Abstract

We systematically study the formation of cavity solitons in a high-finesse, doubly resonant degenerate optical parametric oscillator. Three types of cavity soliton, emanating from different plane-wave critical points, are identified. By means of amplitude equations the bifurcation dynamics of these solutions is studied and classified. We compare cavity solitons calculated from amplitude equations with the full numerical solutions near the critical points and trace their evolution numerically far from bifurcation. We found cavity soliton types previously not identified, namely, dark and oscillating solitons. These numerical studies are complemented by a linear stability analysis of cavity solitons. Various decay situations for unstable cavity solitons are discussed.

© 2002 Optical Society of America

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  1. J. C. Diels and W. Rudolph, Ultrashort Laser Pulse Phenomena (Academic, London, 1996).
  2. L. A. Lugiato, M. Brambilla, and A. Gatti, Optical Pattern Formation, Vol. 40 of Advances in Atomic, Molecular and Optical Physics, B. Bederson and H. Walther, eds. (Academic, New York, 1998), p. 229.
  3. G. L. Oppo, M. Brambilla, and L. A. Lugiato, “Formation and evolution of roll pattern in optical parametric oscillators,” Phys. Rev. A 49, 2028–2032 (1994).
    [CrossRef] [PubMed]
  4. G. L. Oppo, M. Brambilla, D. Camesasca, A. Gatti, and L. A. Lugiato, “Spatiotemporal dynamics of optical parametric oscillators,” J. Mod. Opt. 41, 1151–1162 (1994).
    [CrossRef]
  5. S. Longhi, “Alternating rolls in non-degenerate optical parametric oscillators,” J. Mod. Opt. 41, 1569–1575 (1996).
  6. G. J. de Valcárcel, K. Staliunas, E. Roldán, and V. J. Sánchez-Morcillo, “Transverse patterns in degenerate optical parametric oscillation and degenerate four-wave mixing,” Phys. Rev. A 54, 1609–1624 (1996).
    [CrossRef] [PubMed]
  7. C. Etrich, M. Michaelis, P. Peschel, and F. Lederer, “Short-term stability of patterns in intracavity vectorial second-harmonic generation,” Phys. Rev. E 58, 4005–4008 (1998).
    [CrossRef]
  8. M. Santagiustina, P. Colet, M. San Miguel, and D. Walgraef, “Walk-off and pattern selection in optical parametric oscillators,” Opt. Lett. 23, 1167–1169 (1998).
    [CrossRef]
  9. K. Staliunas and V. Sánchez-Morcillo, “Localized structures in degenerate optical parametric oscillators,” Opt. Commun. 139, 306–312 (1997).
    [CrossRef]
  10. S. Longhi, “Localized structures in optical parametric oscillation,” Phys. Scr. 56, 611–618 (1997).
    [CrossRef]
  11. D. V. Skryabin and W. J. Firth, “Interaction of cavity solitons in degenerate optical parametric oscillators,” Opt. Lett. 24, 1056–1058 (1999).
    [CrossRef]
  12. S. Trillo and M. Haelterman, “Excitation and bistability of self-trapped signal beams in optical parametric oscillators,” Opt. Lett. 23, 1514–1516 (1998).
    [CrossRef]
  13. D. V. Skryabin, “Instabilities of cavity solitons in optical parametric oscillators,” Phys. Rev. E 60, R3508–R3511 (1999).
    [CrossRef]
  14. S. Trillo, M. Haelterman, and A. Sheppard, “Stable topological spatial solitons in optical parametric oscillators,” Opt. Lett. 22, 970–972 (1997).
    [CrossRef] [PubMed]
  15. K. Staliunas and V. J. Sánchez-Morcillo, “Spatial-localised structures in degenerate optical parametric oscillators,” Phys. Rev. A 57, 1454–1457 (1998).
    [CrossRef]
  16. G. L. Oppo, A. J. Scroggie, and W. J. Firth, “From domain walls to localized structures in degenerate optical parametric oscillators,” J. Opt. B 1, 133–138 (1999).
    [CrossRef]
  17. G. L. Oppo, A. J. Scroggie, and W. J. Firth, “Characterization, dynamics and stabilization of diffractive domain walls and dark ring cavity solitons in parametric oscillators,” Phys. Rev. E 63, 066209 1–16 (2001).
    [CrossRef]
  18. D. Michaelis, U. Peschel, and F. Lederer, “Oscillating dark cavity solitons,” Opt. Lett. 23, 1814–1816 (1998).
    [CrossRef]
  19. M. Tlidi, P. Mandel, and M. Haelterman, “Spatiotemporal patterns and localized structures in nonlinear optics,” Phys. Rev. E 56, 6524–6530 (1997).
    [CrossRef]
  20. M. Tlidi and P. Mandel, “Space-time localized structures in degenerate optical parametric oscillators,” Phys. Rev. A 59, R2575–R2578 (1999).
    [CrossRef]
  21. V. J. Sánchez-Morcillo, E. Roldán, G. J. de Valcárcel, and K. Staliunas, “Generalized complex Swift–Hohenberg equation for optical parametric oscillators,” Phys. Rev. A 56, 3237–3244 (1997).
    [CrossRef]
  22. S. Longhi, “Spatial solitary waves in nondegenerate optical parametric oscillators near an inverted bifurcation,” Opt. Commun. 149, 335–340 (1998).
    [CrossRef]
  23. D. V. Skryabin, A. R. Champneys, and W. J. Firth, “Frequency selection by soliton excitation in the nondegenerate intracavity down-conversion,” Phys. Rev. Lett. 84, 463–466 (2000).
    [CrossRef] [PubMed]
  24. G. de Valcárcel, E. Roldán, and K. Staliunas, “Cavity solitons in nondegenerate optical parametric oscillation,” Opt. Commun. 181, 207–213 (2000).
    [CrossRef]
  25. G. Izus, M. San Miguel, and M. Santagiustina, “Bloch domain walls in type II optical parametric oscillators,” Opt. Lett. 25, 1454–1456 (2000).
    [CrossRef]
  26. M. Le Berre, E. Ressayre, and A. Tallet, “Role of the phase mismatch on the structures generated in an optical parametric oscillator,” J. Opt. B 1, 107–113 (1999).
    [CrossRef]
  27. M. Le Berre, D. Leduc, E. Ressayre, and A. Tallet, “Striped and circular domain walls in the DOPO,” J. Opt. B 1, 153–160 (1999).
    [CrossRef]
  28. M. Tlidi, M. Le Berre, E. Ressayre, A. Tallet, and L. Di Menza, “High-intensity localised structures in the degenerate optical parametric oscillator: comparison between the propagation and the mean-field models,” Phys. Rev. A 61, 043806 1–7 (2000).
    [CrossRef]
  29. N. P. Pettiaux, R.-D. Li, and P. Mandel, “Instabilities of the degenerate optical parametric oscillator,” Opt. Commun. 72, 256–260 (1989).
    [CrossRef]
  30. N. N. Rozanov, “Hysteresis phenomena in distributed optical systems,” Sov. Phys. JETP 53, 47–53 (1981).
  31. S. Longhi and A. Geraci, “Swift–Hohenberg equation for optical parametric oscillators,” Phys. Rev. A 54, 4581–4584 (1996).
    [CrossRef] [PubMed]

