Abstract

The existence of macroscopic noise-sustained structures in nonlinear optics is theoretically predicted and numerically observed, in the regime of convective instability. The advection-like term, necessary to turn the instability to convective for the parameter region where advection overwhelms the growth, can stem from pump beam tilting or birefringence induced walk-off. The growth dynamics of both noise-sustained and deterministic patterns is exemplified by means of movies. This allows to observe the process of formation of these structures and to confirm the analytical predictions. The amplification of quantum noise by several orders of magnitude is predicted. The qualitative analysis of the near- and far-field is given. It suffices to distinguish noise-sustained from deterministic structures; quantitative informations can be obtained in terms of the statistical properties of the spectra.

©1998 Optical Society of America

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References

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  1. L. A. Lugiato, S. Barnett, A. Gatti, I. Marzoli, G. L. Oppo, and H. Wiedemann, “Quantum aspects of nonlinear optical patterns,” Coherence and Quantum Optics VII (Plenum Press, New York, 1996), p 5.
  2. A. Gatti, H. Wiedemann, L.A. Lugiato, I. Marzoli, G- L. Oppo, and S.M. Barnett, “A Langevin approach to quantum fluctuations and optical patterns formation,” Phys. Rev. A 56, 877 (1997).
    [Crossref]
  3. A. Gatti, L. A. Lugiato, G-L. Oppo, R. Martin, P. Di Trapani, and A. Berzanskis, “From quantum to classical images,” opt. Express 1, 21 (1997).
    [Crossref] [PubMed]
  4. A. Gatti and L. A. Lugiato, “Quantum images and critical fluctuations in the optical parametric oscillator below threshold,” Phys. Rev. A 52, 1675 (1995).
    [Crossref] [PubMed]
  5. M. Hoyuelos, P. Colet, and M. San Miguel, “Fluctuations and correlations in polarization patterns of a Kerr medium,” to be published, Phys. Rev. E (1998).
    [Crossref]
  6. R. J. Deissler,“Noise-sustained structure, intermittency, and the Ginzburg-Landau equation,” J. Stat. Phys. 40376 (1985).
    [Crossref]
  7. M. Santagiustina, P. Colet, M. San Miguel, and D. Walgraef, “Convective noise-sustained structures in nonlinear optics,” Phys. Rev. Lett. 79, 3363 (1997).
    [Crossref]
  8. L. A. Lugiato and R. Lefever, “Spatial dissipative structures in passive optical systems,” Phys. Rev. A 58, 2209 (1987).
  9. W. J. Firth, A. G. Scroggie, G. S. McDonald, and L. A. Lugiato, “Hexagonal patterns in optical bistability,” Phys. Rev. A 46, R3609 (1992).
    [Crossref] [PubMed]
  10. L. A. Lugiato and F. Castelli, “Quantum noise reduction in a spatial dissipative structure,” Phys. Rev. Lett. 68, 3284 (1992).
    [Crossref] [PubMed]
  11. K. Staliunas, “Optical vortices during three-wave nonlinear coupling,” Opt. Commun. 91, 82 (1992).
    [Crossref]
  12. G-L. Oppo, M. Brambilla, and L. A. Lugiato, “Formation and evolution of roll patterns in optical parametric oscillators,” Phys. Rev. A 49, 2028 (1994).
    [Crossref] [PubMed]
  13. L. A. Wu, H. J. Kimble, J. Hall, and H. Wu, “Generation of squeezed states by parametric down conversion,” Phys. Rev. Lett. 57, 2520 (1986).
    [Crossref] [PubMed]
  14. M. Santagiustina, P. Colet, M. San Miguel, and D. Walgraef, “Two-dimensional noise-sustained structures in optical parametric oscillators,” to be published Phys. Rev. E (1998).
  15. M. Haelterman and G. Vitrant, “Drift instability and spatiotemporal dissipative structures in a nonlinear Fabry-Perot resonator under oblique incidence,” J. Opt. Soc. Am. B 9, 1563 (1992).
    [Crossref]
  16. G. Grynberg, “Drift instability and light-induced spin waves in an alkali vapor with feedback mirror,” Opt. Commun. 109, 483 (1994).
    [Crossref]
  17. A. Petrossian, L. Dambly, and G. Grynberg, “Drift instability for a laser beam transmitted through a rubidium cell with feedback mirror,” Europhys. Lett. 29, 209 (1995).
    [Crossref]
  18. N. Bloembergen, Nonlinear Optics, (Benjamin Inc. Publ., Reading, 1965), Chapter 4.2. item Y. R. Shen, The principles of nonlinear optics, (Wiley, New York, 1984), Chapter 6.9.
  19. P. D. Drummond, K. J. Mc Neil, and D. F. Walls, “Non-equilibrium transitions in sub/second harmonic generation: semiclassical theory,” Opt. Acta 27, 321 (1980).
    [Crossref]

