Abstract

Manifestations of the geometric properties of evolutionary laser-field phases are considered for some laser systems. It is shown that a bidirectional ring laser provides an effective opportunity to control geometric phases.

© 1998 Optical Society of America

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References

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  1. M. V. Berry, “Quantal phase factors accompanying adiabatic changes,” Proc. R. Soc. London, Ser. A 392, 45–57 (1984); for reviews on the geometric phases see S. I. Vinitsky, V. L. Derbov, V. N. Dubovik, B. L. Markovski, and Yu. P. Stepanovskii, “Topological phases in quantum mechanics and polarization optics,” Sov. Phys. Usp. 33, 403–428 (1990); D. J. Moore, “The calculation of nonadiabatic Berry phases,” Phys. Rep. PRPLCM 210, 1–43 (1991); J. W. Zwanziger, M. Koenig, and A. Pines, “Berry’s phase,” Annu. Rev. Phys. Chem. ARPLAP 41, 601–646 (1990).
    [CrossRef]
  2. R. Vilaseca, G. J. de Valcarcel, and E. Roldan, “Physical interpretation of laser phase dynamics,” Phys. Rev. A 41, 5269–5272 (1990).
    [CrossRef] [PubMed]
  3. C. Z. Ning and H. Haken, “Quasiperiodicity involving twin oscillations in the complex Lorenz equations describing a detuned laser,” Z. Phys. B 81, 457–461 (1990).
    [CrossRef]
  4. C. Z. Ning and H. Haken, “Geometrical phase and amplitude accumulations in dissipative systems with cyclic attractors,” Phys. Rev. Lett. 68, 2109–2112 (1992); “The geometric phase in nonlinear dissipative systems,” Mod. Phys. Lett. B 6, 1541–1568 (1992).
    [CrossRef] [PubMed]
  5. V. Yu. Toronov and V. L. Derbov, “Geometric phases in lasers and liquid flows,” Phys. Rev. A 49, 1392–1399 (1994).
    [CrossRef] [PubMed]
  6. H. Haken, Laser Theory, Vol. XXV of Encyclopedia of Physics (Springer-Verlag, Berlin, 1970).
  7. J. Samuel and R. Bhandari, “General setting for Berry’s phase,” Phys. Rev. Lett. 60, 2339–2342 (1988).
    [CrossRef] [PubMed]
  8. C. Z. Ning and H. Haken, “Detuned lasers and the complex Lorenz equations: subcritical and supercritical Hopf bifurcations,” Phys. Rev. A 41, 3826–3837 (1990).
    [CrossRef] [PubMed]
  9. A. C. Fowler, J. D. Gibbon, and M. J. McGuinness, “The complex Lorenz equations,” Physica D 4, 139–163 (1982); J. D. Gibbon and M. J. McGuinness, “The real and complex Lorenz equations in rotating fluids and lasers,” Physica D 5, 108–122 (1982).
    [CrossRef]
  10. H. Zeghlache and P. Mandel, “Influence of detuning on the properties of laser equations,” J. Opt. Soc. Am. B 2, 18–22 (1985).
    [CrossRef]
  11. E. Roldan, G. J. de Valcarcel, R. Vilaseca, and P. Mandel, “Single-mode laser phase dynamics,” Phys. Rev. A 48, 591–598 (1993).
    [CrossRef]
  12. C. O. Weiss, N. B. Abraham, and U. Hubner, “Homoclinic and heteroclinic chaos in a single-mode laser,” Phys. Rev. Lett. 61, 1587–1590 (1988).
    [CrossRef] [PubMed]
  13. A. G. Vladimirov, V. Yu. Toronov, and V. L. Derbov, “On the complex Lorenz model,” Izv. Vuzov. Prikladnaya Nelneynaya Dinamika 3, 51–63 (1995).
  14. C. Sparrow, The Lorenz Equations: Bifurcations, Chaos and Strange Attractors (Springer-Verlag, Berlin, 1982).
  15. M. Sargent III, M. O. Scully, and W. E. Lamb, Jr., Laser Physics (Addison-Wesley, London, 1974).
  16. Yu. L. Klimontovich, V. N. Kuryatov, and P. S. Landa, “On the wave synchronization in a gas laser with the ring cavity,” JETP 51, 3–7 (1966).
  17. T. H. Chyba, “Phase-jump instability in the bidirectional ring laser with backscattering,” Phys. Rev. A 40, 6327–6333 (1989).
    [CrossRef] [PubMed]
  18. D. V. Skryabin, A. D. Vladimirov, and A. M. Radin, “Spontaneous phase symmetry breaking due to cavity detuning in a class-A bidirectional ring laser,” Opt. Commun. 116, 109–114 (1995).
    [CrossRef]
  19. This fact can be understood directly from the equations of motion in R written with the spherical coordinates. One can verify that ρ does not enter the equations for the angular coordinates that therefore completely determine the dynamics of the system. Thus the phase space can be reduced to a two-dimensional sphere, which can possess only limit cycles and fixed points as attractors.
  20. N. B. Abraham and C. O. Weiss, “Dynamical frequency shifts and intensity pulsations in a FIR bidirectional ring laser,” Opt. Commun. 66, 437 (1988).
    [CrossRef]
  21. T. Bitter and D. Dubbers, “Manifestation of Berry’s topological phase on neutron spin rotation,” Phys. Rev. Lett. 59, 251–254 (1987).
    [CrossRef] [PubMed]
  22. R. Bhandari and J. Samuel, “Observation of topological phase by use of a laser interferometer,” Phys. Rev. Lett. 60, 1211–1213 (1988); R. Bhandari, “Evolution of light beams in polarization and direction,” Physica B 175, 111–122 (1991).
    [CrossRef] [PubMed]
  23. H. Zeglache, P. Mandel, N. B. Abraham, L. M. Hoffer, G. L. Lippi, and T. Mello, “Bidirectional ring laser: stability analysis and time-dependent solutions,” Phys. Rev. A 37, 470–497 (1988).
    [CrossRef]
  24. L. M. Hoffer, G. L. Lippi, N. B. Abraham, and P. Mandel, “Phase and frequency jumps in a bidirectional ring laser,” Opt. Commun. 66, 219 (1988).
    [CrossRef]
  25. D. Arovas, “Geometric phases in condensed matter physics,” lecture notes for a course on geometric phases (International Center for Theoretical Physics, Trieste, 1993).