2001 (1)

G. L. Oppo, A. J. Scroggie, and W. J. Firth, “Characterization, dynamics and stabilization of diffractive domain walls and dark ring cavity solitons in parametric oscillators,” Phys. Rev. E 63, 066209 1–16 (2001).
[CrossRef]

2000 (4)

D. V. Skryabin, A. R. Champneys, and W. J. Firth, “Frequency selection by soliton excitation in the nondegenerate intracavity down-conversion,” Phys. Rev. Lett. 84, 463–466 (2000).
[CrossRef] [PubMed]

G. de Valcárcel, E. Roldán, and K. Staliunas, “Cavity solitons in nondegenerate optical parametric oscillation,” Opt. Commun. 181, 207–213 (2000).
[CrossRef]

M. Tlidi, M. Le Berre, E. Ressayre, A. Tallet, and L. Di Menza, “High-intensity localised structures in the degenerate optical parametric oscillator: comparison between the propagation and the mean-field models,” Phys. Rev. A 61, 043806 1–7 (2000).
[CrossRef]

G. Izus, M. San Miguel, and M. Santagiustina, “Bloch domain walls in type II optical parametric oscillators,” Opt. Lett. 25, 1454–1456 (2000).
[CrossRef]

1999 (6)

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

G. L. Oppo, A. J. Scroggie, and W. J. Firth, “From domain walls to localized structures in degenerate optical parametric oscillators,” J. Opt. B 1, 133–138 (1999).
[CrossRef]

M. Le Berre, E. Ressayre, and A. Tallet, “Role of the phase mismatch on the structures generated in an optical parametric oscillator,” J. Opt. B 1, 107–113 (1999).
[CrossRef]

M. Le Berre, D. Leduc, E. Ressayre, and A. Tallet, “Striped and circular domain walls in the DOPO,” J. Opt. B 1, 153–160 (1999).
[CrossRef]

M. Tlidi and P. Mandel, “Space-time localized structures in degenerate optical parametric oscillators,” Phys. Rev. A 59, R2575–R2578 (1999).
[CrossRef]

D. V. Skryabin, “Instabilities of cavity solitons in optical parametric oscillators,” Phys. Rev. E 60, R3508–R3511 (1999).
[CrossRef]

1998 (6)

K. Staliunas and V. J. Sánchez-Morcillo, “Spatial-localised structures in degenerate optical parametric oscillators,” Phys. Rev. A 57, 1454–1457 (1998).
[CrossRef]

C. Etrich, M. Michaelis, P. Peschel, and F. Lederer, “Short-term stability of patterns in intracavity vectorial second-harmonic generation,” Phys. Rev. E 58, 4005–4008 (1998).
[CrossRef]