1998 (1)

M. Hoyuelos, P. Colet, and M. San Miguel, “Fluctuations and correlations in polarization patterns of a Kerr medium,” to be published, Phys. Rev. E (1998).
[Crossref]

1997 (3)

M. Santagiustina, P. Colet, M. San Miguel, and D. Walgraef, “Convective noise-sustained structures in nonlinear optics,” Phys. Rev. Lett. 79, 3363 (1997).
[Crossref]

A. Gatti, H. Wiedemann, L.A. Lugiato, I. Marzoli, G- L. Oppo, and S.M. Barnett, “A Langevin approach to quantum fluctuations and optical patterns formation,” Phys. Rev. A 56, 877 (1997).
[Crossref]

A. Gatti, L. A. Lugiato, G-L. Oppo, R. Martin, P. Di Trapani, and A. Berzanskis, “From quantum to classical images,” opt. Express 1, 21 (1997).
[Crossref] [PubMed]

1995 (2)

A. Gatti and L. A. Lugiato, “Quantum images and critical fluctuations in the optical parametric oscillator below threshold,” Phys. Rev. A 52, 1675 (1995).
[Crossref] [PubMed]

A. Petrossian, L. Dambly, and G. Grynberg, “Drift instability for a laser beam transmitted through a rubidium cell with feedback mirror,” Europhys. Lett. 29, 209 (1995).
[Crossref]

1994 (2)

G. Grynberg, “Drift instability and light-induced spin waves in an alkali vapor with feedback mirror,” Opt. Commun. 109, 483 (1994).
[Crossref]

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

1992 (4)

W. J. Firth, A. G. Scroggie, G. S. McDonald, and L. A. Lugiato, “Hexagonal patterns in optical bistability,” Phys. Rev. A 46, R3609 (1992).
[Crossref] [PubMed]

L. A. Lugiato and F. Castelli, “Quantum noise reduction in a spatial dissipative structure,” Phys. Rev. Lett. 68, 3284 (1992).
[Crossref] [PubMed]

K. Staliunas, “Optical vortices during three-wave nonlinear coupling,” Opt. Commun. 91, 82 (1992).
[Crossref]

M. Haelterman and G. Vitrant, “Drift instability and spatiotemporal dissipative structures in a nonlinear Fabry-Perot resonator under oblique incidence,” J. Opt. Soc. Am. B 9, 1563 (1992).
[Crossref]

1987 (1)

L. A. Lugiato and R. Lefever, “Spatial dissipative structures in passive optical systems,” Phys. Rev. A 58, 2209 (1987).

1986 (1)

L. A. Wu, H. J. Kimble, J. Hall, and H. Wu, “Generation of squeezed states by parametric down conversion,” Phys. Rev. Lett. 57, 2520 (1986).
[Crossref] [PubMed]

1985 (1)

R. J. Deissler,“Noise-sustained structure, intermittency, and the Ginzburg-Landau equation,” J. Stat. Phys. 40376 (1985).
[Crossref]

1980 (1)

P. D. Drummond, K. J. Mc Neil, and D. F. Walls, “Non-equilibrium transitions in sub/second harmonic generation: semiclassical theory,” Opt. Acta 27, 321 (1980).
[Crossref]

Barnett, S.

L. A. Lugiato, S. Barnett, A. Gatti, I. Marzoli, G. L. Oppo, and H. Wiedemann, “Quantum aspects of nonlinear optical patterns,” Coherence and Quantum Optics VII (Plenum Press, New York, 1996), p 5.

Barnett, S.M.

A. Gatti, H. Wiedemann, L.A. Lugiato, I. Marzoli, G- L. Oppo, and S.M. Barnett, “A Langevin approach to quantum fluctuations and optical patterns formation,” Phys. Rev. A 56, 877 (1997).
[Crossref]

Berzanskis, A.

