Abstract

The existence and competition of a novel class of hexagonal patterns in a nonlinear optical system are reported. These states are found in a mean-field model of a doubly resonant frequency divide-by-3 optical parametric oscillator 3ω2ω+ω in which the multistep parametric process, 2ω=ω+ω, is weakly phase matched. A generalized Swift–Hohenberg equation and a set of amplitude equations are derived to describe the coexistence of hexagonal patterns formed by the superposition of either three or six phase-locked tilted waves.

© 2001 Optical Society of America

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References

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  1. G. J. de Valcárel, E. Roldan, and R. Vilaseca, eds., Patterns in Nonlinear Optical Systems, J. Opt. B1, 1–197 (1999).
  2. M. C. Cross and P. C. Hohenberg, Rev. Mod. Phys. 65, 851 (1993).
    [CrossRef]
  3. G.-L. Oppo, M. Brambilla, and L. A. Lugiato, Phys. Rev. A 49, 2028 (1994).
    [CrossRef] [PubMed]
  4. G. J. De Valcarcel, K. Staliunas, E. Roldan, and V. J. Sanchez-Morcillo, Phys. Rev. A 54, 1609 (1996).
    [CrossRef]
  5. V. J. Sanchez-Morcillo, E. Roldan, G. J. De Valcarcel, and K. Staliunas, Phys. Rev. A 56, 3237 (1997).
    [CrossRef]
  6. S. Longhi, Phys. Rev. A 53, 4488 (1996).
    [CrossRef] [PubMed]
  7. S. Longhi, J. Mod. Opt. 43, 1569 (1996).
  8. S. Longhi and A. Geraci, Phys. Rev. A 54, 4581 (1996).
    [CrossRef] [PubMed]
  9. P. Lodahl and M. Saffman, Phys. Rev. A 60, 3251 (1999).
    [CrossRef]
  10. P. Lodahl, M. Bache, and M. Saffman, Opt. Lett. 25, 654 (2000).
    [CrossRef]
  11. K. Koynov and S. Saltiel, Opt. Commun. 152, 96 (1998).
    [CrossRef]
  12. P. T. Nee and N. C. Wong, Opt. Lett. 23, 46 (1998).
    [CrossRef]
  13. Y. S. Kivshar, T. J. Alexander, and S. Saltiel, Opt. Lett. 24, 759 (1999).
    [CrossRef]
  14. See, e.g., A. Douillet and J.-J. Zondy, Opt. Lett. 23, 1259 (1998).
    [CrossRef]
  15. S. Longhi, J. Mod. Opt. 43, 1089 (1996).
  16. S. Ciliberto, P. Coullet, J. Lega, E. Pampaloni, and C. Perez-Garcia, Phys. Rev. Lett. 65, 2370 (1990).
    [CrossRef] [PubMed]
  17. Hexagonal patterns corresponding to mixed states are generally unstable in the framework of the usual amplitude equations that describe subcritical hexagon formation.2,16 It is rather remarkable that the amplitude equations reported here can stabilize these unusual hexagonal states.

2000 (1)

1999 (2)

1998 (3)

1997 (1)

V. J. Sanchez-Morcillo, E. Roldan, G. J. De Valcarcel, and K. Staliunas, Phys. Rev. A 56, 3237 (1997).
[CrossRef]

1996 (5)

S. Longhi, Phys. Rev. A 53, 4488 (1996).
[CrossRef] [PubMed]

S. Longhi, J. Mod. Opt. 43, 1569 (1996).

S. Longhi and A. Geraci, Phys. Rev. A 54, 4581 (1996).
[CrossRef] [PubMed]

G. J. De Valcarcel, K. Staliunas, E. Roldan, and V. J. Sanchez-Morcillo, Phys. Rev. A 54, 1609 (1996).
[CrossRef]

S. Longhi, J. Mod. Opt. 43, 1089 (1996).

1994 (1)

G.-L. Oppo, M. Brambilla, and L. A. Lugiato, Phys. Rev. A 49, 2028 (1994).
[CrossRef] [PubMed]

1993 (1)

M. C. Cross and P. C. Hohenberg, Rev. Mod. Phys. 65, 851 (1993).
[CrossRef]

1990 (1)

S. Ciliberto, P. Coullet, J. Lega, E. Pampaloni, and C. Perez-Garcia, Phys. Rev. Lett. 65, 2370 (1990).
[CrossRef] [PubMed]

Alexander, T. J.

