Abstract

We extend our previous analysis of the sodium beacon [J. Opt. Soc. Am. A 15, 217 (1998)] to the case of continuous-wave excitation. Various effects that could be ignored in the case of pulsed excitation, such as the geomagnetic field, the recoil of the sodium atoms upon absorption and emission, and collisions of the sodium atoms with other mesospheric species, are included. Spin-relaxation collisions are among the most important of these effects for the cases considered. Analytical approximations to numerical results are presented, and using a semi-empirical estimate for Na-O2 spin relaxation, we compute photon returns in good agreement with recently reported measurements at the Steward Observatory.

© 1999 Optical Society of America

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  1. See, for instance, the following reviews and references therein: M. C. Roggemann, B. M. Welsh, R. Q. Fugate, “Improving the resolution of ground-based telescopes,” Rev. Mod. Phys. 69, 437–505 (1997);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).
    [CrossRef] [PubMed]
  2. M. Lloyd-Hart, J. R. P. Angel, B. Jacobsen, D. Wittman, R. Dekany, D. McCarthy, E. Kibblewhite, W. Wild, B. Carter, J. Beletic, “Adaptive optics experiments using sodium laser guide stars,” Astrophys. J. 439, 455–473 (1995);C. E. Max, S. S. Olivier, H. W. Friedman, J. An, K. Avicola, B. V. Beeman, H. D. Bissinger, J. M. Brase, G. V. Erbert, D. T. Gavel, K. Kanz, M. C. Liu, B. Macintosh, K. P. Neeb, J. Patience, K. E. Waltjen, “Image improvement from a sodium-layer guide star adaptive optics system,” Science 277, 1649–1652 (1997);M. Lloyd-Hart, J. R. P. Angel, T. D. Groesbeck, T. Martinez, B. P. Jacobsen, B. A. McLeod, D. W. McCarthy, E. J. Hooper, E. K. Hege, D. G. Sandler, “First astronomical images sharpened with adaptive optics using a sodium laser guide star,” Astrophys. J. 493, 950–954 (1998).
    [CrossRef]
  3. P. W. Milonni, R. Q. Fugate, J. M. Telle, “Analysis of measured photon returns from sodium beacons,” J. Opt. Soc. Am. A 15, 217–233 (1998).
    [CrossRef]
  4. J. Ge, B. P. Jacobsen, J. R. P. Angel, P. C. McGuire, T. Roberts, B. A. McLeod, M. Lloyd-Hart, “Simultaneous measurements of sodium column density and laser guide star brightness,” in SPIE Proceedings on Adaptive Optical System Technologies, D. Bonaccini, R. K. Tyson, eds., Proc. SPIE3353, 242–254 (1998).
    [CrossRef]
  5. W. Hanle, “Über magnetische Beeinflussung der Polarisation der Resonanzfluoreszenz,” Z. Phys. 30, 93–105 (1924);A. Kastler, “Quelques suggestions concernant la production optique et la détection optique d’une inégalité de population des niveaux de quantification spatiale des atomes. Application à l’expérience de Stern et Gerlach et à la résonance magnétique,” J. Phys. Radium 11, 255–265 (1950);A. Kastler, “La détection optique de la résonance électronique paramagnétique par la mesure de la polarisation rotatoire paramagnétique d’une radiation visible,” Compt. Rendu 232, 953–957 (1951).
    [CrossRef]
  6. W. Happer, “Optical Pumping,” Rev. Mod. Phys. 44, 169–249 (1972).
    [CrossRef]
  7. J. J. McClelland, M. H. Kelley, “Detailed look at aspects of optical pumping in sodium,” Phys. Rev. A 31, 3704–3710 (1985).
    [CrossRef] [PubMed]
  8. L. C. Bradley, “Pulse-train excitation of sodium for use as a synthetic beacon,” J. Opt. Soc. Am. B 9, 1931–1944 (1992).
    [CrossRef]
  9. 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]
  10. B. Jacobsen, T. Martinez, J. R. P. Angel, M. Lloyd-Hart, S. Benda, D. Middleton, H. W. Friedman, “Field evaluation of two new continuous-wave dye laser systems optimized for sodium beacon excitation,” in SPIE Proceedings on Adaptive Optics in Astronomy, M. A. Ealey, F. Merkle, eds., Proc. SPIE2201, 342–351 (1994).
    [CrossRef]
  11. An unpublished report detailing the calculation of the dipole matrix elements and radiative transition rates used in this work is available upon request.
  12. E. Arimondo, M. Inguscio, P. Violino, “Experimental determination of the hyperfine structure in alkali atoms,” Rev. Mod. Phys. 49, 31–75 (1977).
    [CrossRef]
  13. See, for instance, K. Gottfried, Quantum Mechanics (Benjamin, Reading, Mass., 1977), p. 269.
  14. A. R. Edmonds, Angular Momentum in Quantum Mechanics (Princeton U. Press, Princeton, N.J., 1996).
  15. S. Berman, M. Jacob, “Systematics of angular and polarization distributions in three-body decays,” Phys. Rev. 139, B1023 (1965).
    [CrossRef]
  16. See E. J. Chernosky, P. F. Fougere, R. O. Hutchinson, “The geomagnetic field,” in Handbook of Geophysics and Space Environments, S. L. Valley, ed. (McGraw-Hill, New York, 1965), Chap. 11.
  17. See, for instance, J. A. Abate, “Preparation of atomic sodium as a two-level atom,” Opt. Commun. 10, 269–272 (1974).
    [CrossRef]
  18. Equation (9) follows from the expression R=σI/hν, where σ=3λ2AS(ν)/8π is the stimulated emission/absorption cross section and S(ν)=(4 ln 2/π)1/2/δνD is the Doppler lineshape function at line center.
  19. A. T. Ramsey, L. W. Anderson, “Spin relaxation in an optically pumped sodium vapor,” Nuovo Cimento 32, 1151–1157 (1964).
    [CrossRef]
  20. N. W. Ressler, R. H. Sands, T. E. Stark, “Measurement of the spin-exchange cross sections for Cs133,Rb87,Rb85,K39, and Na23,” Phys. Rev. 184, 102–118 (1969);A. Moretti, F. Strumia, “Hyperfine optical pumping of sodium vapor,” Phys. Rev. A 3, 349–354 (1971).
    [CrossRef]
  21. The suggestion to use the Na-Na spin-relaxation cross section as a rough estimate for the unknown Na-O2 cross section is due to W. Happer, Department of Physics, Princeton University, Princeton, New Jersey 08544 (personal communication, June1997).
  22. J. Ge, Lawrence Livermore National Laboratory, 7000 East Ave., Livermore, Calif. 94550 (personal communication, April1998).

