Abstract

We present a rigorous theory for the mean and standard deviation of speckle intensity for an arbitrary number of contributing phasors and an arbitrary level of phase fluctuations. This framework contains the classic models of speckle as the special cases, and it is compatible with the recently proposed models of Rician distribution of speckle intensity and of the Strehl ratio variability in adaptive optics images. We demonstrate that applicability of that or another distribution law depends not only on the level of aberrations in the pupil and the number of the phasors, but also on the observation point in the focal plane.

© 2011 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. J. C. Dainty, ed., Laser Speckle and Related Phenomana, 2nd ed. (Springer, 1984).
  2. J. Goodman, Speckle Phenomena in Optics: Theory and Applications (Roberts, 2006).
  3. A. Boccaletti, P. Riaud, and D. Rouan, “Speckle symmetry with high-contrast coronagraphs,” Publ. Astron. Soc. Pac. 114, 132–136 (2002).
    [CrossRef]
  4. C. Marois, D. Lafreniere, B. Macintosh, and R. Doyon, “Confidence level and sensitivity limits in high-contrast imaging,” Astrophys. J. 673, 647–656 (2008).
    [CrossRef]
  5. R. Soummer, A. Ferrari, C. Aime, and L. Jolissaint, “Speckle noise and dynamic range in coronagraphic images,” Astrophys. J. 669, 642–656 (2007).
    [CrossRef]
  6. S. Gladysz and J. Christou, “Reference-less detection, astrometry, and photometry of faint companions with adaptive optics,” Astrophys. J. 698, 28–42 (2009).
    [CrossRef]
  7. S. Gladysz, N. Yaitskova, and J. Christou, “Statistics of intensity in adaptive-optics images and their usefulness for detection and photometry of exoplanets,” J. Opt. Soc. Am. A 27, A64–A75(2010).
    [CrossRef]
  8. J. Uozumi and T. Asakura, “The first-order statistics of partially developed non-Gaussian speckle patterns,” J. Opt. 12, 177–186(1981).
    [CrossRef]
  9. J. Ohtsubo and T. Asakura, “Statistical properties of laser speckle produced in the diffraction field,” Appl. Opt. 16, 1742–1753 (1977).
    [CrossRef] [PubMed]
  10. J. Ohtsubo and T. Asakura, “Statistical properties of the sum of partially developed speckle pattern,” Opt. Lett. 1, 98–100(1977).
    [CrossRef] [PubMed]
  11. J. Ohtsubo and T. Asakura, “Statistical properties of speckle patterns produced by coherent light at the image and defocus planes,” Optik 45, 65–72 (1976).
  12. J. C. Dainty, “Coherent addition of a uniform beam to a speckle pattern,” J. Opt. Soc. Am. 62, 595–596 (1972).
    [CrossRef]
  13. V. Canales and M. Cagigal, “Rician distribution to describe speckle statistics in adaptive optics,” Appl. Opt. 38, 766–771(1999).
    [CrossRef]
  14. S. Gladysz, J. Christou, and M. Redfern, “Characterization of the Lick adaptive optics point spread function,” Proc. SPIE 6272, 62720J (2006).
    [CrossRef]
  15. R. Soummer and A. Ferrari, “The Strehl ratio in adaptive optics images: statistics and estimation,” Astrophys. J. 663, L49–L52(2007).
    [CrossRef]
  16. S. Gladysz, J. C. Christou, L. W. Bradford, and L. C. Roberts, Jr., “Temporal variability and the statistics of the Strehl ratio in adaptive-optics images,” Publ. Astron. Soc. Pac. 120, 1132–1143(2008).
    [CrossRef]
  17. C. Aime and R. Soummer, “The usefulness and limits of coronagraphy in the presence of pinned speckles,” Astrophys. J. 612, L85–L88 (2004).
    [CrossRef]
  18. M. Fitzgerald and J. Graham, “Speckle statistics in adaptively corrected images,” Astrophys. J. 637, 541–547 (2006).
    [CrossRef]
  19. N. Yaitskova, K. Dohlen, and P. Dierickx, “Analytical study of diffraction effects in extremely large segmented telescopes,” J. Opt. Soc. Am. A 20, 1563–1575 (2003).
    [CrossRef]
  20. N. Yaitskova and S. Gladysz, “Telling planets from speckles created by telescope segmentation,” Proc. SPIE 7733, 773351(2010).
    [CrossRef]
  21. E. Aller-Carpentier, M. Kasper, P. Martinez, E. Vernet, E. Fedrigo, C. Soenke, S. Tordo, N. Hubin, C. Verinaud, S. Esposito, E. Pinna, A. Puglisi, A. Tozzi, F. Quiros, A. Basden, S. Goodsell, G. Love, and R. Myers,  “High order test bench for extreme adaptive optics system optimization,” Proc. SPIE 7015, 70153Z(2008).
    [CrossRef]
  22. N. Yaitskova and S. Gladysz, are preparing a manuscript to be called “How many cells in a wavefront?” (nyaitsko@eso.org).
  23. H. Yura and D. Fried, “Variance of the Strehl ratio of an adaptive optics system,” J. Opt. Soc. Am. A 15, 2107–2110 (1998).
    [CrossRef]
  24. T. Fusco and J.-M. Conan, “On- and off-axis statistical behavior of adaptive-optics-corrected short-exposure Strehl ratio,” J. Opt. Soc. Am. A 21, 1277–1289 (2004).
    [CrossRef]
  25. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1986).
  26. N. Yaitskova, “Influence of irregular gaps between primary mirror segments on telescope image quality,” J. Opt. Soc. Am. A 24, 2558–2567 (2007).
    [CrossRef]
  27. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 2005) p. 446.
  28. R. Gilmozzi and J. Spyromilio, “The 42.5 m European ELT: status,” Proc. SPIE 7012, 701219 (2008).
    [CrossRef]
  29. E. Bloemhof, R. Dekany, M. Troy, and B. Oppenheimer, “Behavior of remnant speckles in an adaptively corrected imaging system,” Astrophys. J. Lett. 558, L71–L74 (2001).
    [CrossRef]
  30. M. Spiegel, Theory and Problems of Probability and Statistics (McGraw-Hill, 1992).
  31. A. Labeyrie, “Images of exo-planets obtainable from dark speckles in adaptive telescopes,” Astron. Astrophys. 298, 544–548(1995).
  32. S. Gladysz, J. C. Christou, N. M. Law, R. Dekany, M. Redfern, and C. D. Mackay, “Lucky imaging and speckle discrimination for the detection of faint companions with adaptive optics,” Proc. SPIE 7105, 70152H (2008).
    [CrossRef]

