Abstract

We discuss the emergence of spatial structures in a ring laser model with transverse effects. The emphasis of this study is on the development of a description that can capture the essential features of transverse dynamics without the need for large-scale numerical efforts. We introduce an extension of the uniform field limit and derive a set of modal equations that we solve with conventional numerical methods. Our solutions show evidence of transverse mode competition in the laser dynamics, leading to both time-dependent and multimode stationary (cooperative frequency-locking) behaviors. In the time-dependent regime we analyze the resulting spatial structures and suggest a scheme for the investigation and characterization of spatial complexity.

© 1990 Optical Society of America

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  1. A. M. Turing, Phil. Trans. R. Soc. London Ser. B 237, 37 (1952); H. Haken, Synergetics—An Introduction (Springer-Verlag, Berling, 1974); G. Nicolis and I. Prigogine, Self-organization in Non-equilibrium Systems (Wiley, New York, 1974).
    [Crossref]
  2. E. N. Lorenz, J. Atmos. Sci. 20, 130 (1963).
    [Crossref]
  3. S. Grossmann and S. Thomae, Z. Naturforsch. 32A, 1353 (1977); M. J. Feigenbaum, J. Stat. Phys. 19, 25 (1978); P. Coullet and C. Tresser, J. Phys. (Paris) Colloq. C5, 39 (1978).
    [Crossref]
  4. H. M. Gibbs, F. A. Hopf, D. L. Kaplan, and R. L. Shoemaker, Phys. Rev. Lett. 46, 474 (1981).
    [Crossref]
  5. F. T. Arecchi, R. Meucci, G. P. Puccioni, and J. R. Tredicce, Phys. Rev. Lett. 49, 1217 (1982).
    [Crossref]
  6. R. S. Gioggia and N. B. Abraham, Phys. Rev. Lett. 51, 650 (1983).
    [Crossref]
  7. A. M. Albano, J. Abounadi, T. H. Chyba, C. E. Searle, S. Yong, R. S. Gioggia, and N. B. Abraham, J. Opt. Soc. Am. B 2, 47 (1985).
    [Crossref]
  8. G. P. Puccioni, A. Poggi, W. Gadomski, J. R. Tredicce, and F. T. Arecchi, Phys. Rev. Lett. 55, 339 (1985).
    [Crossref]
  9. See, for example, F. Heslot, B. Castaign, and A. Libchaber, Phys. Rev. A 36, 5870 (1987).
    [Crossref]
  10. M. Le Berre, E. Ressayre, and A. Tallet, Phys. Rev. A 25, 1604 (1982).
    [Crossref]
  11. D. W. McLaughlin, J. V. Moloney, and A. C. Newell, Phys. Rev. Lett. 51, 75 (1983).
    [Crossref]
  12. W. J. Firth, in Instabilities an Chaos in Quantum Optics II, N. B. Abraham, F. T. Arecchi, and L. A. Lugiato, eds. (Prenum, New York, 1988), p. 219.
    [Crossref]
  13. L. A. Lugiato and R. Lefever, Phys. Rev. lett. 58, 220 (1987); L. A. Lugiato and C. Oldano, Phys. Rev. A 37, 3896 (1988).
    [Crossref]
  14. J. V. Moloney and H. M. Gibbs, Phys. Rev. Lett. 58, 467 (1982).
  15. J. V. Moloney, H. Adachihara, D. W. McLaughlin, and A. C. Newell, in Chaos, Noise and Fractals, E. R. Pike and L. A. Lugiato, eds. (Hilger, London, 1988), p. 137.
  16. L. A. Lugiato, F. Prati, L. M. Narducci, and G.-L. Oppo, Opt. Commun. 69, 387 (1989).
    [Crossref]
  17. G. Grynberg, E. Le Bihan, P. Verkerk, P. Simoneau, J. R. R. Leite, D. Bloch, S. Le Boiteux, and M. Ducloy, Opt. Commun. 67, 363 (1988).
    [Crossref]
  18. G. Giusfredi, J. F. Valley, R. Pon, G. Khitrova, and H. M. Gibbs, J. Opt. Soc. Am. B 5, 1181 (1988).
    [Crossref]
  19. L. A. Lugiato, C. Oldano, and L. M. Narducci, J. Opt. Soc. Am. B 5, 879 (1987).
    [Crossref]
  20. L. A. Lugiato, G.-L. Oppo, M. A. Pernigo, J. R. Tredicce, L. M. Narducci, and D. K. Bandy, Opt. Commun. 68, 63 (1988).
    [Crossref]
  21. J. R. Tredicce, E. J. Quel, A. M. Ghazzawi, C. Green, M. A. Pernigo, L. M. Narducci, and L. A. Lugiato, Phys. Rev. Lett. 62, 1274 (1989).
    [Crossref]
  22. C. Tamm, Phys. Rev. A 38, 5960 (1988).
    [Crossref] [PubMed]
  23. L. A. Lugiato, L. M. Narducci, E. V. Eschenazi, D. K. Bandy, and N. B. Abraham, Phys. Rev. A 32, 1563 (1985); L. M. Narducci, J. R. Tredicce, L. A. Lugiato, N. B. Abraham, and D. K. Bandy, Phys. Rev. A 33, 1842 (1986).
    [Crossref] [PubMed]
  24. L. A. Lugiato, F. Prati, L. M. Narducci, P. Ru, J. R. Tredicce, and D. K. Bandy, Phys. Rev. A 37, 3847 (1988).
    [Crossref]
  25. A. F. Suchkov, Zh. Eksp. Teor. Fiz. 49, 1495 (1965) [Sov. Phys. JETP 22, 1026 (1966)].
  26. R. G. Allakhverdyan, A. N. Oraevsky, and A. F. Suchkov, Fiz. Tekh. Probl. Razrab. Polezn. Iskop. 4, 341 (1970) [Sov. Phys. Semicond. 4, 227 (1970)].
  27. V. E. Kuzin and A. F. Suchkov, Kvantovaya Elektron. (Moscow) 3, 52 (1972) [Sov. J. Quantum Electron. 2, 236 (1972)].
  28. I. M. Belousova, G. N. Vinokurov, O. B. Danilov, and N. N. Rozanov, Zh. Eksp. Teor. Fiz. 57, 1146 (1967) [Sov. Phys. JETP 25, 761 (1967)].
  29. P. W. Smith, Appl. Phys. Lett. 13, 235 (1968).
    [Crossref]
  30. D. H. Auston, IEEE J. Quantum Electron. 5, 471 (1968).
    [Crossref]
  31. P. Hollinger, C. Jung, and H. Weber, Department of Physics, University of Kaiserslautern, D-6750 Kaiserslautern, Federal Republic of Germany (personal communication).
  32. P. Hollinger and C. Jung, J. Opt. Soc. Am. B 2, 218 (1985); F. Hollinger, C. Jung, and H. Weber, in Optical Instabilities, R. W. Boyd, M. G. Raymer, and L. M. Narducci, eds. (Cambridge U. Press, Cambridge, 1986), p. 260; see also M. L. Shih and P. W. Milonni, Opt. Commun. 49, 155 (1984).
    [Crossref]
  33. C. O. Weiss and H. King, Opt. Commun. 44, 59 (1982).
    [Crossref]
  34. N. J. Halas, S. N. Liu, and N. B. Abraham, Phys. Rev. A 28, 2915 (1983).
    [Crossref]
  35. D. J. Biswas and R. G. Harrison, Phys. Rev. A 32, 3835 (1985).
    [Crossref] [PubMed]
  36. P. Coullet, L. Gil, and J. Lega, Phys. Rev. Lett. 62, 1619 (1989).
    [Crossref] [PubMed]
  37. P. Ru, L. M. Narducci, J. R. Tredicce, D. K. Bandy, and L. A. Lugiato, Opt. Commun. 63, 310 (1987).
    [Crossref]
  38. See, for example, A. Yariv, Optical Electronics, 3rd ed. (Holt, Rinehart & Winston, New York, 1985), p. 32.