1998 (1)

1997 (1)

See, for instance, the following reviews and references therein: M. C. Roggemann, B. M. Welsh, R. Q. Fugate, “Improving the resolution of ground-based telescopes,” Rev. Mod. Phys. 69, 437–505 (1997);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).
[CrossRef] [PubMed]

1995 (1)

M. Lloyd-Hart, J. R. P. Angel, B. Jacobsen, D. Wittman, R. Dekany, D. McCarthy, E. Kibblewhite, W. Wild, B. Carter, J. Beletic, “Adaptive optics experiments using sodium laser guide stars,” Astrophys. J. 439, 455–473 (1995);C. E. Max, S. S. Olivier, H. W. Friedman, J. An, K. Avicola, B. V. Beeman, H. D. Bissinger, J. M. Brase, G. V. Erbert, D. T. Gavel, K. Kanz, M. C. Liu, B. Macintosh, K. P. Neeb, J. Patience, K. E. Waltjen, “Image improvement from a sodium-layer guide star adaptive optics system,” Science 277, 1649–1652 (1997);M. Lloyd-Hart, J. R. P. Angel, T. D. Groesbeck, T. Martinez, B. P. Jacobsen, B. A. McLeod, D. W. McCarthy, E. J. Hooper, E. K. Hege, D. G. Sandler, “First astronomical images sharpened with adaptive optics using a sodium laser guide star,” Astrophys. J. 493, 950–954 (1998).
[CrossRef]

1994 (1)

1992 (1)

1985 (1)

J. J. McClelland, M. H. Kelley, “Detailed look at aspects of optical pumping in sodium,” Phys. Rev. A 31, 3704–3710 (1985).
[CrossRef] [PubMed]

1977 (1)

E. Arimondo, M. Inguscio, P. Violino, “Experimental determination of the hyperfine structure in alkali atoms,” Rev. Mod. Phys. 49, 31–75 (1977).
[CrossRef]

1974 (1)

See, for instance, J. A. Abate, “Preparation of atomic sodium as a two-level atom,” Opt. Commun. 10, 269–272 (1974).
[CrossRef]

1972 (1)

W. Happer, “Optical Pumping,” Rev. Mod. Phys. 44, 169–249 (1972).
[CrossRef]

1969 (1)

N. W. Ressler, R. H. Sands, T. E. Stark, “Measurement of the spin-exchange cross sections for Cs133,Rb87,Rb85,K39, and Na23,” Phys. Rev. 184, 102–118 (1969);A. Moretti, F. Strumia, “Hyperfine optical pumping of sodium vapor,” Phys. Rev. A 3, 349–354 (1971).
[CrossRef]

1965 (1)

S. Berman, M. Jacob, “Systematics of angular and polarization distributions in three-body decays,” Phys. Rev. 139, B1023 (1965).
[CrossRef]

1964 (1)

A. T. Ramsey, L. W. Anderson, “Spin relaxation in an optically pumped sodium vapor,” Nuovo Cimento 32, 1151–1157 (1964).
[CrossRef]

1924 (1)

W. Hanle, “Über magnetische Beeinflussung der Polarisation der Resonanzfluoreszenz,” Z. Phys. 30, 93–105 (1924);A. Kastler, “Quelques suggestions concernant la production optique et la détection optique d’une inégalité de population des niveaux de quantification spatiale des atomes. Application à l’expérience de Stern et Gerlach et à la résonance magnétique,” J. Phys. Radium 11, 255–265 (1950);A. Kastler, “La détection optique de la résonance électronique paramagnétique par la mesure de la polarisation rotatoire paramagnétique d’une radiation visible,” Compt. Rendu 232, 953–957 (1951).
[CrossRef]

Abate, J. A.

See, for instance, J. A. Abate, “Preparation of atomic sodium as a two-level atom,” Opt. Commun. 10, 269–272 (1974).
[CrossRef]

Anderson, L. W.

A. T. Ramsey, L. W. Anderson, “Spin relaxation in an optically pumped sodium vapor,” Nuovo Cimento 32, 1151–1157 (1964).
[CrossRef]

Angel, J. R. P.