2010 (2)

S. Gladysz, N. Yaitskova, and J. Christou, “Statistics of intensity in adaptive-optics images and their usefulness for detection and photometry of exoplanets,” J. Opt. Soc. Am. A 27, A64–A75(2010).
[CrossRef]

N. Yaitskova and S. Gladysz, “Telling planets from speckles created by telescope segmentation,” Proc. SPIE 7733, 773351(2010).
[CrossRef]

2009 (1)

S. Gladysz and J. Christou, “Reference-less detection, astrometry, and photometry of faint companions with adaptive optics,” Astrophys. J. 698, 28–42 (2009).
[CrossRef]

2008 (5)

S. Gladysz, J. C. Christou, L. W. Bradford, and L. C. Roberts, Jr., “Temporal variability and the statistics of the Strehl ratio in adaptive-optics images,” Publ. Astron. Soc. Pac. 120, 1132–1143(2008).
[CrossRef]

C. Marois, D. Lafreniere, B. Macintosh, and R. Doyon, “Confidence level and sensitivity limits in high-contrast imaging,” Astrophys. J. 673, 647–656 (2008).
[CrossRef]

E. Aller-Carpentier, M. Kasper, P. Martinez, E. Vernet, E. Fedrigo, C. Soenke, S. Tordo, N. Hubin, C. Verinaud, S. Esposito, E. Pinna, A. Puglisi, A. Tozzi, F. Quiros, A. Basden, S. Goodsell, G. Love, and R. Myers,  “High order test bench for extreme adaptive optics system optimization,” Proc. SPIE 7015, 70153Z(2008).
[CrossRef]

R. Gilmozzi and J. Spyromilio, “The 42.5 m European ELT: status,” Proc. SPIE 7012, 701219 (2008).
[CrossRef]

S. Gladysz, J. C. Christou, N. M. Law, R. Dekany, M. Redfern, and C. D. Mackay, “Lucky imaging and speckle discrimination for the detection of faint companions with adaptive optics,” Proc. SPIE 7105, 70152H (2008).
[CrossRef]

2007 (3)

R. Soummer and A. Ferrari, “The Strehl ratio in adaptive optics images: statistics and estimation,” Astrophys. J. 663, L49–L52(2007).
[CrossRef]

N. Yaitskova, “Influence of irregular gaps between primary mirror segments on telescope image quality,” J. Opt. Soc. Am. A 24, 2558–2567 (2007).
[CrossRef]

R. Soummer, A. Ferrari, C. Aime, and L. Jolissaint, “Speckle noise and dynamic range in coronagraphic images,” Astrophys. J. 669, 642–656 (2007).
[CrossRef]

2006 (3)

J. Goodman, Speckle Phenomena in Optics: Theory and Applications (Roberts, 2006).

M. Fitzgerald and J. Graham, “Speckle statistics in adaptively corrected images,” Astrophys. J. 637, 541–547 (2006).
[CrossRef]

S. Gladysz, J. Christou, and M. Redfern, “Characterization of the Lick adaptive optics point spread function,” Proc. SPIE 6272, 62720J (2006).
[CrossRef]

2005 (1)

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 2005) p. 446.

2004 (2)

T. Fusco and J.-M. Conan, “On- and off-axis statistical behavior of adaptive-optics-corrected short-exposure Strehl ratio,” J. Opt. Soc. Am. A 21, 1277–1289 (2004).
[CrossRef]

C. Aime and R. Soummer, “The usefulness and limits of coronagraphy in the presence of pinned speckles,” Astrophys. J. 612, L85–L88 (2004).
[CrossRef]

2003 (1)

N. Yaitskova, K. Dohlen, and P. Dierickx, “Analytical study of diffraction effects in extremely large segmented telescopes,” J. Opt. Soc. Am. A 20, 1563–1575 (2003).
[CrossRef]

2002 (1)

A. Boccaletti, P. Riaud, and D. Rouan, “Speckle symmetry with high-contrast coronagraphs,” Publ. Astron. Soc. Pac. 114, 132–136 (2002).
[CrossRef]

2001 (1)

E. Bloemhof, R. Dekany, M. Troy, and B. Oppenheimer, “Behavior of remnant speckles in an adaptively corrected imaging system,” Astrophys. J. Lett. 558, L71–L74 (2001).
[CrossRef]

1999 (1)

V. Canales and M. Cagigal, “Rician distribution to describe speckle statistics in adaptive optics,” Appl. Opt. 38, 766–771(1999).
[CrossRef]

1998 (1)

H. Yura and D. Fried, “Variance of the Strehl ratio of an adaptive optics system,” J. Opt. Soc. Am. A 15, 2107–2110 (1998).
[CrossRef]

1995 (1)

A. Labeyrie, “Images of exo-planets obtainable from dark speckles in adaptive telescopes,” Astron. Astrophys. 298, 544–548(1995).

1992 (1)

M. Spiegel, Theory and Problems of Probability and Statistics (McGraw-Hill, 1992).

1986 (1)

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1986).

1984 (1)

J. C. Dainty, ed., Laser Speckle and Related Phenomana, 2nd ed. (Springer, 1984).

1981 (1)

J. Uozumi and T. Asakura, “The first-order statistics of partially developed non-Gaussian speckle patterns,” J. Opt. 12, 177–186(1981).
[CrossRef]

1977 (2)

J. Ohtsubo and T. Asakura, “Statistical properties of laser speckle produced in the diffraction field,” Appl. Opt. 16, 1742–1753 (1977).
[CrossRef] [PubMed]

J. Ohtsubo and T. Asakura, “Statistical properties of the sum of partially developed speckle pattern,” Opt. Lett. 1, 98–100(1977).
[CrossRef] [PubMed]

1976 (1)

J. Ohtsubo and T. Asakura, “Statistical properties of speckle patterns produced by coherent light at the image and defocus planes,” Optik 45, 65–72 (1976).

1972 (1)

J. C. Dainty, “Coherent addition of a uniform beam to a speckle pattern,” J. Opt. Soc. Am. 62, 595–596 (1972).
[CrossRef]

Aime, C.

R. Soummer, A. Ferrari, C. Aime, and L. Jolissaint, “Speckle noise and dynamic range in coronagraphic images,” Astrophys. J. 669, 642–656 (2007).
[CrossRef]

C. Aime and R. Soummer, “The usefulness and limits of coronagraphy in the presence of pinned speckles,” Astrophys. J. 612, L85–L88 (2004).
[CrossRef]

Aller-Carpentier, E.

