Abstract

We show the relevance of walk-off effects in pattern formation in a type II optical parametric oscillator at frequency degeneracy. With walk-off neglected only phase patterns are formed, and the intensity distribution is homogeneous. Walk-off changes the instability from absolute to convective for some parameter range. In the absolutely unstable regime it induces for each polarization component of light a competition between two phase stripe patterns (traveling waves) of different wavelength. Phase stripe patterns at each of the wavelengths are equally likely to be selected, and, after a transient regime of coexistence, one of them takes over. In the convectively unstable regime the existence of intensity patterns sustained by noise is shown. The patterns arise from the interference between traveling waves that are generated by the dynamical amplification of noise.

© 1999 Optical Society of America

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References

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  1. L. A. Lugiato, M. Brambilla, and A. Gatti, “Optical pattern formation,” in Vol. 40 of Advances in Atomic Molecular and Optical Physics, B. Bederson and H. Walther, eds. (Academic, New York, 1999), pp. 229–306, and references cited therein.
  2. R. A. Fuerst, D-M. Baboiu, B. Lawrence, W. E. Torruelas, G. I. Stegeman, and S. Trillo, “Spatial modulational instability and multisolitonlike generation in a quadratically nonlinear optical medium,” Phys. Rev. Lett. 78, 2756–2759 (1997).
    [CrossRef]
  3. K. Staliunas, “Optical vortices during three-wave nonlinear coupling,” Opt. Commun. 91, 82–86 (1992); G-L. Oppo, M. Brambilla, and L. A. Lugiato, “Formation and evolution of roll patterns in optical parametric oscillators,” Phys. Rev. A 49, 2028–2032 (1994); S. Longhi, “Hydrodynamic equation model for degenerate optical parametric oscillators,” J. Mod. Opt. JMOPEW 43, 1089–1094 (1996); G. J. de Varcarcel, K. Staliunas, E. Roldan, and V. J. Sanchez-Morcillo, “Transverse patterns in degenerate optical parametric oscillation and degenerate four-wave mixing,” Phys. Rev. A PLRAAN 54, 1609–1624 (1996).
    [CrossRef] [PubMed]
  4. C. Schwob, P. F. Cohadon, C. Fabre, M. A. M. Marte, H. Ritsch, A. Gatti, and L. A. Lugiato, “Transverse effects and mode-coupling in OPOS,” Appl. Phys. B 66, 685–699 (1998).
    [CrossRef]
  5. H. J. Kimble, “Quantum fluctuations in quantum optics—squeezing and related phenomena,” in Fundamental Systems in Quantum Optics, J. Dalibard, J. M. Raimond, and J. Zinn-Justine, eds. (Elsevier Science, Amsterdam, 1992), pp. 545–673.
  6. S. Longhi, “Localized structures in optical parametric oscillators,” Phys. Scr. 57, 611–635 (1997); K. Staliunas and V. J. Sanchez-Morcillo, “Spatial-localized structures in degenerate optical parametric oscillators,” Phys. Rev. A 57, 1454–1457 (1998); S. Trillo and M. Haelterman, “Excitation and bistability of self-trapped signal beams in optical parametric oscillators,” Opt. Lett. OPLEDP 23, 1514–1516 (1998); G. L. Oppo, A. J. Scroggie, and W. J. Firth, “From domain walls to localised structures in degenerate optical parametric oscillators,” J. Opt. B ZZZZZZ 1, 133–138 (1999); M. Le Berre, D. Leduc, E. Ressayre, and A. Tallet, “Striped and circular domain walls in the DOPO,” J. Opt. B ZZZZZZ 1, 153–160 (1999).
    [CrossRef]
  7. 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]
  8. M. Santagiustina, P. Colet, M. San Miguel, and D. Walgraef, “Two-dimensional noise-sustained structures in optical parametric oscillators,” Phys. Rev. E 58, 3843–3853 (1998).
    [CrossRef]
  9. M. Santagiustina, P. Colet, M. San Miguel, and D. Walgraef, “Growth dynamics of noise-sustained structures in nonlinear optical resonators,” Opt. Express 3, 63–70 (1998), http://epubs.osa.org/opticsexpress.
    [CrossRef] [PubMed]
  10. H. Ward, M. N. Ouarzazi, M. Taki, and P. Glorieux, “Transverse dynamics of optical parametric oscillators in presence of walk-off,” Eur. Phys. J. D 3, 275–288 (1998).
    [CrossRef]
  11. S. Longhi, “Traveling-wave states and secondary instabili-ties in optical parametric oscillators,” Phys. Rev. A 53, 4488–4499 (1996).
    [CrossRef] [PubMed]
  12. M. Santagiustina, P. Colet, M. San Miguel, and D. Walgraef, “Noise-sustained convective structures in nonlinear optics,” Phys. Rev. Lett. 79, 3633–3636 (1997).
    [CrossRef]
  13. A. Gatti, H. Wiedemann, L. A. Lugiato, I. Marzoli, G. L. Oppo, and S. M. Barnett, “Langevin treatment of quantum fluctuations and optical patterns in optical parametric oscillators below threshold,” Phys. Rev. A 56, 877–897 (1997).
    [CrossRef]
  14. C. Richy, K. I. Petsas, E. Giacobino, C. Fabre, and L. Lugiato, “Observation of bistability and delayed bifurcation in a triply resonant optical parametric oscillator,” J. Opt. Soc. Am. B 12, 456–461 (1995).
    [CrossRef]
  15. R. J. Deissler, “Noise-sustained structure, intermittency, and the Ginzburg–Landau equation,” J. Stat. Phys. 40, 371–395 (1985); “External noise and the origin and dynamics of structure in convectively unstable systems,” J. Stat. Phys. 54, 1459–1488 (1989).
    [CrossRef]
  16. J. Garcia-Ojalvo, R. Vilaseca, and M. C. Torrent, “Suppression of Autler–Townes gain splitting in lasers with a planar resonator,” Europhys. Lett. 43, 261–266 (1998).
    [CrossRef]
  17. A. Berzanskis, A. Matijosius, A. Piskarkas, V. Smilgevicius, and A. Stabinis, “Sum-frequency mixing of optical vortices in nonlinear crystals,” Opt. Commun. 150, 372–380 (1998).
    [CrossRef]
  18. G-L. Oppo, A. J. Scroggie, and W. J. Firth, “Domain walls in optical parametric oscillators: dynamics and stabilization,” in European Quantum Electronics Conference (EQEC) (Optical Society of America, Washington, D.C., 1998), paper QFA2.

