Abstract

We describe an approach to the computation of photon returns from mesospheric sodium beacons excited by laser pulse trains and discuss as specific examples the required numbers of photons for adaptive-optical compensation of atmospheric turbulence. Computed photon return signals are compared with reported measurements for pulses that are long, short, or comparable to the D2 radiative lifetime (16 ns). Analytical approximations in good agreement with the numerical computations are derived. The results are consistent with experimental data for the different pulse durations.

© 1998 Optical Society of America

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  1. See, for instance, R. Benedict, J. B. Breckinridge, D. L. Fried, eds., “Feature on atmospheric-compensation technology,” J. Opt. Soc. Am. A 11, 255–451, 779–945 (1994), and references therein; L. A. Thompson, “Adaptive optics in astronomy,” Phys. Today 47(12), 24–31 (1994); R. Q. Fugate, “Laser beacon,” Opt. Photon. News 4(6), 14–19 (1993); R. Q. Fugate, W. Wild, “Untwinkling the stars,” Sky Telescope 87(5), 21–31 (1994); J. M. Beckers, “Adaptive optics for astronomy: principles, performance, and applications,” Annu. Rev. Astron. Astrophys. 31, 13–62 (1993); N. Hubin, L. Noethe, “Active optics, adaptive optics, and laser guide stars,” Science 262, 1390–1394 (1993); T. H. Jeys, “Development of a mesospheric sodium laser beacon for atmospheric adaptive optics,” MIT Lincoln Lab. J. 4, 133–149 (1991); C. S. Gardner, B. M. Welsh, L. A. Thompson, “Design and performance analysis of adaptive optical telescopes using laser guide stars,” Proc. IEEE 78, 1724–1743 (1990);H. W. Babcock, “Adaptive optics revisited,” Science 249, 253–257 (1990).
    [CrossRef] [PubMed]
  2. P. W. Milonni, L. E. Thode, “Theory of mesospheric sodium fluorescence excited by pulse trains,” Appl. Opt. 31, 785–800 (1992).
    [CrossRef] [PubMed]
  3. L. C. Bradley, “Pulse-train excitation of sodium for use as a synthetic beacon,” J. Opt. Soc. Am. B 9, 1931–1944 (1992).
    [CrossRef]
  4. J. R. Morris, “Efficient excitation of a mesospheric sodium laser guide star by intermediate-duration pulses,” J. Opt. Soc. Am. A 11, 832–845 (1994).
    [CrossRef]
  5. See, for instance, P. L. Knight, P. W. Milonni, “The Rabi frequency in optical spectra,” Phys. Rep. 66, 21–107 (1980).
    [CrossRef]
  6. R. D. Cowan, The Theory of Atomic Structure and Spectra (U. California Press, Berkeley, Calif., 1981).
  7. 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]
  8. J. W. Goodman, Statistical Optics (Wiley, New York, 1985), Chap. 8.
  9. For discussions of these error sources, see, for instance, R. R. Parenti, R. J. Sasiela, “Laser-guide-star systems for astronomical applications,” J. Opt. Soc. Am. A 11, 288–309 (1994); D. G. Sandler, S. Stahl, J. R. P. Angel, M. Lloyd-Hart, D. McCarthy, “Adaptive optics for diffraction-limited infrared imaging with 8-m telescopes,” J. Opt. Soc. Am. A 11, 925–945 (1994).
    [CrossRef]
  10. C. D. Morgenstern, R. Q. Fugate, A. C. Slavin, “Summary of optical turbulence measurements at the Starfire Optical Range,” in Atmospheric Propagation and Remote Sensing IV, J. C. Dainty, ed., Proc. SPIE2471, 390–398 (1995).
    [CrossRef]
  11. F. Roddier, “Report on the seeing on Mauna Kea” (U. Hawaii, Honolulu, Hawaii, 1992).
  12. D. L. Fried, J. F. Belsher, “Analysis of fundamental limits to artificial-guide-star adaptive-optics-system performance for astronomical imaging,” J. Opt. Soc. Am. A 11, 277–287 (1994).
    [CrossRef]
  13. B. L. Ellerbroek, “First-order performance evaluation of adaptive-optics systems for atmospheric turbulence compensation in extended-field-of-view astronomical telescopes,” J. Opt. Soc. Am. A 11, 783–805 (1994).
    [CrossRef]
  14. G. A. Tyler, D. L. Fried, “Image-position error associated with a quadrant detector,” J. Opt. Soc. Am. 72, 804–808 (1982).
    [CrossRef]
  15. B. M. Welsh, B. L. Ellerbroek, M. C. Roggemann, T. L. Pennington, “Shot noise performance of Hartmann and shearing interferometer wave front sensors,” in Adaptive Optical Systems and Applications, R. K. Tyson, R. Q. Fugate, eds., Proc. SPIE2534, 277–288 (1995).
    [CrossRef]
  16. B. L. Ellerbroek, Air Force Research Laboratory, 3550 Aberdeen Ave. SE, Kirtland Air Force Base, N. Mex. 87117 (personal communication, 1997).
  17. Collision times in the mesosphere are of the order of 100 µs.
  18. K. Avicola, J. M. Brase, J. R. Morris, H. D. Bissinger, J. M. Duff, H. W. Friedman, D. T. Gavel, C. E. Max, S. S. Olivier, R. W. Presta, D. A. Rapp, J. T. Salmon, K. E. Waltjen, “Sodium-layer guide-star experimental results,” J. Opt. Soc. Am. A 11, 825–831 (1994).
    [CrossRef]
  19. See C. S. Gardner, D. G. Voelz, C. F. Sechrist, A. C. Segal, “Lidar studies of the nighttime layer over Urbana, Illinois. I. Seasonal and nocturnal variations,” J. Geophys. Res. 91, 13,659–13,673 (1986).
    [CrossRef]
  20. The phase modulation ϕ(t) used in the density matrix equations was expressed by an analytical formula provided for us by H. W. Friedman, Lawrence Livermore National Laboratory, P. O. Box 808, Livermore, Calif. 94550 (personal communication, 1996).
  21. H. Friedman, G. Erbert, T. Kuklo, T. Salmon, D. Smauley, G. Thompson, J. Malik, N. Wong, K. Kanz, K. Neeb, “Sodium guide star results at the Lick Observatory,” preprint (Lawrence Livermore National Laboratory, Livermore, Calif., July1996).
  22. C. S. Gardner, “Sodium resonance fluorescence lidar applications in atmospheric science and astronomy,” Proc. IEEE 77, 408–418 (1989).
    [CrossRef]
  23. See, for instance, P. W. Milonni, J. H. Eberly, Lasers (Wiley, New York, 1988), Chap. 8.
  24. M. P. Jelonek, R. Q. Fugate, W. J. Lange, A. C. Slavin, R. E. Ruane, R. A. Cleis, “Characterization of artificial guide stars generated in the mesospheric sodium layer with a sum-frequency laser,” J. Opt. Soc. Am. A 11, 806–812 (1994).
    [CrossRef]
  25. V. A. Yakubovich, V. M. Starzhinskii, Linear Differential Equations with Periodic Coefficients (Keter, Jerusalem, 1975), pp. 88–89.
  26. T. H. Jeys, R. M. Heinrichs, K. F. Wall, J. Korn, T. C. Hotaling, E. Kibblewhite, “Observation of optical pumping of mesospheric sodium,” Opt. Lett. 17, 1143–1145 (1992).
    [CrossRef] [PubMed]
  27. R. States, Electro-Optic Systems Laboratory, University of Illinois, Urbana, Ill. 61801 (personal communication, 1996).
  28. P. W. Milonni, “Saturation of anomalous dispersion in cw HF lasers,” Appl. Opt. 20, 1571–1578 (1981), Eq. (42).
    [CrossRef] [PubMed]
  29. D. J. Link, Science Applications Internatioanl Corporation, 140 Intracoastal Pointe Drive, Suite 213, Jupiter, Fla. 33477 (personal communication, 1996).
  30. The typically small effects of the Earth’s magnetic field and of atomic recoil associated with absorption and emission are currently being investigated, and the results will be submitted by the authors for future publication.