E. Aller-Carpentier, M. Kasper, P. Martinez, E. Vernet, E. Fedrigo, C. Soenke, S. Tordo, N. Hubin, C. Verinaud, S. Esposito, E. Pinna, A. Puglisi, A. Tozzi, F. Quiros, A. Basden, S. Goodsell, G. Love, and R. Myers,  “High order test bench for extreme adaptive optics system optimization,” Proc. SPIE 7015, 70153Z(2008).
[CrossRef]

Asakura, T.

J. Uozumi and T. Asakura, “The first-order statistics of partially developed non-Gaussian speckle patterns,” J. Opt. 12, 177–186(1981).
[CrossRef]

J. Ohtsubo and T. Asakura, “Statistical properties of laser speckle produced in the diffraction field,” Appl. Opt. 16, 1742–1753 (1977).
[CrossRef] [PubMed]

J. Ohtsubo and T. Asakura, “Statistical properties of the sum of partially developed speckle pattern,” Opt. Lett. 1, 98–100(1977).
[CrossRef] [PubMed]

J. Ohtsubo and T. Asakura, “Statistical properties of speckle patterns produced by coherent light at the image and defocus planes,” Optik 45, 65–72 (1976).

Basden, A.

E. Aller-Carpentier, M. Kasper, P. Martinez, E. Vernet, E. Fedrigo, C. Soenke, S. Tordo, N. Hubin, C. Verinaud, S. Esposito, E. Pinna, A. Puglisi, A. Tozzi, F. Quiros, A. Basden, S. Goodsell, G. Love, and R. Myers,  “High order test bench for extreme adaptive optics system optimization,” Proc. SPIE 7015, 70153Z(2008).
[CrossRef]

Bloemhof, E.

E. Bloemhof, R. Dekany, M. Troy, and B. Oppenheimer, “Behavior of remnant speckles in an adaptively corrected imaging system,” Astrophys. J. Lett. 558, L71–L74 (2001).
[CrossRef]

Boccaletti, A.

A. Boccaletti, P. Riaud, and D. Rouan, “Speckle symmetry with high-contrast coronagraphs,” Publ. Astron. Soc. Pac. 114, 132–136 (2002).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 2005) p. 446.

Bradford, L. W.

S. Gladysz, J. C. Christou, L. W. Bradford, and L. C. Roberts, Jr., “Temporal variability and the statistics of the Strehl ratio in adaptive-optics images,” Publ. Astron. Soc. Pac. 120, 1132–1143(2008).
[CrossRef]

Cagigal, M.

V. Canales and M. Cagigal, “Rician distribution to describe speckle statistics in adaptive optics,” Appl. Opt. 38, 766–771(1999).
[CrossRef]

Canales, V.

V. Canales and M. Cagigal, “Rician distribution to describe speckle statistics in adaptive optics,” Appl. Opt. 38, 766–771(1999).
[CrossRef]

Christou, J.

S. Gladysz, N. Yaitskova, and J. Christou, “Statistics of intensity in adaptive-optics images and their usefulness for detection and photometry of exoplanets,” J. Opt. Soc. Am. A 27, A64–A75(2010).
[CrossRef]

S. Gladysz and J. Christou, “Reference-less detection, astrometry, and photometry of faint companions with adaptive optics,” Astrophys. J. 698, 28–42 (2009).
[CrossRef]

S. Gladysz, J. Christou, and M. Redfern, “Characterization of the Lick adaptive optics point spread function,” Proc. SPIE 6272, 62720J (2006).
[CrossRef]

Christou, J. C.

S. Gladysz, J. C. Christou, L. W. Bradford, and L. C. Roberts, Jr., “Temporal variability and the statistics of the Strehl ratio in adaptive-optics images,” Publ. Astron. Soc. Pac. 120, 1132–1143(2008).
[CrossRef]

S. Gladysz, J. C. Christou, N. M. Law, R. Dekany, M. Redfern, and C. D. Mackay, “Lucky imaging and speckle discrimination for the detection of faint companions with adaptive optics,” Proc. SPIE 7105, 70152H (2008).
[CrossRef]

Conan, J.-M.

T. Fusco and J.-M. Conan, “On- and off-axis statistical behavior of adaptive-optics-corrected short-exposure Strehl ratio,” J. Opt. Soc. Am. A 21, 1277–1289 (2004).
[CrossRef]

Dainty, J. C.

J. C. Dainty, ed., Laser Speckle and Related Phenomana, 2nd ed. (Springer, 1984).

J. C. Dainty, “Coherent addition of a uniform beam to a speckle pattern,” J. Opt. Soc. Am. 62, 595–596 (1972).
[CrossRef]

Dekany, R.

S. Gladysz, J. C. Christou, N. M. Law, R. Dekany, M. Redfern, and C. D. Mackay, “Lucky imaging and speckle discrimination for the detection of faint companions with adaptive optics,” Proc. SPIE 7105, 70152H (2008).
[CrossRef]

E. Bloemhof, R. Dekany, M. Troy, and B. Oppenheimer, “Behavior of remnant speckles in an adaptively corrected imaging system,” Astrophys. J. Lett. 558, L71–L74 (2001).
[CrossRef]

Dierickx, P.

N. Yaitskova, K. Dohlen, and P. Dierickx, “Analytical study of diffraction effects in extremely large segmented telescopes,” J. Opt. Soc. Am. A 20, 1563–1575 (2003).
[CrossRef]

Dohlen, K.

N. Yaitskova, K. Dohlen, and P. Dierickx, “Analytical study of diffraction effects in extremely large segmented telescopes,” J. Opt. Soc. Am. A 20, 1563–1575 (2003).
[CrossRef]

Doyon, R.

C. Marois, D. Lafreniere, B. Macintosh, and R. Doyon, “Confidence level and sensitivity limits in high-contrast imaging,” Astrophys. J. 673, 647–656 (2008).
[CrossRef]

Esposito, S.

E. Aller-Carpentier, M. Kasper, P. Martinez, E. Vernet, E. Fedrigo, C. Soenke, S. Tordo, N. Hubin, C. Verinaud, S. Esposito, E. Pinna, A. Puglisi, A. Tozzi, F. Quiros, A. Basden, S. Goodsell, G. Love, and R. Myers,  “High order test bench for extreme adaptive optics system optimization,” Proc. SPIE 7015, 70153Z(2008).
[CrossRef]

Fedrigo, E.

