Abstract

We study theoretically the dynamic behavior of a J=1J=0 laser (J is the angular momentum quantum number) optically pumped by means of a linearly polarized coherent field (coupled to an adjacent transition J=0J=1) when the pump field polarization is rotated at a constant angular velocity Ω and the laser field polarization either is fixed by the cavity or is free (isotropic cavity). Because of a strong pump-induced gain anisotropy, the dynamic behavior is completely different in each case. In the case of fixed laser field polarization, rich amplitude dynamics, which depend on the pump field strength, are found. At slow modulation frequencies (with respect to the molecular and the cavity relaxation rates), the system does not always follow the sequence of stationary and dynamic solutions that correspond to the autonomous laser as a function of the relative orientation angle between the polarizations of the pump and the laser fields. Phenomena such as delayed switching and suppression of chaos and stabilization of unstable steady states are found. In the case of an isotropic cavity, the pump field vector rotation is transferred to the laser field vector and amplitude unstable regimes are also strongly inhibited.

© 1999 Optical Society of America

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References

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  1. Y. I. Khanin, Principles of Laser Dynamics (Elsevier, Amsterdam, 1995), Chap. 6, pp. 245–305.
  2. C. O. Weiss and R. Vilaseca, Dynamics of Lasers (VCH, Deerfield Beach, Fla., 1991), Chap. 7, pp. 137–180.
  3. See, for instance, N. B. Abraham and G. M. Stephan, eds., Polarization Effects in Lasers and Spectroscopy, special issue of Quantum Semiclassic. Opt. 10, No. 1 (1998).
  4. See, for instance, D. Lenstra, “On the theory of polarization effects in gas lasers,” Phys. Rep. 59, 299 (1980).
    [CrossRef]
  5. P. J. Manson, D. M. Warrington, N. L. Moise, and W. J. Sandle, “Observation of complex polarization dynamics in the output of a Ne Raman laser,” Quantum Semiclassic. Opt. 10, 157 (1998).
    [CrossRef]
  6. A. Kul’minskii, R. Vilaseca, and R. Corbalán, “Full polarization chaos in a pump-polarization modulated isotropic cavity laser,” Opt. Lett. 20, 2390 (1995).
    [CrossRef] [PubMed]
  7. V. N. Parygin and A. S. Lipatov, “Internal modulation of the polarization of a gas laser,” Radio Eng. Electron. Phys. 20, 66 (1975).
  8. A. M. Kul’minskii, R. Vilaseca, and R. Corbalán, “Dynamics of a gain anisotropic optically pumped laser with arbitrary angle between linear polarizations of pump and laser fields,” J. Mod. Opt. 42, 2295 (1995).
    [CrossRef]
  9. E. Roldán, G. J. de Valcárcel, R. Vilaseca, and R. Corbalán, “Polarization sensitive population trapping in an optically pumped laser,” Phys. Rev. A 49, 1487 (1994).
    [CrossRef]
  10. C. Serrat, A. Kul’minskii, R. Vilaseca, and R. Corbalán, “Polarization chaos in an optically pumped laser,” Opt. Lett. 20, 1353 (1995).
    [CrossRef] [PubMed]
  11. C. Serrat, R. Vilaseca, G. J. De Valcárcel, and E. Roldán, “Polarization switching in an anisotropic cavity coherently pumped J=1→J=0 laser,” Phys. Rev. A 56, 2327 (1997).
    [CrossRef]
  12. M. Arjona, R. Corbalán, F. Laguarta, J. Pujol, and R. Vilaseca, “Influence of light polarization on the dynamics of optically pumped lasers,” Phys. Rev. A 41, 6559 (1990); R. Corbalán, R. Vilaseca, M. Arjona, J. Pujol, E. Roldán, and G. J. de Valcárcel, “Dynamics of coherently pumped lasers with linearly polarized pump and generated fields,” Phys. Rev. A 48, 1483 (1993).
    [CrossRef] [PubMed]
  13. N. G. Douglas, Millimetre and Submillimetre Wavelength Lasers (Springer-Verlag, Berlin, 1989).
  14. W. Klische and C. O. Weiss, “Instabilities and routes to chaos in a homogeneously broadened 1- and 2-mode ring laser,” Phys. Rev. A 31, 4049 (1985); E. H. M. Hogenboom, W. Klische, C. O. Weiss, and A. Godone, “Instabilities of a homogeneously broadened laser,” Phys. Rev. Lett. 55, 2571 (1985).
    [CrossRef] [PubMed]
  15. C. O. Weiss and J. Brock, “Evidence for Lorenz-type chaos in a laser,” Phys. Rev. Lett. 57, 2804 (1986).
    [CrossRef] [PubMed]
  16. M. Arjona, J. Pujol, and R. Corbalán, “Type-I intermittency in a four-level coherently pumped laser,” Phys. Rev. A 50, 871 (1994).
    [CrossRef] [PubMed]
  17. J. Pujol, M. Arjona, and R. Corbalán, “Type-III intermittency in a four-level coherently pumped laser,” Phys. Rev. A 48, 2251 (1993).
    [CrossRef] [PubMed]
  18. C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg, Atom-Photon Interactions (Wiley, New York, 1992), Chap. 6.
  19. P. Mandel, Theoretical Problems in Cavity Nonlinear Optics (Cambridge U. Press, Cambridge, England, 1997), pp. 15–16.
  20. R. Vilaseca, A. Kul’minskii, and R. Corbalán, “Tracking unstable steady states by large periodic modulation of a control parameter in a nonlinear system,” Phys. Rev. E 54, 82 (1996).
    [CrossRef]
  21. R. Dykstra, A. Rayner, D. Y. Tang, and N. R. Heckenberg, “Experimentally tracking unstable steady states by large periodic modulation,” Phys. Rev. E 57, 397 (1998).
    [CrossRef]
  22. A. N. Pisarchik, V. N. Chizhevsky, R. Corbalán, and R. Vilaseca, “Experimental control of nonlinear dynamics by slow parametric modulation,” Phys. Rev. E 55, 2455 (1998).
    [CrossRef]
  23. Z. Qu, G. Hu, G. Yang, and G. Qin, “Phase effect in taming nonautonomous chaos by weak harmonic perturbations,” Phys. Rev. Lett. 74, 1736 (1995).
    [CrossRef] [PubMed]