S. Longhi, “Spatial solitary waves in nondegenerate optical parametric oscillators near an inverted bifurcation,” Opt. Commun. 149, 335–340 (1998).
[CrossRef]

M. Santagiustina, P. Colet, M. San Miguel, and D. Walgraef, “Walk-off and pattern selection in optical parametric oscillators,” Opt. Lett. 23, 1167–1169 (1998).
[CrossRef]

S. Trillo and M. Haelterman, “Excitation and bistability of self-trapped signal beams in optical parametric oscillators,” Opt. Lett. 23, 1514–1516 (1998).
[CrossRef]

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

1997 (5)

S. Trillo, M. Haelterman, and A. Sheppard, “Stable topological spatial solitons in optical parametric oscillators,” Opt. Lett. 22, 970–972 (1997).
[CrossRef] [PubMed]

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

S. Longhi, “Localized structures in optical parametric oscillation,” Phys. Scr. 56, 611–618 (1997).
[CrossRef]

V. J. Sánchez-Morcillo, E. Roldán, G. J. de Valcárcel, and K. Staliunas, “Generalized complex Swift–Hohenberg equation for optical parametric oscillators,” Phys. Rev. A 56, 3237–3244 (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 (3)

S. Longhi, “Alternating rolls in non-degenerate optical parametric oscillators,” J. Mod. Opt. 41, 1569–1575 (1996).

G. J. de Valcárcel, K. Staliunas, E. Roldán, and V. J. Sánchez-Morcillo, “Transverse patterns in degenerate optical parametric oscillation and degenerate four-wave mixing,” Phys. Rev. A 54, 1609–1624 (1996).
[CrossRef] [PubMed]

S. Longhi and A. Geraci, “Swift–Hohenberg equation for optical parametric oscillators,” Phys. Rev. A 54, 4581–4584 (1996).
[CrossRef] [PubMed]

1994 (2)

G. L. Oppo, M. Brambilla, and L. A. Lugiato, “Formation and evolution of roll pattern in optical parametric oscillators,” Phys. Rev. A 49, 2028–2032 (1994).
[CrossRef] [PubMed]

G. L. Oppo, M. Brambilla, D. Camesasca, A. Gatti, and L. A. Lugiato, “Spatiotemporal dynamics of optical parametric oscillators,” J. Mod. Opt. 41, 1151–1162 (1994).
[CrossRef]

1989 (1)

N. P. Pettiaux, R.-D. Li, and P. Mandel, “Instabilities of the degenerate optical parametric oscillator,” Opt. Commun. 72, 256–260 (1989).
[CrossRef]

1981 (1)

N. N. Rozanov, “Hysteresis phenomena in distributed optical systems,” Sov. Phys. JETP 53, 47–53 (1981).

Brambilla, M.

G. L. Oppo, M. Brambilla, D. Camesasca, A. Gatti, and L. A. Lugiato, “Spatiotemporal dynamics of optical parametric oscillators,” J. Mod. Opt. 41, 1151–1162 (1994).
[CrossRef]

G. L. Oppo, M. Brambilla, and L. A. Lugiato, “Formation and evolution of roll pattern in optical parametric oscillators,” Phys. Rev. A 49, 2028–2032 (1994).
[CrossRef] [PubMed]

Camesasca, D.

G. L. Oppo, M. Brambilla, D. Camesasca, A. Gatti, and L. A. Lugiato, “Spatiotemporal dynamics of optical parametric oscillators,” J. Mod. Opt. 41, 1151–1162 (1994).
[CrossRef]

Champneys, A. R.

D. V. Skryabin, A. R. Champneys, and W. J. Firth, “Frequency selection by soliton excitation in the nondegenerate intracavity down-conversion,” Phys. Rev. Lett. 84, 463–466 (2000).
[CrossRef] [PubMed]

Colet, P.

de Valcárcel, G.

G. de Valcárcel, E. Roldán, and K. Staliunas, “Cavity solitons in nondegenerate optical parametric oscillation,” Opt. Commun. 181, 207–213 (2000).
[CrossRef]

de Valcárcel, G. J.

V. J. Sánchez-Morcillo, E. Roldán, G. J. de Valcárcel, and K. Staliunas, “Generalized complex Swift–Hohenberg equation for optical parametric oscillators,” Phys. Rev. A 56, 3237–3244 (1997).
[CrossRef]

G. J. de Valcárcel, K. Staliunas, E. Roldán, and V. J. Sánchez-Morcillo, “Transverse patterns in degenerate optical parametric oscillation and degenerate four-wave mixing,” Phys. Rev. A 54, 1609–1624 (1996).
[CrossRef] [PubMed]

Di Menza, L.

M. Tlidi, M. Le Berre, E. Ressayre, A. Tallet, and L. Di Menza, “High-intensity localised structures in the degenerate optical parametric oscillator: comparison between the propagation and the mean-field models,” Phys. Rev. A 61, 043806 1–7 (2000).
[CrossRef]

Etrich, C.

