Abstract

The interaction of polarized laser beams and sodium vapor can lead to a variety of intensity and polarization patterns owing to optical-pumping-induced refractive-index variations along and transverse to the beam-propagation direction. While the steady-state behavior has been observed and described in the past, the modeling of the dynamics of the pattern formation processes has become possible only recently. We present the first, to our knowledge, three-dimensional simulations of the dynamics of beam bouncing, beam switching, and beam splitting and the deflection of a laser beam by the inhomogeneous magnetic field of a current-carrying wire. Numerical algorithms used for the calculations and experimental aspects for future observations are discussed.

© 1998 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. N. B. Abraham and W. J. Firth, “Overview of transverse effects in nonlinear-optical systems,” J. Opt. Soc. Am. B 7, 951–962 (1990).
    [CrossRef]
  2. W. Happer, “Optical pumping,” Rev. Mod. Phys. 44, 169–249 (1972).
    [CrossRef]
  3. S. Dangel and R. Holzner, “Semi-classical theory for the interaction dynamics of laser light and sodium atoms including the hyperfine structure,” Phys. Rev. A 56, 3937–3949 (1997).
    [CrossRef]
  4. S. Dangel, P. Eschle, B. Röhricht, U. Rusch, H. Schmid, and R. Holzner, “Dynamics of laser beam switching in sodium vapor,” J. Opt. Soc. Am. B 12, 681–686 (1995).
    [CrossRef]
  5. A. W. McCord and R. J. Ballagh, “Mutual beam reshaping by two interacting radiation modes,” J. Opt. Soc. Am. B 7, 73–83 (1990).
    [CrossRef]
  6. R. Holzner, P. Eschle, A. W. McCord, and D. M. Warrington, “Transverse “bouncing” of polarized laser beams in sodium vapor,” Phys. Rev. Lett. 69, 2192–2195 (1992).
    [CrossRef] [PubMed]
  7. B. Röhricht, U. Rusch, S. Dangel, H. Schmid, P. Eschle, R. Holzner, and W. Sandle, “Optical pumping induced ring structures of polarized laser light propagating through sodium vapor,” J. Opt. Soc. Am. B 12, 1411–1415 (1995).
    [CrossRef]
  8. D. McClelland, H. Bachor, and J. Wang, “Experimental observation of spatial polarization separation by absorptive self-focussing,” Opt. Commun. 84, 184–188 (1991).
    [CrossRef]
  9. A. Wilson, W. Sandle, D. Warrington, R. Ballagh, and A. McCord, “Observation of separated, polarized ring structures induced by nonlinear beam reshaping,” Opt. Commun. 88, 67–72 (1992).
    [CrossRef]
  10. A. Gahl, J. Seipenbusch, A. Aumann, M. Möller, and W. Lange, “Self-induced planar and cylindrical splitting of a laser beam in sodium vapor,” Phys. Rev. A 50, R917–920 (1994).
    [CrossRef] [PubMed]
  11. B. Röhricht, A. W. McCord, M. Brambilla, F. Prati, S. Dangel, P. Eschle, and R. Holzner, “Spatial separation of circularly polarized laser beams in sodium vapor,” Opt. Commun. 118, 601–606 (1995).
    [CrossRef]
  12. D. Suter, “Optically excited Zeeman coherences in atomic ground states: nuclear-spin effects,” Phys. Rev. A 46, 344–350 (1992).
    [CrossRef] [PubMed]
  13. R. Richard, S. Dangel, P. Eschle, B. Röhricht, H. Schmid, U. Rusch, and R. Holzner, “Laser beam switching, splitting and bending in sodium vapor,” in Digest of the International Quantum Electronics Conference (Optical Society of America, Washington, D.C., 1996), pp. 214–215.
  14. R. J. Ballagh and A. W. McCord, “Deflection of a laser beam by a current carrying wire,” in Proceedings of the Seventeenth International Quantum Electronics Conference (IEEE, New York, 1992), pp. 520–522.
  15. R. Holzner, P. Eschle, S. Dangel, R. Richard, H. Schmid, U. Rusch, B. Röhricht, R. J. Ballagh, A. W. McCord, and W. J. Sandle, “Observation of magnetic-field-induced laser beam deflection in sodium vapor,” Phys. Rev. Lett. 78, 3451–54 (1997).
    [CrossRef]
  16. L. C. Balling, “Optical pumping,” Adv. Quantum Electron. 3, 1–167 (1975).
  17. M. Brambilla, L. Lugiato, V. Penna, F. Prati, C. Tamm, and C. Weiss, “Transverse laser patterns. II. Variational principle for pattern selection, spatial multistability, and laser hydrodynamics,” Phys. Rev. A 43, S114–S120 (1991).
    [CrossRef]
  18. I. Stakgold, Green’s Functions and Boundary-Value Problems (Wiley, New York, 1979).
  19. G. D. Smith, Numerical Solution of Partial Differential Equations, 2nd ed. (Clarendon, London, 1978).
  20. P. Swarztrauber, “The methods of cyclic reduction, Fourier analysis and the FACR algorithm for the discrete solution of Poisson’s equation on a rectangle,” SIAM (Soc. Ind. Appl. Math.) Rev. 19, 490–501 (1977).
  21. A. Omont, “Irreducible components of the density matrix,” Prog. Quantum Electron. 5, 69–138 (1977).
    [CrossRef]
  22. M. Ducloy, “Nonlinear effects in optical pumping of atoms by a high-intensity multimode gas laser. General theory,” Phys. Rev. A 8, 1844–1859 (1973).
    [CrossRef]
  23. H. Kogelnik and T. Li, “Laser beams and resonators,” Appl. Opt. 5, 1550–1567 (1966).
    [CrossRef] [PubMed]