1998

R. Dykstra, A. Rayner, D. Y. Tang, and N. R. Heckenberg, “Experimentally tracking unstable steady states by large periodic modulation,” Phys. Rev. E 57, 397 (1998).
[CrossRef]

A. N. Pisarchik, V. N. Chizhevsky, R. Corbalán, and R. Vilaseca, “Experimental control of nonlinear dynamics by slow parametric modulation,” Phys. Rev. E 55, 2455 (1998).
[CrossRef]

P. J. Manson, D. M. Warrington, N. L. Moise, and W. J. Sandle, “Observation of complex polarization dynamics in the output of a Ne Raman laser,” Quantum Semiclassic. Opt. 10, 157 (1998).
[CrossRef]

1997

C. Serrat, R. Vilaseca, G. J. De Valcárcel, and E. Roldán, “Polarization switching in an anisotropic cavity coherently pumped J=1→J=0 laser,” Phys. Rev. A 56, 2327 (1997).
[CrossRef]

1996

R. Vilaseca, A. Kul’minskii, and R. Corbalán, “Tracking unstable steady states by large periodic modulation of a control parameter in a nonlinear system,” Phys. Rev. E 54, 82 (1996).
[CrossRef]

1995

C. Serrat, A. Kul’minskii, R. Vilaseca, and R. Corbalán, “Polarization chaos in an optically pumped laser,” Opt. Lett. 20, 1353 (1995).
[CrossRef] [PubMed]

A. Kul’minskii, R. Vilaseca, and R. Corbalán, “Full polarization chaos in a pump-polarization modulated isotropic cavity laser,” Opt. Lett. 20, 2390 (1995).
[CrossRef] [PubMed]

