Abstract

We present the results of our study of the dynamics of two-dimensional Kerr cavity solitons. The solitons are absolutely stable over a substantial parameter domain. We analyze their dynamics beyond the instability boundary, finding regions of stable oscillation and of fivefold or sixfold azimuthal instability. The Hopf oscillation is surprisingly robust, owing to the influence of a lower-amplitude unstable soliton.

© 2002 Optical Society of America

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    [CrossRef]
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    [CrossRef]
  6. G. S. McDonald and W. J. Firth, “Spatial solitary-wave optical memory,” J. Opt. Soc. Am. B 7, 1328–1335 (1990).
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  7. S. Wabnitz, “Suppression of interactions in a phase-locked soliton optical memory,” Opt. Lett. 18, 601–603 (1993).
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  8. G. S. McDonald and W. J. Firth, “Switching dynamics of spatial solitary wave pixels,” J. Opt. Soc. Am. B 10, 1081–1089 (1993).
    [CrossRef]
  9. M. Tlidi, P. Mandel, and R. Lefever, “Localized structures and localized patterns in optical bistability,” Phys. Rev. Lett. 73, 640–643 (1994).
    [CrossRef] [PubMed]
  10. W. J. Firth and A. J. Scroggie, “Optical bullet holes: robust controllable localized states of a nonlinear cavity,” Phys. Rev. Lett. 76, 1623–1626 (1996).
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  11. W. J. Firth and A. Lord, “Two-dimensional solitons in a Kerr cavity,” J. Mod. Opt. 43, 1071–1077 (1996).
    [CrossRef]
  12. W. J. Firth, A. Lord, and A. J. Scroggie, “Optical bullet holes,” Phys. Scr. T67, 12–16 (1996).
    [CrossRef]
  13. M. Brambilla, L. A. Lugiato, and M. Stefani, “Interaction and control of optical localized structures,” Europhys. Lett. 34, 109–114 (1996).
    [CrossRef]
  14. M. Tlidi and P. Mandel, “Spatial patterns in nascent optical bistability,” Chaos, Solitons Fractals 4, 1475–1485 (1996).
    [CrossRef]
  15. A. J. Scroggie, W. J. Firth, G. S. McDonald, M. Tlidi, R. Lefever, and L. A. Lugiato, “Pattern formation in a passive Kerr cavity,” Chaos, Solitons Fractals 4, 1323–1354 (1996).
    [CrossRef]
  16. S. Longhi, “Dark solitons in degenerate optical parametric oscillators,” Opt. Lett. 21, 860–862 (1996).
    [CrossRef] [PubMed]
  17. G. Steinmeyer, A. Schwache, and F. Mitschke, “Quantitative characterization of turbulence in an optical experiment,” Phys. Rev. E 53, 5399–5402 (1996).
    [CrossRef]
  18. M. Brambilla, L. A. Lugiato, F. Prati, L. Spinelli, and W. J. Firth, “Spatial soliton pixels in semiconductor devices,” Phys. Rev. Lett. 79, 2042–2045 (1997).
    [CrossRef]
  19. L. Spinelli, G. Tissoni, M. Brambilla, F. Prati, and L. A. Lugiato, “Spatial solitons in semiconductor microcavities,” Phys. Rev. A 58, 2542–2559 (1998).
    [CrossRef]
  20. V. B. Taranenko, I. Ganne, R. J. Kuszelewicz, and C. O. Weiss, “Patterns and localized structures in bistable semiconductor resonators,” Phys. Rev. A 61, 0638–18 (2000).
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  21. C. Etrich, U. Peschel, and F. Lederer, “Solitary waves in quadratically nonlinear resonators,” Phys. Rev. Lett. 79, 2454–2457 (1997).
    [CrossRef]
  22. D. Michaelis, U. Peschel, and F. Lederer, “Multistable localized structures and superlattices in semiconductor optical resonators,” Phys. Rev. A 56, R3366–R3373 (1997).
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  24. M. Tlidi, P. Mandel, and M. Haelterman, “Spatiotemporal patterns and localized structures in nonlinear optics,” Phys. Rev. E 56, 6524–6530 (1997).
    [CrossRef]
  25. D. V. Skryabin and W. J. Firth, “Interaction of cavity solitons in degenerate optical parametric oscillators,” Opt. Lett. 24, 1056–1058 (1999).
    [CrossRef]
  26. I. V. Barashenkov and E. V. Zelyanaya, “Stable complexes of parametrically driven, damped nonlinear Schrödinger solitons,” Phys. Rev. Lett. 83, 2568–2571 (1999).
    [CrossRef]
  27. W. J. Firth and G. K. Harkness, “Cavity solitons,” Asian J. Phys. 7, 665–677 (1998).
  28. N. N. Rosanov, D. V. Fedorov, and G. V. Khodova, “Particle-like light structures in the wide-aperture laser with saturating absorption,” J. Exp. Theor. Phys. 107, 376–392 (1995).
  29. B. Schäpers, M. Feldmann, T. Ackemann, and W. Lange, “Interaction of localized structures in an optical pattern-forming system,” Phys. Rev. Lett. 85, 748–751 (2000).
    [CrossRef] [PubMed]
  30. W. J. Firth, “Processing information with arrays of spatial solitons,” in Photonics Devices, and Systems, M. Hrabovsky, M. Miller, and P. Tomanek, eds., Proc. SPIE 4016, 388–394 (2000).
    [CrossRef]
  31. G. Tissoni, L. Spinelli, M. Brambilla, T. Maggipinto, I. Perrini, and L. A. Lugiato, “Cavity solitions in passive bulk semiconductor microcavities. I. Microscopic model and modulational instabilities,” J. Opt. Soc. Am. B 16, 2083–2094 (1999).
    [CrossRef]
  32. G. Tissoni, L. Spinelli, M. Brambilla, T. Maggipinto, I. Perrini, and L. A. Lugiato, “Cavity solitons in passive bulk semiconductor microcavities. II. Dynamical properties and control,” J. Opt. Soc. Am. B 16, 2095–2105 (1999).
    [CrossRef]
  33. T. Maggipinto, M. Brambilla, G. K. Harkness, and W. J. Firth, “Cavity solitons in semiconductor microresonators: existence, stability, and dynamical properties,” Phys. Rev. E 62, 8726–8739 (2000).
    [CrossRef]
  34. A. Schreiber, B. Thüering, M. Kreuzer, and T. Tschudi, “Experimental investigation of solitary structures in a nonlinear optical feedback system,” Opt. Commun. 136, 415–418 (1997).
    [CrossRef]
  35. N. N. Akhmediev and A. Ankiwicz, Solitons: Nonlinear Pulses and Beams (Chapman & Hall, London, 1997).
  36. A. C. Newell, Solitons in Mathematics & Physics (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1987).
  37. D. Gomila, M. Santagiustina, P. Colet, and M. San Miguel, “From hexagons to optical turbulence,” preprint.
  38. L. A. Lugiato and R. Lefever, “Spatial dissipative structures in passive optical systems,” Phys. Rev. Lett. 58, 2209–2211 (1987).
    [CrossRef] [PubMed]
  39. D. V. Skryabin, “Energy of the soliton internal modes and broken symmetries in nonlinear optics,” J. Opt. Soc. Am. B 19, 529–536 (2002).
    [CrossRef]
  40. D. Michaelis, U. Peschel, and F. Lederer, “Oscillating dark cavity solitons,” Opt. Lett. 23, 1814–1816 (1998).
    [CrossRef]
  41. A dynamical simulation may be viewed at http://cnqo.phys.strath.ac.uk/movies/kerrhopf.mpg.