1989 (3)

L. A. Lugiato, F. Prati, L. M. Narducci, and G.-L. Oppo, Opt. Commun. 69, 387 (1989).
[Crossref]

J. R. Tredicce, E. J. Quel, A. M. Ghazzawi, C. Green, M. A. Pernigo, L. M. Narducci, and L. A. Lugiato, Phys. Rev. Lett. 62, 1274 (1989).
[Crossref]

P. Coullet, L. Gil, and J. Lega, Phys. Rev. Lett. 62, 1619 (1989).
[Crossref] [PubMed]

1988 (5)

G. Giusfredi, J. F. Valley, R. Pon, G. Khitrova, and H. M. Gibbs, J. Opt. Soc. Am. B 5, 1181 (1988).
[Crossref]

C. Tamm, Phys. Rev. A 38, 5960 (1988).
[Crossref] [PubMed]

L. A. Lugiato, F. Prati, L. M. Narducci, P. Ru, J. R. Tredicce, and D. K. Bandy, Phys. Rev. A 37, 3847 (1988).
[Crossref]

G. Grynberg, E. Le Bihan, P. Verkerk, P. Simoneau, J. R. R. Leite, D. Bloch, S. Le Boiteux, and M. Ducloy, Opt. Commun. 67, 363 (1988).
[Crossref]

L. A. Lugiato, G.-L. Oppo, M. A. Pernigo, J. R. Tredicce, L. M. Narducci, and D. K. Bandy, Opt. Commun. 68, 63 (1988).
[Crossref]

1987 (4)

L. A. Lugiato and R. Lefever, Phys. Rev. lett. 58, 220 (1987); L. A. Lugiato and C. Oldano, Phys. Rev. A 37, 3896 (1988).
[Crossref]

See, for example, F. Heslot, B. Castaign, and A. Libchaber, Phys. Rev. A 36, 5870 (1987).
[Crossref]

L. A. Lugiato, C. Oldano, and L. M. Narducci, J. Opt. Soc. Am. B 5, 879 (1987).
[Crossref]

P. Ru, L. M. Narducci, J. R. Tredicce, D. K. Bandy, and L. A. Lugiato, Opt. Commun. 63, 310 (1987).
[Crossref]

1985 (5)

P. Hollinger and C. Jung, J. Opt. Soc. Am. B 2, 218 (1985); F. Hollinger, C. Jung, and H. Weber, in Optical Instabilities, R. W. Boyd, M. G. Raymer, and L. M. Narducci, eds. (Cambridge U. Press, Cambridge, 1986), p. 260; see also M. L. Shih and P. W. Milonni, Opt. Commun. 49, 155 (1984).
[Crossref]

A. M. Albano, J. Abounadi, T. H. Chyba, C. E. Searle, S. Yong, R. S. Gioggia, and N. B. Abraham, J. Opt. Soc. Am. B 2, 47 (1985).
[Crossref]

D. J. Biswas and R. G. Harrison, Phys. Rev. A 32, 3835 (1985).
[Crossref] [PubMed]

G. P. Puccioni, A. Poggi, W. Gadomski, J. R. Tredicce, and F. T. Arecchi, Phys. Rev. Lett. 55, 339 (1985).
[Crossref]

L. A. Lugiato, L. M. Narducci, E. V. Eschenazi, D. K. Bandy, and N. B. Abraham, Phys. Rev. A 32, 1563 (1985); L. M. Narducci, J. R. Tredicce, L. A. Lugiato, N. B. Abraham, and D. K. Bandy, Phys. Rev. A 33, 1842 (1986).
[Crossref] [PubMed]

1983 (3)

D. W. McLaughlin, J. V. Moloney, and A. C. Newell, Phys. Rev. Lett. 51, 75 (1983).
[Crossref]

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

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

1982 (4)

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

C. O. Weiss and H. King, Opt. Commun. 44, 59 (1982).
[Crossref]

M. Le Berre, E. Ressayre, and A. Tallet, Phys. Rev. A 25, 1604 (1982).
[Crossref]

J. V. Moloney and H. M. Gibbs, Phys. Rev. Lett. 58, 467 (1982).

1981 (1)

H. M. Gibbs, F. A. Hopf, D. L. Kaplan, and R. L. Shoemaker, Phys. Rev. Lett. 46, 474 (1981).
[Crossref]

1977 (1)

S. Grossmann and S. Thomae, Z. Naturforsch. 32A, 1353 (1977); M. J. Feigenbaum, J. Stat. Phys. 19, 25 (1978); P. Coullet and C. Tresser, J. Phys. (Paris) Colloq. C5, 39 (1978).
[Crossref]

1972 (1)

V. E. Kuzin and A. F. Suchkov, Kvantovaya Elektron. (Moscow) 3, 52 (1972) [Sov. J. Quantum Electron. 2, 236 (1972)].

1970 (1)

R. G. Allakhverdyan, A. N. Oraevsky, and A. F. Suchkov, Fiz. Tekh. Probl. Razrab. Polezn. Iskop. 4, 341 (1970) [Sov. Phys. Semicond. 4, 227 (1970)].

1968 (2)

P. W. Smith, Appl. Phys. Lett. 13, 235 (1968).
[Crossref]

D. H. Auston, IEEE J. Quantum Electron. 5, 471 (1968).
[Crossref]

1967 (1)

I. M. Belousova, G. N. Vinokurov, O. B. Danilov, and N. N. Rozanov, Zh. Eksp. Teor. Fiz. 57, 1146 (1967) [Sov. Phys. JETP 25, 761 (1967)].

1965 (1)

A. F. Suchkov, Zh. Eksp. Teor. Fiz. 49, 1495 (1965) [Sov. Phys. JETP 22, 1026 (1966)].

1963 (1)

E. N. Lorenz, J. Atmos. Sci. 20, 130 (1963).
[Crossref]

1952 (1)

A. M. Turing, Phil. Trans. R. Soc. London Ser. B 237, 37 (1952); H. Haken, Synergetics—An Introduction (Springer-Verlag, Berling, 1974); G. Nicolis and I. Prigogine, Self-organization in Non-equilibrium Systems (Wiley, New York, 1974).
[Crossref]

Abounadi, J.

Abraham, N. B.

A. M. Albano, J. Abounadi, T. H. Chyba, C. E. Searle, S. Yong, R. S. Gioggia, and N. B. Abraham, J. Opt. Soc. Am. B 2, 47 (1985).
[Crossref]

L. A. Lugiato, L. M. Narducci, E. V. Eschenazi, D. K. Bandy, and N. B. Abraham, Phys. Rev. A 32, 1563 (1985); L. M. Narducci, J. R. Tredicce, L. A. Lugiato, N. B. Abraham, and D. K. Bandy, Phys. Rev. A 33, 1842 (1986).
[Crossref] [PubMed]

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

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

Adachihara, H.

J. V. Moloney, H. Adachihara, D. W. McLaughlin, and A. C. Newell, in Chaos, Noise and Fractals, E. R. Pike and L. A. Lugiato, eds. (Hilger, London, 1988), p. 137.

Albano, A. M.

Allakhverdyan, R. G.

R. G. Allakhverdyan, A. N. Oraevsky, and A. F. Suchkov, Fiz. Tekh. Probl. Razrab. Polezn. Iskop. 4, 341 (1970) [Sov. Phys. Semicond. 4, 227 (1970)].

