Abstract

When equipped with adaptive optics, the coming generation of large 6–10-m telescopes can combine huge light grasp with very sharp images. We describe a specific design concept for recovery of diffraction-limited images in the 1.6- and the 2.2-μm atmospheric windows, yielding 0.05-arcsec resolution for an 8-m telescope. Our goal has been to achieve this performance routinely by not requiring above-average atmospheric conditions or the use of unusually bright nearby stars. Atmospheric blurring is sensed with a sodium laser beacon of a few watts. Image motion is sensed by starlight, with a quadrant detector that is sensitive to the broad infrared band in which photon flux is typically largest and the field star has been sharpened by laser-beacon correction that is shared with the science target. A detailed performance analysis shows that for typical conditions Strehl ratios of >25% are expected at 2.2 μm, with the probability of finding a sufficiently bright field star exceeding 50%.

© 1994 Optical Society of America

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    [CrossRef]
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1994

1993

M. Lloyd-Hart, R. Dekany, B. McLeod, D. Wittman, D. Colucci, D. McCarthy, R. Angel, “Direct 75 milliarc-second images from the Multiple Mirror Telescope with adaptive optics,” Astrophys. Lett. 402, L81–L84 (1993).
[CrossRef]

1992

R. J. Wainscoat, M. Cohen, K. Volk, H. J. Walker, D. E. Schwartz, “A model of the 8–25 μm point source infrared sky,” Astrophys. J. Suppl. Ser. 83, 111–146 (1992).
[CrossRef]

1991

R. Q. Fugate, D. L. Fried, G. A. Ameer, B. R. Boeke, S. L. Browne, P. H. Roberts, R. E. Ruane, G. A. Tyler, L. M. Wopat, “Measurements of atmospheric wavefront distortion using scattered light from a laser guide star,” Nature (London) 353, 144–146 (1991).
[CrossRef]

C. A. Primmerman, D. V. Murphy, D. A. Page, G. Zollars, H. T. Barclay, “Compensation of atmospheric optical distortion using a synthetic beacon,” Nature (London) 353, 141–143 (1991).
[CrossRef]

J. C. Fontanella, G. Rousset, P. Lena, “Adaptive optics, a key element of the VLT,” J. Opt. (Paris) 22, 99–101 (1991).
[CrossRef]

F. Roddier, M. Northcott, J. E. Graves, “A simple low-order adaptive optics system for near-infrared applications,” Publ. Astron. Soc. Pac. 103, 131–149 (1991).
[CrossRef]

1990

C. S. Gardner, B. M. Welsh, L. A. Thompson, “Design and performance analysis of adaptive optical telescopes using laser guide stars,” Proc. IEEE 78, 1721–1743 (1990).
[CrossRef]

G. A. Tyler, “Analysis of propagation through turbulence: evaluation of an integral involving the product of three Bessel functions,” J. Opt. Soc. Am. A 7, 1218–1223 (1990).
[CrossRef]

1984

1983

1982

1981

J. N. Bahcall, R. M. Soniera, “Predicated star counts in selected fields and photometric bands: applications to galactic structure, the disk luminosity function, and the detection of a massive halo,” Astrophys. J. Suppl. Ser. 47, 357–401 (1981).
[CrossRef]

1980

1979

1976

1966

1953

H. W. Babcock, “The possibility of compensating astronomical seeing,” Publ. Astron. Soc. Pac. 65, 229–236 (1953).
[CrossRef]

Ameer, G. A.

R. Q. Fugate, D. L. Fried, G. A. Ameer, B. R. Boeke, S. L. Browne, P. H. Roberts, R. E. Ruane, G. A. Tyler, L. M. Wopat, “Measurements of atmospheric wavefront distortion using scattered light from a laser guide star,” Nature (London) 353, 144–146 (1991).
[CrossRef]

Angel, J. R. P.

D. Wittman, J. R. P. Angel, M. Lloyd-Hart, D. Collucci, D. McCarthy, “Optical sensing of infrared wavefronts for adaptive control: a new CCD detector and MMT experiments,” in Vol. 42 of Proceedings of European Southern Observatory Conference on Progress in Telescopeand Instrumentation Technologies, M.-H. Ulrich, ed. (European Southern Observatory, Garching, Germany, 1992), pp. 453–460.

J. R. P. Angel, “Use of natural stars with laser beacons for large telescope adaptive optics,” presented at the Workshop on Laser Guide Star Adaptive Optics, Albuquerque, N.M., (1992).

Angel, R.

M. Lloyd-Hart, R. Dekany, B. McLeod, D. Wittman, D. Colucci, D. McCarthy, R. Angel, “Direct 75 milliarc-second images from the Multiple Mirror Telescope with adaptive optics,” Astrophys. Lett. 402, L81–L84 (1993).
[CrossRef]

Arnold, R.

Azount, M.

F. Roddier, L. Cowie, J. E. Graves, S. Songaila, D. McKenna, J. Vernin, M. Azount, J. L. Caccia, E. Limburg, C. Roddier, D. Salmon, S. Beland, D. Cowley, S. Hill, “Seeing at Mauna Kea: a joint UH-UN-NOAO-CFHT study,” in Advanced Technology Optical Telescope IV, L. D. Barr, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1236, Part 1, 485–491 (1990).
[CrossRef]

Babcock, H. W.

H. W. Babcock, “The possibility of compensating astronomical seeing,” Publ. Astron. Soc. Pac. 65, 229–236 (1953).
[CrossRef]

Bahcall, J. N.

J. N. Bahcall, R. M. Soniera, “Predicated star counts in selected fields and photometric bands: applications to galactic structure, the disk luminosity function, and the detection of a massive halo,” Astrophys. J. Suppl. Ser. 47, 357–401 (1981).
[CrossRef]

Barclay, H. T.

C. A. Primmerman, D. V. Murphy, D. A. Page, G. Zollars, H. T. Barclay, “Compensation of atmospheric optical distortion using a synthetic beacon,” Nature (London) 353, 141–143 (1991).
[CrossRef]

Barrett, T.

Bauer, A.

M. P. Lesser, A. Bauer, L. Ulrickson, D. Ouellette, “Bump bonded back illuminated CCDs,” in High-Resolution Sensors and Hybrid Systems, M. M. Blouke, W. Chang, R. P. Khoslan, L. J. Thorpe, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1656, 508–516 (1992).
[CrossRef]

Beland, R. R.

R. R. Beland, “A decade of balloon microthermal probe measurements of optical turbulence,” in Adaptive Optics for Large Telescopes, Vol. 19 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), pp. 14–16.

Beland, S.

