.

R. H. T. Bates, M. O. Milner, G. I. Lund, A. D. Seager, Opt. Commun. 26, 22 (1978).
[CrossRef]

Namamura, Y.

K. Sasaki, T. Sato, Y. Namamura, IEEE Trans. Sonics Ultrasón. SU-24, 193 (1977).
[CrossRef]

Napier, P. J.

R. H. T. Bates, P. T. Gough, P. J. Napier, Astron. Astrophys. 22, 319 (1973).

Nisenson, P.

D. C. Ehn, P. Nisenson, J. Opt. Soc. Am. 65, 1196 (1975).

Reed, I. S.

I. S. Reed, IRE Trans. Inf. Theory IT-8, 194 (1964).

Roddier, C.

C. Roddier, F. Roddier, J. Vernin, in Proceedings, ESO Conference on Scientific Importance of High Angular Resolution at IR and Optical Wavelengths, M. H. Ulrich, K. Kjär, Eds. (ESO, Garching, Germany, 1981), p. 165.

Roddier, F.

C. Roddier, F. Roddier, J. Vernin, in Proceedings, ESO Conference on Scientific Importance of High Angular Resolution at IR and Optical Wavelengths, M. H. Ulrich, K. Kjär, Eds. (ESO, Garching, Germany, 1981), p. 165.

Sasaki, K.

K. Sasaki, T. Sato, Y. Namamura, IEEE Trans. Sonics Ultrasón. SU-24, 193 (1977).
[CrossRef]

Sato, T.

K. Sasaki, T. Sato, Y. Namamura, IEEE Trans. Sonics Ultrasón. SU-24, 193 (1977).
[CrossRef]

Seager, A. D.

R. H. T. Bates, M. O. Milner, G. I. Lund, A. D. Seager, Opt. Commun. 26, 22 (1978).
[CrossRef]

Silbershatz, G.

Swindell, W.

H. H. Barrett, W. Swindell, Radiological Imaging: Theory and Image Formation, Detection and Processing (Academic, New York, 1981), Vols. 1 and 2.

Thompson, B. J.

K. T. Knox, B. J. Thompson, Astrophys. J. Lett. 193, L45 (1974).
[CrossRef]

Vernin, J.

C. Roddier, F. Roddier, J. Vernin, in Proceedings, ESO Conference on Scientific Importance of High Angular Resolution at IR and Optical Wavelengths, M. H. Ulrich, K. Kjär, Eds. (ESO, Garching, Germany, 1981), p. 165.

von der Heide, K.

K. von der Heide, Astron. Astrophys. 70, 777 (1978).

Walker, J. G.

Weigelt, G.

G. Baier, G. Weigelt, Astron. Astrophys. 121, 137 (1983).

G. Weigelt, Proc. Soc. Photo-Opt. Instrum. Eng. 332, 284 (1982).

G. Weigelt, in Proceedings, ESO Conference on Scientific Importance of High Angular Resolution at IR and Optical Wavelengths,M. H. Ulrich, K. Kjar, Eds. (ESO, Garching, Germany, 1981), p. 95.

G. Weigelt, in Proceedings, Conference on Image Processing in Astronomy,G. Sedmak, M. Capaccioli, R. J. Allen, Eds. (Osservatorio Astronomico di Trieste, Trieste, Italy, 1979), p. 422.

A. W. Lohmann, G. Weigelt, in Proceedings, ESA/ESO Workshop on Astronomical Uses of the Space Telescope,F. Macchetto, F. Pacini, M. Tarenghi, Eds. (ESO, Geneva, 1979), p. 353.

Weigelt, G. P.

Wirnitzer, B.

Worden, S. P.

