Abstract

The existence of a novel kind of soliton in dissipative systems with competing, noninstantaneous nonlinearities has been predicted and experimentally verified. These subcritically bifurcating solitons exist near a supercritical continuous wave bifurcation. We show that their peculiarities originate from different saturation behavior of nonlinear loss and gain with regard to power and energy. Branches of stable and unstable solitary waves have been identified. For the first time to our best knowledge, we have experimentally proved critical slowing down of solitons.

© 2002 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. M. C. Cross and P. C. Hohenberg, “Pattern formation outside of equilibrium,” Rev. Mod. Phys. 65, 851–1112 (1993).
    [CrossRef]
  2. A. Hasegawa and Y. Kodama, Solitons in Optical Communications (Clarendon, Oxford, 1995).
  3. J. C. Diels and W. Rudolph, Ultrashort Laser Pulse Phenomena (Academic, London, 1996).
  4. S. Fauve and O. Thual, “Solitary waves generated by subcritical instabilities in dissipative systems,” Phys. Rev. Lett. 64, 282–284 (1990).
    [CrossRef] [PubMed]
  5. W. van Saarloos and P. C. Hohenberg, “Pulses and fronts in the complex Ginzburg–Landau equation near a subcritical bifurcation,” Phys. Rev. Lett. 64, 749–752 (1990).
    [CrossRef] [PubMed]
  6. S. Longhi, “Localized structures in optical parametric oscillation,” Phys. Scr. 56, 611–618 (1997).
    [CrossRef]
  7. K. Staliunas and V. Sanchez-Morcillo, “Localized structures in degenerate optical parametric oscillators,” Opt. Commun. 139, 306–312 (1997).
    [CrossRef]
  8. M. Tlidi, P. Mandel, and R. Lefever, “Localized structures and localized patterns in optical bistability,” Phys. Rev. Lett. 73, 640–643 (1994).
    [CrossRef] [PubMed]
  9. W. J. Firth and A. J. Scroggie, “Optical bullet holes: robust controllable localized states of a nonlinear cavity,” Phys. Rev. Lett. 76, 1623–1626 (1996).
    [CrossRef] [PubMed]
  10. S. Trillo, M. Haelterman, and A. Sheppard, “Stable topological spatial solitons in optical parametric oscillators,” Opt. Lett. 22, 1514–972 (1997).
    [CrossRef]
  11. K. Staliunas and V. J. Sanchez-Morcillo, “Spatial-localised structures in degenerate optical parametric oscillators,” Phys. Rev. A 57, 1454–1457 (1998).
    [CrossRef]
  12. U. Peschel, D. Michaelis, C. Etrich, and F. Lederer, “Formation, motion and decay of vectorial cavity solitons,” Phys. Rev. E 58, R2745–R2748 (1998).
    [CrossRef]
  13. R. Gallego, M. San Miguel, and R. Toral, “Self-similar domain growth, localized structures, and labyrinthine patterns in vectorial Kerr resonators,” Phys. Rev. E 61, 2241–2244 (2000).
    [CrossRef]
  14. N. Akhmediev and A. Ankiewicz, Solitons, Nonlinear Pulses and Beams (Chapman & Hall, London, 1997).
  15. Y. Silberberg, “All-optical repeater,” Opt. Lett. 11, 392–394 (1986).
    [CrossRef] [PubMed]
  16. A. S. Shcherbakov, “Dynamics of sculpturing picosecond optical solitons in periodically multilayered semiconductor laser structures,” in Nonlinear Guided Waves and Their Applications, Vol. 55 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2001), pp. 106–108.
  17. Z. Bakonyi, G. Onishchukov, C. Knöll, M. Gölles, and F. Lederer, “10Gbit/s RZ transmission over 5000 km with gain-clamped semiconductor optical amplifiers and saturable absorbers,” Electron. Lett. 36, 1790–1791 (2000).
    [CrossRef]
  18. N. N. Akhmediev, M. J. Lederer, and B. Luther-Davies, “Exact localized solution for nonconservative systems with delayed nonlinear response,” Phys. Rev. E 57, 3664–3667 (1998).
    [CrossRef]
  19. F. Mitschke, C. Boden, W. Lange, and P. Mandel, “Exploring the dynamics of the unstable branch of bistable systems,” Opt. Commun. 71, 385–391 (1989).
    [CrossRef]
  20. J. Danckaert, G. Vitrant, R. Reinisch, and M. Georgiou, “Nonlinear dynamics in single-mode resonators,” Phys. Rev. A 48, 2324–2333 (1993).
    [CrossRef] [PubMed]