C. Etrich, M. Michaelis, P. Peschel, and F. Lederer, “Short-term stability of patterns in intracavity vectorial second-harmonic generation,” Phys. Rev. E 58, 4005–4008 (1998).
[CrossRef]

Firth, W. J.

G. L. Oppo, A. J. Scroggie, and W. J. Firth, “Characterization, dynamics and stabilization of diffractive domain walls and dark ring cavity solitons in parametric oscillators,” Phys. Rev. E 63, 066209 1–16 (2001).
[CrossRef]

D. V. Skryabin, A. R. Champneys, and W. J. Firth, “Frequency selection by soliton excitation in the nondegenerate intracavity down-conversion,” Phys. Rev. Lett. 84, 463–466 (2000).
[CrossRef] [PubMed]

G. L. Oppo, A. J. Scroggie, and W. J. Firth, “From domain walls to localized structures in degenerate optical parametric oscillators,” J. Opt. B 1, 133–138 (1999).
[CrossRef]

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

Gatti, A.

G. L. Oppo, M. Brambilla, D. Camesasca, A. Gatti, and L. A. Lugiato, “Spatiotemporal dynamics of optical parametric oscillators,” J. Mod. Opt. 41, 1151–1162 (1994).
[CrossRef]

Geraci, A.

S. Longhi and A. Geraci, “Swift–Hohenberg equation for optical parametric oscillators,” Phys. Rev. A 54, 4581–4584 (1996).
[CrossRef] [PubMed]

Haelterman, M.

Izus, G.

Le Berre, M.

M. Tlidi, M. Le Berre, E. Ressayre, A. Tallet, and L. Di Menza, “High-intensity localised structures in the degenerate optical parametric oscillator: comparison between the propagation and the mean-field models,” Phys. Rev. A 61, 043806 1–7 (2000).
[CrossRef]

M. Le Berre, E. Ressayre, and A. Tallet, “Role of the phase mismatch on the structures generated in an optical parametric oscillator,” J. Opt. B 1, 107–113 (1999).
[CrossRef]

M. Le Berre, D. Leduc, E. Ressayre, and A. Tallet, “Striped and circular domain walls in the DOPO,” J. Opt. B 1, 153–160 (1999).
[CrossRef]

Lederer, F.

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

C. Etrich, M. Michaelis, P. Peschel, and F. Lederer, “Short-term stability of patterns in intracavity vectorial second-harmonic generation,” Phys. Rev. E 58, 4005–4008 (1998).
[CrossRef]

Leduc, D.

M. Le Berre, D. Leduc, E. Ressayre, and A. Tallet, “Striped and circular domain walls in the DOPO,” J. Opt. B 1, 153–160 (1999).
[CrossRef]

Li, R.-D.

N. P. Pettiaux, R.-D. Li, and P. Mandel, “Instabilities of the degenerate optical parametric oscillator,” Opt. Commun. 72, 256–260 (1989).
[CrossRef]

Longhi, S.

S. Longhi, “Spatial solitary waves in nondegenerate optical parametric oscillators near an inverted bifurcation,” Opt. Commun. 149, 335–340 (1998).
[CrossRef]

S. Longhi, “Localized structures in optical parametric oscillation,” Phys. Scr. 56, 611–618 (1997).
[CrossRef]

S. Longhi, “Alternating rolls in non-degenerate optical parametric oscillators,” J. Mod. Opt. 41, 1569–1575 (1996).

S. Longhi and A. Geraci, “Swift–Hohenberg equation for optical parametric oscillators,” Phys. Rev. A 54, 4581–4584 (1996).
[CrossRef] [PubMed]

Lugiato, L. A.

G. L. Oppo, M. Brambilla, and L. A. Lugiato, “Formation and evolution of roll pattern in optical parametric oscillators,” Phys. Rev. A 49, 2028–2032 (1994).
[CrossRef] [PubMed]

G. L. Oppo, M. Brambilla, D. Camesasca, A. Gatti, and L. A. Lugiato, “Spatiotemporal dynamics of optical parametric oscillators,” J. Mod. Opt. 41, 1151–1162 (1994).
[CrossRef]

Mandel, P.

M. Tlidi and P. Mandel, “Space-time localized structures in degenerate optical parametric oscillators,” Phys. Rev. A 59, R2575–R2578 (1999).
[CrossRef]

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

N. P. Pettiaux, R.-D. Li, and P. Mandel, “Instabilities of the degenerate optical parametric oscillator,” Opt. Commun. 72, 256–260 (1989).
[CrossRef]

Michaelis, D.

Michaelis, M.

C. Etrich, M. Michaelis, P. Peschel, and F. Lederer, “Short-term stability of patterns in intracavity vectorial second-harmonic generation,” Phys. Rev. E 58, 4005–4008 (1998).
[CrossRef]

Oppo, G. L.

