Abstract

We consider a laser with an injected signal, in which the polarization can be adiabatically eliminated, we study the stability of the steady-state solutions, and we discuss the time-dependent solutions. For the laser alone, the only possible solution is constant intensity. However, the introduction of an external field, with an amplitude that does not satisfy the injection-locking condition, destabilizes the system. In such a case, numerical results show the existence of a self-Q-switching process, which induces relaxation oscillations. The frequency of the giant pulses is directly related to the amplitude of the external field, whereas the frequency of the relaxation oscillations depends on the damping rates. We show also that, depending on the value assigned to control parameters, the interaction between these frequencies leads to a chaotic behavior through intermittency or period-doubling bifurcations. Finally, topological equivalence between our laser system and a unidimensional circle map is shown for some values of control parameters.

© 1985 Optical Society of America

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  1. H. L. Stover, W. H. Steier, Appl. Phys. Lett. 8, 91 (1966).
    [CrossRef]
  2. L. E. Erickson, A. Szabo, Appl. Phys. Lett. 18, 433 (1981).
    [CrossRef]
  3. C. J. Buczek, R. J. Freiberg, IEEE J. Quantum Electron. QE-8, 643 (1972).
  4. A. Girard, Opt. Commun. 11, 346 (1974).
    [CrossRef]
  5. J. L. Lachambre, P. Lavigne, G. Otis, M. Noel, IEEE J. Quantum Electron. QE-12, 756 (1976).
    [CrossRef]
  6. R. Daigle, P. Belanger, Opt. Commun. 23, 165 (1977).
    [CrossRef]
  7. Y. K. Park, G. Giuliani, R. L. Byer, Opt. Lett. 5, 96 (1980).
    [CrossRef]
  8. P. Burlamacchi, R. Salimbeni, Opt. Commun. 17, 6 (1976).
    [CrossRef]
  9. R. W. Dunn, S. T. Hendow, W. W. Chow, J. Small, Opt. Lett. 8, 319 (1983).
    [CrossRef] [PubMed]
  10. W. Annovazzi, S. Donati, IEEE J. Quantum Electron. QE-16, 859 (1980).
  11. R. Lang, IEEE J. Quantum Electron. QE-18, 979 (1982), and references therein.
  12. R. Adler, Proc. IRE 34, 351 (1946).
    [CrossRef]
  13. M. Spencer, W. Lamb, Phys. Rev. A 5, 884 (1972).
    [CrossRef]
  14. C. J. Buczek, R. J. Freiberg, M. Skolnick, Proc. IEEE 61, 1411 (1973).
    [CrossRef]
  15. W. W. Chow, IEEE J. Quantum Electron. QE-19, 243 (1983).
    [CrossRef]
  16. H. Haken, in Handbuch der Physik, L. Genzel, ed. (Springer-Verlag, New York, 1970), Vol. 25/2c.
  17. F. T. Arecchi, Acta Phys. Austraca. (to be published).
  18. F. T. Arecchi, G. L. Lippi, G. P. Puccioni, J. R. Tredicce, in Coherence and Quantum Optics V, L. Mandel, E. Wolf, eds. (Plenum, New York, 1984), p. 1227;Opt. Commun. (submitted for publication).
  19. T. Yamada, R. Graham, Phys. Rev. Lett. 45, 1322 (1980).
    [CrossRef]
  20. H. Scholz, T. Yamada, H. Brand, R. Graham, Phys. Lett. 82A, 321 (1981).
  21. F. T. Arecchi, R. Meucci, G. P. Puccioni, J. R. Tredicce, Phys. Rev. Lett. 49, 1217 (1982).
    [CrossRef]
  22. E. Brun, B. Derighetti, R. Holzner, D. Meier, Helv. Phys. Acta 56, 825 (1983).
  23. F. Morgensen, G. Jacobsen, A. Olesen, Opt. Quantum Electron. 16, 183 (1984).
    [CrossRef]
  24. J. R. Tredicce, G. L. Lippi, N. B. Abraham, F. T. Arecchi, Proc. R. Soc. (London) (to be published).
  25. L. A. Lugiato, presented at the Workshop on Chaotic and Pulse Instabilities, Schloss Elmau, Federal Republic of Germany, May, 1984.
  26. H. Haken, Phys. Lett. 53A, 77 (1975).
  27. L. A. Lugiato, L. M. Narducci, D. K. Bandy, C. A. Pennise, Opt. Commun. 46, 64 (1983).
    [CrossRef]
  28. D. K. Bandy, L. M. Narducci, C. A. Pennise, L. A. Lugiato, in Coherence and Quantum Optics V, L. Mandel, E. Wolf, eds. (Plenum, New York, 1984), p. 585.
  29. C. O. Weiss, H. King, Opt. Commun. 44, 59 (1982);C. O. Weiss, A. Godone, A. Olafsson, Phys. Rev. A 28, 892 (1983).
    [CrossRef]
  30. N. J. Halas, S.-N. Liu, N. B. Abraham, Phys. Rev. A 28, 2915 (1983).
    [CrossRef]
  31. L. W. Casperson, Phys. Rev. A 21, 911 (1980);Phys. Rev. A 23, 248 (1981).
    [CrossRef]
  32. R. S. Gioggia, N. B. Abraham, Phys. Rev. Lett. 51, 650 (1983).
    [CrossRef]
  33. N. G. Basov, V. S. Znev, P. G. Krjukov, in Lasers, J. Weber, ed. (Gordon & Breach, New York, 1968), Vol. 10/a, p. 257;F. J. McClung, R. W. Hellwarth, Proc. IEEE 51, 46 (1963).
    [CrossRef]
  34. S. Ostlund, D. Rand, J. Sethna, E. Siggia, Physica 8D, 303 (1983).
  35. M. Høgh Jensen, P. Bak, T. Bohr, Phys. Rev. A (to be published).

