Abstract

We introduce the use of Super-Gaussian apodizing functions in the telescope pupil plane and/or the coronagraph Lyot plane to improve the imaging contrast in ground-based coronagraphs. We describe the properties of the Super-Gaussian function, we estimate its second-order moment in the pupil and Fourier planes and we check it as an apodizing function. We then use Super-Gaussian function to apodize the telescope pupil, the coronagraph Lyot plane or both of them. The result is that a proper apodizing masks combination can reduce the exoplanet detection distance up to a 45% with respect to the classic Lyot coronagraph, for moderately aberrated wavefronts. Compared to the prolate spheroidal function the Super-Gaussian apodizing function allows the planet light up to 3 times brighter. An extra help to increase the extinction rate is to perform a frame selection (Lucky Imaging technique). We show that a selection of the 10% best frames will reduce up to a 20% the detection angular distance when using the classic Lyot coronagraph but that the reduction is only around the 5% when using an apodized coronagraph.

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References

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  1. F. J. Harris, “On the use of windows for harmonic analysis with discrete Fourier transform,” Proc. IEEE66(1), 51–83 (1978).
  2. V. F. C. Vidal F. Canales, P. J. V. Pedro J. Valle, J. E. O. Jose E. Oti, and M. P. C. Manuel P. Cagigal, “Pupil apodization for increasing data storage density,” Chin. Opt. Lett.7(8), 720–723 (2009).
    [CrossRef]
  3. F. M. Dickey and S. C. Holswade, Laser Beam Shaping: Theory and Techniques (Marcel Dekker Inc., 2000).
  4. V. F. Canales and M. P. Cagigal, “Pupil filter design by using a Bessel functions basis at the image plane,” Opt. Express14(22), 10393–10402 (2006).
    [CrossRef] [PubMed]
  5. P. Jacquinot and B. Roizen-Dossier, “II Apodisation,” in Progress in Optics, E. Wolf, ed. (Elsevier, 1964), Vol. 3, pp. 29–186.
  6. D. Slepian, “Analytic solution for two apodization problems,” J. Opt. Soc. Am.55(9), 1110–1115 (1965).
    [CrossRef]
  7. R. Soummer, “Apodized pupil Lyot coronagraphs for arbitrary telescope apertures,” Astrophys. J.618(2), L161–L164 (2005).
    [CrossRef]
  8. R. J. Vanderbei, D. N. Spergel, and N. J. Kasdin, “Spider web masks for high-contrast imaging,” Astrophys. J.590(1), 593–603 (2003).
    [CrossRef]
  9. O. Guyon, “Phase-induced amplitude apodization of telescope pupils for extrasolar terrestrial planet imaging,” Astron. Astrophys.404(1), 379–387 (2003).
    [CrossRef]
  10. R. J. Vanderbei and W. A. Traub, “Pupil mapping in two dimensions for high-contrast imaging,” Astrophys. J.626(2), 1079–1090 (2005).