1998 (7)

C. Schwob, P. F. Cohadon, C. Fabre, M. A. M. Marte, H. Ritsch, A. Gatti, and L. A. Lugiato, “Transverse effects and mode-coupling in OPOS,” Appl. Phys. B 66, 685–699 (1998).
[CrossRef]

M. Santagiustina, P. Colet, M. San Miguel, and D. Walgraef, “Two-dimensional noise-sustained structures in optical parametric oscillators,” Phys. Rev. E 58, 3843–3853 (1998).
[CrossRef]

H. Ward, M. N. Ouarzazi, M. Taki, and P. Glorieux, “Transverse dynamics of optical parametric oscillators in presence of walk-off,” Eur. Phys. J. D 3, 275–288 (1998).
[CrossRef]

J. Garcia-Ojalvo, R. Vilaseca, and M. C. Torrent, “Suppression of Autler–Townes gain splitting in lasers with a planar resonator,” Europhys. Lett. 43, 261–266 (1998).
[CrossRef]

A. Berzanskis, A. Matijosius, A. Piskarkas, V. Smilgevicius, and A. Stabinis, “Sum-frequency mixing of optical vortices in nonlinear crystals,” Opt. Commun. 150, 372–380 (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]

M. Santagiustina, P. Colet, M. San Miguel, and D. Walgraef, “Growth dynamics of noise-sustained structures in nonlinear optical resonators,” Opt. Express 3, 63–70 (1998), http://epubs.osa.org/opticsexpress.
[CrossRef] [PubMed]

1997 (3)

R. A. Fuerst, D-M. Baboiu, B. Lawrence, W. E. Torruelas, G. I. Stegeman, and S. Trillo, “Spatial modulational instability and multisolitonlike generation in a quadratically nonlinear optical medium,” Phys. Rev. Lett. 78, 2756–2759 (1997).
[CrossRef]