2002 (1)

2000 (4)

V. B. Taranenko, I. Ganne, R. J. Kuszelewicz, and C. O. Weiss, “Patterns and localized structures in bistable semiconductor resonators,” Phys. Rev. A 61, 0638–18 (2000).
[CrossRef]

B. Schäpers, M. Feldmann, T. Ackemann, and W. Lange, “Interaction of localized structures in an optical pattern-forming system,” Phys. Rev. Lett. 85, 748–751 (2000).
[CrossRef] [PubMed]

W. J. Firth, “Processing information with arrays of spatial solitons,” in Photonics Devices, and Systems, M. Hrabovsky, M. Miller, and P. Tomanek, eds., Proc. SPIE 4016, 388–394 (2000).
[CrossRef]

T. Maggipinto, M. Brambilla, G. K. Harkness, and W. J. Firth, “Cavity solitons in semiconductor microresonators: existence, stability, and dynamical properties,” Phys. Rev. E 62, 8726–8739 (2000).
[CrossRef]

1999 (4)

1998 (3)

D. Michaelis, U. Peschel, and F. Lederer, “Oscillating dark cavity solitons,” Opt. Lett. 23, 1814–1816 (1998).
[CrossRef]

W. J. Firth and G. K. Harkness, “Cavity solitons,” Asian J. Phys. 7, 665–677 (1998).

L. Spinelli, G. Tissoni, M. Brambilla, F. Prati, and L. A. Lugiato, “Spatial solitons in semiconductor microcavities,” Phys. Rev. A 58, 2542–2559 (1998).
[CrossRef]

1997 (6)

M. Brambilla, L. A. Lugiato, F. Prati, L. Spinelli, and W. J. Firth, “Spatial soliton pixels in semiconductor devices,” Phys. Rev. Lett. 79, 2042–2045 (1997).
[CrossRef]

A. Schreiber, B. Thüering, M. Kreuzer, and T. Tschudi, “Experimental investigation of solitary structures in a nonlinear optical feedback system,” Opt. Commun. 136, 415–418 (1997).
[CrossRef]

C. Etrich, U. Peschel, and F. Lederer, “Solitary waves in quadratically nonlinear resonators,” Phys. Rev. Lett. 79, 2454–2457 (1997).
[CrossRef]

D. Michaelis, U. Peschel, and F. Lederer, “Multistable localized structures and superlattices in semiconductor optical resonators,” Phys. Rev. A 56, R3366–R3373 (1997).
[CrossRef]

K. Staliunas and V. J. Sanchez-Morillo, “Localized structures in degenerate parametric oscillators,” Opt. Commun. 139, 306–312 (1997).
[CrossRef]

M. Tlidi, P. Mandel, and M. Haelterman, “Spatiotemporal patterns and localized structures in nonlinear optics,” Phys. Rev. E 56, 6524–6530 (1997).
[CrossRef]