    [CrossRef]
  11. O. Guyon, E. A. Pluzhnik, R. Galicher, R. Martinache, S. T. Ridgway, and R. A. Woodruff, “Exoplanet imaging with a phase-induced amplitude apodization coronagraph. I. Principle,” Astrophys. J.622(1), 744–758 (2005).
    [CrossRef]
  12. R. Soummer, A. Sivaramakrishnan, L. Pueyo, B. Macintosh, and B. R. Oppenheimer, “Apodized pupil Lyot coronagraphs for arbitrary apertures. III. Quasi-achromatic solutions,” Astrophys. J.729(2), 144 (2011).
    [CrossRef]
  13. J. L. Codona and R. Angel, “Imaging extrasolar planets by stellar halo suppression in separately corrected color bands,” Astrophys. J.604(2), L117–L120 (2004).
    [CrossRef]
  14. D. Mawet and ., “Review of small-angle coronagraphic techniques in the wake of ground-based second-generation adaptive optics systems,” http://arxiv.org/abs/1207.5481 (2012).
    [CrossRef]
  15. M. A. Cagigas, P. J. Valle, and M. P. Cagigal, “Coronagraphs adapted to atmosphere conditions,” Opt. Express20(4), 4574–4582 (2012).
    [CrossRef] [PubMed]
  16. S. Bollanti, P. Di Lazzaro, D. Murra, and A. Torre, “Analytical propagation of supergaussian-like beams in the far-field,” Opt. Commun.138(1-3), 35–39 (1997).
    [CrossRef]
  17. H. Weyl, Theory of Groups and Quantum Mechanics (Dover Publications, 1950).
  18. A. Parent, M. Morin, and P. Lavigne, “Propagation of super-Gaussian field distributions,” Opt. Quantum Electron.24(9), S1071–S1079 (1992).
    [CrossRef]
  19. H. J. Landau and H. O. Pollak, “Prolate spheroidal wave functions, Fourier analysis and uncertainty II,” Bell Syst. Tech. J.40, 65–84 (1961).
  20. R. Soummer, C. Aime, A. Ferrari, A. Sivaramakrishnan, B. R. Oppenheimer, R. Makidon, and B. Macintosh “Apodized pupil lyot coronagraphs: concepts and application to the gemini planet imager,” in Direct Imaging of Exoplanets: Science and Techniques, Proceedings IAU Colloquium No. 200,2005, C. Aime and F. Vakili., eds. (Cambridge University, 2006), pp.367–372.
  21. T. Verma, S. Bilbao, and T. H. Y. Meng, “The digital prolate spheroidal window,” in Proceedings of IEEE International Conference on Acoustics, Speech, and Signal Proceeding (ICASSP 1996) Vol. 3, pp. 1351–1354.
  22. M. P. Cagigal and V. F. Canales, “Generalized Fried parameter after adaptive optics partial wave-front compensation,” J. Opt. Soc. Am. A17(5), 903–910 (2000).
    [CrossRef] [PubMed]
  23. J. W. Hardy, Adaptive Optics for Astronomical Telescopes (Oxford University, 1998).
  24. N. Roddier, “Atmospheric wavefront simulation using Zernike polynomials,” Opt. Eng.29(10), 1174–1180 (1990).
    [CrossRef]
  25. J. R. Crepp, A. D. Vanden Heuvel, and J. Ge, “Comparative Lyot coronagraphy with extreme adaptive optics systems,” Astrophys. J.661(2), 1323–1331 (2007).
    [CrossRef]