1984

F. Morgensen, G. Jacobsen, A. Olesen, Opt. Quantum Electron. 16, 183 (1984).
[CrossRef]

1983

E. Brun, B. Derighetti, R. Holzner, D. Meier, Helv. Phys. Acta 56, 825 (1983).

L. A. Lugiato, L. M. Narducci, D. K. Bandy, C. A. Pennise, Opt. Commun. 46, 64 (1983).
[CrossRef]

N. J. Halas, S.-N. Liu, N. B. Abraham, Phys. Rev. A 28, 2915 (1983).
[CrossRef]

W. W. Chow, IEEE J. Quantum Electron. QE-19, 243 (1983).
[CrossRef]

R. S. Gioggia, N. B. Abraham, Phys. Rev. Lett. 51, 650 (1983).
[CrossRef]

S. Ostlund, D. Rand, J. Sethna, E. Siggia, Physica 8D, 303 (1983).

R. W. Dunn, S. T. Hendow, W. W. Chow, J. Small, Opt. Lett. 8, 319 (1983).
[CrossRef] [PubMed]

1982

R. Lang, IEEE J. Quantum Electron. QE-18, 979 (1982), and references therein.

C. O. Weiss, H. King, Opt. Commun. 44, 59 (1982);C. O. Weiss, A. Godone, A. Olafsson, Phys. Rev. A 28, 892 (1983).
[CrossRef]

F. T. Arecchi, R. Meucci, G. P. Puccioni, J. R. Tredicce, Phys. Rev. Lett. 49, 1217 (1982).
[CrossRef]

1981

H. Scholz, T. Yamada, H. Brand, R. Graham, Phys. Lett. 82A, 321 (1981).

L. E. Erickson, A. Szabo, Appl. Phys. Lett. 18, 433 (1981).
[CrossRef]

1980

Y. K. Park, G. Giuliani, R. L. Byer, Opt. Lett. 5, 96 (1980).
[CrossRef]

T. Yamada, R. Graham, Phys. Rev. Lett. 45, 1322 (1980).
[CrossRef]

W. Annovazzi, S. Donati, IEEE J. Quantum Electron. QE-16, 859 (1980).

L. W. Casperson, Phys. Rev. A 21, 911 (1980);Phys. Rev. A 23, 248 (1981).
[CrossRef]

1977

R. Daigle, P. Belanger, Opt. Commun. 23, 165 (1977).
[CrossRef]

1976

P. Burlamacchi, R. Salimbeni, Opt. Commun. 17, 6 (1976).
[CrossRef]

J. L. Lachambre, P. Lavigne, G. Otis, M. Noel, IEEE J. Quantum Electron. QE-12, 756 (1976).
[CrossRef]

1975

H. Haken, Phys. Lett. 53A, 77 (1975).

1974

A. Girard, Opt. Commun. 11, 346 (1974).
[CrossRef]

1973

C. J. Buczek, R. J. Freiberg, M. Skolnick, Proc. IEEE 61, 1411 (1973).
[CrossRef]

1972

M. Spencer, W. Lamb, Phys. Rev. A 5, 884 (1972).
[CrossRef]

C. J. Buczek, R. J. Freiberg, IEEE J. Quantum Electron. QE-8, 643 (1972).

1966

H. L. Stover, W. H. Steier, Appl. Phys. Lett. 8, 91 (1966).
[CrossRef]

1946

R. Adler, Proc. IRE 34, 351 (1946).
[CrossRef]

Abraham, N. B.

R. S. Gioggia, N. B. Abraham, Phys. Rev. Lett. 51, 650 (1983).
[CrossRef]

N. J. Halas, S.-N. Liu, N. B. Abraham, Phys. Rev. A 28, 2915 (1983).
[CrossRef]

J. R. Tredicce, G. L. Lippi, N. B. Abraham, F. T. Arecchi, Proc. R. Soc. (London) (to be published).

Adler, R.

R. Adler, Proc. IRE 34, 351 (1946).
[CrossRef]

Annovazzi, W.

W. Annovazzi, S. Donati, IEEE J. Quantum Electron. QE-16, 859 (1980).

Arecchi, F. T.