E. Aller-Carpentier, M. Kasper, P. Martinez, E. Vernet, E. Fedrigo, C. Soenke, S. Tordo, N. Hubin, C. Verinaud, S. Esposito, E. Pinna, A. Puglisi, A. Tozzi, F. Quiros, A. Basden, S. Goodsell, G. Love, and R. Myers,  “High order test bench for extreme adaptive optics system optimization,” Proc. SPIE 7015, 70153Z(2008).
[CrossRef]

Ferrari, A.

R. Soummer, A. Ferrari, C. Aime, and L. Jolissaint, “Speckle noise and dynamic range in coronagraphic images,” Astrophys. J. 669, 642–656 (2007).
[CrossRef]

R. Soummer and A. Ferrari, “The Strehl ratio in adaptive optics images: statistics and estimation,” Astrophys. J. 663, L49–L52(2007).
[CrossRef]

Fitzgerald, M.

M. Fitzgerald and J. Graham, “Speckle statistics in adaptively corrected images,” Astrophys. J. 637, 541–547 (2006).
[CrossRef]

Fried, D.

H. Yura and D. Fried, “Variance of the Strehl ratio of an adaptive optics system,” J. Opt. Soc. Am. A 15, 2107–2110 (1998).
[CrossRef]

Fusco, T.

T. Fusco and J.-M. Conan, “On- and off-axis statistical behavior of adaptive-optics-corrected short-exposure Strehl ratio,” J. Opt. Soc. Am. A 21, 1277–1289 (2004).
[CrossRef]

Gilmozzi, R.

R. Gilmozzi and J. Spyromilio, “The 42.5 m European ELT: status,” Proc. SPIE 7012, 701219 (2008).
[CrossRef]

Gladysz, S.

N. Yaitskova and S. Gladysz, “Telling planets from speckles created by telescope segmentation,” Proc. SPIE 7733, 773351(2010).
[CrossRef]

S. Gladysz, N. Yaitskova, and J. Christou, “Statistics of intensity in adaptive-optics images and their usefulness for detection and photometry of exoplanets,” J. Opt. Soc. Am. A 27, A64–A75(2010).
[CrossRef]

S. Gladysz and J. Christou, “Reference-less detection, astrometry, and photometry of faint companions with adaptive optics,” Astrophys. J. 698, 28–42 (2009).
[CrossRef]

S. Gladysz, J. C. Christou, L. W. Bradford, and L. C. Roberts, Jr., “Temporal variability and the statistics of the Strehl ratio in adaptive-optics images,” Publ. Astron. Soc. Pac. 120, 1132–1143(2008).
[CrossRef]

S. Gladysz, J. C. Christou, N. M. Law, R. Dekany, M. Redfern, and C. D. Mackay, “Lucky imaging and speckle discrimination for the detection of faint companions with adaptive optics,” Proc. SPIE 7105, 70152H (2008).
[CrossRef]

S. Gladysz, J. Christou, and M. Redfern, “Characterization of the Lick adaptive optics point spread function,” Proc. SPIE 6272, 62720J (2006).
[CrossRef]

N. Yaitskova and S. Gladysz, are preparing a manuscript to be called “How many cells in a wavefront?” (nyaitsko@eso.org).

Goodman, J.

J. Goodman, Speckle Phenomena in Optics: Theory and Applications (Roberts, 2006).

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1986).

Goodsell, S.

E. Aller-Carpentier, M. Kasper, P. Martinez, E. Vernet, E. Fedrigo, C. Soenke, S. Tordo, N. Hubin, C. Verinaud, S. Esposito, E. Pinna, A. Puglisi, A. Tozzi, F. Quiros, A. Basden, S. Goodsell, G. Love, and R. Myers,  “High order test bench for extreme adaptive optics system optimization,” Proc. SPIE 7015, 70153Z(2008).
[CrossRef]

Graham, J.

M. Fitzgerald and J. Graham, “Speckle statistics in adaptively corrected images,” Astrophys. J. 637, 541–547 (2006).
[CrossRef]

Hubin, N.

E. Aller-Carpentier, M. Kasper, P. Martinez, E. Vernet, E. Fedrigo, C. Soenke, S. Tordo, N. Hubin, C. Verinaud, S. Esposito, E. Pinna, A. Puglisi, A. Tozzi, F. Quiros, A. Basden, S. Goodsell, G. Love, and R. Myers,  “High order test bench for extreme adaptive optics system optimization,” Proc. SPIE 7015, 70153Z(2008).
[CrossRef]

Jolissaint, L.

R. Soummer, A. Ferrari, C. Aime, and L. Jolissaint, “Speckle noise and dynamic range in coronagraphic images,” Astrophys. J. 669, 642–656 (2007).
[CrossRef]

Kasper, M.

E. Aller-Carpentier, M. Kasper, P. Martinez, E. Vernet, E. Fedrigo, C. Soenke, S. Tordo, N. Hubin, C. Verinaud, S. Esposito, E. Pinna, A. Puglisi, A. Tozzi, F. Quiros, A. Basden, S. Goodsell, G. Love, and R. Myers,  “High order test bench for extreme adaptive optics system optimization,” Proc. SPIE 7015, 70153Z(2008).
[CrossRef]

Labeyrie, A.

A. Labeyrie, “Images of exo-planets obtainable from dark speckles in adaptive telescopes,” Astron. Astrophys. 298, 544–548(1995).

Lafreniere, D.

C. Marois, D. Lafreniere, B. Macintosh, and R. Doyon, “Confidence level and sensitivity limits in high-contrast imaging,” Astrophys. J. 673, 647–656 (2008).
[CrossRef]

Law, N. M.

S. Gladysz, J. C. Christou, N. M. Law, R. Dekany, M. Redfern, and C. D. Mackay, “Lucky imaging and speckle discrimination for the detection of faint companions with adaptive optics,” Proc. SPIE 7105, 70152H (2008).
[CrossRef]

Love, G.

E. Aller-Carpentier, M. Kasper, P. Martinez, E. Vernet, E. Fedrigo, C. Soenke, S. Tordo, N. Hubin, C. Verinaud, S. Esposito, E. Pinna, A. Puglisi, A. Tozzi, F. Quiros, A. Basden, S. Goodsell, G. Love, and R. Myers,  “High order test bench for extreme adaptive optics system optimization,” Proc. SPIE 7015, 70153Z(2008).
[CrossRef]

Macintosh, B.

C. Marois, D. Lafreniere, B. Macintosh, and R. Doyon, “Confidence level and sensitivity limits in high-contrast imaging,” Astrophys. J. 673, 647–656 (2008).
[CrossRef]

Mackay, C. D.