F. Roddier, L. Cowie, J. E. Graves, S. Songaila, D. McKenna, J. Vernin, M. Azount, J. L. Caccia, E. Limburg, C. Roddier, D. Salmon, S. Beland, D. Cowley, S. Hill, “Seeing at Mauna Kea: a joint UH-UN-NOAO-CFHT study,” in Advanced Technology Optical Telescope IV, L. D. Barr, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1236, Part 1, 485–491 (1990).
[CrossRef]

Blessinger, M.

R. Thompson, M. Rieke, E. Young, D. McCarthy, R. Rasche, M. Blessinger, K. Vural, W. Kleinhaus, “Future directions for NICMOS arrays,” in Proceedings of Third Infrared Detector Technology Workshop, C. R. McCreight, ed., NASA Tech. Memo 102209 (NASA, Washington, D.C., 1989).

Boeke, B. R.

R. Q. Fugate, D. L. Fried, G. A. Ameer, B. R. Boeke, S. L. Browne, P. H. Roberts, R. E. Ruane, G. A. Tyler, L. M. Wopat, “Measurements of atmospheric wavefront distortion using scattered light from a laser guide star,” Nature (London) 353, 144–146 (1991).
[CrossRef]

Browne, S. L.

R. Q. Fugate, D. L. Fried, G. A. Ameer, B. R. Boeke, S. L. Browne, P. H. Roberts, R. E. Ruane, G. A. Tyler, L. M. Wopat, “Measurements of atmospheric wavefront distortion using scattered light from a laser guide star,” Nature (London) 353, 144–146 (1991).
[CrossRef]

Caccia, J. L.

F. Roddier, L. Cowie, J. E. Graves, S. Songaila, D. McKenna, J. Vernin, M. Azount, J. L. Caccia, E. Limburg, C. Roddier, D. Salmon, S. Beland, D. Cowley, S. Hill, “Seeing at Mauna Kea: a joint UH-UN-NOAO-CFHT study,” in Advanced Technology Optical Telescope IV, L. D. Barr, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1236, Part 1, 485–491 (1990).
[CrossRef]

Cleis, R. A.

M. P. Jelonek, R. Q. Fugate, W. J. Lange, A. L. Slavin, R. E. Ruane, R. A. Cleis, “Sodium laser guide star experiments at Starfire Optical Range,” presented at the Workshop on Laser Guide Star Adaptive Optics, Albuquerque, N.M., 1992.

Cohen, M.

R. J. Wainscoat, M. Cohen, K. Volk, H. J. Walker, D. E. Schwartz, “A model of the 8–25 μm point source infrared sky,” Astrophys. J. Suppl. Ser. 83, 111–146 (1992).
[CrossRef]

Collucci, D.

D. Wittman, J. R. P. Angel, M. Lloyd-Hart, D. Collucci, D. McCarthy, “Optical sensing of infrared wavefronts for adaptive control: a new CCD detector and MMT experiments,” in Vol. 42 of Proceedings of European Southern Observatory Conference on Progress in Telescopeand Instrumentation Technologies, M.-H. Ulrich, ed. (European Southern Observatory, Garching, Germany, 1992), pp. 453–460.

Colucci, D.

M. Lloyd-Hart, R. Dekany, B. McLeod, D. Wittman, D. Colucci, D. McCarthy, R. Angel, “Direct 75 milliarc-second images from the Multiple Mirror Telescope with adaptive optics,” Astrophys. Lett. 402, L81–L84 (1993).
[CrossRef]

Cowie, L.

F. Roddier, L. Cowie, J. E. Graves, S. Songaila, D. McKenna, J. Vernin, M. Azount, J. L. Caccia, E. Limburg, C. Roddier, D. Salmon, S. Beland, D. Cowley, S. Hill, “Seeing at Mauna Kea: a joint UH-UN-NOAO-CFHT study,” in Advanced Technology Optical Telescope IV, L. D. Barr, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1236, Part 1, 485–491 (1990).
[CrossRef]

Cowley, D.

F. Roddier, L. Cowie, J. E. Graves, S. Songaila, D. McKenna, J. Vernin, M. Azount, J. L. Caccia, E. Limburg, C. Roddier, D. Salmon, S. Beland, D. Cowley, S. Hill, “Seeing at Mauna Kea: a joint UH-UN-NOAO-CFHT study,” in Advanced Technology Optical Telescope IV, L. D. Barr, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1236, Part 1, 485–491 (1990).
[CrossRef]

Cuellar, L.

Dekany, R.

M. Lloyd-Hart, R. Dekany, B. McLeod, D. Wittman, D. Colucci, D. McCarthy, R. Angel, “Direct 75 milliarc-second images from the Multiple Mirror Telescope with adaptive optics,” Astrophys. Lett. 402, L81–L84 (1993).
[CrossRef]

Ellerbroek, B.

B. Ellerbroek, “Fitting error coefficients for three segmented mirror-Hartmann sensor configurations,” (R&D Associates, Albuquerque, N.M., 1991).

Fontanella, J. C.

J. C. Fontanella, G. Rousset, P. Lena, “Adaptive optics, a key element of the VLT,” J. Opt. (Paris) 22, 99–101 (1991).
[CrossRef]

Fried, D. L.

R. Q. Fugate, D. L. Fried, G. A. Ameer, B. R. Boeke, S. L. Browne, P. H. Roberts, R. E. Ruane, G. A. Tyler, L. M. Wopat, “Measurements of atmospheric wavefront distortion using scattered light from a laser guide star,” Nature (London) 353, 144–146 (1991).
[CrossRef]

D. L. Fried, “Anisoplanatism in adaptive optics,”J. Opt. Soc. Am. 72, 52–61 (1982).
[CrossRef]

G. Tyler, D. L. Fried, “Image position error associated with a quadrant detector,”J. Opt. Soc. Am. 72, 804–808 (1982).
[CrossRef]

D. P. Greenwood, D. L. Fried, “Power spectra requirements for wave-front compensative systems,”J. Opt. Soc. Am. 66, 193–206 (1976).
[CrossRef]

D. L. Fried, “Optical resolution through a randomly inhomogeneous medium for very long and very short exposures,”J. Opt. Soc. Am. 56, 1372–1379 (1966).
[CrossRef]

D. L. Fried, “Analysis of focus anisoplanatism: the fundamental limit in performance of artificial guide star adaptive optics systems,” presented at the Workshop on Laser Guide Star Adaptive Optics, Albuquerque, N.M., 1992.

Friedman, H.

H. Friedman, J. Morris, J. Horton, “System design for high power sodium beacon laser,” presented at the Workshop on Laser Guide Star Adaptive Optics, Albuquerque, N.M., 1992.

Fugate, R. Q.