1994

See, for instance, R. Benedict, J. B. Breckinridge, D. L. Fried, eds., “Feature on atmospheric-compensation technology,” J. Opt. Soc. Am. A 11, 255–451, 779–945 (1994), and references therein; L. A. Thompson, “Adaptive optics in astronomy,” Phys. Today 47(12), 24–31 (1994); R. Q. Fugate, “Laser beacon,” Opt. Photon. News 4(6), 14–19 (1993); R. Q. Fugate, W. Wild, “Untwinkling the stars,” Sky Telescope 87(5), 21–31 (1994); J. M. Beckers, “Adaptive optics for astronomy: principles, performance, and applications,” Annu. Rev. Astron. Astrophys. 31, 13–62 (1993); N. Hubin, L. Noethe, “Active optics, adaptive optics, and laser guide stars,” Science 262, 1390–1394 (1993); T. H. Jeys, “Development of a mesospheric sodium laser beacon for atmospheric adaptive optics,” MIT Lincoln Lab. J. 4, 133–149 (1991); C. S. Gardner, B. M. Welsh, L. A. Thompson, “Design and performance analysis of adaptive optical telescopes using laser guide stars,” Proc. IEEE 78, 1724–1743 (1990);H. W. Babcock, “Adaptive optics revisited,” Science 249, 253–257 (1990).
[CrossRef] [PubMed]

J. R. Morris, “Efficient excitation of a mesospheric sodium laser guide star by intermediate-duration pulses,” J. Opt. Soc. Am. A 11, 832–845 (1994).
[CrossRef]

For discussions of these error sources, see, for instance, R. R. Parenti, R. J. Sasiela, “Laser-guide-star systems for astronomical applications,” J. Opt. Soc. Am. A 11, 288–309 (1994); D. G. Sandler, S. Stahl, J. R. P. Angel, M. Lloyd-Hart, D. McCarthy, “Adaptive optics for diffraction-limited infrared imaging with 8-m telescopes,” J. Opt. Soc. Am. A 11, 925–945 (1994).
[CrossRef]

D. L. Fried, J. F. Belsher, “Analysis of fundamental limits to artificial-guide-star adaptive-optics-system performance for astronomical imaging,” J. Opt. Soc. Am. A 11, 277–287 (1994).
[CrossRef]