F. T. Arecchi, R. Meucci, G. P. Puccioni, J. R. Tredicce, Phys. Rev. Lett. 49, 1217 (1982).
[CrossRef]

J. R. Tredicce, G. L. Lippi, N. B. Abraham, F. T. Arecchi, Proc. R. Soc. (London) (to be published).

F. T. Arecchi, Acta Phys. Austraca. (to be published).

F. T. Arecchi, G. L. Lippi, G. P. Puccioni, J. R. Tredicce, in Coherence and Quantum Optics V, L. Mandel, E. Wolf, eds. (Plenum, New York, 1984), p. 1227;Opt. Commun. (submitted for publication).

Bak, P.

M. Høgh Jensen, P. Bak, T. Bohr, Phys. Rev. A (to be published).

Bandy, D. K.

L. A. Lugiato, L. M. Narducci, D. K. Bandy, C. A. Pennise, Opt. Commun. 46, 64 (1983).
[CrossRef]

D. K. Bandy, L. M. Narducci, C. A. Pennise, L. A. Lugiato, in Coherence and Quantum Optics V, L. Mandel, E. Wolf, eds. (Plenum, New York, 1984), p. 585.

Basov, N. G.

N. G. Basov, V. S. Znev, P. G. Krjukov, in Lasers, J. Weber, ed. (Gordon & Breach, New York, 1968), Vol. 10/a, p. 257;F. J. McClung, R. W. Hellwarth, Proc. IEEE 51, 46 (1963).
[CrossRef]

Belanger, P.

R. Daigle, P. Belanger, Opt. Commun. 23, 165 (1977).
[CrossRef]

Bohr, T.

M. Høgh Jensen, P. Bak, T. Bohr, Phys. Rev. A (to be published).

Brand, H.

H. Scholz, T. Yamada, H. Brand, R. Graham, Phys. Lett. 82A, 321 (1981).

Brun, E.

E. Brun, B. Derighetti, R. Holzner, D. Meier, Helv. Phys. Acta 56, 825 (1983).

Buczek, C. J.

C. J. Buczek, R. J. Freiberg, M. Skolnick, Proc. IEEE 61, 1411 (1973).
[CrossRef]

C. J. Buczek, R. J. Freiberg, IEEE J. Quantum Electron. QE-8, 643 (1972).

Burlamacchi, P.

P. Burlamacchi, R. Salimbeni, Opt. Commun. 17, 6 (1976).
[CrossRef]

Byer, R. L.

Casperson, L. W.

L. W. Casperson, Phys. Rev. A 21, 911 (1980);Phys. Rev. A 23, 248 (1981).
[CrossRef]

Chow, W. W.

Daigle, R.

R. Daigle, P. Belanger, Opt. Commun. 23, 165 (1977).
[CrossRef]

Derighetti, B.

E. Brun, B. Derighetti, R. Holzner, D. Meier, Helv. Phys. Acta 56, 825 (1983).

Donati, S.

W. Annovazzi, S. Donati, IEEE J. Quantum Electron. QE-16, 859 (1980).

Dunn, R. W.

Erickson, L. E.

L. E. Erickson, A. Szabo, Appl. Phys. Lett. 18, 433 (1981).
[CrossRef]

Freiberg, R. J.

C. J. Buczek, R. J. Freiberg, M. Skolnick, Proc. IEEE 61, 1411 (1973).
[CrossRef]

C. J. Buczek, R. J. Freiberg, IEEE J. Quantum Electron. QE-8, 643 (1972).

Gioggia, R. S.

R. S. Gioggia, N. B. Abraham, Phys. Rev. Lett. 51, 650 (1983).
[CrossRef]

Girard, A.

A. Girard, Opt. Commun. 11, 346 (1974).
[CrossRef]

Giuliani, G.

Graham, R.

H. Scholz, T. Yamada, H. Brand, R. Graham, Phys. Lett. 82A, 321 (1981).

T. Yamada, R. Graham, Phys. Rev. Lett. 45, 1322 (1980).
[CrossRef]

Haken, H.

H. Haken, Phys. Lett. 53A, 77 (1975).

H. Haken, in Handbuch der Physik, L. Genzel, ed. (Springer-Verlag, New York, 1970), Vol. 25/2c.

Halas, N. J.

N. J. Halas, S.-N. Liu, N. B. Abraham, Phys. Rev. A 28, 2915 (1983).
[CrossRef]

Hendow, S. T.

Høgh Jensen, M.

M. Høgh Jensen, P. Bak, T. Bohr, Phys. Rev. A (to be published).

Holzner, R.

E. Brun, B. Derighetti, R. Holzner, D. Meier, Helv. Phys. Acta 56, 825 (1983).

Jacobsen, G.

F. Morgensen, G. Jacobsen, A. Olesen, Opt. Quantum Electron. 16, 183 (1984).
[CrossRef]

King, H.