S. Gladysz, J. C. Christou, N. M. Law, R. Dekany, M. Redfern, and C. D. Mackay, “Lucky imaging and speckle discrimination for the detection of faint companions with adaptive optics,” Proc. SPIE 7105, 70152H (2008).
[CrossRef]

Marois, C.

C. Marois, D. Lafreniere, B. Macintosh, and R. Doyon, “Confidence level and sensitivity limits in high-contrast imaging,” Astrophys. J. 673, 647–656 (2008).
[CrossRef]

Martinez, P.

E. Aller-Carpentier, M. Kasper, P. Martinez, E. Vernet, E. Fedrigo, C. Soenke, S. Tordo, N. Hubin, C. Verinaud, S. Esposito, E. Pinna, A. Puglisi, A. Tozzi, F. Quiros, A. Basden, S. Goodsell, G. Love, and R. Myers,  “High order test bench for extreme adaptive optics system optimization,” Proc. SPIE 7015, 70153Z(2008).
[CrossRef]

Myers, R.

E. Aller-Carpentier, M. Kasper, P. Martinez, E. Vernet, E. Fedrigo, C. Soenke, S. Tordo, N. Hubin, C. Verinaud, S. Esposito, E. Pinna, A. Puglisi, A. Tozzi, F. Quiros, A. Basden, S. Goodsell, G. Love, and R. Myers,  “High order test bench for extreme adaptive optics system optimization,” Proc. SPIE 7015, 70153Z(2008).
[CrossRef]

Ohtsubo, J.

J. Ohtsubo and T. Asakura, “Statistical properties of the sum of partially developed speckle pattern,” Opt. Lett. 1, 98–100(1977).
[CrossRef] [PubMed]

J. Ohtsubo and T. Asakura, “Statistical properties of laser speckle produced in the diffraction field,” Appl. Opt. 16, 1742–1753 (1977).
[CrossRef] [PubMed]

J. Ohtsubo and T. Asakura, “Statistical properties of speckle patterns produced by coherent light at the image and defocus planes,” Optik 45, 65–72 (1976).

Oppenheimer, B.

E. Bloemhof, R. Dekany, M. Troy, and B. Oppenheimer, “Behavior of remnant speckles in an adaptively corrected imaging system,” Astrophys. J. Lett. 558, L71–L74 (2001).
[CrossRef]

Pinna, E.

E. Aller-Carpentier, M. Kasper, P. Martinez, E. Vernet, E. Fedrigo, C. Soenke, S. Tordo, N. Hubin, C. Verinaud, S. Esposito, E. Pinna, A. Puglisi, A. Tozzi, F. Quiros, A. Basden, S. Goodsell, G. Love, and R. Myers,  “High order test bench for extreme adaptive optics system optimization,” Proc. SPIE 7015, 70153Z(2008).
[CrossRef]

Puglisi, A.

E. Aller-Carpentier, M. Kasper, P. Martinez, E. Vernet, E. Fedrigo, C. Soenke, S. Tordo, N. Hubin, C. Verinaud, S. Esposito, E. Pinna, A. Puglisi, A. Tozzi, F. Quiros, A. Basden, S. Goodsell, G. Love, and R. Myers,  “High order test bench for extreme adaptive optics system optimization,” Proc. SPIE 7015, 70153Z(2008).
[CrossRef]

Quiros, F.

E. Aller-Carpentier, M. Kasper, P. Martinez, E. Vernet, E. Fedrigo, C. Soenke, S. Tordo, N. Hubin, C. Verinaud, S. Esposito, E. Pinna, A. Puglisi, A. Tozzi, F. Quiros, A. Basden, S. Goodsell, G. Love, and R. Myers,  “High order test bench for extreme adaptive optics system optimization,” Proc. SPIE 7015, 70153Z(2008).
[CrossRef]

Redfern, M.

S. Gladysz, J. C. Christou, N. M. Law, R. Dekany, M. Redfern, and C. D. Mackay, “Lucky imaging and speckle discrimination for the detection of faint companions with adaptive optics,” Proc. SPIE 7105, 70152H (2008).
[CrossRef]

S. Gladysz, J. Christou, and M. Redfern, “Characterization of the Lick adaptive optics point spread function,” Proc. SPIE 6272, 62720J (2006).
[CrossRef]

Riaud, P.

A. Boccaletti, P. Riaud, and D. Rouan, “Speckle symmetry with high-contrast coronagraphs,” Publ. Astron. Soc. Pac. 114, 132–136 (2002).
[CrossRef]

Roberts, L. C.

S. Gladysz, J. C. Christou, L. W. Bradford, and L. C. Roberts, Jr., “Temporal variability and the statistics of the Strehl ratio in adaptive-optics images,” Publ. Astron. Soc. Pac. 120, 1132–1143(2008).
[CrossRef]

Rouan, D.

A. Boccaletti, P. Riaud, and D. Rouan, “Speckle symmetry with high-contrast coronagraphs,” Publ. Astron. Soc. Pac. 114, 132–136 (2002).
[CrossRef]

Soenke, C.

E. Aller-Carpentier, M. Kasper, P. Martinez, E. Vernet, E. Fedrigo, C. Soenke, S. Tordo, N. Hubin, C. Verinaud, S. Esposito, E. Pinna, A. Puglisi, A. Tozzi, F. Quiros, A. Basden, S. Goodsell, G. Love, and R. Myers,  “High order test bench for extreme adaptive optics system optimization,” Proc. SPIE 7015, 70153Z(2008).
[CrossRef]

Soummer, R.

R. Soummer, A. Ferrari, C. Aime, and L. Jolissaint, “Speckle noise and dynamic range in coronagraphic images,” Astrophys. J. 669, 642–656 (2007).
[CrossRef]

R. Soummer and A. Ferrari, “The Strehl ratio in adaptive optics images: statistics and estimation,” Astrophys. J. 663, L49–L52(2007).
[CrossRef]

C. Aime and R. Soummer, “The usefulness and limits of coronagraphy in the presence of pinned speckles,” Astrophys. J. 612, L85–L88 (2004).
[CrossRef]

Spiegel, M.

M. Spiegel, Theory and Problems of Probability and Statistics (McGraw-Hill, 1992).

Spyromilio, J.

R. Gilmozzi and J. Spyromilio, “The 42.5 m European ELT: status,” Proc. SPIE 7012, 701219 (2008).
[CrossRef]

Tordo, S.