Arecchi, F. T.

G. P. Puccioni, A. Poggi, W. Gadomski, J. R. Tredicce, and F. T. Arecchi, Phys. Rev. Lett. 55, 339 (1985).
[Crossref]

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

Auston, D. H.

D. H. Auston, IEEE J. Quantum Electron. 5, 471 (1968).
[Crossref]

Bandy, D. K.

L. A. Lugiato, F. Prati, L. M. Narducci, P. Ru, J. R. Tredicce, and D. K. Bandy, Phys. Rev. A 37, 3847 (1988).
[Crossref]

L. A. Lugiato, G.-L. Oppo, M. A. Pernigo, J. R. Tredicce, L. M. Narducci, and D. K. Bandy, Opt. Commun. 68, 63 (1988).
[Crossref]

P. Ru, L. M. Narducci, J. R. Tredicce, D. K. Bandy, and L. A. Lugiato, Opt. Commun. 63, 310 (1987).
[Crossref]

L. A. Lugiato, L. M. Narducci, E. V. Eschenazi, D. K. Bandy, and N. B. Abraham, Phys. Rev. A 32, 1563 (1985); L. M. Narducci, J. R. Tredicce, L. A. Lugiato, N. B. Abraham, and D. K. Bandy, Phys. Rev. A 33, 1842 (1986).
[Crossref] [PubMed]

Belousova, I. M.

I. M. Belousova, G. N. Vinokurov, O. B. Danilov, and N. N. Rozanov, Zh. Eksp. Teor. Fiz. 57, 1146 (1967) [Sov. Phys. JETP 25, 761 (1967)].

Biswas, D. J.

D. J. Biswas and R. G. Harrison, Phys. Rev. A 32, 3835 (1985).
[Crossref] [PubMed]

Bloch, D.

G. Grynberg, E. Le Bihan, P. Verkerk, P. Simoneau, J. R. R. Leite, D. Bloch, S. Le Boiteux, and M. Ducloy, Opt. Commun. 67, 363 (1988).
[Crossref]

Castaign, B.

See, for example, F. Heslot, B. Castaign, and A. Libchaber, Phys. Rev. A 36, 5870 (1987).
[Crossref]

Chyba, T. H.

Coullet, P.

P. Coullet, L. Gil, and J. Lega, Phys. Rev. Lett. 62, 1619 (1989).
[Crossref] [PubMed]

Danilov, O. B.

I. M. Belousova, G. N. Vinokurov, O. B. Danilov, and N. N. Rozanov, Zh. Eksp. Teor. Fiz. 57, 1146 (1967) [Sov. Phys. JETP 25, 761 (1967)].

Ducloy, M.

G. Grynberg, E. Le Bihan, P. Verkerk, P. Simoneau, J. R. R. Leite, D. Bloch, S. Le Boiteux, and M. Ducloy, Opt. Commun. 67, 363 (1988).
[Crossref]

Eschenazi, E. V.

L. A. Lugiato, L. M. Narducci, E. V. Eschenazi, D. K. Bandy, and N. B. Abraham, Phys. Rev. A 32, 1563 (1985); L. M. Narducci, J. R. Tredicce, L. A. Lugiato, N. B. Abraham, and D. K. Bandy, Phys. Rev. A 33, 1842 (1986).
[Crossref] [PubMed]

Firth, W. J.

W. J. Firth, in Instabilities an Chaos in Quantum Optics II, N. B. Abraham, F. T. Arecchi, and L. A. Lugiato, eds. (Prenum, New York, 1988), p. 219.
[Crossref]

Gadomski, W.

G. P. Puccioni, A. Poggi, W. Gadomski, J. R. Tredicce, and F. T. Arecchi, Phys. Rev. Lett. 55, 339 (1985).
[Crossref]

Ghazzawi, A. M.

J. R. Tredicce, E. J. Quel, A. M. Ghazzawi, C. Green, M. A. Pernigo, L. M. Narducci, and L. A. Lugiato, Phys. Rev. Lett. 62, 1274 (1989).
[Crossref]

Gibbs, H. M.

G. Giusfredi, J. F. Valley, R. Pon, G. Khitrova, and H. M. Gibbs, J. Opt. Soc. Am. B 5, 1181 (1988).
[Crossref]

J. V. Moloney and H. M. Gibbs, Phys. Rev. Lett. 58, 467 (1982).

H. M. Gibbs, F. A. Hopf, D. L. Kaplan, and R. L. Shoemaker, Phys. Rev. Lett. 46, 474 (1981).
[Crossref]

Gil, L.

P. Coullet, L. Gil, and J. Lega, Phys. Rev. Lett. 62, 1619 (1989).
[Crossref] [PubMed]

Gioggia, R. S.

Giusfredi, G.

Green, C.

J. R. Tredicce, E. J. Quel, A. M. Ghazzawi, C. Green, M. A. Pernigo, L. M. Narducci, and L. A. Lugiato, Phys. Rev. Lett. 62, 1274 (1989).
[Crossref]

Grossmann, S.

S. Grossmann and S. Thomae, Z. Naturforsch. 32A, 1353 (1977); M. J. Feigenbaum, J. Stat. Phys. 19, 25 (1978); P. Coullet and C. Tresser, J. Phys. (Paris) Colloq. C5, 39 (1978).
[Crossref]

Grynberg, G.

G. Grynberg, E. Le Bihan, P. Verkerk, P. Simoneau, J. R. R. Leite, D. Bloch, S. Le Boiteux, and M. Ducloy, Opt. Commun. 67, 363 (1988).
[Crossref]

Halas, N. J.

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

Harrison, R. G.

D. J. Biswas and R. G. Harrison, Phys. Rev. A 32, 3835 (1985).
[Crossref] [PubMed]

Heslot, F.

See, for example, F. Heslot, B. Castaign, and A. Libchaber, Phys. Rev. A 36, 5870 (1987).
[Crossref]

Hollinger, P.

Hopf, F. A.

H. M. Gibbs, F. A. Hopf, D. L. Kaplan, and R. L. Shoemaker, Phys. Rev. Lett. 46, 474 (1981).
[Crossref]

Jung, C.

Kaplan, D. L.

H. M. Gibbs, F. A. Hopf, D. L. Kaplan, and R. L. Shoemaker, Phys. Rev. Lett. 46, 474 (1981).
[Crossref]

Khitrova, G.

King, H.

C. O. Weiss and H. King, Opt. Commun. 44, 59 (1982).
[Crossref]

Kuzin, V. E.

V. E. Kuzin and A. F. Suchkov, Kvantovaya Elektron. (Moscow) 3, 52 (1972) [Sov. J. Quantum Electron. 2, 236 (1972)].

Le Berre, M.

M. Le Berre, E. Ressayre, and A. Tallet, Phys. Rev. A 25, 1604 (1982).
[Crossref]

Le Bihan, E.

G. Grynberg, E. Le Bihan, P. Verkerk, P. Simoneau, J. R. R. Leite, D. Bloch, S. Le Boiteux, and M. Ducloy, Opt. Commun. 67, 363 (1988).
[Crossref]

Le Boiteux, S.

G. Grynberg, E. Le Bihan, P. Verkerk, P. Simoneau, J. R. R. Leite, D. Bloch, S. Le Boiteux, and M. Ducloy, Opt. Commun. 67, 363 (1988).
[Crossref]

Lefever, R.

L. A. Lugiato and R. Lefever, Phys. Rev. lett. 58, 220 (1987); L. A. Lugiato and C. Oldano, Phys. Rev. A 37, 3896 (1988).
[Crossref]

Lega, J.