2012

2011

R. Soummer, A. Sivaramakrishnan, L. Pueyo, B. Macintosh, and B. R. Oppenheimer, “Apodized pupil Lyot coronagraphs for arbitrary apertures. III. Quasi-achromatic solutions,” Astrophys. J.729(2), 144 (2011).
[CrossRef]

2009

2007

J. R. Crepp, A. D. Vanden Heuvel, and J. Ge, “Comparative Lyot coronagraphy with extreme adaptive optics systems,” Astrophys. J.661(2), 1323–1331 (2007).
[CrossRef]

2006

2005

R. Soummer, “Apodized pupil Lyot coronagraphs for arbitrary telescope apertures,” Astrophys. J.618(2), L161–L164 (2005).
[CrossRef]

R. J. Vanderbei and W. A. Traub, “Pupil mapping in two dimensions for high-contrast imaging,” Astrophys. J.626(2), 1079–1090 (2005).
[CrossRef]

O. Guyon, E. A. Pluzhnik, R. Galicher, R. Martinache, S. T. Ridgway, and R. A. Woodruff, “Exoplanet imaging with a phase-induced amplitude apodization coronagraph. I. Principle,” Astrophys. J.622(1), 744–758 (2005).
[CrossRef]

2004

J. L. Codona and R. Angel, “Imaging extrasolar planets by stellar halo suppression in separately corrected color bands,” Astrophys. J.604(2), L117–L120 (2004).
[CrossRef]

2003

R. J. Vanderbei, D. N. Spergel, and N. J. Kasdin, “Spider web masks for high-contrast imaging,” Astrophys. J.590(1), 593–603 (2003).
[CrossRef]

O. Guyon, “Phase-induced amplitude apodization of telescope pupils for extrasolar terrestrial planet imaging,” Astron. Astrophys.404(1), 379–387 (2003).
[CrossRef]

2000

1997

S. Bollanti, P. Di Lazzaro, D. Murra, and A. Torre, “Analytical propagation of supergaussian-like beams in the far-field,” Opt. Commun.138(1-3), 35–39 (1997).
[CrossRef]

1992

A. Parent, M. Morin, and P. Lavigne, “Propagation of super-Gaussian field distributions,” Opt. Quantum Electron.24(9), S1071–S1079 (1992).
[CrossRef]

1990

N. Roddier, “Atmospheric wavefront simulation using Zernike polynomials,” Opt. Eng.29(10), 1174–1180 (1990).
[CrossRef]

1978

F. J. Harris, “On the use of windows for harmonic analysis with discrete Fourier transform,” Proc. IEEE66(1), 51–83 (1978).

1965

1961

H. J. Landau and H. O. Pollak, “Prolate spheroidal wave functions, Fourier analysis and uncertainty II,” Bell Syst. Tech. J.40, 65–84 (1961).

Angel, R.

J. L. Codona and R. Angel, “Imaging extrasolar planets by stellar halo suppression in separately corrected color bands,” Astrophys. J.604(2), L117–L120 (2004).
[CrossRef]

Bilbao, S.

T. Verma, S. Bilbao, and T. H. Y. Meng, “The digital prolate spheroidal window,” in Proceedings of IEEE International Conference on Acoustics, Speech, and Signal Proceeding (ICASSP 1996) Vol. 3, pp. 1351–1354.

Bollanti, S.

S. Bollanti, P. Di Lazzaro, D. Murra, and A. Torre, “Analytical propagation of supergaussian-like beams in the far-field,” Opt. Commun.138(1-3), 35–39 (1997).
[CrossRef]

Cagigal, M. P.

Cagigas, M. A.

Canales, V. F.

Codona, J. L.

J. L. Codona and R. Angel, “Imaging extrasolar planets by stellar halo suppression in separately corrected color bands,” Astrophys. J.604(2), L117–L120 (2004).
[CrossRef]

Crepp, J. R.

J. R. Crepp, A. D. Vanden Heuvel, and J. Ge, “Comparative Lyot coronagraphy with extreme adaptive optics systems,” Astrophys. J.661(2), 1323–1331 (2007).
[CrossRef]

Di Lazzaro, P.

S. Bollanti, P. Di Lazzaro, D. Murra, and A. Torre, “Analytical propagation of supergaussian-like beams in the far-field,” Opt. Commun.138(1-3), 35–39 (1997).
[CrossRef]

Galicher, R.

O. Guyon, E. A. Pluzhnik, R. Galicher, R. Martinache, S. T. Ridgway, and R. A. Woodruff, “Exoplanet imaging with a phase-induced amplitude apodization coronagraph. I. Principle,” Astrophys. J.622(1), 744–758 (2005).
[CrossRef]

Ge, J.

J. R. Crepp, A. D. Vanden Heuvel, and J. Ge, “Comparative Lyot coronagraphy with extreme adaptive optics systems,” Astrophys. J.661(2), 1323–1331 (2007).
[CrossRef]

Guyon, O.