G. L. Oppo, A. J. Scroggie, and W. J. Firth, “Characterization, dynamics and stabilization of diffractive domain walls and dark ring cavity solitons in parametric oscillators,” Phys. Rev. E 63, 066209 1–16 (2001).
[CrossRef]

G. L. Oppo, A. J. Scroggie, and W. J. Firth, “From domain walls to localized structures in degenerate optical parametric oscillators,” J. Opt. B 1, 133–138 (1999).
[CrossRef]

G. L. Oppo, M. Brambilla, D. Camesasca, A. Gatti, and L. A. Lugiato, “Spatiotemporal dynamics of optical parametric oscillators,” J. Mod. Opt. 41, 1151–1162 (1994).
[CrossRef]

G. L. Oppo, M. Brambilla, and L. A. Lugiato, “Formation and evolution of roll pattern in optical parametric oscillators,” Phys. Rev. A 49, 2028–2032 (1994).
[CrossRef] [PubMed]

Peschel, P.

C. Etrich, M. Michaelis, P. Peschel, and F. Lederer, “Short-term stability of patterns in intracavity vectorial second-harmonic generation,” Phys. Rev. E 58, 4005–4008 (1998).
[CrossRef]

Peschel, U.

Pettiaux, N. P.

N. P. Pettiaux, R.-D. Li, and P. Mandel, “Instabilities of the degenerate optical parametric oscillator,” Opt. Commun. 72, 256–260 (1989).
[CrossRef]

Ressayre, E.

M. Tlidi, M. Le Berre, E. Ressayre, A. Tallet, and L. Di Menza, “High-intensity localised structures in the degenerate optical parametric oscillator: comparison between the propagation and the mean-field models,” Phys. Rev. A 61, 043806 1–7 (2000).
[CrossRef]

M. Le Berre, E. Ressayre, and A. Tallet, “Role of the phase mismatch on the structures generated in an optical parametric oscillator,” J. Opt. B 1, 107–113 (1999).
[CrossRef]

M. Le Berre, D. Leduc, E. Ressayre, and A. Tallet, “Striped and circular domain walls in the DOPO,” J. Opt. B 1, 153–160 (1999).
[CrossRef]

Roldán, E.

G. de Valcárcel, E. Roldán, and K. Staliunas, “Cavity solitons in nondegenerate optical parametric oscillation,” Opt. Commun. 181, 207–213 (2000).
[CrossRef]

V. J. Sánchez-Morcillo, E. Roldán, G. J. de Valcárcel, and K. Staliunas, “Generalized complex Swift–Hohenberg equation for optical parametric oscillators,” Phys. Rev. A 56, 3237–3244 (1997).
[CrossRef]

G. J. de Valcárcel, K. Staliunas, E. Roldán, and V. J. Sánchez-Morcillo, “Transverse patterns in degenerate optical parametric oscillation and degenerate four-wave mixing,” Phys. Rev. A 54, 1609–1624 (1996).
[CrossRef] [PubMed]

Rozanov, N. N.

N. N. Rozanov, “Hysteresis phenomena in distributed optical systems,” Sov. Phys. JETP 53, 47–53 (1981).

San Miguel, M.

Sánchez-Morcillo, V.

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

Sánchez-Morcillo, V. J.

K. Staliunas and V. J. Sánchez-Morcillo, “Spatial-localised structures in degenerate optical parametric oscillators,” Phys. Rev. A 57, 1454–1457 (1998).
[CrossRef]

V. J. Sánchez-Morcillo, E. Roldán, G. J. de Valcárcel, and K. Staliunas, “Generalized complex Swift–Hohenberg equation for optical parametric oscillators,” Phys. Rev. A 56, 3237–3244 (1997).
[CrossRef]

G. J. de Valcárcel, K. Staliunas, E. Roldán, and V. J. Sánchez-Morcillo, “Transverse patterns in degenerate optical parametric oscillation and degenerate four-wave mixing,” Phys. Rev. A 54, 1609–1624 (1996).
[CrossRef] [PubMed]

Santagiustina, M.

Scroggie, A. J.

G. L. Oppo, A. J. Scroggie, and W. J. Firth, “Characterization, dynamics and stabilization of diffractive domain walls and dark ring cavity solitons in parametric oscillators,” Phys. Rev. E 63, 066209 1–16 (2001).
[CrossRef]

G. L. Oppo, A. J. Scroggie, and W. J. Firth, “From domain walls to localized structures in degenerate optical parametric oscillators,” J. Opt. B 1, 133–138 (1999).
[CrossRef]

Sheppard, A.

Skryabin, D. V.

D. V. Skryabin, A. R. Champneys, and W. J. Firth, “Frequency selection by soliton excitation in the nondegenerate intracavity down-conversion,” Phys. Rev. Lett. 84, 463–466 (2000).
[CrossRef] [PubMed]

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

D. V. Skryabin, “Instabilities of cavity solitons in optical parametric oscillators,” Phys. Rev. E 60, R3508–R3511 (1999).
[CrossRef]

Staliunas, K.

