Abstract

Transverse nonlinear front (or domain wall) propagation in degenerate optical parametric oscillators, for positive detunings and in the presence of walk-off, is investigated. A quintic Ginzburg–Landau equation including diffraction and walk-off is derived close to subcritical bifurcation. A new threshold is found below the linear one, where nonlinear front propagation dominates the dynamics. The velocity and the wave number of these fronts are determined. Nonlinear absolute and convective instabilities are shown to strongly alter the hysteresis cycle, which completely vanishes when the walk-off exceeds some critical value.

© 2000 Optical Society of America

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  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]
  2. 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]
  3. 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); G. Izús, M. Santagiustina, M. San Miguel, and P. Colet, “Pattern formation in presence of walk-off for a type II optical parametric oscillator,” J. Opt. Soc. Am. B 16, 1592–1596 (1999).
    [CrossRef]
  4. J. N. Kutz, T. Erneux, S. Trillo, and M. Haelterman, “Curvature dynamics and stability of topological solitons in the optical parametric oscillator,” J. Opt. Soc. Am. B 16, 1936–1941 (1999); S. Trillo, M. Haelterman, and A. Sheppard, “Stable topological spatial solitons in optical parametric oscillators,” Opt. Lett. 22, 970–972 (1997).
    [CrossRef] [PubMed]
  5. T. Nishikawa and N. Uesugi, “Walk-off and pump energy dependence of transverse beam profiles on traveling wave parametric generation,” Opt. Commun. 140, 277–280 (1997); “Effects of walk-off and group velocity difference on the optical parametric generation in KTiOPO4 crystals,” J. Appl. Phys. 77, 4941–4947 (1995).
    [CrossRef]
  6. 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); K. Staliunas, “Optical vortices during three-wave nonlinear coupling,” Opt. Commun. 58, 82–86 (1992); G. J. de Varcacel, K. Staliunas, E. Roldan, and V. J. Sanchez-Morcillo, “Transverse patterns in degenerate optical parametric oscillators and degenerate four-wave mixing,” Phys. Rev. A PLRAAN 54, 1609–1624 (1996).
    [CrossRef] [PubMed]
  7. S. Longhi, “Spatial-temporal instabilities and threshold conditions in broad-area optical parametric oscillators,” Opt. Commun. 153, 90–94 (1998); “Traveling waves states and secondary instabilities in optical parametric oscillators,” Phys. Scr. 56, 611–618 (1997).
    [CrossRef]
  8. A. Newell and J. Moloney, Nonlinear Optics (Addison-Wesley, Redwood City, Calif., 1992), Chaps. 5 and 6.
  9. P. Coullet, L. Gil, and F. Rocca, “Optical vortices,” Opt. Commun. 73, 403–408 (1989).
    [CrossRef]
  10. A. Bers, Basic Plasma Physics I, A. A. Galeev and R. N. Sudan, eds. (North-Holland, Amsterdam, 1983).
  11. P. A. Monkewitz, P. Huerre, and J. M. Chomaz, “Global linear stability analysis of weakly nonparallel shear flows,” J. Fluid Mech. 251, 1–20 (1993).
    [CrossRef]
  12. H. W. Müller and M. Tveitereid, “Absolute and convective nature of Eckhaus and zigzag instability,” Phys. Rev. Lett. 74, 1582–1585 (1995); K. L. Babcock, G. Ahlers, and D. S. Cannell, “Noise amplification in open Taylor–Couette flow,” Phys. Rev. E 50, 3670–3692 (1994).
    [CrossRef]
  13. W. van Saarloos and P. C. Hohenberg, “Fronts, pulses, sources and sinks in generalized complex Ginzburg–Landau equations,” Physica D 14, 303–367 (1992).
    [CrossRef]
  14. A. Couairon and J. M. Chomaz, “Global instability in nonlinear systems,” Phys. Rev. Lett. 77, 4015–4018 (1996); “Absolute and convective instabilities, front velocities and global modes in nonlinear systems,” Physica D 108, 236–276 (1997).
    [CrossRef] [PubMed]
  15. W. van Saarloos, “Front propagation into unstable states: marginal stability as a dynamical mechanism for velocity selection,” Phys. Rev. A 37, 211–229 (1988); W. van Saarloos, “Front propagation into unstable states: II. Linear versus nonlinear marginal stability and rate of convergence,” Phys. Rev. A 39, 6367–6390 (1989).
    [CrossRef] [PubMed]
  16. M. Taki, M. San Miguel, and M. Santagiustina, “Order parameter description of walk-off effect on pattern selection in degenerate optical parametric oscillators,” Phys. Rev. E 61, 2133–2136 (2000).
    [CrossRef]
  17. S. Longhi, “Spatial solitary waves in nondegenerate optical parametric oscillators near an inverted bifurcation,” Opt. Commun. 149, 335–340 (1998).
    [CrossRef]
  18. J. A. Powell and M. Tabor, “Non-generic connections corresponding to front solutions,” J. Phys. A 25, 3773–3796 (1992); J. A. Powell, A. C. Newell, and C. K. R. T. Jones, “Competition between generic and nongeneric fronts in envelope equations,” Phys. Rev. A 44, 3636–3652 (1991); C. K. R. T. Jones, T. M. Kapitula, and J. A. Powell, “Nearly real fronts in a quintic amplitude equation,” Proc. R. Soc. Edinburgh Sect. A PEAMDU 116, 193–206 (1990).
    [CrossRef] [PubMed]