R. Q. Fugate, D. L. Fried, G. A. Ameer, B. R. Boeke, S. L. Browne, P. H. Roberts, R. E. Ruane, G. A. Tyler, L. M. Wopat, “Measurements of atmospheric wavefront distortion using scattered light from a laser guide star,” Nature (London) 353, 144–146 (1991).
[CrossRef]

M. P. Jelonek, R. Q. Fugate, W. J. Lange, A. L. Slavin, R. E. Ruane, R. A. Cleis, “Sodium laser guide star experiments at Starfire Optical Range,” presented at the Workshop on Laser Guide Star Adaptive Optics, Albuquerque, N.M., 1992.

Gardner, C. S.

C. S. Gardner, B. M. Welsh, L. A. Thompson, “Design and performance analysis of adaptive optical telescopes using laser guide stars,” Proc. IEEE 78, 1721–1743 (1990).
[CrossRef]

Geary, J. C.

J. C. Geary “Rapid-framing low-noise CCD imagers for adaptive optics,” presented at the Workshop on Laser Guide Star Adaptive Optics, Albuquerque, N.M., 1992.

Goodman, J. W.

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

Graves, J. E.

F. Roddier, M. Northcott, J. E. Graves, “A simple low-order adaptive optics system for near-infrared applications,” Publ. Astron. Soc. Pac. 103, 131–149 (1991).
[CrossRef]

J. E. Graves, D. L. McKenna, M. J. Northcott, F. Roddier, “The UH prototype adaptive optics system,” in Adaptive Optics for Large Telescopes, Vol. 19 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), pp. 173–175.

F. Roddier, L. Cowie, J. E. Graves, S. Songaila, D. McKenna, J. Vernin, M. Azount, J. L. Caccia, E. Limburg, C. Roddier, D. Salmon, S. Beland, D. Cowley, S. Hill, “Seeing at Mauna Kea: a joint UH-UN-NOAO-CFHT study,” in Advanced Technology Optical Telescope IV, L. D. Barr, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1236, Part 1, 485–491 (1990).
[CrossRef]

Greenwood, D. P.

Hermann, J.

Hill, S.

F. Roddier, L. Cowie, J. E. Graves, S. Songaila, D. McKenna, J. Vernin, M. Azount, J. L. Caccia, E. Limburg, C. Roddier, D. Salmon, S. Beland, D. Cowley, S. Hill, “Seeing at Mauna Kea: a joint UH-UN-NOAO-CFHT study,” in Advanced Technology Optical Telescope IV, L. D. Barr, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1236, Part 1, 485–491 (1990).
[CrossRef]

Horton, J.

H. Friedman, J. Morris, J. Horton, “System design for high power sodium beacon laser,” presented at the Workshop on Laser Guide Star Adaptive Optics, Albuquerque, N.M., 1992.

Jelonek, M. P.

M. P. Jelonek, R. Q. Fugate, W. J. Lange, A. L. Slavin, R. E. Ruane, R. A. Cleis, “Sodium laser guide star experiments at Starfire Optical Range,” presented at the Workshop on Laser Guide Star Adaptive Optics, Albuquerque, N.M., 1992.

Jeys, T. H.

T. H. Jeys, “Mesopheric laser beacon,” presented at the Workshop on Laser Guide Star Adaptive Optics, Albuquerque, N.M., 1992.

Johnson, P.

Kibblewhite, E.

E. Kibblewhite, “Laser beacons for astronomy,” presented at the Workshop on Laser Guide Star Adaptive Optics, Albuquerque, N.M., 1992.

Kleinhaus, W.

R. Thompson, M. Rieke, E. Young, D. McCarthy, R. Rasche, M. Blessinger, K. Vural, W. Kleinhaus, “Future directions for NICMOS arrays,” in Proceedings of Third Infrared Detector Technology Workshop, C. R. McCreight, ed., NASA Tech. Memo 102209 (NASA, Washington, D.C., 1989).

Lange, W. J.

M. P. Jelonek, R. Q. Fugate, W. J. Lange, A. L. Slavin, R. E. Ruane, R. A. Cleis, “Sodium laser guide star experiments at Starfire Optical Range,” presented at the Workshop on Laser Guide Star Adaptive Optics, Albuquerque, N.M., 1992.

Lefebvre, M.

Lena, P.

J. C. Fontanella, G. Rousset, P. Lena, “Adaptive optics, a key element of the VLT,” J. Opt. (Paris) 22, 99–101 (1991).
[CrossRef]

Lesser, M. P.

M. P. Lesser, A. Bauer, L. Ulrickson, D. Ouellette, “Bump bonded back illuminated CCDs,” in High-Resolution Sensors and Hybrid Systems, M. M. Blouke, W. Chang, R. P. Khoslan, L. J. Thorpe, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1656, 508–516 (1992).
[CrossRef]

Limburg, E.

F. Roddier, L. Cowie, J. E. Graves, S. Songaila, D. McKenna, J. Vernin, M. Azount, J. L. Caccia, E. Limburg, C. Roddier, D. Salmon, S. Beland, D. Cowley, S. Hill, “Seeing at Mauna Kea: a joint UH-UN-NOAO-CFHT study,” in Advanced Technology Optical Telescope IV, L. D. Barr, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1236, Part 1, 485–491 (1990).
[CrossRef]

Lloyd-Hart, M.

M. Lloyd-Hart, R. Dekany, B. McLeod, D. Wittman, D. Colucci, D. McCarthy, R. Angel, “Direct 75 milliarc-second images from the Multiple Mirror Telescope with adaptive optics,” Astrophys. Lett. 402, L81–L84 (1993).
[CrossRef]

D. Wittman, J. R. P. Angel, M. Lloyd-Hart, D. Collucci, D. McCarthy, “Optical sensing of infrared wavefronts for adaptive control: a new CCD detector and MMT experiments,” in Vol. 42 of Proceedings of European Southern Observatory Conference on Progress in Telescopeand Instrumentation Technologies, M.-H. Ulrich, ed. (European Southern Observatory, Garching, Germany, 1992), pp. 453–460.

McCarthy, D.

M. Lloyd-Hart, R. Dekany, B. McLeod, D. Wittman, D. Colucci, D. McCarthy, R. Angel, “Direct 75 milliarc-second images from the Multiple Mirror Telescope with adaptive optics,” Astrophys. Lett. 402, L81–L84 (1993).
[CrossRef]

D. Wittman, J. R. P. Angel, M. Lloyd-Hart, D. Collucci, D. McCarthy, “Optical sensing of infrared wavefronts for adaptive control: a new CCD detector and MMT experiments,” in Vol. 42 of Proceedings of European Southern Observatory Conference on Progress in Telescopeand Instrumentation Technologies, M.-H. Ulrich, ed. (European Southern Observatory, Garching, Germany, 1992), pp. 453–460.

