Abstract

The possibility to produce diffraction-limited images by large telescopes through Adaptive Optics is closely linked to the precision of measurement of the position of the guide star on the wavefront sensor. In the case of laser guide stars, many parameters can lead to a strong distortion on the shape of the LGS spot. Here we study the influence of both the saturation of the sodium layer excited by different types of lasers, the spatial quality of the laser mode at the ground and the influence of the atmospheric turbulence on the upward propagation of the laser beam. Both shape and intensity of the LGS spot are found to depend strongly on these three effects with important consequences on the precision on the wavefront analysis.

© 2009 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. F. Roddier (ed.), Adaptive Optics in Astronomy (Cambridge Univ. Press, 1999).
    [CrossRef]
  2. C. A. Primmerman, D. V. Murphy, D. A. Page, B. G. Zollars, and H. T. Barclay, "Compensation of atmospheric optical distorsions using a synthetic beacon," Nature 353, 141-143 (1991).
    [CrossRef]
  3. L. Thompson and C. Gardner, " Experiments on laser Guide Stars at Mauna Kea Observatory for Adaptive Optics in Astronomy," Nature 328, 229-231 (1987).
    [CrossRef]
  4. A. E. Siegman, Lasers (University Science Books, 1986), 206-208.
  5. L. Michaille, A. D. Cañas, J. C. Dainty, J. Maxwell, T. Gregory, J. C. Quartel, F. C. Reawell, R. W. Wilson, and N. J. Wooder, "A laser Beacon for monitoring the mesospheric sodium layer at La Palma," Mon. Not. R. Astron. Soc. 318, 139-144 (2000).
    [CrossRef]
  6. O. Lai, C. Veillet, D. Salmon, K. Ho, M. R. Baril, G. A. Barrick, J. Thomas, D. Teeple, T. Benedict, J.-P. Pique, and H. Guillet de Chatellus, "VASAO: visible all sky adaptive optics: a new adaptive optics concept for CFHT," Proc. SPIE 7015, 701543 (2008).
    [CrossRef]
  7. C. d’Orgeville, B. J. Bauman, J. W. Catone, B. L. Ellerbroek, D. T. Gavel, and R. A. Buchroeder, "Gemini north and south laser guide star systems requirements and preliminary designs," Proc. SPIE 4494, 302-316 (2002).
    [CrossRef]
  8. R. Holzloehner, D. Bonaccini Calia, and W. Hackenberg, "Physical Optics Modeling and Optimization of Laser Guide Star propagation," Proc. SPIE 7015, 701521 (2008).
    [CrossRef]
  9. J.-P. Pique and S. Farinotti, "Efficient modeless laser for a mesospheric sodium laser guide star," J. Opt. Soc. Am. B 20, 2093-2102 (2003).
    [CrossRef]
  10. H. Guillet de Chatellus, J.-P. Pique, and I. C. Moldovan, "Return flux budget of polychromatic laser guide stars," J. Opt. Soc. Am. A 25, 400-415 (2008).
    [CrossRef]
  11. A. E. Siegman, Lasers (University Science Books, 1986), 626-697.
  12. http://www.aor.com/
  13. Q1. A. N. Kolmogorov, "Dissipation of energy in a locally isotropic turbulence," Dokl. Akad. Nauk SSSR 32, 16-18 (1941), English translation in Proc.R. Soc. Lond. A 434, 15-17 (1991).
  14. Q2. F. Roddier, "The effects of atmospheric turbulence in optical astronomy," Prog. Opt. 19, 281-376 (1981)
    [CrossRef]

2008 (3)

O. Lai, C. Veillet, D. Salmon, K. Ho, M. R. Baril, G. A. Barrick, J. Thomas, D. Teeple, T. Benedict, J.-P. Pique, and H. Guillet de Chatellus, "VASAO: visible all sky adaptive optics: a new adaptive optics concept for CFHT," Proc. SPIE 7015, 701543 (2008).
[CrossRef]

R. Holzloehner, D. Bonaccini Calia, and W. Hackenberg, "Physical Optics Modeling and Optimization of Laser Guide Star propagation," Proc. SPIE 7015, 701521 (2008).
[CrossRef]

H. Guillet de Chatellus, J.-P. Pique, and I. C. Moldovan, "Return flux budget of polychromatic laser guide stars," J. Opt. Soc. Am. A 25, 400-415 (2008).
[CrossRef]