1997 (2)

S. Dangel and R. Holzner, “Semi-classical theory for the interaction dynamics of laser light and sodium atoms including the hyperfine structure,” Phys. Rev. A 56, 3937–3949 (1997).
[CrossRef]

R. Holzner, P. Eschle, S. Dangel, R. Richard, H. Schmid, U. Rusch, B. Röhricht, R. J. Ballagh, A. W. McCord, and W. J. Sandle, “Observation of magnetic-field-induced laser beam deflection in sodium vapor,” Phys. Rev. Lett. 78, 3451–54 (1997).
[CrossRef]

1995 (3)

1994 (1)

A. Gahl, J. Seipenbusch, A. Aumann, M. Möller, and W. Lange, “Self-induced planar and cylindrical splitting of a laser beam in sodium vapor,” Phys. Rev. A 50, R917–920 (1994).
[CrossRef] [PubMed]

1992 (3)

D. Suter, “Optically excited Zeeman coherences in atomic ground states: nuclear-spin effects,” Phys. Rev. A 46, 344–350 (1992).
[CrossRef] [PubMed]

R. Holzner, P. Eschle, A. W. McCord, and D. M. Warrington, “Transverse “bouncing” of polarized laser beams in sodium vapor,” Phys. Rev. Lett. 69, 2192–2195 (1992).
[CrossRef] [PubMed]

A. Wilson, W. Sandle, D. Warrington, R. Ballagh, and A. McCord, “Observation of separated, polarized ring structures induced by nonlinear beam reshaping,” Opt. Commun. 88, 67–72 (1992).
[CrossRef]

1991 (2)

D. McClelland, H. Bachor, and J. Wang, “Experimental observation of spatial polarization separation by absorptive self-focussing,” Opt. Commun. 84, 184–188 (1991).
[CrossRef]

M. Brambilla, L. Lugiato, V. Penna, F. Prati, C. Tamm, and C. Weiss, “Transverse laser patterns. II. Variational principle for pattern selection, spatial multistability, and laser hydrodynamics,” Phys. Rev. A 43, S114–S120 (1991).
[CrossRef]

1990 (2)

1977 (2)

P. Swarztrauber, “The methods of cyclic reduction, Fourier analysis and the FACR algorithm for the discrete solution of Poisson’s equation on a rectangle,” SIAM (Soc. Ind. Appl. Math.) Rev. 19, 490–501 (1977).

A. Omont, “Irreducible components of the density matrix,” Prog. Quantum Electron. 5, 69–138 (1977).
[CrossRef]

1973 (1)

M. Ducloy, “Nonlinear effects in optical pumping of atoms by a high-intensity multimode gas laser. General theory,” Phys. Rev. A 8, 1844–1859 (1973).
[CrossRef]

1972 (1)

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

1966 (1)

Abraham, N. B.

Aumann, A.

A. Gahl, J. Seipenbusch, A. Aumann, M. Möller, and W. Lange, “Self-induced planar and cylindrical splitting of a laser beam in sodium vapor,” Phys. Rev. A 50, R917–920 (1994).
[CrossRef] [PubMed]

Bachor, H.

D. McClelland, H. Bachor, and J. Wang, “Experimental observation of spatial polarization separation by absorptive self-focussing,” Opt. Commun. 84, 184–188 (1991).
[CrossRef]

Ballagh, R.

A. Wilson, W. Sandle, D. Warrington, R. Ballagh, and A. McCord, “Observation of separated, polarized ring structures induced by nonlinear beam reshaping,” Opt. Commun. 88, 67–72 (1992).
[CrossRef]

Ballagh, R. J.

R. Holzner, P. Eschle, S. Dangel, R. Richard, H. Schmid, U. Rusch, B. Röhricht, R. J. Ballagh, A. W. McCord, and W. J. Sandle, “Observation of magnetic-field-induced laser beam deflection in sodium vapor,” Phys. Rev. Lett. 78, 3451–54 (1997).
[CrossRef]

A. W. McCord and R. J. Ballagh, “Mutual beam reshaping by two interacting radiation modes,” J. Opt. Soc. Am. B 7, 73–83 (1990).
[CrossRef]

Brambilla, M.

B. Röhricht, A. W. McCord, M. Brambilla, F. Prati, S. Dangel, P. Eschle, and R. Holzner, “Spatial separation of circularly polarized laser beams in sodium vapor,” Opt. Commun. 118, 601–606 (1995).
[CrossRef]

M. Brambilla, L. Lugiato, V. Penna, F. Prati, C. Tamm, and C. Weiss, “Transverse laser patterns. II. Variational principle for pattern selection, spatial multistability, and laser hydrodynamics,” Phys. Rev. A 43, S114–S120 (1991).
[CrossRef]

Dangel, S.