O. Guyon, E. A. Pluzhnik, R. Galicher, R. Martinache, S. T. Ridgway, and R. A. Woodruff, “Exoplanet imaging with a phase-induced amplitude apodization coronagraph. I. Principle,” Astrophys. J.622(1), 744–758 (2005).
[CrossRef]

O. Guyon, “Phase-induced amplitude apodization of telescope pupils for extrasolar terrestrial planet imaging,” Astron. Astrophys.404(1), 379–387 (2003).
[CrossRef]

Harris, F. J.

F. J. Harris, “On the use of windows for harmonic analysis with discrete Fourier transform,” Proc. IEEE66(1), 51–83 (1978).

Jose E. Oti, J. E. O.

Kasdin, N. J.

R. J. Vanderbei, D. N. Spergel, and N. J. Kasdin, “Spider web masks for high-contrast imaging,” Astrophys. J.590(1), 593–603 (2003).
[CrossRef]

Landau, H. J.

H. J. Landau and H. O. Pollak, “Prolate spheroidal wave functions, Fourier analysis and uncertainty II,” Bell Syst. Tech. J.40, 65–84 (1961).

Lavigne, P.

A. Parent, M. Morin, and P. Lavigne, “Propagation of super-Gaussian field distributions,” Opt. Quantum Electron.24(9), S1071–S1079 (1992).
[CrossRef]

Macintosh, B.

R. Soummer, A. Sivaramakrishnan, L. Pueyo, B. Macintosh, and B. R. Oppenheimer, “Apodized pupil Lyot coronagraphs for arbitrary apertures. III. Quasi-achromatic solutions,” Astrophys. J.729(2), 144 (2011).
[CrossRef]

Manuel P. Cagigal, M. P. C.

Martinache, R.

O. Guyon, E. A. Pluzhnik, R. Galicher, R. Martinache, S. T. Ridgway, and R. A. Woodruff, “Exoplanet imaging with a phase-induced amplitude apodization coronagraph. I. Principle,” Astrophys. J.622(1), 744–758 (2005).
[CrossRef]

Meng, T. H. Y.

T. Verma, S. Bilbao, and T. H. Y. Meng, “The digital prolate spheroidal window,” in Proceedings of IEEE International Conference on Acoustics, Speech, and Signal Proceeding (ICASSP 1996) Vol. 3, pp. 1351–1354.

Morin, M.

A. Parent, M. Morin, and P. Lavigne, “Propagation of super-Gaussian field distributions,” Opt. Quantum Electron.24(9), S1071–S1079 (1992).
[CrossRef]

Murra, D.

S. Bollanti, P. Di Lazzaro, D. Murra, and A. Torre, “Analytical propagation of supergaussian-like beams in the far-field,” Opt. Commun.138(1-3), 35–39 (1997).
[CrossRef]

Oppenheimer, B. R.

R. Soummer, A. Sivaramakrishnan, L. Pueyo, B. Macintosh, and B. R. Oppenheimer, “Apodized pupil Lyot coronagraphs for arbitrary apertures. III. Quasi-achromatic solutions,” Astrophys. J.729(2), 144 (2011).
[CrossRef]

Parent, A.

A. Parent, M. Morin, and P. Lavigne, “Propagation of super-Gaussian field distributions,” Opt. Quantum Electron.24(9), S1071–S1079 (1992).
[CrossRef]

Pedro J. Valle, P. J. V.

Pluzhnik, E. A.

O. Guyon, E. A. Pluzhnik, R. Galicher, R. Martinache, S. T. Ridgway, and R. A. Woodruff, “Exoplanet imaging with a phase-induced amplitude apodization coronagraph. I. Principle,” Astrophys. J.622(1), 744–758 (2005).
[CrossRef]

Pollak, H. O.

H. J. Landau and H. O. Pollak, “Prolate spheroidal wave functions, Fourier analysis and uncertainty II,” Bell Syst. Tech. J.40, 65–84 (1961).

Pueyo, L.