Bloembergen, N.

N. Bloembergen, Nonlinear Optics, (Benjamin Inc. Publ., Reading, 1965), Chapter 4.2. item Y. R. Shen, The principles of nonlinear optics, (Wiley, New York, 1984), Chapter 6.9.

Brambilla, M.

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

Castelli, F.

L. A. Lugiato and F. Castelli, “Quantum noise reduction in a spatial dissipative structure,” Phys. Rev. Lett. 68, 3284 (1992).
[Crossref] [PubMed]

Colet, P.

M. Hoyuelos, P. Colet, and M. San Miguel, “Fluctuations and correlations in polarization patterns of a Kerr medium,” to be published, Phys. Rev. E (1998).
[Crossref]

M. Santagiustina, P. Colet, M. San Miguel, and D. Walgraef, “Convective noise-sustained structures in nonlinear optics,” Phys. Rev. Lett. 79, 3363 (1997).
[Crossref]

M. Santagiustina, P. Colet, M. San Miguel, and D. Walgraef, “Two-dimensional noise-sustained structures in optical parametric oscillators,” to be published Phys. Rev. E (1998).

Dambly, L.

A. Petrossian, L. Dambly, and G. Grynberg, “Drift instability for a laser beam transmitted through a rubidium cell with feedback mirror,” Europhys. Lett. 29, 209 (1995).
[Crossref]

Deissler, R. J.

R. J. Deissler,“Noise-sustained structure, intermittency, and the Ginzburg-Landau equation,” J. Stat. Phys. 40376 (1985).
[Crossref]

Di Trapani, P.

Drummond, P. D.

P. D. Drummond, K. J. Mc Neil, and D. F. Walls, “Non-equilibrium transitions in sub/second harmonic generation: semiclassical theory,” Opt. Acta 27, 321 (1980).
[Crossref]

Firth, W. J.

W. J. Firth, A. G. Scroggie, G. S. McDonald, and L. A. Lugiato, “Hexagonal patterns in optical bistability,” Phys. Rev. A 46, R3609 (1992).
[Crossref] [PubMed]

Gatti, A.

A. Gatti, L. A. Lugiato, G-L. Oppo, R. Martin, P. Di Trapani, and A. Berzanskis, “From quantum to classical images,” opt. Express 1, 21 (1997).
[Crossref] [PubMed]

A. Gatti, H. Wiedemann, L.A. Lugiato, I. Marzoli, G- L. Oppo, and S.M. Barnett, “A Langevin approach to quantum fluctuations and optical patterns formation,” Phys. Rev. A 56, 877 (1997).
[Crossref]

A. Gatti and L. A. Lugiato, “Quantum images and critical fluctuations in the optical parametric oscillator below threshold,” Phys. Rev. A 52, 1675 (1995).
[Crossref] [PubMed]

L. A. Lugiato, S. Barnett, A. Gatti, I. Marzoli, G. L. Oppo, and H. Wiedemann, “Quantum aspects of nonlinear optical patterns,” Coherence and Quantum Optics VII (Plenum Press, New York, 1996), p 5.

Grynberg, G.

A. Petrossian, L. Dambly, and G. Grynberg, “Drift instability for a laser beam transmitted through a rubidium cell with feedback mirror,” Europhys. Lett. 29, 209 (1995).
[Crossref]

G. Grynberg, “Drift instability and light-induced spin waves in an alkali vapor with feedback mirror,” Opt. Commun. 109, 483 (1994).
[Crossref]

Haelterman, M.

Hall, J.

L. A. Wu, H. J. Kimble, J. Hall, and H. Wu, “Generation of squeezed states by parametric down conversion,” Phys. Rev. Lett. 57, 2520 (1986).
[Crossref] [PubMed]

Hoyuelos, M.

M. Hoyuelos, P. Colet, and M. San Miguel, “Fluctuations and correlations in polarization patterns of a Kerr medium,” to be published, Phys. Rev. E (1998).
[Crossref]

Kimble, H. J.

L. A. Wu, H. J. Kimble, J. Hall, and H. Wu, “Generation of squeezed states by parametric down conversion,” Phys. Rev. Lett. 57, 2520 (1986).
[Crossref] [PubMed]

Lefever, R.

