Abstract

The steady-state propagation of spatially nonuniform σ+ and σ beams coupled by means of a homogeneously broadened J = ½↔J = ½ transition is shown to give rise in many cases to a spatial separation of the beams The nature of the nonlinear coupling, which allows the response of one beam to be strongly affected by the presence of the other, and which is the root cause of the phenomenon, is examined in detail. Results for propagation both on and off resonance, with varying initial spatial configurations, demonstrate that the phenomenon of self-induced spatial separation of the copropagating components persists over a wide range of situations. A physical explanation is given in terms of an encoding/diffraction sequence, and the experimental implications are discussed

© 1990 Optical Society of America

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  1. By beam reshaping we mean any self-induced change in the transverse intensity profile of a beam. This definition encompasses self-focusing, self-trapping, self-defocusing, self-bending, filamentation, and other more complicated phenomena.
  2. J. H. Marburger, “Self-focusing: theory,” Prog. Quantum Electron. 4, 35–110 (1975).
    [CrossRef]
  3. M. Le Berre, E. Ressayre, A. Tallet, “Resonant self-focusing of a cw intense light beam,” Phys. Rev. A 29, 2669–2676 (1984).
    [CrossRef]
  4. D. E. McClelland, R. J. Ballagh, W. J. Sandle, “Simple analytic approximation to continuous wave on-resonance beam reshaping,” J. Opt. Soc. Am. B 3, 212–218 (1986).
    [CrossRef]
  5. A. W. McCord, R. J. Ballagh, J. Cooper, “Dispersive self-focusing in atomic media,” J. Opt. Soc. Am. B 5, 1323–1334 (1988).
    [CrossRef]
  6. D. Grischkowsky, “Self-focusing of light by potassium vapor,” Phys. Rev. Lett. 24, 866–869 (1970).
    [CrossRef]
  7. C. H. Skinner, P. D. Kleiber, “Observation of anomalous conical emission from laser-excited barium vapor,” Phys. Rev. A 21, 151–156 (1980).
    [CrossRef]
  8. R. W. Boyd, M. G. Raymer, P. Narum, D. J. Harter, “Four-wave parametric interactions in a strongly driven two-level system,” Phys. Rev. A 24, 411–423 (1981).
    [CrossRef]
  9. D. J. Harter, R. W. Boyd, “Four-wave mixing resonantly enhanced by ac-Stark-split levels in self-trapped filaments of light,” Phys. Rev. A 29, 739–748 (1984).
    [CrossRef]
  10. Y. Shevy, M. Rosenbluh, “Multiple conical emission from a strongly driven atomic system,” J. Opt. Soc. Am. B 5, 116–122 (1988).
    [CrossRef]
  11. G. X. Jin, J. M. Yuan, L. M. Narducci, Y. S. Liu, E. J. Seibert, “Theoretical and experimental studies of conical Stokes emission,” Opt. Commun. 68, 379–384 (1988).
    [CrossRef]
  12. R. W. Boyd, M. S. Malcuit, D. J. Gauthier, K. Rzazewski, “Competition between amplified spontaneous emission and four-wave-mixing process,” Phys. Rev. A 35, 1648–1658 (1987).
    [CrossRef] [PubMed]
  13. D. H. Close, C. R. Giuliano, R. W. Hellwarth, L. D. Hess, F. J. McClung, W. G. Wagner, “The self-focusing of light of different polarizations,” IEEE J. Quantum Electron. QE-2, 553–557 (1966).
    [CrossRef]
  14. M. D. Feit, J. A. Fleck, “Beam nonparaxiality, filament formation, and beam breakup in the self-focusing of optical beams,” J. Opt. Soc. Am. B 5, 633–640 (1988).
    [CrossRef]
  15. M. W. Hamilton, R. J. Ballagh, W. J. Sandle, “Polarization switching in a ring cavity with resonantly driven J= ½↔J= ½ atoms,” Z. Phys.B 49, 263–272 (1982).
    [CrossRef]
  16. W. J. Sandle, M. W. Hamilton, R. J. Ballagh, “Polarization switching with J= ½↔J= ½ atoms in a ring cavity,” in Optical Bistability 2, C. M. Bowden, H. M. Gibbs, S. L. McCall, eds. (Plenum, New York, 1984).