A. M. Kul’minskii, R. Vilaseca, and R. Corbalán, “Dynamics of a gain anisotropic optically pumped laser with arbitrary angle between linear polarizations of pump and laser fields,” J. Mod. Opt. 42, 2295 (1995).
[CrossRef]

Z. Qu, G. Hu, G. Yang, and G. Qin, “Phase effect in taming nonautonomous chaos by weak harmonic perturbations,” Phys. Rev. Lett. 74, 1736 (1995).
[CrossRef] [PubMed]

1994

M. Arjona, J. Pujol, and R. Corbalán, “Type-I intermittency in a four-level coherently pumped laser,” Phys. Rev. A 50, 871 (1994).
[CrossRef] [PubMed]

E. Roldán, G. J. de Valcárcel, R. Vilaseca, and R. Corbalán, “Polarization sensitive population trapping in an optically pumped laser,” Phys. Rev. A 49, 1487 (1994).
[CrossRef]

1993

J. Pujol, M. Arjona, and R. Corbalán, “Type-III intermittency in a four-level coherently pumped laser,” Phys. Rev. A 48, 2251 (1993).
[CrossRef] [PubMed]

1986

C. O. Weiss and J. Brock, “Evidence for Lorenz-type chaos in a laser,” Phys. Rev. Lett. 57, 2804 (1986).
[CrossRef] [PubMed]

1980

See, for instance, D. Lenstra, “On the theory of polarization effects in gas lasers,” Phys. Rep. 59, 299 (1980).
[CrossRef]

1975

V. N. Parygin and A. S. Lipatov, “Internal modulation of the polarization of a gas laser,” Radio Eng. Electron. Phys. 20, 66 (1975).

Arjona, M.

M. Arjona, J. Pujol, and R. Corbalán, “Type-I intermittency in a four-level coherently pumped laser,” Phys. Rev. A 50, 871 (1994).
[CrossRef] [PubMed]

J. Pujol, M. Arjona, and R. Corbalán, “Type-III intermittency in a four-level coherently pumped laser,” Phys. Rev. A 48, 2251 (1993).
[CrossRef] [PubMed]

Brock, J.

C. O. Weiss and J. Brock, “Evidence for Lorenz-type chaos in a laser,” Phys. Rev. Lett. 57, 2804 (1986).
[CrossRef] [PubMed]

Chizhevsky, V. N.

A. N. Pisarchik, V. N. Chizhevsky, R. Corbalán, and R. Vilaseca, “Experimental control of nonlinear dynamics by slow parametric modulation,” Phys. Rev. E 55, 2455 (1998).
[CrossRef]

Corbalán, R.

A. N. Pisarchik, V. N. Chizhevsky, R. Corbalán, and R. Vilaseca, “Experimental control of nonlinear dynamics by slow parametric modulation,” Phys. Rev. E 55, 2455 (1998).
[CrossRef]

R. Vilaseca, A. Kul’minskii, and R. Corbalán, “Tracking unstable steady states by large periodic modulation of a control parameter in a nonlinear system,” Phys. Rev. E 54, 82 (1996).
[CrossRef]

A. M. Kul’minskii, R. Vilaseca, and R. Corbalán, “Dynamics of a gain anisotropic optically pumped laser with arbitrary angle between linear polarizations of pump and laser fields,” J. Mod. Opt. 42, 2295 (1995).
[CrossRef]

C. Serrat, A. Kul’minskii, R. Vilaseca, and R. Corbalán, “Polarization chaos in an optically pumped laser,” Opt. Lett. 20, 1353 (1995).
[CrossRef] [PubMed]

A. Kul’minskii, R. Vilaseca, and R. Corbalán, “Full polarization chaos in a pump-polarization modulated isotropic cavity laser,” Opt. Lett. 20, 2390 (1995).
[CrossRef] [PubMed]

E. Roldán, G. J. de Valcárcel, R. Vilaseca, and R. Corbalán, “Polarization sensitive population trapping in an optically pumped laser,” Phys. Rev. A 49, 1487 (1994).
[CrossRef]

M. Arjona, J. Pujol, and R. Corbalán, “Type-I intermittency in a four-level coherently pumped laser,” Phys. Rev. A 50, 871 (1994).
[CrossRef] [PubMed]

J. Pujol, M. Arjona, and R. Corbalán, “Type-III intermittency in a four-level coherently pumped laser,” Phys. Rev. A 48, 2251 (1993).
[CrossRef] [PubMed]

De Valcárcel, G. J.