2000 (2)

R. Gallego, M. San Miguel, and R. Toral, “Self-similar domain growth, localized structures, and labyrinthine patterns in vectorial Kerr resonators,” Phys. Rev. E 61, 2241–2244 (2000).
[CrossRef]

Z. Bakonyi, G. Onishchukov, C. Knöll, M. Gölles, and F. Lederer, “10Gbit/s RZ transmission over 5000 km with gain-clamped semiconductor optical amplifiers and saturable absorbers,” Electron. Lett. 36, 1790–1791 (2000).
[CrossRef]

1998 (3)

N. N. Akhmediev, M. J. Lederer, and B. Luther-Davies, “Exact localized solution for nonconservative systems with delayed nonlinear response,” Phys. Rev. E 57, 3664–3667 (1998).
[CrossRef]

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

U. Peschel, D. Michaelis, C. Etrich, and F. Lederer, “Formation, motion and decay of vectorial cavity solitons,” Phys. Rev. E 58, R2745–R2748 (1998).
[CrossRef]

1997 (3)

S. Trillo, M. Haelterman, and A. Sheppard, “Stable topological spatial solitons in optical parametric oscillators,” Opt. Lett. 22, 1514–972 (1997).
[CrossRef]

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

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

1996 (1)

W. J. Firth and A. J. Scroggie, “Optical bullet holes: robust controllable localized states of a nonlinear cavity,” Phys. Rev. Lett. 76, 1623–1626 (1996).
[CrossRef] [PubMed]

1994 (1)

M. Tlidi, P. Mandel, and R. Lefever, “Localized structures and localized patterns in optical bistability,” Phys. Rev. Lett. 73, 640–643 (1994).
[CrossRef] [PubMed]

1993 (2)

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

J. Danckaert, G. Vitrant, R. Reinisch, and M. Georgiou, “Nonlinear dynamics in single-mode resonators,” Phys. Rev. A 48, 2324–2333 (1993).
[CrossRef] [PubMed]

1990 (2)

S. Fauve and O. Thual, “Solitary waves generated by subcritical instabilities in dissipative systems,” Phys. Rev. Lett. 64, 282–284 (1990).
[CrossRef] [PubMed]

W. van Saarloos and P. C. Hohenberg, “Pulses and fronts in the complex Ginzburg–Landau equation near a subcritical bifurcation,” Phys. Rev. Lett. 64, 749–752 (1990).
[CrossRef] [PubMed]

1989 (1)

F. Mitschke, C. Boden, W. Lange, and P. Mandel, “Exploring the dynamics of the unstable branch of bistable systems,” Opt. Commun. 71, 385–391 (1989).
[CrossRef]

1986 (1)

Akhmediev, N. N.

N. N. Akhmediev, M. J. Lederer, and B. Luther-Davies, “Exact localized solution for nonconservative systems with delayed nonlinear response,” Phys. Rev. E 57, 3664–3667 (1998).
[CrossRef]

Bakonyi, Z.

Z. Bakonyi, G. Onishchukov, C. Knöll, M. Gölles, and F. Lederer, “10Gbit/s RZ transmission over 5000 km with gain-clamped semiconductor optical amplifiers and saturable absorbers,” Electron. Lett. 36, 1790–1791 (2000).
[CrossRef]

Boden, C.

F. Mitschke, C. Boden, W. Lange, and P. Mandel, “Exploring the dynamics of the unstable branch of bistable systems,” Opt. Commun. 71, 385–391 (1989).
[CrossRef]

Cross, M. C.

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

Danckaert, J.

J. Danckaert, G. Vitrant, R. Reinisch, and M. Georgiou, “Nonlinear dynamics in single-mode resonators,” Phys. Rev. A 48, 2324–2333 (1993).
[CrossRef] [PubMed]

Etrich, C.

U. Peschel, D. Michaelis, C. Etrich, and F. Lederer, “Formation, motion and decay of vectorial cavity solitons,” Phys. Rev. E 58, R2745–R2748 (1998).
[CrossRef]

Fauve, S.