2003 (1)

2002 (1)

C. d’Orgeville, B. J. Bauman, J. W. Catone, B. L. Ellerbroek, D. T. Gavel, and R. A. Buchroeder, "Gemini north and south laser guide star systems requirements and preliminary designs," Proc. SPIE 4494, 302-316 (2002).
[CrossRef]

2000 (1)

L. Michaille, A. D. Cañas, J. C. Dainty, J. Maxwell, T. Gregory, J. C. Quartel, F. C. Reawell, R. W. Wilson, and N. J. Wooder, "A laser Beacon for monitoring the mesospheric sodium layer at La Palma," Mon. Not. R. Astron. Soc. 318, 139-144 (2000).
[CrossRef]

1991 (2)

C. A. Primmerman, D. V. Murphy, D. A. Page, B. G. Zollars, and H. T. Barclay, "Compensation of atmospheric optical distorsions using a synthetic beacon," Nature 353, 141-143 (1991).
[CrossRef]

Q1. A. N. Kolmogorov, "Dissipation of energy in a locally isotropic turbulence," Dokl. Akad. Nauk SSSR 32, 16-18 (1941), English translation in Proc.R. Soc. Lond. A 434, 15-17 (1991).

1987 (1)

L. Thompson and C. Gardner, " Experiments on laser Guide Stars at Mauna Kea Observatory for Adaptive Optics in Astronomy," Nature 328, 229-231 (1987).
[CrossRef]

1981 (1)

Q2. F. Roddier, "The effects of atmospheric turbulence in optical astronomy," Prog. Opt. 19, 281-376 (1981)
[CrossRef]

Barclay, H. T.

C. A. Primmerman, D. V. Murphy, D. A. Page, B. G. Zollars, and H. T. Barclay, "Compensation of atmospheric optical distorsions using a synthetic beacon," Nature 353, 141-143 (1991).
[CrossRef]

Baril, M. R.

O. Lai, C. Veillet, D. Salmon, K. Ho, M. R. Baril, G. A. Barrick, J. Thomas, D. Teeple, T. Benedict, J.-P. Pique, and H. Guillet de Chatellus, "VASAO: visible all sky adaptive optics: a new adaptive optics concept for CFHT," Proc. SPIE 7015, 701543 (2008).
[CrossRef]

Barrick, G. A.

O. Lai, C. Veillet, D. Salmon, K. Ho, M. R. Baril, G. A. Barrick, J. Thomas, D. Teeple, T. Benedict, J.-P. Pique, and H. Guillet de Chatellus, "VASAO: visible all sky adaptive optics: a new adaptive optics concept for CFHT," Proc. SPIE 7015, 701543 (2008).
[CrossRef]

Bauman, B. J.

C. d’Orgeville, B. J. Bauman, J. W. Catone, B. L. Ellerbroek, D. T. Gavel, and R. A. Buchroeder, "Gemini north and south laser guide star systems requirements and preliminary designs," Proc. SPIE 4494, 302-316 (2002).
[CrossRef]

Benedict, T.

O. Lai, C. Veillet, D. Salmon, K. Ho, M. R. Baril, G. A. Barrick, J. Thomas, D. Teeple, T. Benedict, J.-P. Pique, and H. Guillet de Chatellus, "VASAO: visible all sky adaptive optics: a new adaptive optics concept for CFHT," Proc. SPIE 7015, 701543 (2008).
[CrossRef]

Bonaccini Calia, D.

R. Holzloehner, D. Bonaccini Calia, and W. Hackenberg, "Physical Optics Modeling and Optimization of Laser Guide Star propagation," Proc. SPIE 7015, 701521 (2008).
[CrossRef]

Buchroeder, R. A.

C. d’Orgeville, B. J. Bauman, J. W. Catone, B. L. Ellerbroek, D. T. Gavel, and R. A. Buchroeder, "Gemini north and south laser guide star systems requirements and preliminary designs," Proc. SPIE 4494, 302-316 (2002).
[CrossRef]

Cañas, A. D.

L. Michaille, A. D. Cañas, J. C. Dainty, J. Maxwell, T. Gregory, J. C. Quartel, F. C. Reawell, R. W. Wilson, and N. J. Wooder, "A laser Beacon for monitoring the mesospheric sodium layer at La Palma," Mon. Not. R. Astron. Soc. 318, 139-144 (2000).
[CrossRef]

Catone, J. W.