1995 (1)

D. V. Skryabin, A. D. Vladimirov, and A. M. Radin, “Spontaneous phase symmetry breaking due to cavity detuning in a class-A bidirectional ring laser,” Opt. Commun. 116, 109–114 (1995).
[CrossRef]

1994 (1)

V. Yu. Toronov and V. L. Derbov, “Geometric phases in lasers and liquid flows,” Phys. Rev. A 49, 1392–1399 (1994).
[CrossRef] [PubMed]

1993 (1)

E. Roldan, G. J. de Valcarcel, R. Vilaseca, and P. Mandel, “Single-mode laser phase dynamics,” Phys. Rev. A 48, 591–598 (1993).
[CrossRef]

1990 (3)

C. Z. Ning and H. Haken, “Detuned lasers and the complex Lorenz equations: subcritical and supercritical Hopf bifurcations,” Phys. Rev. A 41, 3826–3837 (1990).
[CrossRef] [PubMed]

R. Vilaseca, G. J. de Valcarcel, and E. Roldan, “Physical interpretation of laser phase dynamics,” Phys. Rev. A 41, 5269–5272 (1990).
[CrossRef] [PubMed]

C. Z. Ning and H. Haken, “Quasiperiodicity involving twin oscillations in the complex Lorenz equations describing a detuned laser,” Z. Phys. B 81, 457–461 (1990).
[CrossRef]

1989 (1)

T. H. Chyba, “Phase-jump instability in the bidirectional ring laser with backscattering,” Phys. Rev. A 40, 6327–6333 (1989).
[CrossRef] [PubMed]

1988 (5)

N. B. Abraham and C. O. Weiss, “Dynamical frequency shifts and intensity pulsations in a FIR bidirectional ring laser,” Opt. Commun. 66, 437 (1988).
[CrossRef]

H. Zeglache, P. Mandel, N. B. Abraham, L. M. Hoffer, G. L. Lippi, and T. Mello, “Bidirectional ring laser: stability analysis and time-dependent solutions,” Phys. Rev. A 37, 470–497 (1988).
[CrossRef]