C. O. Weiss, H. King, Opt. Commun. 44, 59 (1982);C. O. Weiss, A. Godone, A. Olafsson, Phys. Rev. A 28, 892 (1983).
[CrossRef]

Krjukov, P. G.

N. G. Basov, V. S. Znev, P. G. Krjukov, in Lasers, J. Weber, ed. (Gordon & Breach, New York, 1968), Vol. 10/a, p. 257;F. J. McClung, R. W. Hellwarth, Proc. IEEE 51, 46 (1963).
[CrossRef]

Lachambre, J. L.

J. L. Lachambre, P. Lavigne, G. Otis, M. Noel, IEEE J. Quantum Electron. QE-12, 756 (1976).
[CrossRef]

Lamb, W.

M. Spencer, W. Lamb, Phys. Rev. A 5, 884 (1972).
[CrossRef]

Lang, R.

R. Lang, IEEE J. Quantum Electron. QE-18, 979 (1982), and references therein.

Lavigne, P.

J. L. Lachambre, P. Lavigne, G. Otis, M. Noel, IEEE J. Quantum Electron. QE-12, 756 (1976).
[CrossRef]

Lippi, G. L.

F. T. Arecchi, G. L. Lippi, G. P. Puccioni, J. R. Tredicce, in Coherence and Quantum Optics V, L. Mandel, E. Wolf, eds. (Plenum, New York, 1984), p. 1227;Opt. Commun. (submitted for publication).

J. R. Tredicce, G. L. Lippi, N. B. Abraham, F. T. Arecchi, Proc. R. Soc. (London) (to be published).

Liu, S.-N.

N. J. Halas, S.-N. Liu, N. B. Abraham, Phys. Rev. A 28, 2915 (1983).
[CrossRef]

Lugiato, L. A.

L. A. Lugiato, L. M. Narducci, D. K. Bandy, C. A. Pennise, Opt. Commun. 46, 64 (1983).
[CrossRef]

L. A. Lugiato, presented at the Workshop on Chaotic and Pulse Instabilities, Schloss Elmau, Federal Republic of Germany, May, 1984.

D. K. Bandy, L. M. Narducci, C. A. Pennise, L. A. Lugiato, in Coherence and Quantum Optics V, L. Mandel, E. Wolf, eds. (Plenum, New York, 1984), p. 585.

Meier, D.

E. Brun, B. Derighetti, R. Holzner, D. Meier, Helv. Phys. Acta 56, 825 (1983).

Meucci, R.

F. T. Arecchi, R. Meucci, G. P. Puccioni, J. R. Tredicce, Phys. Rev. Lett. 49, 1217 (1982).
[CrossRef]

Morgensen, F.

F. Morgensen, G. Jacobsen, A. Olesen, Opt. Quantum Electron. 16, 183 (1984).
[CrossRef]

Narducci, L. M.

L. A. Lugiato, L. M. Narducci, D. K. Bandy, C. A. Pennise, Opt. Commun. 46, 64 (1983).
[CrossRef]

D. K. Bandy, L. M. Narducci, C. A. Pennise, L. A. Lugiato, in Coherence and Quantum Optics V, L. Mandel, E. Wolf, eds. (Plenum, New York, 1984), p. 585.

Noel, M.

J. L. Lachambre, P. Lavigne, G. Otis, M. Noel, IEEE J. Quantum Electron. QE-12, 756 (1976).
[CrossRef]

Olesen, A.

F. Morgensen, G. Jacobsen, A. Olesen, Opt. Quantum Electron. 16, 183 (1984).
[CrossRef]

Ostlund, S.

S. Ostlund, D. Rand, J. Sethna, E. Siggia, Physica 8D, 303 (1983).

Otis, G.

J. L. Lachambre, P. Lavigne, G. Otis, M. Noel, IEEE J. Quantum Electron. QE-12, 756 (1976).
[CrossRef]

Park, Y. K.

Pennise, C. A.

L. A. Lugiato, L. M. Narducci, D. K. Bandy, C. A. Pennise, Opt. Commun. 46, 64 (1983).
[CrossRef]

D. K. Bandy, L. M. Narducci, C. A. Pennise, L. A. Lugiato, in Coherence and Quantum Optics V, L. Mandel, E. Wolf, eds. (Plenum, New York, 1984), p. 585.

Puccioni, G. P.

F. T. Arecchi, R. Meucci, G. P. Puccioni, J. R. Tredicce, Phys. Rev. Lett. 49, 1217 (1982).
[CrossRef]

F. T. Arecchi, G. L. Lippi, G. P. Puccioni, J. R. Tredicce, in Coherence and Quantum Optics V, L. Mandel, E. Wolf, eds. (Plenum, New York, 1984), p. 1227;Opt. Commun. (submitted for publication).

Rand, D.

S. Ostlund, D. Rand, J. Sethna, E. Siggia, Physica 8D, 303 (1983).

Salimbeni, R.

P. Burlamacchi, R. Salimbeni, Opt. Commun. 17, 6 (1976).
[CrossRef]

Scholz, H.