M. Lloyd-Hart, J. R. P. Angel, B. Jacobsen, D. Wittman, R. Dekany, D. McCarthy, E. Kibblewhite, W. Wild, B. Carter, J. Beletic, “Adaptive optics experiments using sodium laser guide stars,” Astrophys. J. 439, 455–473 (1995);C. E. Max, S. S. Olivier, H. W. Friedman, J. An, K. Avicola, B. V. Beeman, H. D. Bissinger, J. M. Brase, G. V. Erbert, D. T. Gavel, K. Kanz, M. C. Liu, B. Macintosh, K. P. Neeb, J. Patience, K. E. Waltjen, “Image improvement from a sodium-layer guide star adaptive optics system,” Science 277, 1649–1652 (1997);M. Lloyd-Hart, J. R. P. Angel, T. D. Groesbeck, T. Martinez, B. P. Jacobsen, B. A. McLeod, D. W. McCarthy, E. J. Hooper, E. K. Hege, D. G. Sandler, “First astronomical images sharpened with adaptive optics using a sodium laser guide star,” Astrophys. J. 493, 950–954 (1998).
[CrossRef]

B. Jacobsen, T. Martinez, J. R. P. Angel, M. Lloyd-Hart, S. Benda, D. Middleton, H. W. Friedman, “Field evaluation of two new continuous-wave dye laser systems optimized for sodium beacon excitation,” in SPIE Proceedings on Adaptive Optics in Astronomy, M. A. Ealey, F. Merkle, eds., Proc. SPIE2201, 342–351 (1994).
[CrossRef]

J. Ge, B. P. Jacobsen, J. R. P. Angel, P. C. McGuire, T. Roberts, B. A. McLeod, M. Lloyd-Hart, “Simultaneous measurements of sodium column density and laser guide star brightness,” in SPIE Proceedings on Adaptive Optical System Technologies, D. Bonaccini, R. K. Tyson, eds., Proc. SPIE3353, 242–254 (1998).
[CrossRef]

Arimondo, E.

E. Arimondo, M. Inguscio, P. Violino, “Experimental determination of the hyperfine structure in alkali atoms,” Rev. Mod. Phys. 49, 31–75 (1977).
[CrossRef]

Beletic, J.

M. Lloyd-Hart, J. R. P. Angel, B. Jacobsen, D. Wittman, R. Dekany, D. McCarthy, E. Kibblewhite, W. Wild, B. Carter, J. Beletic, “Adaptive optics experiments using sodium laser guide stars,” Astrophys. J. 439, 455–473 (1995);C. E. Max, S. S. Olivier, H. W. Friedman, J. An, K. Avicola, B. V. Beeman, H. D. Bissinger, J. M. Brase, G. V. Erbert, D. T. Gavel, K. Kanz, M. C. Liu, B. Macintosh, K. P. Neeb, J. Patience, K. E. Waltjen, “Image improvement from a sodium-layer guide star adaptive optics system,” Science 277, 1649–1652 (1997);M. Lloyd-Hart, J. R. P. Angel, T. D. Groesbeck, T. Martinez, B. P. Jacobsen, B. A. McLeod, D. W. McCarthy, E. J. Hooper, E. K. Hege, D. G. Sandler, “First astronomical images sharpened with adaptive optics using a sodium laser guide star,” Astrophys. J. 493, 950–954 (1998).
[CrossRef]

Benda, S.

B. Jacobsen, T. Martinez, J. R. P. Angel, M. Lloyd-Hart, S. Benda, D. Middleton, H. W. Friedman, “Field evaluation of two new continuous-wave dye laser systems optimized for sodium beacon excitation,” in SPIE Proceedings on Adaptive Optics in Astronomy, M. A. Ealey, F. Merkle, eds., Proc. SPIE2201, 342–351 (1994).
[CrossRef]

Berman, S.

S. Berman, M. Jacob, “Systematics of angular and polarization distributions in three-body decays,” Phys. Rev. 139, B1023 (1965).
[CrossRef]

Bradley, L. C.

Carter, B.

M. Lloyd-Hart, J. R. P. Angel, B. Jacobsen, D. Wittman, R. Dekany, D. McCarthy, E. Kibblewhite, W. Wild, B. Carter, J. Beletic, “Adaptive optics experiments using sodium laser guide stars,” Astrophys. J. 439, 455–473 (1995);C. E. Max, S. S. Olivier, H. W. Friedman, J. An, K. Avicola, B. V. Beeman, H. D. Bissinger, J. M. Brase, G. V. Erbert, D. T. Gavel, K. Kanz, M. C. Liu, B. Macintosh, K. P. Neeb, J. Patience, K. E. Waltjen, “Image improvement from a sodium-layer guide star adaptive optics system,” Science 277, 1649–1652 (1997);M. Lloyd-Hart, J. R. P. Angel, T. D. Groesbeck, T. Martinez, B. P. Jacobsen, B. A. McLeod, D. W. McCarthy, E. J. Hooper, E. K. Hege, D. G. Sandler, “First astronomical images sharpened with adaptive optics using a sodium laser guide star,” Astrophys. J. 493, 950–954 (1998).
[CrossRef]

Chernosky, E. J.

See E. J. Chernosky, P. F. Fougere, R. O. Hutchinson, “The geomagnetic field,” in Handbook of Geophysics and Space Environments, S. L. Valley, ed. (McGraw-Hill, New York, 1965), Chap. 11.

Dekany, R.

M. Lloyd-Hart, J. R. P. Angel, B. Jacobsen, D. Wittman, R. Dekany, D. McCarthy, E. Kibblewhite, W. Wild, B. Carter, J. Beletic, “Adaptive optics experiments using sodium laser guide stars,” Astrophys. J. 439, 455–473 (1995);C. E. Max, S. S. Olivier, H. W. Friedman, J. An, K. Avicola, B. V. Beeman, H. D. Bissinger, J. M. Brase, G. V. Erbert, D. T. Gavel, K. Kanz, M. C. Liu, B. Macintosh, K. P. Neeb, J. Patience, K. E. Waltjen, “Image improvement from a sodium-layer guide star adaptive optics system,” Science 277, 1649–1652 (1997);M. Lloyd-Hart, J. R. P. Angel, T. D. Groesbeck, T. Martinez, B. P. Jacobsen, B. A. McLeod, D. W. McCarthy, E. J. Hooper, E. K. Hege, D. G. Sandler, “First astronomical images sharpened with adaptive optics using a sodium laser guide star,” Astrophys. J. 493, 950–954 (1998).
[CrossRef]

Edmonds, A. R.