1996 (8)

W. J. Firth and A. J. Scroggie, “Optical bullet holes: robust controllable localized states of a nonlinear cavity,” Phys. Rev. Lett. 76, 1623–1626 (1996).
[CrossRef] [PubMed]

W. J. Firth and A. Lord, “Two-dimensional solitons in a Kerr cavity,” J. Mod. Opt. 43, 1071–1077 (1996).
[CrossRef]

W. J. Firth, A. Lord, and A. J. Scroggie, “Optical bullet holes,” Phys. Scr. T67, 12–16 (1996).
[CrossRef]

M. Brambilla, L. A. Lugiato, and M. Stefani, “Interaction and control of optical localized structures,” Europhys. Lett. 34, 109–114 (1996).
[CrossRef]

M. Tlidi and P. Mandel, “Spatial patterns in nascent optical bistability,” Chaos, Solitons Fractals 4, 1475–1485 (1996).
[CrossRef]

A. J. Scroggie, W. J. Firth, G. S. McDonald, M. Tlidi, R. Lefever, and L. A. Lugiato, “Pattern formation in a passive Kerr cavity,” Chaos, Solitons Fractals 4, 1323–1354 (1996).
[CrossRef]

G. Steinmeyer, A. Schwache, and F. Mitschke, “Quantitative characterization of turbulence in an optical experiment,” Phys. Rev. E 53, 5399–5402 (1996).
[CrossRef]

S. Longhi, “Dark solitons in degenerate optical parametric oscillators,” Opt. Lett. 21, 860–862 (1996).
[CrossRef] [PubMed]

1995 (1)

N. N. Rosanov, D. V. Fedorov, and G. V. Khodova, “Particle-like light structures in the wide-aperture laser with saturating absorption,” J. Exp. Theor. Phys. 107, 376–392 (1995).

1994 (2)

L. A. Lugiato, ed., Special issue on nonlinear optical systems, chaos, and noise, Chaos, Solitons Fractals 4, 1251–1258 (1994).
[CrossRef]

M. Tlidi, P. Mandel, and R. Lefever, “Localized structures and localized patterns in optical bistability,” Phys. Rev. Lett. 73, 640–643 (1994).
[CrossRef] [PubMed]

1993 (2)

1990 (2)

1988 (1)

N. N. Rosanov and G. V. Khodova, “Switching of a bistable interferometer on localized inhomogeneities,” Opt. Spectrosc. (USSR) 65, 1399–1401 (1988).

1987 (1)

L. A. Lugiato and R. Lefever, “Spatial dissipative structures in passive optical systems,” Phys. Rev. Lett. 58, 2209–2211 (1987).
[CrossRef] [PubMed]

1983 (1)

D. W. McLaughlin, J. V. Moloney, and A. C. Newell, “Solitary waves as fixed-points of infinite-dimensional maps in an optical bistable ring cavity,” Phys. Rev. Lett. 51, 75–78 (1983).
[CrossRef]

Ackemann, T.

B. Schäpers, M. Feldmann, T. Ackemann, and W. Lange, “Interaction of localized structures in an optical pattern-forming system,” Phys. Rev. Lett. 85, 748–751 (2000).
[CrossRef] [PubMed]

Barashenkov, I. V.

I. V. Barashenkov and E. V. Zelyanaya, “Stable complexes of parametrically driven, damped nonlinear Schrödinger solitons,” Phys. Rev. Lett. 83, 2568–2571 (1999).
[CrossRef]

Brambilla, M.

T. Maggipinto, M. Brambilla, G. K. Harkness, and W. J. Firth, “Cavity solitons in semiconductor microresonators: existence, stability, and dynamical properties,” Phys. Rev. E 62, 8726–8739 (2000).
[CrossRef]

G. Tissoni, L. Spinelli, M. Brambilla, T. Maggipinto, I. Perrini, and L. A. Lugiato, “Cavity solitions in passive bulk semiconductor microcavities. I. Microscopic model and modulational instabilities,” J. Opt. Soc. Am. B 16, 2083–2094 (1999).
[CrossRef]

G. Tissoni, L. Spinelli, M. Brambilla, T. Maggipinto, I. Perrini, and L. A. Lugiato, “Cavity solitons in passive bulk semiconductor microcavities. II. Dynamical properties and control,” J. Opt. Soc. Am. B 16, 2095–2105 (1999).
[CrossRef]

L. Spinelli, G. Tissoni, M. Brambilla, F. Prati, and L. A. Lugiato, “Spatial solitons in semiconductor microcavities,” Phys. Rev. A 58, 2542–2559 (1998).
[CrossRef]

M. Brambilla, L. A. Lugiato, F. Prati, L. Spinelli, and W. J. Firth, “Spatial soliton pixels in semiconductor devices,” Phys. Rev. Lett. 79, 2042–2045 (1997).
[CrossRef]

M. Brambilla, L. A. Lugiato, and M. Stefani, “Interaction and control of optical localized structures,” Europhys. Lett. 34, 109–114 (1996).
[CrossRef]

Etrich, C.

C. Etrich, U. Peschel, and F. Lederer, “Solitary waves in quadratically nonlinear resonators,” Phys. Rev. Lett. 79, 2454–2457 (1997).
[CrossRef]

Fedorov, D. V.

