Abstract

The first analytic verification of the stability of topological solitons corresponding to wave-front solutions of the optical parametric oscillator is given. A translational invariance allows perturbations to the system to shift the front position without affecting the underlying exponential stability of the fronts themselves. For the two-dimensional problem with a positive O(1) signal detuning, the front curvature is shown to be governed by a heat equation, so that the only stable topological solitons supported must be stripes.

© 1999 Optical Society of America

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  1. N. Bloembergen, Nonlinear Optics (Benjamin, New York, 1965).
  2. See, for instance, the special issues on Optical Parametric Oscillation and Amplification, J. Opt. Soc. Am. B 10, 1655–1791 (1993) and Optical Parametric Devices, J. Opt. Soc. Am. B 12, 2083–2320 (1995).
  3. M. C. Cross and P. C. Hohenberg, “Pattern formation outside of equilibrium,” Rev. Mod. Phys. 65, 851–1112 (1993).
  4. 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).
  5. R. A. Fuerst, D. M. Baboiu, B. Lawrence, W. E. Torruellas, G. I. Stegeman, S. Trillo, and S. Wabnitz, “Spatial modulational instability and multisolitonlike generation in a quadratically nonlinear medium,” Phys. Rev. Lett. 78, 2756–2759 (1997).
  6. S. Longhi, “Localized structures in optical parametric oscillation,” Phys. Scr. 56, 611–618 (1997).
  7. K. Staliunas and V. J. Sánchez-Morcillo, “Localized structures in degenerate optical parametric oscillators,” Opt. Commun. 139, 306–312 (1997).
  8. S. Trillo, M. Haelterman, and A. Sheppard, “Stable topological spatial solitons in optical parametric oscillators,” Opt. Lett. 22, 970–972 (1997).
  9. K. Staliunas and V. J. Sánchez-Morcillo, “Spatial-localized structures in degenerate optical parametric oscillators,” Phys. Rev. A 57, 1454–1457 (1998).
  10. S. Longhi, “Spatial solitary waves in nondegenerate optical parametric oscillators near an inverted bifurcation,” Opt. Commun. 149, 335–340 (1998).
  11. 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).
  12. S. Trillo and M. Haelterman, “Excitation and bistability of self-trapped signal beams in optical parametric oscillators,” Opt. Lett. 23, 1514–1516 (1998).
  13. M. Tidli, P. Mandel, and R. Lefever, “Kinetics of localized pattern formation in optical systems,” Phys. Rev. Lett. 81, 979–982 (1998).
  14. C. Etrich, U. Peschel, and F. Lederer, “Solitary waves in quadratically nonlinear resonators,” Phys. Rev. Lett. 79, 2454–2457 (1997).
  15. N. N. Rozanov, in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1996), Vol. 35, pp. 1–60.
  16. K. Staliunas and V. J. Sánchez-Morcillo, “Dynamics of phase domains in the Swift–Hohenberg equation,” Phys. Lett. A 241, 28–31 (1998).
  17. P. Coullet, L. Gil, and D. Repaux, “Defects and subcritical bifurcations,” Phys. Rev. Lett. 62, 2957–2960 (1989).
  18. V. B. Tarenko, K. Staliunas, and C. O. Weiss, “Pattern formation and localized structures in degenerate optical parametric mixing,” Phys. Rev. Lett. 81, 2236–2239 (1998).
  19. C. Richy, K. I. Petsas, E. Giacobino, and C. Fabre, “Observation of bistability and delayed bifurcation in a triply resonant optical parametric oscillators,” J. Opt. Soc. Am. B 12, 456–461 (1995).
  20. G. L. Oppo, A. J. Scroggie, and W. J. Firth, “Stabilization of domain walls in optical parametric oscillators,” special issue on Pattern Formation in Nonlinear Optical System, Quant. Semiclass. Opt. 1, 133–138 (1999).
  21. B. Friedman, Principles and Techniques of Applied Mathematics (Dover, New York, 1990), Chap. 1.
  22. G. T. Dee and W. Van Sarloos, “Bistable system with propagating fronts leading to pattern formation,” Phys. Rev. Lett. 60, 2641–2644 (1988).
  23. M. Weinstein, “Modulational stability of ground states of nonlinear Schrödinger equation,” SIAM (Soc. Ind. Appl. Math.) J. Math. Anal. 16, 472–491 (1985).
  24. J. N. Kutz and W. L. Kath, “Stability of pulses in nonlinear optical fibers using phase-sensitive amplifiers,” SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 56, 611–626 (1996).
  25. D. Gruner, R. Kapral, and A. Lawniczak, “Nucleation, domain growth and fluctuations in a bistable chemical system,” J. Chem. Phys. 99, 3938–3945 (1993).