A. R. Edmonds, Angular Momentum in Quantum Mechanics (Princeton U. Press, Princeton, N.J., 1996).

Fougere, P. F.

See E. J. Chernosky, P. F. Fougere, R. O. Hutchinson, “The geomagnetic field,” in Handbook of Geophysics and Space Environments, S. L. Valley, ed. (McGraw-Hill, New York, 1965), Chap. 11.

Friedman, H. W.

B. Jacobsen, T. Martinez, J. R. P. Angel, M. Lloyd-Hart, S. Benda, D. Middleton, H. W. Friedman, “Field evaluation of two new continuous-wave dye laser systems optimized for sodium beacon excitation,” in SPIE Proceedings on Adaptive Optics in Astronomy, M. A. Ealey, F. Merkle, eds., Proc. SPIE2201, 342–351 (1994).
[CrossRef]

Fugate, R. Q.

P. W. Milonni, R. Q. Fugate, J. M. Telle, “Analysis of measured photon returns from sodium beacons,” J. Opt. Soc. Am. A 15, 217–233 (1998).
[CrossRef]

See, for instance, the following reviews and references therein: M. C. Roggemann, B. M. Welsh, R. Q. Fugate, “Improving the resolution of ground-based telescopes,” Rev. Mod. Phys. 69, 437–505 (1997);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).
[CrossRef] [PubMed]

Ge, J.

J. Ge, Lawrence Livermore National Laboratory, 7000 East Ave., Livermore, Calif. 94550 (personal communication, April1998).

J. Ge, B. P. Jacobsen, J. R. P. Angel, P. C. McGuire, T. Roberts, B. A. McLeod, M. Lloyd-Hart, “Simultaneous measurements of sodium column density and laser guide star brightness,” in SPIE Proceedings on Adaptive Optical System Technologies, D. Bonaccini, R. K. Tyson, eds., Proc. SPIE3353, 242–254 (1998).
[CrossRef]

Gottfried, K.

See, for instance, K. Gottfried, Quantum Mechanics (Benjamin, Reading, Mass., 1977), p. 269.

Hanle, W.

W. Hanle, “Über magnetische Beeinflussung der Polarisation der Resonanzfluoreszenz,” Z. Phys. 30, 93–105 (1924);A. Kastler, “Quelques suggestions concernant la production optique et la détection optique d’une inégalité de population des niveaux de quantification spatiale des atomes. Application à l’expérience de Stern et Gerlach et à la résonance magnétique,” J. Phys. Radium 11, 255–265 (1950);A. Kastler, “La détection optique de la résonance électronique paramagnétique par la mesure de la polarisation rotatoire paramagnétique d’une radiation visible,” Compt. Rendu 232, 953–957 (1951).
[CrossRef]

Happer, W.

W. Happer, “Optical Pumping,” Rev. Mod. Phys. 44, 169–249 (1972).
[CrossRef]

The suggestion to use the Na-Na spin-relaxation cross section as a rough estimate for the unknown Na-O2 cross section is due to W. Happer, Department of Physics, Princeton University, Princeton, New Jersey 08544 (personal communication, June1997).

Hutchinson, R. O.

See E. J. Chernosky, P. F. Fougere, R. O. Hutchinson, “The geomagnetic field,” in Handbook of Geophysics and Space Environments, S. L. Valley, ed. (McGraw-Hill, New York, 1965), Chap. 11.

Inguscio, M.

E. Arimondo, M. Inguscio, P. Violino, “Experimental determination of the hyperfine structure in alkali atoms,” Rev. Mod. Phys. 49, 31–75 (1977).
[CrossRef]

Jacob, M.

S. Berman, M. Jacob, “Systematics of angular and polarization distributions in three-body decays,” Phys. Rev. 139, B1023 (1965).
[CrossRef]

Jacobsen, B.

M. Lloyd-Hart, J. R. P. Angel, B. Jacobsen, D. Wittman, R. Dekany, D. McCarthy, E. Kibblewhite, W. Wild, B. Carter, J. Beletic, “Adaptive optics experiments using sodium laser guide stars,” Astrophys. J. 439, 455–473 (1995);C. E. Max, S. S. Olivier, H. W. Friedman, J. An, K. Avicola, B. V. Beeman, H. D. Bissinger, J. M. Brase, G. V. Erbert, D. T. Gavel, K. Kanz, M. C. Liu, B. Macintosh, K. P. Neeb, J. Patience, K. E. Waltjen, “Image improvement from a sodium-layer guide star adaptive optics system,” Science 277, 1649–1652 (1997);M. Lloyd-Hart, J. R. P. Angel, T. D. Groesbeck, T. Martinez, B. P. Jacobsen, B. A. McLeod, D. W. McCarthy, E. J. Hooper, E. K. Hege, D. G. Sandler, “First astronomical images sharpened with adaptive optics using a sodium laser guide star,” Astrophys. J. 493, 950–954 (1998).
[CrossRef]

B. Jacobsen, T. Martinez, J. R. P. Angel, M. Lloyd-Hart, S. Benda, D. Middleton, H. W. Friedman, “Field evaluation of two new continuous-wave dye laser systems optimized for sodium beacon excitation,” in SPIE Proceedings on Adaptive Optics in Astronomy, M. A. Ealey, F. Merkle, eds., Proc. SPIE2201, 342–351 (1994).
[CrossRef]

Jacobsen, B. P.

J. Ge, B. P. Jacobsen, J. R. P. Angel, P. C. McGuire, T. Roberts, B. A. McLeod, M. Lloyd-Hart, “Simultaneous measurements of sodium column density and laser guide star brightness,” in SPIE Proceedings on Adaptive Optical System Technologies, D. Bonaccini, R. K. Tyson, eds., Proc. SPIE3353, 242–254 (1998).
[CrossRef]

Kelley, M. H.