L. M. Hoffer, G. L. Lippi, N. B. Abraham, and P. Mandel, “Phase and frequency jumps in a bidirectional ring laser,” Opt. Commun. 66, 219 (1988).
[CrossRef]

J. Samuel and R. Bhandari, “General setting for Berry’s phase,” Phys. Rev. Lett. 60, 2339–2342 (1988).
[CrossRef] [PubMed]

C. O. Weiss, N. B. Abraham, and U. Hubner, “Homoclinic and heteroclinic chaos in a single-mode laser,” Phys. Rev. Lett. 61, 1587–1590 (1988).
[CrossRef] [PubMed]

1987 (1)

T. Bitter and D. Dubbers, “Manifestation of Berry’s topological phase on neutron spin rotation,” Phys. Rev. Lett. 59, 251–254 (1987).
[CrossRef] [PubMed]

1985 (1)

1966 (1)

Yu. L. Klimontovich, V. N. Kuryatov, and P. S. Landa, “On the wave synchronization in a gas laser with the ring cavity,” JETP 51, 3–7 (1966).

Abraham, N. B.

C. O. Weiss, N. B. Abraham, and U. Hubner, “Homoclinic and heteroclinic chaos in a single-mode laser,” Phys. Rev. Lett. 61, 1587–1590 (1988).
[CrossRef] [PubMed]

N. B. Abraham and C. O. Weiss, “Dynamical frequency shifts and intensity pulsations in a FIR bidirectional ring laser,” Opt. Commun. 66, 437 (1988).
[CrossRef]

H. Zeglache, P. Mandel, N. B. Abraham, L. M. Hoffer, G. L. Lippi, and T. Mello, “Bidirectional ring laser: stability analysis and time-dependent solutions,” Phys. Rev. A 37, 470–497 (1988).
[CrossRef]

L. M. Hoffer, G. L. Lippi, N. B. Abraham, and P. Mandel, “Phase and frequency jumps in a bidirectional ring laser,” Opt. Commun. 66, 219 (1988).
[CrossRef]

Bhandari, R.

J. Samuel and R. Bhandari, “General setting for Berry’s phase,” Phys. Rev. Lett. 60, 2339–2342 (1988).
[CrossRef] [PubMed]

Bitter, T.

T. Bitter and D. Dubbers, “Manifestation of Berry’s topological phase on neutron spin rotation,” Phys. Rev. Lett. 59, 251–254 (1987).
[CrossRef] [PubMed]

Chyba, T. H.

T. H. Chyba, “Phase-jump instability in the bidirectional ring laser with backscattering,” Phys. Rev. A 40, 6327–6333 (1989).
[CrossRef] [PubMed]

de Valcarcel, G. J.

E. Roldan, G. J. de Valcarcel, R. Vilaseca, and P. Mandel, “Single-mode laser phase dynamics,” Phys. Rev. A 48, 591–598 (1993).
[CrossRef]

R. Vilaseca, G. J. de Valcarcel, and E. Roldan, “Physical interpretation of laser phase dynamics,” Phys. Rev. A 41, 5269–5272 (1990).
[CrossRef] [PubMed]

Derbov, V. L.

V. Yu. Toronov and V. L. Derbov, “Geometric phases in lasers and liquid flows,” Phys. Rev. A 49, 1392–1399 (1994).
[CrossRef] [PubMed]

Dubbers, D.

T. Bitter and D. Dubbers, “Manifestation of Berry’s topological phase on neutron spin rotation,” Phys. Rev. Lett. 59, 251–254 (1987).
[CrossRef] [PubMed]

Haken, H.

C. Z. Ning and H. Haken, “Quasiperiodicity involving twin oscillations in the complex Lorenz equations describing a detuned laser,” Z. Phys. B 81, 457–461 (1990).
[CrossRef]

C. Z. Ning and H. Haken, “Detuned lasers and the complex Lorenz equations: subcritical and supercritical Hopf bifurcations,” Phys. Rev. A 41, 3826–3837 (1990).
[CrossRef] [PubMed]

Hoffer, L. M.

H. Zeglache, P. Mandel, N. B. Abraham, L. M. Hoffer, G. L. Lippi, and T. Mello, “Bidirectional ring laser: stability analysis and time-dependent solutions,” Phys. Rev. A 37, 470–497 (1988).
[CrossRef]

L. M. Hoffer, G. L. Lippi, N. B. Abraham, and P. Mandel, “Phase and frequency jumps in a bidirectional ring laser,” Opt. Commun. 66, 219 (1988).
[CrossRef]

Hubner, U.