R. Holzner, P. Eschle, S. Dangel, R. Richard, H. Schmid, U. Rusch, B. Röhricht, R. J. Ballagh, A. W. McCord, and W. J. Sandle, “Observation of magnetic-field-induced laser beam deflection in sodium vapor,” Phys. Rev. Lett. 78, 3451–54 (1997).
[CrossRef]

S. Dangel and R. Holzner, “Semi-classical theory for the interaction dynamics of laser light and sodium atoms including the hyperfine structure,” Phys. Rev. A 56, 3937–3949 (1997).
[CrossRef]

B. Röhricht, A. W. McCord, M. Brambilla, F. Prati, S. Dangel, P. Eschle, and R. Holzner, “Spatial separation of circularly polarized laser beams in sodium vapor,” Opt. Commun. 118, 601–606 (1995).
[CrossRef]

B. Röhricht, U. Rusch, S. Dangel, H. Schmid, P. Eschle, R. Holzner, and W. Sandle, “Optical pumping induced ring structures of polarized laser light propagating through sodium vapor,” J. Opt. Soc. Am. B 12, 1411–1415 (1995).
[CrossRef]

S. Dangel, P. Eschle, B. Röhricht, U. Rusch, H. Schmid, and R. Holzner, “Dynamics of laser beam switching in sodium vapor,” J. Opt. Soc. Am. B 12, 681–686 (1995).
[CrossRef]

Ducloy, M.

M. Ducloy, “Nonlinear effects in optical pumping of atoms by a high-intensity multimode gas laser. General theory,” Phys. Rev. A 8, 1844–1859 (1973).
[CrossRef]

Eschle, P.

R. Holzner, P. Eschle, S. Dangel, R. Richard, H. Schmid, U. Rusch, B. Röhricht, R. J. Ballagh, A. W. McCord, and W. J. Sandle, “Observation of magnetic-field-induced laser beam deflection in sodium vapor,” Phys. Rev. Lett. 78, 3451–54 (1997).
[CrossRef]

S. Dangel, P. Eschle, B. Röhricht, U. Rusch, H. Schmid, and R. Holzner, “Dynamics of laser beam switching in sodium vapor,” J. Opt. Soc. Am. B 12, 681–686 (1995).
[CrossRef]

B. Röhricht, U. Rusch, S. Dangel, H. Schmid, P. Eschle, R. Holzner, and W. Sandle, “Optical pumping induced ring structures of polarized laser light propagating through sodium vapor,” J. Opt. Soc. Am. B 12, 1411–1415 (1995).
[CrossRef]

B. Röhricht, A. W. McCord, M. Brambilla, F. Prati, S. Dangel, P. Eschle, and R. Holzner, “Spatial separation of circularly polarized laser beams in sodium vapor,” Opt. Commun. 118, 601–606 (1995).
[CrossRef]

R. Holzner, P. Eschle, A. W. McCord, and D. M. Warrington, “Transverse “bouncing” of polarized laser beams in sodium vapor,” Phys. Rev. Lett. 69, 2192–2195 (1992).
[CrossRef] [PubMed]

Firth, W. J.

Gahl, A.

A. Gahl, J. Seipenbusch, A. Aumann, M. Möller, and W. Lange, “Self-induced planar and cylindrical splitting of a laser beam in sodium vapor,” Phys. Rev. A 50, R917–920 (1994).
[CrossRef] [PubMed]

Happer, W.

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

Holzner, R.

R. Holzner, P. Eschle, S. Dangel, R. Richard, H. Schmid, U. Rusch, B. Röhricht, R. J. Ballagh, A. W. McCord, and W. J. Sandle, “Observation of magnetic-field-induced laser beam deflection in sodium vapor,” Phys. Rev. Lett. 78, 3451–54 (1997).
[CrossRef]

S. Dangel and R. Holzner, “Semi-classical theory for the interaction dynamics of laser light and sodium atoms including the hyperfine structure,” Phys. Rev. A 56, 3937–3949 (1997).
[CrossRef]

B. Röhricht, A. W. McCord, M. Brambilla, F. Prati, S. Dangel, P. Eschle, and R. Holzner, “Spatial separation of circularly polarized laser beams in sodium vapor,” Opt. Commun. 118, 601–606 (1995).
[CrossRef]

B. Röhricht, U. Rusch, S. Dangel, H. Schmid, P. Eschle, R. Holzner, and W. Sandle, “Optical pumping induced ring structures of polarized laser light propagating through sodium vapor,” J. Opt. Soc. Am. B 12, 1411–1415 (1995).
[CrossRef]

S. Dangel, P. Eschle, B. Röhricht, U. Rusch, H. Schmid, and R. Holzner, “Dynamics of laser beam switching in sodium vapor,” J. Opt. Soc. Am. B 12, 681–686 (1995).
[CrossRef]

R. Holzner, P. Eschle, A. W. McCord, and D. M. Warrington, “Transverse “bouncing” of polarized laser beams in sodium vapor,” Phys. Rev. Lett. 69, 2192–2195 (1992).
[CrossRef] [PubMed]

Kogelnik, H.

Lange, W.