C. d’Orgeville, B. J. Bauman, J. W. Catone, B. L. Ellerbroek, D. T. Gavel, and R. A. Buchroeder, "Gemini north and south laser guide star systems requirements and preliminary designs," Proc. SPIE 4494, 302-316 (2002).
[CrossRef]

d’Orgeville, C.

C. d’Orgeville, B. J. Bauman, J. W. Catone, B. L. Ellerbroek, D. T. Gavel, and R. A. Buchroeder, "Gemini north and south laser guide star systems requirements and preliminary designs," Proc. SPIE 4494, 302-316 (2002).
[CrossRef]

Dainty, J. C.

L. Michaille, A. D. Cañas, J. C. Dainty, J. Maxwell, T. Gregory, J. C. Quartel, F. C. Reawell, R. W. Wilson, and N. J. Wooder, "A laser Beacon for monitoring the mesospheric sodium layer at La Palma," Mon. Not. R. Astron. Soc. 318, 139-144 (2000).
[CrossRef]

Ellerbroek, B. L.

C. d’Orgeville, B. J. Bauman, J. W. Catone, B. L. Ellerbroek, D. T. Gavel, and R. A. Buchroeder, "Gemini north and south laser guide star systems requirements and preliminary designs," Proc. SPIE 4494, 302-316 (2002).
[CrossRef]

Farinotti, S.

Gardner, C.

L. Thompson and C. Gardner, " Experiments on laser Guide Stars at Mauna Kea Observatory for Adaptive Optics in Astronomy," Nature 328, 229-231 (1987).
[CrossRef]

Gavel, D. T.

C. d’Orgeville, B. J. Bauman, J. W. Catone, B. L. Ellerbroek, D. T. Gavel, and R. A. Buchroeder, "Gemini north and south laser guide star systems requirements and preliminary designs," Proc. SPIE 4494, 302-316 (2002).
[CrossRef]

Gregory, T.

L. Michaille, A. D. Cañas, J. C. Dainty, J. Maxwell, T. Gregory, J. C. Quartel, F. C. Reawell, R. W. Wilson, and N. J. Wooder, "A laser Beacon for monitoring the mesospheric sodium layer at La Palma," Mon. Not. R. Astron. Soc. 318, 139-144 (2000).
[CrossRef]

Guillet de Chatellus, H.

O. Lai, C. Veillet, D. Salmon, K. Ho, M. R. Baril, G. A. Barrick, J. Thomas, D. Teeple, T. Benedict, J.-P. Pique, and H. Guillet de Chatellus, "VASAO: visible all sky adaptive optics: a new adaptive optics concept for CFHT," Proc. SPIE 7015, 701543 (2008).
[CrossRef]

H. Guillet de Chatellus, J.-P. Pique, and I. C. Moldovan, "Return flux budget of polychromatic laser guide stars," J. Opt. Soc. Am. A 25, 400-415 (2008).
[CrossRef]

Hackenberg, W.

R. Holzloehner, D. Bonaccini Calia, and W. Hackenberg, "Physical Optics Modeling and Optimization of Laser Guide Star propagation," Proc. SPIE 7015, 701521 (2008).
[CrossRef]

Ho, K.

O. Lai, C. Veillet, D. Salmon, K. Ho, M. R. Baril, G. A. Barrick, J. Thomas, D. Teeple, T. Benedict, J.-P. Pique, and H. Guillet de Chatellus, "VASAO: visible all sky adaptive optics: a new adaptive optics concept for CFHT," Proc. SPIE 7015, 701543 (2008).
[CrossRef]

Holzloehner, R.

R. Holzloehner, D. Bonaccini Calia, and W. Hackenberg, "Physical Optics Modeling and Optimization of Laser Guide Star propagation," Proc. SPIE 7015, 701521 (2008).
[CrossRef]

Kolmogorov, A. N.

Q1. A. N. Kolmogorov, "Dissipation of energy in a locally isotropic turbulence," Dokl. Akad. Nauk SSSR 32, 16-18 (1941), English translation in Proc.R. Soc. Lond. A 434, 15-17 (1991).

Lai, O.

O. Lai, C. Veillet, D. Salmon, K. Ho, M. R. Baril, G. A. Barrick, J. Thomas, D. Teeple, T. Benedict, J.-P. Pique, and H. Guillet de Chatellus, "VASAO: visible all sky adaptive optics: a new adaptive optics concept for CFHT," Proc. SPIE 7015, 701543 (2008).
[CrossRef]

Maxwell, J.