C. O. Weiss, N. B. Abraham, and U. Hubner, “Homoclinic and heteroclinic chaos in a single-mode laser,” Phys. Rev. Lett. 61, 1587–1590 (1988).
[CrossRef] [PubMed]

Klimontovich, Yu. L.

Yu. L. Klimontovich, V. N. Kuryatov, and P. S. Landa, “On the wave synchronization in a gas laser with the ring cavity,” JETP 51, 3–7 (1966).

Kuryatov, V. N.

Yu. L. Klimontovich, V. N. Kuryatov, and P. S. Landa, “On the wave synchronization in a gas laser with the ring cavity,” JETP 51, 3–7 (1966).

Landa, P. S.

Yu. L. Klimontovich, V. N. Kuryatov, and P. S. Landa, “On the wave synchronization in a gas laser with the ring cavity,” JETP 51, 3–7 (1966).

Lippi, G. L.

H. Zeglache, P. Mandel, N. B. Abraham, L. M. Hoffer, G. L. Lippi, and T. Mello, “Bidirectional ring laser: stability analysis and time-dependent solutions,” Phys. Rev. A 37, 470–497 (1988).
[CrossRef]

L. M. Hoffer, G. L. Lippi, N. B. Abraham, and P. Mandel, “Phase and frequency jumps in a bidirectional ring laser,” Opt. Commun. 66, 219 (1988).
[CrossRef]

Mandel, P.

E. Roldan, G. J. de Valcarcel, R. Vilaseca, and P. Mandel, “Single-mode laser phase dynamics,” Phys. Rev. A 48, 591–598 (1993).
[CrossRef]

H. Zeglache, P. Mandel, N. B. Abraham, L. M. Hoffer, G. L. Lippi, and T. Mello, “Bidirectional ring laser: stability analysis and time-dependent solutions,” Phys. Rev. A 37, 470–497 (1988).
[CrossRef]

L. M. Hoffer, G. L. Lippi, N. B. Abraham, and P. Mandel, “Phase and frequency jumps in a bidirectional ring laser,” Opt. Commun. 66, 219 (1988).
[CrossRef]

H. Zeghlache and P. Mandel, “Influence of detuning on the properties of laser equations,” J. Opt. Soc. Am. B 2, 18–22 (1985).
[CrossRef]

Mello, T.

H. Zeglache, P. Mandel, N. B. Abraham, L. M. Hoffer, G. L. Lippi, and T. Mello, “Bidirectional ring laser: stability analysis and time-dependent solutions,” Phys. Rev. A 37, 470–497 (1988).
[CrossRef]

Ning, C. Z.

C. Z. Ning and H. Haken, “Detuned lasers and the complex Lorenz equations: subcritical and supercritical Hopf bifurcations,” Phys. Rev. A 41, 3826–3837 (1990).
[CrossRef] [PubMed]

C. Z. Ning and H. Haken, “Quasiperiodicity involving twin oscillations in the complex Lorenz equations describing a detuned laser,” Z. Phys. B 81, 457–461 (1990).
[CrossRef]

Radin, A. M.

D. V. Skryabin, A. D. Vladimirov, and A. M. Radin, “Spontaneous phase symmetry breaking due to cavity detuning in a class-A bidirectional ring laser,” Opt. Commun. 116, 109–114 (1995).
[CrossRef]

Roldan, E.

E. Roldan, G. J. de Valcarcel, R. Vilaseca, and P. Mandel, “Single-mode laser phase dynamics,” Phys. Rev. A 48, 591–598 (1993).
[CrossRef]

R. Vilaseca, G. J. de Valcarcel, and E. Roldan, “Physical interpretation of laser phase dynamics,” Phys. Rev. A 41, 5269–5272 (1990).
[CrossRef] [PubMed]

Samuel, J.

J. Samuel and R. Bhandari, “General setting for Berry’s phase,” Phys. Rev. Lett. 60, 2339–2342 (1988).
[CrossRef] [PubMed]

Skryabin, D. V.

D. V. Skryabin, A. D. Vladimirov, and A. M. Radin, “Spontaneous phase symmetry breaking due to cavity detuning in a class-A bidirectional ring laser,” Opt. Commun. 116, 109–114 (1995).
[CrossRef]

Toronov, V. Yu.