A. Gahl, J. Seipenbusch, A. Aumann, M. Möller, and W. Lange, “Self-induced planar and cylindrical splitting of a laser beam in sodium vapor,” Phys. Rev. A 50, R917–920 (1994).
[CrossRef] [PubMed]

Li, T.

Lugiato, L.

M. Brambilla, L. Lugiato, V. Penna, F. Prati, C. Tamm, and C. Weiss, “Transverse laser patterns. II. Variational principle for pattern selection, spatial multistability, and laser hydrodynamics,” Phys. Rev. A 43, S114–S120 (1991).
[CrossRef]

McClelland, D.

D. McClelland, H. Bachor, and J. Wang, “Experimental observation of spatial polarization separation by absorptive self-focussing,” Opt. Commun. 84, 184–188 (1991).
[CrossRef]

McCord, A.

A. Wilson, W. Sandle, D. Warrington, R. Ballagh, and A. McCord, “Observation of separated, polarized ring structures induced by nonlinear beam reshaping,” Opt. Commun. 88, 67–72 (1992).
[CrossRef]

McCord, A. W.

R. Holzner, P. Eschle, S. Dangel, R. Richard, H. Schmid, U. Rusch, B. Röhricht, R. J. Ballagh, A. W. McCord, and W. J. Sandle, “Observation of magnetic-field-induced laser beam deflection in sodium vapor,” Phys. Rev. Lett. 78, 3451–54 (1997).
[CrossRef]

B. Röhricht, A. W. McCord, M. Brambilla, F. Prati, S. Dangel, P. Eschle, and R. Holzner, “Spatial separation of circularly polarized laser beams in sodium vapor,” Opt. Commun. 118, 601–606 (1995).
[CrossRef]

R. Holzner, P. Eschle, A. W. McCord, and D. M. Warrington, “Transverse “bouncing” of polarized laser beams in sodium vapor,” Phys. Rev. Lett. 69, 2192–2195 (1992).
[CrossRef] [PubMed]

A. W. McCord and R. J. Ballagh, “Mutual beam reshaping by two interacting radiation modes,” J. Opt. Soc. Am. B 7, 73–83 (1990).
[CrossRef]

Möller, M.

A. Gahl, J. Seipenbusch, A. Aumann, M. Möller, and W. Lange, “Self-induced planar and cylindrical splitting of a laser beam in sodium vapor,” Phys. Rev. A 50, R917–920 (1994).
[CrossRef] [PubMed]

Omont, A.

A. Omont, “Irreducible components of the density matrix,” Prog. Quantum Electron. 5, 69–138 (1977).
[CrossRef]

Penna, V.

M. Brambilla, L. Lugiato, V. Penna, F. Prati, C. Tamm, and C. Weiss, “Transverse laser patterns. II. Variational principle for pattern selection, spatial multistability, and laser hydrodynamics,” Phys. Rev. A 43, S114–S120 (1991).
[CrossRef]

Prati, F.

B. Röhricht, A. W. McCord, M. Brambilla, F. Prati, S. Dangel, P. Eschle, and R. Holzner, “Spatial separation of circularly polarized laser beams in sodium vapor,” Opt. Commun. 118, 601–606 (1995).
[CrossRef]

M. Brambilla, L. Lugiato, V. Penna, F. Prati, C. Tamm, and C. Weiss, “Transverse laser patterns. II. Variational principle for pattern selection, spatial multistability, and laser hydrodynamics,” Phys. Rev. A 43, S114–S120 (1991).
[CrossRef]

Richard, R.

R. Holzner, P. Eschle, S. Dangel, R. Richard, H. Schmid, U. Rusch, B. Röhricht, R. J. Ballagh, A. W. McCord, and W. J. Sandle, “Observation of magnetic-field-induced laser beam deflection in sodium vapor,” Phys. Rev. Lett. 78, 3451–54 (1997).
[CrossRef]

Röhricht, B.

R. Holzner, P. Eschle, S. Dangel, R. Richard, H. Schmid, U. Rusch, B. Röhricht, R. J. Ballagh, A. W. McCord, and W. J. Sandle, “Observation of magnetic-field-induced laser beam deflection in sodium vapor,” Phys. Rev. Lett. 78, 3451–54 (1997).
[CrossRef]

S. Dangel, P. Eschle, B. Röhricht, U. Rusch, H. Schmid, and R. Holzner, “Dynamics of laser beam switching in sodium vapor,” J. Opt. Soc. Am. B 12, 681–686 (1995).
[CrossRef]

B. Röhricht, A. W. McCord, M. Brambilla, F. Prati, S. Dangel, P. Eschle, and R. Holzner, “Spatial separation of circularly polarized laser beams in sodium vapor,” Opt. Commun. 118, 601–606 (1995).
[CrossRef]

B. Röhricht, U. Rusch, S. Dangel, H. Schmid, P. Eschle, R. Holzner, and W. Sandle, “Optical pumping induced ring structures of polarized laser light propagating through sodium vapor,” J. Opt. Soc. Am. B 12, 1411–1415 (1995).
[CrossRef]

Rusch, U.

Sandle, W.