2000

M. Taki, M. San Miguel, and M. Santagiustina, “Order parameter description of walk-off effect on pattern selection in degenerate optical parametric oscillators,” Phys. Rev. E 61, 2133–2136 (2000).
[CrossRef]

1998

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, “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]

1993

P. A. Monkewitz, P. Huerre, and J. M. Chomaz, “Global linear stability analysis of weakly nonparallel shear flows,” J. Fluid Mech. 251, 1–20 (1993).
[CrossRef]

1992

W. van Saarloos and P. C. Hohenberg, “Fronts, pulses, sources and sinks in generalized complex Ginzburg–Landau equations,” Physica D 14, 303–367 (1992).
[CrossRef]

1989

P. Coullet, L. Gil, and F. Rocca, “Optical vortices,” Opt. Commun. 73, 403–408 (1989).
[CrossRef]

Chomaz, J. M.

P. A. Monkewitz, P. Huerre, and J. M. Chomaz, “Global linear stability analysis of weakly nonparallel shear flows,” J. Fluid Mech. 251, 1–20 (1993).
[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]

Coullet, P.

P. Coullet, L. Gil, and F. Rocca, “Optical vortices,” Opt. Commun. 73, 403–408 (1989).
[CrossRef]

Gil, L.

P. Coullet, L. Gil, and F. Rocca, “Optical vortices,” Opt. Commun. 73, 403–408 (1989).
[CrossRef]

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]

Hohenberg, P. C.

W. van Saarloos and P. C. Hohenberg, “Fronts, pulses, sources and sinks in generalized complex Ginzburg–Landau equations,” Physica D 14, 303–367 (1992).
[CrossRef]

Huerre, P.

P. A. Monkewitz, P. Huerre, and J. M. Chomaz, “Global linear stability analysis of weakly nonparallel shear flows,” J. Fluid Mech. 251, 1–20 (1993).
[CrossRef]

Longhi, S.

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

Monkewitz, P. A.

P. A. Monkewitz, P. Huerre, and J. M. Chomaz, “Global linear stability analysis of weakly nonparallel shear flows,” J. Fluid Mech. 251, 1–20 (1993).
[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]

Rocca, F.

P. Coullet, L. Gil, and F. Rocca, “Optical vortices,” Opt. Commun. 73, 403–408 (1989).
[CrossRef]

San Miguel, M.

M. Taki, M. San Miguel, and M. Santagiustina, “Order parameter description of walk-off effect on pattern selection in degenerate optical parametric oscillators,” Phys. Rev. E 61, 2133–2136 (2000).
[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]

Santagiustina, M.

M. Taki, M. San Miguel, and M. Santagiustina, “Order parameter description of walk-off effect on pattern selection in degenerate optical parametric oscillators,” Phys. Rev. E 61, 2133–2136 (2000).
[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]

Taki, M.