N. N. Rosanov, D. V. Fedorov, and G. V. Khodova, “Particle-like light structures in the wide-aperture laser with saturating absorption,” J. Exp. Theor. Phys. 107, 376–392 (1995).

Feldmann, M.

B. Schäpers, M. Feldmann, T. Ackemann, and W. Lange, “Interaction of localized structures in an optical pattern-forming system,” Phys. Rev. Lett. 85, 748–751 (2000).
[CrossRef] [PubMed]

Firth, W. J.

W. J. Firth, “Processing information with arrays of spatial solitons,” in Photonics Devices, and Systems, M. Hrabovsky, M. Miller, and P. Tomanek, eds., Proc. SPIE 4016, 388–394 (2000).
[CrossRef]

T. Maggipinto, M. Brambilla, G. K. Harkness, and W. J. Firth, “Cavity solitons in semiconductor microresonators: existence, stability, and dynamical properties,” Phys. Rev. E 62, 8726–8739 (2000).
[CrossRef]

D. V. Skryabin and W. J. Firth, “Interaction of cavity solitons in degenerate optical parametric oscillators,” Opt. Lett. 24, 1056–1058 (1999).
[CrossRef]

W. J. Firth and G. K. Harkness, “Cavity solitons,” Asian J. Phys. 7, 665–677 (1998).

M. Brambilla, L. A. Lugiato, F. Prati, L. Spinelli, and W. J. Firth, “Spatial soliton pixels in semiconductor devices,” Phys. Rev. Lett. 79, 2042–2045 (1997).
[CrossRef]

W. J. Firth and A. Lord, “Two-dimensional solitons in a Kerr cavity,” J. Mod. Opt. 43, 1071–1077 (1996).
[CrossRef]

A. J. Scroggie, W. J. Firth, G. S. McDonald, M. Tlidi, R. Lefever, and L. A. Lugiato, “Pattern formation in a passive Kerr cavity,” Chaos, Solitons Fractals 4, 1323–1354 (1996).
[CrossRef]

W. J. Firth, A. Lord, and A. J. Scroggie, “Optical bullet holes,” Phys. Scr. T67, 12–16 (1996).
[CrossRef]

W. J. Firth and A. J. Scroggie, “Optical bullet holes: robust controllable localized states of a nonlinear cavity,” Phys. Rev. Lett. 76, 1623–1626 (1996).
[CrossRef] [PubMed]

G. S. McDonald and W. J. Firth, “Switching dynamics of spatial solitary wave pixels,” J. Opt. Soc. Am. B 10, 1081–1089 (1993).
[CrossRef]

G. S. McDonald and W. J. Firth, “Spatial solitary-wave optical memory,” J. Opt. Soc. Am. B 7, 1328–1335 (1990).
[CrossRef]

Ganne, I.

V. B. Taranenko, I. Ganne, R. J. Kuszelewicz, and C. O. Weiss, “Patterns and localized structures in bistable semiconductor resonators,” Phys. Rev. A 61, 0638–18 (2000).
[CrossRef]

Haelterman, M.

M. Tlidi, P. Mandel, and M. Haelterman, “Spatiotemporal patterns and localized structures in nonlinear optics,” Phys. Rev. E 56, 6524–6530 (1997).
[CrossRef]

Harkness, G. K.

T. Maggipinto, M. Brambilla, G. K. Harkness, and W. J. Firth, “Cavity solitons in semiconductor microresonators: existence, stability, and dynamical properties,” Phys. Rev. E 62, 8726–8739 (2000).
[CrossRef]

W. J. Firth and G. K. Harkness, “Cavity solitons,” Asian J. Phys. 7, 665–677 (1998).

Khodova, G. V.

N. N. Rosanov, D. V. Fedorov, and G. V. Khodova, “Particle-like light structures in the wide-aperture laser with saturating absorption,” J. Exp. Theor. Phys. 107, 376–392 (1995).

N. N. Rosanov and G. V. Khodova, “Diffractive autosolitons in nonlinear interferometers,” J. Opt. Soc. Am. B 7, 1057–1065 (1990).
[CrossRef]

N. N. Rosanov and G. V. Khodova, “Switching of a bistable interferometer on localized inhomogeneities,” Opt. Spectrosc. (USSR) 65, 1399–1401 (1988).

Kreuzer, M.

A. Schreiber, B. Thüering, M. Kreuzer, and T. Tschudi, “Experimental investigation of solitary structures in a nonlinear optical feedback system,” Opt. Commun. 136, 415–418 (1997).
[CrossRef]

Kuszelewicz, R. J.

V. B. Taranenko, I. Ganne, R. J. Kuszelewicz, and C. O. Weiss, “Patterns and localized structures in bistable semiconductor resonators,” Phys. Rev. A 61, 0638–18 (2000).
[CrossRef]

Lange, W.

B. Schäpers, M. Feldmann, T. Ackemann, and W. Lange, “Interaction of localized structures in an optical pattern-forming system,” Phys. Rev. Lett. 85, 748–751 (2000).
[CrossRef] [PubMed]

Lederer, F.