B. Röhricht, U. Rusch, S. Dangel, H. Schmid, P. Eschle, R. Holzner, and W. Sandle, “Optical pumping induced ring structures of polarized laser light propagating through sodium vapor,” J. Opt. Soc. Am. B 12, 1411–1415 (1995).
[CrossRef]

A. Wilson, W. Sandle, D. Warrington, R. Ballagh, and A. McCord, “Observation of separated, polarized ring structures induced by nonlinear beam reshaping,” Opt. Commun. 88, 67–72 (1992).
[CrossRef]

Sandle, W. J.

R. Holzner, P. Eschle, S. Dangel, R. Richard, H. Schmid, U. Rusch, B. Röhricht, R. J. Ballagh, A. W. McCord, and W. J. Sandle, “Observation of magnetic-field-induced laser beam deflection in sodium vapor,” Phys. Rev. Lett. 78, 3451–54 (1997).
[CrossRef]

Schmid, H.

Seipenbusch, J.

A. Gahl, J. Seipenbusch, A. Aumann, M. Möller, and W. Lange, “Self-induced planar and cylindrical splitting of a laser beam in sodium vapor,” Phys. Rev. A 50, R917–920 (1994).
[CrossRef] [PubMed]

Suter, D.

D. Suter, “Optically excited Zeeman coherences in atomic ground states: nuclear-spin effects,” Phys. Rev. A 46, 344–350 (1992).
[CrossRef] [PubMed]

Swarztrauber, P.

P. Swarztrauber, “The methods of cyclic reduction, Fourier analysis and the FACR algorithm for the discrete solution of Poisson’s equation on a rectangle,” SIAM (Soc. Ind. Appl. Math.) Rev. 19, 490–501 (1977).

Tamm, C.

M. Brambilla, L. Lugiato, V. Penna, F. Prati, C. Tamm, and C. Weiss, “Transverse laser patterns. II. Variational principle for pattern selection, spatial multistability, and laser hydrodynamics,” Phys. Rev. A 43, S114–S120 (1991).
[CrossRef]

Wang, J.

D. McClelland, H. Bachor, and J. Wang, “Experimental observation of spatial polarization separation by absorptive self-focussing,” Opt. Commun. 84, 184–188 (1991).
[CrossRef]

Warrington, D.

A. Wilson, W. Sandle, D. Warrington, R. Ballagh, and A. McCord, “Observation of separated, polarized ring structures induced by nonlinear beam reshaping,” Opt. Commun. 88, 67–72 (1992).
[CrossRef]

Warrington, D. M.

R. Holzner, P. Eschle, A. W. McCord, and D. M. Warrington, “Transverse “bouncing” of polarized laser beams in sodium vapor,” Phys. Rev. Lett. 69, 2192–2195 (1992).
[CrossRef] [PubMed]

Weiss, C.

M. Brambilla, L. Lugiato, V. Penna, F. Prati, C. Tamm, and C. Weiss, “Transverse laser patterns. II. Variational principle for pattern selection, spatial multistability, and laser hydrodynamics,” Phys. Rev. A 43, S114–S120 (1991).
[CrossRef]

Wilson, A.

A. Wilson, W. Sandle, D. Warrington, R. Ballagh, and A. McCord, “Observation of separated, polarized ring structures induced by nonlinear beam reshaping,” Opt. Commun. 88, 67–72 (1992).
[CrossRef]

Appl. Opt. (1)

J. Opt. Soc. Am. B (4)

Opt. Commun. (3)

B. Röhricht, A. W. McCord, M. Brambilla, F. Prati, S. Dangel, P. Eschle, and R. Holzner, “Spatial separation of circularly polarized laser beams in sodium vapor,” Opt. Commun. 118, 601–606 (1995).
[CrossRef]

D. McClelland, H. Bachor, and J. Wang, “Experimental observation of spatial polarization separation by absorptive self-focussing,” Opt. Commun. 84, 184–188 (1991).
[CrossRef]

A. Wilson, W. Sandle, D. Warrington, R. Ballagh, and A. McCord, “Observation of separated, polarized ring structures induced by nonlinear beam reshaping,” Opt. Commun. 88, 67–72 (1992).
[CrossRef]

Phys. Rev. A (5)

A. Gahl, J. Seipenbusch, A. Aumann, M. Möller, and W. Lange, “Self-induced planar and cylindrical splitting of a laser beam in sodium vapor,” Phys. Rev. A 50, R917–920 (1994).
[CrossRef] [PubMed]

D. Suter, “Optically excited Zeeman coherences in atomic ground states: nuclear-spin effects,” Phys. Rev. A 46, 344–350 (1992).
[CrossRef] [PubMed]

M. Brambilla, L. Lugiato, V. Penna, F. Prati, C. Tamm, and C. Weiss, “Transverse laser patterns. II. Variational principle for pattern selection, spatial multistability, and laser hydrodynamics,” Phys. Rev. A 43, S114–S120 (1991).
[CrossRef]

M. Ducloy, “Nonlinear effects in optical pumping of atoms by a high-intensity multimode gas laser. General theory,” Phys. Rev. A 8, 1844–1859 (1973).
[CrossRef]

S. Dangel and R. Holzner, “Semi-classical theory for the interaction dynamics of laser light and sodium atoms including the hyperfine structure,” Phys. Rev. A 56, 3937–3949 (1997).
[CrossRef]

Phys. Rev. Lett. (2)