M. Santagiustina, P. Colet, M. San Miguel, and D. Walgraef, “Noise-sustained convective structures in nonlinear optics,” Phys. Rev. Lett. 79, 3633–3636 (1997).
[CrossRef]

A. Gatti, H. Wiedemann, L. A. Lugiato, I. Marzoli, G. L. Oppo, and S. M. Barnett, “Langevin treatment of quantum fluctuations and optical patterns in optical parametric oscillators below threshold,” Phys. Rev. A 56, 877–897 (1997).
[CrossRef]

1996 (1)

S. Longhi, “Traveling-wave states and secondary instabili-ties in optical parametric oscillators,” Phys. Rev. A 53, 4488–4499 (1996).
[CrossRef] [PubMed]

1995 (1)

Baboiu, D-M.

R. A. Fuerst, D-M. Baboiu, B. Lawrence, W. E. Torruelas, G. I. Stegeman, and S. Trillo, “Spatial modulational instability and multisolitonlike generation in a quadratically nonlinear optical medium,” Phys. Rev. Lett. 78, 2756–2759 (1997).
[CrossRef]

Barnett, S. M.

A. Gatti, H. Wiedemann, L. A. Lugiato, I. Marzoli, G. L. Oppo, and S. M. Barnett, “Langevin treatment of quantum fluctuations and optical patterns in optical parametric oscillators below threshold,” Phys. Rev. A 56, 877–897 (1997).
[CrossRef]

Berzanskis, A.

A. Berzanskis, A. Matijosius, A. Piskarkas, V. Smilgevicius, and A. Stabinis, “Sum-frequency mixing of optical vortices in nonlinear crystals,” Opt. Commun. 150, 372–380 (1998).
[CrossRef]

Cohadon, P. F.

C. Schwob, P. F. Cohadon, C. Fabre, M. A. M. Marte, H. Ritsch, A. Gatti, and L. A. Lugiato, “Transverse effects and mode-coupling in OPOS,” Appl. Phys. B 66, 685–699 (1998).
[CrossRef]

Colet, P.

M. Santagiustina, P. Colet, M. San Miguel, and D. Walgraef, “Two-dimensional noise-sustained structures in optical parametric oscillators,” Phys. Rev. E 58, 3843–3853 (1998).
[CrossRef]

M. Santagiustina, P. Colet, M. San Miguel, and D. Walgraef, “Growth dynamics of noise-sustained structures in nonlinear optical resonators,” Opt. Express 3, 63–70 (1998), http://epubs.osa.org/opticsexpress.
[CrossRef] [PubMed]

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]

M. Santagiustina, P. Colet, M. San Miguel, and D. Walgraef, “Noise-sustained convective structures in nonlinear optics,” Phys. Rev. Lett. 79, 3633–3636 (1997).
[CrossRef]

Fabre, C.

C. Schwob, P. F. Cohadon, C. Fabre, M. A. M. Marte, H. Ritsch, A. Gatti, and L. A. Lugiato, “Transverse effects and mode-coupling in OPOS,” Appl. Phys. B 66, 685–699 (1998).
[CrossRef]

C. Richy, K. I. Petsas, E. Giacobino, C. Fabre, and L. Lugiato, “Observation of bistability and delayed bifurcation in a triply resonant optical parametric oscillator,” J. Opt. Soc. Am. B 12, 456–461 (1995).
[CrossRef]

Fuerst, R. A.

R. A. Fuerst, D-M. Baboiu, B. Lawrence, W. E. Torruelas, G. I. Stegeman, and S. Trillo, “Spatial modulational instability and multisolitonlike generation in a quadratically nonlinear optical medium,” Phys. Rev. Lett. 78, 2756–2759 (1997).
[CrossRef]

Garcia-Ojalvo, J.

J. Garcia-Ojalvo, R. Vilaseca, and M. C. Torrent, “Suppression of Autler–Townes gain splitting in lasers with a planar resonator,” Europhys. Lett. 43, 261–266 (1998).
[CrossRef]

Gatti, A.