Bache, M.

Brambilla, M.

G.-L. Oppo, M. Brambilla, and L. A. Lugiato, Phys. Rev. A 49, 2028 (1994).
[CrossRef] [PubMed]

Ciliberto, S.

S. Ciliberto, P. Coullet, J. Lega, E. Pampaloni, and C. Perez-Garcia, Phys. Rev. Lett. 65, 2370 (1990).
[CrossRef] [PubMed]

Coullet, P.

S. Ciliberto, P. Coullet, J. Lega, E. Pampaloni, and C. Perez-Garcia, Phys. Rev. Lett. 65, 2370 (1990).
[CrossRef] [PubMed]

Cross, M. C.

M. C. Cross and P. C. Hohenberg, Rev. Mod. Phys. 65, 851 (1993).
[CrossRef]

De Valcarcel, G. J.

V. J. Sanchez-Morcillo, E. Roldan, G. J. De Valcarcel, and K. Staliunas, Phys. Rev. A 56, 3237 (1997).
[CrossRef]

G. J. De Valcarcel, K. Staliunas, E. Roldan, and V. J. Sanchez-Morcillo, Phys. Rev. A 54, 1609 (1996).
[CrossRef]

Douillet, A.

Geraci, A.

S. Longhi and A. Geraci, Phys. Rev. A 54, 4581 (1996).
[CrossRef] [PubMed]

Hohenberg, P. C.

M. C. Cross and P. C. Hohenberg, Rev. Mod. Phys. 65, 851 (1993).
[CrossRef]

Kivshar, Y. S.

Koynov, K.

K. Koynov and S. Saltiel, Opt. Commun. 152, 96 (1998).
[CrossRef]

Lega, J.

S. Ciliberto, P. Coullet, J. Lega, E. Pampaloni, and C. Perez-Garcia, Phys. Rev. Lett. 65, 2370 (1990).
[CrossRef] [PubMed]

Lodahl, P.

P. Lodahl, M. Bache, and M. Saffman, Opt. Lett. 25, 654 (2000).
[CrossRef]

P. Lodahl and M. Saffman, Phys. Rev. A 60, 3251 (1999).
[CrossRef]

Longhi, S.

S. Longhi, Phys. Rev. A 53, 4488 (1996).
[CrossRef] [PubMed]

S. Longhi, J. Mod. Opt. 43, 1569 (1996).

S. Longhi and A. Geraci, Phys. Rev. A 54, 4581 (1996).
[CrossRef] [PubMed]

S. Longhi, J. Mod. Opt. 43, 1089 (1996).

Lugiato, L. A.

G.-L. Oppo, M. Brambilla, and L. A. Lugiato, Phys. Rev. A 49, 2028 (1994).
[CrossRef] [PubMed]

Nee, P. T.

Oppo, G.-L.

G.-L. Oppo, M. Brambilla, and L. A. Lugiato, Phys. Rev. A 49, 2028 (1994).
[CrossRef] [PubMed]

Pampaloni, E.

S. Ciliberto, P. Coullet, J. Lega, E. Pampaloni, and C. Perez-Garcia, Phys. Rev. Lett. 65, 2370 (1990).
[CrossRef] [PubMed]

Perez-Garcia, C.

S. Ciliberto, P. Coullet, J. Lega, E. Pampaloni, and C. Perez-Garcia, Phys. Rev. Lett. 65, 2370 (1990).
[CrossRef] [PubMed]

Roldan, E.

V. J. Sanchez-Morcillo, E. Roldan, G. J. De Valcarcel, and K. Staliunas, Phys. Rev. A 56, 3237 (1997).
[CrossRef]

G. J. De Valcarcel, K. Staliunas, E. Roldan, and V. J. Sanchez-Morcillo, Phys. Rev. A 54, 1609 (1996).
[CrossRef]

Saffman, M.

P. Lodahl, M. Bache, and M. Saffman, Opt. Lett. 25, 654 (2000).
[CrossRef]

P. Lodahl and M. Saffman, Phys. Rev. A 60, 3251 (1999).
[CrossRef]

Saltiel, S.

Sanchez-Morcillo, V. J.