P. Coullet, L. Gil, and J. Lega, Phys. Rev. Lett. 62, 1619 (1989).
[Crossref] [PubMed]

Leite, J. R. R.

G. Grynberg, E. Le Bihan, P. Verkerk, P. Simoneau, J. R. R. Leite, D. Bloch, S. Le Boiteux, and M. Ducloy, Opt. Commun. 67, 363 (1988).
[Crossref]

Libchaber, A.

See, for example, F. Heslot, B. Castaign, and A. Libchaber, Phys. Rev. A 36, 5870 (1987).
[Crossref]

Liu, S. N.

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

Lorenz, E. N.

E. N. Lorenz, J. Atmos. Sci. 20, 130 (1963).
[Crossref]

Lugiato, L. A.

L. A. Lugiato, F. Prati, L. M. Narducci, and G.-L. Oppo, Opt. Commun. 69, 387 (1989).
[Crossref]

J. R. Tredicce, E. J. Quel, A. M. Ghazzawi, C. Green, M. A. Pernigo, L. M. Narducci, and L. A. Lugiato, Phys. Rev. Lett. 62, 1274 (1989).
[Crossref]

L. A. Lugiato, F. Prati, L. M. Narducci, P. Ru, J. R. Tredicce, and D. K. Bandy, Phys. Rev. A 37, 3847 (1988).
[Crossref]

L. A. Lugiato, G.-L. Oppo, M. A. Pernigo, J. R. Tredicce, L. M. Narducci, and D. K. Bandy, Opt. Commun. 68, 63 (1988).
[Crossref]

L. A. Lugiato and R. Lefever, Phys. Rev. lett. 58, 220 (1987); L. A. Lugiato and C. Oldano, Phys. Rev. A 37, 3896 (1988).
[Crossref]

P. Ru, L. M. Narducci, J. R. Tredicce, D. K. Bandy, and L. A. Lugiato, Opt. Commun. 63, 310 (1987).
[Crossref]

L. A. Lugiato, C. Oldano, and L. M. Narducci, J. Opt. Soc. Am. B 5, 879 (1987).
[Crossref]

L. A. Lugiato, L. M. Narducci, E. V. Eschenazi, D. K. Bandy, and N. B. Abraham, Phys. Rev. A 32, 1563 (1985); L. M. Narducci, J. R. Tredicce, L. A. Lugiato, N. B. Abraham, and D. K. Bandy, Phys. Rev. A 33, 1842 (1986).
[Crossref] [PubMed]

McLaughlin, D. W.

D. W. McLaughlin, J. V. Moloney, and A. C. Newell, Phys. Rev. Lett. 51, 75 (1983).
[Crossref]

J. V. Moloney, H. Adachihara, D. W. McLaughlin, and A. C. Newell, in Chaos, Noise and Fractals, E. R. Pike and L. A. Lugiato, eds. (Hilger, London, 1988), p. 137.

Meucci, R.

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

Moloney, J. V.

D. W. McLaughlin, J. V. Moloney, and A. C. Newell, Phys. Rev. Lett. 51, 75 (1983).
[Crossref]

J. V. Moloney and H. M. Gibbs, Phys. Rev. Lett. 58, 467 (1982).

J. V. Moloney, H. Adachihara, D. W. McLaughlin, and A. C. Newell, in Chaos, Noise and Fractals, E. R. Pike and L. A. Lugiato, eds. (Hilger, London, 1988), p. 137.

Narducci, L. M.

L. A. Lugiato, F. Prati, L. M. Narducci, and G.-L. Oppo, Opt. Commun. 69, 387 (1989).
[Crossref]

J. R. Tredicce, E. J. Quel, A. M. Ghazzawi, C. Green, M. A. Pernigo, L. M. Narducci, and L. A. Lugiato, Phys. Rev. Lett. 62, 1274 (1989).
[Crossref]

L. A. Lugiato, G.-L. Oppo, M. A. Pernigo, J. R. Tredicce, L. M. Narducci, and D. K. Bandy, Opt. Commun. 68, 63 (1988).
[Crossref]

L. A. Lugiato, F. Prati, L. M. Narducci, P. Ru, J. R. Tredicce, and D. K. Bandy, Phys. Rev. A 37, 3847 (1988).
[Crossref]

P. Ru, L. M. Narducci, J. R. Tredicce, D. K. Bandy, and L. A. Lugiato, Opt. Commun. 63, 310 (1987).
[Crossref]

L. A. Lugiato, C. Oldano, and L. M. Narducci, J. Opt. Soc. Am. B 5, 879 (1987).
[Crossref]

L. A. Lugiato, L. M. Narducci, E. V. Eschenazi, D. K. Bandy, and N. B. Abraham, Phys. Rev. A 32, 1563 (1985); L. M. Narducci, J. R. Tredicce, L. A. Lugiato, N. B. Abraham, and D. K. Bandy, Phys. Rev. A 33, 1842 (1986).
[Crossref] [PubMed]

Newell, A. C.

D. W. McLaughlin, J. V. Moloney, and A. C. Newell, Phys. Rev. Lett. 51, 75 (1983).
[Crossref]

J. V. Moloney, H. Adachihara, D. W. McLaughlin, and A. C. Newell, in Chaos, Noise and Fractals, E. R. Pike and L. A. Lugiato, eds. (Hilger, London, 1988), p. 137.

Oldano, C.

Oppo, G.-L.

L. A. Lugiato, F. Prati, L. M. Narducci, and G.-L. Oppo, Opt. Commun. 69, 387 (1989).
[Crossref]

L. A. Lugiato, G.-L. Oppo, M. A. Pernigo, J. R. Tredicce, L. M. Narducci, and D. K. Bandy, Opt. Commun. 68, 63 (1988).
[Crossref]

Oraevsky, A. N.

R. G. Allakhverdyan, A. N. Oraevsky, and A. F. Suchkov, Fiz. Tekh. Probl. Razrab. Polezn. Iskop. 4, 341 (1970) [Sov. Phys. Semicond. 4, 227 (1970)].

Pernigo, M. A.

J. R. Tredicce, E. J. Quel, A. M. Ghazzawi, C. Green, M. A. Pernigo, L. M. Narducci, and L. A. Lugiato, Phys. Rev. Lett. 62, 1274 (1989).
[Crossref]

L. A. Lugiato, G.-L. Oppo, M. A. Pernigo, J. R. Tredicce, L. M. Narducci, and D. K. Bandy, Opt. Commun. 68, 63 (1988).
[Crossref]

Poggi, A.

G. P. Puccioni, A. Poggi, W. Gadomski, J. R. Tredicce, and F. T. Arecchi, Phys. Rev. Lett. 55, 339 (1985).
[Crossref]

Pon, R.

Prati, F.

L. A. Lugiato, F. Prati, L. M. Narducci, and G.-L. Oppo, Opt. Commun. 69, 387 (1989).
[Crossref]

L. A. Lugiato, F. Prati, L. M. Narducci, P. Ru, J. R. Tredicce, and D. K. Bandy, Phys. Rev. A 37, 3847 (1988).
[Crossref]

Puccioni, G. P.

G. P. Puccioni, A. Poggi, W. Gadomski, J. R. Tredicce, and F. T. Arecchi, Phys. Rev. Lett. 55, 339 (1985).
[Crossref]

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

Quel, E. J.

J. R. Tredicce, E. J. Quel, A. M. Ghazzawi, C. Green, M. A. Pernigo, L. M. Narducci, and L. A. Lugiato, Phys. Rev. Lett. 62, 1274 (1989).
[Crossref]

Ressayre, E.

M. Le Berre, E. Ressayre, and A. Tallet, Phys. Rev. A 25, 1604 (1982).
[Crossref]

Rozanov, N. N.