R. Soummer, A. Sivaramakrishnan, L. Pueyo, B. Macintosh, and B. R. Oppenheimer, “Apodized pupil Lyot coronagraphs for arbitrary apertures. III. Quasi-achromatic solutions,” Astrophys. J.729(2), 144 (2011).
[CrossRef]

Ridgway, S. T.

O. Guyon, E. A. Pluzhnik, R. Galicher, R. Martinache, S. T. Ridgway, and R. A. Woodruff, “Exoplanet imaging with a phase-induced amplitude apodization coronagraph. I. Principle,” Astrophys. J.622(1), 744–758 (2005).
[CrossRef]

Roddier, N.

N. Roddier, “Atmospheric wavefront simulation using Zernike polynomials,” Opt. Eng.29(10), 1174–1180 (1990).
[CrossRef]

Sivaramakrishnan, A.

R. Soummer, A. Sivaramakrishnan, L. Pueyo, B. Macintosh, and B. R. Oppenheimer, “Apodized pupil Lyot coronagraphs for arbitrary apertures. III. Quasi-achromatic solutions,” Astrophys. J.729(2), 144 (2011).
[CrossRef]

Slepian, D.

Soummer, R.

R. Soummer, A. Sivaramakrishnan, L. Pueyo, B. Macintosh, and B. R. Oppenheimer, “Apodized pupil Lyot coronagraphs for arbitrary apertures. III. Quasi-achromatic solutions,” Astrophys. J.729(2), 144 (2011).
[CrossRef]

R. Soummer, “Apodized pupil Lyot coronagraphs for arbitrary telescope apertures,” Astrophys. J.618(2), L161–L164 (2005).
[CrossRef]

Spergel, D. N.

R. J. Vanderbei, D. N. Spergel, and N. J. Kasdin, “Spider web masks for high-contrast imaging,” Astrophys. J.590(1), 593–603 (2003).
[CrossRef]

Torre, A.

S. Bollanti, P. Di Lazzaro, D. Murra, and A. Torre, “Analytical propagation of supergaussian-like beams in the far-field,” Opt. Commun.138(1-3), 35–39 (1997).
[CrossRef]

Traub, W. A.

R. J. Vanderbei and W. A. Traub, “Pupil mapping in two dimensions for high-contrast imaging,” Astrophys. J.626(2), 1079–1090 (2005).
[CrossRef]

Valle, P. J.

Vanden Heuvel, A. D.

J. R. Crepp, A. D. Vanden Heuvel, and J. Ge, “Comparative Lyot coronagraphy with extreme adaptive optics systems,” Astrophys. J.661(2), 1323–1331 (2007).
[CrossRef]

Vanderbei, R. J.

R. J. Vanderbei and W. A. Traub, “Pupil mapping in two dimensions for high-contrast imaging,” Astrophys. J.626(2), 1079–1090 (2005).
[CrossRef]

R. J. Vanderbei, D. N. Spergel, and N. J. Kasdin, “Spider web masks for high-contrast imaging,” Astrophys. J.590(1), 593–603 (2003).
[CrossRef]

Verma, T.

T. Verma, S. Bilbao, and T. H. Y. Meng, “The digital prolate spheroidal window,” in Proceedings of IEEE International Conference on Acoustics, Speech, and Signal Proceeding (ICASSP 1996) Vol. 3, pp. 1351–1354.

Vidal F. Canales, V. F. C.

Woodruff, R. A.

O. Guyon, E. A. Pluzhnik, R. Galicher, R. Martinache, S. T. Ridgway, and R. A. Woodruff, “Exoplanet imaging with a phase-induced amplitude apodization coronagraph. I. Principle,” Astrophys. J.622(1), 744–758 (2005).
[CrossRef]

Astron. Astrophys.

O. Guyon, “Phase-induced amplitude apodization of telescope pupils for extrasolar terrestrial planet imaging,” Astron. Astrophys.404(1), 379–387 (2003).
[CrossRef]

Astrophys. J.