R. Holzner, P. Eschle, A. W. McCord, and D. M. Warrington, “Transverse “bouncing” of polarized laser beams in sodium vapor,” Phys. Rev. Lett. 69, 2192–2195 (1992).
[CrossRef] [PubMed]

R. Holzner, P. Eschle, S. Dangel, R. Richard, H. Schmid, U. Rusch, B. Röhricht, R. J. Ballagh, A. W. McCord, and W. J. Sandle, “Observation of magnetic-field-induced laser beam deflection in sodium vapor,” Phys. Rev. Lett. 78, 3451–54 (1997).
[CrossRef]

Prog. Quantum Electron. (1)

A. Omont, “Irreducible components of the density matrix,” Prog. Quantum Electron. 5, 69–138 (1977).
[CrossRef]

Rev. Mod. Phys. (1)

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

SIAM (Soc. Ind. Appl. Math.) Rev. (1)

P. Swarztrauber, “The methods of cyclic reduction, Fourier analysis and the FACR algorithm for the discrete solution of Poisson’s equation on a rectangle,” SIAM (Soc. Ind. Appl. Math.) Rev. 19, 490–501 (1977).

Other (5)

I. Stakgold, Green’s Functions and Boundary-Value Problems (Wiley, New York, 1979).

G. D. Smith, Numerical Solution of Partial Differential Equations, 2nd ed. (Clarendon, London, 1978).

L. C. Balling, “Optical pumping,” Adv. Quantum Electron. 3, 1–167 (1975).

R. Richard, S. Dangel, P. Eschle, B. Röhricht, H. Schmid, U. Rusch, and R. Holzner, “Laser beam switching, splitting and bending in sodium vapor,” in Digest of the International Quantum Electronics Conference (Optical Society of America, Washington, D.C., 1996), pp. 214–215.

R. J. Ballagh and A. W. McCord, “Deflection of a laser beam by a current carrying wire,” in Proceedings of the Seventeenth International Quantum Electronics Conference (IEEE, New York, 1992), pp. 520–522.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (13)

Fig. 1
Fig. 1

Experimental setup for spatial laser-beam patterns in sodium vapor. λ/4, quarter-wave plate; BS, beam splitter; L, lens; PR, polarization rotator; RP, Rochon prism.

Fig. 2
Fig. 2

Laser-beam patterns at the end of the sodium-vapor cell in the steady state: (a) Ring patterns of a single slightly elliptically polarized input beam. (b) Beam bouncing of two circularly polarized beams. (c) Rib-type pattern of one beam in the bouncing area compared with an intensity contour plot of the J =1/2J=1/2 semiclassical beam-propagation model in the steady state; (d) beam splitting of two initially superimposed σ+ and σ- polarized beams at frequency detunings 15 GHz below resonance (bottom), on resonance, 5 GHz above resonance, and 15 GHz above resonance (top). (e) Separation of three initially mostly overlapping linearly polarized beams (top) into six circularly polarized beams (second from top). Three of them are σ+ polarized (third from top), and the other three are σ- polarized (bottom). (f) Comparison of observed (top left, σ+; bottom left, σ-) and calculated (right) patterns of two beams of opposite circular polarization. The more intense center parts penetrate each other, while the less intense wings bounce. The patterns have a typical size in the range of 50–500 μm.

Fig. 3
Fig. 3

Phase diagrams of regions where beam bouncing and the formation of mixed patterns occur in the steady state, as functions of laser-beam power and laser–atom detuning: (a) Experiment; (b) semiclassical J=1/2J=1/2 model.

Fig. 4
Fig. 4

Schematic representation of the sodium D1 atomic transition. Top: Neglecting the hyperfine structure leaves only four states (J=1/2J=1/2 model). Optical pumping by, e.g., σ+ circularly polarized light can occur directly between the mj=-1/2 and mj=+1/2 ground states. Bottom: full model with all hyperfine levels. Optical pumping occurs in several cycles and is about an order of magnitude slower than in the J =1/2J=1/2 model. Relaxations other than the one to the pumped state are not shown.

Fig. 5
Fig. 5

Geometric arrangement: The beam(s) enter the cell at z=0 (input-window plane) and propagate along the z axis; x and y are the transverse axes. All simulations presented here are symmetric with respect to the y axis; the beam offsets occur always along the x axis. The two-dimensional contours shown in the subsequent figures represent the beam intensity at the output window (z=zend) plane and at the (y=0) plane.

Fig. 6
Fig. 6

Time evolution of a beam-bouncing simulation after both beams have been turned on at t=0: (a) The beams still diverge strongly and cross each other, but self focusing has already started; (b) self focusing is complete; (c) initial stage of separation; (d) steady state with full separation of 116 μm between the beam centers at the rear end of the cell. Parameters: sodium-vapor temperature, 201 °C (atomic density 3.7×1018 m-3); cell length, 6.5 cm; input power, 8 mW per beam; beam waist 60 μm at the cell-input window; offset between the beams at the cell-input window, 250 μm; total angle between the beams, 5 mrad; detuning, 1.3 GHz to the blue; orientation decay rate in the lower level, γl=1.6×104 s-1. The calculation grid size is 72×50 21 in the x, y, and z directions, respectively. The resulting intensities have been normalized along the beam-propagation direction to account for the absorption. Also, intensities below the experimental noise threshold have been suppressed.