L. A. Lugiato and R. Lefever, “Spatial dissipative structures in passive optical systems,” Phys. Rev. A 58, 2209 (1987).

Lugiato, L. A.

A. Gatti, L. A. Lugiato, G-L. Oppo, R. Martin, P. Di Trapani, and A. Berzanskis, “From quantum to classical images,” opt. Express 1, 21 (1997).
[Crossref] [PubMed]

A. Gatti and L. A. Lugiato, “Quantum images and critical fluctuations in the optical parametric oscillator below threshold,” Phys. Rev. A 52, 1675 (1995).
[Crossref] [PubMed]

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

L. A. Lugiato and F. Castelli, “Quantum noise reduction in a spatial dissipative structure,” Phys. Rev. Lett. 68, 3284 (1992).
[Crossref] [PubMed]

W. J. Firth, A. G. Scroggie, G. S. McDonald, and L. A. Lugiato, “Hexagonal patterns in optical bistability,” Phys. Rev. A 46, R3609 (1992).
[Crossref] [PubMed]

L. A. Lugiato and R. Lefever, “Spatial dissipative structures in passive optical systems,” Phys. Rev. A 58, 2209 (1987).

L. A. Lugiato, S. Barnett, A. Gatti, I. Marzoli, G. L. Oppo, and H. Wiedemann, “Quantum aspects of nonlinear optical patterns,” Coherence and Quantum Optics VII (Plenum Press, New York, 1996), p 5.

Lugiato, L.A.

A. Gatti, H. Wiedemann, L.A. Lugiato, I. Marzoli, G- L. Oppo, and S.M. Barnett, “A Langevin approach to quantum fluctuations and optical patterns formation,” Phys. Rev. A 56, 877 (1997).
[Crossref]

Martin, R.

Marzoli, I.

A. Gatti, H. Wiedemann, L.A. Lugiato, I. Marzoli, G- L. Oppo, and S.M. Barnett, “A Langevin approach to quantum fluctuations and optical patterns formation,” Phys. Rev. A 56, 877 (1997).
[Crossref]

L. A. Lugiato, S. Barnett, A. Gatti, I. Marzoli, G. L. Oppo, and H. Wiedemann, “Quantum aspects of nonlinear optical patterns,” Coherence and Quantum Optics VII (Plenum Press, New York, 1996), p 5.

Mc Neil, K. J.

P. D. Drummond, K. J. Mc Neil, and D. F. Walls, “Non-equilibrium transitions in sub/second harmonic generation: semiclassical theory,” Opt. Acta 27, 321 (1980).
[Crossref]

McDonald, G. S.

W. J. Firth, A. G. Scroggie, G. S. McDonald, and L. A. Lugiato, “Hexagonal patterns in optical bistability,” Phys. Rev. A 46, R3609 (1992).
[Crossref] [PubMed]

Oppo, G- L.

A. Gatti, H. Wiedemann, L.A. Lugiato, I. Marzoli, G- L. Oppo, and S.M. Barnett, “A Langevin approach to quantum fluctuations and optical patterns formation,” Phys. Rev. A 56, 877 (1997).
[Crossref]

Oppo, G. L.

L. A. Lugiato, S. Barnett, A. Gatti, I. Marzoli, G. L. Oppo, and H. Wiedemann, “Quantum aspects of nonlinear optical patterns,” Coherence and Quantum Optics VII (Plenum Press, New York, 1996), p 5.

Oppo, G-L.

A. Gatti, L. A. Lugiato, G-L. Oppo, R. Martin, P. Di Trapani, and A. Berzanskis, “From quantum to classical images,” opt. Express 1, 21 (1997).
[Crossref] [PubMed]

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

Petrossian, A.

A. Petrossian, L. Dambly, and G. Grynberg, “Drift instability for a laser beam transmitted through a rubidium cell with feedback mirror,” Europhys. Lett. 29, 209 (1995).
[Crossref]

San Miguel, M.

M. Hoyuelos, P. Colet, and M. San Miguel, “Fluctuations and correlations in polarization patterns of a Kerr medium,” to be published, Phys. Rev. E (1998).
[Crossref]

M. Santagiustina, P. Colet, M. San Miguel, and D. Walgraef, “Convective noise-sustained structures in nonlinear optics,” Phys. Rev. Lett. 79, 3363 (1997).
[Crossref]

M. Santagiustina, P. Colet, M. San Miguel, and D. Walgraef, “Two-dimensional noise-sustained structures in optical parametric oscillators,” to be published Phys. Rev. E (1998).