G. de Valcárcel, E. Roldán, and K. Staliunas, “Cavity solitons in nondegenerate optical parametric oscillation,” Opt. Commun. 181, 207–213 (2000).
[CrossRef]

K. Staliunas and V. J. Sánchez-Morcillo, “Spatial-localised structures in degenerate optical parametric oscillators,” Phys. Rev. A 57, 1454–1457 (1998).
[CrossRef]

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

V. J. Sánchez-Morcillo, E. Roldán, G. J. de Valcárcel, and K. Staliunas, “Generalized complex Swift–Hohenberg equation for optical parametric oscillators,” Phys. Rev. A 56, 3237–3244 (1997).
[CrossRef]

G. J. de Valcárcel, K. Staliunas, E. Roldán, and V. J. Sánchez-Morcillo, “Transverse patterns in degenerate optical parametric oscillation and degenerate four-wave mixing,” Phys. Rev. A 54, 1609–1624 (1996).
[CrossRef] [PubMed]

Tallet, A.

M. Tlidi, M. Le Berre, E. Ressayre, A. Tallet, and L. Di Menza, “High-intensity localised structures in the degenerate optical parametric oscillator: comparison between the propagation and the mean-field models,” Phys. Rev. A 61, 043806 1–7 (2000).
[CrossRef]

M. Le Berre, E. Ressayre, and A. Tallet, “Role of the phase mismatch on the structures generated in an optical parametric oscillator,” J. Opt. B 1, 107–113 (1999).
[CrossRef]

M. Le Berre, D. Leduc, E. Ressayre, and A. Tallet, “Striped and circular domain walls in the DOPO,” J. Opt. B 1, 153–160 (1999).
[CrossRef]

Tlidi, M.

M. Tlidi, M. Le Berre, E. Ressayre, A. Tallet, and L. Di Menza, “High-intensity localised structures in the degenerate optical parametric oscillator: comparison between the propagation and the mean-field models,” Phys. Rev. A 61, 043806 1–7 (2000).
[CrossRef]

M. Tlidi and P. Mandel, “Space-time localized structures in degenerate optical parametric oscillators,” Phys. Rev. A 59, R2575–R2578 (1999).
[CrossRef]

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

Trillo, S.

Walgraef, D.

J. Mod. Opt. (2)

G. L. Oppo, M. Brambilla, D. Camesasca, A. Gatti, and L. A. Lugiato, “Spatiotemporal dynamics of optical parametric oscillators,” J. Mod. Opt. 41, 1151–1162 (1994).
[CrossRef]

S. Longhi, “Alternating rolls in non-degenerate optical parametric oscillators,” J. Mod. Opt. 41, 1569–1575 (1996).

J. Opt. B (3)

G. L. Oppo, A. J. Scroggie, and W. J. Firth, “From domain walls to localized structures in degenerate optical parametric oscillators,” J. Opt. B 1, 133–138 (1999).
[CrossRef]

M. Le Berre, E. Ressayre, and A. Tallet, “Role of the phase mismatch on the structures generated in an optical parametric oscillator,” J. Opt. B 1, 107–113 (1999).
[CrossRef]

M. Le Berre, D. Leduc, E. Ressayre, and A. Tallet, “Striped and circular domain walls in the DOPO,” J. Opt. B 1, 153–160 (1999).
[CrossRef]

Opt. Commun. (4)

G. de Valcárcel, E. Roldán, and K. Staliunas, “Cavity solitons in nondegenerate optical parametric oscillation,” Opt. Commun. 181, 207–213 (2000).
[CrossRef]

N. P. Pettiaux, R.-D. Li, and P. Mandel, “Instabilities of the degenerate optical parametric oscillator,” Opt. Commun. 72, 256–260 (1989).
[CrossRef]

S. Longhi, “Spatial solitary waves in nondegenerate optical parametric oscillators near an inverted bifurcation,” Opt. Commun. 149, 335–340 (1998).
[CrossRef]

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

Opt. Lett. (6)

Phys. Rev. A (7)

K. Staliunas and V. J. Sánchez-Morcillo, “Spatial-localised structures in degenerate optical parametric oscillators,” Phys. Rev. A 57, 1454–1457 (1998).
[CrossRef]

M. Tlidi, M. Le Berre, E. Ressayre, A. Tallet, and L. Di Menza, “High-intensity localised structures in the degenerate optical parametric oscillator: comparison between the propagation and the mean-field models,” Phys. Rev. A 61, 043806 1–7 (2000).
[CrossRef]

S. Longhi and A. Geraci, “Swift–Hohenberg equation for optical parametric oscillators,” Phys. Rev. A 54, 4581–4584 (1996).
[CrossRef] [PubMed]

G. J. de Valcárcel, K. Staliunas, E. Roldán, and V. J. Sánchez-Morcillo, “Transverse patterns in degenerate optical parametric oscillation and degenerate four-wave mixing,” Phys. Rev. A 54, 1609–1624 (1996).
[CrossRef] [PubMed]