R. Thompson, M. Rieke, E. Young, D. McCarthy, R. Rasche, M. Blessinger, K. Vural, W. Kleinhaus, “Future directions for NICMOS arrays,” in Proceedings of Third Infrared Detector Technology Workshop, C. R. McCreight, ed., NASA Tech. Memo 102209 (NASA, Washington, D.C., 1989).

McKenna, D.

F. Roddier, L. Cowie, J. E. Graves, S. Songaila, D. McKenna, J. Vernin, M. Azount, J. L. Caccia, E. Limburg, C. Roddier, D. Salmon, S. Beland, D. Cowley, S. Hill, “Seeing at Mauna Kea: a joint UH-UN-NOAO-CFHT study,” in Advanced Technology Optical Telescope IV, L. D. Barr, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1236, Part 1, 485–491 (1990).
[CrossRef]

McKenna, D. L.

J. E. Graves, D. L. McKenna, M. J. Northcott, F. Roddier, “The UH prototype adaptive optics system,” in Adaptive Optics for Large Telescopes, Vol. 19 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), pp. 173–175.

McLeod, B.

M. Lloyd-Hart, R. Dekany, B. McLeod, D. Wittman, D. Colucci, D. McCarthy, R. Angel, “Direct 75 milliarc-second images from the Multiple Mirror Telescope with adaptive optics,” Astrophys. Lett. 402, L81–L84 (1993).
[CrossRef]

Morris, J.

H. Friedman, J. Morris, J. Horton, “System design for high power sodium beacon laser,” presented at the Workshop on Laser Guide Star Adaptive Optics, Albuquerque, N.M., 1992.

Murphy, D. V.

C. A. Primmerman, D. V. Murphy, D. A. Page, G. Zollars, H. T. Barclay, “Compensation of atmospheric optical distortion using a synthetic beacon,” Nature (London) 353, 141–143 (1991).
[CrossRef]

Noll, R. J.

Northcott, M.

F. Roddier, M. Northcott, J. E. Graves, “A simple low-order adaptive optics system for near-infrared applications,” Publ. Astron. Soc. Pac. 103, 131–149 (1991).
[CrossRef]

Northcott, M. J.

J. E. Graves, D. L. McKenna, M. J. Northcott, F. Roddier, “The UH prototype adaptive optics system,” in Adaptive Optics for Large Telescopes, Vol. 19 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), pp. 173–175.

Ouellette, D.

M. P. Lesser, A. Bauer, L. Ulrickson, D. Ouellette, “Bump bonded back illuminated CCDs,” in High-Resolution Sensors and Hybrid Systems, M. M. Blouke, W. Chang, R. P. Khoslan, L. J. Thorpe, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1656, 508–516 (1992).
[CrossRef]

Page, D. A.

C. A. Primmerman, D. V. Murphy, D. A. Page, G. Zollars, H. T. Barclay, “Compensation of atmospheric optical distortion using a synthetic beacon,” Nature (London) 353, 141–143 (1991).
[CrossRef]

Primmerman, C. A.

C. A. Primmerman, D. V. Murphy, D. A. Page, G. Zollars, H. T. Barclay, “Compensation of atmospheric optical distortion using a synthetic beacon,” Nature (London) 353, 141–143 (1991).
[CrossRef]

Rasche, R.

R. Thompson, M. Rieke, E. Young, D. McCarthy, R. Rasche, M. Blessinger, K. Vural, W. Kleinhaus, “Future directions for NICMOS arrays,” in Proceedings of Third Infrared Detector Technology Workshop, C. R. McCreight, ed., NASA Tech. Memo 102209 (NASA, Washington, D.C., 1989).

Rego, A.

Rieke, M.

R. Thompson, M. Rieke, E. Young, D. McCarthy, R. Rasche, M. Blessinger, K. Vural, W. Kleinhaus, “Future directions for NICMOS arrays,” in Proceedings of Third Infrared Detector Technology Workshop, C. R. McCreight, ed., NASA Tech. Memo 102209 (NASA, Washington, D.C., 1989).

Roberts, P. H.

R. Q. Fugate, D. L. Fried, G. A. Ameer, B. R. Boeke, S. L. Browne, P. H. Roberts, R. E. Ruane, G. A. Tyler, L. M. Wopat, “Measurements of atmospheric wavefront distortion using scattered light from a laser guide star,” Nature (London) 353, 144–146 (1991).
[CrossRef]

Roddier, C.

F. Roddier, L. Cowie, J. E. Graves, S. Songaila, D. McKenna, J. Vernin, M. Azount, J. L. Caccia, E. Limburg, C. Roddier, D. Salmon, S. Beland, D. Cowley, S. Hill, “Seeing at Mauna Kea: a joint UH-UN-NOAO-CFHT study,” in Advanced Technology Optical Telescope IV, L. D. Barr, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1236, Part 1, 485–491 (1990).
[CrossRef]

Roddier, F.

F. Roddier, M. Northcott, J. E. Graves, “A simple low-order adaptive optics system for near-infrared applications,” Publ. Astron. Soc. Pac. 103, 131–149 (1991).
[CrossRef]

J. E. Graves, D. L. McKenna, M. J. Northcott, F. Roddier, “The UH prototype adaptive optics system,” in Adaptive Optics for Large Telescopes, Vol. 19 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), pp. 173–175.

F. Roddier, L. Cowie, J. E. Graves, S. Songaila, D. McKenna, J. Vernin, M. Azount, J. L. Caccia, E. Limburg, C. Roddier, D. Salmon, S. Beland, D. Cowley, S. Hill, “Seeing at Mauna Kea: a joint UH-UN-NOAO-CFHT study,” in Advanced Technology Optical Telescope IV, L. D. Barr, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1236, Part 1, 485–491 (1990).
[CrossRef]

Rousset, G.

J. C. Fontanella, G. Rousset, P. Lena, “Adaptive optics, a key element of the VLT,” J. Opt. (Paris) 22, 99–101 (1991).
[CrossRef]

Ruane, R. E.

R. Q. Fugate, D. L. Fried, G. A. Ameer, B. R. Boeke, S. L. Browne, P. H. Roberts, R. E. Ruane, G. A. Tyler, L. M. Wopat, “Measurements of atmospheric wavefront distortion using scattered light from a laser guide star,” Nature (London) 353, 144–146 (1991).
[CrossRef]

M. P. Jelonek, R. Q. Fugate, W. J. Lange, A. L. Slavin, R. E. Ruane, R. A. Cleis, “Sodium laser guide star experiments at Starfire Optical Range,” presented at the Workshop on Laser Guide Star Adaptive Optics, Albuquerque, N.M., 1992.

Salmon, D.