J. J. McClelland, M. H. Kelley, “Detailed look at aspects of optical pumping in sodium,” Phys. Rev. A 31, 3704–3710 (1985).
[CrossRef] [PubMed]

Kibblewhite, E.

M. Lloyd-Hart, J. R. P. Angel, B. Jacobsen, D. Wittman, R. Dekany, D. McCarthy, E. Kibblewhite, W. Wild, B. Carter, J. Beletic, “Adaptive optics experiments using sodium laser guide stars,” Astrophys. J. 439, 455–473 (1995);C. E. Max, S. S. Olivier, H. W. Friedman, J. An, K. Avicola, B. V. Beeman, H. D. Bissinger, J. M. Brase, G. V. Erbert, D. T. Gavel, K. Kanz, M. C. Liu, B. Macintosh, K. P. Neeb, J. Patience, K. E. Waltjen, “Image improvement from a sodium-layer guide star adaptive optics system,” Science 277, 1649–1652 (1997);M. Lloyd-Hart, J. R. P. Angel, T. D. Groesbeck, T. Martinez, B. P. Jacobsen, B. A. McLeod, D. W. McCarthy, E. J. Hooper, E. K. Hege, D. G. Sandler, “First astronomical images sharpened with adaptive optics using a sodium laser guide star,” Astrophys. J. 493, 950–954 (1998).
[CrossRef]

Lloyd-Hart, M.

M. Lloyd-Hart, J. R. P. Angel, B. Jacobsen, D. Wittman, R. Dekany, D. McCarthy, E. Kibblewhite, W. Wild, B. Carter, J. Beletic, “Adaptive optics experiments using sodium laser guide stars,” Astrophys. J. 439, 455–473 (1995);C. E. Max, S. S. Olivier, H. W. Friedman, J. An, K. Avicola, B. V. Beeman, H. D. Bissinger, J. M. Brase, G. V. Erbert, D. T. Gavel, K. Kanz, M. C. Liu, B. Macintosh, K. P. Neeb, J. Patience, K. E. Waltjen, “Image improvement from a sodium-layer guide star adaptive optics system,” Science 277, 1649–1652 (1997);M. Lloyd-Hart, J. R. P. Angel, T. D. Groesbeck, T. Martinez, B. P. Jacobsen, B. A. McLeod, D. W. McCarthy, E. J. Hooper, E. K. Hege, D. G. Sandler, “First astronomical images sharpened with adaptive optics using a sodium laser guide star,” Astrophys. J. 493, 950–954 (1998).
[CrossRef]

B. Jacobsen, T. Martinez, J. R. P. Angel, M. Lloyd-Hart, S. Benda, D. Middleton, H. W. Friedman, “Field evaluation of two new continuous-wave dye laser systems optimized for sodium beacon excitation,” in SPIE Proceedings on Adaptive Optics in Astronomy, M. A. Ealey, F. Merkle, eds., Proc. SPIE2201, 342–351 (1994).
[CrossRef]

J. Ge, B. P. Jacobsen, J. R. P. Angel, P. C. McGuire, T. Roberts, B. A. McLeod, M. Lloyd-Hart, “Simultaneous measurements of sodium column density and laser guide star brightness,” in SPIE Proceedings on Adaptive Optical System Technologies, D. Bonaccini, R. K. Tyson, eds., Proc. SPIE3353, 242–254 (1998).
[CrossRef]

Martinez, T.

B. Jacobsen, T. Martinez, J. R. P. Angel, M. Lloyd-Hart, S. Benda, D. Middleton, H. W. Friedman, “Field evaluation of two new continuous-wave dye laser systems optimized for sodium beacon excitation,” in SPIE Proceedings on Adaptive Optics in Astronomy, M. A. Ealey, F. Merkle, eds., Proc. SPIE2201, 342–351 (1994).
[CrossRef]

McCarthy, D.

M. Lloyd-Hart, J. R. P. Angel, B. Jacobsen, D. Wittman, R. Dekany, D. McCarthy, E. Kibblewhite, W. Wild, B. Carter, J. Beletic, “Adaptive optics experiments using sodium laser guide stars,” Astrophys. J. 439, 455–473 (1995);C. E. Max, S. S. Olivier, H. W. Friedman, J. An, K. Avicola, B. V. Beeman, H. D. Bissinger, J. M. Brase, G. V. Erbert, D. T. Gavel, K. Kanz, M. C. Liu, B. Macintosh, K. P. Neeb, J. Patience, K. E. Waltjen, “Image improvement from a sodium-layer guide star adaptive optics system,” Science 277, 1649–1652 (1997);M. Lloyd-Hart, J. R. P. Angel, T. D. Groesbeck, T. Martinez, B. P. Jacobsen, B. A. McLeod, D. W. McCarthy, E. J. Hooper, E. K. Hege, D. G. Sandler, “First astronomical images sharpened with adaptive optics using a sodium laser guide star,” Astrophys. J. 493, 950–954 (1998).
[CrossRef]

McClelland, J. J.

J. J. McClelland, M. H. Kelley, “Detailed look at aspects of optical pumping in sodium,” Phys. Rev. A 31, 3704–3710 (1985).
[CrossRef] [PubMed]

McGuire, P. C.

J. Ge, B. P. Jacobsen, J. R. P. Angel, P. C. McGuire, T. Roberts, B. A. McLeod, M. Lloyd-Hart, “Simultaneous measurements of sodium column density and laser guide star brightness,” in SPIE Proceedings on Adaptive Optical System Technologies, D. Bonaccini, R. K. Tyson, eds., Proc. SPIE3353, 242–254 (1998).
[CrossRef]

McLeod, B. A.