D. Michaelis, U. Peschel, and F. Lederer, “Oscillating dark cavity solitons,” Opt. Lett. 23, 1814–1816 (1998).
[CrossRef]

D. Michaelis, U. Peschel, and F. Lederer, “Multistable localized structures and superlattices in semiconductor optical resonators,” Phys. Rev. A 56, R3366–R3373 (1997).
[CrossRef]

C. Etrich, U. Peschel, and F. Lederer, “Solitary waves in quadratically nonlinear resonators,” Phys. Rev. Lett. 79, 2454–2457 (1997).
[CrossRef]

Lefever, R.

A. J. Scroggie, W. J. Firth, G. S. McDonald, M. Tlidi, R. Lefever, and L. A. Lugiato, “Pattern formation in a passive Kerr cavity,” Chaos, Solitons Fractals 4, 1323–1354 (1996).
[CrossRef]

M. Tlidi, P. Mandel, and R. Lefever, “Localized structures and localized patterns in optical bistability,” Phys. Rev. Lett. 73, 640–643 (1994).
[CrossRef] [PubMed]

L. A. Lugiato and R. Lefever, “Spatial dissipative structures in passive optical systems,” Phys. Rev. Lett. 58, 2209–2211 (1987).
[CrossRef] [PubMed]

Longhi, S.

Lord, A.

W. J. Firth, A. Lord, and A. J. Scroggie, “Optical bullet holes,” Phys. Scr. T67, 12–16 (1996).
[CrossRef]

W. J. Firth and A. Lord, “Two-dimensional solitons in a Kerr cavity,” J. Mod. Opt. 43, 1071–1077 (1996).
[CrossRef]

Lugiato, L. A.

G. Tissoni, L. Spinelli, M. Brambilla, T. Maggipinto, I. Perrini, and L. A. Lugiato, “Cavity solitons in passive bulk semiconductor microcavities. II. Dynamical properties and control,” J. Opt. Soc. Am. B 16, 2095–2105 (1999).
[CrossRef]

G. Tissoni, L. Spinelli, M. Brambilla, T. Maggipinto, I. Perrini, and L. A. Lugiato, “Cavity solitions in passive bulk semiconductor microcavities. I. Microscopic model and modulational instabilities,” J. Opt. Soc. Am. B 16, 2083–2094 (1999).
[CrossRef]

L. Spinelli, G. Tissoni, M. Brambilla, F. Prati, and L. A. Lugiato, “Spatial solitons in semiconductor microcavities,” Phys. Rev. A 58, 2542–2559 (1998).
[CrossRef]

M. Brambilla, L. A. Lugiato, F. Prati, L. Spinelli, and W. J. Firth, “Spatial soliton pixels in semiconductor devices,” Phys. Rev. Lett. 79, 2042–2045 (1997).
[CrossRef]

A. J. Scroggie, W. J. Firth, G. S. McDonald, M. Tlidi, R. Lefever, and L. A. Lugiato, “Pattern formation in a passive Kerr cavity,” Chaos, Solitons Fractals 4, 1323–1354 (1996).
[CrossRef]

M. Brambilla, L. A. Lugiato, and M. Stefani, “Interaction and control of optical localized structures,” Europhys. Lett. 34, 109–114 (1996).
[CrossRef]

L. A. Lugiato, ed., Special issue on nonlinear optical systems, chaos, and noise, Chaos, Solitons Fractals 4, 1251–1258 (1994).
[CrossRef]

L. A. Lugiato and R. Lefever, “Spatial dissipative structures in passive optical systems,” Phys. Rev. Lett. 58, 2209–2211 (1987).
[CrossRef] [PubMed]

Maggipinto, T.

Mandel, P.

M. Tlidi, P. Mandel, and M. Haelterman, “Spatiotemporal patterns and localized structures in nonlinear optics,” Phys. Rev. E 56, 6524–6530 (1997).
[CrossRef]

M. Tlidi and P. Mandel, “Spatial patterns in nascent optical bistability,” Chaos, Solitons Fractals 4, 1475–1485 (1996).
[CrossRef]

M. Tlidi, P. Mandel, and R. Lefever, “Localized structures and localized patterns in optical bistability,” Phys. Rev. Lett. 73, 640–643 (1994).
[CrossRef] [PubMed]

McDonald, G. S.

McLaughlin, D. W.

D. W. McLaughlin, J. V. Moloney, and A. C. Newell, “Solitary waves as fixed-points of infinite-dimensional maps in an optical bistable ring cavity,” Phys. Rev. Lett. 51, 75–78 (1983).
[CrossRef]

Michaelis, D.

D. Michaelis, U. Peschel, and F. Lederer, “Oscillating dark cavity solitons,” Opt. Lett. 23, 1814–1816 (1998).
[CrossRef]

D. Michaelis, U. Peschel, and F. Lederer, “Multistable localized structures and superlattices in semiconductor optical resonators,” Phys. Rev. A 56, R3366–R3373 (1997).
[CrossRef]

Mitschke, F.

G. Steinmeyer, A. Schwache, and F. Mitschke, “Quantitative characterization of turbulence in an optical experiment,” Phys. Rev. E 53, 5399–5402 (1996).
[CrossRef]

Moloney, J. V.

D. W. McLaughlin, J. V. Moloney, and A. C. Newell, “Solitary waves as fixed-points of infinite-dimensional maps in an optical bistable ring cavity,” Phys. Rev. Lett. 51, 75–78 (1983).
[CrossRef]

Newell, A. C.