E. Aller-Carpentier, M. Kasper, P. Martinez, E. Vernet, E. Fedrigo, C. Soenke, S. Tordo, N. Hubin, C. Verinaud, S. Esposito, E. Pinna, A. Puglisi, A. Tozzi, F. Quiros, A. Basden, S. Goodsell, G. Love, and R. Myers,  “High order test bench for extreme adaptive optics system optimization,” Proc. SPIE 7015, 70153Z(2008).
[CrossRef]

Tozzi, A.

E. Aller-Carpentier, M. Kasper, P. Martinez, E. Vernet, E. Fedrigo, C. Soenke, S. Tordo, N. Hubin, C. Verinaud, S. Esposito, E. Pinna, A. Puglisi, A. Tozzi, F. Quiros, A. Basden, S. Goodsell, G. Love, and R. Myers,  “High order test bench for extreme adaptive optics system optimization,” Proc. SPIE 7015, 70153Z(2008).
[CrossRef]

Troy, M.

E. Bloemhof, R. Dekany, M. Troy, and B. Oppenheimer, “Behavior of remnant speckles in an adaptively corrected imaging system,” Astrophys. J. Lett. 558, L71–L74 (2001).
[CrossRef]

Uozumi, J.

J. Uozumi and T. Asakura, “The first-order statistics of partially developed non-Gaussian speckle patterns,” J. Opt. 12, 177–186(1981).
[CrossRef]

Verinaud, C.

E. Aller-Carpentier, M. Kasper, P. Martinez, E. Vernet, E. Fedrigo, C. Soenke, S. Tordo, N. Hubin, C. Verinaud, S. Esposito, E. Pinna, A. Puglisi, A. Tozzi, F. Quiros, A. Basden, S. Goodsell, G. Love, and R. Myers,  “High order test bench for extreme adaptive optics system optimization,” Proc. SPIE 7015, 70153Z(2008).
[CrossRef]

Vernet, E.

E. Aller-Carpentier, M. Kasper, P. Martinez, E. Vernet, E. Fedrigo, C. Soenke, S. Tordo, N. Hubin, C. Verinaud, S. Esposito, E. Pinna, A. Puglisi, A. Tozzi, F. Quiros, A. Basden, S. Goodsell, G. Love, and R. Myers,  “High order test bench for extreme adaptive optics system optimization,” Proc. SPIE 7015, 70153Z(2008).
[CrossRef]

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 2005) p. 446.

Yaitskova, N.

N. Yaitskova and S. Gladysz, “Telling planets from speckles created by telescope segmentation,” Proc. SPIE 7733, 773351(2010).
[CrossRef]

S. Gladysz, N. Yaitskova, and J. Christou, “Statistics of intensity in adaptive-optics images and their usefulness for detection and photometry of exoplanets,” J. Opt. Soc. Am. A 27, A64–A75(2010).
[CrossRef]

N. Yaitskova, “Influence of irregular gaps between primary mirror segments on telescope image quality,” J. Opt. Soc. Am. A 24, 2558–2567 (2007).
[CrossRef]

N. Yaitskova, K. Dohlen, and P. Dierickx, “Analytical study of diffraction effects in extremely large segmented telescopes,” J. Opt. Soc. Am. A 20, 1563–1575 (2003).
[CrossRef]

N. Yaitskova and S. Gladysz, are preparing a manuscript to be called “How many cells in a wavefront?” (nyaitsko@eso.org).

Yura, H.

H. Yura and D. Fried, “Variance of the Strehl ratio of an adaptive optics system,” J. Opt. Soc. Am. A 15, 2107–2110 (1998).
[CrossRef]

Appl. Opt. (2)

J. Ohtsubo and T. Asakura, “Statistical properties of laser speckle produced in the diffraction field,” Appl. Opt. 16, 1742–1753 (1977).
[CrossRef] [PubMed]

V. Canales and M. Cagigal, “Rician distribution to describe speckle statistics in adaptive optics,” Appl. Opt. 38, 766–771(1999).
[CrossRef]

Astron. Astrophys. (1)

A. Labeyrie, “Images of exo-planets obtainable from dark speckles in adaptive telescopes,” Astron. Astrophys. 298, 544–548(1995).

Astrophys. J. (6)

R. Soummer and A. Ferrari, “The Strehl ratio in adaptive optics images: statistics and estimation,” Astrophys. J. 663, L49–L52(2007).
[CrossRef]

C. Aime and R. Soummer, “The usefulness and limits of coronagraphy in the presence of pinned speckles,” Astrophys. J. 612, L85–L88 (2004).
[CrossRef]

M. Fitzgerald and J. Graham, “Speckle statistics in adaptively corrected images,” Astrophys. J. 637, 541–547 (2006).
[CrossRef]

C. Marois, D. Lafreniere, B. Macintosh, and R. Doyon, “Confidence level and sensitivity limits in high-contrast imaging,” Astrophys. J. 673, 647–656 (2008).
[CrossRef]

R. Soummer, A. Ferrari, C. Aime, and L. Jolissaint, “Speckle noise and dynamic range in coronagraphic images,” Astrophys. J. 669, 642–656 (2007).
[CrossRef]

S. Gladysz and J. Christou, “Reference-less detection, astrometry, and photometry of faint companions with adaptive optics,” Astrophys. J. 698, 28–42 (2009).
[CrossRef]

Astrophys. J. Lett. (1)

E. Bloemhof, R. Dekany, M. Troy, and B. Oppenheimer, “Behavior of remnant speckles in an adaptively corrected imaging system,” Astrophys. J. Lett. 558, L71–L74 (2001).
[CrossRef]

J. Opt. (1)

J. Uozumi and T. Asakura, “The first-order statistics of partially developed non-Gaussian speckle patterns,” J. Opt. 12, 177–186(1981).
[CrossRef]

J. Opt. Soc. Am. (1)

J. C. Dainty, “Coherent addition of a uniform beam to a speckle pattern,” J. Opt. Soc. Am. 62, 595–596 (1972).
[CrossRef]

J. Opt. Soc. Am. A (5)

N. Yaitskova, “Influence of irregular gaps between primary mirror segments on telescope image quality,” J. Opt. Soc. Am. A 24, 2558–2567 (2007).
[CrossRef]

H. Yura and D. Fried, “Variance of the Strehl ratio of an adaptive optics system,” J. Opt. Soc. Am. A 15, 2107–2110 (1998).
[CrossRef]

T. Fusco and J.-M. Conan, “On- and off-axis statistical behavior of adaptive-optics-corrected short-exposure Strehl ratio,” J. Opt. Soc. Am. A 21, 1277–1289 (2004).
[CrossRef]

S. Gladysz, N. Yaitskova, and J. Christou, “Statistics of intensity in adaptive-optics images and their usefulness for detection and photometry of exoplanets,” J. Opt. Soc. Am. A 27, A64–A75(2010).
[CrossRef]

N. Yaitskova, K. Dohlen, and P. Dierickx, “Analytical study of diffraction effects in extremely large segmented telescopes,” J. Opt. Soc. Am. A 20, 1563–1575 (2003).
[CrossRef]

Opt. Lett. (1)

J. Ohtsubo and T. Asakura, “Statistical properties of the sum of partially developed speckle pattern,” Opt. Lett. 1, 98–100(1977).
[CrossRef] [PubMed]

Optik (1)

J. Ohtsubo and T. Asakura, “Statistical properties of speckle patterns produced by coherent light at the image and defocus planes,” Optik 45, 65–72 (1976).