V. Yu. Toronov and V. L. Derbov, “Geometric phases in lasers and liquid flows,” Phys. Rev. A 49, 1392–1399 (1994).
[CrossRef] [PubMed]

Vilaseca, R.

E. Roldan, G. J. de Valcarcel, R. Vilaseca, and P. Mandel, “Single-mode laser phase dynamics,” Phys. Rev. A 48, 591–598 (1993).
[CrossRef]

R. Vilaseca, G. J. de Valcarcel, and E. Roldan, “Physical interpretation of laser phase dynamics,” Phys. Rev. A 41, 5269–5272 (1990).
[CrossRef] [PubMed]

Vladimirov, A. D.

D. V. Skryabin, A. D. Vladimirov, and A. M. Radin, “Spontaneous phase symmetry breaking due to cavity detuning in a class-A bidirectional ring laser,” Opt. Commun. 116, 109–114 (1995).
[CrossRef]

Weiss, C. O.

C. O. Weiss, N. B. Abraham, and U. Hubner, “Homoclinic and heteroclinic chaos in a single-mode laser,” Phys. Rev. Lett. 61, 1587–1590 (1988).
[CrossRef] [PubMed]

N. B. Abraham and C. O. Weiss, “Dynamical frequency shifts and intensity pulsations in a FIR bidirectional ring laser,” Opt. Commun. 66, 437 (1988).
[CrossRef]

Zeghlache, H.

Zeglache, H.

H. Zeglache, P. Mandel, N. B. Abraham, L. M. Hoffer, G. L. Lippi, and T. Mello, “Bidirectional ring laser: stability analysis and time-dependent solutions,” Phys. Rev. A 37, 470–497 (1988).
[CrossRef]

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

JETP (1)

Yu. L. Klimontovich, V. N. Kuryatov, and P. S. Landa, “On the wave synchronization in a gas laser with the ring cavity,” JETP 51, 3–7 (1966).

Opt. Commun. (3)

D. V. Skryabin, A. D. Vladimirov, and A. M. Radin, “Spontaneous phase symmetry breaking due to cavity detuning in a class-A bidirectional ring laser,” Opt. Commun. 116, 109–114 (1995).
[CrossRef]

N. B. Abraham and C. O. Weiss, “Dynamical frequency shifts and intensity pulsations in a FIR bidirectional ring laser,” Opt. Commun. 66, 437 (1988).
[CrossRef]

L. M. Hoffer, G. L. Lippi, N. B. Abraham, and P. Mandel, “Phase and frequency jumps in a bidirectional ring laser,” Opt. Commun. 66, 219 (1988).
[CrossRef]

Phys. Rev. A (6)

H. Zeglache, P. Mandel, N. B. Abraham, L. M. Hoffer, G. L. Lippi, and T. Mello, “Bidirectional ring laser: stability analysis and time-dependent solutions,” Phys. Rev. A 37, 470–497 (1988).
[CrossRef]

T. H. Chyba, “Phase-jump instability in the bidirectional ring laser with backscattering,” Phys. Rev. A 40, 6327–6333 (1989).
[CrossRef] [PubMed]

E. Roldan, G. J. de Valcarcel, R. Vilaseca, and P. Mandel, “Single-mode laser phase dynamics,” Phys. Rev. A 48, 591–598 (1993).
[CrossRef]

R. Vilaseca, G. J. de Valcarcel, and E. Roldan, “Physical interpretation of laser phase dynamics,” Phys. Rev. A 41, 5269–5272 (1990).
[CrossRef] [PubMed]

V. Yu. Toronov and V. L. Derbov, “Geometric phases in lasers and liquid flows,” Phys. Rev. A 49, 1392–1399 (1994).
[CrossRef] [PubMed]

C. Z. Ning and H. Haken, “Detuned lasers and the complex Lorenz equations: subcritical and supercritical Hopf bifurcations,” Phys. Rev. A 41, 3826–3837 (1990).
[CrossRef] [PubMed]

Phys. Rev. Lett. (3)

J. Samuel and R. Bhandari, “General setting for Berry’s phase,” Phys. Rev. Lett. 60, 2339–2342 (1988).
[CrossRef] [PubMed]