F. Roddier, L. Cowie, J. E. Graves, S. Songaila, D. McKenna, J. Vernin, M. Azount, J. L. Caccia, E. Limburg, C. Roddier, D. Salmon, S. Beland, D. Cowley, S. Hill, “Seeing at Mauna Kea: a joint UH-UN-NOAO-CFHT study,” in Advanced Technology Optical Telescope IV, L. D. Barr, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1236, Part 1, 485–491 (1990).
[CrossRef]

Sandler, D.

D. Sandler, “A multiple spot laser beacon for high-order wavefront control: theory and experiment,” presented at the Workshop on Laser Guide Star Adaptive Optics, Albuquerque, N.M., 1992.

Sandler, D. G.

Schwartz, D. E.

R. J. Wainscoat, M. Cohen, K. Volk, H. J. Walker, D. E. Schwartz, “A model of the 8–25 μm point source infrared sky,” Astrophys. J. Suppl. Ser. 83, 111–146 (1992).
[CrossRef]

Slavin, A. L.

M. P. Jelonek, R. Q. Fugate, W. J. Lange, A. L. Slavin, R. E. Ruane, R. A. Cleis, “Sodium laser guide star experiments at Starfire Optical Range,” presented at the Workshop on Laser Guide Star Adaptive Optics, Albuquerque, N.M., 1992.

Smith, G.

Songaila, S.

F. Roddier, L. Cowie, J. E. Graves, S. Songaila, D. McKenna, J. Vernin, M. Azount, J. L. Caccia, E. Limburg, C. Roddier, D. Salmon, S. Beland, D. Cowley, S. Hill, “Seeing at Mauna Kea: a joint UH-UN-NOAO-CFHT study,” in Advanced Technology Optical Telescope IV, L. D. Barr, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1236, Part 1, 485–491 (1990).
[CrossRef]

Soniera, R. M.

J. N. Bahcall, R. M. Soniera, “Predicated star counts in selected fields and photometric bands: applications to galactic structure, the disk luminosity function, and the detection of a massive halo,” Astrophys. J. Suppl. Ser. 47, 357–401 (1981).
[CrossRef]

Southwell, W. H.

Spivey, B.

Strittmatter, P.

P. Strittmatter, “Columbus Project Telescope,” in Advanced Technology Optical Telescopes IV, L. D. Barr, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1236, 71–78 (1990).
[CrossRef]

Tatarski, V. I.

V. I. Tatarski, Wave Propagation in a Turbulent Medium (Dover, New York, 1961).

Taylor, G.

Thompson, L. A.

C. S. Gardner, B. M. Welsh, L. A. Thompson, “Design and performance analysis of adaptive optical telescopes using laser guide stars,” Proc. IEEE 78, 1721–1743 (1990).
[CrossRef]

Thompson, R.

R. Thompson, M. Rieke, E. Young, D. McCarthy, R. Rasche, M. Blessinger, K. Vural, W. Kleinhaus, “Future directions for NICMOS arrays,” in Proceedings of Third Infrared Detector Technology Workshop, C. R. McCreight, ed., NASA Tech. Memo 102209 (NASA, Washington, D.C., 1989).

Tyler, G.

Tyler, G. A.

R. Q. Fugate, D. L. Fried, G. A. Ameer, B. R. Boeke, S. L. Browne, P. H. Roberts, R. E. Ruane, G. A. Tyler, L. M. Wopat, “Measurements of atmospheric wavefront distortion using scattered light from a laser guide star,” Nature (London) 353, 144–146 (1991).
[CrossRef]

G. A. Tyler, “Analysis of propagation through turbulence: evaluation of an integral involving the product of three Bessel functions,” J. Opt. Soc. Am. A 7, 1218–1223 (1990).
[CrossRef]

G. A. Tyler, “Turbulence-induced adaptive-optics performance degradation: evaluation in the time domain,” J. Opt. Soc. Am. A 1, 251–262 (1984).
[CrossRef]

G. A. Tyler, “Bandwidth considerations for tracking through turbulence,” (Optical Sciences Company, Placentia, Calif., 1988).

Ulrickson, L.

M. P. Lesser, A. Bauer, L. Ulrickson, D. Ouellette, “Bump bonded back illuminated CCDs,” in High-Resolution Sensors and Hybrid Systems, M. M. Blouke, W. Chang, R. P. Khoslan, L. J. Thorpe, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1656, 508–516 (1992).
[CrossRef]

Valley, G. C.

Vernin, J.

F. Roddier, L. Cowie, J. E. Graves, S. Songaila, D. McKenna, J. Vernin, M. Azount, J. L. Caccia, E. Limburg, C. Roddier, D. Salmon, S. Beland, D. Cowley, S. Hill, “Seeing at Mauna Kea: a joint UH-UN-NOAO-CFHT study,” in Advanced Technology Optical Telescope IV, L. D. Barr, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1236, Part 1, 485–491 (1990).
[CrossRef]

Volk, K.

R. J. Wainscoat, M. Cohen, K. Volk, H. J. Walker, D. E. Schwartz, “A model of the 8–25 μm point source infrared sky,” Astrophys. J. Suppl. Ser. 83, 111–146 (1992).
[CrossRef]

Vural, K.

R. Thompson, M. Rieke, E. Young, D. McCarthy, R. Rasche, M. Blessinger, K. Vural, W. Kleinhaus, “Future directions for NICMOS arrays,” in Proceedings of Third Infrared Detector Technology Workshop, C. R. McCreight, ed., NASA Tech. Memo 102209 (NASA, Washington, D.C., 1989).

Wainscoat, R. J.

R. J. Wainscoat, M. Cohen, K. Volk, H. J. Walker, D. E. Schwartz, “A model of the 8–25 μm point source infrared sky,” Astrophys. J. Suppl. Ser. 83, 111–146 (1992).
[CrossRef]

Walker, H. J.

R. J. Wainscoat, M. Cohen, K. Volk, H. J. Walker, D. E. Schwartz, “A model of the 8–25 μm point source infrared sky,” Astrophys. J. Suppl. Ser. 83, 111–146 (1992).
[CrossRef]

Walner, E. P.

Wandzura, S. M.

Welsh, B. M.

C. S. Gardner, B. M. Welsh, L. A. Thompson, “Design and performance analysis of adaptive optical telescopes using laser guide stars,” Proc. IEEE 78, 1721–1743 (1990).
[CrossRef]

Wittman, D.

M. Lloyd-Hart, R. Dekany, B. McLeod, D. Wittman, D. Colucci, D. McCarthy, R. Angel, “Direct 75 milliarc-second images from the Multiple Mirror Telescope with adaptive optics,” Astrophys. Lett. 402, L81–L84 (1993).
[CrossRef]

D. Wittman, J. R. P. Angel, M. Lloyd-Hart, D. Collucci, D. McCarthy, “Optical sensing of infrared wavefronts for adaptive control: a new CCD detector and MMT experiments,” in Vol. 42 of Proceedings of European Southern Observatory Conference on Progress in Telescopeand Instrumentation Technologies, M.-H. Ulrich, ed. (European Southern Observatory, Garching, Germany, 1992), pp. 453–460.