R. J. Vanderbei and W. A. Traub, “Pupil mapping in two dimensions for high-contrast imaging,” Astrophys. J.626(2), 1079–1090 (2005).
[CrossRef]

O. Guyon, E. A. Pluzhnik, R. Galicher, R. Martinache, S. T. Ridgway, and R. A. Woodruff, “Exoplanet imaging with a phase-induced amplitude apodization coronagraph. I. Principle,” Astrophys. J.622(1), 744–758 (2005).
[CrossRef]

R. Soummer, A. Sivaramakrishnan, L. Pueyo, B. Macintosh, and B. R. Oppenheimer, “Apodized pupil Lyot coronagraphs for arbitrary apertures. III. Quasi-achromatic solutions,” Astrophys. J.729(2), 144 (2011).
[CrossRef]

J. L. Codona and R. Angel, “Imaging extrasolar planets by stellar halo suppression in separately corrected color bands,” Astrophys. J.604(2), L117–L120 (2004).
[CrossRef]

R. Soummer, “Apodized pupil Lyot coronagraphs for arbitrary telescope apertures,” Astrophys. J.618(2), L161–L164 (2005).
[CrossRef]

R. J. Vanderbei, D. N. Spergel, and N. J. Kasdin, “Spider web masks for high-contrast imaging,” Astrophys. J.590(1), 593–603 (2003).
[CrossRef]

J. R. Crepp, A. D. Vanden Heuvel, and J. Ge, “Comparative Lyot coronagraphy with extreme adaptive optics systems,” Astrophys. J.661(2), 1323–1331 (2007).
[CrossRef]

Bell Syst. Tech. J.

H. J. Landau and H. O. Pollak, “Prolate spheroidal wave functions, Fourier analysis and uncertainty II,” Bell Syst. Tech. J.40, 65–84 (1961).

Chin. Opt. Lett.

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Commun.

S. Bollanti, P. Di Lazzaro, D. Murra, and A. Torre, “Analytical propagation of supergaussian-like beams in the far-field,” Opt. Commun.138(1-3), 35–39 (1997).
[CrossRef]

Opt. Eng.

N. Roddier, “Atmospheric wavefront simulation using Zernike polynomials,” Opt. Eng.29(10), 1174–1180 (1990).
[CrossRef]

Opt. Express

Opt. Quantum Electron.

A. Parent, M. Morin, and P. Lavigne, “Propagation of super-Gaussian field distributions,” Opt. Quantum Electron.24(9), S1071–S1079 (1992).
[CrossRef]

Proc. IEEE

F. J. Harris, “On the use of windows for harmonic analysis with discrete Fourier transform,” Proc. IEEE66(1), 51–83 (1978).

Other

F. M. Dickey and S. C. Holswade, Laser Beam Shaping: Theory and Techniques (Marcel Dekker Inc., 2000).

D. Mawet and ., “Review of small-angle coronagraphic techniques in the wake of ground-based second-generation adaptive optics systems,” http://arxiv.org/abs/1207.5481 (2012).
[CrossRef]

H. Weyl, Theory of Groups and Quantum Mechanics (Dover Publications, 1950).

R. Soummer, C. Aime, A. Ferrari, A. Sivaramakrishnan, B. R. Oppenheimer, R. Makidon, and B. Macintosh “Apodized pupil lyot coronagraphs: concepts and application to the gemini planet imager,” in Direct Imaging of Exoplanets: Science and Techniques, Proceedings IAU Colloquium No. 200,2005, C. Aime and F. Vakili., eds. (Cambridge University, 2006), pp.367–372.

T. Verma, S. Bilbao, and T. H. Y. Meng, “The digital prolate spheroidal window,” in Proceedings of IEEE International Conference on Acoustics, Speech, and Signal Proceeding (ICASSP 1996) Vol. 3, pp. 1351–1354.

P. Jacquinot and B. Roizen-Dossier, “II Apodisation,” in Progress in Optics, E. Wolf, ed. (Elsevier, 1964), Vol. 3, pp. 29–186.