B. L. Ellerbroek, “First-order performance evaluation of adaptive-optics systems for atmospheric turbulence compensation in extended-field-of-view astronomical telescopes,” J. Opt. Soc. Am. A 11, 783–805 (1994).
[CrossRef]

K. Avicola, J. M. Brase, J. R. Morris, H. D. Bissinger, J. M. Duff, H. W. Friedman, D. T. Gavel, C. E. Max, S. S. Olivier, R. W. Presta, D. A. Rapp, J. T. Salmon, K. E. Waltjen, “Sodium-layer guide-star experimental results,” J. Opt. Soc. Am. A 11, 825–831 (1994).
[CrossRef]

M. P. Jelonek, R. Q. Fugate, W. J. Lange, A. C. Slavin, R. E. Ruane, R. A. Cleis, “Characterization of artificial guide stars generated in the mesospheric sodium layer with a sum-frequency laser,” J. Opt. Soc. Am. A 11, 806–812 (1994).
[CrossRef]

1992

1989

C. S. Gardner, “Sodium resonance fluorescence lidar applications in atmospheric science and astronomy,” Proc. IEEE 77, 408–418 (1989).
[CrossRef]

1986

See C. S. Gardner, D. G. Voelz, C. F. Sechrist, A. C. Segal, “Lidar studies of the nighttime layer over Urbana, Illinois. I. Seasonal and nocturnal variations,” J. Geophys. Res. 91, 13,659–13,673 (1986).
[CrossRef]

1982

1981

1980

See, for instance, P. L. Knight, P. W. Milonni, “The Rabi frequency in optical spectra,” Phys. Rep. 66, 21–107 (1980).
[CrossRef]

1966

Avicola, K.

Belsher, J. F.

Bissinger, H. D.

Bradley, L. C.

Brase, J. M.

Cleis, R. A.

Cowan, R. D.

R. D. Cowan, The Theory of Atomic Structure and Spectra (U. California Press, Berkeley, Calif., 1981).

Duff, J. M.

Eberly, J. H.

See, for instance, P. W. Milonni, J. H. Eberly, Lasers (Wiley, New York, 1988), Chap. 8.

Ellerbroek, B. L.

B. L. Ellerbroek, “First-order performance evaluation of adaptive-optics systems for atmospheric turbulence compensation in extended-field-of-view astronomical telescopes,” J. Opt. Soc. Am. A 11, 783–805 (1994).
[CrossRef]

B. M. Welsh, B. L. Ellerbroek, M. C. Roggemann, T. L. Pennington, “Shot noise performance of Hartmann and shearing interferometer wave front sensors,” in Adaptive Optical Systems and Applications, R. K. Tyson, R. Q. Fugate, eds., Proc. SPIE2534, 277–288 (1995).
[CrossRef]

B. L. Ellerbroek, Air Force Research Laboratory, 3550 Aberdeen Ave. SE, Kirtland Air Force Base, N. Mex. 87117 (personal communication, 1997).

Erbert, G.

H. Friedman, G. Erbert, T. Kuklo, T. Salmon, D. Smauley, G. Thompson, J. Malik, N. Wong, K. Kanz, K. Neeb, “Sodium guide star results at the Lick Observatory,” preprint (Lawrence Livermore National Laboratory, Livermore, Calif., July1996).

Fried, D. L.

Friedman, H.

H. Friedman, G. Erbert, T. Kuklo, T. Salmon, D. Smauley, G. Thompson, J. Malik, N. Wong, K. Kanz, K. Neeb, “Sodium guide star results at the Lick Observatory,” preprint (Lawrence Livermore National Laboratory, Livermore, Calif., July1996).

Friedman, H. W.

K. Avicola, J. M. Brase, J. R. Morris, H. D. Bissinger, J. M. Duff, H. W. Friedman, D. T. Gavel, C. E. Max, S. S. Olivier, R. W. Presta, D. A. Rapp, J. T. Salmon, K. E. Waltjen, “Sodium-layer guide-star experimental results,” J. Opt. Soc. Am. A 11, 825–831 (1994).
[CrossRef]

The phase modulation ϕ(t) used in the density matrix equations was expressed by an analytical formula provided for us by H. W. Friedman, Lawrence Livermore National Laboratory, P. O. Box 808, Livermore, Calif. 94550 (personal communication, 1996).

Fugate, R. Q.

M. P. Jelonek, R. Q. Fugate, W. J. Lange, A. C. Slavin, R. E. Ruane, R. A. Cleis, “Characterization of artificial guide stars generated in the mesospheric sodium layer with a sum-frequency laser,” J. Opt. Soc. Am. A 11, 806–812 (1994).
[CrossRef]

C. D. Morgenstern, R. Q. Fugate, A. C. Slavin, “Summary of optical turbulence measurements at the Starfire Optical Range,” in Atmospheric Propagation and Remote Sensing IV, J. C. Dainty, ed., Proc. SPIE2471, 390–398 (1995).
[CrossRef]

Gardner, C. S.