Wopat, L. M.

R. Q. Fugate, D. L. Fried, G. A. Ameer, B. R. Boeke, S. L. Browne, P. H. Roberts, R. E. Ruane, G. A. Tyler, L. M. Wopat, “Measurements of atmospheric wavefront distortion using scattered light from a laser guide star,” Nature (London) 353, 144–146 (1991).
[CrossRef]

Young, E.

R. Thompson, M. Rieke, E. Young, D. McCarthy, R. Rasche, M. Blessinger, K. Vural, W. Kleinhaus, “Future directions for NICMOS arrays,” in Proceedings of Third Infrared Detector Technology Workshop, C. R. McCreight, ed., NASA Tech. Memo 102209 (NASA, Washington, D.C., 1989).

Zollars, G.

C. A. Primmerman, D. V. Murphy, D. A. Page, G. Zollars, H. T. Barclay, “Compensation of atmospheric optical distortion using a synthetic beacon,” Nature (London) 353, 141–143 (1991).
[CrossRef]

Astrophys. J. Suppl. Ser.

J. N. Bahcall, R. M. Soniera, “Predicated star counts in selected fields and photometric bands: applications to galactic structure, the disk luminosity function, and the detection of a massive halo,” Astrophys. J. Suppl. Ser. 47, 357–401 (1981).
[CrossRef]

R. J. Wainscoat, M. Cohen, K. Volk, H. J. Walker, D. E. Schwartz, “A model of the 8–25 μm point source infrared sky,” Astrophys. J. Suppl. Ser. 83, 111–146 (1992).
[CrossRef]

Astrophys. Lett.

M. Lloyd-Hart, R. Dekany, B. McLeod, D. Wittman, D. Colucci, D. McCarthy, R. Angel, “Direct 75 milliarc-second images from the Multiple Mirror Telescope with adaptive optics,” Astrophys. Lett. 402, L81–L84 (1993).
[CrossRef]

J. Opt. (Paris)

J. C. Fontanella, G. Rousset, P. Lena, “Adaptive optics, a key element of the VLT,” J. Opt. (Paris) 22, 99–101 (1991).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Nature (London)

R. Q. Fugate, D. L. Fried, G. A. Ameer, B. R. Boeke, S. L. Browne, P. H. Roberts, R. E. Ruane, G. A. Tyler, L. M. Wopat, “Measurements of atmospheric wavefront distortion using scattered light from a laser guide star,” Nature (London) 353, 144–146 (1991).
[CrossRef]

C. A. Primmerman, D. V. Murphy, D. A. Page, G. Zollars, H. T. Barclay, “Compensation of atmospheric optical distortion using a synthetic beacon,” Nature (London) 353, 141–143 (1991).
[CrossRef]

Proc. IEEE

C. S. Gardner, B. M. Welsh, L. A. Thompson, “Design and performance analysis of adaptive optical telescopes using laser guide stars,” Proc. IEEE 78, 1721–1743 (1990).
[CrossRef]

Publ. Astron. Soc. Pac.

F. Roddier, M. Northcott, J. E. Graves, “A simple low-order adaptive optics system for near-infrared applications,” Publ. Astron. Soc. Pac. 103, 131–149 (1991).
[CrossRef]

H. W. Babcock, “The possibility of compensating astronomical seeing,” Publ. Astron. Soc. Pac. 65, 229–236 (1953).
[CrossRef]

Other

J. R. P. Angel, “Use of natural stars with laser beacons for large telescope adaptive optics,” presented at the Workshop on Laser Guide Star Adaptive Optics, Albuquerque, N.M., (1992).

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

M. P. Jelonek, R. Q. Fugate, W. J. Lange, A. L. Slavin, R. E. Ruane, R. A. Cleis, “Sodium laser guide star experiments at Starfire Optical Range,” presented at the Workshop on Laser Guide Star Adaptive Optics, Albuquerque, N.M., 1992.

T. H. Jeys, “Mesopheric laser beacon,” presented at the Workshop on Laser Guide Star Adaptive Optics, Albuquerque, N.M., 1992.

H. Friedman, J. Morris, J. Horton, “System design for high power sodium beacon laser,” presented at the Workshop on Laser Guide Star Adaptive Optics, Albuquerque, N.M., 1992.

B. Ellerbroek, “Fitting error coefficients for three segmented mirror-Hartmann sensor configurations,” (R&D Associates, Albuquerque, N.M., 1991).

D. L. Fried, “Analysis of focus anisoplanatism: the fundamental limit in performance of artificial guide star adaptive optics systems,” presented at the Workshop on Laser Guide Star Adaptive Optics, Albuquerque, N.M., 1992.

D. Sandler, “A multiple spot laser beacon for high-order wavefront control: theory and experiment,” presented at the Workshop on Laser Guide Star Adaptive Optics, Albuquerque, N.M., 1992.

J. E. Graves, D. L. McKenna, M. J. Northcott, F. Roddier, “The UH prototype adaptive optics system,” in Adaptive Optics for Large Telescopes, Vol. 19 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), pp. 173–175.

R. Thompson, M. Rieke, E. Young, D. McCarthy, R. Rasche, M. Blessinger, K. Vural, W. Kleinhaus, “Future directions for NICMOS arrays,” in Proceedings of Third Infrared Detector Technology Workshop, C. R. McCreight, ed., NASA Tech. Memo 102209 (NASA, Washington, D.C., 1989).

D. Wittman, J. R. P. Angel, M. Lloyd-Hart, D. Collucci, D. McCarthy, “Optical sensing of infrared wavefronts for adaptive control: a new CCD detector and MMT experiments,” in Vol. 42 of Proceedings of European Southern Observatory Conference on Progress in Telescopeand Instrumentation Technologies, M.-H. Ulrich, ed. (European Southern Observatory, Garching, Germany, 1992), pp. 453–460.

J. C. Geary “Rapid-framing low-noise CCD imagers for adaptive optics,” presented at the Workshop on Laser Guide Star Adaptive Optics, Albuquerque, N.M., 1992.

M. P. Lesser, A. Bauer, L. Ulrickson, D. Ouellette, “Bump bonded back illuminated CCDs,” in High-Resolution Sensors and Hybrid Systems, M. M. Blouke, W. Chang, R. P. Khoslan, L. J. Thorpe, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1656, 508–516 (1992).
[CrossRef]

V. I. Tatarski, Wave Propagation in a Turbulent Medium (Dover, New York, 1961).

P. Strittmatter, “Columbus Project Telescope,” in Advanced Technology Optical Telescopes IV, L. D. Barr, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1236, 71–78 (1990).
[CrossRef]

E. Kibblewhite, “Laser beacons for astronomy,” presented at the Workshop on Laser Guide Star Adaptive Optics, Albuquerque, N.M., 1992.