C. O. Weiss, N. B. Abraham, and U. Hubner, “Homoclinic and heteroclinic chaos in a single-mode laser,” Phys. Rev. Lett. 61, 1587–1590 (1988).
[CrossRef] [PubMed]

T. Bitter and D. Dubbers, “Manifestation of Berry’s topological phase on neutron spin rotation,” Phys. Rev. Lett. 59, 251–254 (1987).
[CrossRef] [PubMed]

Z. Phys. B (1)

C. Z. Ning and H. Haken, “Quasiperiodicity involving twin oscillations in the complex Lorenz equations describing a detuned laser,” Z. Phys. B 81, 457–461 (1990).
[CrossRef]

Other (10)

C. Z. Ning and H. Haken, “Geometrical phase and amplitude accumulations in dissipative systems with cyclic attractors,” Phys. Rev. Lett. 68, 2109–2112 (1992); “The geometric phase in nonlinear dissipative systems,” Mod. Phys. Lett. B 6, 1541–1568 (1992).
[CrossRef] [PubMed]

H. Haken, Laser Theory, Vol. XXV of Encyclopedia of Physics (Springer-Verlag, Berlin, 1970).

M. V. Berry, “Quantal phase factors accompanying adiabatic changes,” Proc. R. Soc. London, Ser. A 392, 45–57 (1984); for reviews on the geometric phases see S. I. Vinitsky, V. L. Derbov, V. N. Dubovik, B. L. Markovski, and Yu. P. Stepanovskii, “Topological phases in quantum mechanics and polarization optics,” Sov. Phys. Usp. 33, 403–428 (1990); D. J. Moore, “The calculation of nonadiabatic Berry phases,” Phys. Rep. PRPLCM 210, 1–43 (1991); J. W. Zwanziger, M. Koenig, and A. Pines, “Berry’s phase,” Annu. Rev. Phys. Chem. ARPLAP 41, 601–646 (1990).
[CrossRef]

A. C. Fowler, J. D. Gibbon, and M. J. McGuinness, “The complex Lorenz equations,” Physica D 4, 139–163 (1982); J. D. Gibbon and M. J. McGuinness, “The real and complex Lorenz equations in rotating fluids and lasers,” Physica D 5, 108–122 (1982).
[CrossRef]

A. G. Vladimirov, V. Yu. Toronov, and V. L. Derbov, “On the complex Lorenz model,” Izv. Vuzov. Prikladnaya Nelneynaya Dinamika 3, 51–63 (1995).

C. Sparrow, The Lorenz Equations: Bifurcations, Chaos and Strange Attractors (Springer-Verlag, Berlin, 1982).

M. Sargent III, M. O. Scully, and W. E. Lamb, Jr., Laser Physics (Addison-Wesley, London, 1974).

This fact can be understood directly from the equations of motion in R written with the spherical coordinates. One can verify that ρ does not enter the equations for the angular coordinates that therefore completely determine the dynamics of the system. Thus the phase space can be reduced to a two-dimensional sphere, which can possess only limit cycles and fixed points as attractors.

R. Bhandari and J. Samuel, “Observation of topological phase by use of a laser interferometer,” Phys. Rev. Lett. 60, 1211–1213 (1988); R. Bhandari, “Evolution of light beams in polarization and direction,” Physica B 175, 111–122 (1991).
[CrossRef] [PubMed]

D. Arovas, “Geometric phases in condensed matter physics,” lecture notes for a course on geometric phases (International Center for Theoretical Physics, Trieste, 1993).

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Figures (4)

Fig. 1
Fig. 1

Solution of the CLE, periodic in R, for k=3, b=1, r1=30, r2=0, and δ=1. (a) Phase portrait in the X plane; (b) projection of the trajectory in R onto the unit sphere; (c) evolution of the phases: Pancharatnam’s phase (1), X-variable phase (2), and dynamic phase (3).

Fig. 2
Fig. 2

Phase portraits of the cyclic attractors observed at k=3, b=0.1, and r=15 (a) in the (ζ1, ζ2) plane for δ=±0.01 and δ=0 (± and 0, respectively) and (b) in the (ζ1, ζ3) plane for δ=0; 0 shows the position of the origin.