L. Michaille, A. D. Cañas, J. C. Dainty, J. Maxwell, T. Gregory, J. C. Quartel, F. C. Reawell, R. W. Wilson, and N. J. Wooder, "A laser Beacon for monitoring the mesospheric sodium layer at La Palma," Mon. Not. R. Astron. Soc. 318, 139-144 (2000).
[CrossRef]

Michaille, L.

L. Michaille, A. D. Cañas, J. C. Dainty, J. Maxwell, T. Gregory, J. C. Quartel, F. C. Reawell, R. W. Wilson, and N. J. Wooder, "A laser Beacon for monitoring the mesospheric sodium layer at La Palma," Mon. Not. R. Astron. Soc. 318, 139-144 (2000).
[CrossRef]

Moldovan, I. C.

Murphy, D. V.

C. A. Primmerman, D. V. Murphy, D. A. Page, B. G. Zollars, and H. T. Barclay, "Compensation of atmospheric optical distorsions using a synthetic beacon," Nature 353, 141-143 (1991).
[CrossRef]

Page, D. A.

C. A. Primmerman, D. V. Murphy, D. A. Page, B. G. Zollars, and H. T. Barclay, "Compensation of atmospheric optical distorsions using a synthetic beacon," Nature 353, 141-143 (1991).
[CrossRef]

Pique, J.-P.

O. Lai, C. Veillet, D. Salmon, K. Ho, M. R. Baril, G. A. Barrick, J. Thomas, D. Teeple, T. Benedict, J.-P. Pique, and H. Guillet de Chatellus, "VASAO: visible all sky adaptive optics: a new adaptive optics concept for CFHT," Proc. SPIE 7015, 701543 (2008).
[CrossRef]

H. Guillet de Chatellus, J.-P. Pique, and I. C. Moldovan, "Return flux budget of polychromatic laser guide stars," J. Opt. Soc. Am. A 25, 400-415 (2008).
[CrossRef]

J.-P. Pique and S. Farinotti, "Efficient modeless laser for a mesospheric sodium laser guide star," J. Opt. Soc. Am. B 20, 2093-2102 (2003).
[CrossRef]

Primmerman, C. A.

C. A. Primmerman, D. V. Murphy, D. A. Page, B. G. Zollars, and H. T. Barclay, "Compensation of atmospheric optical distorsions using a synthetic beacon," Nature 353, 141-143 (1991).
[CrossRef]

Quartel, J. C.

L. Michaille, A. D. Cañas, J. C. Dainty, J. Maxwell, T. Gregory, J. C. Quartel, F. C. Reawell, R. W. Wilson, and N. J. Wooder, "A laser Beacon for monitoring the mesospheric sodium layer at La Palma," Mon. Not. R. Astron. Soc. 318, 139-144 (2000).
[CrossRef]

Reawell, F. C.

L. Michaille, A. D. Cañas, J. C. Dainty, J. Maxwell, T. Gregory, J. C. Quartel, F. C. Reawell, R. W. Wilson, and N. J. Wooder, "A laser Beacon for monitoring the mesospheric sodium layer at La Palma," Mon. Not. R. Astron. Soc. 318, 139-144 (2000).
[CrossRef]

Roddier, F.

Q2. F. Roddier, "The effects of atmospheric turbulence in optical astronomy," Prog. Opt. 19, 281-376 (1981)
[CrossRef]

Salmon, D.

O. Lai, C. Veillet, D. Salmon, K. Ho, M. R. Baril, G. A. Barrick, J. Thomas, D. Teeple, T. Benedict, J.-P. Pique, and H. Guillet de Chatellus, "VASAO: visible all sky adaptive optics: a new adaptive optics concept for CFHT," Proc. SPIE 7015, 701543 (2008).
[CrossRef]

Teeple, D.

O. Lai, C. Veillet, D. Salmon, K. Ho, M. R. Baril, G. A. Barrick, J. Thomas, D. Teeple, T. Benedict, J.-P. Pique, and H. Guillet de Chatellus, "VASAO: visible all sky adaptive optics: a new adaptive optics concept for CFHT," Proc. SPIE 7015, 701543 (2008).
[CrossRef]

Thomas, J.