C. Serrat, R. Vilaseca, G. J. De Valcárcel, and E. Roldán, “Polarization switching in an anisotropic cavity coherently pumped J=1→J=0 laser,” Phys. Rev. A 56, 2327 (1997).
[CrossRef]

E. Roldán, G. J. de Valcárcel, R. Vilaseca, and R. Corbalán, “Polarization sensitive population trapping in an optically pumped laser,” Phys. Rev. A 49, 1487 (1994).
[CrossRef]

Dykstra, R.

R. Dykstra, A. Rayner, D. Y. Tang, and N. R. Heckenberg, “Experimentally tracking unstable steady states by large periodic modulation,” Phys. Rev. E 57, 397 (1998).
[CrossRef]

Heckenberg, N. R.

R. Dykstra, A. Rayner, D. Y. Tang, and N. R. Heckenberg, “Experimentally tracking unstable steady states by large periodic modulation,” Phys. Rev. E 57, 397 (1998).
[CrossRef]

Hu, G.

Z. Qu, G. Hu, G. Yang, and G. Qin, “Phase effect in taming nonautonomous chaos by weak harmonic perturbations,” Phys. Rev. Lett. 74, 1736 (1995).
[CrossRef] [PubMed]

Kul’minskii, A.

Kul’minskii, A. M.

A. M. Kul’minskii, R. Vilaseca, and R. Corbalán, “Dynamics of a gain anisotropic optically pumped laser with arbitrary angle between linear polarizations of pump and laser fields,” J. Mod. Opt. 42, 2295 (1995).
[CrossRef]

Lenstra, D.

See, for instance, D. Lenstra, “On the theory of polarization effects in gas lasers,” Phys. Rep. 59, 299 (1980).
[CrossRef]

Lipatov, A. S.

V. N. Parygin and A. S. Lipatov, “Internal modulation of the polarization of a gas laser,” Radio Eng. Electron. Phys. 20, 66 (1975).

Manson, P. J.

P. J. Manson, D. M. Warrington, N. L. Moise, and W. J. Sandle, “Observation of complex polarization dynamics in the output of a Ne Raman laser,” Quantum Semiclassic. Opt. 10, 157 (1998).
[CrossRef]

Moise, N. L.

P. J. Manson, D. M. Warrington, N. L. Moise, and W. J. Sandle, “Observation of complex polarization dynamics in the output of a Ne Raman laser,” Quantum Semiclassic. Opt. 10, 157 (1998).
[CrossRef]

Parygin, V. N.

V. N. Parygin and A. S. Lipatov, “Internal modulation of the polarization of a gas laser,” Radio Eng. Electron. Phys. 20, 66 (1975).

Pisarchik, A. N.

A. N. Pisarchik, V. N. Chizhevsky, R. Corbalán, and R. Vilaseca, “Experimental control of nonlinear dynamics by slow parametric modulation,” Phys. Rev. E 55, 2455 (1998).
[CrossRef]

Pujol, J.

M. Arjona, J. Pujol, and R. Corbalán, “Type-I intermittency in a four-level coherently pumped laser,” Phys. Rev. A 50, 871 (1994).
[CrossRef] [PubMed]

J. Pujol, M. Arjona, and R. Corbalán, “Type-III intermittency in a four-level coherently pumped laser,” Phys. Rev. A 48, 2251 (1993).
[CrossRef] [PubMed]

Qin, G.