C. R. Lynds, S. P. Worden, J. W. Harvey, Astrophys. J. 207, 174 (1976).
[CrossRef]

Appl. Opt. (4)

Astron. Astrophys. (4)

G. Baier, G. Weigelt, Astron. Astrophys. 121, 137 (1983).

K. von der Heide, Astron. Astrophys. 70, 777 (1978).

A. Labeyrie, Astron. Astrophys. 6, 85 (1970).

R. H. T. Bates, P. T. Gough, P. J. Napier, Astron. Astrophys. 22, 319 (1973).

Astrophys. J. (1)

C. R. Lynds, S. P. Worden, J. W. Harvey, Astrophys. J. 207, 174 (1976).
[CrossRef]

Astrophys. J. Lett. (1)

K. T. Knox, B. J. Thompson, Astrophys. J. Lett. 193, L45 (1974).
[CrossRef]

Aust. J. Electr. Electron. Eng. (1)

J. W. Goodman, Aust. J. Electr. Electron. Eng. 2, 140 (1982).

IEEE Trans. Sonics Ultrasón. (1)

K. Sasaki, T. Sato, Y. Namamura, IEEE Trans. Sonics Ultrasón. SU-24, 193 (1977).
[CrossRef]

IRE Trans. Inf. Theory (1)

I. S. Reed, IRE Trans. Inf. Theory IT-8, 194 (1964).

J. Appl. Phys. (1)

H. Gamo, J. Appl. Phys. 34, 875 (1963).
[CrossRef]

J. Opt. Soc. Am. (2)

B. J. Brames, J. C. Dainty, J. Opt. Soc. Am. 71, 1542 (1981).
[CrossRef]

D. C. Ehn, P. Nisenson, J. Opt. Soc. Am. 65, 1196 (1975).

Opt. Commun. (7)

G. P. Weigelt, Opt. Commun. 21, 55 (1977).
[CrossRef]

R. H. T. Bates, M. O. Milner, G. I. Lund, A. D. Seager, Opt. Commun. 26, 22 (1978).
[CrossRef]

C. Y. C. Liu, A. W. Lohmann, Opt. Commun. 8, 372 (1973).
[CrossRef]

A. M. J. Huiser, Opt. Commun. 42, 226 (1982).
[CrossRef]

A. H. Greenaway, Opt. Commun. 42, 157 (1982).
[CrossRef]

R. H. T. Bates, F. M. Cady, Opt. Commun. 32, 365 (1980).
[CrossRef]

J. C. Dainty, Opt. Commun. 7, 129 (1973).
[CrossRef]

Opt. Eng. (1)

P. Kellman, Opt. Eng. 19, 370 (1980).
[CrossRef]

Opt. Lett. (2)

Proc. Soc. Photo-Opt. Instrum. Eng. (4)

J. Cohen, Proc. Soc. Photo-Opt. Instrum. Eng. 180, 134 (1979).

T. R. Bader, Proc. Soc. Photo-Opt. Instrum. Eng. 185, 140 (1979).

G. Weigelt, Proc. Soc. Photo-Opt. Instrum. Eng. 332, 284 (1982).

W. J. Cocke, Proc. Soc. Photo-Opt. Instrum. Eng. 231, 11 (1980).

Other (6)

C. Roddier, F. Roddier, J. Vernin, in Proceedings, ESO Conference on Scientific Importance of High Angular Resolution at IR and Optical Wavelengths, M. H. Ulrich, K. Kjär, Eds. (ESO, Garching, Germany, 1981), p. 165.

G. Weigelt, in Proceedings, Conference on Image Processing in Astronomy,G. Sedmak, M. Capaccioli, R. J. Allen, Eds. (Osservatorio Astronomico di Trieste, Trieste, Italy, 1979), p. 422.

G. Weigelt, in Proceedings, ESO Conference on Scientific Importance of High Angular Resolution at IR and Optical Wavelengths,M. H. Ulrich, K. Kjar, Eds. (ESO, Garching, Germany, 1981), p. 95.

A. W. Lohmann, G. Weigelt, in Proceedings, ESA/ESO Workshop on Astronomical Uses of the Space Telescope,F. Macchetto, F. Pacini, M. Tarenghi, Eds. (ESO, Geneva, 1979), p. 353.

J. C. Dainty, in Laser Speckle and Related Phenomena,J. C. Dainty, Ed. (Springer, Berlin, 1975).

H. H. Barrett, W. Swindell, Radiological Imaging: Theory and Image Formation, Detection and Processing (Academic, New York, 1981), Vols. 1 and 2.