H. Scholz, T. Yamada, H. Brand, R. Graham, Phys. Lett. 82A, 321 (1981).

Sethna, J.

S. Ostlund, D. Rand, J. Sethna, E. Siggia, Physica 8D, 303 (1983).

Siggia, E.

S. Ostlund, D. Rand, J. Sethna, E. Siggia, Physica 8D, 303 (1983).

Skolnick, M.

C. J. Buczek, R. J. Freiberg, M. Skolnick, Proc. IEEE 61, 1411 (1973).
[CrossRef]

Small, J.

Spencer, M.

M. Spencer, W. Lamb, Phys. Rev. A 5, 884 (1972).
[CrossRef]

Steier, W. H.

H. L. Stover, W. H. Steier, Appl. Phys. Lett. 8, 91 (1966).
[CrossRef]

Stover, H. L.

H. L. Stover, W. H. Steier, Appl. Phys. Lett. 8, 91 (1966).
[CrossRef]

Szabo, A.

L. E. Erickson, A. Szabo, Appl. Phys. Lett. 18, 433 (1981).
[CrossRef]

Tredicce, J. R.

F. T. Arecchi, R. Meucci, G. P. Puccioni, J. R. Tredicce, Phys. Rev. Lett. 49, 1217 (1982).
[CrossRef]

J. R. Tredicce, G. L. Lippi, N. B. Abraham, F. T. Arecchi, Proc. R. Soc. (London) (to be published).

F. T. Arecchi, G. L. Lippi, G. P. Puccioni, J. R. Tredicce, in Coherence and Quantum Optics V, L. Mandel, E. Wolf, eds. (Plenum, New York, 1984), p. 1227;Opt. Commun. (submitted for publication).

Weiss, C. O.

C. O. Weiss, H. King, Opt. Commun. 44, 59 (1982);C. O. Weiss, A. Godone, A. Olafsson, Phys. Rev. A 28, 892 (1983).
[CrossRef]

Yamada, T.

H. Scholz, T. Yamada, H. Brand, R. Graham, Phys. Lett. 82A, 321 (1981).

T. Yamada, R. Graham, Phys. Rev. Lett. 45, 1322 (1980).
[CrossRef]

Znev, V. S.

N. G. Basov, V. S. Znev, P. G. Krjukov, in Lasers, J. Weber, ed. (Gordon & Breach, New York, 1968), Vol. 10/a, p. 257;F. J. McClung, R. W. Hellwarth, Proc. IEEE 51, 46 (1963).
[CrossRef]

Appl. Phys. Lett.

H. L. Stover, W. H. Steier, Appl. Phys. Lett. 8, 91 (1966).
[CrossRef]

L. E. Erickson, A. Szabo, Appl. Phys. Lett. 18, 433 (1981).
[CrossRef]

Helv. Phys. Acta

E. Brun, B. Derighetti, R. Holzner, D. Meier, Helv. Phys. Acta 56, 825 (1983).

IEEE J. Quantum Electron.

W. Annovazzi, S. Donati, IEEE J. Quantum Electron. QE-16, 859 (1980).

R. Lang, IEEE J. Quantum Electron. QE-18, 979 (1982), and references therein.

W. W. Chow, IEEE J. Quantum Electron. QE-19, 243 (1983).
[CrossRef]

C. J. Buczek, R. J. Freiberg, IEEE J. Quantum Electron. QE-8, 643 (1972).

J. L. Lachambre, P. Lavigne, G. Otis, M. Noel, IEEE J. Quantum Electron. QE-12, 756 (1976).
[CrossRef]

Opt. Commun.

R. Daigle, P. Belanger, Opt. Commun. 23, 165 (1977).
[CrossRef]

P. Burlamacchi, R. Salimbeni, Opt. Commun. 17, 6 (1976).
[CrossRef]

A. Girard, Opt. Commun. 11, 346 (1974).
[CrossRef]

C. O. Weiss, H. King, Opt. Commun. 44, 59 (1982);C. O. Weiss, A. Godone, A. Olafsson, Phys. Rev. A 28, 892 (1983).
[CrossRef]

L. A. Lugiato, L. M. Narducci, D. K. Bandy, C. A. Pennise, Opt. Commun. 46, 64 (1983).
[CrossRef]

Opt. Lett.

Opt. Quantum Electron.

F. Morgensen, G. Jacobsen, A. Olesen, Opt. Quantum Electron. 16, 183 (1984).
[CrossRef]

Phys. Lett.

H. Scholz, T. Yamada, H. Brand, R. Graham, Phys. Lett. 82A, 321 (1981).

H. Haken, Phys. Lett. 53A, 77 (1975).

Phys. Rev. A

M. Spencer, W. Lamb, Phys. Rev. A 5, 884 (1972).
[CrossRef]

N. J. Halas, S.-N. Liu, N. B. Abraham, Phys. Rev. A 28, 2915 (1983).
[CrossRef]

L. W. Casperson, Phys. Rev. A 21, 911 (1980);Phys. Rev. A 23, 248 (1981).
[CrossRef]

Phys. Rev. Lett.