Z. Qu, G. Hu, G. Yang, and G. Qin, “Phase effect in taming nonautonomous chaos by weak harmonic perturbations,” Phys. Rev. Lett. 74, 1736 (1995).
[CrossRef] [PubMed]

Qu, Z.

Z. Qu, G. Hu, G. Yang, and G. Qin, “Phase effect in taming nonautonomous chaos by weak harmonic perturbations,” Phys. Rev. Lett. 74, 1736 (1995).
[CrossRef] [PubMed]

Rayner, A.

R. Dykstra, A. Rayner, D. Y. Tang, and N. R. Heckenberg, “Experimentally tracking unstable steady states by large periodic modulation,” Phys. Rev. E 57, 397 (1998).
[CrossRef]

Roldán, E.

C. Serrat, R. Vilaseca, G. J. De Valcárcel, and E. Roldán, “Polarization switching in an anisotropic cavity coherently pumped J=1→J=0 laser,” Phys. Rev. A 56, 2327 (1997).
[CrossRef]

E. Roldán, G. J. de Valcárcel, R. Vilaseca, and R. Corbalán, “Polarization sensitive population trapping in an optically pumped laser,” Phys. Rev. A 49, 1487 (1994).
[CrossRef]

Sandle, W. J.

P. J. Manson, D. M. Warrington, N. L. Moise, and W. J. Sandle, “Observation of complex polarization dynamics in the output of a Ne Raman laser,” Quantum Semiclassic. Opt. 10, 157 (1998).
[CrossRef]

Serrat, C.

C. Serrat, R. Vilaseca, G. J. De Valcárcel, and E. Roldán, “Polarization switching in an anisotropic cavity coherently pumped J=1→J=0 laser,” Phys. Rev. A 56, 2327 (1997).
[CrossRef]

C. Serrat, A. Kul’minskii, R. Vilaseca, and R. Corbalán, “Polarization chaos in an optically pumped laser,” Opt. Lett. 20, 1353 (1995).
[CrossRef] [PubMed]

Tang, D. Y.

R. Dykstra, A. Rayner, D. Y. Tang, and N. R. Heckenberg, “Experimentally tracking unstable steady states by large periodic modulation,” Phys. Rev. E 57, 397 (1998).
[CrossRef]

Vilaseca, R.

A. N. Pisarchik, V. N. Chizhevsky, R. Corbalán, and R. Vilaseca, “Experimental control of nonlinear dynamics by slow parametric modulation,” Phys. Rev. E 55, 2455 (1998).
[CrossRef]

C. Serrat, R. Vilaseca, G. J. De Valcárcel, and E. Roldán, “Polarization switching in an anisotropic cavity coherently pumped J=1→J=0 laser,” Phys. Rev. A 56, 2327 (1997).
[CrossRef]

R. Vilaseca, A. Kul’minskii, and R. Corbalán, “Tracking unstable steady states by large periodic modulation of a control parameter in a nonlinear system,” Phys. Rev. E 54, 82 (1996).
[CrossRef]

A. M. Kul’minskii, R. Vilaseca, and R. Corbalán, “Dynamics of a gain anisotropic optically pumped laser with arbitrary angle between linear polarizations of pump and laser fields,” J. Mod. Opt. 42, 2295 (1995).
[CrossRef]

C. Serrat, A. Kul’minskii, R. Vilaseca, and R. Corbalán, “Polarization chaos in an optically pumped laser,” Opt. Lett. 20, 1353 (1995).
[CrossRef] [PubMed]

A. Kul’minskii, R. Vilaseca, and R. Corbalán, “Full polarization chaos in a pump-polarization modulated isotropic cavity laser,” Opt. Lett. 20, 2390 (1995).
[CrossRef] [PubMed]

E. Roldán, G. J. de Valcárcel, R. Vilaseca, and R. Corbalán, “Polarization sensitive population trapping in an optically pumped laser,” Phys. Rev. A 49, 1487 (1994).
[CrossRef]

Warrington, D. M.