S. Fauve and O. Thual, “Solitary waves generated by subcritical instabilities in dissipative systems,” Phys. Rev. Lett. 64, 282–284 (1990).
[CrossRef] [PubMed]

Firth, W. J.

W. J. Firth and A. J. Scroggie, “Optical bullet holes: robust controllable localized states of a nonlinear cavity,” Phys. Rev. Lett. 76, 1623–1626 (1996).
[CrossRef] [PubMed]

Gallego, R.

R. Gallego, M. San Miguel, and R. Toral, “Self-similar domain growth, localized structures, and labyrinthine patterns in vectorial Kerr resonators,” Phys. Rev. E 61, 2241–2244 (2000).
[CrossRef]

Georgiou, M.

J. Danckaert, G. Vitrant, R. Reinisch, and M. Georgiou, “Nonlinear dynamics in single-mode resonators,” Phys. Rev. A 48, 2324–2333 (1993).
[CrossRef] [PubMed]

Gölles, M.

Z. Bakonyi, G. Onishchukov, C. Knöll, M. Gölles, and F. Lederer, “10Gbit/s RZ transmission over 5000 km with gain-clamped semiconductor optical amplifiers and saturable absorbers,” Electron. Lett. 36, 1790–1791 (2000).
[CrossRef]

Haelterman, M.

S. Trillo, M. Haelterman, and A. Sheppard, “Stable topological spatial solitons in optical parametric oscillators,” Opt. Lett. 22, 1514–972 (1997).
[CrossRef]

Hohenberg, P. C.

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

W. van Saarloos and P. C. Hohenberg, “Pulses and fronts in the complex Ginzburg–Landau equation near a subcritical bifurcation,” Phys. Rev. Lett. 64, 749–752 (1990).
[CrossRef] [PubMed]

Knöll, C.

Z. Bakonyi, G. Onishchukov, C. Knöll, M. Gölles, and F. Lederer, “10Gbit/s RZ transmission over 5000 km with gain-clamped semiconductor optical amplifiers and saturable absorbers,” Electron. Lett. 36, 1790–1791 (2000).
[CrossRef]

Lange, W.

F. Mitschke, C. Boden, W. Lange, and P. Mandel, “Exploring the dynamics of the unstable branch of bistable systems,” Opt. Commun. 71, 385–391 (1989).
[CrossRef]

Lederer, F.

Z. Bakonyi, G. Onishchukov, C. Knöll, M. Gölles, and F. Lederer, “10Gbit/s RZ transmission over 5000 km with gain-clamped semiconductor optical amplifiers and saturable absorbers,” Electron. Lett. 36, 1790–1791 (2000).
[CrossRef]

U. Peschel, D. Michaelis, C. Etrich, and F. Lederer, “Formation, motion and decay of vectorial cavity solitons,” Phys. Rev. E 58, R2745–R2748 (1998).
[CrossRef]

Lederer, M. J.

N. N. Akhmediev, M. J. Lederer, and B. Luther-Davies, “Exact localized solution for nonconservative systems with delayed nonlinear response,” Phys. Rev. E 57, 3664–3667 (1998).
[CrossRef]

Lefever, R.

M. Tlidi, P. Mandel, and R. Lefever, “Localized structures and localized patterns in optical bistability,” Phys. Rev. Lett. 73, 640–643 (1994).
[CrossRef] [PubMed]

Longhi, S.

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

Luther-Davies, B.

N. N. Akhmediev, M. J. Lederer, and B. Luther-Davies, “Exact localized solution for nonconservative systems with delayed nonlinear response,” Phys. Rev. E 57, 3664–3667 (1998).
[CrossRef]

Mandel, P.

M. Tlidi, P. Mandel, and R. Lefever, “Localized structures and localized patterns in optical bistability,” Phys. Rev. Lett. 73, 640–643 (1994).
[CrossRef] [PubMed]

F. Mitschke, C. Boden, W. Lange, and P. Mandel, “Exploring the dynamics of the unstable branch of bistable systems,” Opt. Commun. 71, 385–391 (1989).
[CrossRef]

Michaelis, D.

U. Peschel, D. Michaelis, C. Etrich, and F. Lederer, “Formation, motion and decay of vectorial cavity solitons,” Phys. Rev. E 58, R2745–R2748 (1998).
[CrossRef]

Mitschke, F.