V. J. Sanchez-Morcillo, E. Roldan, G. J. De Valcarcel, and K. Staliunas, Phys. Rev. A 56, 3237 (1997).
[CrossRef]

G. J. De Valcarcel, K. Staliunas, E. Roldan, and V. J. Sanchez-Morcillo, Phys. Rev. A 54, 1609 (1996).
[CrossRef]

Staliunas, K.

V. J. Sanchez-Morcillo, E. Roldan, G. J. De Valcarcel, and K. Staliunas, Phys. Rev. A 56, 3237 (1997).
[CrossRef]

G. J. De Valcarcel, K. Staliunas, E. Roldan, and V. J. Sanchez-Morcillo, Phys. Rev. A 54, 1609 (1996).
[CrossRef]

Wong, N. C.

Zondy, J.-J.

J. Mod. Opt. (2)

S. Longhi, J. Mod. Opt. 43, 1569 (1996).

S. Longhi, J. Mod. Opt. 43, 1089 (1996).

Opt. Commun. (1)

K. Koynov and S. Saltiel, Opt. Commun. 152, 96 (1998).
[CrossRef]

Opt. Lett. (4)

Phys. Rev. A (6)

S. Longhi and A. Geraci, Phys. Rev. A 54, 4581 (1996).
[CrossRef] [PubMed]

P. Lodahl and M. Saffman, Phys. Rev. A 60, 3251 (1999).
[CrossRef]

G.-L. Oppo, M. Brambilla, and L. A. Lugiato, Phys. Rev. A 49, 2028 (1994).
[CrossRef] [PubMed]

G. J. De Valcarcel, K. Staliunas, E. Roldan, and V. J. Sanchez-Morcillo, Phys. Rev. A 54, 1609 (1996).
[CrossRef]

V. J. Sanchez-Morcillo, E. Roldan, G. J. De Valcarcel, and K. Staliunas, Phys. Rev. A 56, 3237 (1997).
[CrossRef]

S. Longhi, Phys. Rev. A 53, 4488 (1996).
[CrossRef] [PubMed]

Phys. Rev. Lett. (1)

S. Ciliberto, P. Coullet, J. Lega, E. Pampaloni, and C. Perez-Garcia, Phys. Rev. Lett. 65, 2370 (1990).
[CrossRef] [PubMed]

Rev. Mod. Phys. (1)

M. C. Cross and P. C. Hohenberg, Rev. Mod. Phys. 65, 851 (1993).
[CrossRef]

Other (2)

G. J. de Valcárel, E. Roldan, and R. Vilaseca, eds., Patterns in Nonlinear Optical Systems, J. Opt. B1, 1–197 (1999).

Hexagonal patterns corresponding to mixed states are generally unstable in the framework of the usual amplitude equations that describe subcritical hexagon formation.2,16 It is rather remarkable that the amplitude equations reported here can stabilize these unusual hexagonal states.

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Figures (4)

Fig. 1
Fig. 1

Fixed points of amplitude equations  (3) with relative domains of existence and stability in the η space.

Fig. 2
Fig. 2

Bifurcation diagrams for (a) TWs and type  I hexagons and (b) type  II and type  III hexagons. Solid curves, stable solutions; dashed curves, unstable solutions.

Fig. 3
Fig. 3

Successive frames of (a) far-field and (b) near-field intensity patterns of the signal field, showing the formation of type  I hexagons from noise. Parameter values are γ1=1, γ2=0.5, a1=a2=1, Δ1=Δ2=-1, σ=0.8, and μ=1.08η2.08.

Fig. 4
Fig. 4

Same as Fig.  3 but for a different realization. Here type  II hexagons are the final patterns after transient.

Equations (4)

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

tA1=γ1-1+iΔ1A1+ia12A1+μA2*+σA1*A2-A22A1,
tA2=γ2-1+iΔ2A2+ia22A2+μA1*-σ2A12-A12A2.
τtψ=μ2-1ψ+iΔ2-Δ1-a2-a12ψ-Δ-a22ψ+σ2ψ*2-2ψ2ψ,
τtFl=μ2-1Fl+σ2Fl+2*Fl+4*-22j=16Fj2-Fl2Fl-4Fl-1Fl+2+Fl+1Fl+4Fl+3*,

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