J. W. Hardy, Adaptive Optics for Astronomical Telescopes (Oxford University, 1998).

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Figures (7)

Fig. 1
Fig. 1

Super-Gaussian (SG) functions as the convolution of Gaussian (G) and hard-edge circle (HE) functions. Super-Gaussian order (solid line and left scale) and Super-Gaussian width (dashed line and right scale) as a function of the Gaussian and hard-edge circle widths ratio. The Super-Gaussian and Gaussian widths are normalized with the hard-edge circle width (diameter) that is fixed.

Fig. 2
Fig. 2

(a) Set of SG(n,r) functions with different orders n and hard-edge circle pupil HEP. (b) Square module Fourier transforms of functions in (a).

Fig. 3
Fig. 3

Evolution of nm2(sg) (squares and left scale) and Super-Gaussian width (circles and right scale) as a function of the Super-Gaussian order.

Fig. 4
Fig. 4

(a) Hard-edge circular pupil (HEP), Super-Gaussian apodized pupil (SGP), and prolate spheroidal apodized pupil (PSP). (b) Fourier transformed HEP (dots), SGP (solid), and PSP (dashed); Horizontal axis is in λ/D units.

Fig. 5
Fig. 5

Aberrated PSF for different seeing conditions with and without Super-Gaussian apodization: D/r0 = 1, (a) unapodized and (b) apodized; D/r0 = 5, (c) unapodized and (d) apodized; D/r0 = 9, (e) unapodized and (f) apodized.

Fig. 6
Fig. 6

Normalized second-order moment as a function of the Super-Gaussian order, for different atmosphere conditions: D/r0 = 1,3,5,7, and 9.

Fig. 7
Fig. 7

Contrast curves for D/r0 = 1, 3, 5, and 7 as a function of the angular distance from the optical axis in λ/D units.

Tables (3)

Tables Icon

Table 1 Companion angular distance and peak intensity

Tables Icon

Table 2 Angular distance reduction

Tables Icon

Table 3 Angular distance reduction with image selection

Equations (13)

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

SG( n,r )=exp( | r/σ | n ),
SG( n,r )=HE( r )G( r ) FT sg( w )=A( w )g( w ),
m k ( F )= 0 2π 0 r k | F(r,θ) | 2 rdrdθ.
n m 2 ( F )= m 2 ( F ) m 0 ( F ) = 0 2π 0 r 2 | F(r,θ) | 2 rdrdθ 0 2π 0 | F(r,θ) | 2 rdrdθ .
n m 2 ( f )= m 2 ( f ) m 0 ( f ) = 0 2π 0 w 2 | f(w,ϑ) | 2 wdwdϑ 0 2π 0 | f(w,ϑ) | 2 wdwdϑ .
n m 2 (F)n m 2 (f)1/ (2π) 2 .
m 2 (sg) = 0 2π 0 w 2 | sg(w) | 2 wdwdϑ= = 1 4 π 2 0 2π 0 | dSG(n,r) dr | 2 rdrdθ= n 8π .
m 0 (sg) = m 0 (SG) = 2π σ 2 2 2/n Γ( 2/n ) n .
n m 2 (sg)= 2 2/n 16 π 2 σ 2 n 2 Γ(2/n) .
m 2 (SG(n,r)) = 0 2π 0 r 2 | SG(n,r) | 2 rdrdθ= 2π σ 4 2 4/n Γ( 4/n ) n .
n m 2 (SG)n m 2 (sg)= 1 16 π 2 n 2 Γ(4/n) Γ (2/n) 2 .
SG(n,r)Φ(x,y)=[HE(r)G(r)]Φ(x,y) FT [A(w)g(w)]φ(u,v)[A(w)φ(u,v)]g(w),
C(r)= I(r) I s (0) | M(r) | 2 ,

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