I. M. Belousova, G. N. Vinokurov, O. B. Danilov, and N. N. Rozanov, Zh. Eksp. Teor. Fiz. 57, 1146 (1967) [Sov. Phys. JETP 25, 761 (1967)].

Ru, P.

L. A. Lugiato, F. Prati, L. M. Narducci, P. Ru, J. R. Tredicce, and D. K. Bandy, Phys. Rev. A 37, 3847 (1988).
[Crossref]

P. Ru, L. M. Narducci, J. R. Tredicce, D. K. Bandy, and L. A. Lugiato, Opt. Commun. 63, 310 (1987).
[Crossref]

Searle, C. E.

Shoemaker, R. L.

H. M. Gibbs, F. A. Hopf, D. L. Kaplan, and R. L. Shoemaker, Phys. Rev. Lett. 46, 474 (1981).
[Crossref]

Simoneau, P.

G. Grynberg, E. Le Bihan, P. Verkerk, P. Simoneau, J. R. R. Leite, D. Bloch, S. Le Boiteux, and M. Ducloy, Opt. Commun. 67, 363 (1988).
[Crossref]

Smith, P. W.

P. W. Smith, Appl. Phys. Lett. 13, 235 (1968).
[Crossref]

Suchkov, A. F.

V. E. Kuzin and A. F. Suchkov, Kvantovaya Elektron. (Moscow) 3, 52 (1972) [Sov. J. Quantum Electron. 2, 236 (1972)].

R. G. Allakhverdyan, A. N. Oraevsky, and A. F. Suchkov, Fiz. Tekh. Probl. Razrab. Polezn. Iskop. 4, 341 (1970) [Sov. Phys. Semicond. 4, 227 (1970)].

A. F. Suchkov, Zh. Eksp. Teor. Fiz. 49, 1495 (1965) [Sov. Phys. JETP 22, 1026 (1966)].

Tallet, A.

M. Le Berre, E. Ressayre, and A. Tallet, Phys. Rev. A 25, 1604 (1982).
[Crossref]

Tamm, C.

C. Tamm, Phys. Rev. A 38, 5960 (1988).
[Crossref] [PubMed]

Thomae, S.

S. Grossmann and S. Thomae, Z. Naturforsch. 32A, 1353 (1977); M. J. Feigenbaum, J. Stat. Phys. 19, 25 (1978); P. Coullet and C. Tresser, J. Phys. (Paris) Colloq. C5, 39 (1978).
[Crossref]

Tredicce, J. R.

J. R. Tredicce, E. J. Quel, A. M. Ghazzawi, C. Green, M. A. Pernigo, L. M. Narducci, and L. A. Lugiato, Phys. Rev. Lett. 62, 1274 (1989).
[Crossref]

L. A. Lugiato, F. Prati, L. M. Narducci, P. Ru, J. R. Tredicce, and D. K. Bandy, Phys. Rev. A 37, 3847 (1988).
[Crossref]

L. A. Lugiato, G.-L. Oppo, M. A. Pernigo, J. R. Tredicce, L. M. Narducci, and D. K. Bandy, Opt. Commun. 68, 63 (1988).
[Crossref]

P. Ru, L. M. Narducci, J. R. Tredicce, D. K. Bandy, and L. A. Lugiato, Opt. Commun. 63, 310 (1987).
[Crossref]

G. P. Puccioni, A. Poggi, W. Gadomski, J. R. Tredicce, and F. T. Arecchi, Phys. Rev. Lett. 55, 339 (1985).
[Crossref]

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

Turing, A. M.

A. M. Turing, Phil. Trans. R. Soc. London Ser. B 237, 37 (1952); H. Haken, Synergetics—An Introduction (Springer-Verlag, Berling, 1974); G. Nicolis and I. Prigogine, Self-organization in Non-equilibrium Systems (Wiley, New York, 1974).
[Crossref]

Valley, J. F.

Verkerk, P.

G. Grynberg, E. Le Bihan, P. Verkerk, P. Simoneau, J. R. R. Leite, D. Bloch, S. Le Boiteux, and M. Ducloy, Opt. Commun. 67, 363 (1988).
[Crossref]

Vinokurov, G. N.

I. M. Belousova, G. N. Vinokurov, O. B. Danilov, and N. N. Rozanov, Zh. Eksp. Teor. Fiz. 57, 1146 (1967) [Sov. Phys. JETP 25, 761 (1967)].

Weber, H.

P. Hollinger, C. Jung, and H. Weber, Department of Physics, University of Kaiserslautern, D-6750 Kaiserslautern, Federal Republic of Germany (personal communication).

Weiss, C. O.

C. O. Weiss and H. King, Opt. Commun. 44, 59 (1982).
[Crossref]

Yariv, A.

See, for example, A. Yariv, Optical Electronics, 3rd ed. (Holt, Rinehart & Winston, New York, 1985), p. 32.

Yong, S.

Appl. Phys. Lett. (1)

P. W. Smith, Appl. Phys. Lett. 13, 235 (1968).
[Crossref]

Fiz. Tekh. Probl. Razrab. Polezn. Iskop. (1)

R. G. Allakhverdyan, A. N. Oraevsky, and A. F. Suchkov, Fiz. Tekh. Probl. Razrab. Polezn. Iskop. 4, 341 (1970) [Sov. Phys. Semicond. 4, 227 (1970)].

IEEE J. Quantum Electron. (1)

D. H. Auston, IEEE J. Quantum Electron. 5, 471 (1968).
[Crossref]

J. Atmos. Sci. (1)

E. N. Lorenz, J. Atmos. Sci. 20, 130 (1963).
[Crossref]

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

Kvantovaya Elektron. (Moscow) (1)

V. E. Kuzin and A. F. Suchkov, Kvantovaya Elektron. (Moscow) 3, 52 (1972) [Sov. J. Quantum Electron. 2, 236 (1972)].

Opt. Commun. (5)

L. A. Lugiato, F. Prati, L. M. Narducci, and G.-L. Oppo, Opt. Commun. 69, 387 (1989).
[Crossref]

G. Grynberg, E. Le Bihan, P. Verkerk, P. Simoneau, J. R. R. Leite, D. Bloch, S. Le Boiteux, and M. Ducloy, Opt. Commun. 67, 363 (1988).
[Crossref]

C. O. Weiss and H. King, Opt. Commun. 44, 59 (1982).
[Crossref]

P. Ru, L. M. Narducci, J. R. Tredicce, D. K. Bandy, and L. A. Lugiato, Opt. Commun. 63, 310 (1987).
[Crossref]

L. A. Lugiato, G.-L. Oppo, M. A. Pernigo, J. R. Tredicce, L. M. Narducci, and D. K. Bandy, Opt. Commun. 68, 63 (1988).
[Crossref]

Phil. Trans. R. Soc. London Ser. B (1)

A. M. Turing, Phil. Trans. R. Soc. London Ser. B 237, 37 (1952); H. Haken, Synergetics—An Introduction (Springer-Verlag, Berling, 1974); G. Nicolis and I. Prigogine, Self-organization in Non-equilibrium Systems (Wiley, New York, 1974).
[Crossref]

Phys. Rev. A (7)

See, for example, F. Heslot, B. Castaign, and A. Libchaber, Phys. Rev. A 36, 5870 (1987).
[Crossref]