    [CrossRef]
  17. A. Corney, Atomic and Laser Spectroscopy (Clarendon, Oxford, 1977).
  18. A. Omont, “Irreducible components of the density matrix. Application to optical pumping,” Prog. Quantum Electron. 5, 69–138 (1977).
    [CrossRef]
  19. Corrections for the nonisotropic nature of collisional relaxation have been discussed by Cooper et al.20 and are very small in the regime where the product of a Rabi frequency and a strong collision duration is small.
  20. J. Cooper, R. J. Ballagh, K. Burnett, D. G. Hummer, “On redistribution and the equations for radiative transfer,” As-trophys. J. 260, 299–316 (1982).
    [CrossRef]
  21. A. Yariv, Quantum Electronics, 2nd ed. (Wiley, New York, 1975).
  22. A. W. McCord, “Laser propagation with interacting radiation modes,” PhD. dissertation (University of Otago, Dunedin, New Zealand, 1988).
  23. P. D. Drummond, “Central partial difference propagation algorithm,” Comput. Phys. Commun. 29, 211–225 (1983).
    [CrossRef]
  24. It is clear from the analytic models of Refs. 4 and 5 that the amplitude and phase behavior at the beam edges is critical to a correct description of propagation in saturable media. An efficient numerical method must therefore adequately resolve these off-axis spatial regions but can afford to have fewer computational points in the highly saturated beam core. This was achieved in the present research by implementation of a five-parameter transverse-map function. Two parameters were used to specify an algebraic rezoning to infinity. The remaining three parameters specified an additional tan−1region of high resolution. The position, width, and number of points in this additional region are chosen automatically as the solution develops and shifts to follow the contracting (or expanding) encoding region. Simpler dynamic rezoning techniques have been used previously in Maxwell–Bloch calculations (e.g., see Ref. 25).
  25. F. P. Mattar, M. C. Newstein, “Adaptive stretching and rezoning as effective computational techniques for two-level Maxwell-Bloch simulation,” Comput. Phys. Commun. 20, 139–163 (1980).
    [CrossRef]
  26. C. E. Grosch, S. A. Orszag, “Numerical solutions of problems in unbounded regions: coordinate transforms,” J. Comput. Phys. 25, 273–296 (1977).
    [CrossRef]
  27. J. A. Fleck, “A cubic spline method for solving the wave equation of nonlinear optics,” J. Comput. Phys. 16, 324–341 (1974).
    [CrossRef]
  28. We have verified this self-guiding nature of the final field configuration numerically by turning off the effect of the medium after the pure circularly polarized annulus and beam have formed. As expected, the σ+annulus collapses rapidly back on axis, and the σ−beam spreads diffractively.
  29. D. G. McCartan, J. M. Farr, “Collision broadening of the sodium resonance lines by noble gases,” J. Phys. B 9, 985–994 (1976).
    [CrossRef]
  30. W. J. Sandle, Alan Gallagher, “Optical bistability by an atomic vapor in a focusing Fabry–Perot cavity,” Phys. Rev. A 24, 2107–2028 (1981).
    [CrossRef]
  31. William Happer, “Optical pumping,” Rev. Mod. Phys. 44, 169–249 (1972).
    [CrossRef]
  32. D. E. McClelland, “Magnetic field modification to laser propagation in a J = ½↔J= ½ system,” Ph.D. dissertation (University of Otago, Dunedin, New Zealand, 1987).
  33. A. C. Tarn, W. Happer, “Long-range interactions between cw self-focused laser beams in atomic vapor,” Phys. Rev. Lett. 38, 278–282 (1977).
    [CrossRef]
  34. J. C. Wang, H. A. Bachor, D. E. McClelland, Department of Australian Physics University National Australia Canberra (personal communication, 1989).