O. Lai, C. Veillet, D. Salmon, K. Ho, M. R. Baril, G. A. Barrick, J. Thomas, D. Teeple, T. Benedict, J.-P. Pique, and H. Guillet de Chatellus, "VASAO: visible all sky adaptive optics: a new adaptive optics concept for CFHT," Proc. SPIE 7015, 701543 (2008).
[CrossRef]

Thompson, L.

L. Thompson and C. Gardner, " Experiments on laser Guide Stars at Mauna Kea Observatory for Adaptive Optics in Astronomy," Nature 328, 229-231 (1987).
[CrossRef]

Veillet, C.

O. Lai, C. Veillet, D. Salmon, K. Ho, M. R. Baril, G. A. Barrick, J. Thomas, D. Teeple, T. Benedict, J.-P. Pique, and H. Guillet de Chatellus, "VASAO: visible all sky adaptive optics: a new adaptive optics concept for CFHT," Proc. SPIE 7015, 701543 (2008).
[CrossRef]

Wilson, R. W.

L. Michaille, A. D. Cañas, J. C. Dainty, J. Maxwell, T. Gregory, J. C. Quartel, F. C. Reawell, R. W. Wilson, and N. J. Wooder, "A laser Beacon for monitoring the mesospheric sodium layer at La Palma," Mon. Not. R. Astron. Soc. 318, 139-144 (2000).
[CrossRef]

Wooder, N. J.

L. Michaille, A. D. Cañas, J. C. Dainty, J. Maxwell, T. Gregory, J. C. Quartel, F. C. Reawell, R. W. Wilson, and N. J. Wooder, "A laser Beacon for monitoring the mesospheric sodium layer at La Palma," Mon. Not. R. Astron. Soc. 318, 139-144 (2000).
[CrossRef]

Zollars, B. G.

C. A. Primmerman, D. V. Murphy, D. A. Page, B. G. Zollars, and H. T. Barclay, "Compensation of atmospheric optical distorsions using a synthetic beacon," Nature 353, 141-143 (1991).
[CrossRef]

J. Opt. Soc. Am. A (1)

J. Opt. Soc. Am. B (1)

Mon. Not. R. Astron. Soc. (1)

L. Michaille, A. D. Cañas, J. C. Dainty, J. Maxwell, T. Gregory, J. C. Quartel, F. C. Reawell, R. W. Wilson, and N. J. Wooder, "A laser Beacon for monitoring the mesospheric sodium layer at La Palma," Mon. Not. R. Astron. Soc. 318, 139-144 (2000).
[CrossRef]

Nature (2)

C. A. Primmerman, D. V. Murphy, D. A. Page, B. G. Zollars, and H. T. Barclay, "Compensation of atmospheric optical distorsions using a synthetic beacon," Nature 353, 141-143 (1991).
[CrossRef]

L. Thompson and C. Gardner, " Experiments on laser Guide Stars at Mauna Kea Observatory for Adaptive Optics in Astronomy," Nature 328, 229-231 (1987).
[CrossRef]

Proc. SPIE (3)

O. Lai, C. Veillet, D. Salmon, K. Ho, M. R. Baril, G. A. Barrick, J. Thomas, D. Teeple, T. Benedict, J.-P. Pique, and H. Guillet de Chatellus, "VASAO: visible all sky adaptive optics: a new adaptive optics concept for CFHT," Proc. SPIE 7015, 701543 (2008).
[CrossRef]

C. d’Orgeville, B. J. Bauman, J. W. Catone, B. L. Ellerbroek, D. T. Gavel, and R. A. Buchroeder, "Gemini north and south laser guide star systems requirements and preliminary designs," Proc. SPIE 4494, 302-316 (2002).
[CrossRef]

R. Holzloehner, D. Bonaccini Calia, and W. Hackenberg, "Physical Optics Modeling and Optimization of Laser Guide Star propagation," Proc. SPIE 7015, 701521 (2008).
[CrossRef]

Prog. Opt. (1)

Q2. F. Roddier, "The effects of atmospheric turbulence in optical astronomy," Prog. Opt. 19, 281-376 (1981)
[CrossRef]

R. Soc. Lond. A (1)

Q1. A. N. Kolmogorov, "Dissipation of energy in a locally isotropic turbulence," Dokl. Akad. Nauk SSSR 32, 16-18 (1941), English translation in Proc.R. Soc. Lond. A 434, 15-17 (1991).