C. Schwob, P. F. Cohadon, C. Fabre, M. A. M. Marte, H. Ritsch, A. Gatti, and L. A. Lugiato, “Transverse effects and mode-coupling in OPOS,” Appl. Phys. B 66, 685–699 (1998).
[CrossRef]

A. Gatti, H. Wiedemann, L. A. Lugiato, I. Marzoli, G. L. Oppo, and S. M. Barnett, “Langevin treatment of quantum fluctuations and optical patterns in optical parametric oscillators below threshold,” Phys. Rev. A 56, 877–897 (1997).
[CrossRef]

Giacobino, E.

Glorieux, P.

H. Ward, M. N. Ouarzazi, M. Taki, and P. Glorieux, “Transverse dynamics of optical parametric oscillators in presence of walk-off,” Eur. Phys. J. D 3, 275–288 (1998).
[CrossRef]

Lawrence, B.

R. A. Fuerst, D-M. Baboiu, B. Lawrence, W. E. Torruelas, G. I. Stegeman, and S. Trillo, “Spatial modulational instability and multisolitonlike generation in a quadratically nonlinear optical medium,” Phys. Rev. Lett. 78, 2756–2759 (1997).
[CrossRef]

Longhi, S.

S. Longhi, “Traveling-wave states and secondary instabili-ties in optical parametric oscillators,” Phys. Rev. A 53, 4488–4499 (1996).
[CrossRef] [PubMed]

Lugiato, L.

Lugiato, L. A.

C. Schwob, P. F. Cohadon, C. Fabre, M. A. M. Marte, H. Ritsch, A. Gatti, and L. A. Lugiato, “Transverse effects and mode-coupling in OPOS,” Appl. Phys. B 66, 685–699 (1998).
[CrossRef]

A. Gatti, H. Wiedemann, L. A. Lugiato, I. Marzoli, G. L. Oppo, and S. M. Barnett, “Langevin treatment of quantum fluctuations and optical patterns in optical parametric oscillators below threshold,” Phys. Rev. A 56, 877–897 (1997).
[CrossRef]

Marte, M. A. M.

C. Schwob, P. F. Cohadon, C. Fabre, M. A. M. Marte, H. Ritsch, A. Gatti, and L. A. Lugiato, “Transverse effects and mode-coupling in OPOS,” Appl. Phys. B 66, 685–699 (1998).
[CrossRef]

Marzoli, I.

A. Gatti, H. Wiedemann, L. A. Lugiato, I. Marzoli, G. L. Oppo, and S. M. Barnett, “Langevin treatment of quantum fluctuations and optical patterns in optical parametric oscillators below threshold,” Phys. Rev. A 56, 877–897 (1997).
[CrossRef]

Matijosius, A.

A. Berzanskis, A. Matijosius, A. Piskarkas, V. Smilgevicius, and A. Stabinis, “Sum-frequency mixing of optical vortices in nonlinear crystals,” Opt. Commun. 150, 372–380 (1998).
[CrossRef]

Oppo, G. L.

A. Gatti, H. Wiedemann, L. A. Lugiato, I. Marzoli, G. L. Oppo, and S. M. Barnett, “Langevin treatment of quantum fluctuations and optical patterns in optical parametric oscillators below threshold,” Phys. Rev. A 56, 877–897 (1997).
[CrossRef]

Ouarzazi, M. N.

H. Ward, M. N. Ouarzazi, M. Taki, and P. Glorieux, “Transverse dynamics of optical parametric oscillators in presence of walk-off,” Eur. Phys. J. D 3, 275–288 (1998).
[CrossRef]

Petsas, K. I.

Piskarkas, A.

A. Berzanskis, A. Matijosius, A. Piskarkas, V. Smilgevicius, and A. Stabinis, “Sum-frequency mixing of optical vortices in nonlinear crystals,” Opt. Commun. 150, 372–380 (1998).
[CrossRef]

Richy, C.

Ritsch, H.

C. Schwob, P. F. Cohadon, C. Fabre, M. A. M. Marte, H. Ritsch, A. Gatti, and L. A. Lugiato, “Transverse effects and mode-coupling in OPOS,” Appl. Phys. B 66, 685–699 (1998).
[CrossRef]

San Miguel, M.