M. Taki, M. San Miguel, and M. Santagiustina, “Order parameter description of walk-off effect on pattern selection in degenerate optical parametric oscillators,” Phys. Rev. E 61, 2133–2136 (2000).
[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]

van Saarloos, W.

W. van Saarloos and P. C. Hohenberg, “Fronts, pulses, sources and sinks in generalized complex Ginzburg–Landau equations,” Physica D 14, 303–367 (1992).
[CrossRef]

Walgraef, D.

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]

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]

Eur. Phys. J. D

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. Fluid Mech.

P. A. Monkewitz, P. Huerre, and J. M. Chomaz, “Global linear stability analysis of weakly nonparallel shear flows,” J. Fluid Mech. 251, 1–20 (1993).
[CrossRef]

Opt. Commun.

P. Coullet, L. Gil, and F. Rocca, “Optical vortices,” Opt. Commun. 73, 403–408 (1989).
[CrossRef]

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

Phys. Rev. E

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. Taki, M. San Miguel, and M. Santagiustina, “Order parameter description of walk-off effect on pattern selection in degenerate optical parametric oscillators,” Phys. Rev. E 61, 2133–2136 (2000).
[CrossRef]

Physica D

W. van Saarloos and P. C. Hohenberg, “Fronts, pulses, sources and sinks in generalized complex Ginzburg–Landau equations,” Physica D 14, 303–367 (1992).
[CrossRef]

Other

A. Couairon and J. M. Chomaz, “Global instability in nonlinear systems,” Phys. Rev. Lett. 77, 4015–4018 (1996); “Absolute and convective instabilities, front velocities and global modes in nonlinear systems,” Physica D 108, 236–276 (1997).
[CrossRef] [PubMed]

W. van Saarloos, “Front propagation into unstable states: marginal stability as a dynamical mechanism for velocity selection,” Phys. Rev. A 37, 211–229 (1988); W. van Saarloos, “Front propagation into unstable states: II. Linear versus nonlinear marginal stability and rate of convergence,” Phys. Rev. A 39, 6367–6390 (1989).
[CrossRef] [PubMed]

J. A. Powell and M. Tabor, “Non-generic connections corresponding to front solutions,” J. Phys. A 25, 3773–3796 (1992); J. A. Powell, A. C. Newell, and C. K. R. T. Jones, “Competition between generic and nongeneric fronts in envelope equations,” Phys. Rev. A 44, 3636–3652 (1991); C. K. R. T. Jones, T. M. Kapitula, and J. A. Powell, “Nearly real fronts in a quintic amplitude equation,” Proc. R. Soc. Edinburgh Sect. A PEAMDU 116, 193–206 (1990).
[CrossRef] [PubMed]

A. Bers, Basic Plasma Physics I, A. A. Galeev and R. N. Sudan, eds. (North-Holland, Amsterdam, 1983).

H. W. Müller and M. Tveitereid, “Absolute and convective nature of Eckhaus and zigzag instability,” Phys. Rev. Lett. 74, 1582–1585 (1995); K. L. Babcock, G. Ahlers, and D. S. Cannell, “Noise amplification in open Taylor–Couette flow,” Phys. Rev. E 50, 3670–3692 (1994).
[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); G. Izús, M. Santagiustina, M. San Miguel, and P. Colet, “Pattern formation in presence of walk-off for a type II optical parametric oscillator,” J. Opt. Soc. Am. B 16, 1592–1596 (1999).
[CrossRef]

J. N. Kutz, T. Erneux, S. Trillo, and M. Haelterman, “Curvature dynamics and stability of topological solitons in the optical parametric oscillator,” J. Opt. Soc. Am. B 16, 1936–1941 (1999); S. Trillo, M. Haelterman, and A. Sheppard, “Stable topological spatial solitons in optical parametric oscillators,” Opt. Lett. 22, 970–972 (1997).
[CrossRef] [PubMed]

T. Nishikawa and N. Uesugi, “Walk-off and pump energy dependence of transverse beam profiles on traveling wave parametric generation,” Opt. Commun. 140, 277–280 (1997); “Effects of walk-off and group velocity difference on the optical parametric generation in KTiOPO4 crystals,” J. Appl. Phys. 77, 4941–4947 (1995).
[CrossRef]