1988

1987

R. W. Boyd, M. S. Malcuit, D. J. Gauthier, K. Rzazewski, “Competition between amplified spontaneous emission and four-wave-mixing process,” Phys. Rev. A 35, 1648–1658 (1987).
[CrossRef] [PubMed]

1986

1984

M. Le Berre, E. Ressayre, A. Tallet, “Resonant self-focusing of a cw intense light beam,” Phys. Rev. A 29, 2669–2676 (1984).
[CrossRef]

D. J. Harter, R. W. Boyd, “Four-wave mixing resonantly enhanced by ac-Stark-split levels in self-trapped filaments of light,” Phys. Rev. A 29, 739–748 (1984).
[CrossRef]

1983

P. D. Drummond, “Central partial difference propagation algorithm,” Comput. Phys. Commun. 29, 211–225 (1983).
[CrossRef]

1982

M. W. Hamilton, R. J. Ballagh, W. J. Sandle, “Polarization switching in a ring cavity with resonantly driven J= ½↔J= ½ atoms,” Z. Phys.B 49, 263–272 (1982).
[CrossRef]

J. Cooper, R. J. Ballagh, K. Burnett, D. G. Hummer, “On redistribution and the equations for radiative transfer,” As-trophys. J. 260, 299–316 (1982).
[CrossRef]

1981

R. W. Boyd, M. G. Raymer, P. Narum, D. J. Harter, “Four-wave parametric interactions in a strongly driven two-level system,” Phys. Rev. A 24, 411–423 (1981).
[CrossRef]

W. J. Sandle, Alan Gallagher, “Optical bistability by an atomic vapor in a focusing Fabry–Perot cavity,” Phys. Rev. A 24, 2107–2028 (1981).
[CrossRef]

1980

F. P. Mattar, M. C. Newstein, “Adaptive stretching and rezoning as effective computational techniques for two-level Maxwell-Bloch simulation,” Comput. Phys. Commun. 20, 139–163 (1980).
[CrossRef]

C. H. Skinner, P. D. Kleiber, “Observation of anomalous conical emission from laser-excited barium vapor,” Phys. Rev. A 21, 151–156 (1980).
[CrossRef]

1977

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

C. E. Grosch, S. A. Orszag, “Numerical solutions of problems in unbounded regions: coordinate transforms,” J. Comput. Phys. 25, 273–296 (1977).
[CrossRef]

A. C. Tarn, W. Happer, “Long-range interactions between cw self-focused laser beams in atomic vapor,” Phys. Rev. Lett. 38, 278–282 (1977).
[CrossRef]

1976

D. G. McCartan, J. M. Farr, “Collision broadening of the sodium resonance lines by noble gases,” J. Phys. B 9, 985–994 (1976).
[CrossRef]

1975

J. H. Marburger, “Self-focusing: theory,” Prog. Quantum Electron. 4, 35–110 (1975).
[CrossRef]

1974

J. A. Fleck, “A cubic spline method for solving the wave equation of nonlinear optics,” J. Comput. Phys. 16, 324–341 (1974).
[CrossRef]

1972

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

1970

D. Grischkowsky, “Self-focusing of light by potassium vapor,” Phys. Rev. Lett. 24, 866–869 (1970).
[CrossRef]

1966

D. H. Close, C. R. Giuliano, R. W. Hellwarth, L. D. Hess, F. J. McClung, W. G. Wagner, “The self-focusing of light of different polarizations,” IEEE J. Quantum Electron. QE-2, 553–557 (1966).
[CrossRef]

Bachor, H. A.

J. C. Wang, H. A. Bachor, D. E. McClelland, Department of Australian Physics University National Australia Canberra (personal communication, 1989).

Ballagh, R. J.

A. W. McCord, R. J. Ballagh, J. Cooper, “Dispersive self-focusing in atomic media,” J. Opt. Soc. Am. B 5, 1323–1334 (1988).
[CrossRef]

D. E. McClelland, R. J. Ballagh, W. J. Sandle, “Simple analytic approximation to continuous wave on-resonance beam reshaping,” J. Opt. Soc. Am. B 3, 212–218 (1986).
[CrossRef]

M. W. Hamilton, R. J. Ballagh, W. J. Sandle, “Polarization switching in a ring cavity with resonantly driven J= ½↔J= ½ atoms,” Z. Phys.B 49, 263–272 (1982).
[CrossRef]

J. Cooper, R. J. Ballagh, K. Burnett, D. G. Hummer, “On redistribution and the equations for radiative transfer,” As-trophys. J. 260, 299–316 (1982).
[CrossRef]

W. J. Sandle, M. W. Hamilton, R. J. Ballagh, “Polarization switching with J= ½↔J= ½ atoms in a ring cavity,” in Optical Bistability 2, C. M. Bowden, H. M. Gibbs, S. L. McCall, eds. (Plenum, New York, 1984).
[CrossRef]

Boyd, R. W.

R. W. Boyd, M. S. Malcuit, D. J. Gauthier, K. Rzazewski, “Competition between amplified spontaneous emission and four-wave-mixing process,” Phys. Rev. A 35, 1648–1658 (1987).
[CrossRef] [PubMed]

D. J. Harter, R. W. Boyd, “Four-wave mixing resonantly enhanced by ac-Stark-split levels in self-trapped filaments of light,” Phys. Rev. A 29, 739–748 (1984).
[CrossRef]

R. W. Boyd, M. G. Raymer, P. Narum, D. J. Harter, “Four-wave parametric interactions in a strongly driven two-level system,” Phys. Rev. A 24, 411–423 (1981).
[CrossRef]

Burnett, K.