M. Santagiustina, P. Colet, M. San Miguel, and D. Walgraef, “Two-dimensional noise-sustained structures in optical parametric oscillators,” Phys. Rev. E 58, 3843–3853 (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]

M. Santagiustina, P. Colet, M. San Miguel, and D. Walgraef, “Growth dynamics of noise-sustained structures in nonlinear optical resonators,” Opt. Express 3, 63–70 (1998), http://epubs.osa.org/opticsexpress.
[CrossRef] [PubMed]

M. Santagiustina, P. Colet, M. San Miguel, and D. Walgraef, “Noise-sustained convective structures in nonlinear optics,” Phys. Rev. Lett. 79, 3633–3636 (1997).
[CrossRef]

Santagiustina, M.

M. Santagiustina, P. Colet, M. San Miguel, and D. Walgraef, “Two-dimensional noise-sustained structures in optical parametric oscillators,” Phys. Rev. E 58, 3843–3853 (1998).
[CrossRef]

M. Santagiustina, P. Colet, M. San Miguel, and D. Walgraef, “Growth dynamics of noise-sustained structures in nonlinear optical resonators,” Opt. Express 3, 63–70 (1998), http://epubs.osa.org/opticsexpress.
[CrossRef] [PubMed]

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]

M. Santagiustina, P. Colet, M. San Miguel, and D. Walgraef, “Noise-sustained convective structures in nonlinear optics,” Phys. Rev. Lett. 79, 3633–3636 (1997).
[CrossRef]

Schwob, C.

C. Schwob, P. F. Cohadon, C. Fabre, M. A. M. Marte, H. Ritsch, A. Gatti, and L. A. Lugiato, “Transverse effects and mode-coupling in OPOS,” Appl. Phys. B 66, 685–699 (1998).
[CrossRef]

Smilgevicius, V.

A. Berzanskis, A. Matijosius, A. Piskarkas, V. Smilgevicius, and A. Stabinis, “Sum-frequency mixing of optical vortices in nonlinear crystals,” Opt. Commun. 150, 372–380 (1998).
[CrossRef]

Stabinis, A.

A. Berzanskis, A. Matijosius, A. Piskarkas, V. Smilgevicius, and A. Stabinis, “Sum-frequency mixing of optical vortices in nonlinear crystals,” Opt. Commun. 150, 372–380 (1998).
[CrossRef]

Stegeman, G. I.

R. A. Fuerst, D-M. Baboiu, B. Lawrence, W. E. Torruelas, G. I. Stegeman, and S. Trillo, “Spatial modulational instability and multisolitonlike generation in a quadratically nonlinear optical medium,” Phys. Rev. Lett. 78, 2756–2759 (1997).
[CrossRef]

Taki, M.

H. Ward, M. N. Ouarzazi, M. Taki, and P. Glorieux, “Transverse dynamics of optical parametric oscillators in presence of walk-off,” Eur. Phys. J. D 3, 275–288 (1998).
[CrossRef]

Torrent, M. C.

J. Garcia-Ojalvo, R. Vilaseca, and M. C. Torrent, “Suppression of Autler–Townes gain splitting in lasers with a planar resonator,” Europhys. Lett. 43, 261–266 (1998).
[CrossRef]

Torruelas, W. E.

R. A. Fuerst, D-M. Baboiu, B. Lawrence, W. E. Torruelas, G. I. Stegeman, and S. Trillo, “Spatial modulational instability and multisolitonlike generation in a quadratically nonlinear optical medium,” Phys. Rev. Lett. 78, 2756–2759 (1997).
[CrossRef]

Trillo, S.

R. A. Fuerst, D-M. Baboiu, B. Lawrence, W. E. Torruelas, G. I. Stegeman, and S. Trillo, “Spatial modulational instability and multisolitonlike generation in a quadratically nonlinear optical medium,” Phys. Rev. Lett. 78, 2756–2759 (1997).
[CrossRef]

Vilaseca, R.

J. Garcia-Ojalvo, R. Vilaseca, and M. C. Torrent, “Suppression of Autler–Townes gain splitting in lasers with a planar resonator,” Europhys. Lett. 43, 261–266 (1998).
[CrossRef]

Walgraef, D.