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Figures (9)

Fig. 1
Fig. 1

Illustration of the imaging processing steps of the astronomical speckle-masking method.

Fig. 2
Fig. 2

Illustration of the triple correlation O(3)(x,x′ = s) = [O(x) · O(x + s)] ⊗ O(x). If the masking vector s is suitably selected, a true image of the object is obtained from its triple correlation. How the object triple correlation O(3)(x,x′) is obtained by evaluating speckle interferograms of the object is discussed in the text.

Fig. 3
Fig. 3

Illustration of some steps in the theory of speckle masking. The raw data for these experiments were recorded with the ESO 3.6-m telescope. We produced 1-D projections of the 2-D speckle interferograms in order to avoid a 4-D transfer function P n ( 3 ) . (a) Generalized speckle-masking transfer function P n ( 3 ) ( u , υ ) calculated from 1000 speckle interferograms. (b) Average (2-D) bispectrum Ĩ n ( 3 ) ( u , υ ) of 300 speckle interferograms of the spectroscopic double-star Omega Leonis. (c) Bispectrum Õ ( 3 ) ( u , υ ) = Ĩ n ( 3 ) ( u , υ ) / P n ( 3 ) ( u , υ ) of Omega Leonis. In all three figures the moduli of the complex values are displayed.

Fig. 4
Fig. 4

Computer simulation of (a) a double star, (b) the modulus of its bispectrum, and (c) its triple correlation.

Fig. 5
Fig. 5

Gaussian speckle-masking experiment of the close spectroscopic double-star Omega Leonis: (a) is one of the 300 evaluated speckle interferograms; (b) show the reconstruced true image of Omega Leonis. We measured (epoch: 1980.019); separation, 0.463 ± 0.004 sec of arc; position angle, 17.0 ± 1°; intensity ratio, 1.6 ± 0.3.

Fig. 6
Fig. 6

Complex bispectrum Õ p , q ( 3 ) of an object contains complete information about the modulus and the phase of the object spectrum |Õr| (exp [r]). The modulus information |Õr| can be reconstructed from one of the axes p = 0, q = 0, or p = q. The phase information (exp[r[) is contained in the area in between these distinguished axes. Because of the eightfold symmetry of the bispectrum of a real function, only one octant of the bispectrum contains nonredundant information, as indicated by the shaded area.

Fig. 7
Fig. 7

Result of the recursive image reconstruction (see III.B) of the spectroscopic double-star Omega Leonis (see also Fig. 5) from its bispectrum [see Fig. 3(c)]. How the recursive image reconstruction method is extended to 2-D objects is described in the text.

Fig. 8
Fig. 8

Generalized speckle-masking transfer function P n ( 3 ) ( u , υ ) is nonzero within the hexagonal area whose size is determined by the telescope cutoff frequency. The highest values are within the small hexagon surrounding the origin. This portion of the frequency domain is also accessible by long exposures. The three stripes along the two axes and along the (−45°) diagonal are medium high. The information (modulus or phase) contained in these three stripes is also obtainable by double correlations of short exposures as exploited by speckle interferometry1 and by Knox-Thompson's algorithm.6 The rest of the hexagon is a plateau at low level. This rest portion is the essential of the generalized speckle-masking transfer function, it permits the Fourier phase of the object to transmit. Since the phase information is obtained from a large area it has a reasonable signal-to-noise ratio.

Fig. 9
Fig. 9

Function B(3)(x,x′) which is defined in the text.