J. Cooper, R. J. Ballagh, K. Burnett, D. G. Hummer, “On redistribution and the equations for radiative transfer,” As-trophys. J. 260, 299–316 (1982).
[CrossRef]

Close, D. H.

D. H. Close, C. R. Giuliano, R. W. Hellwarth, L. D. Hess, F. J. McClung, W. G. Wagner, “The self-focusing of light of different polarizations,” IEEE J. Quantum Electron. QE-2, 553–557 (1966).
[CrossRef]

Cooper, J.

A. W. McCord, R. J. Ballagh, J. Cooper, “Dispersive self-focusing in atomic media,” J. Opt. Soc. Am. B 5, 1323–1334 (1988).
[CrossRef]

J. Cooper, R. J. Ballagh, K. Burnett, D. G. Hummer, “On redistribution and the equations for radiative transfer,” As-trophys. J. 260, 299–316 (1982).
[CrossRef]

Corney, A.

A. Corney, Atomic and Laser Spectroscopy (Clarendon, Oxford, 1977).

Drummond, P. D.

P. D. Drummond, “Central partial difference propagation algorithm,” Comput. Phys. Commun. 29, 211–225 (1983).
[CrossRef]

Farr, J. M.

D. G. McCartan, J. M. Farr, “Collision broadening of the sodium resonance lines by noble gases,” J. Phys. B 9, 985–994 (1976).
[CrossRef]

Feit, M. D.

Fleck, J. A.

M. D. Feit, J. A. Fleck, “Beam nonparaxiality, filament formation, and beam breakup in the self-focusing of optical beams,” J. Opt. Soc. Am. B 5, 633–640 (1988).
[CrossRef]

J. A. Fleck, “A cubic spline method for solving the wave equation of nonlinear optics,” J. Comput. Phys. 16, 324–341 (1974).
[CrossRef]

Gallagher, Alan

W. J. Sandle, Alan Gallagher, “Optical bistability by an atomic vapor in a focusing Fabry–Perot cavity,” Phys. Rev. A 24, 2107–2028 (1981).
[CrossRef]

Gauthier, D. J.

R. W. Boyd, M. S. Malcuit, D. J. Gauthier, K. Rzazewski, “Competition between amplified spontaneous emission and four-wave-mixing process,” Phys. Rev. A 35, 1648–1658 (1987).
[CrossRef] [PubMed]

Giuliano, C. R.

D. H. Close, C. R. Giuliano, R. W. Hellwarth, L. D. Hess, F. J. McClung, W. G. Wagner, “The self-focusing of light of different polarizations,” IEEE J. Quantum Electron. QE-2, 553–557 (1966).
[CrossRef]

Grischkowsky, D.

D. Grischkowsky, “Self-focusing of light by potassium vapor,” Phys. Rev. Lett. 24, 866–869 (1970).
[CrossRef]

Grosch, C. E.

C. E. Grosch, S. A. Orszag, “Numerical solutions of problems in unbounded regions: coordinate transforms,” J. Comput. Phys. 25, 273–296 (1977).
[CrossRef]

Hamilton, M. W.

M. W. Hamilton, R. J. Ballagh, W. J. Sandle, “Polarization switching in a ring cavity with resonantly driven J= ½↔J= ½ atoms,” Z. Phys.B 49, 263–272 (1982).
[CrossRef]

W. J. Sandle, M. W. Hamilton, R. J. Ballagh, “Polarization switching with J= ½↔J= ½ atoms in a ring cavity,” in Optical Bistability 2, C. M. Bowden, H. M. Gibbs, S. L. McCall, eds. (Plenum, New York, 1984).
[CrossRef]

Happer, W.

A. C. Tarn, W. Happer, “Long-range interactions between cw self-focused laser beams in atomic vapor,” Phys. Rev. Lett. 38, 278–282 (1977).
[CrossRef]

Happer, William

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

Harter, D. J.

D. J. Harter, R. W. Boyd, “Four-wave mixing resonantly enhanced by ac-Stark-split levels in self-trapped filaments of light,” Phys. Rev. A 29, 739–748 (1984).
[CrossRef]

R. W. Boyd, M. G. Raymer, P. Narum, D. J. Harter, “Four-wave parametric interactions in a strongly driven two-level system,” Phys. Rev. A 24, 411–423 (1981).
[CrossRef]

Hellwarth, R. W.

D. H. Close, C. R. Giuliano, R. W. Hellwarth, L. D. Hess, F. J. McClung, W. G. Wagner, “The self-focusing of light of different polarizations,” IEEE J. Quantum Electron. QE-2, 553–557 (1966).
[CrossRef]

Hess, L. D.