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]

M. Santagiustina, P. Colet, M. San Miguel, and D. Walgraef, “Growth dynamics of noise-sustained structures in nonlinear optical resonators,” Opt. Express 3, 63–70 (1998), http://epubs.osa.org/opticsexpress.
[CrossRef] [PubMed]

M. Santagiustina, P. Colet, M. San Miguel, and D. Walgraef, “Two-dimensional noise-sustained structures in optical parametric oscillators,” Phys. Rev. E 58, 3843–3853 (1998).
[CrossRef]

M. Santagiustina, P. Colet, M. San Miguel, and D. Walgraef, “Noise-sustained convective structures in nonlinear optics,” Phys. Rev. Lett. 79, 3633–3636 (1997).
[CrossRef]

Ward, H.

H. Ward, M. N. Ouarzazi, M. Taki, and P. Glorieux, “Transverse dynamics of optical parametric oscillators in presence of walk-off,” Eur. Phys. J. D 3, 275–288 (1998).
[CrossRef]

Wiedemann, H.

A. Gatti, H. Wiedemann, L. A. Lugiato, I. Marzoli, G. L. Oppo, and S. M. Barnett, “Langevin treatment of quantum fluctuations and optical patterns in optical parametric oscillators below threshold,” Phys. Rev. A 56, 877–897 (1997).
[CrossRef]

Appl. Phys. B (1)

C. Schwob, P. F. Cohadon, C. Fabre, M. A. M. Marte, H. Ritsch, A. Gatti, and L. A. Lugiato, “Transverse effects and mode-coupling in OPOS,” Appl. Phys. B 66, 685–699 (1998).
[CrossRef]

Eur. Phys. J. D (1)

H. Ward, M. N. Ouarzazi, M. Taki, and P. Glorieux, “Transverse dynamics of optical parametric oscillators in presence of walk-off,” Eur. Phys. J. D 3, 275–288 (1998).
[CrossRef]

Europhys. Lett. (1)

J. Garcia-Ojalvo, R. Vilaseca, and M. C. Torrent, “Suppression of Autler–Townes gain splitting in lasers with a planar resonator,” Europhys. Lett. 43, 261–266 (1998).
[CrossRef]

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

Opt. Commun. (1)

A. Berzanskis, A. Matijosius, A. Piskarkas, V. Smilgevicius, and A. Stabinis, “Sum-frequency mixing of optical vortices in nonlinear crystals,” Opt. Commun. 150, 372–380 (1998).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

Phys. Rev. A (2)

S. Longhi, “Traveling-wave states and secondary instabili-ties in optical parametric oscillators,” Phys. Rev. A 53, 4488–4499 (1996).
[CrossRef] [PubMed]

A. Gatti, H. Wiedemann, L. A. Lugiato, I. Marzoli, G. L. Oppo, and S. M. Barnett, “Langevin treatment of quantum fluctuations and optical patterns in optical parametric oscillators below threshold,” Phys. Rev. A 56, 877–897 (1997).
[CrossRef]

Phys. Rev. E (1)

M. Santagiustina, P. Colet, M. San Miguel, and D. Walgraef, “Two-dimensional noise-sustained structures in optical parametric oscillators,” Phys. Rev. E 58, 3843–3853 (1998).
[CrossRef]

Phys. Rev. Lett. (2)

M. Santagiustina, P. Colet, M. San Miguel, and D. Walgraef, “Noise-sustained convective structures in nonlinear optics,” Phys. Rev. Lett. 79, 3633–3636 (1997).
[CrossRef]

R. A. Fuerst, D-M. Baboiu, B. Lawrence, W. E. Torruelas, G. I. Stegeman, and S. Trillo, “Spatial modulational instability and multisolitonlike generation in a quadratically nonlinear optical medium,” Phys. Rev. Lett. 78, 2756–2759 (1997).
[CrossRef]

Other (6)