Equations (49)

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I n ( x ) = O ( x ) * P n ( x ) n = 1 , 2 , , N ,
| Ĩ n ( u ) | 2 = | Õ ( u ) | 2 | P n ( u ) | 2 ,
I n ( x ) I n ( x ) O ( x ) O ( x )
I n ( 3 ) ( x , x ) : = [ I n ( x ) I n ) x + x ) ] I n ( x ) = I n ( x ) I n ( x + x ) I n ( x + x ) d x
I n ( 3 ) ( x , x ) O ( 3 ) ( x , x ) ,
[ O ( x ) O ( x + s ) ] O ( x ) O ( x )
Ĩ n ( 3 ) ( u , v ) = Ĩ n ( u ) Ĩ n ( v ) Ĩ n ( u v ) ,
Ĩ n ( 3 ) ( u , v ) = Õ ( 3 ) ( u , v ) P n ( 3 ) ( u , v ) ,
phase { Ĩ n ( 3 ) ( u , v ) } = phase { Õ ( 3 ) ( u , v ) } .
Õ ( 3 ) ( u , v ) = Ĩ n ( 3 ) ( u , v ) / P n ( 3 ) ( u , v ) .
I n ( 3 ) ( x , x = s ) I nmm ( 3 ) ( x , x = s ) I nmn ( x , x = s ) I nnm ( 3 ) ( x , x = s ) + 2 I nmk ( 3 ) ( x , x = s ) = O ( 3 ) ( x , x = s ) O ( x ) ,
Õ p , q ( 3 ) = Õ p Õ q Õ p q p , q = N + N ,
Õ p = | Õ p | exp [ i φ p ] .
Õ O , q ( 3 ) = Õ O Õ q Õ q = const | Õ q | 2 ,
exp [ i φ r ] = exp [ i ( φ q + φ r q β r q , q ) ] ,
exp [ i φ r ] = exp [ i ( φ 1 + φ r 1 β r 1 , 1 ) ] , φ 0 = φ 1 = 0 , r = 2 N ,
φ 2 = 2 φ 1 β 1 , 1 , φ 3 = φ 2 + φ 1 β 2 , 1 = 3 φ 1 β 1 , 1 β 2 , 1 , φ r = r φ 1 β 1 , 1 β 2 , 1 β r 1 , 1 .
exp [ i φ r ] = const O < q r / 2 exp [ i ( φ q + φ r q β r q , q ) ] , φ 0 = φ 1 = 0 , r = 2 N ,
I nmk ( 3 ) ( x , x ) = I n ( x ) I m ( x + x ) I k ( x + x ) d x .
Ĩ nmk ( 3 ) ( u , v ) = I nmk ( 3 ) ( x , x ) exp [ 2 π i ( u x + v x ) ] d x d x .
Ĩ nmk ( 3 ) ( u , v ) = Ĩ n ( u ) Ĩ m ( v ) Ĩ k ( u v ) ,
Ĩ n ( 3 ) ( u , v ) = Ĩ nnn ( 3 ) ( u , v ) ,
Ĩ n ( 3 ) ( u , v ) = Ĩ n ( u ) Ĩ n ( v ) Ĩ n ( u v ) .
Ĩ n ( 3 ) ( u , v ) = Ĩ n ( 3 ) ( v , u ) ,
Ĩ n ( 3 ) ( u , v ) = Ĩ n ( 3 ) ( u v , v ) , Ĩ n ( 3 ) ( u , v ) = Ĩ n ( 3 ) * ( u , v ) if I n ( x ) is real .
I n ( x ) = O ( x ) * P n ( x ) = O ( x ) P n ( x x ) d x ,
Ĩ n ( 3 ) ( u , v ) = Õ ( 3 ) ( u , v ) P n ( 3 ) ( u , v ) .
I n ( 3 ) ( x , x ) = O ( 3 ) ( x , x ) * * P n ( 3 ) ( x , x ) .
P n ( u ) = H ̂ ( ξ ) H ̂ * ( ξ + ξ ) d ξ = H ̂ ( λ f u ) H ̂ * ( λ f u + λ f u ) d ( λ f u ) .
H ( u ) = H 0 ( u ) A ( u ) ,
P n ( u ) = A ( u ) A * ( u + u ) H 0 ( u ) H 0 * ( u + u ) d u ,