J. Ge, B. P. Jacobsen, J. R. P. Angel, P. C. McGuire, T. Roberts, B. A. McLeod, M. Lloyd-Hart, “Simultaneous measurements of sodium column density and laser guide star brightness,” in SPIE Proceedings on Adaptive Optical System Technologies, D. Bonaccini, R. K. Tyson, eds., Proc. SPIE3353, 242–254 (1998).
[CrossRef]

Middleton, D.

B. Jacobsen, T. Martinez, J. R. P. Angel, M. Lloyd-Hart, S. Benda, D. Middleton, H. W. Friedman, “Field evaluation of two new continuous-wave dye laser systems optimized for sodium beacon excitation,” in SPIE Proceedings on Adaptive Optics in Astronomy, M. A. Ealey, F. Merkle, eds., Proc. SPIE2201, 342–351 (1994).
[CrossRef]

Milonni, P. W.

Morris, J. R.

Ramsey, A. T.

A. T. Ramsey, L. W. Anderson, “Spin relaxation in an optically pumped sodium vapor,” Nuovo Cimento 32, 1151–1157 (1964).
[CrossRef]

Ressler, N. W.

N. W. Ressler, R. H. Sands, T. E. Stark, “Measurement of the spin-exchange cross sections for Cs133,Rb87,Rb85,K39, and Na23,” Phys. Rev. 184, 102–118 (1969);A. Moretti, F. Strumia, “Hyperfine optical pumping of sodium vapor,” Phys. Rev. A 3, 349–354 (1971).
[CrossRef]

Roberts, T.

J. Ge, B. P. Jacobsen, J. R. P. Angel, P. C. McGuire, T. Roberts, B. A. McLeod, M. Lloyd-Hart, “Simultaneous measurements of sodium column density and laser guide star brightness,” in SPIE Proceedings on Adaptive Optical System Technologies, D. Bonaccini, R. K. Tyson, eds., Proc. SPIE3353, 242–254 (1998).
[CrossRef]

Roggemann, M. C.

See, for instance, the following reviews and references therein: M. C. Roggemann, B. M. Welsh, R. Q. Fugate, “Improving the resolution of ground-based telescopes,” Rev. Mod. Phys. 69, 437–505 (1997);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).
[CrossRef] [PubMed]

Sands, R. H.

N. W. Ressler, R. H. Sands, T. E. Stark, “Measurement of the spin-exchange cross sections for Cs133,Rb87,Rb85,K39, and Na23,” Phys. Rev. 184, 102–118 (1969);A. Moretti, F. Strumia, “Hyperfine optical pumping of sodium vapor,” Phys. Rev. A 3, 349–354 (1971).
[CrossRef]

Stark, T. E.

N. W. Ressler, R. H. Sands, T. E. Stark, “Measurement of the spin-exchange cross sections for Cs133,Rb87,Rb85,K39, and Na23,” Phys. Rev. 184, 102–118 (1969);A. Moretti, F. Strumia, “Hyperfine optical pumping of sodium vapor,” Phys. Rev. A 3, 349–354 (1971).
[CrossRef]

Telle, J. M.

Violino, P.

E. Arimondo, M. Inguscio, P. Violino, “Experimental determination of the hyperfine structure in alkali atoms,” Rev. Mod. Phys. 49, 31–75 (1977).
[CrossRef]

Welsh, B. M.

See, for instance, the following reviews and references therein: M. C. Roggemann, B. M. Welsh, R. Q. Fugate, “Improving the resolution of ground-based telescopes,” Rev. Mod. Phys. 69, 437–505 (1997);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).
[CrossRef] [PubMed]

Wild, W.

M. Lloyd-Hart, J. R. P. Angel, B. Jacobsen, D. Wittman, R. Dekany, D. McCarthy, E. Kibblewhite, W. Wild, B. Carter, J. Beletic, “Adaptive optics experiments using sodium laser guide stars,” Astrophys. J. 439, 455–473 (1995);C. E. Max, S. S. Olivier, H. W. Friedman, J. An, K. Avicola, B. V. Beeman, H. D. Bissinger, J. M. Brase, G. V. Erbert, D. T. Gavel, K. Kanz, M. C. Liu, B. Macintosh, K. P. Neeb, J. Patience, K. E. Waltjen, “Image improvement from a sodium-layer guide star adaptive optics system,” Science 277, 1649–1652 (1997);M. Lloyd-Hart, J. R. P. Angel, T. D. Groesbeck, T. Martinez, B. P. Jacobsen, B. A. McLeod, D. W. McCarthy, E. J. Hooper, E. K. Hege, D. G. Sandler, “First astronomical images sharpened with adaptive optics using a sodium laser guide star,” Astrophys. J. 493, 950–954 (1998).
[CrossRef]

Wittman, D.

M. Lloyd-Hart, J. R. P. Angel, B. Jacobsen, D. Wittman, R. Dekany, D. McCarthy, E. Kibblewhite, W. Wild, B. Carter, J. Beletic, “Adaptive optics experiments using sodium laser guide stars,” Astrophys. J. 439, 455–473 (1995);C. E. Max, S. S. Olivier, H. W. Friedman, J. An, K. Avicola, B. V. Beeman, H. D. Bissinger, J. M. Brase, G. V. Erbert, D. T. Gavel, K. Kanz, M. C. Liu, B. Macintosh, K. P. Neeb, J. Patience, K. E. Waltjen, “Image improvement from a sodium-layer guide star adaptive optics system,” Science 277, 1649–1652 (1997);M. Lloyd-Hart, J. R. P. Angel, T. D. Groesbeck, T. Martinez, B. P. Jacobsen, B. A. McLeod, D. W. McCarthy, E. J. Hooper, E. K. Hege, D. G. Sandler, “First astronomical images sharpened with adaptive optics using a sodium laser guide star,” Astrophys. J. 493, 950–954 (1998).
[CrossRef]

Astrophys. J. (1)