Fig. 7
Fig. 7

Time evolution of a beam-splitting simulation. Both beams have been turned on at t=0. Only one beam is shown in the stages (a)–(e); the other beam always evolves symmetrically, as can be seen in (f) the last stage. Parameters: sodium-vapor temperature, 205 °C (atomic density 4.4×1018 m-3); cell length, 6.5 cm; input power, 10 mW per beam; beam waist, 40 μm at the cell-input window; initial offset, 3.2 μm; parallel beams, detuning, 1.0 GHz to the blue; orientation decay rate in the lower level, γl=1.6×104 s-1. The calculation grid size, normalization, and suppression threshold are as in Fig. 6.

Fig. 8
Fig. 8

Still frame of a video animation produced with data from a simulation of the wire-bouncing effect. (The complete animation can be seen at http://www.physik.unizh.ch/groups/laser/wire.mpeg.) The beam enters the cell at the rear end (input window) and propagates toward the viewer. The wire can be seen at the side of the beam. Before the current is turned on at t =20 μs, the beam has reached its undeflected stationary state. Immediately afterwards, at t=20.05 μs (the situation shown in this frame), the deflection at the cell-output window is 60 μm, which is best seen from the projection of the beam and the wire onto the bottom of the three-dimensional box. Video animations are an efficient tool for showing the overall qualitative behavior of three-dimensional dynamical systems.

Fig. 9
Fig. 9

Time evolution of the transmitted power and beam deflection at the end of a sodium-vapor cell in a wire-bouncing simulation. The beam has been turned on at t=0 and pumps itself through the medium until it reaches the almost stationary state at t=20 μs, when the current in the wire is turned on. Both the output power (top) and the deflection (bottom) then oscillate at the Larmor frequency of 2 MHz, owing to the transverse magnetic field of approximately 1 G at the beam center (within the J=1/2J=1/2 model, the Larmor frequency is four times larger than the correct value of 700 kHz/G obtained from the hyperfine model). The deflection occurs very fast, at the time scale of a Larmor precession, and even overshoots the stationary value of 34 μm. At t=40 μs the electric current through the wire is again turned off and the beam slowly relaxes back to its undeflected position at the time scale of optical pumping. The discrete jumps in the graph showing the position of the deflected beam (bottom) are an artifact resulting from our method of determining the amount of deflection: At the cell-output window, the grid points (or camera pixels in the experiment) with intensities higher than or equal to half the maximum intensity are counted and their unweighted mean distances to the wire are plotted. Since the number of such points varies with time, the discretization steps are not constant. Parameters: sodium-vapor temperature, 205 °C (atomic density 4.4 ×1018 m-3); cell length, 5.7 cm; input power, 6 mW; beam waist, 50 μm at the beginning of the wire; wire length, 3.5 cm; wire diameter, 20 μm; initial transverse distance from beam center to wire, 90 μm; electric current, 60 mA; detuning, 1.7 GHz to the blue; orientation decay rate in the lower level, γl=4.7 ×104 s-1. The calculation grid size is 60×50×27.

Fig. 10
Fig. 10

Simulated laser-beam switching-frequency-doubler mechanism. Left column: Unnormalized contours through the beam in the (y=0) plane. The beams enter the cell from the bottom. Center column: The same contour levels at the cell-output window. Right column: The power of each individual beam as a function of the propagation distance through the cell. (a) The first beam has been turned on at t=0 and is almost stationary. (b) Situation 0.58 μs after the second beam has been turned on at t=10.00 μs. The total output power drops to approximately half its value (compare with Fig. 11). (c) Almost stationary state with two beams turned on. The total output power is about the same as in (a). (d) The second beam has been turned off at t=20 μs, which decreases the total output power by one half. Parameters: sodium-vapor temperature 200 °C (atomic density 3.38×1018 m-3); cell length, 6.0 cm; input power, 20 mW per beam; beam waist, 260 μm at the cell-input window; beam offset 180 μm; detuning, 0 GHz; orientation decay rate in the lower level, γl=1.6×104 s-1; calculation grid size, 60×50×21.

Fig. 11
Fig. 11

Total transmitted power (bottom) for the same switching-frequency-doubler simulation as a function of time. Beam 1 (top) is switched on at t=0, while beam 2 (center) is switched on and off. The output oscillates at twice the frequency of beam 2.

Fig. 12
Fig. 12

Two steps that form the basic algorithm to solve the one-dimensional transport equation (t=cz)E0=f.

Fig. 13
Fig. 13

Fluctuating beams, in (left) simulation and (right) experiment, seen at the cell-output window. Simulated patterns: (a) at 1.23 μs after the beams have been switched on; (b) at 5.53 μs. The corresponding experimental patterns were observed at arbitrary times. In contrast to the experiment, the simulation reaches a steady state after 35 μs, since no fluctuations have been added to the model. Parameters: sodium-vapor temperature, 255 °C (atomic density 4.63×1019 m-3); cell length, 6.5 cm; input power, 25 mW per beam; beam waist, 120 μm at the cell-input window; x offset at input window, 56 μm; y offset at input window, 43 μm, x axis angle between beams, 2.09 mrad; y axis angle between beams, 0.94 mrad; detuning, 6.0 GHz to the blue; orientation decay rate in the lower level, γl=1.6 104 s-1; calculation grid size, 72×72×43.