R. S. Gioggia, N. B. Abraham, Phys. Rev. Lett. 51, 650 (1983).
[CrossRef]

T. Yamada, R. Graham, Phys. Rev. Lett. 45, 1322 (1980).
[CrossRef]

F. T. Arecchi, R. Meucci, G. P. Puccioni, J. R. Tredicce, Phys. Rev. Lett. 49, 1217 (1982).
[CrossRef]

Physica

S. Ostlund, D. Rand, J. Sethna, E. Siggia, Physica 8D, 303 (1983).

Proc. IEEE

C. J. Buczek, R. J. Freiberg, M. Skolnick, Proc. IEEE 61, 1411 (1973).
[CrossRef]

Proc. IRE

R. Adler, Proc. IRE 34, 351 (1946).
[CrossRef]

Other

H. Haken, in Handbuch der Physik, L. Genzel, ed. (Springer-Verlag, New York, 1970), Vol. 25/2c.

F. T. Arecchi, Acta Phys. Austraca. (to be published).

F. T. Arecchi, G. L. Lippi, G. P. Puccioni, J. R. Tredicce, in Coherence and Quantum Optics V, L. Mandel, E. Wolf, eds. (Plenum, New York, 1984), p. 1227;Opt. Commun. (submitted for publication).

J. R. Tredicce, G. L. Lippi, N. B. Abraham, F. T. Arecchi, Proc. R. Soc. (London) (to be published).

L. A. Lugiato, presented at the Workshop on Chaotic and Pulse Instabilities, Schloss Elmau, Federal Republic of Germany, May, 1984.

M. Høgh Jensen, P. Bak, T. Bohr, Phys. Rev. A (to be published).

D. K. Bandy, L. M. Narducci, C. A. Pennise, L. A. Lugiato, in Coherence and Quantum Optics V, L. Mandel, E. Wolf, eds. (Plenum, New York, 1984), p. 585.

N. G. Basov, V. S. Znev, P. G. Krjukov, in Lasers, J. Weber, ed. (Gordon & Breach, New York, 1968), Vol. 10/a, p. 257;F. J. McClung, R. W. Hellwarth, Proc. IEEE 51, 46 (1963).
[CrossRef]

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Figures (12)

Fig. 1
Fig. 1

Qualitative plot of the frequency relations among atomic resonance (homogeneous width γ) centered at ω0, cavity resonance (width K) centered at ωc, and injected field at ω1.

Fig. 2
Fig. 2

Steady-state solution for LIS equations. Output intensity versus intensity of the injected field for constant value of the pump parameter (z0) and different values of the detuning (ω1ωL). The dashed line shows the unstable region and the solid line shows the region where the steady-state solutions are stable. For each curve, the critical value of the intensity of the injected field is marked with Aj2. For A2 larger than Aj2, there is no instability.

Fig. 3
Fig. 3

Phase diagram in the parameter space; detuning versus amplitude of the external field. The limits in the unstable region indicate the different kinds of solutions as shown in Figs. 410.

Fig. 4
Fig. 4

Behavior of the system in the subregion 1 of Fig. 3 [(θδ) = −0.0214; A = 0.063]. (a) Intensity as a function of time: giant pulses induce damped relaxation oscillations. The frequency of the giant pulses (Ω1) is much smaller than the relaxation-oscillation frequency, and the peak intensity is regular. (b) Phases ( ϕ ˙ ) as a function of time: the external field locks the frequency of the system during a long period but is not strong enough to maintain this situation forever. When the system unlocks, a rapid change in the phase by a factor 2π occurs. (c) Imaginary part of the field (y) versus the real part of the field (x). Each point is taken at a constant time interval; the different densities of points in the figure indicate the rapid change in phase and the long time during which the systems remains locked. Considering that the intensity is the distance from these points to the origin, the loop in the upper side shows the existence of relaxation oscillations. (d) Population inversion versus time. This curve shows how the process is similar to Q switching. The population inversion grows until it transfers the energy to the field, giving rise to a giant pulse.

Fig. 5
Fig. 5

Intensity versus time with (θδ) = −0.0175 (subregions 2 and 4 of Fig. 3). As the amplitude of the external field decreases, the frequency of the self-Q switching (Ω1) increases, and the relaxation oscillations became undamped. (a) For A = 0.0485, Ω1 is around six times larger than the relaxation-oscillation frequency Ω2. We observe five undamped pulses between the successive giant pulses. (b) For A = 0.0475, Ω1 increased and is around five times Ω2. (c) For θ = 0.0480, Ω1 and Ω2 cannot lock precisely at an integer ratio, and we observe a subharmonic bifurcation in the lowest frequency.

Fig. 6
Fig. 6

(θδ) = 0.035 and A = 0.021 (subregion 5 in Fig. 3). (a) Intensity versus time. Owing to the interaction between Ω1 and Ω2, which are of the same order, a third, low-frequency Ω3 unequal to Ω1 appears. (b) The corresponding Poincaré map, taken for = 0 in the plane xy, shows the inequality of the frequencies by giving a continuous curve indicating the presence of a two-torus as the underlying attractor.