D. H. Close, C. R. Giuliano, R. W. Hellwarth, L. D. Hess, F. J. McClung, W. G. Wagner, “The self-focusing of light of different polarizations,” IEEE J. Quantum Electron. QE-2, 553–557 (1966).
[CrossRef]

Hummer, D. G.

J. Cooper, R. J. Ballagh, K. Burnett, D. G. Hummer, “On redistribution and the equations for radiative transfer,” As-trophys. J. 260, 299–316 (1982).
[CrossRef]

Jin, G. X.

G. X. Jin, J. M. Yuan, L. M. Narducci, Y. S. Liu, E. J. Seibert, “Theoretical and experimental studies of conical Stokes emission,” Opt. Commun. 68, 379–384 (1988).
[CrossRef]

Kleiber, P. D.

C. H. Skinner, P. D. Kleiber, “Observation of anomalous conical emission from laser-excited barium vapor,” Phys. Rev. A 21, 151–156 (1980).
[CrossRef]

Le Berre, M.

M. Le Berre, E. Ressayre, A. Tallet, “Resonant self-focusing of a cw intense light beam,” Phys. Rev. A 29, 2669–2676 (1984).
[CrossRef]

Liu, Y. S.

G. X. Jin, J. M. Yuan, L. M. Narducci, Y. S. Liu, E. J. Seibert, “Theoretical and experimental studies of conical Stokes emission,” Opt. Commun. 68, 379–384 (1988).
[CrossRef]

Malcuit, M. S.

R. W. Boyd, M. S. Malcuit, D. J. Gauthier, K. Rzazewski, “Competition between amplified spontaneous emission and four-wave-mixing process,” Phys. Rev. A 35, 1648–1658 (1987).
[CrossRef] [PubMed]

Marburger, J. H.

J. H. Marburger, “Self-focusing: theory,” Prog. Quantum Electron. 4, 35–110 (1975).
[CrossRef]

Mattar, F. P.

F. P. Mattar, M. C. Newstein, “Adaptive stretching and rezoning as effective computational techniques for two-level Maxwell-Bloch simulation,” Comput. Phys. Commun. 20, 139–163 (1980).
[CrossRef]

McCartan, D. G.

D. G. McCartan, J. M. Farr, “Collision broadening of the sodium resonance lines by noble gases,” J. Phys. B 9, 985–994 (1976).
[CrossRef]

McClelland, D. E.

D. E. McClelland, R. J. Ballagh, W. J. Sandle, “Simple analytic approximation to continuous wave on-resonance beam reshaping,” J. Opt. Soc. Am. B 3, 212–218 (1986).
[CrossRef]

J. C. Wang, H. A. Bachor, D. E. McClelland, Department of Australian Physics University National Australia Canberra (personal communication, 1989).

D. E. McClelland, “Magnetic field modification to laser propagation in a J = ½↔J= ½ system,” Ph.D. dissertation (University of Otago, Dunedin, New Zealand, 1987).

McClung, F. J.

D. H. Close, C. R. Giuliano, R. W. Hellwarth, L. D. Hess, F. J. McClung, W. G. Wagner, “The self-focusing of light of different polarizations,” IEEE J. Quantum Electron. QE-2, 553–557 (1966).
[CrossRef]

McCord, A. W.

A. W. McCord, R. J. Ballagh, J. Cooper, “Dispersive self-focusing in atomic media,” J. Opt. Soc. Am. B 5, 1323–1334 (1988).
[CrossRef]

A. W. McCord, “Laser propagation with interacting radiation modes,” PhD. dissertation (University of Otago, Dunedin, New Zealand, 1988).

Narducci, L. M.

G. X. Jin, J. M. Yuan, L. M. Narducci, Y. S. Liu, E. J. Seibert, “Theoretical and experimental studies of conical Stokes emission,” Opt. Commun. 68, 379–384 (1988).
[CrossRef]

Narum, P.

R. W. Boyd, M. G. Raymer, P. Narum, D. J. Harter, “Four-wave parametric interactions in a strongly driven two-level system,” Phys. Rev. A 24, 411–423 (1981).
[CrossRef]

Newstein, M. C.

F. P. Mattar, M. C. Newstein, “Adaptive stretching and rezoning as effective computational techniques for two-level Maxwell-Bloch simulation,” Comput. Phys. Commun. 20, 139–163 (1980).
[CrossRef]

Omont, A.

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

Orszag, S. A.