K. Staliunas, “Optical vortices during three-wave nonlinear coupling,” Opt. Commun. 91, 82–86 (1992); G-L. Oppo, M. Brambilla, and L. A. Lugiato, “Formation and evolution of roll patterns in optical parametric oscillators,” Phys. Rev. A 49, 2028–2032 (1994); S. Longhi, “Hydrodynamic equation model for degenerate optical parametric oscillators,” J. Mod. Opt. JMOPEW 43, 1089–1094 (1996); G. J. de Varcarcel, K. Staliunas, E. Roldan, and V. J. Sanchez-Morcillo, “Transverse patterns in degenerate optical parametric oscillation and degenerate four-wave mixing,” Phys. Rev. A PLRAAN 54, 1609–1624 (1996).
[CrossRef] [PubMed]

H. J. Kimble, “Quantum fluctuations in quantum optics—squeezing and related phenomena,” in Fundamental Systems in Quantum Optics, J. Dalibard, J. M. Raimond, and J. Zinn-Justine, eds. (Elsevier Science, Amsterdam, 1992), pp. 545–673.

S. Longhi, “Localized structures in optical parametric oscillators,” Phys. Scr. 57, 611–635 (1997); K. Staliunas and V. J. Sanchez-Morcillo, “Spatial-localized structures in degenerate optical parametric oscillators,” Phys. Rev. A 57, 1454–1457 (1998); S. Trillo and M. Haelterman, “Excitation and bistability of self-trapped signal beams in optical parametric oscillators,” Opt. Lett. OPLEDP 23, 1514–1516 (1998); G. L. Oppo, A. J. Scroggie, and W. J. Firth, “From domain walls to localised structures in degenerate optical parametric oscillators,” J. Opt. B ZZZZZZ 1, 133–138 (1999); M. Le Berre, D. Leduc, E. Ressayre, and A. Tallet, “Striped and circular domain walls in the DOPO,” J. Opt. B ZZZZZZ 1, 153–160 (1999).
[CrossRef]

R. J. Deissler, “Noise-sustained structure, intermittency, and the Ginzburg–Landau equation,” J. Stat. Phys. 40, 371–395 (1985); “External noise and the origin and dynamics of structure in convectively unstable systems,” J. Stat. Phys. 54, 1459–1488 (1989).
[CrossRef]

L. A. Lugiato, M. Brambilla, and A. Gatti, “Optical pattern formation,” in Vol. 40 of Advances in Atomic Molecular and Optical Physics, B. Bederson and H. Walther, eds. (Academic, New York, 1999), pp. 229–306, and references cited therein.

G-L. Oppo, A. J. Scroggie, and W. J. Firth, “Domain walls in optical parametric oscillators: dynamics and stabilization,” in European Quantum Electronics Conference (EQEC) (Optical Society of America, Washington, D.C., 1998), paper QFA2.

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Figures (3)

Fig. 1
Fig. 1

Absolute threshold |Fa| versus detuning (dashed curves) for several values of ρ. The convective threshold |Fc| is the solid line. The shadowed region is the convective unstable regime for ρ=0.15. In the inset we show the most unstable modes for both polarization components.

Fig. 2
Fig. 2

Transient competition between the most unstable modes: (a) Re(A1), (b) |A1|2. In (c) we show Re(A1) for the final state, where the longest-wavelength structure has been selected. In (d) we show Re(A1) for the final state, where the shortest-wavelength structure has been selected. Parameters are 2a0=a=0.25, γ=1, Δ0=0, Δ=-0.15, K0=1, ρ=0.15, and Emax=1.05.

Fig. 3
Fig. 3

Intensity of polarization component A1 in (a) the near field and in (b) the far field in the convectively unstable regime. Parameters are as in Fig. 2, except for 0=1=2=210-13, Δ=-0.2, and Emax=1.009.

Equations (6)

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tA0=γ0[-(1+iΔ0)A0+E0+ia02A0+2iK0A1A2]+0ξ0(r, t),
tA1=γ1[-(1+iΔ1)A1+ia12A1+iK0A2*A0]+1ξ1(r, t),
tA2=γ2[-(1+iΔ2)A2+ρyA2+ia22A2+iK0A1*A0]+2ξ2(r, t),
|F|21+q2a˜+Δ˜+γ2ρqyγ1+γ22.
kλ(k)|ks=0,Re[k2λ(k)|ks]>0,
λ(k)=2γ-1+ρky2+|F|2-a2R2+kx2+ky+iρ4a221/2.

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