1999

G. L. Oppo, A. J. Scroggie, and W. J. Firth, “Stabilization of domain walls in optical parametric oscillators,” special issue on Pattern Formation in Nonlinear Optical System, Quant. Semiclass. Opt. 1, 133–138 (1999).

1998

K. Staliunas and V. J. Sánchez-Morcillo, “Dynamics of phase domains in the Swift–Hohenberg equation,” Phys. Lett. A 241, 28–31 (1998).

V. B. Tarenko, K. Staliunas, and C. O. Weiss, “Pattern formation and localized structures in degenerate optical parametric mixing,” Phys. Rev. Lett. 81, 2236–2239 (1998).

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

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

M. Tidli, P. Mandel, and R. Lefever, “Kinetics of localized pattern formation in optical systems,” Phys. Rev. Lett. 81, 979–982 (1998).

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).

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

1997

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

C. Etrich, U. Peschel, and F. Lederer, “Solitary waves in quadratically nonlinear resonators,” Phys. Rev. Lett. 79, 2454–2457 (1997).

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

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

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

1996

J. N. Kutz and W. L. Kath, “Stability of pulses in nonlinear optical fibers using phase-sensitive amplifiers,” SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 56, 611–626 (1996).

1995

1994

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).

1993

M. C. Cross and P. C. Hohenberg, “Pattern formation outside of equilibrium,” Rev. Mod. Phys. 65, 851–1112 (1993).

D. Gruner, R. Kapral, and A. Lawniczak, “Nucleation, domain growth and fluctuations in a bistable chemical system,” J. Chem. Phys. 99, 3938–3945 (1993).

1989

P. Coullet, L. Gil, and D. Repaux, “Defects and subcritical bifurcations,” Phys. Rev. Lett. 62, 2957–2960 (1989).

1988

G. T. Dee and W. Van Sarloos, “Bistable system with propagating fronts leading to pattern formation,” Phys. Rev. Lett. 60, 2641–2644 (1988).

1985

M. Weinstein, “Modulational stability of ground states of nonlinear Schrödinger equation,” SIAM (Soc. Ind. Appl. Math.) J. Math. Anal. 16, 472–491 (1985).

Baboiu, D. M.

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

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, 2028–2032 (1994).

Colet, P.

Coullet, P.

P. Coullet, L. Gil, and D. Repaux, “Defects and subcritical bifurcations,” Phys. Rev. Lett. 62, 2957–2960 (1989).

Cross, M. C.

M. C. Cross and P. C. Hohenberg, “Pattern formation outside of equilibrium,” Rev. Mod. Phys. 65, 851–1112 (1993).

Dee, G. T.

G. T. Dee and W. Van Sarloos, “Bistable system with propagating fronts leading to pattern formation,” Phys. Rev. Lett. 60, 2641–2644 (1988).

Etrich, C.

C. Etrich, U. Peschel, and F. Lederer, “Solitary waves in quadratically nonlinear resonators,” Phys. Rev. Lett. 79, 2454–2457 (1997).

Fabre, C.

Firth, W. J.

G. L. Oppo, A. J. Scroggie, and W. J. Firth, “Stabilization of domain walls in optical parametric oscillators,” special issue on Pattern Formation in Nonlinear Optical System, Quant. Semiclass. Opt. 1, 133–138 (1999).

Fuerst, R. A.

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

Giacobino, E.