C. E. Grosch, S. A. Orszag, “Numerical solutions of problems in unbounded regions: coordinate transforms,” J. Comput. Phys. 25, 273–296 (1977).
[CrossRef]

Raymer, M. G.

R. W. Boyd, M. G. Raymer, P. Narum, D. J. Harter, “Four-wave parametric interactions in a strongly driven two-level system,” Phys. Rev. A 24, 411–423 (1981).
[CrossRef]

Ressayre, E.

M. Le Berre, E. Ressayre, A. Tallet, “Resonant self-focusing of a cw intense light beam,” Phys. Rev. A 29, 2669–2676 (1984).
[CrossRef]

Rosenbluh, M.

Rzazewski, K.

R. W. Boyd, M. S. Malcuit, D. J. Gauthier, K. Rzazewski, “Competition between amplified spontaneous emission and four-wave-mixing process,” Phys. Rev. A 35, 1648–1658 (1987).
[CrossRef] [PubMed]

Sandle, W. J.

D. E. McClelland, R. J. Ballagh, W. J. Sandle, “Simple analytic approximation to continuous wave on-resonance beam reshaping,” J. Opt. Soc. Am. B 3, 212–218 (1986).
[CrossRef]

M. W. Hamilton, R. J. Ballagh, W. J. Sandle, “Polarization switching in a ring cavity with resonantly driven J= ½↔J= ½ atoms,” Z. Phys.B 49, 263–272 (1982).
[CrossRef]

W. J. Sandle, Alan Gallagher, “Optical bistability by an atomic vapor in a focusing Fabry–Perot cavity,” Phys. Rev. A 24, 2107–2028 (1981).
[CrossRef]

W. J. Sandle, M. W. Hamilton, R. J. Ballagh, “Polarization switching with J= ½↔J= ½ atoms in a ring cavity,” in Optical Bistability 2, C. M. Bowden, H. M. Gibbs, S. L. McCall, eds. (Plenum, New York, 1984).
[CrossRef]

Seibert, E. J.

G. X. Jin, J. M. Yuan, L. M. Narducci, Y. S. Liu, E. J. Seibert, “Theoretical and experimental studies of conical Stokes emission,” Opt. Commun. 68, 379–384 (1988).
[CrossRef]

Shevy, Y.

Skinner, C. H.

C. H. Skinner, P. D. Kleiber, “Observation of anomalous conical emission from laser-excited barium vapor,” Phys. Rev. A 21, 151–156 (1980).
[CrossRef]

Tallet, A.

M. Le Berre, E. Ressayre, A. Tallet, “Resonant self-focusing of a cw intense light beam,” Phys. Rev. A 29, 2669–2676 (1984).
[CrossRef]

Tarn, A. C.

A. C. Tarn, W. Happer, “Long-range interactions between cw self-focused laser beams in atomic vapor,” Phys. Rev. Lett. 38, 278–282 (1977).
[CrossRef]

Wagner, W. G.

D. H. Close, C. R. Giuliano, R. W. Hellwarth, L. D. Hess, F. J. McClung, W. G. Wagner, “The self-focusing of light of different polarizations,” IEEE J. Quantum Electron. QE-2, 553–557 (1966).
[CrossRef]

Wang, J. C.

J. C. Wang, H. A. Bachor, D. E. McClelland, Department of Australian Physics University National Australia Canberra (personal communication, 1989).

Yariv, A.

A. Yariv, Quantum Electronics, 2nd ed. (Wiley, New York, 1975).

Yuan, J. M.

G. X. Jin, J. M. Yuan, L. M. Narducci, Y. S. Liu, E. J. Seibert, “Theoretical and experimental studies of conical Stokes emission,” Opt. Commun. 68, 379–384 (1988).
[CrossRef]

As-trophys. J.

J. Cooper, R. J. Ballagh, K. Burnett, D. G. Hummer, “On redistribution and the equations for radiative transfer,” As-trophys. J. 260, 299–316 (1982).
[CrossRef]

Comput. Phys. Commun.

P. D. Drummond, “Central partial difference propagation algorithm,” Comput. Phys. Commun. 29, 211–225 (1983).
[CrossRef]

F. P. Mattar, M. C. Newstein, “Adaptive stretching and rezoning as effective computational techniques for two-level Maxwell-Bloch simulation,” Comput. Phys. Commun. 20, 139–163 (1980).
[CrossRef]

IEEE J. Quantum Electron.