P. J. Manson, D. M. Warrington, N. L. Moise, and W. J. Sandle, “Observation of complex polarization dynamics in the output of a Ne Raman laser,” Quantum Semiclassic. Opt. 10, 157 (1998).
[CrossRef]

Weiss, C. O.

C. O. Weiss and J. Brock, “Evidence for Lorenz-type chaos in a laser,” Phys. Rev. Lett. 57, 2804 (1986).
[CrossRef] [PubMed]

Yang, G.

Z. Qu, G. Hu, G. Yang, and G. Qin, “Phase effect in taming nonautonomous chaos by weak harmonic perturbations,” Phys. Rev. Lett. 74, 1736 (1995).
[CrossRef] [PubMed]

J. Mod. Opt.

A. M. Kul’minskii, R. Vilaseca, and R. Corbalán, “Dynamics of a gain anisotropic optically pumped laser with arbitrary angle between linear polarizations of pump and laser fields,” J. Mod. Opt. 42, 2295 (1995).
[CrossRef]

Opt. Lett.

Phys. Rep.

See, for instance, D. Lenstra, “On the theory of polarization effects in gas lasers,” Phys. Rep. 59, 299 (1980).
[CrossRef]

Phys. Rev. A

E. Roldán, G. J. de Valcárcel, R. Vilaseca, and R. Corbalán, “Polarization sensitive population trapping in an optically pumped laser,” Phys. Rev. A 49, 1487 (1994).
[CrossRef]

C. Serrat, R. Vilaseca, G. J. De Valcárcel, and E. Roldán, “Polarization switching in an anisotropic cavity coherently pumped J=1→J=0 laser,” Phys. Rev. A 56, 2327 (1997).
[CrossRef]

M. Arjona, J. Pujol, and R. Corbalán, “Type-I intermittency in a four-level coherently pumped laser,” Phys. Rev. A 50, 871 (1994).
[CrossRef] [PubMed]

J. Pujol, M. Arjona, and R. Corbalán, “Type-III intermittency in a four-level coherently pumped laser,” Phys. Rev. A 48, 2251 (1993).
[CrossRef] [PubMed]

Phys. Rev. E

R. Vilaseca, A. Kul’minskii, and R. Corbalán, “Tracking unstable steady states by large periodic modulation of a control parameter in a nonlinear system,” Phys. Rev. E 54, 82 (1996).
[CrossRef]

R. Dykstra, A. Rayner, D. Y. Tang, and N. R. Heckenberg, “Experimentally tracking unstable steady states by large periodic modulation,” Phys. Rev. E 57, 397 (1998).
[CrossRef]

A. N. Pisarchik, V. N. Chizhevsky, R. Corbalán, and R. Vilaseca, “Experimental control of nonlinear dynamics by slow parametric modulation,” Phys. Rev. E 55, 2455 (1998).
[CrossRef]

Phys. Rev. Lett.

Z. Qu, G. Hu, G. Yang, and G. Qin, “Phase effect in taming nonautonomous chaos by weak harmonic perturbations,” Phys. Rev. Lett. 74, 1736 (1995).
[CrossRef] [PubMed]

C. O. Weiss and J. Brock, “Evidence for Lorenz-type chaos in a laser,” Phys. Rev. Lett. 57, 2804 (1986).
[CrossRef] [PubMed]

Quantum Semiclassic. Opt.

P. J. Manson, D. M. Warrington, N. L. Moise, and W. J. Sandle, “Observation of complex polarization dynamics in the output of a Ne Raman laser,” Quantum Semiclassic. Opt. 10, 157 (1998).
[CrossRef]

Radio Eng. Electron. Phys.

V. N. Parygin and A. S. Lipatov, “Internal modulation of the polarization of a gas laser,” Radio Eng. Electron. Phys. 20, 66 (1975).