Gil, L.

P. Coullet, L. Gil, and D. Repaux, “Defects and subcritical bifurcations,” Phys. Rev. Lett. 62, 2957–2960 (1989).

Gruner, D.

D. Gruner, R. Kapral, and A. Lawniczak, “Nucleation, domain growth and fluctuations in a bistable chemical system,” J. Chem. Phys. 99, 3938–3945 (1993).

Haelterman, M.

Hohenberg, P. C.

M. C. Cross and P. C. Hohenberg, “Pattern formation outside of equilibrium,” Rev. Mod. Phys. 65, 851–1112 (1993).

Kapral, R.

D. Gruner, R. Kapral, and A. Lawniczak, “Nucleation, domain growth and fluctuations in a bistable chemical system,” J. Chem. Phys. 99, 3938–3945 (1993).

Kath, W. L.

J. N. Kutz and W. L. Kath, “Stability of pulses in nonlinear optical fibers using phase-sensitive amplifiers,” SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 56, 611–626 (1996).

Kutz, J. N.

J. N. Kutz and W. L. Kath, “Stability of pulses in nonlinear optical fibers using phase-sensitive amplifiers,” SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 56, 611–626 (1996).

Lawniczak, A.

D. Gruner, R. Kapral, and A. Lawniczak, “Nucleation, domain growth and fluctuations in a bistable chemical system,” J. Chem. Phys. 99, 3938–3945 (1993).

Lawrence, B.

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

Lederer, F.

C. Etrich, U. Peschel, and F. Lederer, “Solitary waves in quadratically nonlinear resonators,” Phys. Rev. Lett. 79, 2454–2457 (1997).

Lefever, R.

M. Tidli, P. Mandel, and R. Lefever, “Kinetics of localized pattern formation in optical systems,” Phys. Rev. Lett. 81, 979–982 (1998).

Longhi, S.

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

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

Lugiato, L. A.

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).

Mandel, P.

M. Tidli, P. Mandel, and R. Lefever, “Kinetics of localized pattern formation in optical systems,” Phys. Rev. Lett. 81, 979–982 (1998).

Oppo, G. L.

G. L. Oppo, A. J. Scroggie, and W. J. Firth, “Stabilization of domain walls in optical parametric oscillators,” special issue on Pattern Formation in Nonlinear Optical System, Quant. Semiclass. Opt. 1, 133–138 (1999).

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).

Peschel, U.

C. Etrich, U. Peschel, and F. Lederer, “Solitary waves in quadratically nonlinear resonators,” Phys. Rev. Lett. 79, 2454–2457 (1997).

Petsas, K. I.

Repaux, D.

P. Coullet, L. Gil, and D. Repaux, “Defects and subcritical bifurcations,” Phys. Rev. Lett. 62, 2957–2960 (1989).

Richy, C.

San Miguel, M.

Sánchez-Morcillo, V. J.

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

K. Staliunas and V. J. Sánchez-Morcillo, “Dynamics of phase domains in the Swift–Hohenberg equation,” Phys. Lett. A 241, 28–31 (1998).

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

Santagiustina, M.

Scroggie, A. J.

G. L. Oppo, A. J. Scroggie, and W. J. Firth, “Stabilization of domain walls in optical parametric oscillators,” special issue on Pattern Formation in Nonlinear Optical System, Quant. Semiclass. Opt. 1, 133–138 (1999).

Sheppard, A.

Staliunas, K.

K. Staliunas and V. J. Sánchez-Morcillo, “Dynamics of phase domains in the Swift–Hohenberg equation,” Phys. Lett. A 241, 28–31 (1998).

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

V. B. Tarenko, K. Staliunas, and C. O. Weiss, “Pattern formation and localized structures in degenerate optical parametric mixing,” Phys. Rev. Lett. 81, 2236–2239 (1998).

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

Stegeman, G. I.

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

Tarenko, V. B.

V. B. Tarenko, K. Staliunas, and C. O. Weiss, “Pattern formation and localized structures in degenerate optical parametric mixing,” Phys. Rev. Lett. 81, 2236–2239 (1998).