C. S. Gardner, “Sodium resonance fluorescence lidar applications in atmospheric science and astronomy,” Proc. IEEE 77, 408–418 (1989).
[CrossRef]

See C. S. Gardner, D. G. Voelz, C. F. Sechrist, A. C. Segal, “Lidar studies of the nighttime layer over Urbana, Illinois. I. Seasonal and nocturnal variations,” J. Geophys. Res. 91, 13,659–13,673 (1986).
[CrossRef]

Gavel, D. T.

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley, New York, 1985), Chap. 8.

Heinrichs, R. M.

Hotaling, T. C.

Jelonek, M. P.

Jeys, T. H.

Kanz, K.

H. Friedman, G. Erbert, T. Kuklo, T. Salmon, D. Smauley, G. Thompson, J. Malik, N. Wong, K. Kanz, K. Neeb, “Sodium guide star results at the Lick Observatory,” preprint (Lawrence Livermore National Laboratory, Livermore, Calif., July1996).

Kibblewhite, E.

Knight, P. L.

See, for instance, P. L. Knight, P. W. Milonni, “The Rabi frequency in optical spectra,” Phys. Rep. 66, 21–107 (1980).
[CrossRef]

Korn, J.

Kuklo, T.

H. Friedman, G. Erbert, T. Kuklo, T. Salmon, D. Smauley, G. Thompson, J. Malik, N. Wong, K. Kanz, K. Neeb, “Sodium guide star results at the Lick Observatory,” preprint (Lawrence Livermore National Laboratory, Livermore, Calif., July1996).

Lange, W. J.

Link, D. J.

D. J. Link, Science Applications Internatioanl Corporation, 140 Intracoastal Pointe Drive, Suite 213, Jupiter, Fla. 33477 (personal communication, 1996).

Malik, J.

H. Friedman, G. Erbert, T. Kuklo, T. Salmon, D. Smauley, G. Thompson, J. Malik, N. Wong, K. Kanz, K. Neeb, “Sodium guide star results at the Lick Observatory,” preprint (Lawrence Livermore National Laboratory, Livermore, Calif., July1996).

Max, C. E.

Milonni, P. W.

P. W. Milonni, L. E. Thode, “Theory of mesospheric sodium fluorescence excited by pulse trains,” Appl. Opt. 31, 785–800 (1992).
[CrossRef] [PubMed]

P. W. Milonni, “Saturation of anomalous dispersion in cw HF lasers,” Appl. Opt. 20, 1571–1578 (1981), Eq. (42).
[CrossRef] [PubMed]

See, for instance, P. L. Knight, P. W. Milonni, “The Rabi frequency in optical spectra,” Phys. Rep. 66, 21–107 (1980).
[CrossRef]

See, for instance, P. W. Milonni, J. H. Eberly, Lasers (Wiley, New York, 1988), Chap. 8.

Morgenstern, C. D.

C. D. Morgenstern, R. Q. Fugate, A. C. Slavin, “Summary of optical turbulence measurements at the Starfire Optical Range,” in Atmospheric Propagation and Remote Sensing IV, J. C. Dainty, ed., Proc. SPIE2471, 390–398 (1995).
[CrossRef]

Morris, J. R.

Neeb, K.

H. Friedman, G. Erbert, T. Kuklo, T. Salmon, D. Smauley, G. Thompson, J. Malik, N. Wong, K. Kanz, K. Neeb, “Sodium guide star results at the Lick Observatory,” preprint (Lawrence Livermore National Laboratory, Livermore, Calif., July1996).

Olivier, S. S.

Parenti, R. R.

Pennington, T. L.

B. M. Welsh, B. L. Ellerbroek, M. C. Roggemann, T. L. Pennington, “Shot noise performance of Hartmann and shearing interferometer wave front sensors,” in Adaptive Optical Systems and Applications, R. K. Tyson, R. Q. Fugate, eds., Proc. SPIE2534, 277–288 (1995).
[CrossRef]

Presta, R. W.

Rapp, D. A.

Roddier, F.

F. Roddier, “Report on the seeing on Mauna Kea” (U. Hawaii, Honolulu, Hawaii, 1992).

Roggemann, M. C.

B. M. Welsh, B. L. Ellerbroek, M. C. Roggemann, T. L. Pennington, “Shot noise performance of Hartmann and shearing interferometer wave front sensors,” in Adaptive Optical Systems and Applications, R. K. Tyson, R. Q. Fugate, eds., Proc. SPIE2534, 277–288 (1995).
[CrossRef]

Ruane, R. E.

Salmon, J. T.

Salmon, T.

H. Friedman, G. Erbert, T. Kuklo, T. Salmon, D. Smauley, G. Thompson, J. Malik, N. Wong, K. Kanz, K. Neeb, “Sodium guide star results at the Lick Observatory,” preprint (Lawrence Livermore National Laboratory, Livermore, Calif., July1996).

Sasiela, R. J.

Sechrist, C. F.

See C. S. Gardner, D. G. Voelz, C. F. Sechrist, A. C. Segal, “Lidar studies of the nighttime layer over Urbana, Illinois. I. Seasonal and nocturnal variations,” J. Geophys. Res. 91, 13,659–13,673 (1986).
[CrossRef]

Segal, A. C.

See C. S. Gardner, D. G. Voelz, C. F. Sechrist, A. C. Segal, “Lidar studies of the nighttime layer over Urbana, Illinois. I. Seasonal and nocturnal variations,” J. Geophys. Res. 91, 13,659–13,673 (1986).
[CrossRef]

Slavin, A. C.