Other

Y. I. Khanin, Principles of Laser Dynamics (Elsevier, Amsterdam, 1995), Chap. 6, pp. 245–305.

C. O. Weiss and R. Vilaseca, Dynamics of Lasers (VCH, Deerfield Beach, Fla., 1991), Chap. 7, pp. 137–180.

See, for instance, N. B. Abraham and G. M. Stephan, eds., Polarization Effects in Lasers and Spectroscopy, special issue of Quantum Semiclassic. Opt. 10, No. 1 (1998).

M. Arjona, R. Corbalán, F. Laguarta, J. Pujol, and R. Vilaseca, “Influence of light polarization on the dynamics of optically pumped lasers,” Phys. Rev. A 41, 6559 (1990); R. Corbalán, R. Vilaseca, M. Arjona, J. Pujol, E. Roldán, and G. J. de Valcárcel, “Dynamics of coherently pumped lasers with linearly polarized pump and generated fields,” Phys. Rev. A 48, 1483 (1993).
[CrossRef] [PubMed]

N. G. Douglas, Millimetre and Submillimetre Wavelength Lasers (Springer-Verlag, Berlin, 1989).

W. Klische and C. O. Weiss, “Instabilities and routes to chaos in a homogeneously broadened 1- and 2-mode ring laser,” Phys. Rev. A 31, 4049 (1985); E. H. M. Hogenboom, W. Klische, C. O. Weiss, and A. Godone, “Instabilities of a homogeneously broadened laser,” Phys. Rev. Lett. 55, 2571 (1985).
[CrossRef] [PubMed]

C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg, Atom-Photon Interactions (Wiley, New York, 1992), Chap. 6.

P. Mandel, Theoretical Problems in Cavity Nonlinear Optics (Cambridge U. Press, Cambridge, England, 1997), pp. 15–16.

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Figures (11)

Fig. 1
Fig. 1

Unidirectional cavity and fields involved in an optically pumped laser. E2 and E1 represent the pump and laser field vectors, respectively, whose orientation with respect to the x axis is given by the angles θ and θ.

Fig. 2
Fig. 2

Phase diagram of the unmodulated laser, in the plane defined by the pump amplitude β and the pump polarization angle θ, for θ=0 (fixed laser polarization). 0, no emission; ST, steady-state emission; P(1)1 and P(2)1, periodic regimes. The ST regime is stable above the dotted curve, and for a large domain of values of β and θ it coexists with periodic or chaotic solutions. (Reproduced by permission from Ref. 8.)

Fig. 3
Fig. 3

(a) Laser field amplitude and (b) J=0 level population as a function of the pump polarization angle θ for β=0.03γ. Dashed curves, unmodulated laser; solid curves, modulated laser, for which θ=Ωt with Ω=0.02γ. E1 and t are given in all the figures in units of γ and γ-1, respectively.

Fig. 4
Fig. 4

Same as Fig. 3(a), but for β=0.05γ and (a) Ω=0.05γ and (b) Ω=0.0745γ. (c) Attractor projection on the plane defined by the laser field amplitude E1 and the polarization induced in the amplifying medium, Im ρ01, associated with the two solutions in (a) (dashed and continuous curves). (d) Same as (c) except associated with (b). (e) Same as in (c) and (d), but for Ω=0.9γ.

Fig. 5
Fig. 5

Same as Figs. 3(a), 4(a), and 4(b), but for β=0.1γ. (a) The solid curves correspond to Ω=0.2γ, and the dashed–dotted curve to Ω=0.43γ. (b) As in Figs. 4(c)4(e), attractors associated with the solution in (a); ST, steady states. (c) The same as (b), but for Ω=0.95γ. (d) Bifurcation diagram showing the different types of solutions appearing when the rotation frequency Ω is increased. P, periodic behaviors; CH, chaotic behaviors; I1 and I3, intermittency type-I and type-III, respectively; F, Feigenbaum cascades. (e) Peak-intensity return maps (intensity of the nth peak versus intensity of the (n+1)th peak) for the unmodulated laser (case Ω=0, θ=π/2) and the two cases in (d).