G. A. Tyler, “Bandwidth considerations for tracking through turbulence,” (Optical Sciences Company, Placentia, Calif., 1988).

F. Roddier, L. Cowie, J. E. Graves, S. Songaila, D. McKenna, J. Vernin, M. Azount, J. L. Caccia, E. Limburg, C. Roddier, D. Salmon, S. Beland, D. Cowley, S. Hill, “Seeing at Mauna Kea: a joint UH-UN-NOAO-CFHT study,” in Advanced Technology Optical Telescope IV, L. D. Barr, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1236, Part 1, 485–491 (1990).
[CrossRef]

R. R. Beland, “A decade of balloon microthermal probe measurements of optical turbulence,” in Adaptive Optics for Large Telescopes, Vol. 19 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), pp. 14–16.

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Figures (16)

Fig. 1
Fig. 1

Conceptual layout of the adaptive-optical system for correcting 8-m telescopes at λ = 1.6 to λ = 2.2 μm. DSP, digital signal processor. Refer to the text for an explanation of the components.

Fig. 2
Fig. 2

Turbulent index-of-refraction profiles used in our analysis and simulations. Gray curve, MK model derived from data in Ref. 21. Solid black curve, MMK model with the 4-km MK layer moved to 10 km. Dashed curve, version of the Hufnagel–Valley model used in Ref. 23.

Fig. 3
Fig. 3

Rms off-axis tilt error, and corresponding Strehl ratios versus field-star angle, for three wavelengths. The curves are for D = 8 m and the MK turbulence model. (a) The rms angular error (one axis) is in units of λ/D. (b) The Strehl ratios were computed with Eq. (9).

Fig. 4
Fig. 4

Plots corresponding to those shown in Fig. 3, for the MMK turbulence model.

Fig. 5
Fig. 5

Simulated images of guide stars at a 25-arcsec field angle for (a) λ = 2 μm, (b) λ = 1.5 μm, and (c) λ = 0.7 μm, obtained with the MK turbulence model. The stars were imaged through an 8-m telescope that was precorrected in the target-object direction with a sodium laser guide star.

Fig. 6
Fig. 6

Same data and conditions as for Fig. 5, obtained with the MMK turbulence model.

Fig. 7
Fig. 7

Simulated 1.45-μm images of field stars at (a) 20-arcsec and (b) 50-arcsec separation from the science target, for D = 8 m and MK turbulence. A sodium laser beacon in the direction of the science target has been used to correct the telescope. For comparison, (c) the diffraction-limited image at 1.45 μm and (d) the uncorrected short-exposure image are also shown.

Fig. 8
Fig. 8

Simulated 1.45-μm images of field stars at (a) 10-arcsec and (b) 25-arcsec separation from the science target, for D = 8 m and MMK turbulence. A sodium laser beacon in the direction of the science target has been used to correct the telescope. For comparison, (c) the diffraction-limited image at 1.45 μm and (d) the uncorrected short-exposure image are also shown.

Fig. 9
Fig. 9

Strehl ratios corresponding to the angular decorrelation of focus and astigmatism over an 8-m telescope, as a function of separation of the field star from the science target, for λ = 0.5, λ = 1, and λ = 2 μm. (a) MK turbulence, (b) MMK turbulence.

Fig. 10
Fig. 10

Strehl ratios, resulting from finite beacon height, for the cone effect corresponding to the residual wave-front error given in Eq. (28), with D = 8 m and h = 90 km, as a function of science wavelength. Results are shown for the MK, the MMK, and the HV turbulence models.

Fig. 11
Fig. 11

Zernike decomposition of the residual cone error for λ = 0.5 μm and D = 8 m versus mode number. We used the mode ordering given in Ref. 18. The results for the MK, the MMK, and the HV turbulence profiles are shown.

Fig. 12
Fig. 12

Arrangement of the square 1-m subapertures that were used for our simulations of wave-front correction. There are eight subapertures across the 8-m telescope, giving a total of 52. Enlargements of adjacent subapertures ij and i + l, j, with the measured wave-front slopes in the x direction being indicated by the dashed lines, are also shown.

Fig. 13
Fig. 13

Contours of phase distortion corresponding to a Monte Carlo simulation of wave-front correction for D = 8 m and λ = 2 μm. (a) Kolmogorov distortion across the telescope (0.4 wave rms) before adaptive correction. (b) The reconstructed wavefront corresponding to tilt and piston phasing of 1-m subapertures. (c) The residual wave-front error after correction (0.11 wave rms), obtained by differencing (a) and (b).

Fig. 14
Fig. 14

Three-dimensional plots of the phase distortion in radians, corresponding to the phase contours shown in Fig. 13. (a) Uncorrected phase; (b) reconstructed phase; (c) difference of (a) and (b).

Fig. 15
Fig. 15

Correlation parameter Γn of Eqs. (A6) and (A7) for Zernike modes over an 8-m aperture as a function of off-axis field-star angle, for (a) MK turbulence and (b) MMK turbulence. The correlation parameters for the tilt, the focus, and the coma at λ = 0.5 μm, λ = 1.0 μm, and λ = 2.0 μm are shown.

Fig. 16
Fig. 16

Value of the noise propagator G, given in Eq. (29), for phase reconsruction over the subaperture arrangement that is shown in Fig. 12, as a function of increasing Zernike-mode index.

Tables (4)

Tables Icon

Table 1 Atmospheric Parameters for the MK and the MMK Turbulence Models at λ = 0.5, λ = 1.6, λ = 2.2 μm

Tables Icon

Table 2 Tilt Errors from the Combined Effects of Temporal Decorrelation and Measurement Noisea

Tables Icon

Table 3 Error Sources for Correcting the High-Order Wave-Front across an 8-m Telescope at λ = 1.6 μm and λ = 2.2 μma

Tables Icon

Table 4 Contributions to the Total Strehl Ratio for the H (λ = 1.6 μm) and the K (λ = 2.2 μm) Wave Bandsa

Equations (61)