Fig. 7
Fig. 7

For the same value of (θδ) as in Fig. 6 and increasing A = 0.02295, the two frequencies lock at a ratio Ω31 = 14. (a) Intensity versus time. (b) Poincaré map showing a discrete number of pointers (14).

Fig. 8
Fig. 8

(a) A further increase in A causes the system to enter a region of intermittency. The laminar period decreases as A increases until no periodicity can be found. For sake of simplicity, we plot only the intensity maxima versus the time for different values of A. (b) Iteration map of the intensity maxima In+14 versus In. Note the near transparency of the map to the fixed point line (main diagonal), which is a well-known indication of intermittency.

Fig. 9
Fig. 9

Poincaré map in the plane xy for = 0 when the system enters the chaotic region. The nonregion structure of the strange attractor can be easily seen.

Fig. 10
Fig. 10

(θδ) = −0.0214 (subregion 6 in Fig. 3). Plotting intensity as a function of time for different values of A, we observe a transition to chaos through period-doubling bifurcations.

Fig. 11
Fig. 11

Poincaré section in the plane zn for, in the region where Ω1 and Ω3 are unequal.

Fig. 12
Fig. 12

Return map for the phase, corresponding to the Poincare section of Fig. 11. The dashed line shows the return map obtained by resolving Eqs. (10) numerically. It is clear that the system can be described by a circle map. For comparison, we show by means of a continuous line the well-known sine map using (ω1ωL)/[K γ(z0 − 1)]1/2 and A as parameters.

Equations (45)