M. P. Jelonek, R. Q. Fugate, W. J. Lange, A. C. Slavin, R. E. Ruane, R. A. Cleis, “Characterization of artificial guide stars generated in the mesospheric sodium layer with a sum-frequency laser,” J. Opt. Soc. Am. A 11, 806–812 (1994).
[CrossRef]

C. D. Morgenstern, R. Q. Fugate, A. C. Slavin, “Summary of optical turbulence measurements at the Starfire Optical Range,” in Atmospheric Propagation and Remote Sensing IV, J. C. Dainty, ed., Proc. SPIE2471, 390–398 (1995).
[CrossRef]

Smauley, D.

H. Friedman, G. Erbert, T. Kuklo, T. Salmon, D. Smauley, G. Thompson, J. Malik, N. Wong, K. Kanz, K. Neeb, “Sodium guide star results at the Lick Observatory,” preprint (Lawrence Livermore National Laboratory, Livermore, Calif., July1996).

Starzhinskii, V. M.

V. A. Yakubovich, V. M. Starzhinskii, Linear Differential Equations with Periodic Coefficients (Keter, Jerusalem, 1975), pp. 88–89.

States, R.

R. States, Electro-Optic Systems Laboratory, University of Illinois, Urbana, Ill. 61801 (personal communication, 1996).

Thode, L. E.

Thompson, G.

H. Friedman, G. Erbert, T. Kuklo, T. Salmon, D. Smauley, G. Thompson, J. Malik, N. Wong, K. Kanz, K. Neeb, “Sodium guide star results at the Lick Observatory,” preprint (Lawrence Livermore National Laboratory, Livermore, Calif., July1996).

Tyler, G. A.

Voelz, D. G.

See C. S. Gardner, D. G. Voelz, C. F. Sechrist, A. C. Segal, “Lidar studies of the nighttime layer over Urbana, Illinois. I. Seasonal and nocturnal variations,” J. Geophys. Res. 91, 13,659–13,673 (1986).
[CrossRef]

Wall, K. F.

Waltjen, K. E.

Welsh, B. M.

B. M. Welsh, B. L. Ellerbroek, M. C. Roggemann, T. L. Pennington, “Shot noise performance of Hartmann and shearing interferometer wave front sensors,” in Adaptive Optical Systems and Applications, R. K. Tyson, R. Q. Fugate, eds., Proc. SPIE2534, 277–288 (1995).
[CrossRef]

Wong, N.

H. Friedman, G. Erbert, T. Kuklo, T. Salmon, D. Smauley, G. Thompson, J. Malik, N. Wong, K. Kanz, K. Neeb, “Sodium guide star results at the Lick Observatory,” preprint (Lawrence Livermore National Laboratory, Livermore, Calif., July1996).

Yakubovich, V. A.

V. A. Yakubovich, V. M. Starzhinskii, Linear Differential Equations with Periodic Coefficients (Keter, Jerusalem, 1975), pp. 88–89.

Appl. Opt.

J. Geophys. Res.

See C. S. Gardner, D. G. Voelz, C. F. Sechrist, A. C. Segal, “Lidar studies of the nighttime layer over Urbana, Illinois. I. Seasonal and nocturnal variations,” J. Geophys. Res. 91, 13,659–13,673 (1986).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

D. L. Fried, J. F. Belsher, “Analysis of fundamental limits to artificial-guide-star adaptive-optics-system performance for astronomical imaging,” J. Opt. Soc. Am. A 11, 277–287 (1994).
[CrossRef]

B. L. Ellerbroek, “First-order performance evaluation of adaptive-optics systems for atmospheric turbulence compensation in extended-field-of-view astronomical telescopes,” J. Opt. Soc. Am. A 11, 783–805 (1994).
[CrossRef]

See, for instance, R. Benedict, J. B. Breckinridge, D. L. Fried, eds., “Feature on atmospheric-compensation technology,” J. Opt. Soc. Am. A 11, 255–451, 779–945 (1994), and references therein; L. A. Thompson, “Adaptive optics in astronomy,” Phys. Today 47(12), 24–31 (1994); R. Q. Fugate, “Laser beacon,” Opt. Photon. News 4(6), 14–19 (1993); R. Q. Fugate, W. Wild, “Untwinkling the stars,” Sky Telescope 87(5), 21–31 (1994); J. M. Beckers, “Adaptive optics for astronomy: principles, performance, and applications,” Annu. Rev. Astron. Astrophys. 31, 13–62 (1993); N. Hubin, L. Noethe, “Active optics, adaptive optics, and laser guide stars,” Science 262, 1390–1394 (1993); T. H. Jeys, “Development of a mesospheric sodium laser beacon for atmospheric adaptive optics,” MIT Lincoln Lab. J. 4, 133–149 (1991); C. S. Gardner, B. M. Welsh, L. A. Thompson, “Design and performance analysis of adaptive optical telescopes using laser guide stars,” Proc. IEEE 78, 1724–1743 (1990);H. W. Babcock, “Adaptive optics revisited,” Science 249, 253–257 (1990).
[CrossRef] [PubMed]