Proc. SPIE (5)

S. Gladysz, J. Christou, and M. Redfern, “Characterization of the Lick adaptive optics point spread function,” Proc. SPIE 6272, 62720J (2006).
[CrossRef]

N. Yaitskova and S. Gladysz, “Telling planets from speckles created by telescope segmentation,” Proc. SPIE 7733, 773351(2010).
[CrossRef]

E. Aller-Carpentier, M. Kasper, P. Martinez, E. Vernet, E. Fedrigo, C. Soenke, S. Tordo, N. Hubin, C. Verinaud, S. Esposito, E. Pinna, A. Puglisi, A. Tozzi, F. Quiros, A. Basden, S. Goodsell, G. Love, and R. Myers,  “High order test bench for extreme adaptive optics system optimization,” Proc. SPIE 7015, 70153Z(2008).
[CrossRef]

R. Gilmozzi and J. Spyromilio, “The 42.5 m European ELT: status,” Proc. SPIE 7012, 701219 (2008).
[CrossRef]

S. Gladysz, J. C. Christou, N. M. Law, R. Dekany, M. Redfern, and C. D. Mackay, “Lucky imaging and speckle discrimination for the detection of faint companions with adaptive optics,” Proc. SPIE 7105, 70152H (2008).
[CrossRef]

Publ. Astron. Soc. Pac. (2)

S. Gladysz, J. C. Christou, L. W. Bradford, and L. C. Roberts, Jr., “Temporal variability and the statistics of the Strehl ratio in adaptive-optics images,” Publ. Astron. Soc. Pac. 120, 1132–1143(2008).
[CrossRef]

A. Boccaletti, P. Riaud, and D. Rouan, “Speckle symmetry with high-contrast coronagraphs,” Publ. Astron. Soc. Pac. 114, 132–136 (2002).
[CrossRef]

Other (6)

J. C. Dainty, ed., Laser Speckle and Related Phenomana, 2nd ed. (Springer, 1984).

J. Goodman, Speckle Phenomena in Optics: Theory and Applications (Roberts, 2006).

N. Yaitskova and S. Gladysz, are preparing a manuscript to be called “How many cells in a wavefront?” (nyaitsko@eso.org).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1986).

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 2005) p. 446.

M. Spiegel, Theory and Problems of Probability and Statistics (McGraw-Hill, 1992).

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.


Figures (7)

Fig. 1
Fig. 1

Top: reconstructed wavefront from the High-Order Testbench after AO compensation [21]. Bottom: pupil of the EELT composed of 894 hexagonal segments [28].

Fig. 2
Fig. 2

Four simulated speckle images in the linear scale with increasing standard deviation of the OPD: 0.4, 0.45, 0.5, and 0.55 μm . The wavelength is 1.6 μm . Note how from one image to another the central peak gradually disappears in the speckles.

Fig. 3
Fig. 3

Key functions of the theory: PSF 0 ( w ) (solid), square of g f ( w ) (dashed-dotted), and square of t ( w ) (dashed). The pupil is composed of 61 hexagonal cells; d is the inner diameter of a hexagon. The peaks of the grid function are the replicas of PSF 0 with the period 2 π · λ z / π d .

Fig. 4
Fig. 4

Speckle contrast as a function of phase standard deviation for different locations in the image: on top of the ring of the PSF 0 (solid), close to the zero of the PSF 0 (dashed), in-between the first two cases (dash-dot), and in the center of the image (dash-dot-dot). The locations of the points are given in the text. (a) Full range of the vertical axis. (b) Shortened range of the vertical axis. Dotted line: asymptote when PSF 0 is set to zero. N = 61 .

Fig. 5
Fig. 5

Standard deviation of the intensity. Solid curve: exact expression [Eq. (26)], dashed curve: third-order approximation, dash-dotted curve: modified Rician approximation ( I S 2 + 2 I C I S ). Dash and dash-dotted curves in this case coincide. (a) Function of a distance from the image center, σ = π / 8 ; dotted curve: approximation for the halo speckles from Eq. (37). (b) Function of the phase standard deviation, w = 0.64 λ z / π d ; dotted curve: linear approximation from Eq. (36). N = 61 .

Fig. 6
Fig. 6

Standard deviation of the central peak intensity. Solid: exact curve, dash-dotted: quadratic approximation, Eq. (41), dotted: standard deviation calculated from the Rician formula.

Fig. 7
Fig. 7

Comparison of the statistical moments for the central peak intensity obtained in the exact model (solid curves) and the gamma-based model (dashed curves). (a) Intensity mean and (b) intensity standard deviation. N = 61 .

Equations (54)