D. W. McLaughlin, J. V. Moloney, and A. C. Newell, “Solitary waves as fixed-points of infinite-dimensional maps in an optical bistable ring cavity,” Phys. Rev. Lett. 51, 75–78 (1983).
[CrossRef]

Perrini, I.

Peschel, U.

D. Michaelis, U. Peschel, and F. Lederer, “Oscillating dark cavity solitons,” Opt. Lett. 23, 1814–1816 (1998).
[CrossRef]

D. Michaelis, U. Peschel, and F. Lederer, “Multistable localized structures and superlattices in semiconductor optical resonators,” Phys. Rev. A 56, R3366–R3373 (1997).
[CrossRef]

C. Etrich, U. Peschel, and F. Lederer, “Solitary waves in quadratically nonlinear resonators,” Phys. Rev. Lett. 79, 2454–2457 (1997).
[CrossRef]

Prati, F.

L. Spinelli, G. Tissoni, M. Brambilla, F. Prati, and L. A. Lugiato, “Spatial solitons in semiconductor microcavities,” Phys. Rev. A 58, 2542–2559 (1998).
[CrossRef]

M. Brambilla, L. A. Lugiato, F. Prati, L. Spinelli, and W. J. Firth, “Spatial soliton pixels in semiconductor devices,” Phys. Rev. Lett. 79, 2042–2045 (1997).
[CrossRef]

Rosanov, N. N.

N. N. Rosanov, D. V. Fedorov, and G. V. Khodova, “Particle-like light structures in the wide-aperture laser with saturating absorption,” J. Exp. Theor. Phys. 107, 376–392 (1995).

N. N. Rosanov and G. V. Khodova, “Diffractive autosolitons in nonlinear interferometers,” J. Opt. Soc. Am. B 7, 1057–1065 (1990).
[CrossRef]

N. N. Rosanov and G. V. Khodova, “Switching of a bistable interferometer on localized inhomogeneities,” Opt. Spectrosc. (USSR) 65, 1399–1401 (1988).

Sanchez-Morillo, V. J.

K. Staliunas and V. J. Sanchez-Morillo, “Localized structures in degenerate parametric oscillators,” Opt. Commun. 139, 306–312 (1997).
[CrossRef]

Schäpers, B.

B. Schäpers, M. Feldmann, T. Ackemann, and W. Lange, “Interaction of localized structures in an optical pattern-forming system,” Phys. Rev. Lett. 85, 748–751 (2000).
[CrossRef] [PubMed]

Schreiber, A.

A. Schreiber, B. Thüering, M. Kreuzer, and T. Tschudi, “Experimental investigation of solitary structures in a nonlinear optical feedback system,” Opt. Commun. 136, 415–418 (1997).
[CrossRef]

Schwache, A.

G. Steinmeyer, A. Schwache, and F. Mitschke, “Quantitative characterization of turbulence in an optical experiment,” Phys. Rev. E 53, 5399–5402 (1996).
[CrossRef]

Scroggie, A. J.

A. J. Scroggie, W. J. Firth, G. S. McDonald, M. Tlidi, R. Lefever, and L. A. Lugiato, “Pattern formation in a passive Kerr cavity,” Chaos, Solitons Fractals 4, 1323–1354 (1996).
[CrossRef]

W. J. Firth, A. Lord, and A. J. Scroggie, “Optical bullet holes,” Phys. Scr. T67, 12–16 (1996).
[CrossRef]

W. J. Firth and A. J. Scroggie, “Optical bullet holes: robust controllable localized states of a nonlinear cavity,” Phys. Rev. Lett. 76, 1623–1626 (1996).
[CrossRef] [PubMed]

Skryabin, D. V.

Spinelli, L.

Staliunas, K.

K. Staliunas and V. J. Sanchez-Morillo, “Localized structures in degenerate parametric oscillators,” Opt. Commun. 139, 306–312 (1997).
[CrossRef]

Stefani, M.

M. Brambilla, L. A. Lugiato, and M. Stefani, “Interaction and control of optical localized structures,” Europhys. Lett. 34, 109–114 (1996).
[CrossRef]

Steinmeyer, G.

G. Steinmeyer, A. Schwache, and F. Mitschke, “Quantitative characterization of turbulence in an optical experiment,” Phys. Rev. E 53, 5399–5402 (1996).
[CrossRef]

Taranenko, V. B.

V. B. Taranenko, I. Ganne, R. J. Kuszelewicz, and C. O. Weiss, “Patterns and localized structures in bistable semiconductor resonators,” Phys. Rev. A 61, 0638–18 (2000).
[CrossRef]

Thüering, B.

A. Schreiber, B. Thüering, M. Kreuzer, and T. Tschudi, “Experimental investigation of solitary structures in a nonlinear optical feedback system,” Opt. Commun. 136, 415–418 (1997).
[CrossRef]

Tissoni, G.

Tlidi, M.