K. Avicola, J. M. Brase, J. R. Morris, H. D. Bissinger, J. M. Duff, H. W. Friedman, D. T. Gavel, C. E. Max, S. S. Olivier, R. W. Presta, D. A. Rapp, J. T. Salmon, K. E. Waltjen, “Sodium-layer guide-star experimental results,” J. Opt. Soc. Am. A 11, 825–831 (1994).
[CrossRef]

For discussions of these error sources, see, for instance, R. R. Parenti, R. J. Sasiela, “Laser-guide-star systems for astronomical applications,” J. Opt. Soc. Am. A 11, 288–309 (1994); D. G. Sandler, S. Stahl, J. R. P. Angel, M. Lloyd-Hart, D. McCarthy, “Adaptive optics for diffraction-limited infrared imaging with 8-m telescopes,” J. Opt. Soc. Am. A 11, 925–945 (1994).
[CrossRef]

J. R. Morris, “Efficient excitation of a mesospheric sodium laser guide star by intermediate-duration pulses,” J. Opt. Soc. Am. A 11, 832–845 (1994).
[CrossRef]

M. P. Jelonek, R. Q. Fugate, W. J. Lange, A. C. Slavin, R. E. Ruane, R. A. Cleis, “Characterization of artificial guide stars generated in the mesospheric sodium layer with a sum-frequency laser,” J. Opt. Soc. Am. A 11, 806–812 (1994).
[CrossRef]

J. Opt. Soc. Am. B

Opt. Lett.

Phys. Rep.

See, for instance, P. L. Knight, P. W. Milonni, “The Rabi frequency in optical spectra,” Phys. Rep. 66, 21–107 (1980).
[CrossRef]

Proc. IEEE

C. S. Gardner, “Sodium resonance fluorescence lidar applications in atmospheric science and astronomy,” Proc. IEEE 77, 408–418 (1989).
[CrossRef]

Other

See, for instance, P. W. Milonni, J. H. Eberly, Lasers (Wiley, New York, 1988), Chap. 8.

The phase modulation ϕ(t) used in the density matrix equations was expressed by an analytical formula provided for us by H. W. Friedman, Lawrence Livermore National Laboratory, P. O. Box 808, Livermore, Calif. 94550 (personal communication, 1996).

H. Friedman, G. Erbert, T. Kuklo, T. Salmon, D. Smauley, G. Thompson, J. Malik, N. Wong, K. Kanz, K. Neeb, “Sodium guide star results at the Lick Observatory,” preprint (Lawrence Livermore National Laboratory, Livermore, Calif., July1996).

R. States, Electro-Optic Systems Laboratory, University of Illinois, Urbana, Ill. 61801 (personal communication, 1996).

V. A. Yakubovich, V. M. Starzhinskii, Linear Differential Equations with Periodic Coefficients (Keter, Jerusalem, 1975), pp. 88–89.

D. J. Link, Science Applications Internatioanl Corporation, 140 Intracoastal Pointe Drive, Suite 213, Jupiter, Fla. 33477 (personal communication, 1996).

The typically small effects of the Earth’s magnetic field and of atomic recoil associated with absorption and emission are currently being investigated, and the results will be submitted by the authors for future publication.

R. D. Cowan, The Theory of Atomic Structure and Spectra (U. California Press, Berkeley, Calif., 1981).

C. D. Morgenstern, R. Q. Fugate, A. C. Slavin, “Summary of optical turbulence measurements at the Starfire Optical Range,” in Atmospheric Propagation and Remote Sensing IV, J. C. Dainty, ed., Proc. SPIE2471, 390–398 (1995).
[CrossRef]

F. Roddier, “Report on the seeing on Mauna Kea” (U. Hawaii, Honolulu, Hawaii, 1992).

J. W. Goodman, Statistical Optics (Wiley, New York, 1985), Chap. 8.

B. M. Welsh, B. L. Ellerbroek, M. C. Roggemann, T. L. Pennington, “Shot noise performance of Hartmann and shearing interferometer wave front sensors,” in Adaptive Optical Systems and Applications, R. K. Tyson, R. Q. Fugate, eds., Proc. SPIE2534, 277–288 (1995).
[CrossRef]

B. L. Ellerbroek, Air Force Research Laboratory, 3550 Aberdeen Ave. SE, Kirtland Air Force Base, N. Mex. 87117 (personal communication, 1997).

Collision times in the mesosphere are of the order of 100 µs.

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Figures (14)

Fig. 1
Fig. 1

Twenty-four magnetic substates of the different hyperfine levels of the sodium D2 line. The dashed line indicates the two-state atom produced after sufficiently long irradiation by circularly polarized light inducing Δm=+1 absorptive transitions.

Fig. 2
Fig. 2

Computed absorption cross section for the Doppler-broadened sodium D2 line at T=200 K. The frequencies are shown relative to the 3S1/2(F=2)3P3/2(F=3) resonance. The computation was performed for low-power unpolarized radiation. Therefore no optical pumping or saturation occurred.

Fig. 3
Fig. 3

Average short-exposure intensity profiles for (a) the near-field and (b) the far-field limits obtained by use of the Fried model with circular aperture diameter DA=13 cm, λ=589 nm; z=90 km and r0=15  cm (solid curves), 10 cm (dashed curves), and 5 cm (dotted curves).