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

F ( x ) = n = 1 N θ n ( x r n ) exp ( i ϕ n ) ,
U ( w ) = 1 λ z F ( x ) exp ( i 2 π λ z x · w ) d 2 x = 1 λ z n = 1 N θ n ( x r n ) exp ( i ϕ n ) exp ( i 2 π λ z x · w ) d 2 x ,
U ( w ) = A N λ z · 1 A θ ( ξ ) exp ( i 2 π λ z w · ξ ) d 2 ξ · 1 N n = 1 N exp ( i ϕ n ) exp ( i 2 π λ z w · r n ) ,
A = θ ( ξ ) d 2 ξ .
t ( w ) = 1 A θ ( ξ ) exp ( i 2 π λ z w · ξ ) d 2 ξ .
U ( w ) = t ( w ) 1 N n = 1 N exp ( i ϕ n ) exp ( i 2 π λ z w · r n ) .
U ( 0 ) = 1 N n = 1 N exp ( i ϕ n ) .
I ( w ) = | t ( w ) | 2 1 N 2 n = 1 N m = 1 N exp [ i ( ϕ n ϕ m ) ] exp [ i 2 π λ z w · ( r n r m ) ] .
exp [ i ( ϕ n ϕ m ) ] = { 1 , when     n = m | exp ( i ϕ n ) | 2 , when     n m .
a 1 = exp ( i ϕ n ) ,
I ( w ) = a 1 2 | t ( w ) | 2 | 1 N n = 1 N exp ( i 2 π λ z w · r n ) | 2 + 1 a 1 2 N | t ( w ) | 2 .
I ( w ) ϕ = 0 = | t ( w ) | 2 | 1 N n = 1 N exp ( i 2 π λ z w · r n ) | 2 .
g f ( w ) = 1 N n = 1 N exp ( i 2 π λ z w · r n ) ,
PSF 0 ( w ) = | t ( w ) | 2 | g f ( w ) | 2 .
I C ( w ) = a 1 2 PSF 0 ( w ) .
I S ( w ) = 1 a 1 2 N | t ( w ) | 2 .
I ( w ) = I C ( w ) + I S ( w ) .
I ( w ) | t ( w ) | 2 N .
I 2 ( w ) = | t ( w ) | 4 1 N 4 n = 1 N m = 1 N p = 1 N q = 1 N exp [ i ( ϕ n ϕ m + ϕ p ϕ q ) ] exp ( i 2 π λ z w · r n ) exp ( i 2 π λ z w · r m ) exp ( i 2 π λ z w · r p ) exp ( i 2 π λ z w · r q ) .
σ I 2 ( w ) = | t ( w ) | 4 N { α 0 N + α 1 | g f ( w ) | 2 + α 2 | g f ( 2 w ) | 2 + α 3 Re [ g f 2 ( w ) g f * ( 2 w ) ] } ,
α 0 = ( 1 a 1 2 ) 2 + 2 ( a 2 a 1 4 ) [ ( 1 + a 2 ) 2 a 1 2 ] 2 N , α 1 = 2 a 1 2 { 1 a 1 2 2 [ ( 1 + a 2 ) 2 a 1 2 ] N } , α 2 = ( a 2 a 1 2 ) 2 N , α 3 = 2 a 1 2 ( a 2 a 1 2 ) .
a 2 = exp ( i 2 ϕ n ) = M ϕ ( 2 ) .
σ I 2 ( 0 ) = 1 N ( α 0 N + α 1 + α 2 + α 3 ) = N 1 N 3 [ 1 + 2 a 1 2 ( N 2 ) ( a 2 + 1 ) + a 2 2 2 a 1 4 ( 2 N 3 ) ] .
σ I ( w ) | t ( w ) | 2 N 1 1 N .
a 1 = exp ( σ 2 / 2 ) and a 2 = exp ( 2 σ 2 ) = a 1 4 .
σ I 2 ( w ) = I S 2 ( w ) [ 1 ( 1 a 1 2 ) 2 N ] + 2 I C ( w ) I S ( w ) ( 1 2 1 a 1 2 N ) + I C 2 ( w ) ( 1 1 a 1 2 N g f ( 2 w ) g f 2 ( w ) ) 2 I C 2 ( w ) .
C = σ I ( w ) I ( w ) .
C ( w ) 1 1 N , σ 1.
C ( w ) = 1 [ 1 exp ( σ 2 ) ] 2 N , PSF 0 ( w ) = 0.
I C ( w ) ( 1 σ 2 + σ 4 / 2 ) PSF 0 ( w ) , I S ( w ) N 1 ( σ 2 σ 4 / 2 ) | t ( w ) | 2 .
C ( w ) σ 2 N | t ( w ) | 2 PSF 0 ( w ) , σ 1 , PSF 0 ( w ) 0 , w 0.
C ( w ) 1 , σ 1 , PSF 0 ( w ) = 0.
C ( 0 ) σ 2 2 N 1 N 2 , σ 1 , w = 0.
σ I 2 ( w ) I S 2 ( w ) [ 1 ( 1 a 1 2 ) 2 N ] + 2 I C ( w ) I S ( w ) .
σ I 2 ( w ) PSF 0 ( w ) | t ( w ) | 2 N ( 2 σ 2 3 σ 4 ) + | t ( w ) | 4 N 2 σ 4 , σ 1.
σ I , pinned ( w ) σ 2 PSF 0 ( w ) | t ( w ) | 2 N , σ 1.
σ I , halo ( w ) σ 2 | t ( w ) | 2 N , σ 1.
I C ( 0 ) = a 1 2 , I S ( 0 ) = 1 a 1 2 N .
I ( 0 ) = I C + I S , σ I 2 ( 0 ) = I S 2 N I S 4 + 2 I C I S ( 1 2 I S ) 2 I C 2 I S + I C 2 I S 2 ,
I C = I C ( 0 ) = exp ( σ 2 ) , I S = I S ( 0 ) = [ 1 exp ( σ 2 ) ] / N .
σ I 2 ( 0 ) 2 N 1 N 2 σ 4 , σ 1 ,
σ peak = ln ( 3 4 N + 1 + 1 4 9 N 2 16 N ) 1.048 + 0.424 N .
σ I ( 0 ) max 0.554 1 N ( 1 1 6 N ) .
σ S , Yura 2 ( σ 2 ) 2 ( 2 β δ ϕ + 2 α δ ϕ 2 4 γ δ ϕ ) ,
N = ( D d s ) 2 .
σ S , Fusco 2 σ 4 1.04 ( n 0 + 1 ) 2 .
N = 1.92 ( n 0 + 1 ) 2 2 [ 1 + 1 4 1.92 ( n 0 + 1 ) 2 ] 1.92 ( n 0 + 1 ) 2 1.
σ ^ 2 = 1 N n = 1 N ( ϕ n ϕ ¯ n ) 2 ,
I ( 0 ) = exp ( σ ^ 2 ) .
I Gamma = ( 1 + N 1 2 σ 2 ) 1 N 2 , σ I , Gamma 2 = ( 1 + N 1 4 σ 2 ) 1 N 2 ( 1 + N 1 2 σ 2 ) 1 N .
R n ( w ) = exp ( i 2 π λ z w · r n ) .
n = 1 N R n ( w ) = N g f ( w ) , n = 1 N R n 2 ( w ) = N g f ( 2 w ) , | R n ( w ) | 2 = 1 , R n ( 0 ) = 1.
C n m p q = exp [ i ( δ n δ m + δ p δ q ) ] .
1 N 4 n = 1 N m = 1 N p = 1 N q = 1 N C n m p q R n ( w ) R m * ( w ) R p ( w ) R q * ( w ) .

Metrics