G. L. Oppo, M. Brambilla, and L. A. Lugiato, “Formation and evolution of roll patterns in optical parametric oscillators,” Phys. Rev. A 49, 2028–2032 (1994); K. Staliunas, “Optical vortices during three-wave nonlinear coupling,” Opt. Commun. 58, 82–86 (1992); G. J. de Varcacel, K. Staliunas, E. Roldan, and V. J. Sanchez-Morcillo, “Transverse patterns in degenerate optical parametric oscillators and degenerate four-wave mixing,” Phys. Rev. A PLRAAN 54, 1609–1624 (1996).
[CrossRef] [PubMed]

S. Longhi, “Spatial-temporal instabilities and threshold conditions in broad-area optical parametric oscillators,” Opt. Commun. 153, 90–94 (1998); “Traveling waves states and secondary instabilities in optical parametric oscillators,” Phys. Scr. 56, 611–618 (1997).
[CrossRef]

A. Newell and J. Moloney, Nonlinear Optics (Addison-Wesley, Redwood City, Calif., 1992), Chaps. 5 and 6.

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Figures (2)

Fig. 1
Fig. 1

(a) Pump parameter evolution of the absolute instability thresholds with the walk-off αs. μAL (dashed curve) and μANL (solid curve) are the linear and the nonlinear absolute thresholds, respectively. αsL sets the upper limit of the region in which the nonlinear criteria prevail. The linear (nonlinear) convective instability limit is μc (μM, the Maxwell point). Below this value μc (μM) of the pump parameter the system is linearly (nonlinearly) stable. When the critical walk-off value αsc is reached the nonlinear absolute threshold coincides with the critical (convective) linear threshold. See Fig. 2 caption for meaning of filled diamonds (1) and (2). (b), (c) The evolution of the homogeneous stationary solution is plotted versus the pump parameter μ for two values of the signal walk-off αs. (b) αs<αsc: The hysteresis cycle persists. (c) αs>αsc: Disappearance of the cycle. The solution, marked by ×’s on the curves, is not reached asymptotically. MS (triangles), AS, NLC, and NLA (thick lines) refer to metastable, absolutely stable, nonlinear convective, and absolute regions, respectively. The filled points on the x axis of (b) and the dashed curves in (b) and (c) indicate the linearly unstable solution. Here we have taken Δp=Δs=2.0, as=0.1=2ap, γp=γs=1.

Fig. 2
Fig. 2

Nonlinear front evolution obtained by numerical integration of the 2D DOPO mean-field model [Eqs. (1)]. (a) Transverse evolution of the signal in the nonlinear convective domain [filled diamond (1) in Fig. 1(a)]. (b) Transverse evolution of the signal in the nonlinear absolute domain [filled diamond (2) in Fig. 1(a)].

Equations (11)

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

tAp=γp[-(1+iΔp)Ap+iap2Ap-As2+E(x, y)],
tAs=γs[-(1+iΔs)As+ias2As+ApAs*-αsxAs],
T2φ=μcμ2γsφ+γsasΔs2φ-α1γsXφ+C1+C2+C3,
C1=-γs23(1+Δp2)2[(ΔpΔs-1)+(ΔpΔs-1)2]φ5,
C2=-γs2γsγp4(1+Δp2)2(ΔpΔs-1)+2 (ΔpΔs-1)2(1+Δp2)φ5,
C3=-γs21(1+Δp2)2[(1+Δp2)+(ΔpΔs-1)]φ5.
τψ=μc(μ-μc)ψ+asΔs2ψ-αsxψ+b3ψ3-b5ψ5,
vf=v=b3asΔs3b51/221+4μcb32b5(μ-μc)1/2-1forμM<μ<μLv*=2[asΔsμc(μ-μc)]1/2forμ>μL,
ki=-b32(3asΔsb5)1/21+1+4b5μcb32(μ-μc)2,
ki*=-μcasΔs(μ-μc)1/2,
μA=μc+3αs2/(16asΔsμc)+(3/16μc)[2b3αs/(3asΔsb5)1/2-b32/b5]forαs<αsLμc+αs2/(4asΔsμc)forαs>αsL .

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