M. Le Berre, E. Ressayre, and A. Tallet, Phys. Rev. A 25, 1604 (1982).
[Crossref]

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

D. J. Biswas and R. G. Harrison, Phys. Rev. A 32, 3835 (1985).
[Crossref] [PubMed]

C. Tamm, Phys. Rev. A 38, 5960 (1988).
[Crossref] [PubMed]

L. A. Lugiato, L. M. Narducci, E. V. Eschenazi, D. K. Bandy, and N. B. Abraham, Phys. Rev. A 32, 1563 (1985); L. M. Narducci, J. R. Tredicce, L. A. Lugiato, N. B. Abraham, and D. K. Bandy, Phys. Rev. A 33, 1842 (1986).
[Crossref] [PubMed]

L. A. Lugiato, F. Prati, L. M. Narducci, P. Ru, J. R. Tredicce, and D. K. Bandy, Phys. Rev. A 37, 3847 (1988).
[Crossref]

Phys. Rev. Lett. (8)

P. Coullet, L. Gil, and J. Lega, Phys. Rev. Lett. 62, 1619 (1989).
[Crossref] [PubMed]

D. W. McLaughlin, J. V. Moloney, and A. C. Newell, Phys. Rev. Lett. 51, 75 (1983).
[Crossref]

H. M. Gibbs, F. A. Hopf, D. L. Kaplan, and R. L. Shoemaker, Phys. Rev. Lett. 46, 474 (1981).
[Crossref]

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

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

J. R. Tredicce, E. J. Quel, A. M. Ghazzawi, C. Green, M. A. Pernigo, L. M. Narducci, and L. A. Lugiato, Phys. Rev. Lett. 62, 1274 (1989).
[Crossref]

G. P. Puccioni, A. Poggi, W. Gadomski, J. R. Tredicce, and F. T. Arecchi, Phys. Rev. Lett. 55, 339 (1985).
[Crossref]

L. A. Lugiato and R. Lefever, Phys. Rev. lett. 58, 220 (1987); L. A. Lugiato and C. Oldano, Phys. Rev. A 37, 3896 (1988).
[Crossref]

J. V. Moloney and H. M. Gibbs, Phys. Rev. Lett. 58, 467 (1982).

Z. Naturforsch. (1)

S. Grossmann and S. Thomae, Z. Naturforsch. 32A, 1353 (1977); M. J. Feigenbaum, J. Stat. Phys. 19, 25 (1978); P. Coullet and C. Tresser, J. Phys. (Paris) Colloq. C5, 39 (1978).
[Crossref]

Zh. Eksp. Teor. Fiz. (2)

A. F. Suchkov, Zh. Eksp. Teor. Fiz. 49, 1495 (1965) [Sov. Phys. JETP 22, 1026 (1966)].

I. M. Belousova, G. N. Vinokurov, O. B. Danilov, and N. N. Rozanov, Zh. Eksp. Teor. Fiz. 57, 1146 (1967) [Sov. Phys. JETP 25, 761 (1967)].

Other (4)

See, for example, A. Yariv, Optical Electronics, 3rd ed. (Holt, Rinehart & Winston, New York, 1985), p. 32.

P. Hollinger, C. Jung, and H. Weber, Department of Physics, University of Kaiserslautern, D-6750 Kaiserslautern, Federal Republic of Germany (personal communication).

W. J. Firth, in Instabilities an Chaos in Quantum Optics II, N. B. Abraham, F. T. Arecchi, and L. A. Lugiato, eds. (Prenum, New York, 1988), p. 219.
[Crossref]

J. V. Moloney, H. Adachihara, D. W. McLaughlin, and A. C. Newell, in Chaos, Noise and Fractals, E. R. Pike and L. A. Lugiato, eds. (Hilger, London, 1988), p. 137.

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Figures (8)

Fig. 1
Fig. 1

Schematic representation of a ring laser cavity with two spherical mirrors of radius of curvature R0. The length of the ring is Λ; the distance between the two curved reflectors is L. η = 0 indicates the origin of the longitudinal coordinate system, and η measures various longitudinal positions of relevance to our discussion, in units of Λ. The active medium is confined within the rectangular region of length LA.

Fig. 2
Fig. 2

Illustrating the significance of the two classes of transverse modes. The mode (n = N − 1, p = 3) is nearly degenerate with the reference mode (n = N, p = 0). We have omitted the angular indices m for clarity of illustration.

Fig. 3
Fig. 3

The shift of the laser carrier from the reference frequency in a range of parameters where the competition between the p = 0 and p = 1 modes leads to stationary solutions. (a) Plot of the curve [Eq. (3.11)] obtained through perturbative expansion of the steady-state equations. (b) numerical integration of Eqs. (3.4) for κ = 1.0. The other parameters are fixed to the values reported in the text.

Fig. 4
Fig. 4

Scan of the total flux as a function of the modal spacing a1 for (a) κ = 1.0, (b) κ = 0.3. The other parameters are selected as stated in the text. For each value of a1 we plot the maximum and minimum excursion of the oscillations after the transients have died out.

Fig. 5
Fig. 5

Time evolution of the first three modal amplitudes during the scan of Fig. 4 for (a1)–(a3) κ = 1.0 and (b1)–(b3) κ = 0.3. The other parameters are selected as stated in the text.

Fig. 6
Fig. 6

Spatiotemporal evolution of the modulus of the output field for κ’ = 1.0; (a) a1 = 0.34, (b) a1 = 0.32, and (c) a1 = 0.25. The other parameters are selected as stated in the text.

Fig. 7
Fig. 7

Radial dependence of the correlation function Cmax(0, ρ) for the spatiotemporal evolutions of Fig. 6.

Fig. 8
Fig. 8

Schematic representation of the empty resonator. The meaning of the symbols is the same as in Fig. 1.

Equations (126)