Equations (44)

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

z+1c t-ic2ω 2x2+2y2E0(r, t)
=iω2c0 P0(r, t),
ρt=-i [H, ρ]+ρtRelax,
H=H0+Hlaser+Hmagnet.
2E-1c2 2Et2=1ε0c2 2Pt2
E(r, t)=Re{E0(r, t)exp[-i(ωt-kz)]}=12 {E0(r, t)exp[-i(ωt-kz)]+c.c.},
I=12 ε0c|E0|2.
2E0+2ik E0z+2iωc2 E0t-1c2 2E0t2
=-k2ε0 P0-2iωε0c2 P0t+1ε0c2 2P0t2.
z+1c t-ic2ω 2x2+2y2E0(r, t)
=iω2cε0 P0(r, t).
2E0x2=2E0y2=0,
c E0z+E0t=iω2ε0 P0=:f(z, t).
E0(z, t)=0tdtf[z-c(t-t), t]+E0(z-ct, 0),
E0(z, t+Δt)Δt2 [f(z-cΔt, t)+f(z, t+Δt)]+E0(z-cΔt, t).
E0z=ic2ω 2x2+2y2E0(r, t).
E0(a, z)=ibi(z)Bi(a, z),
E0(a, z, t)=ibi(z, t)Bi(a, z),
(cz+t)E^0=iω2ε0 P^0+c22iω (kx2+ky2)E^0=:fˆ,
E^0(kx, ky, z, t+Δt)Δt2 [fˆ(kx, ky, z-cΔt, t)+fˆ(kx, ky, z, t+Δt)]+E^0(kx, ky, z-cΔt, t)=11+iΔtc24ω (kx2+ky2)×iΔtω4ε0 [P^0(kx, ky, z-Δt, t)+P^0(kx, ky, z, t+Δt)]+Δtc24iω (kx2+ky2)+1×E^0(kx, ky, z-cΔt, t).
E0(1)(z, t+Δt)=E0(z-cΔt, t)+Δtf(z-cΔt, t).
E0(2)(z, t+Δt)=E0(z-cΔt, t)+Δt2 [f(z-cΔt, t)+f(1)(z, t+Δt)]
P^0(t+nΔt)=P^0(t)+n P^0(t+sΔt)-P^0(t)s,
lρ˙00=γnat12-lρ00+Im(F- luρ˜-11+F+ luρ˜11),
lρ˙01=-γl lρ01-γnat3uρ01+Im(F+ luρ˜11-F- luρ˜-11)-iθl2 (lρ-11+lρ11),
lρ˙11=-(γl+iλl)lρ11-γnat3uρ11-iθl2lρ01+iF-2 (luρ˜00+luρ˜01)+iF+*2 (luρ˜00*-luρ˜01*),
uρ˙01=-γu uρ01+Im(F+ luρ˜11-F- luρ˜-11)-iθu2 (uρ-11+uρ11),
uρ˙11=-(γu+iλu)uρ11-iθu2uρ01-iF-2 (luρ˜00-luρ˜01)-iF+*2 (luρ˜00*+luρ˜01*),
luρ˜.00=-(iΔω+γlu)luρ˜00-iλ- luρ˜01+iθ-2 (luρ˜11-luρ˜-11)-iF+*2 (uρ-11-lρ-11)-iF-*2 (uρ11-lρ11),
luρ˜.01=-(iΔω+γlu)luρ˜01-iλ- luρ˜00-iθ+2 (luρ˜-11+luρ˜11)-iF+*2 (lρ-11+uρ-11)+iF-*2 (lρ11+uρ11),
luρ˜.11=-(iΔω+γlu+iλ+)luρ˜11+iθ-2luρ˜00-iθ+2luρ˜01+iF+*2 12-2lρ00-lρ01-uρ01,
luρ˜.-11=-(iΔω+γlu-iλ+)luρ˜-11-iθ-2luρ˜00-iθ+2luρ˜01+iF-*2 12-2lρ00+lρ01+uρ01,
P0±=2nd1/23luρ˜±11*,
d1/2=6γnatπc3ε0ω03.
e±1=12 (ex±iey),e0=ez,
E0=-E0-e+-E0+e-,P0=-P0-e+-P0+e-,
θl,u=gl,uμBohrBtrans/,θ±=(θl±θu)/2,
λl,u=gl,uμBohrBlong/,λ±=(λl+λu)/2,
E(x, y, z, t)=E(x=y=z=0, t) w0w(z)×exp-x2+y2w(z)2×expik x2+y22R(z)+arctan zzR×exp[-i(ωt-kz)],
zR=πw02λ (Rayleighlength),
w(z)=w01+zzR2 (spotsize),
R(z)=z1+zRz2(radiusofwave-frontcurvature).
zz+αx,yy,xx-αz,
E(x, y, z=0, t)=E(x=y=z=0, t)1+αxzR2 exp-x2+y2w021+αxzR2×expik x2+y22αx1+zRαx2+arctan αxzR×exp(ikαx)exp[-i(ωt-kz)].

Metrics