M. Tlidi, P. Mandel, and M. Haelterman, “Spatiotemporal patterns and localized structures in nonlinear optics,” Phys. Rev. E 56, 6524–6530 (1997).
[CrossRef]

M. Tlidi and P. Mandel, “Spatial patterns in nascent optical bistability,” Chaos, Solitons Fractals 4, 1475–1485 (1996).
[CrossRef]

A. J. Scroggie, W. J. Firth, G. S. McDonald, M. Tlidi, R. Lefever, and L. A. Lugiato, “Pattern formation in a passive Kerr cavity,” Chaos, Solitons Fractals 4, 1323–1354 (1996).
[CrossRef]

M. Tlidi, P. Mandel, and R. Lefever, “Localized structures and localized patterns in optical bistability,” Phys. Rev. Lett. 73, 640–643 (1994).
[CrossRef] [PubMed]

Tschudi, T.

A. Schreiber, B. Thüering, M. Kreuzer, and T. Tschudi, “Experimental investigation of solitary structures in a nonlinear optical feedback system,” Opt. Commun. 136, 415–418 (1997).
[CrossRef]

Wabnitz, S.

Weiss, C. O.

V. B. Taranenko, I. Ganne, R. J. Kuszelewicz, and C. O. Weiss, “Patterns and localized structures in bistable semiconductor resonators,” Phys. Rev. A 61, 0638–18 (2000).
[CrossRef]

Zelyanaya, E. V.

I. V. Barashenkov and E. V. Zelyanaya, “Stable complexes of parametrically driven, damped nonlinear Schrödinger solitons,” Phys. Rev. Lett. 83, 2568–2571 (1999).
[CrossRef]

Asian J. Phys. (1)

W. J. Firth and G. K. Harkness, “Cavity solitons,” Asian J. Phys. 7, 665–677 (1998).

Chaos, Solitons Fractals (3)

M. Tlidi and P. Mandel, “Spatial patterns in nascent optical bistability,” Chaos, Solitons Fractals 4, 1475–1485 (1996).
[CrossRef]

A. J. Scroggie, W. J. Firth, G. S. McDonald, M. Tlidi, R. Lefever, and L. A. Lugiato, “Pattern formation in a passive Kerr cavity,” Chaos, Solitons Fractals 4, 1323–1354 (1996).
[CrossRef]

L. A. Lugiato, ed., Special issue on nonlinear optical systems, chaos, and noise, Chaos, Solitons Fractals 4, 1251–1258 (1994).
[CrossRef]

Europhys. Lett. (1)

M. Brambilla, L. A. Lugiato, and M. Stefani, “Interaction and control of optical localized structures,” Europhys. Lett. 34, 109–114 (1996).
[CrossRef]

J. Exp. Theor. Phys. (1)

N. N. Rosanov, D. V. Fedorov, and G. V. Khodova, “Particle-like light structures in the wide-aperture laser with saturating absorption,” J. Exp. Theor. Phys. 107, 376–392 (1995).

J. Mod. Opt. (1)

W. J. Firth and A. Lord, “Two-dimensional solitons in a Kerr cavity,” J. Mod. Opt. 43, 1071–1077 (1996).
[CrossRef]

J. Opt. Soc. Am. B (6)

Opt. Commun. (2)

A. Schreiber, B. Thüering, M. Kreuzer, and T. Tschudi, “Experimental investigation of solitary structures in a nonlinear optical feedback system,” Opt. Commun. 136, 415–418 (1997).
[CrossRef]

K. Staliunas and V. J. Sanchez-Morillo, “Localized structures in degenerate parametric oscillators,” Opt. Commun. 139, 306–312 (1997).
[CrossRef]

Opt. Lett. (4)

Opt. Spectrosc. (USSR) (1)

N. N. Rosanov and G. V. Khodova, “Switching of a bistable interferometer on localized inhomogeneities,” Opt. Spectrosc. (USSR) 65, 1399–1401 (1988).

Phys. Rev. A (3)

D. Michaelis, U. Peschel, and F. Lederer, “Multistable localized structures and superlattices in semiconductor optical resonators,” Phys. Rev. A 56, R3366–R3373 (1997).
[CrossRef]

L. Spinelli, G. Tissoni, M. Brambilla, F. Prati, and L. A. Lugiato, “Spatial solitons in semiconductor microcavities,” Phys. Rev. A 58, 2542–2559 (1998).
[CrossRef]

V. B. Taranenko, I. Ganne, R. J. Kuszelewicz, and C. O. Weiss, “Patterns and localized structures in bistable semiconductor resonators,” Phys. Rev. A 61, 0638–18 (2000).
[CrossRef]

Phys. Rev. E (3)

M. Tlidi, P. Mandel, and M. Haelterman, “Spatiotemporal patterns and localized structures in nonlinear optics,” Phys. Rev. E 56, 6524–6530 (1997).
[CrossRef]

G. Steinmeyer, A. Schwache, and F. Mitschke, “Quantitative characterization of turbulence in an optical experiment,” Phys. Rev. E 53, 5399–5402 (1996).
[CrossRef]

T. Maggipinto, M. Brambilla, G. K. Harkness, and W. J. Firth, “Cavity solitons in semiconductor microresonators: existence, stability, and dynamical properties,” Phys. Rev. E 62, 8726–8739 (2000).
[CrossRef]

Phys. Rev. Lett. (8)