Fig. 6
Fig. 6

Similar to Figs. 3(a), 4(a), 4(b), and 5(a): Time evolution of the laser intensity IE12, as a function of θ(=Ωt), for β=0.2γ and (a) dashed curve, Ω=0, solid curve, Ω=0.01γ and (b) Ω=0.08γ. Note the suppression of chaos and tracking of the steady state in (a). In an unmodulated laser the steady states are stable below the dotted line and unstable above it.

Fig. 7
Fig. 7

Domain in the (β, Ω) plane where suppression of chaos [as in Fig. 6(a)] is achieved. The upper and lower curves denote the limits of this domain.

Fig. 8
Fig. 8

Same as Fig. 6, but for β=0.55γ and Ω=0.02γ; reproduced by permission from Ref. 8.

Fig. 9
Fig. 9

Case of an isotropic cavity. Solid curves, angle θ-θ (between pump and laser field vectors) as a function of Ω; dashed curves, laser field amplitude E1 (right-hand-side vertical axis) as a function of Ω. Curves 1, β=0.06γ; 2, β=0.09γ; 3, β=0.2γ.

Fig. 10
Fig. 10

(a) Time evolution of the relative polarization angle θ-θ (upper curve, right-hand-side scale) and laser emission intensity IE12 (lower curve, left-hand-side scale) as a function of time, for β=0.6γ and θ=Ωt, with Ω=γ. (b) Intensity pulsations of an unmodulated laser for β=0.6γ. Note that the self-pulsing frequency of the modulated laser in (a) is not defined by Ω but by the self-pulsing frequency of the autonomous laser in (b).

Fig. 11
Fig. 11

Time dependence of the laser emission intensity IE12 for β=0.6γ. At t=0, pump polarization rotation θ=Ωt, with Ω=0.05γ, is switched on and stabilizes the system’s evolution.

Equations (13)

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E1±(z, t)=E1±(t)exp(ik1z)+c.c.=eˆ± μ±1α±(t)exp{i[k1z-ω1t-ϕ1±(t)]}+c.c.,
E2±(z, t)=E2±(t)exp(ik2z)+c.c.=eˆ± μ±2β± exp{i[k2z-ω2t-φ2±(t)]}+c.c.,
ρ˙11=-2(α+ Im ρ1++α- Im ρ1-)-γ1ρ11+λ1,
ρ˙22=i[β+ρ2+ exp(-iθ-)-β+ρ+2 exp(iθ-)-β-ρ-2 exp(iθ+)+β-ρ2- exp(-iθ+)]-γ2ρ22+λ2,
ρ˙±±=iβ±[ρ±2 exp(iθ)-ρ2± exp(-iθ)]+2α± Im ρ1±-γ±ρ±±+λ±,
ρ˙2+=-i[-ρ2++β+d+2 exp(iθ-)+β-ρ-+ exp(iθ+)-α+ρ21]-Γ2+ρ2+,
ρ˙2-=-i[ρ2-+β-d-2 exp(iθ+)+β+ρ+- exp(iθ-)-α-ρ21 exp(-iψ)]-Γ2-ρ2-,
ρ˙1+=-i{[ϕ˙1+(t)-]ρ1++α+d+1+α-ρ-+ exp(-iψ)-β+ρ12 exp(iθ-)}-γρ1+,
ρ˙1-=-i{(ϕ˙1-(t)+)ρ1-+α-d-1+α+ρ+- exp(iψ)-β-ρ12 exp(iθ+)exp(iψ)}-γρ1-,
ρ˙21=i[ϕ˙1+(t)ρ21+α+ρ2++α-ρ2- exp(iψ)-β+ρ+1 exp(iθ-)-β-ρ-1 exp(iθ+)exp(iψ)]-Γ21ρ21,
ρ˙+-=-i[2ρ+-+β+ρ2- exp(-iθ-)+α+ρ1- exp(-iψ)-β-ρ+2 exp(iθ+)-α-ρ+1 exp(-iψ)]-Γ+-ρ+-,
α˙±=-γc±α±/2-g Im ρ1±,
ϕ˙1±=-Δ1c+g Re ρ1±/α±,

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