M. Lloyd-Hart, J. R. P. Angel, B. Jacobsen, D. Wittman, R. Dekany, D. McCarthy, E. Kibblewhite, W. Wild, B. Carter, J. Beletic, “Adaptive optics experiments using sodium laser guide stars,” Astrophys. J. 439, 455–473 (1995);C. E. Max, S. S. Olivier, H. W. Friedman, J. An, K. Avicola, B. V. Beeman, H. D. Bissinger, J. M. Brase, G. V. Erbert, D. T. Gavel, K. Kanz, M. C. Liu, B. Macintosh, K. P. Neeb, J. Patience, K. E. Waltjen, “Image improvement from a sodium-layer guide star adaptive optics system,” Science 277, 1649–1652 (1997);M. Lloyd-Hart, J. R. P. Angel, T. D. Groesbeck, T. Martinez, B. P. Jacobsen, B. A. McLeod, D. W. McCarthy, E. J. Hooper, E. K. Hege, D. G. Sandler, “First astronomical images sharpened with adaptive optics using a sodium laser guide star,” Astrophys. J. 493, 950–954 (1998).
[CrossRef]

J. Opt. Soc. Am. A (2)

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

Nuovo Cimento (1)

A. T. Ramsey, L. W. Anderson, “Spin relaxation in an optically pumped sodium vapor,” Nuovo Cimento 32, 1151–1157 (1964).
[CrossRef]

Opt. Commun. (1)

See, for instance, J. A. Abate, “Preparation of atomic sodium as a two-level atom,” Opt. Commun. 10, 269–272 (1974).
[CrossRef]

Phys. Rev. (2)

S. Berman, M. Jacob, “Systematics of angular and polarization distributions in three-body decays,” Phys. Rev. 139, B1023 (1965).
[CrossRef]

N. W. Ressler, R. H. Sands, T. E. Stark, “Measurement of the spin-exchange cross sections for Cs133,Rb87,Rb85,K39, and Na23,” Phys. Rev. 184, 102–118 (1969);A. Moretti, F. Strumia, “Hyperfine optical pumping of sodium vapor,” Phys. Rev. A 3, 349–354 (1971).
[CrossRef]

Phys. Rev. A (1)

J. J. McClelland, M. H. Kelley, “Detailed look at aspects of optical pumping in sodium,” Phys. Rev. A 31, 3704–3710 (1985).
[CrossRef] [PubMed]

Rev. Mod. Phys. (3)

See, for instance, the following reviews and references therein: M. C. Roggemann, B. M. Welsh, R. Q. Fugate, “Improving the resolution of ground-based telescopes,” Rev. Mod. Phys. 69, 437–505 (1997);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).
[CrossRef] [PubMed]

W. Happer, “Optical Pumping,” Rev. Mod. Phys. 44, 169–249 (1972).
[CrossRef]

E. Arimondo, M. Inguscio, P. Violino, “Experimental determination of the hyperfine structure in alkali atoms,” Rev. Mod. Phys. 49, 31–75 (1977).
[CrossRef]

Z. Phys. (1)

W. Hanle, “Über magnetische Beeinflussung der Polarisation der Resonanzfluoreszenz,” Z. Phys. 30, 93–105 (1924);A. Kastler, “Quelques suggestions concernant la production optique et la détection optique d’une inégalité de population des niveaux de quantification spatiale des atomes. Application à l’expérience de Stern et Gerlach et à la résonance magnétique,” J. Phys. Radium 11, 255–265 (1950);A. Kastler, “La détection optique de la résonance électronique paramagnétique par la mesure de la polarisation rotatoire paramagnétique d’une radiation visible,” Compt. Rendu 232, 953–957 (1951).
[CrossRef]

Other (9)

The suggestion to use the Na-Na spin-relaxation cross section as a rough estimate for the unknown Na-O2 cross section is due to W. Happer, Department of Physics, Princeton University, Princeton, New Jersey 08544 (personal communication, June1997).

J. Ge, Lawrence Livermore National Laboratory, 7000 East Ave., Livermore, Calif. 94550 (personal communication, April1998).

J. Ge, B. P. Jacobsen, J. R. P. Angel, P. C. McGuire, T. Roberts, B. A. McLeod, M. Lloyd-Hart, “Simultaneous measurements of sodium column density and laser guide star brightness,” in SPIE Proceedings on Adaptive Optical System Technologies, D. Bonaccini, R. K. Tyson, eds., Proc. SPIE3353, 242–254 (1998).
[CrossRef]

See, for instance, K. Gottfried, Quantum Mechanics (Benjamin, Reading, Mass., 1977), p. 269.

A. R. Edmonds, Angular Momentum in Quantum Mechanics (Princeton U. Press, Princeton, N.J., 1996).

B. Jacobsen, T. Martinez, J. R. P. Angel, M. Lloyd-Hart, S. Benda, D. Middleton, H. W. Friedman, “Field evaluation of two new continuous-wave dye laser systems optimized for sodium beacon excitation,” in SPIE Proceedings on Adaptive Optics in Astronomy, M. A. Ealey, F. Merkle, eds., Proc. SPIE2201, 342–351 (1994).
[CrossRef]

An unpublished report detailing the calculation of the dipole matrix elements and radiative transition rates used in this work is available upon request.

See E. J. Chernosky, P. F. Fougere, R. O. Hutchinson, “The geomagnetic field,” in Handbook of Geophysics and Space Environments, S. L. Valley, ed. (McGraw-Hill, New York, 1965), Chap. 11.

Equation (9) follows from the expression R=σI/hν, where σ=3λ2AS(ν)/8π is the stimulated emission/absorption cross section and S(ν)=(4 ln 2/π)1/2/δνD is the Doppler lineshape function at line center.