D. H. Close, C. R. Giuliano, R. W. Hellwarth, L. D. Hess, F. J. McClung, W. G. Wagner, “The self-focusing of light of different polarizations,” IEEE J. Quantum Electron. QE-2, 553–557 (1966).
[CrossRef]

J. Comput. Phys.

C. E. Grosch, S. A. Orszag, “Numerical solutions of problems in unbounded regions: coordinate transforms,” J. Comput. Phys. 25, 273–296 (1977).
[CrossRef]

J. A. Fleck, “A cubic spline method for solving the wave equation of nonlinear optics,” J. Comput. Phys. 16, 324–341 (1974).
[CrossRef]

J. Opt. Soc. Am. B

J. Phys. B

D. G. McCartan, J. M. Farr, “Collision broadening of the sodium resonance lines by noble gases,” J. Phys. B 9, 985–994 (1976).
[CrossRef]

Opt. Commun.

G. X. Jin, J. M. Yuan, L. M. Narducci, Y. S. Liu, E. J. Seibert, “Theoretical and experimental studies of conical Stokes emission,” Opt. Commun. 68, 379–384 (1988).
[CrossRef]

Phys. Rev. A

R. W. Boyd, M. S. Malcuit, D. J. Gauthier, K. Rzazewski, “Competition between amplified spontaneous emission and four-wave-mixing process,” Phys. Rev. A 35, 1648–1658 (1987).
[CrossRef] [PubMed]

C. H. Skinner, P. D. Kleiber, “Observation of anomalous conical emission from laser-excited barium vapor,” Phys. Rev. A 21, 151–156 (1980).
[CrossRef]

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[CrossRef]

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Corrections for the nonisotropic nature of collisional relaxation have been discussed by Cooper et al.20 and are very small in the regime where the product of a Rabi frequency and a strong collision duration is small.

It is clear from the analytic models of Refs. 4 and 5 that the amplitude and phase behavior at the beam edges is critical to a correct description of propagation in saturable media. An efficient numerical method must therefore adequately resolve these off-axis spatial regions but can afford to have fewer computational points in the highly saturated beam core. This was achieved in the present research by implementation of a five-parameter transverse-map function. Two parameters were used to specify an algebraic rezoning to infinity. The remaining three parameters specified an additional tan−1region of high resolution. The position, width, and number of points in this additional region are chosen automatically as the solution develops and shifts to follow the contracting (or expanding) encoding region. Simpler dynamic rezoning techniques have been used previously in Maxwell–Bloch calculations (e.g., see Ref. 25).

A. Yariv, Quantum Electronics, 2nd ed. (Wiley, New York, 1975).

A. W. McCord, “Laser propagation with interacting radiation modes,” PhD. dissertation (University of Otago, Dunedin, New Zealand, 1988).

D. E. McClelland, “Magnetic field modification to laser propagation in a J = ½↔J= ½ system,” Ph.D. dissertation (University of Otago, Dunedin, New Zealand, 1987).

We have verified this self-guiding nature of the final field configuration numerically by turning off the effect of the medium after the pure circularly polarized annulus and beam have formed. As expected, the σ+annulus collapses rapidly back on axis, and the σ−beam spreads diffractively.

J. C. Wang, H. A. Bachor, D. E. McClelland, Department of Australian Physics University National Australia Canberra (personal communication, 1989).

By beam reshaping we mean any self-induced change in the transverse intensity profile of a beam. This definition encompasses self-focusing, self-trapping, self-defocusing, self-bending, filamentation, and other more complicated phenomena.

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Figures (10)

Fig. 1
Fig. 1

σ+ and σ transitions (solid arrows) for the J = ½↔J = ½ atom are coupled by spontaneous decay (shaded arrows) and collisional mixing of Zeeman level ⇔.

Fig. 2
Fig. 2

The atomic response |η++| for Δ = 0 and β = 103 as a function of I+ and I. The diagonal line on the surface indicates the response for linearly polarized light.

Fig. 3
Fig. 3

Spatial distribution of (a) I+ and (b) I for the diffraction-free propagation of an initial elliptically polarized Gaussian beam with I+00 = 36,I−00 = 25, F± = 60, and β = 103.

Fig. 4
Fig. 4

Intensities I+(z) (solid line) and I(z) (dashed line) for the plane-wave propagation of an elliptically polarized field with initial intensities I+(0) = 36 and I(0) = 25. Here β = 103, Δ+ = Δ = 0, and the dotted line is the difference I+(z) − I(z).

Fig. 5
Fig. 5

Spatial intensity profiles (a) I+(ρ, z) and (b) I(ρ, z) for the full propagation of a resonant initially Gaussian beam of uniform ellipticity with I+00 = 36, I−00 = 25, F± = 60, and β = 103. The insets show vertically expanded views. Vertical scales for pairs of σ+ and σ plots are the same.