P n ( 3 ) ( u , υ ) = P n ( u ) P n ( υ ) P n ( u υ ) .
P N ( 3 ) ( u , υ ) = H 0 ( u ) H 0 * ( u + u ) H 0 ( υ ) H 0 * ( υ + υ ) H 0 ( ω ) H 0 * ( w u υ ) A ( u ) A * ( u + u ) A ( υ ) A * ( υ + υ ) A ( ω ) A * ( w u υ ) d u d υ d w .
A 1 A 2 * A 3 A 4 * A 5 A 6 * = A 1 A 2 * A 3 A 4 * A 5 A 6 * + A 1 A 2 * A 3 A 4 * A 5 A 6 * + A 1 A 2 * A 5 A 6 * A 3 A 4 * 2 A 1 A 2 * A 3 A 4 * A 5 A 6 * + A 1 A 4 * A 3 A 6 * A 5 A 2 * + A 1 A 6 * A 3 A 2 * A 5 A 4 * ,
A ( u ) A * ( u + u ) = C A ( u ) .
C A ( u ) = const δ ( u ) .
P n ( 3 ) ( u , υ ) = P n ( u ) P n ( υ ) P n ( u υ ) + [ 1 ] P n ( u ) P n ( υ ) P n ( u υ ) + [ 2 ] P n ( υ ) P n ( u ) P n ( u υ ) [ 3 ] 2 P n ( u ) P n ( υ ) P n ( u υ ) + [ 4 ] const B ( 3 ) ( u , υ ) , [ 5 ]
B ( 3 ) ( u , υ ) = | H 0 ( w ) | 2 | H 0 ( w + u + υ ) | 2 × [ | H 0 ( w + u ) | 2 + | H 0 ( w + υ ) | 2 ] 2 d w .
B ( 3 ) ( u , υ ) = H 0 ( u ) H 0 * ( u + u ) H 0 ( υ ) H 0 * ( υ + υ ) H 0 ( w ) H 0 * ( w u υ ) [ C A ( u υ + u ) C A ( υ w + υ ) C A ( u + w u υ ) + C A ( u w + u ) C A ( u + υ + υ ) C A ( υ + w u υ ) ] d u d υ d w .
Õ ( 3 ) ( u , υ ) B ( 3 ) ( u , υ ) = Ĩ n ( 3 ) ( u , υ ) Ĩ nmm ( 3 ) ( u , υ ) Ĩ nnm ( 3 ) ( u , υ ) Ĩ nmn ( 3 ) ( u , υ ) + 2 Ĩ nmk ( 3 ) ( u , υ ) Õ ( 3 ) ( u , υ ) .
O ( 3 ) ( x , x ) * * B ( 3 ) ( x , x ) = I n 3 ( x , x ) I nmm ( 3 ) ( x , x ) I nnm ( x , x ) I nmn ( 3 ) ( x , x ) + 2 I nmk ( x , x ) O ( 3 ) ( x , x ) .
J ( x ) = I ( x ) + N ( x ) ,
N ( x ) = N = const , N ( x ) N ( x + x ) = N ( 2 ) ( x ) .
J ( 3 ) ( x , x ) = I ( 3 ) ( x , x ) + [ 0 ] N [ I ( 2 ) ( x ) + I ( 2 ) ( x ) + I ( 2 ) ( x x ) ] + [ 1 ] Ī [ N ( 2 ) ( x ) + N ( 2 ) ( x ) + N ( 2 ) ( x x ) ] + [ 2 ] N ( 3 ) ( x , x ) , [ 3 ]
I ( 2 ) ( x ) = I ( x 1 ) I ( x 1 + x ) d x 1 , Ī = I ( x ) d x .
J n ( 3 ) J nmm ( 3 ) + J nnm ( 3 ) + J nmn ( 3 ) 2 J nmk ( 3 ) = I n ( 3 ) I nmm ( 3 ) + I nnm ( 3 ) + I nmn ( 3 ) 2 I nmk ( 3 ) + [ 1 ] N n ( 3 ) N nmm ( 3 ) + N nnm ( 3 ) + N nmn ( 3 ) 2 N nmk ( 3 ) , [ 2 ]
N n ( x ) = const term [ 2 ] = 0 ,
N n ( x ) = 0 term [ 2 ] = N n ( 3 ) ( x , x ) .
N n ( x ) = const term [ 2 ] = N n ( 3 ) ( x , x ) .

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