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

Ψ ( x ) = 2 π λ Θ · x + Φ ( x ) ,
θ = Θ - Θ ^ ,
ϕ ( x ) = Φ ( x ) - Φ ^ ( x ) .
ψ ( x ) = 2 π λ θ · x + ϕ ( x ) .
S = I ( r c ) I d l ( r c ) ,
S = 16 π 2 D 4 exp [ - 1 2 D ψ ( x , x ) ] w ( x ) w ( x ) d x d x ,
w ( x ) = { 1 if x < D / 2 0 otherwise
D ψ ( x , x ) [ ψ ( x ) - ψ ( x ) ] 2
S θ = 1 1 + π 2 2 ( σ θ λ / D ) 2 .
σ θ 2 = 1 2 [ ( σ θ x ) 2 + ( σ θ y ) 2 ] ,
D ϕ ( x , x ) = 2 σ ϕ 2 - 2 ϕ ( x ) ϕ ( x ) ,
σ ϕ = [ 4 π D 2 w ( x ) ϕ 2 ( x ) d x ] 1 / 2 .
S ϕ = exp ( - σ ϕ 2 ) + 16 π 2 D 4 exp [ - ϕ ( x ) ϕ ( x ) ] × w ( x ) w ( x ) d x d x
S ϕ = exp ( - σ ϕ 2 ) .
S = S θ S ϕ .
σ θ 2 = ( σ θ iso ) 2 + ( σ θ noise ) 2 + ( σ θ time ) 2 ,
σ ϕ 2 = ( σ ϕ cone ) 2 + ( σ ϕ rec ) 2 + ( σ ϕ time ) 2 + ( σ ϕ iso ) 2 .
r 0 - 5 / 3 = 0.423 k 2 sec ( ξ ) C n 2 ( z ) d z ,
σ Φ = 0.366 ( D r 0 ) 5 / 6 .
σ Θ = 0.43 λ D ( D r 0 ) 5 / 6 .
σ ψ iso = ( θ θ 0 ) 5 / 6
θ 0 - 5 / 3 = 2.91 k 2 sec 8 / 3 ( ξ ) C n 2 ( z ) z 5 / 3 d z .
σ ψ time = ( Δ t t 0 ) 5 / 6 ,
t 0 = 0.31 r 0 v ¯ w
v ¯ w 5 / 3 = C n 2 ( z ) v w 5 / 3 ( z ) d z C n 2 ( z ) d z
( σ θ iso ) 2 = 8 π 2 ( 0.448 ) ( 1 - Γ 2 ) ( D r 0 ) 5 / 3 ( λ D ) 2 ,
σ θ noise = α w λ D 1 N ( 1 + 4 n 2 N ) 1 / 2 ,
σ θ time = 0.177 T t 0 ( r 0 D ) 1 / 6 λ D .
σ ϕ cone = ( D d 0 ) 5 / 6 ,
( σ ϕ rec ) 2 = G σ pd 2 ,
σ pd = 2 π σ θ SH λ 0 / d λ 0 λ ,
( σ ϕ rec ) 2 = 4 π 2 G α 2 w 2 N ( 1 + 4 n 2 N ) ( λ 0 λ ) 2 .
N = 300 τ P A η ,
( σ ϕ rec ) 2 = 2.4 P τ ( 1 + 0.25 P τ ) ( 0.589 λ ) 2 ,
σ ϕ fit = c fit ( d r 0 ) 5 / 6 ,
( σ Ψ iso ) 2 = 4 π D 2 w ( x ) [ Ψ ( x ) - Ψ ^ ( x ) ] 2 d x ,
( σ Ψ iso ) 2 = 2 ( 1 - Γ ) σ Ψ 2 ,
Γ = 4 π D 2 σ Ψ 2 w ( x ) Ψ ( x ) Ψ ^ ( x ) d x .
Ψ ( r ) = a n Z n ( r ) ,
Ψ ^ ( r ) = b n Z n ( r ) .
Δ n = 2 ( 1 - Γ n ) a n 2
Γ n ( θ ) = γ n ( z θ ) C n 2 ( z ) d z C n 2 ( z ) d z ,
γ n ( z θ ) = C j m j m ( Δ ) ± C j - m j m ( Δ ) C j m j m ( 0 ) ,
C j m j m ( Δ ) = ( - ) j + j 8 π 2 C R β ɛ ( m , m ) [ ( 2 j + m + 1 ) × ( 2 j + m + 1 ) ] 1 / 2 0 d z z - β - 3 J m - m ( 2 z Δ ) × J 2 j + m + 1 ( z ) J 2 j + m + 1 ( z ) ,
Δ 2 = 2 ( 0.448 ) ( 1 - Γ 2 ) ( D r 0 ) 5 / 3 ,
( σ θ x ) 2 ( λ / D ) 2 = 0.0472 ( θ θ 0 ) 2 ( D r 0 ) - 1 / 3 - 0.0107 ( θ θ 0 ) 4 ( D r 0 ) - 7 / 3 ,
( σ θ y ) 2 ( λ / D ) 2 = 0.0157 ( θ θ 0 ) 2 ( D r 0 ) - 1 / 3 - 0.00214 ( θ θ 0 ) 4 ( D r 0 ) - 7 / 3 .
σ θ noise = λ / D 4 SNR 0 1 H ( x D , 0 ) d x ,
SNR = N ( 4 n 2 + N ) 1 / 2 ,
H ( x D , 0 ) = 2 π exp [ - 3.44 ( x D r 0 ) 5 / 3 ( 1 - x 1 / 3 ) ] × [ cos - 1 ( x ) - x ( 1 - x 2 ) 1 / 2 ] .
σ θ noise = α w λ D 1 N ( 1 + 4 n 2 N ) 1 / 2 .
Δ ϕ i j x = 4 π d w i j ( r ) x Φ ( r ) d r + n i j ,
Φ i + 1 , j - Φ i j = d 2 ( s i + 1 , j x + s i j x ) .
i j [ Φ ^ i + 1 , j - Φ ^ i j - d 2 ( s i + 1 , j x + s i j x ) ] 2 + [ Φ ^ i , j + 1 - Φ ^ i j - d 2 ( s i , j + 1 y + s i j y ) ] 2
2 Φ ^ i j = · s i j ,
Φ ^ i j ( t + 1 ) = Φ ^ i j ( t ) + ω [ Φ ¯ i j ( t ) + b i j / g i j - Φ ^ i j ( t ) ] ,
Φ ¯ i j ( t ) = [ Φ ^ i + 1 , j ( t ) + Φ ^ i - 1 , j ( t ) + Φ ^ i , j + 1 ( t ) + Φ ^ i , j - 1 ( t ) ] ,
b i j = ( s i - 1 , j x - s i j x + s i . j - 1 y - s i j y ) ,
g i j = 2 ,             i = 1             or             n i , j = 1             or             n j , = 3 { i = 1 or n i , j = 2 to n j - 1 , i = 2 to n i - 1 , j = 1 or n j = 4             otherwise .
σ ϕ rec = [ 1 N i j ( Φ ^ i j - Φ i j ) 2 ] 1 / 2 .
( σ ϕ rec ) 2 = G σ pd 2 .

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