Fig. 6
Fig. 6

Spatial distribution of (a) |M+| (ρ, z) and (b) |M| (ρ, z) for the full diffractive solution of Fig. 5.

Fig. 7
Fig. 7

Spatial intensity profiles (a) I+(ρ, z) and (b) I(ρ, z) for a Gaussian beam with the same initial intensity profiles and the same medium as Fig. 5 but with nonzero detuning Δ+ = Δ = 2.

Fig. 8
Fig. 8

Spatial intensity profiles (a) I(ρ, z) of a pure σ Gaussian beam propagating alone in the medium and (b) I+(ρ, z) of a σ+ annulus propagating alone. The parameters are F± = 30, Δ± = 0, A−00 = 2, A+ann = 1.5, ρann = 2.5, and H = 20.

Fig. 9
Fig. 9

Spatial intensity profiles (a) I(ρ, z) and (b) I+(ρ, z) of the initially Gaussian σ beam and the σ+ annulus propagating together in the medium. The parameters are the same as in Fig. 8.

Fig. 10
Fig. 10

Spatial intensity profiles (a) I+(ρ, z) and (b) I(ρ, z) for the coupled propagation of offset σ+ and σ annular beams. Here F± = 200, β = 103, Δ± = 0, A+ann = A−ann = 1.5, and ρann+ = 2.50, ρann− = 2.49, and H = 20.

Equations (37)

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E ( r , t ) = E sat [ + ( r ) ê 1 * + ( r ) ê + 1 * ] exp [ i ω l ( z / c t ) ] + c . c . ,
Δ ± ω l ω ± Γ l u .
P ( r , t ) = N d u l 3 q ê q [ ( ) q ρ q 1 ( l u , r t ) * ρ q 1 ( l u , r t ) ] ,
P ( r , t ) = [ P + ( r ) ê 1 * + P ( r ) ê + 1 * ] exp [ i ω l ( z / c t ) ] + c . c . ,
P + ( r ) = i c 0 α 0 ω l η ± E sat ± ( r ) ,
α 0 = N π γ Γ l u c 2 ω u l 2 ,
E sat = d u l 6 Γ l u γ .
η ± = ( 1 + i Δ ± 1 + Δ ± 2 ) [ 1 + 8 I ± + 2 ( β 2 ) I ± I 1 + 4 β I ] 1 ,
I ± ± * ( r ) ± ( r ) 1 + Δ ± 2 ,
η ± [ 1 + 8 I ± + 2 ( β 2 ) I ± I 1 + 4 β I ] 1 ,
η ± = 1 + 4 β I D ,
D = 1 + ( 2 β + 4 ) ( I + + I ) + 32 β I + I .
β γ Γ l Γ u ( Γ l + Γ u γ / 3 )
η + I | I + = ( 2 β 4 ) ( 1 + 4 β I + ) D 2 .
η ± = 1 1 + 4 ( I + + I ) ,
I β + 2 8 β ,
η + ( 2 β β + 2 ) 1 1 + 16 β β + 2 I + ( σ strong ) ,
η + 2 1 1 + 16 I + .
η ± 1 8 I ± ( σ + , σ strong ) .
I 1 4 β + 8 ,
η + 1 1 + ( 2 β + 4 ) I + ( σ weak ) ,
η ± = 1 1 + 8 I ± ,
± ( ρ , z ) z = i t 2 ± ( ρ , z ) + M ± ( ρ , z ) ,
M ± ( ρ , z ) 2 F ± ( 1 + i Δ ± ) η ± ( ρ , z ) ± ( ρ , z ) .
t 2 = 2 ρ 2 + 1 ρ ρ .
F ± α 0 z R 1 + Δ ± 2 ,
= I + I I + + I ,
I ± 00 = | A ± 00 | 2 1 + Δ ± 2 ,
D ( z ) = I + ( z ) I ( z ) ,
d D ( z ) d z 4 F D D ( z ) .
Δ > 4 F ln 16 I + 1
Δ > 8 F ln ( 2 β + 4 ) I + 1
( ρ , z = 0 ) = A 00 exp ( ρ 2 )
+ ( ρ , z = 0 ) = A + ann exp [ H ( ρ ρ ann ) 2 ] ,
z max ρ ann 8 H
+ ( ρ , z = 0 ) = A + ann exp [ H ( ρ ρ ann + ) 2 ] , ( ρ , z = 0 ) = A ann exp { H ( ρ ρ ann ) 2 } ,
Γ l eff Γ l + ω B 2 Γ l ,

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