F. Mitschke, C. Boden, W. Lange, and P. Mandel, “Exploring the dynamics of the unstable branch of bistable systems,” Opt. Commun. 71, 385–391 (1989).
[CrossRef]

Onishchukov, G.

Z. Bakonyi, G. Onishchukov, C. Knöll, M. Gölles, and F. Lederer, “10Gbit/s RZ transmission over 5000 km with gain-clamped semiconductor optical amplifiers and saturable absorbers,” Electron. Lett. 36, 1790–1791 (2000).
[CrossRef]

Peschel, U.

U. Peschel, D. Michaelis, C. Etrich, and F. Lederer, “Formation, motion and decay of vectorial cavity solitons,” Phys. Rev. E 58, R2745–R2748 (1998).
[CrossRef]

Reinisch, R.

J. Danckaert, G. Vitrant, R. Reinisch, and M. Georgiou, “Nonlinear dynamics in single-mode resonators,” Phys. Rev. A 48, 2324–2333 (1993).
[CrossRef] [PubMed]

San Miguel, M.

R. Gallego, M. San Miguel, and R. Toral, “Self-similar domain growth, localized structures, and labyrinthine patterns in vectorial Kerr resonators,” Phys. Rev. E 61, 2241–2244 (2000).
[CrossRef]

Sanchez-Morcillo, V.

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

Sanchez-Morcillo, V. J.

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

Scroggie, A. J.

W. J. Firth and A. J. Scroggie, “Optical bullet holes: robust controllable localized states of a nonlinear cavity,” Phys. Rev. Lett. 76, 1623–1626 (1996).
[CrossRef] [PubMed]

Sheppard, A.

S. Trillo, M. Haelterman, and A. Sheppard, “Stable topological spatial solitons in optical parametric oscillators,” Opt. Lett. 22, 1514–972 (1997).
[CrossRef]

Silberberg, Y.

Staliunas, K.

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

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

Thual, O.

S. Fauve and O. Thual, “Solitary waves generated by subcritical instabilities in dissipative systems,” Phys. Rev. Lett. 64, 282–284 (1990).
[CrossRef] [PubMed]

Tlidi, M.

M. Tlidi, P. Mandel, and R. Lefever, “Localized structures and localized patterns in optical bistability,” Phys. Rev. Lett. 73, 640–643 (1994).
[CrossRef] [PubMed]

Toral, R.

R. Gallego, M. San Miguel, and R. Toral, “Self-similar domain growth, localized structures, and labyrinthine patterns in vectorial Kerr resonators,” Phys. Rev. E 61, 2241–2244 (2000).
[CrossRef]

Trillo, S.

S. Trillo, M. Haelterman, and A. Sheppard, “Stable topological spatial solitons in optical parametric oscillators,” Opt. Lett. 22, 1514–972 (1997).
[CrossRef]

van Saarloos, W.

W. van Saarloos and P. C. Hohenberg, “Pulses and fronts in the complex Ginzburg–Landau equation near a subcritical bifurcation,” Phys. Rev. Lett. 64, 749–752 (1990).
[CrossRef] [PubMed]

Vitrant, G.

J. Danckaert, G. Vitrant, R. Reinisch, and M. Georgiou, “Nonlinear dynamics in single-mode resonators,” Phys. Rev. A 48, 2324–2333 (1993).
[CrossRef] [PubMed]

Electron. Lett. (1)

Z. Bakonyi, G. Onishchukov, C. Knöll, M. Gölles, and F. Lederer, “10Gbit/s RZ transmission over 5000 km with gain-clamped semiconductor optical amplifiers and saturable absorbers,” Electron. Lett. 36, 1790–1791 (2000).
[CrossRef]

Opt. Commun. (2)

F. Mitschke, C. Boden, W. Lange, and P. Mandel, “Exploring the dynamics of the unstable branch of bistable systems,” Opt. Commun. 71, 385–391 (1989).
[CrossRef]

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

Opt. Lett. (2)

Y. Silberberg, “All-optical repeater,” Opt. Lett. 11, 392–394 (1986).
[CrossRef] [PubMed]

S. Trillo, M. Haelterman, and A. Sheppard, “Stable topological spatial solitons in optical parametric oscillators,” Opt. Lett. 22, 1514–972 (1997).
[CrossRef]