M. Brambilla, L. A. Lugiato, F. Prati, L. Spinelli, and W. J. Firth, “Spatial soliton pixels in semiconductor devices,” Phys. Rev. Lett. 79, 2042–2045 (1997).
[CrossRef]

M. Tlidi, P. Mandel, and R. Lefever, “Localized structures and localized patterns in optical bistability,” Phys. Rev. Lett. 73, 640–643 (1994).
[CrossRef] [PubMed]

W. J. Firth and A. J. Scroggie, “Optical bullet holes: robust controllable localized states of a nonlinear cavity,” Phys. Rev. Lett. 76, 1623–1626 (1996).
[CrossRef] [PubMed]

D. W. McLaughlin, J. V. Moloney, and A. C. Newell, “Solitary waves as fixed-points of infinite-dimensional maps in an optical bistable ring cavity,” Phys. Rev. Lett. 51, 75–78 (1983).
[CrossRef]

C. Etrich, U. Peschel, and F. Lederer, “Solitary waves in quadratically nonlinear resonators,” Phys. Rev. Lett. 79, 2454–2457 (1997).
[CrossRef]

I. V. Barashenkov and E. V. Zelyanaya, “Stable complexes of parametrically driven, damped nonlinear Schrödinger solitons,” Phys. Rev. Lett. 83, 2568–2571 (1999).
[CrossRef]

L. A. Lugiato and R. Lefever, “Spatial dissipative structures in passive optical systems,” Phys. Rev. Lett. 58, 2209–2211 (1987).
[CrossRef] [PubMed]

B. Schäpers, M. Feldmann, T. Ackemann, and W. Lange, “Interaction of localized structures in an optical pattern-forming system,” Phys. Rev. Lett. 85, 748–751 (2000).
[CrossRef] [PubMed]

Phys. Scr. (1)

W. J. Firth, A. Lord, and A. J. Scroggie, “Optical bullet holes,” Phys. Scr. T67, 12–16 (1996).
[CrossRef]

Proc. SPIE (1)

W. J. Firth, “Processing information with arrays of spatial solitons,” in Photonics Devices, and Systems, M. Hrabovsky, M. Miller, and P. Tomanek, eds., Proc. SPIE 4016, 388–394 (2000).
[CrossRef]

Other (5)

N. N. Akhmediev and A. Ankiwicz, Solitons: Nonlinear Pulses and Beams (Chapman & Hall, London, 1997).

A. C. Newell, Solitons in Mathematics & Physics (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1987).

D. Gomila, M. Santagiustina, P. Colet, and M. San Miguel, “From hexagons to optical turbulence,” preprint.

A dynamical simulation may be viewed at http://cnqo.phys.strath.ac.uk/movies/kerrhopf.mpg.

J. V. Moloney and A. C. Newell, Nonlinear Optics (Addison-Wesley, Redwood City, Calif., 1992).

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Figures (8)

Fig. 1
Fig. 1

Typical cavity soliton, showing a bright peak on a darker homogeneous background, with a few weak diffraction rings. Plotted is the modulus of the intracavity field as a function of the transverse coordinates x and y; parameters are θ=1.2 and I=0.9.

Fig. 2
Fig. 2

Maximum amplitude of stationary solutions versus I for θ=1.0. Solutions shown are the 2D KCS solutions (solid curve) and the hexagonal pattern (dashed curve). The slight irregularity in the hexagon curve is a numerical artifact associated with the existence of a continuum of hexagonal patterns with different lattice spacings.

Fig. 3
Fig. 3

Stability of the upper- (solid curves) and lower- (dashed curves) branch cavity solitons: how the eigenvalues with largest real parts change with I|Es|2. The detuning is θ=1.3.

Fig. 4
Fig. 4

Stability of 2D KCSs in the θ, I plane. Solitons exist above the lowest of the curves and are stable in the unshaded region. In the darkest area they are unstable to a Hopf mode with m=0, whereas in the other two shaded regions they are unstable to m=5 and m=6 modes (see text).

Fig. 5
Fig. 5

Dynamics of oscillating 2D KCS beyond the Hopf bifurcation: |A(0)|, as a function of time (θ=1.3, I=0.9). Dashed lines show |A(0)| for the upper- and lower-branch solitons. Inset, amplitude |a0| of Hopf-unstable eigenmode v0(r), where A(r, t)Au(r)+a0(t)v0(r); Au is the Hopf-unstable KCS and the dashed line is the gradient predicted by the corresponding eigenvalue from the linear stability analysis.

Fig. 6
Fig. 6

Dynamics of an oscillating 2D KCS beyond the Hopf bifurcation (θ=1.3, I=0.9): cross section |E(x, y=0, t)| relative to x and time. (The oscillation preserves cylindrical symmetry.)

Fig. 7
Fig. 7

Phase-plane representation of the dynamics of an oscillating 2D KCS (θ=1.3, I=0.9). Dashed curves, phase portrait of A(r) at the extrema of the oscillation; dotted curves, A(0) at intermediate times. Arrows, directions of rotation. Upper and lower solid curves (Hopf-unstable) upper and (amplitude unstable) lower KCSs, respectively.

Fig. 8
Fig. 8

Eigenmode corresponding to an azimuthal m=5 instability. θ=1.2, I=0.95.

Equations (1)

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i Et+2E+|E|2E=iε(-E-iθE+Ein),

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