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Ė = K E ( 1 + i θ ) + g 2 E Δ N γ ( 1 + i δ ) , Δ ˙ N = γ ( Δ N Δ N 0 ) 4 g 2 | E | 2 Δ N γ ( 1 + δ 2 ) ,
θ = ( ω c ω L ) / K , δ = ( ω 0 ω L ) / ,
g 2 = ω μ 2 / 0 υ .
| Ē | 2 = γ γ 4 g 2 [ g 2 Δ N 0 γ K ( 1 + δ 2 ) ] , Δ N ¯ = γ K g 2 ( 1 + δ 2 ) ,
θ = δ
ω c ω L K = ω L ω 0 γ .
λ = γ z 0 2 ± [ γ 2 z 0 2 4 2 K γ ( z 0 1 ) ] 1 / 2 ,
z 0 = g 2 Δ N 0 γ K ( 1 + δ 2 )
4 K γ [ 8 K γ ( 2 K γ 1 ) ] 1 / 2 < z 0 , z 0 < 4 K γ + [ 8 K γ ( 2 K γ 1 ) ] 1 / 2 .
λ = λ R + i λ I γ z 0 2 + i [ 2 K γ ( z 0 1 ) ] 1 / 2 .
Ė = K E ( 1 + i θ ) + g 2 γ E Δ N 1 + i δ + K E 0 e i α K t , Δ ˙ = γ ( Δ N Δ N 0 ) 4 g 2 γ | E | 2 Δ N ( 1 + δ 2 ) ,
α = ω 1 ω K , θ = ω c ω K , δ = ω 0 ω γ .
= ( z 1 ) x + ( θ δ z ) y + A , = ( θ δ z ) x + ( z 1 ) y , K γ ż = z x 2 + y 2 1 + δ 2 z + z 0 ,
x = 2 g ( γ γ ) 1 / 2 E R , y = 2 g ( γ γ ) 1 / 2 E I , z = g 2 Δ N γ K ( 1 + δ 2 ) , A = 2 g E 0 ( γ γ ) 1 / 2 ,
z ¯ = z 0 1 + x ¯ 2 + y ¯ 2 1 + δ 2
A 2 = [ ( z ¯ 1 ) 2 + ( θ δ z ¯ ) 2 ] ( x ¯ 2 + y ¯ 2 ) .
( θ δ ) = K + γ K γ ( ω 1 ω L ) .
λ 3 + [ γ K ( I + 1 ) 2 ( z ¯ 1 ) ] λ 2 + [ ( z 1 ) 2 + ( θ δ z ) 2 2 γ K ( z 1 ) ( I + 1 ) + 2 γ K z I ] λ + { γ K ( I + 1 ) [ ( z ¯ 1 ) 2 + ( θ δ z ¯ ) 2 ] + 2 γ K z ¯ I [ δ ( θ δ z ¯ ) ( z ¯ 1 ) ] } = 0
γ K ( I + 1 ) [ ( z ¯ 1 ) 2 + ( θ δ z ¯ ) 2 ] + 2 γ K z ¯ I [ δ ( θ δ z ¯ ) ( z ¯ 1 ) ] = 0 .
z ¯ 1 1 ; I 1 z 0 1 , z ¯ 2 z 0 4 [ 1 + ( 1 + 8 / z 0 ) 1 / 2 ] , I 2 4 1 + ( 1 + 8 / z 0 ) 1 / 2 1 , z ¯ 3 z 0 4 [ 1 ( 1 + 8 / z 0 ) 1 / 2 ] , I 3 4 1 ( 1 + 8 / z 0 ) 1 / 2 1 .
λ 2 , 3 = λ R ( 2 , 3 ) ± i λ I ( 2 , 3 ) = ± i λ I ( 2 , 3 ) .
[ ( z ¯ 1 ) 2 + ( θ δ z ¯ ) 2 2 γ K ( z ¯ 1 ) ( I + 1 ) + 2 γ K z ¯ I ] = { γ K ( I + 1 ) [ ( z ¯ 1 ) 2 + ( θ δ z ¯ ) 2 ] + 2 γ K z ¯ I [ δ ( θ δ z ¯ ) ( z ¯ 1 ) ] } [ γ K ( I + 1 ) 2 ( z ¯ 1 ) ] . .
0 < z ¯ 1 , I 0 = z 0 1 I , λ 1 < 0 ,
1 z ¯ z 0 4 [ 1 + ( 1 + 8 / z 0 ) 1 / 2 z 0 2 , 4 1 + ( 1 + 8 / z 0 ) 1 / 2 1 < I z 0 1 , λ 1 > 0 ,
z 0 2 z 0 4 [ 1 + ( 1 + 8 / z 0 ) ] 1 / 2 z ¯ < z 0 , 0 < I 4 ( 1 + 1 + 8 / z 0 ) 1 / 2 1 1 , λ 1 < 0 .
2 γ K ( I + 1 ) ( z ¯ 1 ) + 2 γ K z ¯ I + ( z ¯ 1 ) 2 = ( z ¯ 1 ) [ γ K ( I + 1 ) ( z ¯ 1 ) + 2 γ K z ¯ I ] 2 ( z ¯ 1 ) γ K ( I + 1 ) .
λ 2 + [ γ K ( I + 1 ) 2 ( z ¯ 1 ) ] λ + [ ( z ¯ 1 ) 2 + ( θ δ z ¯ ) 2 2 γ K ( z ¯ 1 ) ( I + 1 ) + 2 γ K z ¯ I ] = 0 ,
λ 2 , 3 = [ ( z ¯ 1 ) γ 2 K ( I + 1 ) ] + [ γ 2 4 K 2 ( I + 1 ) 2 + γ K ( z ¯ 1 ) ( I + 1 ) ( θ δ z ¯ ) 2 2 γ K z ¯ I ] 1 / 2 .
λ 2 , 3 γ 2 K z 0 ± i [ 2 γ K ( z 0 1 ) + ( θ δ ) 2 ] 1 / 2 ;
Ω 1 [ 2 γ K ( z 0 1 ) + ( θ δ ) 2 ] 1 / 2 .
λ 1 λ 2 λ 3 = γ K ( z ¯ 1 ) [ ( z ¯ 1 ) ( I + 1 ) + 2 z ¯ I ] , λ 2 λ 3 + λ 1 ( λ 2 + λ 3 ) = ( z ¯ 1 ) 2 + ( θ δ z ¯ ) 2 2 γ K [ ( z ¯ 1 ) ( I + 1 ) + z ¯ I ] , λ 1 + λ 2 + λ 3 = 2 ( z ¯ 1 ) γ K ( I + 1 ) ,
λ 1 + λ 2 + λ 3 = 2 ( z ¯ 1 ) , λ 2 λ 3 + λ 1 ( λ 2 + λ 3 ) = ( z ¯ 1 ) 2 , λ 1 λ 2 λ 3 = ( z ¯ 1 ) [ γ K ( z ¯ 1 ) ( I + 1 ) + 2 γ K z ¯ I ] ,
λ 3 = λ 2 * .
z ¯ = z 0 4 ( 1 + ( 1 + 8 / z 0 ) 1 / 2 I = 4 1 + ( 1 + 8 / z 0 ) 1 / 2 1 .
λ 2 , 3 = { z 0 4 [ 1 + ( 1 + 8 / z 0 ) 1 / 2 ] 1 } ± i { θ δ z 0 4 [ 1 + ( 1 + 8 / z 0 ) 1 / 2 ] } .
Ω 2 = θ δ z 0 4 ( 1 + 8 / z 0 ) 1 / 2 .
A 0 2 ( θ δ ) 2 ( z 0 1 ) A 0 z 0 1 ( θ δ ) .
ω inst = ω 1 ϕ ,
ϕ = arg ( x + i y )
ϕ ˙ = d ϕ d ( K t ) = x y x 2 + y 2
θ n + 1 = f ( θ n ) = θ n + Ω + g ( θ n ) ,
g ( θ n ) = g ( θ n + 1 ) mod 1 .
θ n + 1 = f ( θ n ) = θ n + Ω K 2 π sin ( 2 π θ n ) .
W = lim n f n ( θ ) θ n .
θ n + Q = θ n + P ( rational W = P / Q ) ,

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