M. Tlidi and P. Mandel, “Space-time localized structures in degenerate optical parametric oscillators,” Phys. Rev. A 59, R2575–R2578 (1999).
[CrossRef]

V. J. Sánchez-Morcillo, E. Roldán, G. J. de Valcárcel, and K. Staliunas, “Generalized complex Swift–Hohenberg equation for optical parametric oscillators,” Phys. Rev. A 56, 3237–3244 (1997).
[CrossRef]

G. L. Oppo, M. Brambilla, and L. A. Lugiato, “Formation and evolution of roll pattern in optical parametric oscillators,” Phys. Rev. A 49, 2028–2032 (1994).
[CrossRef] [PubMed]

Phys. Rev. E (4)

D. V. Skryabin, “Instabilities of cavity solitons in optical parametric oscillators,” Phys. Rev. E 60, R3508–R3511 (1999).
[CrossRef]

G. L. Oppo, A. J. Scroggie, and W. J. Firth, “Characterization, dynamics and stabilization of diffractive domain walls and dark ring cavity solitons in parametric oscillators,” Phys. Rev. E 63, 066209 1–16 (2001).
[CrossRef]

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

C. Etrich, M. Michaelis, P. Peschel, and F. Lederer, “Short-term stability of patterns in intracavity vectorial second-harmonic generation,” Phys. Rev. E 58, 4005–4008 (1998).
[CrossRef]

Phys. Rev. Lett. (1)

D. V. Skryabin, A. R. Champneys, and W. J. Firth, “Frequency selection by soliton excitation in the nondegenerate intracavity down-conversion,” Phys. Rev. Lett. 84, 463–466 (2000).
[CrossRef] [PubMed]

Phys. Scr. (1)

S. Longhi, “Localized structures in optical parametric oscillation,” Phys. Scr. 56, 611–618 (1997).
[CrossRef]

Sov. Phys. JETP (1)

N. N. Rozanov, “Hysteresis phenomena in distributed optical systems,” Sov. Phys. JETP 53, 47–53 (1981).

Other (2)

J. C. Diels and W. Rudolph, Ultrashort Laser Pulse Phenomena (Academic, London, 1996).

L. A. Lugiato, M. Brambilla, and A. Gatti, Optical Pattern Formation, Vol. 40 of Advances in Atomic, Molecular and Optical Physics, B. Bederson and H. Walther, eds. (Academic, New York, 1998), p. 229.

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Figures (12)

Fig. 1
Fig. 1

Bifurcation diagrams displaying the different critical points on the plane-wave branches for Δ1=1, γ=0.5, and (a) Δ2=0 (EB supercritical), (b) Δ2=1.5 (EB subcritical). Thin solid curves, stable solutions; thin dashed curves, homogeneously unstable solutions; bold curves, modulationally unstable but otherwise homogeneously stable solutions.

Fig. 2
Fig. 2

(a) Bifurcation diagram (case A) displaying plane-wave solutions (thin curves) and cavity solitons (bold curves; FH amplitude at center) for Δ1=-1, Δ2=-3, and γ=0.5. Solid curves, stable solutions (homogeneously stable for plane-wave solutions); dashed curves, unstable solution; bold dotted curve, cavity soliton solutions of the amplitude equations (7). (b) Shapes of a stable and (c) a higher-order unstable cavity soliton on a trivial plane-wave background (FH, solid curve; SH, dashed curve).

Fig. 3
Fig. 3

Decay of an unstable cavity soliton on a trivial plane-wave background with m=5 for E=4.17, Δ1=-1, Δ2=-3, and γ=0.5 (FH, left; SH, right).

Fig. 4
Fig. 4

(a) Bifurcation diagram (cases A and C) displaying plane-wave solutions (thin curves) and cavity solitons (bold curves, FH amplitude at center) for Δ1=-0.7, Δ2=-1.3, and γ=0.5. Solid curves, stable solutions (homogeneously stable for plane-wave solutions); dashed curves, unstable solutions. (b) Cavity solitons developing into a switching wave for E=1.674, E=1.671, and E=1.67 (left to right).

Fig. 5
Fig. 5

Bifurcation diagrams (case B) comparing branches of cavity solitons emanating from EM2 (bold curves, FH amplitude at center) of (a) the Swift–Hohenberg equation (8) and (b) the full equation (5) for Δ1=0.1, Δ2=0, and γ=0.5. Thin curves, plane-wave solutions; solid curves, stable solutions (homogeneously stable for plane-wave solutions); dashed curves, unstable solutions. (c) Shapes of unstable cavity solitons (FH) near the bifurcation point for E=0.5038 [Swift–Hohenberg equation (dashed curve) and complete equations (solid curve)].

Fig. 6
Fig. 6

Bifurcation diagrams (case B) displaying different branches of cavity solitons (bold curves, FH amplitude at center) for Δ1=1, Δ2=0, and γ=0.5. Solid curves, stable solutions; dashed curves, unstable solutions; thin solid curves, homogeneously stable plane-wave solutions. The filled circles mark the maximum and minimum amplitudes of oscillating cavity solitons.