Fig. 3
Fig. 3

Phase portraits of the chaotic attractors observed at k=3, b=1, and r=30 (a) in the (ζ1, ζ2) plane for δ=±0.01 and δ=0 and (b) in the (ζ1, ζ3) plane for δ=0.

Fig. 4
Fig. 4

Bidirectional ring laser. External mirror M couples the phases of counterpropagating waves in a way similar to effective intracavity local-loss source L.

Equations (58)

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

i Et=(ω0/2-i2πσ)E+c22ω0 2E+2πω0P,
i Pt=-iγabP-|p|2 ED,
Dt=D0-DT+2i (E*P-EP*),
|ψη=EηPη,
i(/t)|ψ=Hˆ(D)|ψ,
ψα|ψβ=W(Eα*Eβ+Pα*Pβ)d3r,
γ=arg[ψ(0)|ψ(t)],
γd(t)=0th(t)dt,
h(t)=ψ(t)|ψ(t)-1 Re[ψ(t)|Hˆ(t)|ψ(t)],
γg(t)=-ΓTAsds,
As=Im[ψ(s)|d/ds|ψ(s)]/ψ(s)|ψ(s).
Im(Ψ|t|Ψ)=0.
|ψ(t)=(2R)1/2 cos(θ/2)exp(iφ/2)(2R)1/2 sin(θ/2)exp(-iφ/2)exp(iΘ),
As=-dΘ/ds+sin2(θ/2)dφ/ds.
γg(t)arg[Ψ(t)|Ψ(0)]=ΓT sin2(θ/2)dφ.
dx/dt=-k(x-y),
dy/dt=-(1-iδ)y+(r-z)x,
dz/dt=-bz+1/2(x*y+xy*),
|ψ=xy.
Hˆ(t)=i-kkr-z-1+iδ.
ψ1|ψ2=x1*x2+y1*y2
h(t)=[δ|y|2-i2 (k+z-r)(x*y-xy*)]×(|x|2+|y|2)-1.
|Ψ(t)=exp[-iα(t)]|ψ(t)=XY,
dX/dt=-(k-ih)X+kY,
dY/dt=-(1-iδ-ih)Y+(r-z)X,
dz/dt=-bz+1/2(X*Y+XY*).
ζ1=(x*y+xy*)/2,
ζ2=-i(x*y-xy*)/2,
ζ3=(|x|2-|y|2)/2.
dζ1dt=-(k+1)ζ1-δζ2-(k-r+z)ζ3+(k+r-z)R,
dζ2dt=-(k+1)ζ2+δζ1,
dζ3dt=-(k+1)ζ3+(k-r+z)ζ1-(k-1)R,
dzdt=-bz+ζ2,
ζ1=I,ζ2=I δk+1,
ζ3=-I δ22(k+1)2,z(t)=I/b,
I=|x|2=b(r-rc)
ω=kδ/(k+1).
u=(|E+|2-|E-|2)/2,v+iw=E+*E-.
u=ρ cos θ,v=ρ sin θ cos ϕ,
w=ρ sin θ sin ϕ.
ρ=(|E+|2+|E-|2)/2,θ=2a tan(|E+|/|E-|),
ϕ=arg(E-)-arg(E+).
ddt E±=[η±+iζ±-μ|E±|2-ξ|E±|2]E±+g±E,
u˙=Λu+Δρ-4μuρ,
q˙=[Λ-iδ-2(μ+ξ)ρ+2i(μ-ξ)u]q+2ρ exp(-ikz),
φ˙+=-|E-|2|E+|2 φ˙-.
u=ρ cos θ=Δρ/(4μρ-Λ),
q=|q|exp(iφ)=2ρ[Λ-2(μ+ξ)ρ]-1×exp(-ikz).
cos θ=uρ=ϑ-1ϑ+1,
2π(1-cos θ)=4π/(ϑ+1),
γg=2π/(ϑ+1).
ν+ν=11+ϑ Vc.
ν-ν=-ϑ ν+ν=-ϑ1+ϑ Vc.
E+(t)=A exp(iν+t),E-(t)=B exp(iν-t),
iν+=[η+-μ|A|2-ξ|B|2]+B/A,
iν-=[η--μ|B|2-ξ|A|2]+A/B.
ν+=Im(B/A),ν-=Im(A/B).
Δ2(4μρ-Λ)2+4[Λ-2(μ+ξ)ρ]2+(kV)2=1,

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