Santagiustina, M.

M. Santagiustina, P. Colet, M. San Miguel, and D. Walgraef, “Convective noise-sustained structures in nonlinear optics,” Phys. Rev. Lett. 79, 3363 (1997).
[Crossref]

M. Santagiustina, P. Colet, M. San Miguel, and D. Walgraef, “Two-dimensional noise-sustained structures in optical parametric oscillators,” to be published Phys. Rev. E (1998).

Scroggie, A. G.

W. J. Firth, A. G. Scroggie, G. S. McDonald, and L. A. Lugiato, “Hexagonal patterns in optical bistability,” Phys. Rev. A 46, R3609 (1992).
[Crossref] [PubMed]

Shen, Y. R.

N. Bloembergen, Nonlinear Optics, (Benjamin Inc. Publ., Reading, 1965), Chapter 4.2. item Y. R. Shen, The principles of nonlinear optics, (Wiley, New York, 1984), Chapter 6.9.

Staliunas, K.

K. Staliunas, “Optical vortices during three-wave nonlinear coupling,” Opt. Commun. 91, 82 (1992).
[Crossref]

Vitrant, G.

Walgraef, D.

M. Santagiustina, P. Colet, M. San Miguel, and D. Walgraef, “Convective noise-sustained structures in nonlinear optics,” Phys. Rev. Lett. 79, 3363 (1997).
[Crossref]

M. Santagiustina, P. Colet, M. San Miguel, and D. Walgraef, “Two-dimensional noise-sustained structures in optical parametric oscillators,” to be published Phys. Rev. E (1998).

Walls, D. F.

P. D. Drummond, K. J. Mc Neil, and D. F. Walls, “Non-equilibrium transitions in sub/second harmonic generation: semiclassical theory,” Opt. Acta 27, 321 (1980).
[Crossref]

Wiedemann, H.

A. Gatti, H. Wiedemann, L.A. Lugiato, I. Marzoli, G- L. Oppo, and S.M. Barnett, “A Langevin approach to quantum fluctuations and optical patterns formation,” Phys. Rev. A 56, 877 (1997).
[Crossref]

L. A. Lugiato, S. Barnett, A. Gatti, I. Marzoli, G. L. Oppo, and H. Wiedemann, “Quantum aspects of nonlinear optical patterns,” Coherence and Quantum Optics VII (Plenum Press, New York, 1996), p 5.

Wu, H.

L. A. Wu, H. J. Kimble, J. Hall, and H. Wu, “Generation of squeezed states by parametric down conversion,” Phys. Rev. Lett. 57, 2520 (1986).
[Crossref] [PubMed]

Wu, L. A.

L. A. Wu, H. J. Kimble, J. Hall, and H. Wu, “Generation of squeezed states by parametric down conversion,” Phys. Rev. Lett. 57, 2520 (1986).
[Crossref] [PubMed]

Europhys. Lett. (1)

A. Petrossian, L. Dambly, and G. Grynberg, “Drift instability for a laser beam transmitted through a rubidium cell with feedback mirror,” Europhys. Lett. 29, 209 (1995).
[Crossref]

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

J. Stat. Phys. (1)

R. J. Deissler,“Noise-sustained structure, intermittency, and the Ginzburg-Landau equation,” J. Stat. Phys. 40376 (1985).
[Crossref]

Opt. Acta (1)

P. D. Drummond, K. J. Mc Neil, and D. F. Walls, “Non-equilibrium transitions in sub/second harmonic generation: semiclassical theory,” Opt. Acta 27, 321 (1980).
[Crossref]

Opt. Commun. (2)

G. Grynberg, “Drift instability and light-induced spin waves in an alkali vapor with feedback mirror,” Opt. Commun. 109, 483 (1994).
[Crossref]

K. Staliunas, “Optical vortices during three-wave nonlinear coupling,” Opt. Commun. 91, 82 (1992).
[Crossref]

opt. Express (1)

Phys. Rev. A (5)

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

L. A. Lugiato and R. Lefever, “Spatial dissipative structures in passive optical systems,” Phys. Rev. A 58, 2209 (1987).

W. J. Firth, A. G. Scroggie, G. S. McDonald, and L. A. Lugiato, “Hexagonal patterns in optical bistability,” Phys. Rev. A 46, R3609 (1992).
[Crossref] [PubMed]