Fig. 7
Fig. 7

Amplitude versus radius of a stable cavity soliton for E=2.5, Δ1=1, Δ2=0, and γ=0.5 (FH, solid curve; SH, dashed curve).

Fig. 8
Fig. 8

(a) Maximum and (b) minimum FH of an oscillating cavity soliton for E=4, Δ1=1, Δ2=0, and γ=0.5.

Fig. 9
Fig. 9

Time propagation of an oscillating cavity soliton: (a) FH and (b) SH for E=4, Δ1=1, Δ2=0, and γ=0.5 (cross sections).

Fig. 10
Fig. 10

Decay of an unstable cavity soliton on a nontrivial plane-wave background with m=3 for E=1.7, Δ1=1, Δ2=0, and γ=0.5 (FH, left; SH, right).

Fig. 11
Fig. 11

Bifurcation diagram (case C) displaying plane-wave solutions (thin curves) and cavity solitons (bold curves; FH amplitude at center) for Δ1=1, Δ2=1.5, and γ=0.5. Solid curves, stable solutions (homogeneously stable for plane-wave solutions); dashed curves, unstable solutions; bold dotted curve, cavity soliton solutions of the amplitude equation (9). The bold dotted curve marks cavity soliton solutions of amplitude equation (10).

Fig. 12
Fig. 12

Shape of stable cavity solitons for E=2.05, Δ1=1, Δ2=1.5, and γ=0.5 (FH, solid curves; SH, dashed curves, from first four stable branches of Fig. 10.

Equations (47)

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i AT+2AX2+2AY2+(Δ1+i)A+A*B=0,
i BT+α2BX2+2BY2+(Δ2+iγ)B+A2=E,
A0=0,|B0|2=E2Δ22+γ2,
|A0|2=Δ1Δ2-γ±[E2-(γΔ1+Δ2)2]1/2
|B0|2=Δ12+1.
λ1,2=-1±E2Δ22+γ2-(k2-Δ1)21/2,
λ3,4=-γ±ik22-Δ2,
(Δ1+Δ2)(Δ1Δ2-γ)-γ(γΔ1+Δ2)=0
d2AdR2+1R dAdR+(Δ1+i)A+A*B=0,
12d2BdR2+1R dBdR+(Δ2+iγ)B+A2=E,
A1=A0+εA1+ε2A2+ε3A3+,
B=B0+εB1+ε2B2+ε3B3+
|A1|t+Δ1L|A1|-1Δ22+γ2
×[EBE2+(Δ1Δ2-γ)|A1|2]|A1|=0,
|A1|t+12(L+Δ11)2|A1|
-1Δ22+γ2(EBE2-γ|A1|2)|A1|=0,
α0 |A1|t-α1L|A1|±(β1|A1|2-β0E2)=0,
A1=A0+A1+2A2+3A3+,
B=B0+B1+2B2+3B3+
λ=-1+1+2EBδEΔ22+γ2+2Δ1k2-k41/2.
t=2T,x=X,y=Y
(Δ1+i)A1+B0A1*=0,
(Δ2+iγ)B1=0,
A1A1*=-1EB[Δ1Δ2-γ-i(Δ1γ+Δ2)].
(Δ1+i)A2+B0A2*=0,
(Δ2+iγ)B2=E2-A12,
(Δ1+i)A3+B0A3*=-A1*B2-i A1t-LA1,
(Δ2+iγ)B3=-2A1A2-i B1t-12LB1,
|A1|t+Δ1L|A1|-1Δ22+γ2
×[EBE2+(Δ1Δ2-γ)|A1|2]|A1|=0.
t=2T,x=X,y=Y.
A1A1*=1EB0(γ+iΔ2),
|A1|t+12(L+Δ11)2|A1|
-1Δ22+γ2(EB0E2-γ|A1|2)|A1|=0,
t=T,x=X,y=Y.
(Δ1+i)A1+B0A1*+A0*B1=0,
(Δ2+iγ)B1+2A0A1=0.
Δ1+i-2|A0|2Δ2+iγA1+B0A1*=0,
A1A1*=i [Δ1Δ2-γ+i(Δ1γ+Δ2)]2(Δ12+1)(Δ22+γ2),
(Δ1+i)A2+B0A2*+A0*B2
=-A1*B1-i A1t-LA1,
(Δ2+iγ)B2+2A0A2=E2-A12-i B1t-12LB1.
α0 |A1|t-α1L|A1|±(β1|A1|2-β0E2)=0,
α0=2(γ+1)(Δ1Δ2-γ)+Δ22+γ2,
α1=(Δ1+Δ2)(Δ1Δ2-γ)-γ(γΔ1+Δ2),
β0=(Δ1Δ2-γ)1/2[(Δ12+1)(Δ22+γ2)]1/2,
β1=2(Δ1Δ2-γ)3/2|γΔ1+Δ2|[(Δ12+1)(Δ22+γ2)]1/2,

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