Tidli, M.

M. Tidli, P. Mandel, and R. Lefever, “Kinetics of localized pattern formation in optical systems,” Phys. Rev. Lett. 81, 979–982 (1998).

Torruellas, W. E.

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

Trillo, S.

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

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

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

Van Sarloos, W.

G. T. Dee and W. Van Sarloos, “Bistable system with propagating fronts leading to pattern formation,” Phys. Rev. Lett. 60, 2641–2644 (1988).

Wabnitz, S.

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

Walgraef, D.

Weinstein, M.

M. Weinstein, “Modulational stability of ground states of nonlinear Schrödinger equation,” SIAM (Soc. Ind. Appl. Math.) J. Math. Anal. 16, 472–491 (1985).

Weiss, C. O.

V. B. Tarenko, K. Staliunas, and C. O. Weiss, “Pattern formation and localized structures in degenerate optical parametric mixing,” Phys. Rev. Lett. 81, 2236–2239 (1998).

J. Chem. Phys.

D. Gruner, R. Kapral, and A. Lawniczak, “Nucleation, domain growth and fluctuations in a bistable chemical system,” J. Chem. Phys. 99, 3938–3945 (1993).

J. Opt. Soc. Am. B

Opt. Commun.

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

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

Opt. Lett.

Phys. Lett. A

K. Staliunas and V. J. Sánchez-Morcillo, “Dynamics of phase domains in the Swift–Hohenberg equation,” Phys. Lett. A 241, 28–31 (1998).

Phys. Rev. A

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

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).

Phys. Rev. Lett.

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

M. Tidli, P. Mandel, and R. Lefever, “Kinetics of localized pattern formation in optical systems,” Phys. Rev. Lett. 81, 979–982 (1998).

C. Etrich, U. Peschel, and F. Lederer, “Solitary waves in quadratically nonlinear resonators,” Phys. Rev. Lett. 79, 2454–2457 (1997).

P. Coullet, L. Gil, and D. Repaux, “Defects and subcritical bifurcations,” Phys. Rev. Lett. 62, 2957–2960 (1989).

V. B. Tarenko, K. Staliunas, and C. O. Weiss, “Pattern formation and localized structures in degenerate optical parametric mixing,” Phys. Rev. Lett. 81, 2236–2239 (1998).

G. T. Dee and W. Van Sarloos, “Bistable system with propagating fronts leading to pattern formation,” Phys. Rev. Lett. 60, 2641–2644 (1988).

Phys. Scr.

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

Quant. Semiclass. Opt.

G. L. Oppo, A. J. Scroggie, and W. J. Firth, “Stabilization of domain walls in optical parametric oscillators,” special issue on Pattern Formation in Nonlinear Optical System, Quant. Semiclass. Opt. 1, 133–138 (1999).

Rev. Mod. Phys.

M. C. Cross and P. C. Hohenberg, “Pattern formation outside of equilibrium,” Rev. Mod. Phys. 65, 851–1112 (1993).

SIAM (Soc. Ind. Appl. Math.) J. Appl. Math.

J. N. Kutz and W. L. Kath, “Stability of pulses in nonlinear optical fibers using phase-sensitive amplifiers,” SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 56, 611–626 (1996).

SIAM (Soc. Ind. Appl. Math.) J. Math. Anal.

M. Weinstein, “Modulational stability of ground states of nonlinear Schrödinger equation,” SIAM (Soc. Ind. Appl. Math.) J. Math. Anal. 16, 472–491 (1985).

Other

B. Friedman, Principles and Techniques of Applied Mathematics (Dover, New York, 1990), Chap. 1.

N. Bloembergen, Nonlinear Optics (Benjamin, New York, 1965).

See, for instance, the special issues on Optical Parametric Oscillation and Amplification, J. Opt. Soc. Am. B 10, 1655–1791 (1993) and Optical Parametric Devices, J. Opt. Soc. Am. B 12, 2083–2320 (1995).

N. N. Rozanov, in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1996), Vol. 35, pp. 1–60.

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