Phys. Rev. A (2)

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

J. Danckaert, G. Vitrant, R. Reinisch, and M. Georgiou, “Nonlinear dynamics in single-mode resonators,” Phys. Rev. A 48, 2324–2333 (1993).
[CrossRef] [PubMed]

Phys. Rev. E (3)

N. N. Akhmediev, M. J. Lederer, and B. Luther-Davies, “Exact localized solution for nonconservative systems with delayed nonlinear response,” Phys. Rev. E 57, 3664–3667 (1998).
[CrossRef]

U. Peschel, D. Michaelis, C. Etrich, and F. Lederer, “Formation, motion and decay of vectorial cavity solitons,” Phys. Rev. E 58, R2745–R2748 (1998).
[CrossRef]

R. Gallego, M. San Miguel, and R. Toral, “Self-similar domain growth, localized structures, and labyrinthine patterns in vectorial Kerr resonators,” Phys. Rev. E 61, 2241–2244 (2000).
[CrossRef]

Phys. Rev. Lett. (4)

S. Fauve and O. Thual, “Solitary waves generated by subcritical instabilities in dissipative systems,” Phys. Rev. Lett. 64, 282–284 (1990).
[CrossRef] [PubMed]

W. van Saarloos and P. C. Hohenberg, “Pulses and fronts in the complex Ginzburg–Landau equation near a subcritical bifurcation,” Phys. Rev. Lett. 64, 749–752 (1990).
[CrossRef] [PubMed]

M. Tlidi, P. Mandel, and R. Lefever, “Localized structures and localized patterns in optical bistability,” Phys. Rev. Lett. 73, 640–643 (1994).
[CrossRef] [PubMed]

W. J. Firth and A. J. Scroggie, “Optical bullet holes: robust controllable localized states of a nonlinear cavity,” Phys. Rev. Lett. 76, 1623–1626 (1996).
[CrossRef] [PubMed]

Phys. Scr. (1)

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

Rev. Mod. Phys. (1)

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

Other (4)

A. Hasegawa and Y. Kodama, Solitons in Optical Communications (Clarendon, Oxford, 1995).

J. C. Diels and W. Rudolph, Ultrashort Laser Pulse Phenomena (Academic, London, 1996).

N. Akhmediev and A. Ankiewicz, Solitons, Nonlinear Pulses and Beams (Chapman & Hall, London, 1997).

A. S. Shcherbakov, “Dynamics of sculpturing picosecond optical solitons in periodically multilayered semiconductor laser structures,” in Nonlinear Guided Waves and Their Applications, Vol. 55 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2001), pp. 106–108.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (4)

Fig. 1
Fig. 1

Schematic of the system.

Fig. 2
Fig. 2

(a) Numerically determined bifurcation diagram; (b) zoomed view of the bifurcation points.

Fig. 3
Fig. 3

Experimental observation of solitons: (a) bifurcation diagram, (b) autocorrelation function for two values of the net gain, (c) relaxation of perturbed pulse trains.

Fig. 4
Fig. 4

Experimental results showing (a) soliton relaxation, (b) soliton decay, and (c) dependence of relaxation parameter on net gain.

Equations (8)

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

iz-D2+igt2+χKerr|u|2+iα-(i+hSOA)NSOA+(i+hSA)NSAu=0,
tNSOA=-1EQ[exp(2NSOA)-1]exp(-2α)|u|2-NSOA-ΠSOAτQ,
tNSA=-[1-exp(-2NSA)]exp(2NSOA-2α)|u|2-(NSA-ΠSA).
χKerr=2πn2EsatSA[1-exp(-αfLf)]λ0AeffTSAαf,
iZ-D2+igT2-iδG+χKerr|u1|2+(i+hSOA)×[exp(2ΠSOA)-1]exp(-2α)PQ|u1|2-(i+hSA)×[1-exp(-2ΠSA)]exp(-2α+2ΠSOA)|u1|2u1=0.
|ucw|2δG exp(-2ΠSA)[1-exp(-2ΠSOA)]/PQ-[1-exp(-2ΠSA)].
PQ<1-exp(-2ΠSOA)1-exp(-2ΠSA)
z-gt2-δG+K(EQ)-tu2dt+L(EQ)×-tu2dt2u=0.

Metrics