Fig. 4
Fig. 4

Mean square wave-front phase error owing to measurement noise versus number of detected electrons per subaperture of a Shack–Hartmann sensor for the satellite imaging (solid curve) and IR astronomy (dashed curve) applications, based on the adaptive-optics system parameters of Tables 1 and 2.

Fig. 5
Fig. 5

Return signal versus zenith angle predicted by Eq. (34) for two beacon diameters.

Fig. 6
Fig. 6

Predicted photon returns versus average laser power based on Eq. (45) (solid curve), together with experimental data points (squares) taken from Fig. 9 of Avicola et al.18

Fig. 7
Fig. 7

Results of density matrix computations of NB for parameters in the experiments of Avicola et al.18 (circles), together with an analytical fit to these results (solid curve).

Fig. 8
Fig. 8

Computed photon returns versus average laser power in experiments of Avicola et al.18 (solid curve), together with experimental data points. The computations are based on the full density matrix code including the effect of atmospheric turbulence on the propagation of the laser beam to the mesosphere.

Fig. 9
Fig. 9

Computed x (solid curve) and y (dashed curve) intensity profiles after propagation to the mesosphere, showing effects of diffraction and turbulence under the assumption of a 4 cm×8 cm uniphase flat-topped beam at launch. The computation is based on the Fried model with a single phase sheet as described in the text. These intensity profiles were used to compute the photon returns shown in Fig. 8.

Fig. 10
Fig. 10

Sum of the occupation probabilities of the 3S1/2(M =2) and 3P3/2(M=3) states just before each micropulse of a macropulse consisting of 4800 micropulses.

Fig. 11
Fig. 11

Sum of the occupation probabilities of the three 3S1/2(F=1) ground states just before each micropulse of a macropulse consisting of 4800 micropulses.

Fig. 12
Fig. 12

Computed fluorescence signal (relative units) from an idealized CCD camera as a function of radial distance from the center of the fluorescence image in the mesosphere.

Fig. 13
Fig. 13

Experimental data of States27 for percent saturation at the peak offset frequency (squares) and at the crossover frequency (circles). The solid and dashed curves are obtained from relation (76) with Isat=0.40 and Isat=0.48 W/cm2, respectively.

Fig. 14
Fig. 14

As in Fig. 13, except that the solid and dashed theoretical curves are here based on numerical solutions of the density matrix equations.

Tables (2)

Tables Icon

Table 1 Atmospheric and System Parameters and Wave-Front-Error Contributions for Satellite Imaging at the Starfire Optical Range and Infrared Astronomy at Mauna Kea

Tables Icon

Table 2 Representative System Parameters for Satellite Imaging at the Starfire Optical Range and Infrared Astronomy at Mauna Kea

Equations (80)