Other (4)

A. E. Siegman, Lasers (University Science Books, 1986), 626-697.

http://www.aor.com/

A. E. Siegman, Lasers (University Science Books, 1986), 206-208.

F. Roddier (ed.), Adaptive Optics in Astronomy (Cambridge Univ. Press, 1999).
[CrossRef]

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 (9)

Fig. 1.
Fig. 1.

Methodology of the present study. The calculation is based on two steps: i) propagation of the laser beam to the mesosphere, ii) calculation of the fluorescence spot (i.e. LGS) from the laser intensity spot. The size of each map is 300cm × 300cm. Data are normalized according to the color scale (right).

Fig. 2.
Fig. 2.

Laser spectra considered in this study. SM, PM and ML stand respectively for single mode, phase modulated and modeless lasers. The parameters of the phase modulation function are given in [10]. The black solid line is the profile of the sodium D2 line at the temperature of the mesosphere.

Fig. 3.
Fig. 3.

Number of photons emitted per steradian, per sodium atom and per laser pulse as a function of the laser power density for different laser types (50 ns pulse duration, linear polarization). The numerical values are given by the density matrix code Beacon [10].

Fig. 4.
Fig. 4.

Evolution of the spot broadening with the laser power for different laser formats (SM: single-mode, PM: phase modulation and ML: modeless laser).

Fig. 5.
Fig. 5.

Variation of the figure of merit of the LGS with the laser power for different laser format (SM: single-mode, PM: phase modulation and ML: modeless laser).

Fig. 6.
Fig. 6.

Variation of the HWHM of the LGS spot with the laser pulse energy for a laser with M2 =1. Solid lines: single mode laser, dashed lines: phase-modulated laser, dotted lines: modeless laser. Green data: r0 = 5 cm, red data: r0 = 10 cm and black data: r0 infinite. Error bars correspond to the standard deviation of the data defined in Table 1.

Fig. 7.
Fig. 7.

Variation of the HWHM of the LGS spot with the laser pulse energy for a laser with M2 =1.3. Solid lines: single mode laser, dashed lines: phase-modulated laser, dotted lines: modeless laser. Green data: r0 = 5 cm, red data: r0 = 10 cm and black data: r0 infinite. Error bars correspond to the standard deviation of the data defined in Table 1.

Fig. 8.
Fig. 8.

Variation of the intensity of the LGS spot with the laser pulse energy for a laser with M2 =1. Solid lines: single mode laser, dashed lines: phase-modulated laser, dotted lines: modeless laser. Green data: r0 = 5 cm, red data: r0 = 10 cm and black data: r0 infinite.

Fig. 9.
Fig. 9.

Variation of the intensity of the LGS spot with the laser pulse energy for a laser for M2 =1 (black) and M2 = 1.3 (blue). r0 is infinite. Solid lines: single mode laser, dashed lines: phase-modulated laser, dotted lines: modeless laser.

Tables (3)

Tables Icon

Table 1. Summary of the different simulations performed for an accurate statistical treatment for each laser type (SP, PM and ML) and each pulse energy (0.05, 0.1, 0.2, 0.5, 1 and 2 mJ). The pulse width is fixed to 50 ns.

Tables Icon

Table 2. Surface broadening for 2mJ/50ns laser pulses for different laser formats and seeing conditions.

Tables Icon

Table 3. Return flux at the ground per m2 of telescope produced per 2 mJ laser pulses for different laser formats and seeing conditions. The atmospheric transmission is supposed to be 100%.

Equations (9)

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

M 2 = π r 1 / e 2 θ λ
σ ( I ) 2 = + + I ( x , y ) [ ( x x 0 ) 2 + ( y y 0 ) 2 ] dxdy + + I ( x , y ) dxdy
A ( x , y ) = A 0 e x 2 + y 2 r 1 / e 2 2
I ( x , y ) = A ( x , y ) A * ( x , y ) = I 0 e 2 x 2 + y 2 r 1 / e 2 2 = I 0 e log ( 2 ) x 2 + y 2 r 1 / 2 2
r 1 / e 2 = r 1 / 2 2 log ( 2 )
S = π r 1 / 2 2
σ ( I ) = r 1 / e 2 2 = r 1 / 2 2 log ( 2 )
W 2 ( ρ ) = 0.023 e ρ 2 L i 2 r 0 5 / 3 ( ρ 2 + 1 L 0 2 ) 11 / 6
Φ = d Na s D 2 i = 1 512 j = 1 512 n ( i , j )

Metrics