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

i 4 2 F + F η + 1 υ F τ = i δ Ω υ F α Λ P ,
P τ = [ F D + ( 1 + i Δ ) P ] ,
D τ = γ [ 1 2 ( F * P + F P * ) + D χ ]
η = z Λ , ρ = ( π Λ λ ) 1 / 2 r , τ = γ t
υ = c Λ γ .
E ( r , φ , z , t ) = ћ γ γ 2 μ 1 2 { F ( r , φ , z , t ) exp [ i ( k 0 z ω 0 t ) ] × exp ( i δ Ω t ) + c . c . } ,
F ( ρ , φ , η , τ ) = p , m , i A p m ( i ) ( ρ , φ , η ) f p m ( i ) ( η , τ ) ,
f p m ( i ) η + 1 υ f p m ( i ) τ = i Ω υ f p m ( i ) α Λ 0 d ρ ρ 0 2 π d φ A p m ( i ) * ( ρ , φ , η ) P ( ρ , φ , η , τ ) .
f p m ( i ) ( 1 2 f A , τ ) = R exp ( i δ p m ) exp ( i δ Ω γ Λ L A c ) × f p m ( i ) ( 1 2 f A , τ γ Λ L A c ) ,
δ p m = ω n p m ω n 00 c / Λ
η = η f A z L A ,
τ = τ + γ Λ L A c ( η f A + 1 2 )
f p m ( i ) ( η , τ ) = φ p m ( i ) ( η , τ ) × exp [ ( ln R i δ p m + i δ Ω γ Λ L A c ) ( η + 1 / 2 ) ] .
φ p m ( i ) η ( ln R i δ p m + i δ Ω γ Λ c ) φ p m ( i ) + γ Λ c φ p m ( i ) τ = α L A exp [ ( ln R i δ p m + i δ Ω γ Λ L A c ) ( η + 1 / 2 ) ] × 0 d ρ ρ 0 2 π d φ A p m ( i ) * P ,
φ p m ( i ) ( 1 2 , τ ) = φ p m ( i ) ( 1 2 , τ ) .
α L A 1 , T 1 R 1 ,
α L A / T 2 C
δ Ω γ Λ c = O ( T ) .
| δ p m 2 π M p m | = O ( T ) .
γ Λ c T 1
κ = c T Λ γ
φ p m ( i ) ( η , τ ) = Φ p m ( i ) ( η , τ ) exp [ 2 π i M p m ( η + 1 / 2 ) ] ,
Φ p m ( i ) η = O ( T ) ,
Φ p m ( i ) η ( ln R i δ ¯ p m + i δ Ω γ Λ c ) Φ p m ( i ) + γ Λ c Φ p m ( i ) τ = α L A exp [ ( ln R i δ ¯ p m + i δ Ω γ Λ L A c ) ( η + 1 / 2 ) ] × 0 d ρ ρ 0 2 π d φ A p m ( i ) * P ,
δ ¯ p m = δ p m 2 π M p m .
Ψ p m ( i ) ( τ ) = 1 / 2 + 1 / 2 d η Φ p m ( i ) ( η , τ ) .
Ψ p m ( i ) τ = κ ( 1 + i a p m i δ Ω / κ ) Ψ p m ( i ) 2 C κ 1 / 2 + 1 / 2 d η 0 d ρ ρ 0 2 π d φ A p m ( i ) * ( ρ , φ , η ) P ,
a p m = δ ¯ p m / T
δ Ω κ = δ A C 1 + κ = Δ .
L A Λ η 1 ,
A p m ( i ) ( ρ , φ , η ) A p m ( i ) ( ρ , φ , 0 ) A p m ( i ) ( ρ , φ ) ,
Ψ p m ( i ) τ = κ ( 1 + i a p m Δ ) Ψ p m ( i ) 2 C κ 0 d ρ ρ 0 2 π d φ A p m ( i ) ( ρ , φ ) P ( ρ , φ , τ ) ,
P τ = [ F D + ( 1 + i Δ ) P ] ,
D τ = γ [ 1 2 ( F * P + F P * ) + D χ ( ρ ) ] ,
F ( ρ , φ , τ ) = p m i A p m ( i ) ( ρ , φ ) Ψ p m ( i ) ( τ ) .
ρ = ρ / η 1 ,
Ψ ¯ p m ( i ) = Ψ p m ( i ) η 1
A ¯ p m ( i ) ( ρ , φ ) = 2 ( 2 ρ 2 ) m / 2 [ p ! ( p + m ) ! ] 1 / 2 L p m ( 2 ρ 2 ) × exp ( ρ 2 ) B m ( φ ) = η 1 A p m ( i ) ( ρ , φ ) ,
Ψ ¯ p m ( i ) τ = κ ( 1 + i a p m i Δ ) Ψ ¯ p m ( i ) 2 C k 0 d ρ ρ 0 2 π d φ A ¯ p m ( i ) P ,
P τ = [ F D + ( 1 + i Δ ) P ] ,
D τ = γ [ 1 2 ( F * P + F P * ) + D χ ( ρ ) ] ,
F ( ρ , φ , τ ) = p m i A ¯ p m ( i ) ( ρ , φ ) Ψ ¯ p m ( i ) ( τ )
0 d ρ ρ 0 2 π d φ A ¯ p m ( i ) * ( ρ , φ ) A ¯ p m ( i ) ( ρ , φ ) = δ p p δ m m δ i i .
( 1 4 2 ρ 2 ) A ¯ p m ( i ) = ( 2 p + m + 1 ) A ¯ p m ( i ) ,
2 = 2 ρ 2 + 1 ρ ρ + 1 ρ 2 2 φ 2 .
F τ = κ [ 1 i Δ i a 2 ( 1 4 2 ρ 2 + 1 ) ] × F 2 C κ P ( ρ , φ , τ ) ,
a = 4 T ( tan 1 1 f A 2 η 2 + tan 1 f A 2 η 1 ) .
P ( ρ , φ , τ ) = p m i A ¯ p m ( i ) ( ρ , φ ) p p m ( i ) ( τ ) ,
D ( ρ , φ , τ ) = p m i A ¯ p m ( i ) ( ρ , φ ) d p m ( i ) ( τ ) ,
Ψ ¯ p m ( i ) τ = κ ( 1 + i a p m i Δ ) Ψ ¯ p m ( i ) 2 C κ p p m ( i ) ,
p p m ( i ) τ = [ p m i p m i Γ ( σ , σ , σ ) Ψ ¯ p m ( i ) d p m ( i ) + ( 1 + i Δ ) p p m ( i ) ] ,
d p m ( i ) τ = γ { 1 2 [ p m i p m i Γ ( σ , σ , σ ) × Ψ ¯ p m ( i ) * p p m ( i ) + c . c . ] + d p m ( i ) χ p m ( i ) } ,
Γ ( σ , σ , σ ) = 0 d ρ ρ 0 2 π d φ A ¯ p m ( i ) A ¯ p m ( i ) A ¯ p m ( i ) ,
χ p m ( i ) = 0 d ρ ρ 0 2 π d φ A ¯ p m ( i ) χ ( ρ ) .
Ψ ¯ p τ = κ Ψ ¯ p i κ ( a p Δ ) Ψ ¯ p 2 C κ p p ,
p p τ = [ p p Γ p p p Ψ ¯ p d p ( 1 + i Δ ) p p ] ,
d p τ = γ [ 1 2 ( p p Γ p p p Ψ ¯ p * p p + c . c . ) + d p χ p ] .
A ¯ p ( ρ ) = 2 L p ( 2 ρ 2 ) exp ( ρ 2 ) ,
0 d ρ ρ A ¯ p ( ρ ) A ¯ q ( ρ ) = δ p q .
κ = κ g p , g p = 1 + β p 4 ( p = 0 , 1 , 2 , ) ,
χ p = ( 1 ) p .
a p = a 1 p ,
a 1 = δ 1 2 π .
Re ( Ψ ¯ p st ) = ( 1 I p g p { 2 C g p [ 1 + ( Δ ) 2 ] } ) ,
Im ( Ψ ¯ p st ) = 0 ,
= κ a p 1 + κ ,
I p = q Γ p p q Γ q p p = 4 0 d x e 2 x [ L p ( x ) ] 4 ,
= κ ( 2 C 1 Δ 2 ) ( a 1 + Δ β ) 4 ( 1 + κ ) ( 1 + Δ 2 ) ( a 1 2 + β 2 ) .
Φ ( t ) = 2 π 0 d ρ ρ | F ( ρ , t ) | 2 = 2 π p | Ψ ¯ p ( t ) | 2 ,