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E(t)=12E0(t)exp{-i[ωt+ϕ(t)]}+12E0*(t)exp{i[ωt+ϕ(t)]},
E(t)=A0(t)xˆ cos(ωt+ϕ).
E(t)=12A0(t)[xˆ cos(ωt+ϕ)±yˆ sin(ωt+ϕ)].
σij(t)=Sij(t)exp[-i(ωt+ϕ)](i<j),
S˙ij=-i(ωji-ω-ϕ˙-iβij)Sij+i2dji·E0(σii-σjj)(i<j).
σ˙ii=-Aiσii+j>iAjiσjj+i2k<idik·E0Ski*+i2k>idik·E0*Sik+c.c.
δνD=δωD2π2λ2RT ln 2M1/21GHz.
P˙=-AP+σ(ν)Ihν
σ(ν)=hνAIP¯.
σ(ν)=λ2A8πg2g158S1(ν)+38S2(ν).
Sj(ν)=1δνD4 ln 2π1/2 exp[-4(ν-ν0j)2 ln 2/δνD2]
(j=1, 2).
N=ρLdxdyA0Tdtjβjp¯j(x, y, t),
R=RpT0Aa4πz2N=T0Aa4πz2RpρLdxdyA0Tdtjβjp¯j(x, y, t)=T0Csz2RpAadxdyNB(x, y),
NB(x, y)=A4π0Tdtjβjp¯j(x, y, t).
E(r, z)=d2rh(r, r)E(r, 0)=d2rh(r, r)A(r)exp[iϕ(r)],
h(r, r)=-iλzexp(2πiz/λ)exp(iπr2/λz)×exp(-2πir·r/λz).
τ(K)=d2r|E(r, z)|2exp(2πiK·r),
|E(r, z)|2=d2Kτ(K)exp(-2πiK·r).
exp{i[ϕ(r)-ϕ(r)]}=exp{-12[ϕ(r)-ϕ(r)]2}=exp[-12Ds(|r-r|)],
τ(K)=d2rd2rA*(r)A(r)exp[-12Ds(|r-r|)]×d2rh*(r, r)h(r, r)exp(2πiK·r)=d2rA*(r-λzK)A(r)exp[-12Ds(λzK)]τ0(K)exp[-12Ds(λzK)].
ϕ(r)=2πλ0zdzn(r, z),
[n(r, 0)-n(0, z)]2=Cn2R2/3.
Ds(r)=2.91Cn22πλ2r5/3z;
Ds(r)=115r5/3λ20zdξCn2(ξ).
r0=0.185λ20zdξCn2(ξ)3/5
τ(K)=τ0(K)exp[-3.44(λzK/r0)5/3].
Ds(r)=6.88(r/r0)5/3[1-(r/DA)1/3](DAλz)
Ds(r)=6.88(r/r0)5/3[1-12(r/DA)1/3](DAλz)
τ0(K)=|A0|2 2πcos-1λzKDA-λzKDA1-λzKDA21/2
SR=exp[-(En2+Ef2+Es2+EFA2)],
En2=α0.09 ln(nsa)σθ2ds22πλimg2
σθ=πλbds3162+θb/24λb/ds21/21ns+4ne2ns21/2,
R/Aa=ns/τd(ms)/ds2(cm2)/T/η=4.3photons/cm2/ms(satelliteimaging)
=0.037photons/cm2/ms(IRastronomy).
Rmax=0.25A(10-3)Cssec ψ(πz2dϕ2/4)(T0)sec ψ4π(z sec ψ)2=2.3×10-8Csdϕ2T0sec ψ cos ψ,
jp¯j=(1/2)Ip/Isat1+Ip/Isat,
Rst=0dνσ(ν)hνI(ν)=σ¯hν0dνI(ν)σ¯hνIp,
Isat=hνA2σ¯.
NBA4π(1/2)Ip/Isat1+Ip/Isat0τpdt+0dt exp(-At)=Aτp8π(1+1/Aτp)Ip/Isat1+Ip/IsatKIp/Isat1+Ip/Isat.
R/Aa=T0Csz2RpdxdyNB(x, y)=KT0Csz2RpdxdyIp(x, y)/Isat1+Ip(x, y)/Isat.
dxdyNB(x, y)=2πK(I0/Isat)0drrexp(-2r2/w2)1+(I0/Isat)exp(-2r2/w2)=Kπw22ln(1+I0/Isat),
R/Aa=T0Csz2πw22RpK ln(1+I0/Isat).
P=RpI0dxdy exp(-2r2/w2)×-dt exp[-(4 ln 2)t2/τp2]=π4 ln 23/2RpI0a2τp,
I0=T04 ln 2π3/2 PRpa2τp,
R/Aa=π4 ln 2 18π(Aτp+1)×T0Csz2Rpa2 ln1+4 ln 2π3/2×T0PRpIsata2τp,
R/Aa=1.0×10-9Cs ln[1+3.0×10-3P]
(photons/cm2/ms),
R/AaT0Csz2RpKIsatdxdyIp(x, y)=4 ln 2π1/2 T02PKCsz2τpIsat,
R/Aa=4.2×10-12CsP(photons/cm2/ms),
R/Aa=4.0×10-12CsP(photons/cm2/ms)
n=T02 Aa4πz2λJhcαL,
θ=d-dtE0(t)=d8πcIp1/2-dt exp[-(2 ln 2)t2/τp2]=λ3Aτp2Ip8πc ln 21/2.
p2(x, y)=sin2 λ3Aτp232πc ln 2Ip(x, y)1/2.
n=T0Csz2AadxdyNB(x, y).
NB(x, y)14π(τMRp)β2p2(x, y),
NB(x, y)1.54π(τMRp)sin2λ3Aτp232πc ln 2Ip(x, y)1/21.54π(τMRp)λ3Aτp232πc ln 2Ip(x, y)
J=τMRpτpπ4 ln 21/2dxdyIp(x, y)
n/J/AaT02 1.54πz2Csλ3Aτp16πc1π ln 21/2
n/Aa/J80T02Csτp(photons/cm2/J).
X˙=A(t)X,
X(t)=C(t)X(0),
x˙=A(t)x,
x(t)=C(t)x(0).
Xn=Cn(T)X(0),
P=RM(RpτM)π4 ln 23/2I0τpa2,
NB=3.50Ip+10.0Ip2
NB=3.55Ip-15.0Ip2
Xsat(θ)=1-R(θ)Runsat
w2(z)=4λ2π2θ2+θ2z24θ2z24,
Ip(0, z)T0Ip(0, 0)w02w2(z)T0Ip(0, 0)z2z02=8T0πz2θ2πw022Ip(0, 0),
E=2π0drr-dtIp(0, 0)×exp(-2r2/w02)exp[-(4 ln 2)t2/τp2]=πw022Ip(0, 0)τpπ4 ln 21/2.
Ip(0, z)=8T0πz21θ2Eτp4 ln 2π1/2
R(θ)=T0CsKRp4πz22π0drr×Ip(0, z)/Isatexp[2r2/w2(z)]+Ip(0, z)/Isat=Aθ2 ln1+Bθ2,
AT0CsKRp32×10-6,
B8T0Eπz2τp4 ln 2π1/2 1Isat×106,
Xsat(θ)=1-θ2 ln(1+B/θ2)(1.7)2 ln[1+B/(1.7)2],
Xsat(θ)=1-θ2 ln(1+0.052/Isatθ2)(1.7)2 ln[1+0.052/Isat(1.7)2].
Xsat(θ)0.026Isatθ2
I˜s=1(π ln 2)1/2δνDδν0Is,

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