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Figures (11)

Fig. 1
Fig. 1

Occupation probabilities of (a) state 3S1/2 (F=2, M=2) and (b) state 3P3/2 (F=3, M=3) for θ=30° and 90° (see text) as a function of the normalized time t/τ, τ=4π/(μBB), assuming in each case that the occupation probability is 1.0 at t=0.

Fig. 2
Fig. 2

Magnetic substates of the sodium D2 line. The dashed line indicates the two-state atom that can be produced by irradiation with circularly polarized light inducing Δm=+1 absorptive transitions.

Fig. 3
Fig. 3

Doppler-averaged occupation probability of (a) the 3S1/2 (F=2, M=2) state and (b) the 3P3/2 (F=3, M=3) state, with and without the geomagnetic field (B=0.5 G and θ=30°), for a cw circularly polarized laser with intensity I=10 mW/cm2.

Fig. 4
Fig. 4

Modification of the results of Fig. 3 when collisions are included with an assumed collision time of 100 μs. Except for the inclusion of collisions as described in the text, the assumed parameters are identical to those used in obtaining Fig. 3.

Fig. 5
Fig. 5

Data of Jacobsen et al.10 for sodium backscatter versus relative frequency with respect to the D2 hyperfine peak frequency, for circular polarization. The solid curve is a fit to the data, and the inset histogram is the distribution function for the laser frequency jitter from Ge et al.4 (From Ref. 4 with permission.)

Fig. 6
Fig. 6

(a) Computed steady-state value of NB versus intensity for four assumed values of Rcollτrad, where τrad(=16 ns) is the radiative lifetime of the 3P3/2 states: long-dashed line, Rcollτrad=2.5×10-5; solid line, Rcollτrad=5.0×10-5; short-dashed line, Rcollτrad=1.0×10-4; dotted line, Rcollτrad=2.0×10-4. The laser is circularly polarized, and its frequency is equal to the 3S1/2 (F=2)3P3/2 (F=3) transition frequency. (b) As in (a), but with the laser frequency matched to that corresponding to the smaller peak in Fig. 5, 1.772 GHz higher than that corresponding to the larger peak.

Fig. 7
Fig. 7

Computed curve of sodium backscatter versus frequency detuning from the D2 absorption peak for the experimental conditions of Fig. 5.

Fig. 8
Fig. 8

Computed absolute photon returns versus 1/Rcoll at the two peaks of Fig. 7. As 1/Rcoll0, the upper and lower curves asymptote to 110 and 66, respectively, giving the peak ratio 5/3 associated with thermal equilibrium.

Fig. 9
Fig. 9

Steady-state populations, averaged over both the Doppler distribution and the spatial beam profile, of the magnetic sublevels of the 3S1/2 (F=1) level (squares) and the 3S1/2 (F=2) level (circles). The parameters assumed are appropriate to the experiments of Ge et al.4 with circularly polarized light inducing ΔMF=+1 transitions.

Fig. 10
Fig. 10

As in Fig. 7, but for linear polarization.

Fig. 11
Fig. 11

As in Fig. 8, but for linear polarization.

Equations (34)

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S˙ij=-i(ωji-ω-ϕ˙-iβij)Sij+i2 djiE0(σii-σjj)(i<j),
σ˙ii=-Aiσii+j>iAjiσjj+i2 k<idikE0Ski*+i2 k>idikE0*Sik+complexconjugate.
Hmag=gFμBBF,
I=3/2,
gF=gJ F(F+1)+J(J+1)-I(I+1)2F(F+1),
gJ=3/2(3P3/2)=1.334,gJ=1/2(3S1/2)=2.
[σ˙M1M2(F)]mag=-iM3VM2M3(F)σM1M3(F)+iM3VM3M1(F)σM3M2(F),
VM2M3(F)=1 M2|g(F)μBBF|M3
VM2M3(F)=1 g(F)μBBMMM2|M; RM; R|M3.
VM2M3(F)=1 g(F)μBBMMdM2M(F)(θ)dMM3(F)(-θ),
R=3λ3A16π2c1δνD 4 ln 2π I=7.2×107I(W/cm2)sec-1
m d2Rd t2=(dR)E(R, t).
dvdt=-km Rei ij>idijE0*Sij,
dvdt=-km i(σ˙ii)ind,
v(t)=v(0)-km i0tdt[σ˙ii(t)]ind.
ωji-ω+kv(t)=ωji-ω+kv(0)-k2m i0tdt[σ˙ii(t)]ind,
ν˙recoil=12π k2m λ22π T0Phν 4 ln 2πa2=T0λ2πmc 4 ln 2πa2 P70T0P/a2 MHz/μs,
Rcoll(Na-N2)=NN2σ8RTπ 1MN2+1MNa=2.4×10-19NN2
Rcoll(Na-O2)=5.6×10-10NO2,
(σ˙ii)coll=-Rcollσii-18 j=18σjj.
tt+v x,
pt+v px=σI0hν f(x)-Ap,
p˜t+A+1TDp˜=σI0hν,
Rcw=T0Cs4πz2 Adxdyβp¯=1.5 T02Cs4πz2 3λ3A16π2c 1δνD 4 ln 2π P(watts),
Rcw=8.9×10-8CsP(W)photons/cm2/s,
p¯j=2πIc2A 1δνD 4 ln 2π 18 k<j(djk)q2,
k<j(djk)q2=3λ3A32π3 M A(F, M+12, M)A,
Rcw=3T02CsA512π3z2 λ3Pc 1δνD 4 ln 2π×FMβF,M+1 A(F, M+12, M)A.
Rcw=3×10-8CsT02P(W).
Rcw=ToCsz2 dxdyNB(x, y),
NB=A4π jβjp¯j(x, y),
NB=0.625×10-5=2400I(W/cm2)photons/atom/ms/sr
Rcw=T0Csz2 (2400)dxdyI(x, y)=T0Csz2 (2400)P(W),
Rcw/P=110photons/cm2/s/W

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