A. Gatti and L. A. Lugiato, “Quantum images and critical fluctuations in the optical parametric oscillator below threshold,” Phys. Rev. A 52, 1675 (1995).
[Crossref] [PubMed]

A. Gatti, H. Wiedemann, L.A. Lugiato, I. Marzoli, G- L. Oppo, and S.M. Barnett, “A Langevin approach to quantum fluctuations and optical patterns formation,” Phys. Rev. A 56, 877 (1997).
[Crossref]

Phys. Rev. E (2)

M. Hoyuelos, P. Colet, and M. San Miguel, “Fluctuations and correlations in polarization patterns of a Kerr medium,” to be published, Phys. Rev. E (1998).
[Crossref]

M. Santagiustina, P. Colet, M. San Miguel, and D. Walgraef, “Two-dimensional noise-sustained structures in optical parametric oscillators,” to be published Phys. Rev. E (1998).

Phys. Rev. Lett. (3)

M. Santagiustina, P. Colet, M. San Miguel, and D. Walgraef, “Convective noise-sustained structures in nonlinear optics,” Phys. Rev. Lett. 79, 3363 (1997).
[Crossref]

L. A. Lugiato and F. Castelli, “Quantum noise reduction in a spatial dissipative structure,” Phys. Rev. Lett. 68, 3284 (1992).
[Crossref] [PubMed]

L. A. Wu, H. J. Kimble, J. Hall, and H. Wu, “Generation of squeezed states by parametric down conversion,” Phys. Rev. Lett. 57, 2520 (1986).
[Crossref] [PubMed]

Other (2)

N. Bloembergen, Nonlinear Optics, (Benjamin Inc. Publ., Reading, 1965), Chapter 4.2. item Y. R. Shen, The principles of nonlinear optics, (Wiley, New York, 1984), Chapter 6.9.

L. A. Lugiato, S. Barnett, A. Gatti, I. Marzoli, G. L. Oppo, and H. Wiedemann, “Quantum aspects of nonlinear optical patterns,” Coherence and Quantum Optics VII (Plenum Press, New York, 1996), p 5.

Supplementary Material (4)

» Media 1: MOV (1814 KB)     
» Media 2: MOV (690 KB)     
» Media 3: MOV (2258 KB)     
» Media 4: MOV (2041 KB)     

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Figures (6)

Movie 1.
Movie 1. Near field (left) and far field (right) growth dynamics in the absolutely unstable regime. [Media 1]

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Movie 2.
Movie 2. Near field (left) and far field (right) growth dynamics in the convectively unstable regime. [Media 2]

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Fig. 1.
Fig. 1. Stability diagram for the OPO. Shadowed regions are: stable (green), convectively unstable (red). The white region indicates the absolute instability. Absolute threshold shifts upwards by increasing α 0 (red dashed curves).

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Movie 3.
Movie 3. Near field (left) and far field (right) growth dynamics in the absolutely unstable regime. [Media 3]

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Fig. 2.
Fig. 2. Snapshots of the near(far)-field at time t = 2000 on the left (right) hand side. Parameters of the top, middle and bottom images correspond respectively to (*, +, X) of Fig. 1.

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Movie 4.
Movie 4. Near field (left) and far field (right) growth dynamics in the convectively unstable regime. [Media 4]

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Equations (6)

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

ℜ [ λ ( k → s , F a ) ] = 0
ℜ [ ∇ k → 2 λ ( k → = k → s ) , F a ] ≥ 0
∇ k → λ ( k → = k → s , F a ) = 0
∂ t A − 2 α 0 ∂ x A = i ∂ x 2 A − [ 1 + iη ( Δ − ∣ A ∣ 2 ) ] A + E 0 + ∈ ξ x t ,
∂ t A 0 = γ 0 [ − ( 1 + i Δ 0 ) A 0 + E 0 + i a 0 ∇ 2 A 0 + 2 i K 0 A 1 2 ] + ∈ 0 ξ 0 x y t
∂ t A 1 = γ 1 [ − ( 1 + i Δ 1 ) A 1 + ρ 1 ∂ y A 1 + i a 1 ∇ 2 A 1 + i K 0 A 1 * A 0 ] + ∈ 1 ξ 1 x y t

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