C max ( 0 , ρ ) = Max ( δ τ ) × I ( 0 , τ ) I ( ρ , τ + δ τ ) I ( 0 , τ ) I ( ρ , τ ) σ ( 0 ) σ ( ρ ) ,
2 E ( r , z , t ) 1 c 2 2 t 2 E ( r , z , t ) = 0 ,
2 2 r 2 + 1 r r + 2 z 2
E ( r , z , t ) = A ( r , z ) exp [ i ( k z ω t ) ] ,
( 2 r 2 + 1 r r ) A ( r , z ) + 2 i k z A ( r , z ) = 0
η = z Λ , ρ = ( π Λ λ ) 1 / 2 r ,
A η = i 4 ( 2 ρ 2 + 1 ρ ρ ) A .
A ( 1 ) ( ρ , η ) = 2 υ ( η ) L p [ 2 ρ 2 υ 2 ( η ) ] exp [ ρ 2 υ 2 ( η ) + i ρ 2 u ( η ) ] × exp [ i ( 2 p + 1 ) tan 1 η / η 1 ] ,
υ ( η ) = η 1 [ 1 + ( η / η 1 ) 2 ] 1 / 2
u ( η ) 1 η ( η 1 2 + η 2 )
η 1 = υ ( 0 )
A ( 2 ) ( ρ , η ) = 2 α ( η ) L p [ 2 ρ 2 α 2 ( η ) ] exp [ ρ 2 α 2 ( η ) + i ρ β ( η ) ] × exp [ i ( 2 p + 1 ) tan 1 η 1 / 2 η 2 ] exp ( i φ ) ,
α ( η ) = η 2 [ 1 + ( η 1 / 2 η 2 ) 2 ] 1 / 2 ,
β ( η ) = 1 η 1 / 2 [ ( η 1 / 2 ) 2 + η 2 2 ] ,
1 q ( η ) = 1 u ( η ) + i 1 υ 2 ( η ) .
q 2 = q 1 + d
q 2 = q 1 2 ρ 0 q 1 + 1 ,
f = L / Λ ,
q ( η = 0 ) = i υ 2 ( 0 ) = i η 1 ,
q ( η = 1 2 f ) = q ( η = 0 ) + 1 2 f
q ( η = 1 2 f + ) = q ( η = 1 2 f + ) 2 ρ 0 q ( η = 1 2 f ) + 1 ,
q ( η = 1 ) = q ( η = 1 1 2 f + ) + 1 2 f q ( η = 0 ) ,
η 1 = 1 2 [ ( ρ 0 f ) ( ρ 0 + f 2 f ) ρ 0 + f 1 ] 1 / 2 ,
η 2 = 1 ρ 0 2 η 1 [ η 1 2 ( 1 f ρ 0 ) 2 + 1 4 ( ρ 0 + f 2 f ) 2 ] .
R = exp ( i 2 ρ 2 ρ 0 ) .
R E ( 1 ) ( η = 1 2 f , ρ , t ) = E ( 2 ) ( η = 1 2 f + , ρ , t ) .
υ 2 ( η = 1 2 f ) = α 2 ( η = 1 2 f + ) ,
2 ρ 0 + 1 u ( η = 1 2 f ) = 1 β ( η = 1 2 f + ) .
φ = ( 2 p + 1 ) ( tan 1 f 2 η 1 + tan 1 1 f 2 η 2 ) ,
R E ( 2 ) ( η = 1 2 f , ρ , t ) = E ( 1 ) ( η = 1 2 f + , ρ , t ) ,
ω n , p = 2 π c Λ n + 2 c Λ ( 2 p + 1 ) ( tan 1 1 f 2 η 2 + tan 1 f 2 η 1 ) ( n , p = 0 , 1 , 2 , ) .
2 = 2 ρ 2 + 1 ρ ρ + 1 ρ 2 2 φ 2 ,
A p m ( ρ , φ , η ) = B m ( φ ) C p m ( ρ , η ) exp [ i θ p m ( ρ , η ) ] ,
B m ( φ ) = { 1 2 π m = 0 1 π sin m φ m > 0 , 1 π cos m φ m > 0
C p m ( ρ , η ) = 2 υ ( η ) [ 2 ρ 2 υ 2 ( η ) ] m / 2 [ p ! ( p + m ) ! ] 1 / 2 × L p m [ 2 ρ 2 υ 2 ( η ) ] exp [ ρ 2 υ 2 ( η ) ] ,
θ p m ( ρ , η ) = ρ 2 u ( η ) ( 2 p + m + 1 ) tan 1 η / η 1 ( p = 0 , 1 , 2 , ; m = 0 , 1 , 2 , ) .
0 2 π d φ 0 d ρ ρ A p m ( i ) * ( ρ , φ , η ) A p m ( i ) ( ρ , φ , η ) = δ p p , δ m m δ i i .
ω n , p , m = 2 π c Λ n + 2 c Λ ( 2 p + m + 1 ) × [ tan 1 1 f 2 η 2 + tan 1 f 2 η 1 ] ,
n , p , m = 0 , 1 , 2 ,
ω n , p , m = π c / Λ [ 2 ( n + p ) + m + 1 ] .
Γ p , p , p = 0 d ξ ξ A ¯ p A ¯ p , A ¯ p ,
A ¯ p ( ξ ) = 2 L p ( 2 ξ 2 ) exp ( ξ 2 ) .
Γ p , p , p = 2 0 d x exp ( 3 2 x ) L p ( x ) L p ( x ) L p ( x ) .
n L n ( x ) = ( 2 n 1 x ) L n 1 ( x ) ( n 1 ) L n 2 ( x ) ( n 1 )
x d L n ( x ) d x = n L n ( x ) n L n 1 ( x ) .
Γ p , p , p = 2 0 d x exp ( 3 2 x ) L p L p × ( 2 p 1 p L p 1 x p L p 1 p 1 p L p 2 ) ,
Γ p , p , p = 2 p 1 p Γ p , p , p 1 p 1 p Γ p , p , p 2 1 p × 2 0 d x exp ( 3 2 x ) x L p L p L p 1 .
0 d x exp ( 3 2 x ) x L p L p L p 1 = 2 3 0 d x exp ( 3 2 x ) × ( x d L p d x L p L p 1 + x d L p d x L p L p 1 + x d L p 1 d x L p L p + L p L p L p 1 ) ,
Γ p , p , p = 2 p + 2 p 4 p + 3 3 p Γ p , p , p 1 1 3 p 1 p Γ p , p , p 2 + 2 3 p p Γ p , p 1 , p 1 + 2 3 p p Γ p 1 , p , p 1 ( p 1 ) .
Γ p , p Γ p , p , 0 = 2 0 d x exp ( 3 2 x ) L p L p .
Γ p , p = 2 p 4 p + 3 3 p Γ p , p 1 1 3 p 1 p Γ p , p 2 + 2 3 p p Γ p 1 , p 1 ( p 1 ) .
Γ p , 0 = 4 / 3 p + 1 .
Ψ ¯ p st = e i Ψ ¯ p = x p + i y p ,
x 0 ( 1 + 2 x 0 2 + 2 x 0 x 1 + x 1 2 ) + [ ( a 1 κ Δ ) y 1 g 1 x 1 ] × ( x 0 2 + 2 x 0 x 1 + x 1 2 ) + x 0 ( κ + Δ ) ( Δ + x 0 y 1 + x 1 y 1 ) [ ( a 1 κ Δ ) x 1 + g 1 y 1 ] ( x 0 y 1 + 1 2 x 1 y 1 ) + 2 C x 0 = 0 ,
x 0 ( Δ + x 0 y 1 + x 1 y 1 ) + x 0 ( κ + Δ ) ( 1 + y 1 2 ) + [ ( a 1 κ Δ ) y 1 + g 1 x 1 ] ( x 0 y 1 + 1 2 x 1 y 1 ) y 1 2 2 [ ( a 1 κ Δ ) x 1 + g 1 y 1 ] = 0 ,
x 0 ( x 0 2 + 2 x 0 x 1 + 1 2 x 1 2 ) + x 0 y 1 ( κ + Δ ) ( x 0 + 1 2 x 1 ) + ( 1 + x 0 2 + x 0 x 1 + x 1 2 ) [ ( a 1 κ Δ ) y 1 g 1 x 1 ] ( Δ + 1 2 x 0 y 1 + x 1 y 1 ) [ ( a 1 κ Δ ) x 1 + g 1 y 1 ] + 2 C x 1 = 0 ,
x 0 y 1 ( x 0 + 1 2 x 1 ) + 1 2 x 0 y 1 2 ( κ + Δ ) + ( Δ + 1 2 x 0 y 1 + x 1 y 1 ) [ ( a 1 κ Δ ) y 1 g 1 x 1 ] ( 1 + y 1 2 ) [ ( a 1 κ Δ ) x 1 + g 1 y 1 ] + 2 C y 1 = 0.

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